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arXiv:1608.03703v2 [] 26 Apr 2017 Template estimation in computational anatomy: Fréchet means in top and quotient spaces are not consistent Loïc Devilliers∗, Stéphanie Allassonnière†, Alain Trouvé‡, and Xavier Pennec§ April 27, 2017 Abstract In this article, we study the consistency of the template estimation with the Fréchet mean in quotient spaces. The Fréchet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases. Keyword : Template, Fréchet mean, group action, quotient space, inconsistency, consistency bias, empirical Fréchet mean, Hilbert space, manifold ∗ Université Côte d’Azur, Inria, France, loic.devilliers@inria.fr Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ‡ CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France § Université Côte d’Azur, Inria, France † CMAP, 1 Contents 1 Introduction 3 2 Definitions, notations and generative model 5 3 Inconsistency for finite group when the template is point 3.1 Presence of inconsistency . . . . . . . . . . . . . . . . 3.2 Upper bound of the consistency bias . . . . . . . . . . 3.3 Study of the consistency bias in a simple example . . . . . . . . . . . . . . . . . . . . . 4 Inconsistency for any group when the template is point 4.1 Presence of an inconsistency . . . . . . . . . . . . . . 4.2 Analysis of the condition in theorem 4.1 . . . . . . . 4.3 Lower bound of the consistency bias . . . . . . . . . 4.4 Upper bound of the consistency bias . . . . . . . . . 4.5 Empirical Fréchet mean . . . . . . . . . . . . . . . . 4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Action of translation on L2 (R/Z) . . . . . . . 4.6.2 Action of discrete translation on RZ/NZ . . . . 4.6.3 Action of rotations on Rn . . . . . . . . . . . . . . . . . . . . 5 Fréchet means top and quotient spaces the template is a fixed point 5.1 Result . . . . . . . . . . . . . . . . . . 5.2 Proofs of these theorems . . . . . . . . 5.2.1 Proof of theorem 5.1 . . . . . . 5.2.2 Proof of theorem 5.2 . . . . . . 6 Conclusion and discussion a regular 8 9 12 13 not a fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 15 18 20 22 22 23 23 23 are not consistent when . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 26 26 28 28 A Proof of theorems for finite groups’ setting 29 A.1 Proof of theorem 3.2: differentiation of the variance in the quotient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2 Proof of theorem 3.1: the gradient is not zero at the template . . 32 A.3 Proof of theorem 3.3: upper bound of the consistency bias . . . . 32 A.4 Proof of proposition 3.2: inconsistency in R2 for the action of translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B Proof of lemma 5.1: differentiation of the variance in the top space 35 2 1 Introduction In Kendall’s shape space theory [Ken89], in computational anatomy [GM98], in statistics on signals, or in image analysis, one often aims at estimating a template. A template stands for a prototype of the data. The data can be the shape of an organ studied in a population [DPC+ 14] or an aircraft [LAJ+ 12], an electrical signal of the human body, a MR image etc. To analyse the observations, one assumes that these data follow a statistical model. One often models observations as random deformations of the template with additional noise. This deformable template model proposed in [GM98] is commonly used in computational anatomy. The concept of deformation introduces the notion of group action: the deformations we consider are elements of a group which acts on the space of observations, called here the top space. Since the deformations are unknown, one usually considers equivalent classes of observations under the group action. In other words, one considers the quotient space of the top space (or ambient space) by the group. In this particular setting, the template estimation is most of the time based on the minimisation of the empirical variance in the quotient space (for instance [KSW11, JDJG04, SBG08] among many others). The points that minimise the empirical variance are called the empirical Fréchet mean. The Fréchet means introduced in [Fré48] is comprised of the elements minimising the variance. This generalises the notion of expected value in non linear spaces. Note that the existence or uniqueness of Fréchet mean is not ensured. But sufficient conditions may be given in order to reach existence and uniqueness (for instance [Kar77] and [Ken90]). Several group actions are used in practice: some signals can be shifted in time compared to other signals (action of translations [HCG+ 13]), landmarks can be transformed rigidly [Ken89], shapes can be deformed by diffeomorphisms [DPC+ 14], etc. In this paper we restrict to transformation which leads the norm unchanged. Rotations for instance leave the norm unchanged, but it may seem restrictive. In fact, the square root trick detailed in section 5, allows to build norms which are unchanged, for instance by reparametrization of curves with a diffeomorphism, where our work can be applied. We raise several issues concerning the estimation of the template. 1. Is the Fréchet mean in the quotient space equal to the original template projected in the quotient space? In other words, is the template estimation with the Fréchet mean in quotient space consistent? 2. If there is an inconsistency, how large is the consistency bias? Indeed, we may expect the consistency bias to be negligible in many practicable cases. 3. If one gets only a finite sample, one can only estimate the empirical Fréchet mean. How far is the empirical Fréchet mean from the original template? These issues originated from an example exhibited by Allassonnière, Amit and Trouvé [AAT07]: they took a step function as a template and they added some 3 noise and shifted in time this function. By repeating this process they created a data sample from this template. With this data sample, they tried to estimate the template with the empirical Fréchet mean in the quotient space. In this example, minimising the empirical variance did not succeed in estimating well the template when the noise added to the template increases, even with a large sample size. One solution to ensure convergence to the template is to replace this estimation method with a Bayesian paradigm ([AKT10, BG14] or [ZSF13]). But there is a need to have a better understanding of the failure of the template estimation with the Fréchet mean. One can studied the inconsistency of the template estimation. Bigot and Charlier [BC11] first studied the question of the template estimation with a finite sample in the case of translated signals or images by providing a lower bound of the consistency bias. This lower bound was unfortunately not so informative as it is converging to zero asymptotically when the dimension of the space tends to infinity. Miolane et al. [MP15, MHP16] later provided a more general explanation of why the template is badly estimated for a general group action thanks to a geometric interpretation. They showed that the external curvature of the orbits is responsible for the inconsistency. This result was further quantified with Gaussian noise. In this article, we provide sufficient conditions on the noise for which inconsistency appears and we quantify the consistency bias in the general (non necessarily Gaussian) case. Moreover, we mostly consider a vector space (possibly infinite dimensional) as the top space while the article of Miolane et al. is restricted to finite dimensional manifolds. In a preliminary unpublished version of this work [ADP15], we proved the inconsistency when the transformations come from a finite group acting by translation. The current article extends these results by generalizing to any isometric action of finite and non-finite groups. This article is organised as follows. Section 2 details the mathematical terms that we use and the generative model. In sections 3 and 4, we exhibit sufficient condition that lead to an inconsistency when the template is not a fixed point under the group action. This sufficient condition can be roughly understand as follows: with a non zero probability, the projection of the random variable on the orbit of the template is different from the template itself. This condition is actually quite general. In particular, this condition it is always fulfilled with the Gaussian noise or with any noise whose support is the whole space. Moreover we quantify the consistency bias with lower and upper bounds. We restrict our study to Hilbert spaces and isometric actions. This means that the space is linear, the group acts linearly and leaves the norm (or the dot product) unchanged. Section 3 is dedicated to finite groups. Then we generalise our result in section 4 to non-finite groups. To complete this study, we extend in section 5 the result when the template is a fixed point under the group action and when the top space is a manifold. As a result we show that the inconsistency exists for almost all noises. Although the bias can be neglected when the noise level is sufficiently small, its linear asymptotic behaviour with respect to the noise level show that it becomes unavoidable for large noises. 4 2 Definitions, notations and generative model We denote by M the top space, which is the image/shape space, and G the group acting on M . The action is a map: G×M (g, m) → M 7→ g · m satisfying the following properties: for all g, g 0 ∈ G, m ∈ M (gg 0 )·m = g ·(g 0 ·m) and eG · m = m where eG is the neutral element of G. For m ∈ M we note by [m] the orbit of m (or the class of m). This is the set of points reachable from m under the group action: [m] = {g · m, g ∈ G}. Note that if we take two orbits [m] and [n] there are two possibilities: 1. The orbits are equal: [m] = [n] i.e. ∃g ∈ G s.t. n = g · m. 2. The orbits have an empty intersection: [m] ∩ [n] = ∅. We call quotient of M by the group G the set all orbits. This quotient is noted by: Q = M/G = {[m], m ∈ M }. The orbit of an element m ∈ M can be seen as the subset of M of all elements g · m for g ∈ G or as a point in the quotient space. In this article we use these two ways. We project an element m of the top space M into the quotient by taking [m]. Now we are interested in adding a structure on the quotient from an existing structure in the top space: take M a metric space, with dM its distance. Suppose that dM is invariant under the group action which means that ∀g ∈ G, ∀a, b ∈ M dM (a, b) = dM (g · a, g · b). Then we obtain a pseudo-distance on Q defined by: dQ ([a], [b]) = inf dM (g · a, b). (1) g∈G We remind that a distance on M is a map dM : M × M 7→ R+ such that for all m, n, p ∈ M : 1. dM (m, n) = dM (n, m) (symmetry). 2. dM (m, n) ≤ dM (m, p) + dM (p, n) (triangular inequality). 3. dM (m, m) = 0. 4. dM (m, n) = 0 ⇐⇒ m = n. A pseudo-distance satisfies only the first three conditions. If we suppose that all the orbits are closed sets of M , then one can show that dQ is a distance. In this article, we assume that dQ is always a distance, even if a pseudo-distance would be sufficient. dQ ([a], [b]) can be interpreted as the distance between the shapes a and b, once one has removed the parametrisation by the group G. In other words, a and b have been registered. In this article, except in section 5, we 5 suppose that the the group acts isometrically on an Hilbert space, this means that the map x 7→ g ·x is linear, and that the norm associated to the dot product is conserved: kg · xk = kxk. Then dM (a, b) = ka − bk is a particular case of invariant distance. We now introduce the generative model used in this article for M a vector space. Let us take a template t0 ∈ M to which we add a unbiased noise : X = t0 + . Finally we transform X with a random shift S of G. We assume that this variable S is independent of X and the only observed variable is: Y = S · X = S · (t0 + ), with E() = 0, (2) while S, X and  are hidden variables. Note that it is not the generative model defined by Grenander and often used in computational anatomy. Where the observed variable is rather Y 0 = S · t0 + 0 . But when the noise is isotropic and the action is isometric, one can show that the two models have the same law, since S ·  and  have the same probability distribution. As a consequence, the inconsistency of the template estimation with the Fréchet mean in quotient space with one model implies the inconsistency with the other model. Because the former model (2) leads to simpler computation we consider only this model. We can now set the inverse problem: given the observation Y , how to estimate the template t0 in M ? This is an ill-posed problem. Indeed for some element group g ∈ G, the template t0 can be replaced by the translated g ·t0 , the shift S by Sg −1 and the noise  by g, which leads to the same observation Y . So instead of estimating the template t0 , we estimate its orbit [t0 ]. By projecting the observation Y in the quotient space we obtain [Y ]. Although the observation Y = S · X and the noisy template X are different random variables in the top space, their projections on the quotient space lead to the same random orbit [Y ] = [X]. That is why we consider the generative model (2): the projection in the quotient space remove the transformation of the group G. From now on, we use the random orbit [X] in lieu of the random orbit of the observation [Y ]. The variance of the random orbit [X] (sometimes called the Fréchet functional or the energy function) at the quotient point [m] ∈ Q is the expected value of the square distance between [m] and the random orbit [X], namely: Q 3 [m] 7→ E(dQ ([m], [X])2 ) (3) An orbit [m] ∈ Q which minimises this map is called a Fréchet mean of [X]. If we have an i.i.d sample of observations Y1 , . . . , Yn we can write the empirical quotient variance: n Q 3 [m] 7→ n 1X 1X dQ ([m], [Yi ])2 = inf km − gi · Yi k2 . n i=1 n i=1 gi ∈G (4) Thanks to the equality of the quotient variables [X] and [Y ], an element which minimises this map is an empirical Fréchet mean of [X]. 6 In order to minimise the empirical quotient variance (4), the max-max algon P rithm1 alternatively minimises the function J(m, (gi )i ) = n1 km−gi ·Yi k2 over i=1 a point m of the orbit [m] and over the hidden transformation (gi )1≤i≤n ∈ Gn . With these notations we can reformulate our questions as: 1. Is the orbit of the template [t0 ] a minimiser of the quotient variance defined in (3)? If not, the Fréchet mean in quotient space is an inconsistent estimator of [t0 ]. 2. In this last case, can we quantify the quotient distance between [t0 ] and a Fréchet mean of [X]? 3. Can we quantify the distance between [t0 ] and an empirical Fréchet mean of a n-sample? This article shows that the answer to the first question is usually "no" in the framework of an Hilbert space M on which a group G acts linearly and isometrically. The only exception is theorem 5.1 where the top space M is a manifold. In order to prove inconsistency, an important notion in this framework is the isotropy group of a point m in the top space. This is the subgroup which leaves this point unchanged: Iso(m) = {g ∈ G, g · m = m}. We start in section 3 with the simple example where the group is finite and the isotropy group of the template is reduced to the identity element (Iso(t0 ) = {eG }, in this case t0 is called a regular point). We turn in section 4 to the case of a general group and an isotropy group of the template which does not cover the whole group (Iso(t0 ) 6= G) i.e t0 is not a fixed point under the group action. To complete the analysis, we assume in section 5 that the template t0 is a fixed point which means that Iso(t0 ) = G. In sections 3 and 4 we show lower and upper bounds of the consistency bias which we define as the quotient distance between the template orbit and the Fréchet mean in quotient space. These results give an answer to the second question. In section 4, we show a lower bound for the case of the empirical Fréchet mean which answers to the third question. As we deal with different notions whose name or definition may seem similar, we use the following vocabulary: 1. The variance of the noisy template X in the top space is the function E : m ∈ M 7→ E(km − Xk2 ). The unique element which minimises this function is the Fréchet mean of X in the top space. With our assumptions it is the template t0 itself. 2. We call variability (or noise level) of the template the value of the variance at this minimum: σ 2 = E(kt0 − Xk2 ) = E(t0 ). 1 The term max-max algorithm is used for instance in [AAT07], and we prefer to keep the same name, even if it is a minimisation. 7 3. The variance of the random orbit [X] in the quotient space is the function F : m 7→ E(dQ ([m], [X])2 ). Notice that we define this function from the top space and not from the quotient space. With this definition, an orbit [m? ] is a Fréchet mean of [X] if the point m? is a global minimiser of F . In sections 3 and 4, we exhibit a sufficient condition for the inconsistency, which is: the noisy template X takes value with a non zero probability in the set of points which are strictly closer to g · t0 for some g ∈ G than the template t0 itself. This is linked to the folding of the distribution of the noisy template when it is projected to the quotient space. The points for which the distance to the template orbit in the quotient space is equal to the distance to the template in the top space are projected without being folded. If the support of the distribution of the noisy template contains folded points (we only assume that the probability measure of X, noted P, is a regular measure), then there is inconsistency. The support of the noisy template X is defined by the set of points x such that P(X ∈ B(x, r)) > 0 for all r > 0. For different geometries of the orbit of the template, we show that this condition is fulfilled as soon as the support of the noise is large enough. The recent article of Cleveland et al. [CWS16] may seem contradictory with our current work. Indeed the consistency of the template estimation with the Fréchet mean in quotient space is proved under hypotheses which seem to satisfy our framework: the norm is unchanged under their group action (isometric action) and a noise is present in their generative model. However we believe that the noise they consider might actually not be measurable. Indeed, their top space is:   Z 1 L2 ([0, 1]) = f : [0, 1] → R such that f is measurable and f 2 (t)dt < +∞ . 0 The noise e is supposed to be in L2 ([0, 1]) such that for all t, s ∈ [0, 1], E(e(t)) = 0 and E(e(t)e(s)) = σ 2 1s=t , for σ > 0. This means that e(t) and e(s) are chosen without correlation as soon as s 6= t. In this case, it is not clear for us that the resulting function e is measurable, and thus that its Lebesgue integration makes sense. Thus, the existence of such a random process should be established before we can fairly compare the results of both works. 3 Inconsistency for finite group when the template is a regular point In this Section, we consider a finite group G acting isometrically and effectively on M = Rn a finite dimensional space equipped with the euclidean norm k k, associated to the dot product h , i. We say that the action is effective if x 7→ g · x is the identity map if and only if g = eG . Note that if the action is not effective, we can define a new effective action by simply quotienting G by the subgroup of the element g ∈ G such that x 7→ g · x is the identity map. 8 The template is assumed to be a regular point which means that the isotropy group of the template is reduced to the neutral element of G. Note that the measure of singular points (the points which are not regular) is a null set for the Lebesgue measure (see item 1 in appendix A.1). Example 3.1. The action of translation on coordinates: this action is a simplified setting for image registration, where images can be obtained by the translation of one scan to another due to different poses. More precisely, we take the vector space M = RT where G = T = (Z/N Z)D is the finite torus in Ddimension. An element of RT is seen as a function m : T → R, where m(τ ) is the grey value at pixel τ . When D = 1, m can be seen like a discretised signal with N points, when D = 2, we can see m like an image with N × N pixels etc. We then define the group action of T on RT by: τ ∈ T, m ∈ RT τ · m : σ 7→ m(σ + τ ). This group acts isometrically and effectively on M = RT . In this setting, if E(kXk2 ) < +∞ then the variance of [X] is well defined: F : m ∈ M 7→ E(dQ ([X], [m])2 ). In this framework, F is non-negative and continuous. Schwarz inequality we have: lim F (m) ≥ kmk→∞ (5) Thanks to Cauchy- lim kmk2 − 2kmkE(kXk) + E(kXk2 ) = +∞. kmk→∞ Thus for some R > 0 we have: for all m ∈ M if kmk > R then F (m) ≥ F (0) + 1. The closed ball B(0, R) is a compact set (because M is a finite vector space) then F restricted to this ball reached its minimum m? . Then for all m ∈ M , if m ∈ B(0, R), F (m? ) ≤ F (m), if kmk > R then F (m) ≥ F (0) + 1 > F (0) ≥ F (m? ). Therefore [m? ] is a Fréchet mean of [X] in the quotient Q = M/G. Note that this ensure the existence but not the uniqueness. In this Section, we show that as soon as the support of the distribution of X is big enough, the orbit of the template is not a Fréchet mean of [X]. We provide a upper bound of the consistency bias depending on the variability of X and an example of computation of this consistency bias. 3.1 Presence of inconsistency The following theorem gives a sufficient condition on the random variable X for an inconsistency: Theorem 3.1. Let G be a finite group acting on M = Rn isometrically and effectively. Assume that the random variable X is absolutely continuous with respect to the Lebesgue’s measure, with E(kXk2 ) < +∞. We assume that t0 = E(X) is a regular point. 9 g · t0 0 Cone(t0 ) t0 g 0 · t0 Figure 1: Planar representation of a part of the orbit of the template t0 . The lines are the hyperplanes whose points are equally distant of two distinct elements of the orbit of t0 , Cone(t0 ) represented in points is the set of points closer from t0 than any other points in the orbit of t0 . Theorem 3.1 states that if the support (the dotted disk) of the random variable X is not included in this cone, then there is an inconsistency. We define Cone(t0 ) as the set of points closer from t0 than any other points of the orbit [t0 ], see fig. 1 or item 6 in appendix A.1 for a formal definition. In other words, Cone(t0 ) is defined as the set of points already registered with t0 . Suppose that: P (X ∈ / Cone(t0 )) > 0, (6) then [t0 ] is not a Fréchet mean of [X]. The proof of theorem 3.1 is based on two steps: first, differentiating the variance F of [X]. Second, showing that the gradient at the template is not zero, therefore the template can not be a minimum of F . Theorem 3.2 makes the first step. Theorem 3.2. The variance F of [X] is differentiable at any regular points. For m0 a regular point, we define g(x, m0 ) as the almost unique g ∈ G minimising km0 − g · xk (in other words, g(x, m0 ) · x ∈ Cone(m0 )). This allows us to compute the gradient of F at m0 : ∇F (m0 ) = 2(m0 − E(g(X, m0 ) · X)). (7) This Theorem is proved in appendix A.1. Then we show that the gradient of F at t0 is not zero. To ensure that F is differentiable at t0 we suppose in the assumptions of theorem 3.1 that t0 = E(X) is a regular point. Thanks to theorem 3.2 we have: ∇F (t0 ) = 2(t0 − E(g(X, t0 ) · X)). Therefore ∇F (t0 )/2 is the difference between two terms, which are represented on fig. 2: on fig. 2a there is a mass under the two hyperplanes outside 10 g · t0 0 g · t0 Cone(t0 ) t0 0 g 0 · t0 Cone(t0 ) t0 Z g 0 · t0 (a) Graphic representation of the template t0 = E(X) mean of points of the support of X. (b) Graphic representation of Z = E(g(X, t0 ) · X). The points X which were outside Cone(t0 ) are now in Cone(t0 ) (thanks to g(X, t0 )). This part, in grid-line, represents the points which have been folded. Figure 2: Z is the mean of points in Cone(t0 ) where Cone(t0 ) is the set of points closer from t0 than g · t0 for g ∈ G \ eG . Therefore it seems that Z is higher that t0 , therefore ∇F (t0 ) = 2(t0 − Z) 6= 0. Cone(t0 ), so this mass is nearer from gt0 for some g ∈ G than from t0 . In the following expression Z = E(g(X, t0 ) · X), for X ∈ / Cone(t0 ), g(X, t0 )X ∈ Cone(t0 ) such points are represented in grid-line on fig. 2. This suggests that the point Z = E(g(X, t0 ) · X) which is the mean of points in Cone(t0 ) is further away from 0 than t0 . Then ∇F (t0 )/2 = t0 − Z should be not zero, and t0 = E(X) is not a critical point of the variance of [X]. As a conclusion [t0 ] is not a Fréchet mean of [X]. This is turned into a rigorous proof in appendix A.2. In the proof of theorem 3.1, we took M an Euclidean space and we work with the Lebesgue’s measure in order to have P(X ∈ H) = 0 for every hyperplane H. Therefore the proof of theorem 3.1 can be extended immediately to any Hilbert space M , if we make now the assumption that P(X ∈ H) = 0 for every hyperplane H, as long as we keep a finite group acting isometrically and effectively on M . Figure 2 illustrates the condition of theorem 3.1: if there is no mass beyond the hyperplanes, then the two terms in ∇F (t0 ) are equal (because almost surely g(X, t0 ) · X = X). Therefore in this case we have ∇F (t0 ) = 0. This do not prove necessarily that there is no inconsistency, just that the template t0 is a critical point of F . Moreover this figure can give us an intuition on what the consistency bias (the distance between [t0 ] and the set of all Fréchet mean in the quotient space) depends: for t0 a fixed regular point, when the variability of X (defined by E(kX − t0 k2 )) increases the mass beyond the hyperplanes on fig. 2 also increases, the distance between E(g(X, t0 ) · X) and t0 (i.e. the norm of ∇F (t0 )) augments. Therefore q the Fréchet mean should be further from t0 , (because at this point one should have ∇F (q) = 0 or q is a singular 11 point). Therefore the consistency bias appears to increase with the variability of X. By establishing a lower and upper bound of the consistency bias and by computing the consistency bias in a very simple case, sections 3.2, 3.3, 4.3 and 4.4 investigate how far this hypothesis is true. We can also wonder if the converse of theorem 3.1 is true: if the support is included in Cone(t0 ), is there consistency? We do not have a general answer to that. In the simple example section 3.3 it happens that condition (6) is necessary and sufficient. More generally the following proposition provides a partial converse: Cone(y) g · t0 y t0 O Cone(t0 ) g 0 · t0 Figure 3: y 7→ Cone(y) is continuous. When the support of the X is bounded and included in the interior of Cone(t0 ) the hatched cone. For y sufficiently close to the template t0 , the support of the X (the ball in red) is still included in Cone(y) (in grey), then F (y) = (E(kX − yk2 ). Therefore in this case, [t0 ] is at least a Karcher mean of [X]. Proposition 3.1. If the support of X is a compact set included in the interior of Cone(t0 ), then the orbit of the template [t0 ] is at least a Karcher mean of [X] (a Karcher mean is a local minimum of the variance). Proof. If the support of X is a compact set included in the interior of Cone(t0 ) then we know that X-almost surely: dQ ([X], [t0 ]) = kX −t0 k. Thus the variance at t0 in the quotient space is equal to the variance at t0 in the top space. Now by continuity of the distance map (see fig. 3) for y in a small neighbourhood of t0 , the support of X is still included in the interior of Cone(y). We still have dQ ([X], [y]) = kX − yk X-almost surely. In other words, locally around t0 , the variance in the quotient space is equal to the variance in the top space. Moreover we know that t0 = E(X) is the only global minimiser of the variance of X: m 7→ E(km − Xk2 ) = E(m). Therefore t0 is a local minimum of F the variance in the quotient space (since the two variances are locally equal). Therefore [t0 ] is at least a Karcher mean of [X] in this case. 3.2 Upper bound of the consistency bias In this Subsection we show an explicit upper bound of the consistency bias. 12 Theorem 3.3. When G is a finite group acting isometrically on M = Rn , we denote |G| the cardinal of the group G. If X is Gaussian vector: X ∼ N (t0 , s2 IdRn ), and m? ∈ argmin F , then we have the upper bound of the consistency bias: p (8) dQ ([t0 ], [m? ]) ≤ s 8 log(|G|). The proof is postponed in appendix A.3. When X ∼ N (t0 , s2 Idn ) the variability of X is σ 2 = E(||X − t0 ||2 )p= ns2 and we can write the upper bound of the bias: dQ ([t0 ], [m? ]) ≤ √σn 8 log |G|. This Theorem shows that the consistency bias is low when the variability of X is small, which tends to confirm our hypothesis in section 3.1. It is important to notice that this upper bound explodes when the cardinal of the group tends to infinity. 3.3 Study of the consistency bias in a simple example In this Subsection, we take a particular case of example 3.1: the action of translation with T = Z/2Z. We identify RT with R2 and we note by (u, v)T an element of RT . In this setting, one can completely describe the action of T on RT : 0 · (u, v)T = (u, v)T and 1 · (u, v)T = (v, u)T . The set of singularities is the line L = {(u, u)T , u ∈ R}. We note HPA = {(u, v)T , v > u} the half-plane above L and HPB the half-plane below L. This simple example will allow us to provide necessary and sufficient condition for an inconsistency at regular and singular points. Moreover we can compute exactly the consistency bias, and exhibit which parameters govern the bias. We can then find an equivalent of the consistency bias when the noise tends to zero or infinity. More precisely, we have the following theorem proved in appendix A.4: Proposition 3.2. Let X be a random variable such that E(kXk2 ) < +∞ and t0 = E(X). 1. If t0 ∈ L, there is no inconsistency if and only if the support of X is included in the line L = {(u, u), u ∈ R}. If t0 ∈ HPA (respectively in HPB ), there is no inconsistency if and only if the support of X is included in HPA ∪ L (respectively in HPB ∪ L). 2. If X is Gaussian: X ∼ N (t0 , s2 Id2 ), then the Fréchet mean of [X] exists and is unique. This Fréchet mean [m? ] is on the line passing through E(X) and perpendicular to L and the consistency bias ρ̃ = dQ ([t0 ], [m? ]) is the function of s and d = dist(t0 , L) given by:  2   Z 2 +∞ 2 r d ρ̃(d, s) = s r exp − g dr, (9) π ds 2 rs where g is a non-negative function on [0, 1] defined by g(x) = sin(arccos(x))− x arccos(x). (a) If d > 0 then s 7→ ρ̃(d, s) has an asymptotic linear expansion:  2 Z 2 +∞ 2 r ρ̃(d, s) ∼ s r exp − dr. s→∞ π 0 2 13 (10) (b) If d > 0, then ρ̃(d, s) = o(sk ) when s → 0, for all k ∈ N. (c) s → 7 ρ̃(0, s) is linear with respect to s (for d = 0 the template is a fixed point). Remark 3.1. Here, contrarily to the case of the action of rotation in [MHP16], it is not the ratio kE(X)k over the noise which matters to estimate the consistency bias. Rather the ratio dist(E(X), L) over the noise. However in both cases we measure the distance between the signal and the singularities which was {0} in [MHP16] for the action of rotations, L in this case. 4 Inconsistency for any group when the template is not a fixed point In section 3 we exhibited sufficient condition to have an inconsistency, restricted to the case of finite group acting on an Euclidean space. We now generalize this analysis to Hilbert spaces of any dimension included infinite. Let M be such an Hilbert space with its dot product noted by h , i and its associated norm k k. In this section, we do not anymore suppose that the group G is finite. In the following, we prove that there is an inconsistency in a large number of situations, and we quantify the consistency bias with lower and upper bounds. Example 4.1. The action of continuous translation: We take G = (R/Z)D acting on M = L2 ((R/Z)D , R) with: ∀τ ∈ G ∀f ∈ M (τ · f ) : t 7→ f (t + τ ) This isometric action is the continuous version of the example 3.1: the elements of M are now continuous images in dimension D. 4.1 Presence of an inconsistency We state here a generalization of theorem 3.1: Theorem 4.1. Let G be a group acting isometrically on M an Hilbert space, and X a random variable in M , E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: P (dQ ([t0 ], [X]) < kt0 − Xk) > 0, (11)   P sup hg · X, t0 i > hX, t0 i > 0. (12) or equivalently: g∈G Then [t0 ] is not a Fréchet mean of [X] in Q = M/G. The condition of this Theorem is the same condition of theorem 3.1: the support of the law of X contains points closer from gt0 for some g than t0 . Thus the condition (12) is equivalent to E(dQ ([X], [t0 ])2 ) < E(kX − t0 k2 ). In other words, the variance in the quotient space at t0 is strictly smaller than the variance in the top space at t0 . 14 Proof. First the two conditions are equivalent by definition of the quotient distance and by expansion of the square norm of kt0 − Xk and of kt0 − gXk for g ∈ G. As above, we define the variance of [X] by:   2 F (m) = E inf kg · X − mk . g∈G In order to prove this Theorem, we find a point m such that F (m) < F (t0 ), which directly implies that [t0 ] is not be a Fréchet mean of [X]. In the proof of theorem 3.1, we showed that under condition (6) we had h∇F (t0 ), t0 i < 0. This leads us to study F restricted to R+ t0 : we define for a ∈ R+ f (a) = F (at0 ) = E(inf g∈G kg · X − ak2 ). Thanks to the isometric action we can expand f (a) by:   f (a) = a2 kt0 k2 − 2aE sup hg · X, t0 i + E(kXk2 ), (13) g∈G and explicit the unique element of R+ which minimises f :   E sup hg · X, t0 i g∈G . a? = kt0 k2 (14) For all x ∈ M , we have sup hg · x, t0 i ≥ hx, t0 i and thanks to condition (12) we g∈G get: E(sup hg · X, t0 i) > E(hX, t0 i) = hE(X), t0 i = kt0 k2 , (15) g∈G which implies a? > 1. Then F (a? t0 ) < F (t0 ).
Note that kt0 k2 (a? − 1) = E supg∈G hg · X, t0 i − E(hX, t0 i) (which is positive) is exactly − h∇F (t0 ), t0 i /2 in the case of finite group, see Equation (44). Here we find the same expression without having to differentiate the variance F , which may be not possible in the current setting. 4.2 Analysis of the condition in theorem 4.1 We now look for general cases when we are sure that Equation (12) holds which implies the presence of inconsistency. We saw in section 3 that when the group was finite, it is possible to have no inconsistency only if the support of the law is included in a cone delimited by some hyperplanes. The hyperplanes were defined as the set of points equally distant of the template t0 and g ·t0 for g ∈ G. Therefore if the cardinal of the group becomes more and more important, one could think that in order to have no inconsistency the space where X should takes value becomes smaller and smaller. At the limit it leaves only at most an hyperplane. In the following, we formalise this idea to make it rigorous. We show that the cases where theorem 4.1 cannot be applied are not generic cases. 15 First we can notice that it is not possible to have the condition (12) if t0 is a fixed point under the action of G. Indeed in this case hg · X, t0 i = X, g −1 t0 = hX, t0 i). So from now, we suppose that t0 is not a fixed point. Now let us see some settings when we have the condition (11) and thus condition (12). Proposition 4.1. Let G be a group acting isometrically on an Hilbert space M , and X a random variable in M , with E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: 1. [t0 ] \ {t0 } is a dense set in [t0 ]. 2. There exists η > 0 such that the support of X contains a ball B(t0 , η). Then condition (12) holds, and the estimator is inconsistent according to theorem 4.1. B(t0 , η) O t0 g · t0 [t0 ] Figure 4: The smallest disk is included in the support of X and the points in that disk is closer from g · t0 than from t0 . According to theorem 4.1 there is an inconsistency. Proof. By density, one takes g · t0 ∈ B(t0 , η) \ {t0 } for some g ∈ G, now if we take r < min(kg ·t0 −t0 k/2, η −kg ·t0 −t0 k) then B(g ·t0 , r) ⊂ B(t0 , ). Therefore by the assumption we made on the support one has P(X ∈ B(g · t0 , r)) > 0. For y ∈ B(g · t0 , r) we have that kgt0 − yk < kt0 − yk (see fig. 4). Then we have: P (dQ ([X], [t0 ]) < kX − t0 k) ≥ P(X ∈ B(g · t0 , r)) > 0. Then we verify condition (12), and we can apply theorem 4.1. Proposition 4.1 proves that there is a large number of cases where we can ensure the presence of an inconsistency. For instance when M is a finite dimensional vector space and the random variable X has a continuous positive density (for the Lebesgue’s measure) at t0 , condition 2 of Proposition 4.1 is fulfilled. Unfortunately this proposition do not cover the case where there is no mass at the expected value t0 = E(X). This situation could appear if X has two modes for instance. The following proposition deals with this situation: 16 Proposition 4.2. Let G be a group acting isometrically on M . Let X be a random variable in M , such that E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: 1. ∃ϕ s.t. ϕ : (−a, a) → [t0 ] is C 1 with ϕ(0) = t0 , ϕ0 (0) = v 6= 0. 2. The support of X is not included in the hyperplane v ⊥ : P(X ∈ / v ⊥ ) > 0. Then condition (12) is fulfilled, which leads to an inconsistency thanks to Theorem 4.1. Proof. Thanks to the isometric action: ht0 , vi = 0. We choose y ∈ / v ⊥ in the support of X and make a Taylor expansion of the following square distance (see also Figure 5) at 0: kϕ(x) − yk2 = kt0 + xv + o(x) − yk2 = kt0 − yk2 − 2x hy, vi + o(x). Then: ∃x? ∈ (−a, a) s.t. kx? k < a, x hy, vi > 0 and kϕ(x? ) − yk < kt0 − yk. For some g ∈ G, ϕ(x? ) = g · t0 . By continuity of the norm we have: ∃r > 0 s.t. ∀z ∈ B(y, r) kg · t0 − zk < kt0 − zk. Then P(kg·t0 −Xk < kt0 −Xk) ≥ P(X ∈ B(y, r)) > 0. Theorem 4.1 applies. Proposition 4.2 was a sufficient condition on inconsistency in the case of an orbit which contains a curve. This brings us to extend this result for orbits which are manifolds: Proposition 4.3. Let G be a group acting isometrically on an Hilbert space M , X a random variable in M , with E(kXk2 ) < +∞. Assume X = t0 + σ, where t0 6= 0 and E() = 0, and E(kk) = 1. We suppose that [t0 ] is a sub-manifold of M and write Tt0 [t0 ] the linear tangent space of [t0 ] at t0 . If: P(X ∈ / Tt0 [t0 ]⊥ ) > 0, (16) P( ∈ / Tt0 [t0 ]⊥ ) > 0, (17) which is equivalent to: then there is an inconsistency. Proof. First t0 ⊥ Tt0 [t0 ] (because the action is isometric), Tt0 [t0 ]⊥ = t0 + Tt0 [t0 ]⊥ , then the event {X ∈ Tt0 [t0 ]⊥ } is equal to { ∈ Tt0 [t0 ]⊥ }. This proves that equations (16) and (17) are equivalent. Thanks to assumption (16), we can choose y in the support of X such that y ∈ / Tt0 [t0 ]⊥ . Let us take v ∈ Tt0 [t0 ] 1 such that hy, vi = 6 0 and choose ϕ a C curve in [t0 ], such that ϕ(0) = t0 and ϕ0 (0) = v. Applying proposition 4.2 we get the inconsistency. Note that Condition (16) is very weak, because Tt0 [t0 ] is a strict linear subspace of M . 17 [t0 ] Tt0 [t0 ] y g · t0 O t0 Tt0 [t0 ]⊥ Figure 5: y ∈ / Tt0 [t0 ]⊥ therefore y is closer from g · t0 for some g ∈ G than t0 itself. In conclusion, if y is in the support of X, there is an inconsistency. 4.3 Lower bound of the consistency bias Under the assumption of Theorem 4.1, we have an element a? t0 such that F (a? t0 ) < F (t0 ) where F is the variance of [X]. From this element, we deduce lower bounds of the consistency bias: Theorem 4.2. Let δ be the unique positive solution of the following equation: δ 2 + 2δ (kt0 k + EkXk) − kt0 k2 (a? − 1)2 = 0. Let δ? be the unique positive solution of the following equation:   p δ 2 + 2δkt0 k 1 + 1 + σ 2 /kt0 k2 − kt0 k2 (a? − 1)2 = 0, (18) (19) where σ 2 = E(kX − t0 k2 ) is the variability of X. Then δ and δ? are two lower bounds of the consistency bias. Proof. In order to prove this Theorem, we exhibit a ball around t0 such that the points on this ball have a variance bigger than the variance at the point a? t0 , where a? was defined in Equation (14): thanks to the expansion of the function f we did in (13) we get : F (t0 ) − F (a? t0 ) = kt0 k2 (a? − 1)2 > 0, (20) Moreover we can show (exactly like equation (43)) that for all x ∈ M :   2 2 |F (t0 ) − F (x)| ≤ E inf kg · X − t0 k − inf kg · X − xk g∈G g∈G ≤ kx − t0 k (2kt0 k + kx − t0 k + E(k2Xk)) . (21) With Equations (20) and (21), for all x ∈ B(t0 , δ) we have F (x) > F (a? t0 ). No point in that ball mapped in the quotient space is a Fréchet mean of [X]. So 18 δ is a lower bound of the consistency bias. Now by usingthe fact that E(kXk) ≤ p p kt0 k2 + σ 2 , we get: 2|F (t0 )−F (x)| ≤ 2kx−t0 k×kt0 k 1 + 1 + σ 2 /kt0 k2 + kx − t0 k2 . This proves that δ? is also a lower bound of the consistency bias. δ? is smaller than δ, but the variability of X intervenes in δ? . Therefore we propose to study the asymptotic behaviour of δ? when the variability tends to infinity. We have the following proposition: Proposition 4.4. Under the hypotheses of Theorem 4.2, we write X = t0 + σ, with E() = 0, and E(kk2 ) = 1 and note ν = E(supg∈G hg, t0 /kt0 ki) ∈ (0, 1], we have that: p δ? ∼ σ( 1 + ν 2 − 1), σ→+∞ In particular, the consistency bias explodes when the variability of X tends to infinity. Proof. First, let us prove that that ν ∈ (0, 1] under the condition (12). We have ν ≥ E(h, t0 /kt0 ki = 0. By a reductio ad absurdum: if ν = 0, then sup hg, t0 i = h, t0 i almost surely. We have then almost surely: hX, t0 i ≤ g∈G supg∈G hgX, t0 i ≤ kt0 k2 + supg∈G σ hg, t0 i = kt0 k2 + σ h, t0 i ≤ hX, t0 i , which p is in contradiction with (12). Besides ν ≤ E(kk) ≤ Ekk2 = 1 Second, we exhibit equivalent of the terms in equation (19) when σ → +∞:   p 2kt0 k 1 + 1 + σ 2 /kt0 k2 ∼ 2σ. (22) Now by definition of a? in Equation (14) and the decomposition of X = t0 + σ we get:   1 E sup (hg · t0 , t0 i + hg · σ, t0 i) − kt0 k kt0 k(a? − 1) = kt0 k g∈G   1 kt0 k(a? − 1) ≤ E sup hg · σ, t0 i = σν (23) kt0 k g∈G   1 kt0 k(a? − 1) ≥ E sup hg · σ, t0 i − 2kt0 k = σν − 2kt0 k, (24) kt0 k g∈G The lower bound and the upper bound of kt0 k(a? −1) found in (23) and (24) are both equivalent to σν, when σ → +∞. Then the constant term of the quadratic Equation (19) has an equivalent: − kt0 k2 (a? − 1)2 ∼ −σ 2 ν 2 . (25) Finallye if we solve the quadratic Equation (19), we write δ? as a function of the coefficients of the quadratic equation (19). We use the equivalent of each of these terms thanks to equation (22) and (25), this proves proposition 4.4. 19 Remark 4.1. Thanks to inequality (24), if ktσ0 k < ν2 , then kt0 k2 (1 − a? )2 ≥ (σν −2kt0 k)2 , then we write δ? as a function of the coefficients of Equation (19), we obtain a lower bound of the inconsistency bias as a function of kt0 k, σ and ν for σ > 2kt0 k/ν: q p p δ? 2 2 ≥ −(1 + 1 + σ /kt0 k ) + (1 + 1 + σ 2 /kt0 k2 )2 + (σν/kt0 k − 2)2 . kt0 k Although the constant ν intervenes in this lower bound, it is not an explicit term. We now explicit its behaviour depending on t0 . We remind that:   1 ν= E sup hg, t0 i . kt0 k g∈G To this end, we first note that the set of fixed points under the action of G is a closed linear space, (because we can write it as an intersection of the kernel of the continuous and linear functions: x 7→ g · x − x for all g ∈ G). We denote by p the orthogonal projection on the set of fixed points Fix(M ). Then for x ∈ M , we have: dist(x, Fix(M )) = kx − p(x)k. Which yields: hg, t0 i = hg, t0 − p(t0 )i + h, p(t0 )i . (26) The right hand side of Equation (26) does not depend on g as p(t0 ) ∈ Fix(M ). Then:   kt0 kν = E sup hg, t0 − p(t0 )i + hE(), p(t0 )i . g∈G Applying the Cauchy-Schwarz inequality and using E() = 0, we can conclude that: ν≤ 1 dist(t0 , Fix(M ))E(kk) = dist(t0 /kt0 k, Fix(M ))E(kk). kt0 k (27) This leads to the following comment: our lower bound of the consistency bias is smaller when our normalized template t0 /kt0 k is closer to the set of fixed points. 4.4 Upper bound of the consistency bias In this Section, we find a upper bound of the consistency bias. More precisely we have the following Theorem: Proposition 4.5. Let X be a random variable in M , such that X = t0 + σ where σ > 0, E() = 0 and E(||||2 ) = 1. We suppose that [m? ] is a Fréchet mean of [X]. Then we have the following upper bound of the quotient distance between the orbit of the template t0 and the Fréchet mean of [X]: p dQ ([m? ], [t0 ]) ≤ σν(m∗ −m0 )+ σ 2 ν(m∗ − m0 )2 + 2dist(t0 , Fix(M ))σν(m∗ − m0 ), (28) where we have noted ν(m) = E(supg hg, m/kmki) ∈ [0, 1] if m 6= 0 and ν(0) = 0, and m0 the orthogonal projection of t0 on F ix(M ). 20 Note that we made no hypothesis on the template pin this proposition. We deduce from Equation (28) that √ dQ ([m? ], [t0 ]) ≤ σ + σ 2 + 2σdist(t0 , Fix(M )) is a O(σ) when σ → ∞, but a O( σ) when σ → 0, in particular the consistency bias can be neglected when σ is small. Proof. First we have: F (m? ) ≤ F (t0 ) = E(inf ||t0 − g(t0 + σ)||2 ) ≤ E(||σ||2 ) = σ 2 . g (29) Secondly we have for all m ∈ M , (in particular for m? ): F (m) = E(inf (km − gt0 k2 + σ 2 kk2 − 2hgσ, m − gt0 i)) ≥ dQ ([m], [t0 ])2 + σ 2 − 2E(suphσ, gmi). g (30) g With Inequalities (29) and (30) one gets: dQ ([m∗ ], [t0 ])2 ≤ 2E(sup hσ, gm? i) = 2σν(m? )||m? ||, g note that at this point, if m? = 0 then E(supg hσ, gm? i) = 0 and ν(m? ) = 0 although Equation (4.4) is still true even if m? = 0. Moreover with the triangular inequality applied at [m? ], [0] and [t0 ], one gets: km? k ≤ kt0 k + dQ ([m? ], [t0 ]) and then: dQ ([m∗ ], [t0 ])2 ≤ 2σν(m? )(dQ ([m∗ ], [t0 ]) + kt0 k). (31) We can solve inequality (31) and we get: p dQ ([m? ], [t0 ]) ≤ σν(m? ) + σ 2 ν(m? )2 + 2kt0 kσν(m? ), (32) We note by FX instead of F the variance in the quotient space of [X], and we want to apply inequality (32) to X − m0 . As m0 is a fixed point:   2 FX (m) = E inf kX − m0 − g · (m − m0 )k = FX−m0 (m − m0 ) g∈G Then m? minimises FX if and only if m? − m0 minimises FX−m0 . We apply Equation (32) to X − m0 , with E(X − m0 ) = t0 − m0 and [m? − m0 ] a Fréchet mean of [X − m0 ]. We get: p dQ ([m? −m0 ], [t0 −m0 ]) ≤ σν(m∗ −m0 )+ σ 2 ν(m∗ − m0 )2 + 2kt0 − m0 kσν(m∗ − m0 ). Moreover dQ ([m? ], [t0 ]) = dQ ([m? − m0 ], [t0 − m0 ]), which concludes the proof. 21 4.5 Empirical Fréchet mean In practice, we never compute the Fréchet mean in quotient space, only the empirical Fréchet mean in quotient space when the size of a sample is supposed to be large enough. If the empirical Fréchet in the quotient space means converges to the Fréchet mean in the quotient space then we can not use these empirical Fréchet mean in order to estimate the template. In [BB08], it has been proved that the empirical Fréchet mean converges to the Fréchet mean with a √1n convergence speed, however the law of the random variable is supposed to be included in a ball whose radius depends on the geometry on the manifold. Here we are not in a manifold, indeed the quotient space contains singularities, moreover we do not suppose that the law is necessarily bounded. However in [Zie77] the empirical Fréchet means is proved to converge to the Fréchet means but no convergence rate is provided. We propose now to prove that the quotient distance between the template and the empirical Fréchet mean in quotient space have an lower bound which is the asymptotic of the one lower bound of the consistency bias found in (18). Take X, X1 , . . . , Xn independent and identically distributed (with t0 = E(X) not a fixed point). We define the empirical variance of [X] by: n m ∈ M 7→ Fn (m) = n 1X 1X dQ ([m], [Xi ])2 = inf km − g · Xi k2 , n i=1 n i=1 g∈G and we say that [mn? ] is a empirical Fréchet mean of [X] if mn? is a global minimiser of Fn . Proposition 4.6. Let X, X1 , . . . , Xn independent and identically distributed random variables, with t0 = E(X). Let be [mn? ] be an empirical Fréchet mean of [X]. Then δn is a lower bound of the quotient distance between the orbit of the template and [mn? ], where δn is the unique positive solution of: ! n 1X 2 kXi k δ − kt0 k2 (an? − 1)2 = 0. δ + 2 ||t0 || + n i=1 an? is defined like a? in section 4.1 by: n P 1 sup hg · Xi , t0 i n i=1g∈G an? = . kt0 k2 We have that δn → δ by the law of large numbers. The proof is a direct application of theorem 4.2, but applied to the empirical law of X given by the realization of X1 , . . . , Xn . 4.6 Examples In this Subsection, we discuss, in some examples, the application of theorem 4.1 and see the behaviour of the constant ν. This constant intervened in lower bound of the consistency bias. 22 4.6.1 Action of translation on L2 (R/Z) We take an orbit O = [f0 ], where f0 ∈ C 2 (R/Z), non constant. We show easily that O is a manifold of dimension 1 and the tangent space at f0 is2 Rf00 . Therefore a sufficient condition on X such that E(X) = f0 to have an inconsistency is: P(X ∈ / f00⊥ ) > 0 according to proposition 4.3. Now if we denote by 1 the constant function on R/Z equal to 1. We have in this setting: that the set of fixed points under the action of G is the set of constant functions: Fix(M ) = R1 and: s 2 Z 1 Z 1 f0 (t) − f0 (s)ds dt. dist(f0 , Fix(M )) = kf0 − hf0 , 1i 1k = 0 0 This distance to the fixed points is used in the upper bound of the constant ν in Equation (27). Note that if f0 is not differentiable, then [f0 ] is not necessarily a manifold, and (4.3) does not apply. However proposition 4.1 does: if f0 is not a constant function, then [f0 ] \ {f0 } is dense in [f0 ]. Therefore as soon as the support of X contains a ball around f0 , there is an inconsistency. 4.6.2 Action of discrete translation on RZ/NZ We come back on example 3.1, with D = 1 (discretised signals). For some signal t0 , ν previously defined is:   1 ν= E max h, τ · t0 i . kt0 k τ ∈Z/NZ Therefore if we have a sample of size I of  iid, then: ν= I 1X 1 lim max hi , τi · t0 i , kt0 k I→+∞ I i=1 τi ∈Z/N Z By an exhaustive research, we can find the τi ’s which maximise the dot product, then with this sample and t0 we can approximate ν. We have done this approximation for several signals t0 on fig. 6. According the previous results, the bigger ν is, the more important the lower bound of the consistency bias is. We remark that the ν estimated is small, ν  1 for different signals. 4.6.3 Action of rotations on Rn Now we consider the action of rotations on Rn with a Gaussian noise. Take X ∼ N (t0 , s2 Idn ) then the variability of X is ns2 , then X has a decomposition: ] − 21 , 12 [ → O is a local parametrisation of O: f0 = ϕ(0), and we t 7→ f0 (. − t) 0 check that: lim kϕ(x) − ϕ(0) − xf0 kL2 = 0 with Taylor-Lagrange inequality at the order 2 Indeed ϕ : x→0 2. As a conclusion ϕ is differentiable at 0, and it is an immersion (since f00 6= 0), and D0 ϕ : x 7→ xf00 , then O is a manifold of dimension 1 and the tangent space of O at f0 is: Tf0 O = D0 ϕ(R) = Rf00 . 23 nu value for each signal 0.4 0.14456 0.082143 0.24981 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 0.2 0.4 0.6 0.8 1 Figure 6: Different signals and their ν approximated with a sample of size 103 in RZ/100Z .  is here a Gaussian noise in RZ/100Z , such that E() = 0 and E(kk2 ) = 1. For instance the blue signal is a signal defined randomly, and when we approximate the ν which corresponds to that t0 we find ' 0.25. √ X = t0 + ns with E() = 0 and E(kk2 ) = 1. According to proposition 4.4 we have by noting δ? the lower bound of the consistency bias when s → ∞: p √ δ? → n(−1 + 1 + ν 2 ). s Now ν = E(supg∈G hg, t0 )i /kt0 k = E(kk) → 1 when n tends to infinity (expected value of the Chi distribution) we have that for n large enough: √ √ δ? ' n( 2 − 1). s→∞ s We compare this result with the exact computation of the consistency bias (noted here CB) made by Miolane et al. [MHP16], which writes with our current notations: CB √ Γ((n + 1)/2) lim = 2 . s→∞ s Γ(n/2) lim Using a standard Taylor expansion on the Gamma function, we have that for n large enough: CB √ lim ' n. s→∞ s As a conclusion, when the dimension of the space is large enough our lower bound and the exact computation of the √ bias have the same asymptotic behaviour. It differs only by the constant 2 − 1 ' 0.4 in our lower bound, 1 in the work of Miolane et al. [MP15]. 24 5 Fréchet means top and quotient spaces are not consistent when the template is a fixed point In this Section, we do not assume that the top space M is a vector space, but rather a manifold. We need then to rewrite the generative model likewise: let t0 ∈ M , and X any random variable of M such as t0 is a Fréchet mean of X. Then Y = S · X is the observed variable where S is a random variable whose value are in G. In this Section we make the assumption that the template t0 is a fixed point under the action of G. 5.1 Result Let X be a random variable on M and define the variance of X as: E(m) = E(dM (m, X)2 ). We say that t0 is a Fréchet mean of X if t0 is a global minimiser of the variance E. We prove the following result: Theorem 5.1. Assume that M is a complete finite dimensional Riemannian manifold and that dM is the geodesic distance on M . Let X be a random variable on M , with E(d(x, X)2 ) < +∞ for some x ∈ M . We assume that t0 is a fixed point and a Fréchet mean of X and that P(X ∈ C(t0 )) = 0 where C(t0 ) is the cut locus of t0 . Suppose that there exists a point in the support of X which is not a fixed point nor in the cut locus of t0 . Then [t0 ] is not a Fréchet mean of [X]. The previous result is finite dimensional and does not cover interesting infinite dimensional setting concerning curves for instance. However, a simple extension to the previous result can be stated when M is a Hilbert vector space since then the space is flat and some technical problems like the presence of cut locus point do not occur. Theorem 5.2. Assume that M is a Hilbert space and that dM is given by the Hilbert norm on M . Let X be a random variable on M , with E(kXk2 ) < +∞. We assume that t0 = E(X). Suppose that there exists a point in the support of the law of X that is not a fixed point for the action of G. Then [t0 ] is not a Fréchet mean of [X]. Note that the reciprocal is true: if all the points in the support of the law of X are fixed points, then almost surely, for all m ∈ M and for all g ∈ G we have: dM (X, m) = dM (g · X, m) = dQ ([X], [m]). Up to the projection on the quotient, we have that the variance of X is equal to the variance of [X] in M/G, therefore [t0 ] is a Fréchet mean of [X] if and only if t0 is a Fréchet mean of X. There is no inconsistency in that case. 25 Example 5.1. Theorem 5.2 covers the interesting case of the Fisher Rao metric on functions: F = {f : [0, 1] → R | f is absolutely continuous}. Then considering for G the group of smooth diffeomorphisms γ on [0, 1] such that γ(0) = 0 and γ(1) = 1, we have a right group action G × F → F given by γ · f = f ◦ γ. The Fisher Rao metric is built as a pull back metric q of the 2 2 ˙ L ([0, 1], R) space through the map Q : F → L given by: Q(f ) = f / |f˙|. This square root trick is often used, see for instance [KSW11]. Note that in this case, Rt Q is a bijective mapping with inverse given by q 7→ f with √ f (t) = 0 q(s)|q(s)|ds. We can define a group action on M = L2 as: γ · q = q ◦ γ γ̇, for which one can check easily by a change of variable that: p p kγ · q − γ · q 0 k2 = kq ◦ γ γ̇ − q 0 ◦ γ γ̇k2 = kq − q 0 k2 . So up to the mapping Q, the Fisher Rao metric on curve corresponds to the situation M where theorem 5.2 applies. Note that in this case the set of fixed points under the action of G corresponds in the space F to constant functions. We can also provide an computation of the consistency bias in this setting: Proposition 5.1. Under the assumptions of theorem 5.2, we write X = t0 + σ where t0 is a fixed point, σ > 0, E() = 0 and E(kk2 ) = 1, if there is a Fréchet mean of [X], then the consistency bias is linear with respect to σ and it is equal to: σ sup E(sup hv, g · i). kvk=1 g∈G Proof. For λ > 0 and kvk = 1, we compute the variance F in the quotient space of [X] at the point t0 + λv. Since t0 is a fixed point we get: F (t0 +λv) = E( inf kt0 +λv−gXk2 ) = E(kXk2 )−kt0 k2 −2λE(sup hv, g(X − t0 )i)+λ2 . g∈G g Then we minimise F with respect to λ, and after we minimise with respect to v (with kvk = 1). Which concludes. 5.2 5.2.1 Proofs of these theorems Proof of theorem 5.1 We start with the following simple result, which aims to differentiate the variance of X. This classical result (see [Pen06] for instance) is proved in appendix B in order to be the more self-contained as possible: Lemma 5.1. Let X a random variable on M such that E(d(x, X)2 ) < +∞ for some x ∈ M . Then the variance m 7→ E(m) = E(dM (m, X)2 ) is a continuous 26 function which is differentiable at any point m ∈ M such that P(X ∈ C(m)) = 0 where C(m) is the cut locus of m. Moreover at such point one has: ∇E(m) = −2E(logm (X)), where logm : M \ C(m) → Tm M is defined for any x ∈ M \ C(m) as the unique u ∈ Tm M such that expm (u) = x and kukm = dM (x, m). We are now ready to prove theorem 5.1. Proof. (of theorem 5.1) Let m0 be a point in the support of M which is not a fixed point and not in the cut locus of t0 . Then there exists g0 ∈ G such that m1 = g0 m0 6= m0 . Note that since x 7→ g0 x is a symmetry (the distance is equivariant under the action of G) have that m1 = g0 m0 ∈ / C(g0 t0 ) = C(t0 ) (t0 is a fixed point under the action of G). Let v0 = logt0 (m0 ) and v1 = logt0 (m1 ). We have v0 6= v1 and since C(t0 ) is closed and the logt0 is continuous application on M \ C(t0 ) we have: lim →0 P(X 1 E(1X∈B(m0 ,) logt0 (X)) = v0 . ∈ B(m0 , )) (we use here the fact that since m0 is in the support of the law of X, P(X ∈ B(m0 , )) > 0 for any  > 0 so that the denominator does not vanish and the fact that since M is a complete manifold, it is a locally compact space (the closed balls are compacts) and logt0 is locally bounded). Similarly: lim →0 P(X 1 E(1X∈B(m0 ,) logt0 (g0 X)) = v1 . ∈ B(m0 , )) Thus for sufficiently small  > 0 we have (since v0 6= v1 ): E(logt0 (X)1X∈B(m0 ,) ) 6= E(logt0 (g0 X)1X∈B(m0 ,) ). (33) By using using a reductio ad absurdum, we suppose that [t0 ] is a Fréchet mean of [X] and we want to find a contradiction with (33). In order to do that we introduce simple functions as the function x 7→ 1x∈B(m0 ,) which intervenes in Equation (33). Let s : M → G be a simple function (i.e. a measurable function with finite number of values in G). Then x 7→ h(x) = s(x)x is a measurable function3 . Now, let Es (x) = E(d(x, s(X)X)2 ) be the variance of the variable s(X)X. Note that (and this is the main point): ∀g ∈ G 3 Indeed if: s = dM (t0 , x) = dM (gt0 , gx) = dM (t0 , gx) = dQ ([t0 ], [x]), n P gi 1Ai where (Ai )1≤i≤n is a partition of M (such that the sum is always i=1 defined). Then for any Borel set B ⊂ M we have: h−1 (B) = n S gi−1 (B) ∩ Ai is a measurable i=1 set since x 7→ gi x is a measurable function. 27 we have: Es (t0 ) = E(t0 ). Assume now that [t0 ] a Fréchet mean for [X] on the quotient space and let us show that Es has a global minimum at t0 . Indeed for any m, we have: Es (m) = E(dM (m, s(X)X)2 ) ≥ E(dQ ([m], [X])2 ) ≥ E(dQ ([t0 ], [X])2 ) = Es (t0 ). Now, we want to apply lemma 5.1 to the random variables s(X)X and X at the point t0 . Since we assume that X ∈ / C(t0 ) almost surely and X ∈ / C(t0 ) implies s(X)X ∈ / C(t0 ) we get P(s(X)X ∈ C(t0 )) = 0 and the lemma 5.1 applies. As t0 is a minimum, we already know that the differential of Es (respectively E) at t0 should be zero. We get: E(logt0 (X)) = E(logt0 (s(X)X)) = 0. (34) Now we apply Equation (34) to a particular simple function defined by s(x) = g0 1x∈B(m0 ,) + eG 1x∈B(m . We split the two expected values in (34) into two / 0 ,) parts: E(logt0 (X)1X∈B(m0 ,) ) + E(logt0 (X)1X ∈B(m ) = 0, (35) / 0 ,) ) = 0. E(logt0 (g0 X)1X∈B(m0 ,) ) + E(logt0 (X)1X ∈B(m / 0 ,) (36) By substrating (35) from (36), one gets: E(logt0 (X)1X∈B(m0 ,) ) = E(logt0 (g0 X)1X∈B(m0 ,) ), which is a contradiction with (33). Which concludes. 5.2.2 Proof of theorem 5.2 Proof. The extension to theorem 5.2 is quite straightforward. In this setting many things are now explicit since d(x, y) = kx − yk , ∇x d(x, y)2 = 2(x − y), logx (y) = y − x and the cut locus is always empty. It is then sufficient to go along the previous proof and to change the quantity accordingly. Note that the local compactness of the space is not true in infinite dimension. However this was only used to prove that the log was locally bounded but this last result is trivial in this setting. 6 Conclusion and discussion In this article, we exhibit conditions which imply that the template estimation with the Fréchet mean in quotient space is inconsistent. These conditions are rather generic. As a result, without any more information, a priori there is inconsistency. The behaviour of the consistency bias is summarized in table 1. Surely future works could improve these lower and upper bounds. In a more general case: when we take an infinite-dimensional vector space quotiented by a non isometric group action, is there always an inconsistency? An important example of such action is the action of diffeomorphisms. Can we estimate the consistency bias? In this setting, one estimates the template (or 28 Table 1: Behaviour of the consistency bias with respect to σ 2 the variability of X = t0 + σ. The constants Ki ’s depend on the kind of noise, on the template t0 and on the group action. Consistency bias : CB G is any group Supplementary properties for G a finite group √ Upper bound of CB CB ≤ σ + 2 σ 2 + K1 σ CB ≤ K2 σ (theorem 3.3) (proposition 4.5) Lower bound of CB for σ → ∞ CB ≥ L ∼ K3 σ (proposition 4.4) σ→∞ when the template is not a fixed point √ Behavior of CB for σ → 0 when CB ≤ U ∼ K4 σ CB = o(σ k ), ∀k ∈ N in the σ→0 0 the template is not a fixed point section 3.3, can we extend this result for finite group? CB = σ sup E(supg∈G hv, gi) (proposition 5.1) CB when the template is a fixed point kvk=1 an atlas), but does not exactly compute the Fréchet mean in quotient space, because a regularization term is added. In this setting, can we ensure that the consistency bias will be small enough to estimate the original template? Otherwise, one has to reconsider the template estimation with stochastic algorithms as in [AKT10] or develop new methods. A Proof of theorems for finite groups’ setting A.1 Proof of theorem 3.2: differentiation of the variance in the quotient space In order to show theorem 3.2 we proceed in three steps. First we see some following properties and definitions which will be used. Most of these properties are the consequences of the fact that the group G is finite. Then we show that the integrand of F is differentiable. Finally we show that we can permute gradient and integral signs. 1. The set of singular points in Rn , is a null set (for the Lebesgue’s measure), since it is equal to: [ ker(x 7→ g · x − x), g6=eG a finite union of strict linear subspaces of Rn thanks to the linearity and effectively of the action and to the finite group. 2. If m is regular, then for g, g 0 two different elements of G, we pose: H(g · m, g 0 · m) = {x ∈ Rn , kx − g · mk = kx − g 0 · mk}. Moreover H(g · m, g 0 · m) = (g · m − g 0 · m)⊥ is an hyperplane. 29 3. For m a regular point we define the set of points which are equally distant from two different points of the orbit of m: [ H(g · m, g 0 · m). Am = g6=g 0 Then Am is a null set. For m regular and x ∈ / Am the minimum in the definition of the quotient distance : dQ ([m], [x]) = minkm − g · xk, g∈G (37) is reached at a unique g ∈ G, we call g(x, m) this unique element. 4. By expansion of the squared norm: g minimises km − g · xk if and only if g maximises hm, g · xi. 5. If m is regular and x ∈ / Am then: ∀g ∈ G \ {g(x, m)}, km − g(x, m) · xk < km − g · xk, by continuity of the norm and by the fact that G is a finite group, we can find α > 0, such that for µ ∈ B(m, α) and y ∈ B(x, α): ∀g ∈ G \ {g(x, m)} kµ − g(x, m) · yk < kµ − g · yk. (38) Therefore for such y and µ we have: g(x, m) = g(y, µ). 6. For m a regular point, we define Cone(m) the convex cone of Rn : Cone(m) = {x ∈ Rn / ∀g ∈ G kx − mk ≤ kx − g · mk} (39) n = {x ∈ R / ∀g ∈ G hm, xi ≥ hgm, xi}. This is the intersection of |G| − 1 half-spaces: each half space is delimited by H(m, gm) for g 6= eG (see fig. 1). Cone(m) is the set of points whose projection on [m] is m, (where the projection of one point p on [m] is one point g · m which minimises the set {kp − g · mk, g ∈ G}). 7. Taking a regular T point m allows us to see the T quotient. For every point x ∈ Rn we have: [x] Cone(m) 6= ∅, card([x] Cone(m)) ≥ 2 if and only if x ∈ Am . The borders of the cone is Cone(m)\Int(Cone(m)) = Cone(m)∩ Am (we denote by Int(A) the interior of a part A). Therefore Q = Rn /G can be seen like Cone(m) whose border have been glued together. The proof of theorem 3.2 is the consequence of the following lemmas. The first lemma studies the differentiability of the integrand, and the second allows us to permute gradient and integral sign. Let us denote by f the integrand of F: 30 ∀ m, x ∈ M f (x, m) = minkm − g · xk2 . (40) g∈G Thus we have: F (m) = E(f (X, m)). The min of differentiable functions is not necessarily differentiable, however we prove the following result: Lemma A.1. Let m0 be a regular point, if x ∈ / Am0 then m 7→ f (x, m) is differentiable at m0 , besides we have: ∂f (x, m0 ) = 2(m0 − g(x, m0 ) · x) ∂m (41) Proof. If m0 is regular and x ∈ / Am0 then we know from the item 5 of the appendix A.1 that g(x, m0 ) is locally constant. Therefore around m0 , we have: f (x, m) = km − g(x, m0 ) · xk2 , which can differentiate with respect to m at m0 . This proves the lemma A.1. Now we want to prove that we can permute the integral and the gradient sign. The following lemma provides us a sufficient condition to permute integral and differentiation signs thanks to the dominated convergence theorem: Lemma A.2. For every m0 ∈ M we have the existence of an integrable function Φ : M → R+ such that: ∀m ∈ B(m0 , 1), ∀x ∈ M |f (x, m0 ) − f (x, m)| ≤ km − m0 kΦ(x). (42) Proof. For all g ∈ G, m ∈ M we have: kg · x − m0 k2 − kg · x − mk2 = hm − m0 , 2g · x − (m0 + m)i ≤ km − m0 k × (km0 + mk + k2xk) 2 minkg · x − m0 k ≤ km − m0 k (km0 + mk + k2xk) + kg · x − mk2 g∈G minkg · x − m0 k2 ≤ km − m0 k (km0 + mk + k2xk) + minkg · x − mk2 g∈G 2 g∈G 2 minkg · x − m0 k − minkg · x − mk ≤ km − m0 k (2km0 k + km − m0 k + k2xk) g∈G g∈G By symmetry we get also the same control of f (x, m) − f (x, m0 ), then: |f (x, m0 ) − f (x, m)| ≤ km0 − mk (2km0 k + km − m0 k + k2xk) (43) The function Φ should depend on x or m0 , but not on m. That is why we take only m ∈ B(m0 , 1), then we replace km−m0 k by 1 in (43), which concludes. 31 A.2 Proof of theorem 3.1: the gradient is not zero at the template To prove it, we suppose that ∇F (t0 ) = 0, and we take the dot product with t0 : h∇F (t0 ), t0 i = 2E(hX, t0 i − hg(X, t0 ) · X, t0 i) = 0. (44) The item 4 of (x, m) 7→ g(x, m) seen at appendix A.1 leads to: hX, t0 i − hg(X, t0 ) · X, t0 i ≤ 0 almost surely. So the expected value of a non-positive random variable is null. Then hX, t0 i − hg(X, t0 ) · X, t0 i = 0 almost surely hX, t0 i = hg(X, t0 ) · X, t0 i almost surely. Then g = eG maximizes the dot product almost surely. Therefore (as we know that g(X, t0 ) is unique almost surely, since t0 is regular): g(X, t0 ) = eG almost surely, which is a contradiction with Equation (6). A.3 Proof of theorem 3.3: upper bound of the consistency bias In order to show this Theorem, we use the following lemma: Lemma A.3. We write X = t0 + where E() = 0 and we make the assumption that the noise  is a subgaussian random variable. This means that it exists c > 0 such that:   2 s kmk2 . (45) ∀m ∈ M = Rn , E(exp(h, mi)) ≤ c exp 2 If for m ∈ M we have: p ρ̃ := dQ ([m], [t0 ]) ≥ s 2 log(c|G|), (46) p ρ̃2 − ρ̃s 8 log(c|G|) ≤ F (m) − E(kk2 ). (47) then we have: Proof. (of lemma A.3) First we expand the right member of the inequality (47):   E(kk2 ) − F (m) = E max(kX − t0 k2 − kX − gmk2 ) g∈G We use the formula kAk2 − kA + Bk2 = −2 hA, Bi − kBk2 with A = X − t0 and B = t0 − gm:  
E(kk2 ) − F (m) = E max −2 hX − t0 , t0 − gmi − kt0 − gmk2 = E(max ηg ), g∈G g∈G (48) 32 with ηg = −kt0 − gmk2 + 2 h, gm − t0 i. Our goal is to find a lower bound of F (m) − E(kk2 ), that is why we search an upper bound of E(maxηg ) with the g∈G Jensen’s inequality. We take x > 0 and we get by using the assumption (45):   X exp(xE(max ηg )) ≤ E(exp(max xηg )) ≤ E  exp(xηg ) g∈G g∈G ≤ X g∈G 2 exp(−xkt0 − gmk )E(exp(h, 2x(gm − t0 )i) g X ≤c exp(−xkt0 − gmk2 ) exp(2s2 x2 kgm − t0 k2 ) g X ≤c exp(kgm − t0 k2 (−x + 2x2 s2 )) (49) g Now if (−x + 2t2 x2 ) < 0, we can take an upper bound of the sum sign in (49) by taking the smallest value in the sum sign, which is reached when g minimizes kg · m − t0 k multiplied by the number of elements summed. Moreover (−x + 2x2 s) < 0 ⇐⇒ 0 < x < 2s12 . Then we have: exp(xE(max ηg )) ≤ c|G| exp(ρ̃2 (−x + 2x2 s2 )) as soon as 0 < x < g∈G 1 . 2s2 Then by taking the log: E(maxηg ) ≤ g∈G log c|G| + (2xs2 − 1)ρ̃2 . x (50) Now we find the x which optimizes inequality (50).p By differentiation, the right member of inequality (50) is minimal for x? = log c|G|/2/(sρ̃) which is a valid choice because x? ∈ (0, 2s12 ) by using the assumption (46). With the equations (48) and (50) and x? we get the result. Proof. (of theorem 3.3) We take m? ∈ argmin F , ρ̃ = dQ ([m? ], [t0 ]), and  = 2 2 X − tp 0 . We have: F (m? ) ≤ F (t0 ) ≤ E(kk ) then F (m? ) − E(kk ) ≤ 0. If ρ̃ > s 2 log(|G|) then we can apply lemma A.3 with c = 1. Thus: p ρ̃2 − ρ̃s 8 log(|G|) ≤ 2F (m? ) − E(kk2 ) ≤ 0, p p which yields to ρ̃ ≤ s 8 log(|G|). If ρ̃ ≤ s 2 log(|G|), we have nothing to prove. Note that the proof of this upper bound does not use the fact that the action is isometric, therefore this upper bound is true for every finite group action. 33 A.4 Proof of proposition 3.2: inconsistency in R2 for the action of translation Proof. We suppose that E(X) ∈ HPA ∪ L. In this setting we call τ (x, m) one of element of the group G = T which minimises kτ · x − mk see (37) instead of g(x, m). The variance in the quotient space at the point m is:   F (m) = E min kτ · X − mk2 = E(kτ (X, m) · X − mk2 ). τ ∈Z/2Z As we want to minimize F and F (1 · m) = F (m), we can suppose that m ∈ HPA ∪ L. We can completely write what take τ (x, m) for x ∈ M : • If x ∈ HPA ∪ L we can set τ (x, m) = 0 (because in this case x, m are on the same half plane delimited by L the perpendicular bisector of m and −m). • If x ∈ HPB then we can set τ (x, m) = 1 (because in this case x, m are not on the same half plane delimited by L the perpendicular bisector of m and −m). This allows use to write the variance at the point m ∈ HPA :


F (m) = E kX − mk2 1{X∈HPA ∪L} + E k1 · X − mk2 1{X∈HPB } Then we define the random variable Z by: Z = X1X∈HPA ∪L + 1 · X1X∈HPB , such that for m ∈ HPA we have: F (m) = E(kZ − mk2 ) and F (m) = F (1 · m). Thus if m? is a global minimiser of F , then m? = E(Z) or m? = 1 · E(Z). So the Fréchet mean of [X] is [E(Z)]. Here instead of using theorem 3.1, we can work explicitly: Indeed there is no inconsistency if and only if E(Z) = E(X), (E(Z) = 1 · E(X) would be another possibility, but by assumption E(Z), E(X) ∈ HPA ), by writing X = X1X∈HPA + X1X∈HPB ∪L , we have: E(Z) = E(X) ⇐⇒ E(1 · X1X∈HPB ∪L ) = E(X1X∈HPB ∪L ) ⇐⇒ 1 · E(X1X∈HPB ∪L ) = E(X1X∈HPB ∪L ) ⇐⇒ E(X1X∈HPB ∪L ) ∈ L ⇐⇒ P(X ∈ HPB ) = 0, Therefore there is an inconsistency if and only if P(X ∈ HPB ) > 0 (we remind that we made the assumption that E(X) ∈ HPA ∪ L). If E(X) is regular (i.e. E(X) ∈ / L), then there is an inconsistency if and only if X takes values in HPB , (this is exactly the condition of theorem 3.1, but in this particular case, this is a necessarily and sufficient condition). This proves point 1. Now we make the assumption that X follows a Gaussian noise in order compute E(Z) (note that we could take another noise, as long as we are able to compute E(Z)). For that we convert to polar coordinates: (u, v)T = E(X) + (r cos θ, r sin θ)T where r > 0 et θ ∈ [0, 2π]. We also define: d = dist(E(X), L), E(X) is a regular point if 34 and only if d > 0. We still suppose that E(X) = (α, β)T ∈ HPA ∪ L. First we parametrise in function of (r, θ) the points which are in HPB : v < u ⇐⇒ β + r sin θ < α + r cos θ ⇐⇒ β−α √ π < 2 cos(θ + ) r 4 d π < cos(θ + ) r h 4 i π π ⇐⇒ θ ∈ − − arccos(d/r), − + arccos(d/r) and d < r 4 4 ⇐⇒ Then we compute E(Z): E(Z) =E(X1X∈HPA ) + E(1 · X1X∈HPB )    exp − r2 Z d Z 2π  2s2 α + r cos θ rdθdr E(Z) = 2 β + r sin θ 2πs 0 0    exp − r2 Z +∞ Z 2π− π4 −arccos( dr )  2 2s α + r cos θ rdrdθ + 2 β + r sin θ d π 2πs arccos( r )− 4 d  2 r  Z +∞ Z − π4 +arccos( dr )  β + r sin θ exp − 2s2 + rdrdθ α + r cos θ d 2πs2 d −π 4 −arccos( r )   Z +∞ 2 r2 √ r exp(− 2s d 2) =E(X) + 2g dr × (−1, 1)T , 2 πs r d We compute ρ̃ = dQ ([E(X)], [E(Z)]) where dQ is the distance in the quotient space defined in (1). As we know that E(X), E(Z) are in the same half-plane delimited by L, we have: ρ̃ = dQ ([E(Z)], [E(X)]) = kE(Z) − E(X)k. This proves eq. (9), note that items 2a to 2c are the direct consequence of eq. (9) and basic analysis. B Proof of lemma 5.1: differentiation of the variance in the top space Proof. By triangle inequality it is easy to show that E is finite and continuous everywhere. Moreover, it is a well known fact that x 7→ dM (x, z)2 is differentiable at any m ∈ M \ C(z) (i.e. z ∈ / C(m)) with derivative −2 logm (z). Now since: |dM (x, z)2 − dM (y, z)2 | = |dM (x, z) − dM (y, z)kdM (x, z) + dM (y, z)| ≤ dM (x, y)(2dM (x, z) + dM (y, x)), we get in a local chart φ : U → V ⊂ Rn at t = φ(m) we have locally around t that: h 7→ dM (φ−1 (t), φ−1 (t + h)), 35 is smooth and |dM (φ−1 (t), φ−1 (t+h))| ≤ C|h| for a C > 0. Hence for sufficiently small h, |dM (φ−1 (t), z)2 − dM (φ−1 (t + h), z)2 | ≤ C|h|(2dM (m, z) + 1). We get the result from dominated convergence Lebesgue theorem with E(dM (m, X)) ≤ E(dM (m, X)2 + 1) < +∞. References [AAT07] Stéphanie Allassonnière, Yali Amit, and Alain Trouvé. Towards a coherent statistical framework for dense deformable template estimation. 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Tiny SSD: A Tiny Single-shot Detection Deep Convolutional Neural Network for Real-time Embedded Object Detection arXiv:1802.06488v1 [] 19 Feb 2018 Alexander Wong, Mohammad Javad Shafiee, Francis Li, Brendan Chwyl Dept. of Systems Design Engineering University of Waterloo, DarwinAI {a28wong, mjshafiee}@uwaterloo.ca, {francis, brendan}@darwinai.ca Abstract—Object detection is a major challenge in computer vision, involving both object classification and object localization within a scene. While deep neural networks have been shown in recent years to yield very powerful techniques for tackling the challenge of object detection, one of the biggest challenges with enabling such object detection networks for widespread deployment on embedded devices is high computational and memory requirements. Recently, there has been an increasing focus in exploring small deep neural network architectures for object detection that are more suitable for embedded devices, such as Tiny YOLO and SqueezeDet. Inspired by the efficiency of the Fire microarchitecture introduced in SqueezeNet and the object detection performance of the singleshot detection macroarchitecture introduced in SSD, this paper introduces Tiny SSD, a single-shot detection deep convolutional neural network for real-time embedded object detection that is composed of a highly optimized, non-uniform Fire subnetwork stack and a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers designed specifically to minimize model size while maintaining object detection performance. The resulting Tiny SSD possess a model size of 2.3MB (∼26X smaller than Tiny YOLO) while still achieving an mAP of 61.3% on VOC 2007 (∼4.2% higher than Tiny YOLO). These experimental results show that very small deep neural network architectures can be designed for real-time object detection that are well-suited for embedded scenarios. Keywords-object detection; deep neural network; embedded; real-time; single-shot I. I NTRODUCTION Object detection can be considered a major challenge in computer vision, as it involves a combination of object classification and object localization within a scene (see Figure 1). The advent of modern advances in deep learning [7], [6] has led to significant advances in object detection, with the majority of research focuses on designing increasingly more complex object detection networks for improved accuracy such as SSD [9], R-CNN [1], Mask R-CNN [2], and other extended variants of these networks [4], [8], [15]. Despite the fact that such object detection networks have showed stateof-the-art object detection accuracies beyond what can be achieved by previous state-of-the-art methods, such networks are often intractable for use for embedded applications due to computational and memory constraints. In fact, even faster variants of these networks such as Faster R-CNN [13] are only Figure 1. Tiny SSD results on the VOC test set. The bounding boxes, categories, and confidences are shown. capable of single-digit frame rates on a high-end graphics processing unit (GPU). As such, more efficient deep neural networks for real-time embedded object detection is highly desired given the large number of operational scenarios that such networks would enable, ranging from smartphones to aerial drones. Recently, there has been an increasing focus in exploring small deep neural network architectures for object detection that are more suitable for embedded devices. For example, Redmon et al. introduced YOLO [11] and YOLOv2 [12], which were designed with speed in mind and was able to achieve real-time object detection performance on a high-end Nvidia Titan X desktop GPU. However, the model size of YOLO and YOLOv2 remains very large in size (753 MB and 193 MB, respectively), making them too large from a memory perspective for most embedded devices. Furthermore, their object detection speed drops considerably when running on embedded chips [14]. To address this issue, Tiny YOLO [10] was introduced where the network architecture was reduced considerably to greatly reduce model size (60.5 MB) as well as greatly reduce the number of floating point operations required (just 6.97 billion operations) at a cost of object detection accuracy (57.1% on the twenty-category VOC 2017 test set). Similarly, Wu et al. introduced SqueezeDet [16], a fully convolutional neural network that leveraged the efficient Fire microarchitecture introduced in SqueezeNet [5] within an end-to-end object detection network architecture. Given that the Fire microarchitecture is highly efficient, the resulting SqueezeDet had a reduced model size specifically for the purpose of autonomous driving. However, SqueezeDet has only been demonstrated for objection detection with limited object categories (only three) and thus its ability to handle larger number of categories have not been demonstrated. As such, the design of highly efficient deep neural network architectures that are well-suited for real-time embedded object detection while achieving improved object detection accuracy on a variety of object categories is still a challenge worth tackling. In an effort to achieve a fine balance between object detection accuracy and real-time embedded requirements (i.e., small model size and real-time embedded inference speed), we take inspiration by both the incredible efficiency of the Fire microarchitecture introduced in SqueezeNet [5] and the powerful object detection performance demonstrated by the single-shot detection macroarchitecture introduced in SSD [9]. The resulting network architecture achieved in this paper is Tiny SSD, a single-shot detection deep convolutional neural network designed specifically for realtime embedded object detection. Tiny SSD is composed of a non-uniform highly optimized Fire sub-network stack, which feeds into a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers, designed specifically to minimize model size while retaining object detection performance. This paper is organized as follows. Section 2 describes the highly optimized Fire sub-network stack leveraged in the Tiny SSD network architecture. Section 3 describes the highly optimized sub-network stack of SSD-based convolutional feature layers used in the Tiny SSD network architecture. Section 4 presents experimental results that evaluate the efficacy of Tiny SSD for real-time embedded object detection. Finally, conclusions are drawn in Section 5. II. O PTIMIZED F IRE S UB - NETWORK S TACK The overall network architecture of the Tiny SSD network for real-time embedded object detection is composed of two main sub-network stacks: i) a non-uniform Fire sub-network stack, and ii) a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers, with the first sub-network stack feeding into the second subnetwork stack. In this section, let us first discuss in detail the design philosophy behind the first sub-network stack of the Tiny SSD network architecture: the optimized fire sub-network stack. A powerful approach to designing smaller deep neural network architectures for embedded inference is to take a more principled approach and leverage architectural design strategies to achieve more efficient deep neural network microarchitectures [3], [5]. A very illustrative example of such a principled approach is the SqueezeNet [5] network architecture, where three key design strategies were leveraged: 1) reduce the number of 3 × 3 filters as much as possible, Figure 2. An illustration of the Fire microarchitecture. The output of previous layer is squeezed by a squeeze convolutional layer of 1 × 1 filters, which reduces the number of input channels to 3 × 3 filters. The result of the squeeze convolutional layers is passed into the expand convolutional layer which consists of both 1 × 1 and 3 × 3 filters. 2) reduce the number of input channels to 3 × 3 filters where possible, and 3) perform downsampling at a later stage in the network. This principled designed strategy led to the design of what the authors referred to as the Fire module, which consists of a squeeze convolutional layer of 1x1 filters (which realizes the second design strategy of effectively reduces the number of input channels to 3 × 3 filters) that feeds into an expand convolutional layer comprised of both 1 × 1 filters and 3 × 3 filters (which realizes the first design strategy of effectively reducing the number of 3 × 3 filters). An illustration of the Fire microarchitecture is shown in Figure 2. Inspired by the elegance and simplicity of the Fire microarchitecture design, we design the first sub-network stack of the Tiny SSD network architecture as a standard convolutional layer followed by a set of highly optimized Fire modules. One of the key challenges to designing this sub-network stack is to determine the ideal number of Fire modules as well as the ideal microarchitecture of each of the Fire modules to achieve a fine balance between object detection performance and model size as well as inference speed. First, it was determined empirically that 10 Fire modules in the optimized Fire sub-network stack provided strong object detection performance. In terms of the ideal microarchitecture, the key design parameters of the Fire microarchitecture are the number of filters of each size (1 × 1 or 3 × 3) that form this microarchitecture. In the SqueezeNet network architecture that first introduced the Fire microarchitecture [5], the microarchitectures of the Fire modules are largely uniform, with many of the modules sharing the same microarchitecture configuration. In an effort to achieve more optimized Fire microarchitectures on a permodule basis, the number of filters of each size in each Fire Table I T HE OPTIMIZED F IRE SUB - NETWORK STACK OF THE T INY SSD NETWORK ARCHITECTURE . T HE NUMBER OF FILTERS AND INPUT SIZE TO EACH LAYER ARE REPORTED FOR THE CONVOLUTIONAL LAYERS AND F IRE MODULES . E ACH F IRE MODULE IS REPORTED IN ONE ROW FOR A BETTER REPRESENTATION . ”x@S – y@E1 – z@E3" STANDS FOR x NUMBERS OF 1 × 1 FILTERS IN THE SQUEEZE CONVOLUTIONAL LAYER , y NUMBERS OF 1 × 1 FILTERS AND z NUMBERS OF 3 × 3 FILTERS IN THE EXPAND CONVOLUTIONAL LAYER . Type / Stride Conv1 / s2 Pool1 / s2 Fire1 Fire2 Figure 3. An illustration of the network architecture of the second sub-network stack of Tiny SSD. The output of three Fire modules and two auxiliary convolutional feature layers, all with highly optimized microarchitecture configurations, are combined together for object detection. module is optimized to have as few parameters as possible while still maintaining the overall object detection accuracy. As a result, the optimized Fire sub-network stack in the Tiny SSD network architecture is highly non-uniform in nature for an optimal sub-network architecture configuration. Table I shows the overall architecture of the highly optimized Fire sub-network stack in Tiny SSD, and the number of parameters in each layer of the sub-network stack. III. O PTIMIZED S UB - NETWORK S TACK OF SSD- BASED C ONVOLUTIONAL F EATURE L AYERS In this section, let us first discuss in detail the design philosophy behind the second sub-network stack of the Tiny SSD network architecture: the sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers. One of the most widely-used and effective object detection network macroarchitectures in recent years has been the single-shot multibox detection (SSD) macroarchitecture [9]. The SSD macroarchitecture augments a base feature extraction network architecture with a set of auxiliary convolutional feature layers and convolutional predictors. The auxiliary convolutional feature layers are designed such that they decrease in size in a progressive manner, thus enabling the flexibility of detecting objects within a scene across different scales. Each of the auxiliary convolutional feature layers can then be leveraged to obtain either: i) a confidence score for a object category, or ii) a shape offset relative to default bounding box coordinates [9]. As a result, a number of object detections can be obtained per object category in this manner in a powerful, end-to-end single-shot manner. Inspired by the powerful object detection performance and multi-scale flexibility of the SSD macroarchitecture [9], the second sub-network stack of Tiny SSD is comprised of a set of auxiliary convolutional feature layers and convo- Pool3 / s2 Fire3 Fire4 Pool5 / s2 Fire5 Fire6 Fire7 Fire8 Pool9 / s2 Fire 9 Pool10 / s2 Fire10 Filter Shapes 3 × 3 × 57 3×3 15@S – 49@E1 – 53@E3 Concat1 15@S – 54@E1 – 52@E3 Concat2 3×3 29@S – 92@E1 – 94@E3 Concat3 29@S – 90@E1 – 83@E3 Concat4 3×3 44@S – 166@E1 – 161@E3 Concat5 45@S – 155@E1 – 146@E3 Concat6 49@S – 163@E1 – 171@E3 Concat7 25@S – 29@E1 – 54@E3 Concat8 3×3 37@S – 45@E1 – 56@E3 Concat9 3×3 38@S – 41@E1 – 44@E3 Concat10 Input Size 300 × 300 149 × 149 74 × 74 74 × 74 74 × 74 37 × 37 37 × 37 37 × 37 18 × 18 18 × 18 18 × 18 18 × 18 18 × 18 9×9 4×4 lutional predictors with highly optimized microarchitecture configurations (see Figure 3). As with the Fire microarchitecture, a key challenge to designing this sub-network stack is to determine the ideal microarchitecture of each of the auxiliary convolutional feature layers and convolutional predictors to achieve a fine balance between object detection performance and model size as well as inference speed. The key design parameters of the auxiliary convolutional feature layer microarchitecture are the number of filters that form this microarchitecture. As such, similar to the strategy taken for constructing the highly optimized Fire sub-network stack, the number of filters in each auxiliary convolutional feature layer is optimized to minimize the number of parameters while preserving overall object detection accuracy of the full Tiny SSD network. As a result, the optimized sub-network stack of auxiliary convolutional feature layers in the Tiny SSD network architecture is highly non-uniform in nature for an optimal sub-network architecture configuration. Table II shows the overall architecture of the optimized sub-network stack of the auxiliary convolutional feature layers within the Tiny SSD network architecture, along with the number of Table II T HE OPTIMIZED SUB - NETWORK STACK OF THE AUXILIARY CONVOLUTIONAL FEATURE LAYERS WITHIN THE T INY SSD NETWORK ARCHITECTURE . T HE INPUT SIZES TO EACH CONVOLUTIONAL LAYER AND KERNEL SIZES ARE REPORTED . Type / Stride Conv12-1 / s2 Conv12-2 Conv13-1 Conv13-2 Fire5-mbox-loc Fire5-mbox-conf Fire9-mbox-loc Fire9-mbox-conf Fire10-mbox-loc Fire10-mbox-conf Fire11-mbox-loc Fire11-mbox-conf Conv12-2-mbox-loc Conv12-2-mbox-conf Conv13-2-mbox-loc Conv13-2-mbox-conf Filter Shape 3 × 3 × 51 3 × 3 × 46 3 × 3 × 55 3 × 3 × 85 3 × 3 × 16 3 × 3 × 84 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 16 3 × 3 × 84 Input Size 4×4 4×4 2×2 2×2 37 × 37 37 × 37 18 × 18 18 × 18 9×9 9×9 4×4 4×4 2×2 2×2 1×1 1×1 parameters in each layer. Model size 60.5MB 2.3MB mAP (VOC 2007) 57.1% 61.3% Table IV R ESOURCE USAGE OF T INY SSD. Model Name Tiny SSD V. E XPERIMENTAL R ESULTS AND D ISCUSSION To study the utility of Tiny SSD for real-time embedded object detection, we examine the model size, object detection accuracies, and computational operations on the VOC2007/2012 datasets. For evaluation purposes, the Tiny YOLO network [10] was used as a baseline reference comparison given its popularity for embedded object detection, and was also demonstrated to possess one of the smallest model sizes in literature for object detection on the VOC 2007/2012 datasets (only 60.5MB in size and requiring just 6.97 billion operations). The VOC2007/2012 datasets consist of natural images that have been annotated with 20 different types of objects, with illustrative examples shown in Figure 4. The tested deep neural networks were trained using the VOC2007/2012 training datasets, and the mean average precision (mAP) was computed on the VOC2007 test dataset to evaluate the object detection accuracy of the deep neural networks. A. Training Setup Table III O BJECT DETECTION ACCURACY RESULTS OF T INY SSD ON VOC 2007 TEST SET. T INY YOLO RESULTS ARE PROVIDED AS A BASELINE COMPARISON . Model Name Tiny YOLO [10] Tiny SSD reductions while having a negligible effect on object detection accuracy. Total number of Parameters 1.13M Total number of MACs 571.09M IV. PARAMETER P RECISION O PTIMIZATION In this section, let us discuss the parameter precision optimization strategy for Tiny SSD. For embedded scenarios where the computational requirements and memory requirements are more strict, an effective strategy for reducing computational and memory footprint of deep neural networks is reducing the data precision of parameters in a deep neural network. In particular, modern CPUs and GPUs have moved towards accelerated mixed precision operations as well as better handling of reduced parameter precision, and thus the ability to take advantage of these factors can yield noticeable improvements for embedded scenarios. For Tiny SSD, the parameters are represented in half precision floating-point, thus leading to further deep neural network model size The proposed Tiny SSD network was trained for 220,000 iterations in the Caffe framework with training batch size of 24. RMSProp was utilized as the training policy with base learning rate set to 0.00001 and γ = 0.5. B. Discussion Table III shows the model size and the object detection accuracy of the proposed Tiny SSD network on the VOC 2007 test dataset, along with the model size and the object detection accuracy of Tiny YOLO. A number of interesting observations can be made. First, the resulting Tiny SSD possesses a model size of 2.3MB, which is ∼26X smaller than Tiny YOLO. The significantly smaller model size of Tiny SSD compared to Tiny YOLO illustrates its efficacy for greatly reducing the memory requirements for leveraging Tiny SSD for real-time embedded object detection purposes. Second, it can be observed that the resulting Tiny SSD was still able to achieve an mAP of 61.3% on the VOC 2007 test dataset, which is ∼4.2% higher than that achieved using Tiny YOLO. Figure 5 demonstrates several example object detection results produced by the proposed Tiny SSD compared to Tiny YOLO. It can be observed that Tiny SSD has comparable object detection results as Tiny YOLO in some cases, while in some cases outperforms Tiny YOLO in assigning more accurate category labels to detected objects. For example, in the first image case, Tiny SSD is able to detect the chair in the scene, while Tiny YOLO misses the chair. In the third image case, Tiny SSD is able to identify the dog in the scene while Tiny YOLO detects two bounding boxes around the dog, with one of the bounding boxes incorrectly labeling it as cat. This significant improvement Figure 4. Example images from the Pascal VOC dataset. The ground-truth bounding boxes and object categories are shown for each image. in object detection accuracy when compared to Tiny YOLO illustrates the efficacy of Tiny SSD for providing more reliable embedded object detection performance. Furthermore, as seen in Table IV, Tiny SSD requires just 571.09 million MAC operations to perform inference, making it well-suited for real-time embedded object detection. These experimental results show that very small deep neural network architectures can be designed for real-time object detection that are wellsuited for embedded scenarios. VI. C ONCLUSIONS In this paper, a single-shot detection deep convolutional neural network called Tiny SSD is introduced for real-time embedded object detection. Composed of a highly optimized, non-uniform Fire sub-network stack and a non-uniform subnetwork stack of highly optimized SSD-based auxiliary convolutional feature layers designed specifically to minimize model size while maintaining object detection performance, Tiny SSD possesses a model size that is ∼26X smaller than Tiny YOLO, requires just 571.09 million MAC operations, while still achieving an mAP of that is ∼4.2% higher than Tiny YOLO on the VOC 2007 test dataset. These results demonstrates the efficacy of designing very small deep neural network architectures such as Tiny SSD for real-time object detection in embedded scenarios. ACKNOWLEDGMENT The authors thank Natural Sciences and Engineering Research Council of Canada, Canada Research Chairs Program, DarwinAI, and Nvidia for hardware support. 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1 An overview of deep learning based methods for unsupervised and semi-supervised anomaly detection in videos arXiv:1801.03149v2 [] 30 Jan 2018 B Ravi Kiran, Dilip Mathew Thomas, Ranjith Parakkal Abstract—Videos represent the primary source of information for surveillance applications and are available in large amounts but in most cases contain little or no annotation for supervised learning. This article reviews the state-of-the-art deep learning based methods for video anomaly detection and categorizes them based on the type of model and criteria of detection. We also perform simple studies to understand the different approaches and provide the criteria of evaluation for spatio-temporal anomaly detection. Index Terms—Unsupervised methods, Anomaly detection, Representation learning, Autoencoders, LSTMs, Generative adversarial networks, Variational Autoencoders, Predictive models. F 1 I NTRODUCTION Unsupervised representation learning has become an important domain with the advent of deep generative models which include the variational autoencoder (VAE) [1] , generative adversarial networks (GANs) [2], Long Short Term memory networks (LSTMs) [3] , and others. Anomaly detection is a well-known sub-domain of unsupervised learning in the machine learning and data mining community. Anomaly detection for images and videos are challenging due to their high dimensional structure of the images, combined with the nonlocal temporal variations across frames. We focus on reviewing firstly, deep convolution architectures for feature or representation learnt “end-to-end” and secondly, predictive and generative models specifically for the task of video anomaly detection. Anomaly detection is an unsupervised learning task where the goal is to identify abnormal patterns or motions in data that are by definition infrequent or rare events. Furthermore, anomalies are rarely annotated and labeled data rarely available to train a deep convolutional network to separate normal class from the anomalous class. This is a fairly complex task since the class of normal points includes frequently occurring objects and regular foreground movements while the anomalous class include various types of rare events and unseen objects that could be summarized as a consistent class. Long streams of videos containing no anomalies are made available using which one is required to build a representation for a moving window over the video stream that estimates the normal behavior class while detecting anomalous movements and appearance, such as unusual objects in the scene. • • • Preprint of an article under review at MDPI Journal of Imaging Bangalore Ravi Kiran is with Uncanny Vision Solutions, Bangalore and CRIStAL Lab, UMR 9189, Université Charles de Gaulle, Lille 3. Email: beedotkiran@gmail.com Dilip Mathew Thomas and Ranjith Parakkal, are with Uncanny Vision Solutions, Bangalore. Email: dilip@uncannyvision.com, ranjith@uncannyvision.com Given a set of training samples containing no anomalies, the goal of anomaly detection is to design or learn a feature representation, that captures “normal” motion and spatial appearance patterns. Any deviations from this normal can be identified by measuring the approximation error either geometrically in a vector space or the posterior probability of a given model which fits training sample representation vectors or by modeling the conditional probability of future samples given their past values and measuring the prediction error of test samples by training a predictive model, thus accounting for temporal structure in videos. 1.1 Anomaly detection Anomaly detection is an unsupervised pattern recognition task that can be defined under different statistical models. In this study we will explore models that perform linear approximations by PCA, non-linear approximation by various types of autoencoders and finally deep generative models. Intuitively, a complex system under the action of various transformations is observed, the normal behavior is described through a few samples and a statistical model is built using the said normal behavior samples that is capable of generalizing well on unseen samples. The normal class distribution D is estimated using the training samples x i ∈ X train , by building a representation f θ : X train → R which minimizes model prediction loss X X θ ∗ = arg min L D (θ ; x i ) = arg min k f θ (x i ) − x i k2 θ x i ∈ X train θ x i ∈ X train (1) error over all the training samples, over all i , is evaluated. Now the deviation of the test samples x j ∈ X test under this representation f θ∗ is evaluated as the anomaly score, a(x j ) = k f θ∗ (x j ) − x j k2 is used as a measure of deviation. For said models, the anomalous points are samples that are poorly approximated by the estimated model f θ∗ . Detection is achieved by evaluating a threshold on the anomaly score a j > Tthresh . The threshold is a parameter of the detection 2 algorithm and the variation of the threshold w.r.t detection performance is discussed under the Area under ROC section. For probabilistic models, anomalous points can be defined as samples that lie in low density or concentration regions of the domain of an input training distribution P (x|θ ). Representation learning automates feature extraction for video data for tasks such as action recognition, action similarity, scene classification, object recognition, semantic video segmentation [4], human pose estimation, human behavior recognition and various other tasks. Unsupervised learning tasks in video include anomaly detection [5], [6], unsupervised representation learning [7], generative models for video [8], and video prediction [9]. 1.2 Datasets We now define the video anomaly detection problem setup. The videos considered come from a surveillance camera where the background remains static, while the foreground constitutes of moving objects such as pedestrians, traffic and so on. The anomalous events are the change in appearance and motion patterns that deviate from the normal patterns observed in the training set. We see a few examples demonstrated in figure 1 : Here we list the frequently evaluated datasets, though this is not exhaustive. The UCSD dataset [5] consists of pedestrian videos where anomalous time instances correspond to the appearance of objects like a cyclist, a wheelchair, and a car in the scene that is usually populated with pedestrians walking along the roads. People walking in unusual locations are also considered anomalous. In CUHK Avenue Dataset [10] anomalies correspond to strange actions such as a person throwing papers or bag, moving in unusual directions, and appearance of unusual objects like bags and bicycle. In the Subway entry and exit datasets people moving in the wrong direction, loitering and so on are considered as anomalies. UMN dataset [11] consists of videos showing unusual crowd activity, and is a particular case of the video anomaly detection problem. The Train dataset [12] contains moving people in a train. The anomalous events are mainly due to unusual movements of people in the train. And finally the Queen Mary University of London U-turn dataset [13] contains normal traffic with anomalous events such as jaywalking and movement of a fire engine. More recently, a controlled environment based LV dataset has been introduced by [14], with challenging examples for the task of online video anomaly detection. 2 R EPRESENTATION LEARNING A NOMALY D ETECTION (VAD) FOR V IDEO Videos are high dimensional signals with both spatialstructure, as well as local temporal variations. An important problem of anomaly detection in videos is to learn a representation of input sample space f θ : X → Rd , to d -dimensional vectors. The idea of feature learning is to automate the process of finding a good representation of the input space, that takes into account important prior information about the problem [15]. This follows from the No-Free-Lunch-Theorem which states that no universal learner exists for every training distribution D . Following work already established for video anomaly detection, the task concretely consists in detecting deviations from models of static background, normal crowd appearance and motion from optical flow, change in trajectory and other priors. Representation learning consists of building a parameterized model f θ : X → Z → X , and in this study we focus on representations that reconstruct the input, while the latent space Z is constrained to be invariant to changes in the input, such as change in luminance, translations of objects in the scene that don’t deviate normal movement patterns, and others. This provides a way to introduce prior information to reconstruct normal samples. 2.1 Taxonomy The goal of this survey is to provide a compact review of the state of the art in video anomaly detection based on unsupervised and semi-supervised deep learning architectures. The survey characterizes the underlying video representation or model as one of the following : 1) 2) 3) Representation learning for reconstruction : Methods such as Principal component analysis (PCA), Autoencoders (AEs) are used to represent the different linear and non-linear transformations to the appearance (image) or motion (flow), that model the normal behavior in surveillance videos. Anomalies represent any deviations that are poorly reconstructed. Predictive modeling : Video frames are viewed as temporal patterns or time series, and the goal is to model the conditional distribution P (x t |(x t−1 , x t−2 , ..., x t− p )). In contrast to reconstruction, where the goal is to learn a generative model that can successfully reconstruct frames of a video, the goal here is to predict the current frame or its encoded representation using the past frames. Examples include autoregressive models and convolutional Long-Short-Term-Memory models. Generative models : Variational Autoencoders (VAE), Generative Adversarial Networks (GAN) and Adversarially trained AutoEncoders (AAE), are used for the purpose of modeling the likelihood of normal video samples in an end-to-end deep learning framework. An important common aspect in all these models is the problem of representation learning, which refers to the feature extraction or transformation of input training data for the task of anomaly detection. We shall also remark the other secondary feature transformations performed in each of these different models and their purposes. 2.2 Context of the review A short review on the subject of video anomaly detection is provided here [16]. To the best of our knowledge, there has not been a systematic study of deep architectures for video anomaly detection, which is characterized by abnormal appearance and motion features, that occur rarely. We cite below the other domains which do not fall under this study. • A detailed review of abnormal human behavior and crowd motion analysis is provided in [17] and [18]. This includes deep architectures such as Social-LSTM [19] based on the social force model [20] where the goal is to predict pedestrian motion taking into account the movement of neighboring pedestrians. 3 Training Samples Test Samples Anomaly Mask Fig. 1. UCSD dataset (top two rows) : Unlike normal training video streams, anomalies consist of a person on a bicycle and a skating board, ground truth detection shown in anomaly mask. Avenue dataset (bottom row) : Unlike normal training video streams, anomalies consist of a person throwing papers. Other examples include walking with an abnormal object (bicycle), a person running (strange action), and a person walking in the wrong direction. • • Action recognition is an important domain in computer vision which requires supervised learning of efficient representations of appearance and motion for the purpose of classification [21]. Convolutional networks were employed to classify various actions in video quite early [22]. Recent work involves fusing feature maps evaluated on different frames (over time) of video [23] yielding state of the art results. Finally, convolutional networks using 3-D filters (C3D) have become a recent base-line for action recognition [24]. Unsupervised representation learning is a wellestablished domain and the readers are directed towards a complete review of the topic in [25], as well as the deep learning book [26]. In this review, we shall mainly focus on the taxonomy provided and restrict our review to deep convolutional networks and deep generative models that enable end-to-end spatiotemporal representation learning for the task of anomaly detection in videos. We also aim to provide an understanding of what aspects of detection do these different models target. Apart from the taxonomy being addressed in this study, there have been many other approaches. One could cite work on anomaly detection based on K-Nearest Neighbors [27], unsupervised clustering [28], and object speed and size [29]. We briefly review the set of hand-engineered features used for the task of video anomaly detection, though our focus still remains deep learning based architectures. Mixture of dynamic textures (MDT) is a generative mixture model defined for each spatio-temporal windows or cubes of the raw training video [5], [6]. It models appearance and motion features and thus detects both spatial and temporal anomalies. Histogram of oriented optical flow and oriented gradients [30] is a baseline used in anomaly detection and crowd analysis, [31], [32], [33]. Tracklets are representation of movements in videos and have been applied to abnormal crowd motion analysis [34], [35]. More recently, there has been work on developing optical flow acceleration features for motion description [36]. Problem setup : Given a training sequence of images from a video, X train ∈ R Ntrain ×r× c , which contains only “normal motion patterns” and no anomalies, and given a test sequence X test ∈ R Ntest ×r× c , which is susceptible to contain anomalies, the task consists in associating each frame with an anomaly score for the temporal variation, as well as a spatial score to localize the anomaly in space. This is demonstrated in figure 2. The anomaly detection task is usually considered unsupervised when there is no direct information or labels available about the positive rare class. However the samples in the study with no anomalies are available, and thus is a semisupervised learning problem. For Z = {x i , yi }, i ∈ [1, N ] we have samples only with yi = 0. The goal thus of anomaly detection is two-fold : first, find the representation of the input feature f θ (x i ), for example, using convolutional neural networks (CNNs), and then the decision rule s( f θ (x i )) ∈ {0, 1} that detects anomalies, whose detection rate can be parameterized as per the application. 4 the training set, and evaluate reconstruction error on the test optical flow magnitude. This serves a baseline for our study. A refined version was implemented and evaluated in [37] which evaluated the atomic movement patterns using a probabilistic PCA [38], over rectangular regions over the image domain. Optical flow estimation is a costly step of this algorithm, and there has been large progress in the improving its evaluating speed. Authors [39], propose to trade of accuracy of for fast approximation of optical flow using PCA to interpolate flow fields. Input layer Hidden layer Output layer X1 X̂ 1 X2 h(1) 1 X̂ 2 X3 h(1) 2 X̂ 3 X4 h(1) 3 X̂ 4 X5 +1 X̂ 5 Fig. 2. Visualizing anomalous regions and temporal anomaly score. X6 3 We begin with an input training video X train ∈ R N ×d , with N frames and d = r × c pixels per frame, which represents the dimensionality of each vector. In this section, we shall focus on reducing the expected reconstruction error by different methods. We shall describe the Principal Component Analysis (PCA), Convolutional AutoEncoder (ConvAE), and Contractive AutoEncoders (CtractAE), and their setup for dimensionality reduction and reconstruction. 3.1 X̂ 6 R ECONSTRUCTION M ODELS Principal Component Analysis PCA finds the directions of maximal variance in the training data. In the case of videos, we are aiming to model the spatial correlation between pixel values which are components of the vector representing a frame at a particular time instant. With input training matrix X , which has zero mean, we are looking for a set of orthogonal projections that whiten/decorrelate the features in the training set : min k X − ( X W )W T k2F = k X − X̂ k2F W T W =I (2) where, W ∈ Rd ×k , with the constraint W T W = I representing an orthonormal reconstruction of the input X . The projection X W is a vector in a lower dimensional subspace, with fewer components than the vectors from X . This reduction in dimensionality is used to capture the anomalous behavior, as samples that are not well reconstructed. The anomaly score is given by the Mahalanobis distance between the input and the reconstruction, or the variance scaled reconstruction error : A = ( X − X̂ )Σ−1 ( X − X̂ )T (3) We associate each frame with continual optical flow magnitude, and learn atomic motion patterns with standard PCA on +1 Fig. 3. Autoencoder with single hidden layer. 3.2 Autoencoders An Autoencoder is a neural network trained by backpropagation and provides an alternative to PCA to perform dimensionality reduction by reducing the reconstruction error on the training set, shown in figure 3. It takes an input x ∈ Rd and maps it to the latent space representation z ∈ Rk , by a deterministic application, z = σ(W x + b). Unlike the PCA the autoencoder (AE) performs a nonlinear point-wise transform of the input σ : R → R, which is required to be a differentiable function. It is usually a rectified linear unit (ReLU) (σ( x) = max(0, x)) or Sigmoid (σ( x) = (1 + e− x )−1 ). Thus we can write a similar reconstruction of the input matrix given by : min k X − σ( XU )V k2F U,V (4) The low-dimensional representation is given by σ( XU ∗ ), where U ∗ represents the optimal linear encoding that minimizes the reconstruction loss above. There are multiple ways of regularizing the parameters U, V . One of the constraints is the average value of the activations in the hidden layer, this enforces sparsity. 3.3 Convolutional AutoEncoders (CAEs) Autoencoders in their original form do view the input as a signal decomposed as the sum of other signals. Convolutional 5 AutoEncoders (CAEs) [40], makes this decomposition explicit by weighting the result of the convolution operator. For a single channel input x (for example a gray-scale image), the latent representation of the kth filter would be : h k = σ(x ∗ W k ) + b k (5) The reconstruction is obtained by mapping back to the original image domain with the latent maps H and the gk : decoding convolutional filter W X k gk + c) x̂ = σ( h ∗W (6) k∈ H where σ : R → R is a point-wise non-linearity like the sigmoid or hyperbolic tangent function. A single bias value is broadcast to each component of a latent map. These k-output maps can be used as an input to the next layer of the CAE. Several CAEs can be stacked into a deep hierarchy, which we again refer as a CAE to simplify the naming convention. We represent the stack of such operations as a single function f W : Rr× c× p → Rr× c× p where the convolutional weights and biases are together represented by the weights W . In retrospect, the PCA, and traditional AE, ignore the spatial structure and location of pixels in the image. This is also termed as being permutation invariant. It is important to note that when working with image frames of few 100 × 100 pixels, these methods introduce large redundancy in network parameters W , and furthermore span the entire visual receptive field. CAEs have fewer parameters on account of their weights being shared across many input locations/pixels. 3.4 CAEs for Video anomaly detection In the recent work by [41], a deep convolutional autoencoder was trained to reconstruct an input sequence of frames from a training video set. We call this a Spatio-Temporal Stacked frame AutoEncoder (STSAE), to avoid confusion with similar names in the rest of the article. The STSAE in [41] stacks p frames x i = [ X i , X i−1 , ..., X i− p+1 ] with each time slice treated as a different channel in the input tensor to a convolutional autoencoder. The model is regularized by augmenting the loss function with L2-norm of the model weights : 1 X kx i − f W (x i )k22 + νkW k22 L (W ) = 2N i (7) where the tensor x i ∈ Rr× c× p is a cuboid with spatial dimensions r, c are the spatial dimensions and p is the number of frames temporally back into the past, with hyper-parameter ν which balances the reconstruction error and norm of the parameters, and N is the mini-batch size. The architecture of the convolutional autoencoder is reproduced in figure 4. The image or tensor xbi reconstructed by the autoencoder enforces temporal regularity since the convolutional (weights) representation along with the bottleneck architecture of the autoencoder compresses information. The spatio-temporal autoencoder in [42] is shown in the right panel of figure 4. The reconstruction error map at frame t is given by E t = | X t − X̂ t |, while the temporal regularity score is given by the inverted, normalized reconstruction error : P (x,y) E t − min(x,y) (E t ) s( t) = 1 − (8) max(x,y) (E t ) P where the , min and max operators are across the spatial indices’s ( x, y). In other models, the normalized reconstruction error is directly used as the anomaly score. One could envisage the use of Mahalanobis distance here since the task is to evaluate the distance between test points from the points from the normal ones. This is evaluated as the error between the original tensor and the reconstruction from the autoencoder. Robust versions of Convolutional AutoEncoders (RCAE) are studied in [43], where the goal is to evaluate anomalies in images by imposing L2-constraints on parameters W as well as adding a bias term. A video-patch (spatio-temporal) based autoencoder was employed by [44] to reconstruct patches, with a sparse autoencoder whose average activations were set to parameter ρ , enforcing sparseness, following the work in [45]. 3.5 Contractive Autoencoders Contractive autoencoders explicitly create invariance by adding the Jacobian of the latent space representation w.r.t the input of the autoencoder, to the reconstruction loss L(x, r (x)). This forces the latent space representation to remain the same for small changes in the input [46]. Let us consider the autoencoder with the encoder mapping the input image to the latent space z = f (x) and the decoder mapping back to the input image space r (x) = g( f (x)). The regularized loss function is : ° ° h ° ∂ f (x) ° i ° L (W ) = Ex∼ X train L(x, r (x)) + λ ° (9) ° ∂x ° Authors in [47] describe what regularized autoencoders learn from the data generating the density function, and show for contractive and denoising encoders that this corresponds to the direction in which density is increasing the most. Regularization forces the autoencoders to become less sensitive to input variation, though enforcing minimal reconstruction error keeps it sensitive to variations along the manifold having high density. Contractive autoencoders capture variations on the manifold, while mostly ignoring variations orthogonal to it. Contractive autoencoder estimates the tangent plane of the data manifold [26]. 3.6 Other deep models De-noising AutoEncoders (DAE) and Stacked DAEs (SDAEs) are well-known robust feature extraction methods in the domain of unsupervised learning [48], where the reconstruction error minimization criteria is augmented with that of reconstructing from corrupted inputs. SDAEs are used to learn representations from a video using both appearance, i.e. raw values, and motion information, i.e. optical flow between consecutive frames [49]. Correlations between optical flow and raw image values are modeled by coupling these two SDAE pipelines to learn a joint representation. Deep belief networks (DBNs) are generative models, created by stacking multiple hidden layer units, which are usually trained greedily to perform unsupervised feature learning. They are generative models in the sense that they can reconstruct the original inputs. They have been discriminatively trained using back-propagation [50] to achieve improved accuracies for supervised learning tasks. The DBNs have been used to perform a raw image value based representation learning in [51]. 6 Input : 10 × 227 × 227 Input : 10 × 227 × 227 Conv : 11 × 11, 512 Filters, Stride 4(10, 512, 55, 55) Conv : 11 × 11, 128 Filters, Stride 4 (10, 128, 55, 55) Max Pool : 2 × 2, (10, 512, 27, 27) Conv : 5 × 5, 64 Filters, Stride 2 (10, 64, 26, 26) Conv : 5 × 5, 256 Filters, (10, 256, 27, 27) Max Pool : 2 × 2, (10, 256, 13, 13) ConvLSTM2D : 3 × 3, 64 filters (10, 64, 26, 26) Conv : 3 × 3, 128 Filters, (10, 128, 13, 13) ConvLSTM2D : 3 × 3, 32 filters (10, 32, 26, 26) Deconv : 3 × 3, 128 Filters, (10, 128, 13, 13) ConvLSTM2D : 3 × 3, 64 filters (10, 64, 26, 26) Un Pool : 2 × 2, (10, 128, 27, 27) Deconv : 5 × 5, 256 Filters, (10, 256, 27, 27) Deconv : 5 × 5, 128 Filters, Stride 2 (10, 128, 55, 55) Un Pool : 2 × 2, (10, 256, 55, 55) Deconv : 5 × 5, 128 Filters, Stride 2(10, 1, 227, 227) Deconv : 11 × 11, 512 Filters, (10, 512, 55, 55) Output : 10 × 227 × 227 Output : 10 × 227 × 227 Spatio-Temporal Stacked frame AutoEncoder (STSAE) Convolutional LSTM based autoencoder (CLSTM-AE) Fig. 4. Spatio-Temporal Stacked frame AutoEncoder (left) : A sequence of 10 frames are being reconstructed by a convolutional autoencoder, image reproduced from [41]. Convolutional LSTM based autoencoder (right) : A sequence of 10 frames are being reconstructed by a spatiotemporal autoencoder, [42]. The Convolutional LSTM layers are predictive models that model the spatio-temporal correlation of pixels in the video. This is described in the predictive model section. An early application of autoencoders to anomaly detection was performed in [52], on non-visual data. The Replicating neural network [52], constitutes of a feed-forward multi-layer perceptron with three hidden layers, trained to map the training dataset to itself and anomalies correspond to large reconstruction error over test datasets. This is an autoencoder setup with a staircase like non-linearity applied at the middle hidden layer. The activation levels of this hidden units are thus quantized into N discrete values, 0, N1−1 , N1−2 , ..., 1. The step-wise activation function used for the middle hidden layer divides the continuously distributed data points into a number of discrete-valued vectors. The staircase non-linearity quantizes data points into clusters. This approach identifies cluster labels for each sample, and this often helps interpret resulting outliers. A rejection cascade over spatio-temporal cubes was generated to improve the performance speed of Deep-CNN based video anomaly detection framework by authors in [53]. Videos can be viewed as a special case of spatio-temporal processes. A direct approach to video anomaly detection can be estimating the spatio-temporal mean and covariance. A major issue is estimating the spatio-temporal covariance matrix due to its large size n2 (where n = N pixels × p frames). In [54], space-time pixel covariance for crowd videos were represented as a sum of Kronecker products using only a few Kronecker P factors, Σn×n u ri=1 T i ⊗ S i . To evaluate the anomaly score, the Mahanalobis distance for clips longer than the learned covariance needs to be evaluated. The inverse of the larger covariance matrix needs to be inferred from the estimated one, by block Toeplitz extension [55]. It is to be noted that this study [54] only evaluates performance on the UMN dataset. 4 P REDICTIVE MODELING Predictive models aim to model the current output frame X t as a function of the past p frames [ X t−1 , X t−2 , ..., X t− p+1 ]. This is well-known in time series analysis under auto-regressive models, where the function over the past is linear. Recurrent neural networks (RNN) model this function as a recurrence relationship, frequently involving a non-linearity such as a sigmoid function. LSTM is the standard model for sequence prediction. It learns a gating function over the classical RNN architecture to prevent the vanishing gradient problem during backpropagation through time (BPTT) [3]. Recently there have also been attempts to perform efficient video prediction using feed-forward convolutional networks for video prediction by minimizing the mean-squared error (MSE) between predicted and future frames [9]. Similar efforts were performed in [56] using a CNN-LSTM-deCNN framework while combining MSE and an adversarial loss. 4.1 Composite Model : Reconstruction and prediction This composite LSTM model in [7], combines an autoencoder model and predictive LSTM model, see figure 5. Autoencoders suffer learning trivial representations of input, by memorization, while memorization is not useful for predicting future frames. On the other hand, the future predictor’s role requires memory of temporally past few frames, though this would not be compatible with the autoencoder loss which is more global. The composite model was used to extract features from video data for the tasks of action recognition. The composite LSTM model is defined using a fully connected LSTM (FC-LSTM) layer. 4.2 Convolutional LSTM Convolutional long short-term memory (ConvLSTM) model [57] is a composite LSTM based encoder-decoder model. FCLSTM does not take spatial correlation into consideration and is permutation invariant to pixels, while a ConvLSTM has convolutional layers instead of fully connected layers, thus modeling spatio-temporal correlations. The ConvLSTM as described in equations 10 evaluates future states of cells in 7 Input reconstruction Xˆ3 Xˆ2 Prediction Encoding n/w Xˆ1 ConvLSTM2 W2 ConvLSTM4 copy W2 Input sequence W1 W2 Xˆ3 copy Xˆ2 Future prediction X1 X2 ConvLSTM3 ConvLSTM1 X3 Xˆ5 Xˆ4 W3 Input Forecasting n/w Xˆ6 Fig. 6. A convolutional LSTM architecture for spatio-temporal prediction. W3 4.3 Xˆ4 Xˆ5 Fig. 5. Composite LSTM module [7]. The LSTM model weights W represent fully connected layers. a spatial grid as a function of the inputs and past states of its local neighbors. Authors in [57], consider a spatial grid, with each grid cell containing multiple spatial measurements, which they aim to forecast for the next K future frames, given J observations in the past. The spatio-temporal correlations are used as input to a recurrent model, the convolutional LSTM. The equations for input, gating and the output are presented below. L rec (W ) = i t = σ(Wxi ∗ X t + Whi ∗ H t−1 + Wci ◦ C t−1 + b i ) f t = σ(Wx f ∗ X t + Wh f ∗ H t−1 + Wc f ◦ C t−1 + b f ) C t = f t ◦ C t−1 + i t ◦ tanh(Wxc + X t + Whc ∗ H t−1 + b c ) 3D-Autoencoder and Predictor As remarked by authors in [61], while 2D-ConvNets are appropriate representations learnt for image recognition and detection tasks, they are incapable of capturing the temporal information encoded in consecutive frames for video analysis problems. 3-D convolutional architectures are known to perform well for action recognition [24], and are used in the form of an autoencoder. Such a 3D autoencoder learns representations that are invariant to spatio-temporal changes (movement) encoded by the 3-D convolutional feature maps. Authors in [61] propose to use a 3D kernel by stacking Tframes together as in [41]. The output feature map of each kernel is a 3D tensor including the temporal dimension and are aimed to summarize motion information. The reconstruction branch follows an autoencoder loss: (10) o t = σ(Wxo ∗ X t + Who ∗ H t−1 + Wco ◦ C t + b o ) N 1 X W k X i − f rec ( X i )k22 N i=1 (11) The prediction branch loss is inversely weighted by moving window’s length that falls off symmetrically w.r.t the current frame, to reduce the effect of past frames on the predicted frame : H t = o t ◦ tanh(C t ) Here, ∗ refers to the convolution operation while ◦ refers to the Hadamard product, the element-wise product of matrices. Encoding network compresses the input sequence into a hidden state tensor while the forecasting network unfolds the hidden state tensor to make a prediction. The hidden representations can be used to represent moving objects in the scene, a larger transitional kernel captures faster motions compared to smaller kernels [57]. The ConvLSTM model was used as a unit within the composite LSTM model [7] following an encoder-decoder, with a branch for reconstruction and another for prediction. This architecture was applied for video anomaly detection by [58], [59], with promising results. In [60], a convolutional representation of the input video is used as input to the convolutional LSTM and a de-convolution to reconstruct the ConvLSTM output to the original resolution. The authors call this a ConvLSTM Autoencoder, though fundamentally it is not very different from a ConvLSTM. L pred (W ) = N 1 X T 1 X W (T − t)k X i+T − f pred ( X i )k22 N i=1 T 2 t=1 (12) Thus the final optimization objective minimized is : L (W ) = L rec (W ) + L pred (W ) + λkW k22 (13) Anomalous regions where spatio-temporal blocks, that even when poorly reconstructed by the autoencoder branch, would be well predicted by the prediction branch. The prediction loss was designed to enforce local temporal coherence by tracking spatio-temporal correlation, and not for the prediction of the appearance of new objects in the relatively longterm future. 4.4 Slow Feature Analysis (SFA) Slow feature analysis [62] is an unsupervised representation learning method which aims at extracting slowly varying 8 Input : 16 × 1 × 128 × 128(T, C, X , Y ) 3D-Conv-BN-Lrelu-3DPool : (8, 32, 64, 64) 3D-Conv-BN-Lrelu-3DPool : (4, 48, 64, 64) 3D-Conv-BN-Lrelu-3DPool : (2, 64, 16, 16) 3D-Conv-BN-Lrelu-3DPool : (2, 64, 16, 16) 3D-Conv-BN-Lrelu-3DPool : (4, 48, 32, 32) 3D-Conv-BN-Lrelu-3DPool : (4, 48, 32, 32) 3D-Conv-BN-Lrelu-3DPool : (8, 32, 64, 64) 3D-Conv-BN-Lrelu-3DPool : (8, 32, 64, 64) 3D-Conv-BN-Lrelu-3DPool : (16, 32, 128, 128) 3D-Conv-BN-Lrelu-3DPool : (16, 32, 128, 128) 3D-Conv-Sigmoid : (16:, 1 , 128 3D-Conv-BN-Lrelu-3DPool (16 , 32, ,128) 128, 128) 3D-Conv-Sigmoid : (16:, 1 , 128 3D-Conv-BN-Lrelu-3DPool (16 , 32, ,128) 128, 128) Reconstruction Prediction Fig. 7. Spatio-temporal autoencoder architecture from [61] with reconstruction and prediction branches, following the composite model in [7]. Batch Normalization(BN) is applied at each layer, following which a leaky Relu non-linearity is applied, finally followed by a 3D max-pooling operation. representations of rapidly varying high dimensional input. The SFA is based on the slowness principle, which states that the responses of individual receptors or pixel variations are highly sensitive to local variations in the environment, and thus vary much faster, while the higher order internal visual representations vary on a slow timescale. From a predictive modeling perspective SFA extracts a representation y( t) of the high dimensional input x t that maximizes information on the next time sample x t+1 . Given a high dimensional input varying over time, [x1 , x2 , ..., xT ], t ∈ [ t 0 , t 1 ] SFA extracts a representation y = f θ (x) which is a solution to the following optimization problem [63] : arg min E t [y t ]=0, E t [y2j ]=1, E t [y j y j0 ]=0]=1, E t [y t+1 − y t ] (14) As seen in the constraints, the representation is enforced to have, zero mean to ensure a unique solution, while unit covariance to avoid trivial zero solution. Feature de-correlation removes redundancy across the features. SFA has been well known in pattern recognition and has been applied to the problem of activity recognition [64], [65]. Authors in [66] propose an incremental application of SFA that updates slow features incrementally. The SFA is calculated using batch PCA, iterated twice. The first PCA to whiten the inputs. The second PCA is applied on the derivative of the normalized input to evaluate the flow features. To achieve a computationally tractable solution, a two-layer localized SFA architecture is proposed by authors [67] for the task of online slow feature extraction and consequent anomaly detection. Other Predictive models : A convolutional feature representation was fed into an LSTM model to predict the latent space representation and its prediction error was used to evaluate anomalies in a robotics application [68]. A recurrent autoencoder using an LSTM that models temporal dependence between patches from a sequence of input frames is used to detect video forgery [69]. 5 5.1 D EEP GENERATIVE M ODELS Generative Vs Discriminative Let us consider a supervised learning setup ( X i , yi ) ∈ Rd × {C j }Kj=1 , where i indexes the number of samples i = 1 : N in the dataset. Generative models estimate class conditional posterior distribution P ( X | y), which can be difficult if the input data are high dimensional images or spatio-temporal tensors. A discriminative model evaluates the class probability P ( y| X ) directly from the data to classify the samples X into different classes {C j }Kj=1 . Deep generative models that can learn via the principle of maximum likelihood differ with respect to how they represent or approximate the likelihood. The explicit models are ones where the density p model (x, θ ) is evaluated explicitly and the likelihood maximized. In this section, we will review the stochastic autoencoders; the variational autoencoder and the adversarial autoencoder, and their applications to the problem of anomaly detection. And finally the generative adversarial networks to anomaly detection in images and videos. 5.2 Variational Autoencoders (VAEs) Variational Autoencoders [1] are generative models that approximate the data distribution P ( X ) of a high dimensional input X , an image or video. Variational approximation of the latent space is achieved using an autoencoder architecture, with a probabilistic encoder q φ (x|z) that produces Gaussian distribution in the latent space, and a probabilistic decoder p θ (z|x), which given a code produces distribution over the input space. The motivation behind variational methods is to pick a family of distributions over the latent variables with its own variational parameters q φ (z), and estimate the parameters for this family so that it approaches q φ . The loss function constitutes of the KL-Divergence regularization term, and the expected negative reconstruction error with an additional KL-divergence term between the latent space vector and the representation with a mean vector and a standard deviation vector, that optimizes the 9 variational lower bound on the marginal log-likelihood of each observation. NX train standard autoencoder, where latent variables are defined by deterministic mappings. L 5.4 Generative Adversarial Networks (GANs) 1 X (log p θ (x(i) |z(i,l) )) L l =1 A GAN [2] consists of a generator G , usually a decoder, and i =1 a discriminator D , usually an binary classifier that assigns where z(i,l) = g φ (²(i,l) , x(i) ) and ²(l) ∼ p(²) (15) a probability of an image being generated (fake), or sampled from the training data (real). The generator G in fact learns a The function g φ maps sample x(i) and noise vector ²(l) distribution p g over data x via a mapping G (z) of samples z, to a sample from the approximate posterior for that data- 1D vectors of uniformly distributed input noise sampled from point z(i,l) = g φ (²(l) , x(i) ) where, z(i,l) ∼ q φ (z|x(i) ). To solve this latent space Z , to 2D images in the image space manifold X sampling problem authors [1] propose the reparameterization , which is populated by normal examples. In this setting, the trick. The random variable z ∼ q φ (z|x) is expressed as function network architecture of the generator G is equivalent to a conof a deterministic variable z = g φ (², z) where ² is an auxiliary volutional decoder that utilizes a stack of strided convolutions. variable with independent marginal p(²). This reparameter- The discriminator D is a standard CNN that maps a 2D image ization rewrites an expectation w.r.t q φ (z|x) such that the to a single scalar value D (·). The discriminator output D (·) can Monte Carlo estimate of the expectation is differentiable w.r.t. be interpreted as the probability that the given input to the φ. A valid reparameterization was the unit-Gaussian case discriminator D was a real image x sampled from training z(i,l) = µ(i,l) + σ(i) ¯ ²(l) where ²(l) ∼ N (0, I ). data X or a generated image using G (z) by the generator G . Specifically for a VAE, the goal is to learn a low dimen- D and G are simultaneously optimized through the following sional representation z by modeling p θ (x|z) with a simpler two-player minimax game with value function V (G, D ) : distribution, a centered isotropic multivariate Gaussian, i.e. p θ (z) = N (z, 0, I ). In this model both the prior p θ (z), and max V (D,G ) = Ex∼ pdata [log D (x)] + Ez∼ pz (z) [log(1 − D (G (z)))] q φ (z|x) are Gaussian; and the resulting loss function was min D G (17) described in equation 15. The discriminator is trained to maximize the probability of assigning real training examples the “real” and samples from p g the “fake” label. The generator G is simultaneously trained to fool D via minimizing V (G ) = log(1 − D (G (z))), which is z φ θ equivalent to maximizing V (G ) = D (G (z)). During adversarial training, the generator improves in generating realistic images and the discriminator progresses in correctly identifying real and generated images. GANs are implicit models [71], that sample directly from the distribution represented by the x model. L (θ , φ, x(i) ) = −D K L ( q φ (z|x(i) )|| p θ (z)) + N 5.5 Fig. 8. Graphical model for the VAE : Solid lines denote the generative model p θ (z) p θ (x|z), dashed lines denote the variational approximation q φ (z|x) to the intractable posterior p θ (z|x) [1]. 5.3 Anomaly detection using VAE Anomaly detection using the VAE framework has been studied in [70]. Authors define the reconstruction probability as E qφ (z|x) [log p θ (x|z)]. Once the VAE is trained, for a new test sample x(i) , one first evaluates the mean and standard deviation vectors with the probabislistic encoder, (µ z(i) , σz(i) ) = f θ (z|x(i) ). Then samples L latent space vectors, z(i,l) ∼ N (µ z(i) , σ z(i) ). The parameters of the input distribution are reconstructed using these L samples, µ x̂(i,l) , σx̂(i,l) = g φ (x|z(i,l) ) then the reconstruction probability for test sample x(i) is given by : Precon (x(i) ) = ³ ´ L 1 X p θ x(i) |µx̂(i,l) , σx̂(i,l) L l =1 (16) Multiple samples drawn from the latent variable distribution, lets Precon (x(i) ) take into account the variability of the latent variable space, which is one of the essential distinctions between the stochastic variational autoencoder and a GANs for anomaly detection in Images This section reviews work done by authors [72] who apply a GAN model for the task of anomaly detection in medical images. GANs are generative models that best produce a set of training data points x ∼ Pdata (x) where Pdata represents the probability density of the training data points. The basic idea in anomaly detection is to be able to evaluate the density function of the normal vectors in the training set containing no anomalies while for the test set we evaluate a negative loglikelihood score which serves as the final anomaly score. The score corresponds to the test sample’s posterior probability of being generated from the same generative model representing the training data points. GANs provide a generative model that minimizes the distance between the training data distribution and the generative model samples without explicitly defining a parametric function, which is why it is called an implicit generative model [71]. Thus to be successfully used in an anomaly detection framework the authors [72] evaluate the mapping x → z, i.e. Image domain → latent representation. This was done by choosing the closest point zγ using backpropagation. Once done the residual loss in the image space P was defined as L R (zγ ) = |x − G (zγ )|. GANs are generative models and to evaluate a likelihood one requires a mapping from the image domain to the latent 10 space. This is achieved by authors in [72], which we shall shortly describe here. Given a query image x ∼ p test , the authors aim to find a point z in the latent space that corresponds to an image G ( z) that is visually most similar to the query image x and that is located on the manifold X . The degree of similarity of x and G (z) depends on to which extent the query image follows the data distribution p g that was used for training of the generator. To find the best z, one starts randomly sampling z1 from the latent space distribution Z and feeds it into the trained generator which yields the generated image G (z1 ). Based on the generated image G (z1 ) we can define a loss function, which provides gradients for the update of the coefficients of z1 resulting in an updated position in the latent space, z2 . In order to find the most similar image G (zΓ ), the location of z in the latent space Z is optimized in an iterative process via γ = 1, 2, ..., Γ back-propagation steps. G F →O ,G O→F are trained to map training frames and their optical flow to their cross-channel counterparts. The goal is to force a poor cross-channel prediction on test video frames containing an anomaly so that the trained discriminators shall provide a low probability score. The trained discriminators D F →O , D O→F are patchdiscriminators that produce scores S O , S F on a grid with resolution smaller than the image. These scores do not require the reconstruction of the different channels to be evaluated. The final score is S = S O + S F which is normalized between [0, 1] based on the maximum value of individual scores for each frame. The U-net uses the Markovian structure present spatially by the skip connections shown between the input and the output of the generators in figure 9. Cross-channel prediction aims at modeling the spatio-temporal correlation present across channels in the context of video anomaly detection. 5.6 Adversarial Discriminators using Cross-channel prediction Here we shall review the work done in [73] applied to anomaly detection in videos. The anomaly detection problem in this paper is formulated as a cross-channel prediction task, where the two channels are the raw-image values F t and the optical flow vectors O t for frames F t , F t−1 in the videos. This work combines two architectures, the pixel-GAN architecture by [74] to model the normal/training data distribution, and the Split-Brain Autoencoders [75]. The SplitBrain architectures aims at predicting a multi-channel output by building cross-channel autoencoders. That is, given training examples X ∈ RH ×W ×C , we split data into X 1 ∈ RH ×W ×C1 and X 2 ∈ RH ×W ×C2 , where C 1 , C 2 ⊂ C , and the authors train f2 = F1 ( X 1 ) and X f1 = F2 ( X 2 ), multiple deep representations X which when concatenated provided a reconstruction of the input tensor X , just like an autoencoder. Various manners of aggregating these predictors have been explored in [75]. In the same spirit as the cross-channel autoencoders [75], Conditional GANs were developed [74] to learn a generative model that learns a mapping from one input domain to the other. The authors [73] train two networks much in the spirit of the conditional GAN [74] where : N O→F which generates the raw image frames from the optical flow and N F →O which generates the optical flow from the raw images. F t are image frames with RGB channels and O t are vertical and horizontal optical flow vector arrays. The input to discriminator D is thus a 6-D tensor. We now describe the adaption of the crosschannel autoencoders for the task of anomaly detection. • N F →O : Training set is X = {(F t , O t )} N t=1 . The L1 loss function with x = F t , y = O t : L L1 ( x, y) = k y − G ( x, z)k1 (18) with the conditional adversarial loss being L cG AN (G, D ) =E(x,y)∈ X [log D ( x, y)] +E x∈{F t },z∈Z log(1 − D ( x,G ( x, z))) • Conversely in N {(O t , F t )} N t=1 . O →F N ∆S O →F Ot F̂ t − ∆o − Ft Ô t N F →O Fig. 9. The cross-channel prediction conditional GAN architecture in [73]. There are two GAN models : flow→RGB predictor (N O→F ) and RGB→Flow predictor (N F →O ). Each of the generators shown has a U-net architecture which uses the common underlying structure in the image RGB channels and optical flow between two frames. 5.7 Adversarial Autoencoders (AAEs) Adversarial Autoencoders are probabilistic autoencoders that use GANs to perform variational approximation of the aggregated posterior of the latent space representation [76] using an arbitrary prior. AAEs were applied to the problem of anomalous event detection over images by authors [77]. In figure 10, x denotes input vectors from training distribution, q(z|x) the encoder’s posterior distribution, p(z) the prior that the user wants to impose on the latent space vectors z. The latent space distribution is given by Z q(z) = q(z|x) p d (x) d x (20) x∈ X train (19) the training set changes to X = The generators/discriminators follow a U-net architecture as in [74] with skip connections. The two generators where p d (x) represents the training data distribution. In an AAE, the encoder acts like the generator of the adversarial network, and it tries to fool the discriminator into believing that q(z) comes from the actual data distribution p(z). During the joint training, the encoder is updated to improve the reconstruction error in the autoencoder path, while it is updated by the discriminator of the adversarial network to make the 11 In similar work by [80], discriminative autoencoders aim at learning low-dimensional discriminative representations for positive ( X + ) and negative ( X − ) classes of data. The discriminative autoencoders build a latent space representation under the constraint that the positive data should be better reconstructed than the negative data. This is done by minimizing the reconstruction error for positive examples while ensuring that those of the negative class are pushed away from the manifold. q(z|x) z ∼ q(z) x b x − p(z) + Ld (X + ∪ X −) = 1 1+ e− x Input latent space distribution approach the imposed prior. As prior distribution for the generator of the network, authors [77] use the Gaussian distribution of 256 dimensions, with the dropout set to 0.5 probability. The method achieves close to state of the art performance. As the authors remark themselves, the AAE does not take into account the temporal structure in the video sequences. Controlling reconstruction for anomaly detection One of the common problems using deep autoencoders is their capability to produce low reconstruction errors for test samples, even over anomalous events. This is due to the way autoencoders are trained in a semi-supervised way on videos with no anomalies, but with sufficient training samples, they are able to approximate most test samples well. In [78], the authors propose to limit the reconstruction capability of the generative adversarial networks by learning conflicting objectives for the normal and anomalous data. They use negative examples to enforce explicit poor reconstruction. Thus this setup is weakly supervised, not requiring labels. Given two random variables X , Y with samples {x}K , {y} Jj=1 , we want the network to reconstruct the input i =1 distribution X while poorly reconstruct Y . This was achieved by maximizing the following objective function: K X i =1 log p θ (xˆi |x i ) + J X log p θ (yˆi |y j ) (22) In the above loss function, t(x) = {−1, +1} denotes as the label e k the distance of that example of the sample, and e(x) = kx − x to the manifold. Minimizing the hinge loss in equation 22 achieves reconstruction such that the discriminative autoencoders build a latent space representation of data that better reconstructs positive data compared to the negative data 6 p θ ( X̂ | X ) − p θ (Ŷ |Y ) = max(0, t(x) · (kx̂ − xk − 1)) x∈ X + ∪ X − Fig. 10. Two paths in the Adversarial Autoencoder : Top path refers to the standard autoencoder configuration that minimizes reconstruction error. The bottom path constitutes of an adversarial network that ensures an approximation of the user input defined samples from distribution p(z), and the latent or code vector distribution, provided by q(z|x). 5.8 X (21) j =1 where θ refers to the autoencoders parameters. This setup assumes strong class imbalance, i.e.very few samples of the anomalous class Y are available compared to the normal class X . The motivation for negative learning using anomalous examples is to consistently provide poor reconstruction of anomalous samples. During the training phase, authors [68] reconstruct positive samples by minimizing the reconstruction error between samples, while negative samples are forced to have a bad reconstruction by maximizing the error. This last step was termed as negative learning. The datasets evaluated were the reconstruction of the images from MNIST and Japanese highway video patches [79]. E XPERIMENTS There are two large classes of experiments : first, recone t, structing the input video on a single frame basis X t → X e second the reconstruction of a stack of frames X t− p:t → X t− p:t . These reconstruction schemes are performed either on raw frame values, or on the optical flow between consequent frame pairs. Reconstructing raw image values modeled the back-ground image, since minimizing the reconstruction error was in fact evaluating the background. Convolutional autoencoders reconstructing a sequence of frames captured temporal appearance changes as described by [41]. When learning feature representations on optical flow we indirectly operate on two frames, since each optical flow map evaluates the relative motion between two consequent frame pairs. In the case of predictive models the current frame X t was predicted after observing the past p frames. This provides a different temporal structure as compared to a simple reconstruction e t− p:t , where the temporal of a sequence of frames X t− p:t → X coherence results from enforcing a bottleneck in the autoencoder architectures. The goal of these experiments were not evaluate the best performing model, and were intended as a tool to understand how background estimation and temporal appearance were approximated by the different models. A complete detailed study is beyond the scope of this review. In this section, we evaluate the performance of the following classes of models on the UCSD and CUHK-Avenue datasets. As a baseline, we use the reconstruction of a dense optical flow calculated using the Farneback method in OpenCV 3, by principal component analysis, with around 150 components. For predictive models, as a baseline we use a vector autoregressive model (VAR), referred to as LinPred. The coefficients of the model are estimated on a lower dimensional, random projection of the raw image or optical flow maps from the input training video stream. The random projection avoids badly conditioned and expensive matrix inversion. We compare the performance of Contractive autoencoders, simple 3D autoencoders based on C3D [24] CNNs (C3D-AE), the ConvLSTM and ConvLSTM autoencoder from the predictive model family and finally the VAE from the generative models family. The VAE’s loss function consists of the binary cross-entropy (similar to a reconstruction error) 12 between the model prediction and the input image, and the KL-divergence D K L [Q ( z| X )kP ( z)], between the encoded latent space vectors and P ( z) = N (0, I ) multivariate unit-Gaussian. These models were built in Keras [81] with Tensorflow backend and executed on a K-80 GPU. 6.1 Architectures Our Contractive and Variational AE (VAE) constitutes of a random projection to reduce the dimensionality to 2500 from an input frame of size 200 × 200. The Contractive AE constitutes of one fully connected hidden layer of size 1250 which map back to the reconstruction of the randomly projected vector of size 2500. While the VAE contains two hidden layers (dimensions: 1024, 32), which maps back to the output of 2500 dimensions. We use the latent space representation of the variational autoencoders to fit a multivariate 1-Gaussian on the training dataset and evaluate the negative-log probability for the test samples. 6.2 Observations and Issues The results are summarized in the table 1 and 2. The performance measures reported are the Area Under ReceiverOutput-Characteristics plot (AU-ROC), Area Under PrecisionRecall plot. These scores are calculated when the input channels correspond to the raw image (raw) and the optical flow (flow), each of which has been normalized by the maximum value. The final temporal anomaly score is given by equation 8. These measures are described in the next section. We also describe the utility of these measures under different frequencies of occurrences of the anomalous positive class. Reconstruction model issues : Deep autoencoders identify anomalies by poor reconstruction of objects that have never appeared in the training set, when raw image pixels are used as input. It is difficult to achieve this in practice due to a stable reconstruction of new objects by deep autoencoders. This pertains to the high capacity of autoencoders, and their tendency to well approximate even the anomalous objects. Controlling reconstruction using negative examples could be a possible solution. This holds true, but to a lower extent, when reconstructing a sequence of frames (spatio-temporal block). AUC-ROC vs AUC-PR : The anomalies in the UCSD pedestrian dataset have a duration of several hundred frames on average, compared to the anomalies in the CUHK avenue dataset which occur only for a few tens of frames. This makes the anomalies statistically less probable. This can be seen by looking at the AU-PR table 2, where the average scores for CUHK-avenue are much lower than for UCSD pedestrian datasets. It is important to note that this does not mean the performance over the CUHK-Avenue dataset is lower, but just the fact that the positive anomalous class is rarer in occurrence. Rescaling image size : The models used across different experiments in the articles that were reviewed, varied in the input image size. In some cases the images were resized to sizes (128,128), (224, 224), (227, 227). We have tried to fix this to be uniformly (200,200). Though it is essential to note that there is a substantial change in performance when this image is resized to certain sizes. Generating augmented video clips : Training the convolutional LSTM for video anomaly detection takes a large number of epochs. Furthermore, the training video data required is much higher and data augmentation for video anomaly detection requires careful thinking. Translations and rotations may be transformations to which the anomaly detection algorithm requires to be sensitive to, based on the surveillance application. Performance of models : VAEs perform consistently as well as or better than PCA on optical flow. It is still left as a future study to understand clearly, why the performance of a stochastic autoencoder such as VAE is better. Convolutional LSTM on raw image values follow closely behind as the first predictive model performing as good as PCA but sometimes poorer. Convolutional LSTM-AE is a similar architecture with similar performance. Finally, the 3D convolutional autoencoder, based on the work by [24], performs as well as PCA on optical flow, while modeling local motion patterns. To evaluate the specific advantages of each of these models, a larger number of real world, video surveillance examples are required demonstrating the representation or feature that is most discriminant. In experiments, we have also observed that application of PCA on the random projection of individual frames performed well in avenue dataset, indicating that very few frames were sufficient to identify the anomaly; while the PCA performed poorly on UCSD pedestrian datasets, where motion patterns were key to detect the anomalies. For single frame input based models, optical flow served as a good input since it already encoded part of the predictive information in the training videos. On the other hand, convolutional LSTMs and linear predictive models required p = [2, 10] input raw image values, in the training videos, to predict the current frame raw image values. 6.3 Evaluation measures The anomaly detection task is a single class estimation task, where 0 is assigned to samples with likelihood (or reconstruction error) above (below) a certain threshold, and 1 assigned to detected anomalies with low likelihood (high reconstruction error). Statistically, the anomalies are a rare class and occurs less frequently compared to the normal class. The most common characterization of this behavior is the expected frequency of occurrence of anomalies. We briefly review the anomaly detection evaluation procedure as well as the performance measures that were used across different studies. For a complete treatment of the subject, the reader is referred to [82]. The final anomaly score is a value that is treated as a probability which lies in s( t) ∈ [0, 1], ∀ t, t ∈ [1, T ], T being the maximum time index. For various level sets or thresholds of the anomaly score a 1:T , one can evaluate the True Positives (TP, the samples which are truly anomalous and detected as anomalous), True Negatives (TN, the samples that are truly normal and detected as normal), False Positives (FP, the samples which are truly normal samples but detected as anomalous) and finally False Negatives (FN, the samples which are truly anomalous but detected as normal). P TP True positive rate (TPR) or Recall = TP+FN P FP (23) False positive rate(FPR) = TP+FN P TP Precision = TP+FP 13 TABLE 1 Area Under-ROC (AUROC) Methods Feature PCA flow LinPred (raw, flow) C3D-AE (raw, flow) ConvLSTM (raw, flow) ConvLSTM-AE (raw, flow) CtractAE (raw, flow) VAE (raw, flow) UCSDped1 UCSDped2 CUHK-avenue 0.75 0.80 0.82 (0.71, 0.71) (0.73, 0.78) (0.74, 0.84) (0.70, 0.70) (0.64, 0.81) (0.86, 0.18) (0.67, 0.71) (0.77, 0.80) (0.84, 0.18) (0.43, 0.74) (0.25, 0.81) (0.50, 0.84) (0.66, 0.75) (0.65, 0.79) (0.83, 0.84) (0.63, 0.72) (0.72, 0.86) (0.78, 0.80) TABLE 2 Area Under-Precision Recall (AUPR) Methods Feature PCA flow LinPred (raw, flow) C3D-AE (raw, flow) ConvLSTM (raw, flow) ConvLSTM-AE (raw, flow) CtractAE (raw, flow) VAE (raw, flow) UCSDped1 UCSDped2 CUHK-avenue 0.78 0.95 0.66 (0.71, 0.71) (0.92, 0.94) (0.49, 0.65) (0.75, 0.77) (0.88, 0.95) (0.65, 0.15) (0.67, 0.71) (0.94, 0.95) (0.66, 0.15) (0.52, 0.81) (0.74, 0.95) (0.34, 0.70) (0.70, 0.81) (0.88, 0.95) (0.69, 0.70) (0.66, 0.76) (0.92, 0.97) (0.54, 0.68) We require these measures to evaluate the ROC curve, which measures the performance of the detection at various False positive rates. That is ROC plots TPR vs FPR, while the PR plots precision vs recall. The performance of anomaly detection task is evaluated based on an important criterion, the probability of occurrence of the anomalous positive class. Based on this value, different performance curves are useful. We define the two commonly used performance curves : Precision-Recall (PR) curves and Receiver-Operator-Characteristics (ROC) curves. The area under the PR curve (AU-PR) is useful when true negatives are much more common than true positives (i.e., TN » TP). The precision recall curve only focuses on predictions around the positive (rare) class. This is good for anomaly detection because predicting true negatives (TN) is easy in anomaly detection. The difficulty is in predicting the rare true positive events. Precision is directly influenced by class (im)balance since FP is affected, whereas TPR only depends on positives. This is why ROC curves do not capture such effects. Precisionrecall curves are better to highlight differences between models for highly imbalanced data sets. For this reason, if one would like to evaluate models under imbalanced class settings, AU-PR scores would exhibit larger differences than the area under the ROC curve. 7 C ONCLUSION In this review paper, we have focused on categorizing the different unsupervised learning models for the task of anomaly detection in videos into three classes based on the prior information used to build the representations to characterize anomalies. They are reconstruction based, spatio-temporal predictive models, and generative models. Reconstruction based models build representations that minimize the reconstruction error of training samples from the normal distribution. Spatiotemporal predictive models take into account the spatiotemporal correlation by viewing videos as a spatio-temporal time series. Such models are trained to minimize the prediction error on spatio-temporal sequences from the training series, where the length of the time window is a parameter. Finally, the generative models learn to generate samples from the training distribution, while minimizing the reconstruction error as well as distance between generated and training distribution, where the focus is on modeling the distance between sample and distributions. Each of these methods focuses on learning certain prior information that is useful for constructing the representation for the video anomaly detection task. One key concept which occurs in various architectures for video anomaly detection is how temporal coherence is implemented. Spatio-temporal autoencoders and Convolutional LSTM learn a reconstruction based or spatio-temporal predictive model that both use some form of (not explicitly defined) spatio-temporal regularity assumptions. We can conclude from our study that evaluating how sensitive the learned representation is to certain transformations such as time warping, viewpoint, applied to the input training video stream, is an important modeling criterion. Certain invariances are as well defined by the choice of the representation (translation, rotation) either due to reusing convolutional architectures or imposing a predictive structure. A final component in the design of the video anomaly detection system is the choice of thresholds for the anomaly score, which was not covered in this review. The performance of the detection systems were evaluated using ROC plots which evaluated performance across all thresholds. Defining a spatially variant threshold is an important but non-trivial problem. Finally, as more data is acquired and annotated in a video-surveillance setup, the assumption of having no labeled anomalies progressively turns false, partly discussed in the section on controlling reconstruction for anomaly detection. Certain anomalous points with well defined spatio-temporal regularities become a second class that can be estimated well; and methods to include the positive anomalous class information into detection algorithms becomes essential. Handling class imbalance becomes essential in such a case. Another problem of interest in videos is the variation in temporal scale of motion patterns across different surveillance videos, sharing a similar background and foreground. Learning a representation that is invariant to such time warping would be of practical interest. There are various additional components of the stochastic 14 gradient descent algorithm that were not covered in this review. 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Energy Clustering Guilherme França∗ and Joshua T. Vogelstein† Johns Hopkins University Abstract arXiv:1710.09859v1 [stat.ML] 26 Oct 2017 Energy statistics was proposed by Székely in the 80’s inspired by the Newtonian gravitational potential from classical mechanics, and it provides a hypothesis test for equality of distributions. It was further generalized from Euclidean spaces to metric spaces of strong negative type, and more recently, a connection with reproducing kernel Hilbert spaces (RKHS) was established. Here we consider the clustering problem from an energy statistics theory perspective, providing a precise mathematical formulation yielding a quadratically constrained quadratic program (QCQP) in the associated RKHS, thus establishing the connection with kernel methods. We show that this QCQP is equivalent to kernel k-means optimization problem once the kernel is fixed. These results imply a first principles derivation of kernel k-means from energy statistics. However, energy statistics fixes a family of standard kernels. Furthermore, we also consider a weighted version of energy statistics, making connection to graph partitioning problems. To find local optimizers of such QCQP we propose an iterative algorithm based on Hartigan’s method, which in this case has the same computational cost as kernel k-means algorithm, based on Lloyd’s heuristic, but usually with better clustering quality. We provide carefully designed numerical experiments showing the superiority of the proposed method compared to kernel k-means, spectral clustering, standard k-means, and Gaussian mixture models in a variety of settings. ∗ guifranca@gmail.com † jovo@jhu.edu 1 I. INTRODUCTION Energy statistics [1] is based on a notion of statistical potential energy between probability distributions, in close analogy to Newton’s gravitational potential in classical mechanics. It provides a model-free hypothesis test for equality of distributions which is achieved under minimum energy. When probability distributions are different the statistical potential energy diverges as sample size increases, while tends to a nondegenerate limit distribution when probability distributions are equal. Energy statistics has been applied to several goodnessof-fit hypothesis tests, multi-sample tests of equality of distributions, analysis of variance [2], nonlinear dependence tests through distance covariance and distance correlation, which generalizes the Pearson correlation coefficient, and hierarchical clustering [3] by extending Ward’s method of minimum variance. Moreover, in Euclidean spaces, an application of energy statistics to clustering was already proposed [4]. We refer the reader to [1], and references therein, for an overview of energy statistics theory and its applications. In its original formulation, energy statistics has a compact representation in terms of expectations of pairwise Euclidean distances, providing straightforward empirical estimates. The notion of distance covariance was further generalized from Euclidean spaces to metric spaces of strong negative type [5]. Furthermore, the missing link between energy distance based tests and kernel based tests has been recently resolved [6], establishing an equivalence between generalized energy distances to maximum mean discrepancies (MMD), which are distances between embeddings of distributions in reproducing kernel Hilbert spaces (RKHS). This equivalence immediately relates energy statistics to kernel methods often used in machine learning, and form the basis of our approach. Clustering has such a long history in machine learning, making it impossible to mention all important contributions in a short space. Perhaps, the most used method is k-means [7– 9], which is based on Lloyd’s heuristic [7] of assigning a data point to the cluster with closest center. The only statistical information about each cluster comes from its mean, making it sensitive to outliers. Nevertheless, k-means works very well when data is linearly separable in Euclidean space. Gaussian mixture models (GMM) is another very common approach, providing more flexibility than k-means, however, it still makes strong assumptions about the distribution of the data. To account for nonlinearities, kernel methods were introduced [10, 11]. A mercer kernel 2 [12] is used to implicitly map data points to a RKHS, then clustering can be performed in the associated Hilbert space by using its inner product. However, the kernel choice remains the biggest challenge since there is no principled theory to construct a kernel for a given dataset, and usually a kernel introduces hyperparameters that need to be carefully chosen. A well-known kernel based clustering method is kernel k-means, which is precisely k-means formulated in the feature space [11]. Furthermore, kernel k-means algorithm [13, 14] is still based on Loyd’s heuristic [7] of grouping points that are closer to a cluster center. We refer the reader to [15] for a survey of clustering methods. Although clustering from energy statistics, in Euclidean spaces, was considered in [4], the precise optimization problem behind this approach remains elusive, as well as the connection with kernel methods. The main theoretical contribution of this paper is to fill this gap. Since the statistical potential energy is minimum when distributions are equal, the principle behind clustering is to maximize the statistical energy, enforcing probability distributions associated to each cluster to be different from one another. We provide a precise mathematical formulation to this statement, leading to a quadratically constrained quadratic program (QCQP) in the associated RKHS. This immediately establishes the connection between energy statistics based clustering, or energy clustering for short, with kernel methods. Moreover, our formulation holds for general semimetric spaces of negative type. We also show that such QCQP is equivalent to kernel k-means optimization problem, however, the kernel is fixed by energy statistics. The equivalence between kernel k-means, spectral clustering, and graph partitioning problems is well-known [13, 14]. We further demonstrate how these relations arise from a weighted version of energy statistics. Our main algorithmic contribution is to use Hartigan’s method [16] to find local solutions of the above mentioned QCQP, which is NP-hard in general. Hartigan’s method was also used in [4], but without any connection to kernels. More importantly, the advantages of Hartigan’s over Lloyd’s method was already demonstrated in some simple settings [17, 18], but apparently this method did not receive the deserved attention. To the best of our knowledge, Hartigan’s method was not previously employed together with kernel methods. We provide a fully kernel based Hartigan’s algorithm for clustering, where the kernel is fixed by energy statistics. We make clear the advantages of this proposal versus Lloyd’s method, which kernel k-means is based upon and will also be used to solve our QCQP. We show that both algorithms have the same time complexity, but Hartigan’s method in kernel spaces 3 offer several advantages. Furthermore, in the examples considered in this paper, it also provides superior performance compared spectral clustering, which is more expensive and in fact solves a relaxed version of our QCQP. Our numerical results provide compelling evidence that Hartigan’s method applied to energy clustering is more accurate and robust than kernel k-means algorithm. Furthermore, our experiments illustrate the flexibility of energy clustering, showing that it is able to perform accurately on data coming from very different distributions, contrary to k-means and GMM for instance. More specifically, the proposed method performs closely to k-means and GMM on normally distributed data, however, it is significantly better on data that is not normally distributed. Its superiority in high dimensions is striking, being more accurate than k-means and GMM even on Gaussian settings. II. BACKGROUND ON ENERGY STATISTICS AND RKHS In this section we introduce the main concepts from energy statistics and its relation to RKHS which form the basis of our work. For more details we refer the reader to [1] and [6]. iid iid Consider random variables in RD such that X, X 0 ∼ P and Y, Y 0 ∼ Q, where P and Q are cumulative distribution functions with finite first moments. The quantity E(P, Q) ≡ 2EkX − Y k − EkX − X 0 k − EkY − Y 0 k, (1) called energy distance [1], is rotationally invariant and nonnegative, E(P, Q) ≥ 0, where equality to zero holds if and only if P = Q. Above, k · k denotes the Euclidean norm in RD . Energy distance provides a characterization of equality of distributions, and E 1/2 is a metric on the space of distributions. The energy distance can be generalized as, for instance, Eα (P, Q) ≡ 2EkX − Y kα − EkX − X 0 kα − EkY − Y 0 kα (2) where 0 < α ≤ 2. This quantity is also nonnegative, Eα (P, Q) ≥ 0. Furthermore, for 0 < α < 2 we have that Eα (P, Q) = 0 if and only if P = Q, while for α = 2 we have E2 (P, Q) = 2kE(X) − E(Y )k2 which shows that equality to zero only requires equality of the means, and thus E2 (P, Q) = 0 does not imply equality of distributions. The energy distance can be even further generalized. Let X, Y ∈ X where X is an arbitrary space endowed with a semimetric of negative type ρ : X × X → R, which is 4 required to satisfy n X i,j=1 where Xi ∈ X and ci ∈ R such that ci cj ρ(Xi , Xj ) ≤ 0, Pn i=1 ci = 0. Then, X is called a space of negative type. We can thus replace RD → X and kX − Y k → ρ(X, Y ) in the definition (1), obtaining the generalized energy distance E(P, Q) ≡ 2Eρ(X, Y ) − Eρ(X, X 0 ) − Eρ(Y, Y 0 ). (3) For spaces of negative type there exists a Hilbert space H and a map ϕ : X → H such that ρ(X, Y ) = kϕ(X) − ϕ(Y )k2H . This allows us to compute quantities related to probability distributions over X in the associated Hilbert space H. Even though the semimetric ρ may not satisfy the triangle inequality, ρ1/2 does since it can be shown to be a proper metric. Our energy clustering formulation, proposed in the next section, will be based on the generalized energy distance (3). There is an equivalence between energy distance, commonly used in statistics, and distances between embeddings of distributions in RKHS, commonly used in machine learning. This equivalence was established in [6]. Let us first recall the definition of RKHS. Let H be a Hilbert space of real-valued functions over X . A function K : X × X → R is a reproducing kernel of H if it satisfies the following two conditions: 1. hx ≡ K(·, x) ∈ H for all x ∈ X . 2. hhx , f iH = f (x) for all x ∈ X and f ∈ H. In other words, for any x ∈ X and any function f ∈ H, there is a unique hx ∈ H that reproduces f (x) through the inner product of H. If such a kernel function K exists, then H is called a RKHS. The above two properties immediately imply that K is symmetric and positive definite. Indeed, notice that hhx , hy i = hy (x) = K(x, y), and by definition hhx , hy i∗ = hhy , hx i, but since the inner product is real we have hhy , hx i = hhx , hy i, or P equivalently K(y, x) = K(x, y). Moreover, for any w ∈ H we can write w = ni=1 ci hxi P where {hxi }ni=1 is a basis of H. It follows that hw, wiH = ni,j=1 ci cj K(xi , xj ) ≥ 0, showing that the kernel is positive definite. If G is a matrix with elements Gij = K(xi , xj ) this is equivalent to G being positive semidefinite, i.e. v > G v ≥ 0 for any vector v ∈ Rn . 5 The Moore-Aronszajn theorem [19] establishes the converse of the above paragraph. For every symmetric and positive definite function K : X ×X → R, there is an associated RKHS HK with reproducing kernel K. The map ϕ : x 7→ hx ∈ HK is called the canonical feature map. Given a kernel K, this theorem enables us to define an embedding of a probability R measure P into the RKHS as follows: P 7→ hP ∈ HK such that f (x)dP (x) = hf, hP i for R all f ∈ HK , or alternatively, hP ≡ K( · , x)dP (x). We can now introduce the notion of distance between two probability measures using the inner product of HK , which is called the maximum mean discrepancy (MMD) and is given by γK (P, Q) ≡ khP − hQ kHK . (4) 2 γK (P, Q) = EK(X, X 0 ) + EK(Y, Y 0 ) − 2EK(X, Y ) (5) This can also be written as [20] iid iid where X, X 0 ∼ P and Y, Y 0 ∼ Q. From the equality between (4) and (5) we also have hhP , hQ iHK = E K(X, Y ). Thus, in practice, we can estimate the inner product between embedded distributions by averaging the kernel function over sampled data. The following important result shows that semimetrics of negative type and symmetric positive definite kernels are closely related [21]. Let ρ : X × X → R and x0 ∈ X an arbitrary but fixed point. Define K(x, y) ≡ 21 [ρ(x, x0 ) + ρ(y, x0 ) − ρ(x, y)] . (6) Then, it can be shown that K is positive definite if and only if ρ is a semimetric of negative type. We have a family of kernels, one for each choice of x0 . Conversely, if ρ is a semimetric of negative type and K is a kernel in this family, then ρ(x, y) = K(x, x) + K(y, y) − 2K(x, y) = khx − hy k2HK (7) and the canonical feature map ϕ : x 7→ hx is injective [6]. When these conditions are satisfied we say that the kernel K generates the semimetric ρ. If two different kernels generate the same ρ they are said to be equivalent kernels. 6 Now we can state the equivalence between the generalized energy distance (3) and inner products on RKHS, which is one of the main results of [6]. If ρ is a semimetric of negative type and K a kernel that generates ρ, then replacing (7) into (3), and using (5), yields 2 (P, Q). E(P, Q) = 2 [E K(X, X 0 ) + E K(Y, Y 0 ) − 2E K(X, Y )] = 2γK Due to (4) we can compute the energy distance E(P, Q) between two probability distributions using the inner product of HK . Finally, let us recall the main formulas from generalized energy statistics for the test statistic of equality of distributions [1]. Assume we have data X = {x1 , . . . , xn } where S xi ∈ X , and X is a space of negative type. Consider a disjoint partition X = kj=1 Cj , with Ci ∩ Cj = ∅. Each expectation in the generalized energy distance (3) can be computed through the function g(Ci , Cj ) ≡ 1 XX ρ(x, y), ni nj x∈C y∈C i (8) j where ni = |Ci | is the number of elements in partition Ci . The within energy dispersion is defined by W ≡ k X nj j=1 2 g(Cj , Cj ), (9) and the between-sample energy statistic is defined by S≡ where n = Pk j=1 X ni nj [2g(Ci , Cj ) − g(Ci , Ci ) − g(Cj , Cj )] , 2n 1≤i<j≤k (10) nj . Given a set of distributions {Pj }kj=1 , where x ∈ Cj if and only if x ∼ Pj , the quantity S provides a test statistic for equality of distributions [1]. When the sample size is large enough, n → ∞, under the null hypothesis H0 : P1 = P2 = · · · = Pk we have that S → 0, and under the alternative hypothesis H1 : Pi 6= Pj for at least two i 6= j, we have that S → ∞. Note that this test does not make any assumptions about the distributions Pj , thus it is said to be non-parametric or distribution-free. One can make a physical analogy by thinking that points x ∈ Cj form a massive body whose total mass is characterized by the distribution function Pj . The quantity S is thus a potential energy of the from S(P1 , . . . , Pk ) which measures how different the distribution of these masses are, and achieves the ground state S = 0 when all bodies have the same mass distribution. The potential energy S increases as bodies have different mass distributions. 7 III. CLUSTERING BASED ON ENERGY STATISTICS This section contains the main theoretical results of this paper, where we formulate an optimization problem for clustering based on energy statistics and RKHS introduced in the previous section. Due to the previous test statistic for equality of distributions, the obvious criterion for clustering data is to maximize S which makes each cluster as different as possible from the other ones. In other words, given a set of points coming from different probability distributions, the test statistic S should attain a maximum when each point is correctly classified as belonging to the cluster associated to its probability distribution. The following straightforward result shows that maximizing S is, however, equivalent to minimizing W which has a more convenient form. Lemma 1. Let X = {x1 , . . . , xn } where each data point xi lives in a space X endowed with a S semimetric ρ : X × X → R of negative type. For a fixed integer k, the partition X = kj=1 Cj , where Ci ∩ Cj = ∅ for all i 6= j, maximizes the between-sample statistic S, defined in equation (10), if and only if min W (C1 , . . . , Ck ), C1 ,...,Ck (11) where the within energy dispersion W is defined by (9). Proof. From (9) and (10) we have  k k  k X 1 X 1 X nj ni g(Ci , Ci ) ni nj g(Ci , Cj ) + n− S+W = 2n i,j=1 2n i=1 j6=i=1 i6=j = k 1 X 1 XX n ni nj g(Ci , Cj ) = ρ(x, y) = g(X, X). 2n i,j=1 2n x∈X y∈X 2 Note that the right hand side of this equation only depends on the pooled data, so it is a constant independent of the choice of partition. Therefore, maximizing S over the choice of partition is equivalent to minimizing W . For a given k, the clustering problem amounts to finding the best partition of the data by minimizing W . Notice that this is a hard clustering problem as partitions are disjoint. The optimization problem (11) based on energy statistics was already proposed in [4]. However, it is important to note that this is equivalent to maximizing S, which is the test statistic for 8 equality of distributions. In this current form, the relation with kernels and other clustering methods is obscure. In the following, we show what is the explicit optimization problem behind (11) in the corresponding RKHS, establishing the connection with kernel methods. Based on the relation between kernels and semimetrics of negative type, assume that the kernel K : X × X → R generates ρ. Define the Gram matrix    K(x1 , x1 ) K(x1 , x2 ) · · · K(x1 , xn )     K(x2 , x1 ) K(x2 , x2 ) · · · K(x2 , xn )  . G≡ .. .. ..   .. .   . . .   K(xn , x1 ) K(xn , x2 ) · · · K(xn , xn ) (12) Let Z ∈ {0, 1}n×k be the label matrix, with only one nonvanishing entry per row, indicating to which cluster (column) each point (row) belongs to. This matrix satisfies Z > Z = D, where the diagonal matrix D = diag(n1 , . . . , nk ) contains the number of points in each cluster. We also introduce the rescaled matrix Y ≡ ZD−1/2 . In component form they are given by Zij ≡   1 if xi ∈ Cj Yij ≡  0 otherwise    √1 nj  0 if xi ∈ Cj . (13) otherwise Throughout the paper, we use the notation Mi• to denote the ith row of a matrix M , and M•j denotes its jth column. Our next result shows that the optimization problem (11) is NP-hard since it is a quadratically constrained quadratic program (QCQP) in the RKHS. Proposition 2. The optimization problem (11) is equivalent to max Tr Y > G Y Y
s.t. Y ≥ 0, Y > Y = I, Y Y > e = e, (14) where e = (1, 1, . . . , 1)> ∈ Rn is the all-ones vector, and G is the Gram matrix (12). Proof. From (7), (8), and (9) we have  k k X X 1X 1 X 1 X W (C1 , . . . , Ck ) = K(x, x) − ρ(x, y) = K(x, y) . 2 j=1 nj x,y∈C nj y∈C j=1 x∈C j j (15) j Note that the first term is global so it does not contribute to the optimization problem. Therefore, minimizing (15) is equivalent to k X 1 X max K(x, y). C1 ,...,Ck n j=1 j x,y∈C j 9 (16) But X K(x, y) = n X n X Zpj Zqj Gpq = (Z > G Z)jj , p=1 q=1 x,y∈Cj −1 where we used the definitions (12) and (13). Notice that n−1 = Djj , where the diagonal j matrix D = diag(n1 , . . . , nk ) contains the number of points in each cluster, thus the objective

P −1 function in (16) is equal to kj=1 Djj Z > GZ jj = Tr D−1 Z > GZ . Now we can use the cyclic property of the trace, and by the definition of the matrix Z in (13), we obtain the following integer programing problem: max Tr Z  ZD−1/2
> G ZD−1/2
 s.t. Zij ∈ {0, 1}, Pk j=1 Zij = 1, Pn i=1 Zij = nj . (17) Now we write this in terms of the matrix Y = ZD−1/2 . The objective function immedi
ately becomes Tr Y > G Y . Notice that the above constraints imply that Z T Z = D, which in turn gives D−1/2 Y T Y D−1/2 = D, or Y > Y = I. Also, every entry of Y is positive by definition, Y ≥ 0. Now it only remains to show the last constraint in (14), which comes from the last constraint in (17). In matrix form this reads Z T e = De. Replacing Z = Y D1/2 we have Y > e = D1/2 e. Multiplying this last equation on the left by Y , and noticing that Y D1/2 e = Ze = e, we finally obtain Y Y > e = e. Therefore, the optimization problem (17) is equivalent to (14) . Based on Proposition 2, to group data X = {x1 , . . . , xn } into k clusters we first compute the Gram matrix G and then solve the optimization problem (14) for Y ∈ Rn×k . The ith row of Y will contain a single nonzero element in some jth column, indicating that xi ∈ Cj . This optimization problem is nonconvex, and also NP-hard, thus a direct approach is computational prohibitive even for small datasets. However, one can find approximate solutions by relaxing some of the constraints, or obtaining a relaxed SDP version of it. For instance, the relaxed problem max Tr Y > G Y Y
s.t. Y > Y = I has a well-known closed form solution Y ? = U R, where the columns of U ∈ Rn×k contain the top k eigenvectors of G corresponding to the k largest eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λk , and R ∈ Rk×k is an arbitrary orthogonal matrix. The resulting optimal objective function
P assumes the value max Tr Y ? > G Y ? = ki=1 λi . Spectral clustering is based on the above 10 approach, where one further normalize the rows of Y ? , then cluster the resulting rows as data points. A procedure on these lines was proposed in the seminal papers [22, 23]. Note that the optimization problem (14) based on energy statistics is valid for data living in an arbitrary space of negative type, where a semimetric ρ, and thus the kernel K, are assumed to be known. The standard energy distance (2) fixes a family of choices in Euclidean spaces given by ρα (x, y) = kx − ykα , Kα (x, y) = 21 (kxkα + kykα − kx − ykα ) , for 0 < α ≤ 2 and we fix x0 = 0 in (6). The same would be valid for data living in a more general semimetric space (X , ρ) where ρ fixes the kernel. In practice, the clustering quality strongly depend on the choice of a suitable ρ. Nevertheless, if prior information is available to make this choice, it can be immediately incorporated in the optimization problem (14). Relation to Kernel k-Means One may wonder how energy clustering relates to the well-known kernel k-means problem1 which is extensively used in machine learning. For a positive semidefinite Gram matrix G, as defined in (12), there exists a map ϕ : X → HK such that K(x, y) = hϕ(x), ϕ(y)i. Kernel k-means optimization problem, in feature space, is defined by   k X X 2 min J(C1 , . . . , Ck ) ≡ kϕ(x) − ϕ(µj )k C1 ,...,Ck where µj = 1 nj P x∈Cj (18) j=1 x∈Cj x is the mean of cluster Cj in the ambient space. Notice that the above objective function is strongly tied to the idea of minimizing distances between points and cluster centers, which arises from k-means objective function based on Lloyd’s method [7]. It is known [13, 14] that problem (18) can be cast into a trace maximization in the same form as (14). The next result makes this explicit, showing that (11) and (18) are actually equivalent. Proposition 3. For a fixed kernel, the clustering optimization problem (11) based on energy statistics is equivalent to the kernel k-means optimization problem (18), and both are equivalent to (14). 1 When we refer to kernel k-means problem we mean specifically the optimization problem (18), which should not be confused with kernel k-means algorithm that is just one possible recipe to solve (18). The distinction should also be clear from the context. 11 Proof. Notice that kϕ(x)−ϕ(µj )k2 = hϕ(x), ϕ(x)i−2hϕ(x), ϕ(µj )i+hϕ(µj ), ϕ(µj )i, therefore  k X X 1 X 2 X K(x, y) + 2 K(y, z) . J= K(x, x) − nj y∈C nj y,z∈C j=1 x∈C j j (19) j The first term is global so it does not contribute to the optimization problem. Notice that P P P the third term gives x∈Cj n12 y,z∈Cj K(y, z) = n1j y,z∈Cj K(y, z), which is the same as the j second term. Thus, problem (18) is equivalent to k X 1 X K(x, y) C1 ,...,Ck n j j=1 x,y∈C max j which is exactly the same as (16) from the energy statistics formulation. Therefore, once the kernel K is fixed, the function W given by (9) is the same as J in (18). The remaining of the proof proceeds as already shown in the proof of Proposition 2, leading to the optimization problem (14). The above result shows that kernel k-means optimization problem is equivalent to the clustering problem formulated in the energy statistics framework, when operating on the same kernel. Notice, however, that energy statistics is valid for arbitrary semimetric spaces of negative type, fixing the kernel function in the associated RKHS, which is guaranteed to be positive definite. On the other hand, kernel k-means (18) by itself is just an heuristic approach that does not make any explicit mention to the kernel. Based on Proposition 3 one may view kernel k-means as being derived from the energy statistics framework. Kernel k-means, spectral clustering, and graph partitioning problems such as ratio association, ratio cut, and normalized cut are all equivalent to a QCQP of the form (14) [13, 14]. One can thus use kernel k-means algorithm to solve these problems as well. This correspondence involves a weighted version of problem (14), that will be demonstrated in the following from the perspective of energy statistics. IV. CLUSTERING BASED ON WEIGHTED ENERGY STATISTICS We now generalize energy statistics to incorporate weights associated to each data point. Let w(x) be a weight function associated to point x ∈ X . Define g(Ci , Cj ) ≡ 1 XX w(x)w(y)ρ(x, y), si sj x∈C y∈C i j 12 si ≡ X x∈Ci w(x). (20) Replace this function in the formulas (9) and (10), with ni → si and n → s, where s = Pk j=1 sj . With these changes Proposition 1 remains the unaltered, so the clustering problem becomes   k X sj g(Cj , Cj ) min W (C1 , . . . , Ck ) ≡ C1 ,...,Ck 2 j=1 where now g is given by (20). Define the following matrices and vector:    √1 if xi ∈ Cj sj Yij ≡ , W ≡ diag(w1 , . . . , wn ), H ≡ W 1/2 Y,  0 otherwise (21) ω ≡ We, (22) where wi = w(xi ) and e ∈ Rn is the all-ones vector.The analogous of Proposition 2 is as follows. Proposition 4. The weighted energy clustering given by problem (21) is equivalent to  max Tr H > (W 1/2 GW 1/2 )H s.t. H ≥ 0, H > H = I, HH > ω = ω, H (23) where G is the Gram matrix (12), ω = (w1 , . . . , wn )T contains the weights of each point, and W = diag(ω). Proof. Replacing (7) and eliminating the global terms which do not contribute, the optimization problem (21) becomes k X 1 XX w(x)w(y)K(x, y). C1 ,...,Ck s j j=1 x∈C y∈C max j j This objective function can be written as √ k n n k X n X n >√ X X Zjp wp 1/2 wq Zqj 1 XX 1/2 wp wq Zpj Zqj Gpq = wp Gpq wq √ √ s sj sj j=1 j p=1 q=1 j=1 p=1 q=1 = k X j=1 H > W 1/2 GW 1/2 H
jj
= Tr H > W 1/2 GW 1/2 H . To obtain the constraints, note that Hij ≥ 0 by definition, and (H > H)ij = n n X 1 δij X w` Z`i Z`j = w` Z`i = δij , Y`i W`` Y`j = √ √ s s i sj i `=1 `=1 `=1 n X 13 where δij = 1 if i = j and δij = 0 if i 6= j is the Kronecker delta. Therefore, H > H = I. This is a constraint on the rows of H. To obtain a condition on its columns observe that √ wp wq  k  if both xp , xq ∈ Ci X
Z Z √ pj qj si = H > H pq = wp wq  sj 0 j=1 otherwise. Therefore, (H > HW 1/2 )pq = √ wp wq s−1 if both points xp and xq belong to the same cluster, i which we denote by Ci for some i ∈ {1, . . . , k}, and (H > HW 1/2 )pq = 0 otherwise. Thus, the pth line of this matrix is nonzero only on entries corresponding to points that are in the same Pn √ √ wp , cluster as xp . If we sum over the columns of this line we obtain wp s−1 i q=1 wq Zqi = or equivalently HH > W 1/2 e = W 1/2 e, which gives the constraint HH > ω = ω. Connection with Graph Partitioning The relation between kernel k-means and graph partitioning problems is known [13, 14]. For conciseness, we repeat a similar analysis due to the relation of these problems to energy statistics and RKHS, which provides a different perspective. Consider a graph G = (V, E, A) where V is the set of vertices, E the set of edges, and A is an affinity matrix of the graph, which measures the similarities between pairs of nodes. Thus, Aij 6= 0 if (i, j) ∈ E, and Aij = 0 otherwise. We also associate weights to every vertex, P wi = w(i) for i ∈ V, and let sj = i∈Cj wi , where Cj ⊆ V is one partition of V. Let links(C` , Cm ) ≡ X Aij . i∈C` ,j∈Cm We want to partition the set of vertices V into k disjoint subsets, V = generalized ratio association problem is given by max Ci ,...,Ck Sk j=1 k X links(Cj , Cj ) j=1 sj Cj . The (24) and maximizes the within cluster association. The generalized ratio cut problem min Ci ,...,Ck k X links(Cj , V\Cj ) sj j=1 (25) minimizes the cut between clusters. These two problems are equivalent, in analogous way as minimizing (9) is equivalent to maximizing (10) as shown in Proposition 1. Here this is 14 due to the equality links(Cj , V\Cj ) = links(Cj , V) − links(Cj , Cj ). Several graph partitioning methods [22, 24–26] can be seen as a particular case of (24) or (25). Consider the ratio association problem (24), whose objective function can be written as k X k n X n > X X
Zjp Zqj 1 XX > Apq = √ Apq √ = Tr Y AY , s sj sj j=1 p=1 q=1 j=1 j p∈C q∈C j j with Z defined in (13) and Y in (22). Therefore, the ratio association problem can be written in the form (23), i.e. max Tr H > W −1/2 AW −1/2 H H
s.t. H ≥ 0, H > H = I, HH > ω = ω. This is exactly the same problem as weighted energy clustering with G = W −1 AW −1 . Assuming this matrix is positive semidefinite, this generates a semimetric (7) for graphs given by ρ(i, j) = Aii Ajj 2Aij + 2 − 2 wi wj wi wj or ρ(i, j) = − 2Aij wi wj (26) for vertices i, j ∈ V, and where in the second equation we assume the graph has no self-loops, i.e. Aii = 0. Using (26) in the energy statistics formulation allows one to make inference on graphs. Above, the weight wi = w(i) of node i ∈ V can be, for instance, its degree wi = d(i). V. TWO-CLASS PROBLEM IN ONE DIMENSION Before stating a general algorithm to solve the optimization problem (14) we first consider the simplest possible case which is one-dimensional data and a two-class problem. This will be useful to test energy clustering on a simple setting. Fixing ρ(x, y) = |x − y| according to the standard energy distance, we can actually compute the function (8) in O(n log n) and minimize W directly. This is done by noting that |x − y| = (x − y)1x≥y − (x − y)1x<y = x (1x≥y − 1x<y ) + y (1y>x − 1y≤x ) where we have the indicator function defined by 1A = 1 if A is true, and 1A = 0 otherwise. Let C be a partition with n elements. Using the above distance we have g (C, C) = 1 XX x (1x≥y + 1y>x − 1x≥y − 1x<y ) . n2 x∈C y∈C 15 Algorithm 1 E 1D -clustering algorithm to find local solutions to the optimization problem (11) for a two-class problem in one dimension. input data X output label matrix Z e = [x1 , . . . , xn ] 1: sort X obtaining X 2: for j ∈ [1, . . . , n] do 3: 4: Ce1,j ← [xi : i = 1, . . . , j], and Ce2,j ← [xi : i = j + 1, . . . , n]
W (j) ← W Ce1,j , Ce2,j , from (27) 5: end for 6: j ? ← arg minj W (j) 7: Zj• ← (1, 0) if j ≤ j ? , and Zj• ← (0, 1) otherwise, for j = 1, . . . , n The sum over y can be eliminated since each term in the parenthesis is simply counting the number of elements in C that satisfy the condition of the indicator function. Assuming that we first order the data in C, obtaining Ce = [xj ∈ C : x1 ≤ x2 ≤ · · · ≤ xn ], we get n
2 X e e (2` − 1 − n)x` . g C, C = 2 n `=1
e Ce is O(n) and the cost of sorting the data is at the Note that the cost of computing g C, S most O(n log n). Assuming that each partition is ordered, X = kj=1 Cej , the within energy dispersion can be written explicitly as
W Ce1 , . . . , Cek = nj k X X 2` − 1 − nj j=1 `=1 nj x` . (27) For a two-class problem we can use the formula (27) to cluster the data through a simple e Then we compute (27) algorithm as follows. We first order the entire dataset, X → X. e and pick the point which gives the minimum value of W . This for each possible split of X procedure is described in Algorithm 1 and called E 1D -clustering. Note that this algorithm is deterministic, however, it only works for one-dimensional data with Euclidean distance. The total complexity of E 1D -clustering is O(n log n + n2 ) = O(n2 ). Assuming the true label matrix Z is available, a direct measure of how different the 16 estimated matrix Ẑ is from Z, up to label permutations, is given by n k 1 XX accuracy(Ẑ) ≡ max Ẑiσ(j) Zij σ n i=1 j=1 (28) where σ is a permutation of the k cluster groups. The accuracy is always between [0, 1], where 1 corresponds to all points correctly clustered, and 0 to all points wrongly clustered. For a balanced two-class problem the value 1/2 correspond to chance. We now consider two simple experiments where we sample n points from a two-class mixture. We plot the average accuracy (28) versus n, with error bars indicating standard error. The data is clustered using E 1D -clustering algorithm, GMM and k-means. For GMM and k-means we use the implementations from the well-known scikit-learn library in Python [27], where k-means is initialized through k-means++ procedure [28], and GMM is initialized with the output of k-means. We run both algorithms 5 times with different initializations and pick the answer with best objective function value. Notice that E 1D -clustering does not require random initialization so we only run it once. For each n we use use 100 Monte Carlo runs. In Fig. 1a we have the results for data sampled from the Gaussian mixture

iid µ1 = 1.5, σ1 = 0.3, µ2 = 0, σ2 = 1.5. x ∼ 21 N µ1 , σ12 + 12 N µ2 , σ22 , (29) In this case the optimal accuracy obtained from Bayes classification error is ≈ 0.88, indicated by the dashed line in the plot. The three methods perform closely, with a slight advantage of GMM, as expected since it is a consistent model to the data, and E 1D -clustering performs slightly better than k-means. In Fig. 1c we show a density estimation from clustering 1000 points from this mixture using the three algorithms. Notice that all of them are able to distinguish the two classes. On the other hand, in Fig. 1b we consider a mixture of lognormal distributions, 

iid x ∼ 21 exp N µ1 , σ12 + 12 exp N µ2 , σ22 , µ1 = 1.5, σ1 = 0.3, µ2 = 0, σ2 = 1.5. (30) The optimal Bayes accuracy is again ≈ 0.88. We can now see that E 1D -clustering is still very accurate, while GMM and k-means basically cluster at chance. Density estimation after clustering 1000 points this mixture using the three algorithms are is shown in Fig. 1d. Note that only E 1D -clustering was able to distinguish the two classes. k-means and GMM put most of the points in a single cluster, and points on the tail of the second component of (30) in the other cluster. The experiments of Fig. 1 illustrate how energy clustering is more flexible compared to k-means and GMM. 17 0.90 0.90 0.80 0.85 accuracy accuracy 0.85 1D E -clustering k-means GMM 0.80 0.75 E 1D -clustering k-means GMM 0.70 0.65 0.60 0.55 0.75 100 200 300 400 500 600 700 800 0.50 100 200 300 0.8 truth E 1D k-means GMM 700 800 0.5 0.4 0.3 0.4 0.2 0.2 0.1 −4 truth E 1D k-means GMM 0.6 0.6 0.0 −6 600 0.7 1.4 1.0 500 (b) (a) 1.2 400 # points # points −2 0 2 4 6 x 0.0 −2 0 2 4 6 8 10 12 14 x (c) (d) FIG. 1. E 1D -clustering versus k-means and GMM. (a,b) We plot the mean accuracy (28) over 100 Monte Carlo trials, versus the number of sampled points. Error bars are standard error. The dashed line indicates Bayes accuracy (≈ 0.88 in both cases). (a) Clustering results for data normally distributed as in (29). (b) Data lognormally distributed as in (30). (c) Density estimation of each component in the mixture (29) after clustering 1000 sampled points using the three algorithms, compared to the ground truth. (d) The same but for lognormal data (30). VI. ITERATIVE ALGORITHMS FOR ENERGY CLUSTERING In this section we introduce an iterative algorithm to find a local maximizer of the optimization problem (14). Due to Proposition 3 we can also find an approximate solution by the well-known kernel k-means algorithm based on Lloyd’s heuristic [13, 14], which for convenience will also be restated in the present context. Consider the optimization problem (16) written as max {C1 ,...,Ck }   k X Qj Q= , nj j=1 18 Qj ≡ X x,y∈Cj K(x, y), (31) where Qj represents an internal energy cost of cluster Cj , and Q is the total energy cost where each Qj is weighted by the inverse of the number of points in Cj . For a data point xi we denote its own energy cost with the entire cluster C` by Q` (xi ) ≡ X y∈C` K(xi , y) = Gi• · Z•` , where we recall that Gi• (G•i ) denotes the ith row (column) of matrix G. Lloyd’s Method for Energy Clustering To optimize kernel k-means objective function (19) we remove the global term and define the function J (`) (xi ) ≡ 2 1 Q − Q` (xi ). 2 ` n` n` (32) We are thus solving min Z n X k X Zi` J (`) (xi ). i=1 `=1 One possible strategy is to assign xi to cluster Cj ? according to j ? = arg min J (`) (xi ). `=1,...,k This is done for every data point xi and repeated until convergence, i.e. until no new assignments are made. The entire procedure is described in Algorithm 2, which we name E L -clustering to emphasize that we are optimizing the within energy function W based on Lloyd’s method [7]. It can be shown that this algorithm converges provided G is positive semidefinite. E L -clustering is precisely kernel k-means algorithm [13, 14] but written more concisely and with the kernel induced by energy statistics. Indeed, recalling that K(x, y) = hϕ(x), ϕ(y)i where ϕ : X → HK is the feature map, we have from (32) that J (`) (xi ) = hϕ(µ` ), ϕ(µ` )i − 2hϕ(xi ), ϕ(µ` )i = kϕ(xi ) − ϕ(µ` )k2 − kϕ(xi )k2 , where µ` = 1 n` P x∈C` x is the mean of cluster C` . Therefore, min` J (`) (xi ) = min` kϕ(xi ) − ϕ(µ` )k2 , i.e. we are assigning xi to the cluster with closest center (in feature space), which is the familiar Lloyd’s heuristic approach that kernel k-means is based upon. 19 Algorithm 2 E L -clustering is Lloyd’s method for energy clustering, which is precisely kernel k-means algorithm, with the kernel induced by energy statistics. This procedure finds local solutions to the optimization problem (14). input number of clusters k, Gram matrix G, initial label matrix Z ← Z0 output label matrix Z 1: q ← (Q1 , . . . , Qk )> have the costs of each cluster, defined in (31) 2: n ← (n1 , . . . , nk )> have the number of points in each cluster 3: repeat 4: for i = 1, . . . , n do 5: let j be such that xi ∈ Cj 6: j ? ← arg min`=1,...,k J (`) (xi ), where J (`) (xi ) is defined in (32) 7: if j ? 6= j then 8: move xi to Cj ? : Zij ← 0 and Zij ? ← 1 9: update n: nj ← nj − 1 and nj ? ← nj ? + 1 10: update q: qj ← qj − 2Qj (xi ) and qj ? ← qj ? + 2Qj ? (xi ) 11: end if 12: end for 13: until convergence To check the complexity of E L -clustering, notice that to compute the second term of J (`) (xi ) in (32) requires O(n` ) operations, and although the first term requires O(n2` ) it only needs to be computed once outside loop through data points (step 1 of Algorithm 2). Therefore, the time complexity of E L -clustering is O(nk max` n` ) = O(kn2 ). For a sparse Gram matrix G having n0 nonzero elements this complexity can be further reduced to O(kn0 ). Hartigan’s Method for Energy Clustering We now consider Hartigan’s method [16] applied to the optimization problem in the form (31), which gives a local solution to the QCQP defined in (14). The method is based in computing the maximum change in the total cost function Q when moving each data 20 point to another cluster. More specifically, suppose point xi is currently assigned to cluster Cj yielding a total cost function denoted by Q(j) . Moving xi to cluster C` yields another total cost function denoted by Q(`) . We are interested in computing the maximum change ∆Qj→` (xi ) ≡ Q(`) − Q(j) , for ` 6= j. From (31), by explicitly writing the costs related to these two cluster we obtain − ∆Q j→` Qj Qj Q` Q+ ` + − − (xi ) = n` + 1 nj − 1 nj n` − where Q+ ` denote the cost of the new `th cluster with the point xi added to it, and Qj is the cost of new jth cluster with xi removed from it. Noting that Q+ ` = Q` + 2Q` (xi ) + Gii and Q− j = Qj − 2Qj (xi ) + Gii , we get the formula     Qj 1 Q` 1 j→` − 2Qj (xi ) + Gii − − 2Q` (xi ) − Gii . ∆Q (xi ) = nj − 1 nj n` + 1 n` (33) Therefore, if ∆Qj→` (xi ) > 0 we get closer to a maximum of (31) by moving xi to C` , otherwise we keep xi in Cj . We thus propose the following algorithm. We start with an initial configuration for the label matrix Z, then for each point xi we compute the cost of moving it to another cluster C` , i.e. ∆Qj→` (xi ) for ` = 1, . . . , k with ` 6= j, where j denotes the index of its current partition, x ∈ Cj . Hence, we choose j ? = arg max ∆j→` (xi ). `=1,...,k | `6=j ? If ∆Qj→j (xi ) > 0 we move xi to cluster Cj ? , otherwise we keep xi in its original cluster Cj . This process is repeated until no points are assigned to new clusters. The entire procedure is explicitly described in Algorithm 3, which we denote E H -clustering to emphasize that it is based on Hartigan’s method. This method automatically ensures that the objective function is monotonically increasing at each iteration, and consequently the algorithm converges in a finite number of steps. The complexity analysis of E H -clustering is the following. Computing the Gram matrix G requires O(Dn2 ) operations, where D is the dimension of each data point and n is the data size. However, both algorithms E L - and E H -clustering assume that G is given. There are more efficient methods to compute G, specially if it is sparse, but we will not consider this further and just assume that G is given. The computation of each cluster cost Qj has complexity O(n2j ), and overall to compute q we have O(n21 + · · · + n2k ) = O(k maxj n2j ). 21 Algorithm 3 E H -clustering is Hartigan’s method for energy clustering. This algorithm finds local solutions to the optimization problem (14). The steps 6 and 10 are different than E L -clustering described in Algorithm 2. input number of clusters k, Gram matrix G, initial label matrix Z ← Z0 output label matrix Z 1: q ← (Q1 , . . . , Qk )> have the energy costs of each cluster, defined in (31) 2: n ← (n1 , . . . , nk )> have the number of points in each cluster 3: repeat 4: for i = 1, . . . , n do 5: let j be such that xi ∈ Cj 6: j ? ← arg max`=1,...,k | `6=j ∆Qj→` (xi ) using (33) 7: if ∆Qj→j (xi ) > 0 then ? 8: move xi to Cj ? : Zij ← 0 and Zij ? ← 1 9: update n: nj ← nj − 1 and nj ? ← nj ? + 1 10: update q: qj ← qj − 2Qj (xi ) + Gii and qj ? ← qj ? + 2Qj ? (xi ) + Gii 11: end if 12: end for 13: until convergence These operations only need to be performed a single time. For each point xi we need to compute Qj (xi ) once, which is O(nj ), and we need to compute Q` (xi ) for each ` 6= j. The cost of computing Q` (xi ) is O(n` ), thus the cost of step 6 in Algorithm 3 is O(k max` n` ) for ` = 1, . . . , k. For the entire dataset this gives a time complexity of O(nk max` n` ) = O(kn2 ). Note that this is the same cost as in E L -clustering, or kernel k-means algorithm. Again, if G is sparse this can be reduced to O(kn0 ) where n0 is the number of nonzero entries of G. In the following we mention some important known results about Hartigan’s method. Theorem 5 (Telgarsky-Vattani [17]). Hartigan’s method has the cost function strictly decreasing in each iteration. Moreover, if n > k then 1. the resulting partition has no empty clusters, and 22 2. the resulting partition has distinct means. Neither of these two conditions are guaranteed to be satisfied by Lloyd’s method, and consequently by E L -clustering algorithm. The next result indicates that Hartigan’s method can potentially escape local optima of Lloyd’s method. Theorem 6 (Telgarsky-Vattani [17]). The set of local optima of Hartigan’s method is a (possibly strict) subset of local optima of Lloyd’s method. The above theorem implies that E L -clustering cannot improve on a local optima of E H - clustering. On the other hand, E H might improve on a local optima of E L . Lloyd’s method forms Voronoi partitions, while Hartigan’s method groups data in regions formed by the intersection of spheres called circlonoi cells. It can be shown that the circlonoi cells are contained within a smaller volume of a Voronoi cell, and this excess volume grows exponentially with the dimension of X [17, Theorems 2.4 and 3.1]. Points in this excess volume force Hartigan’s method to iterate, contrary to Lloyd’s method. Therefore, Hartigan’s can escape local optima of Lloyd’s. Moreover, this improvement should be more prominent as dimension increases. Also, the improvement grows as the number of clusters k increases. The empirical results of [17] show that an implementation of Hartigan’s method has comparable execution time to an implementation of Lloyd’s method, but no explicit complexity was provided. We show that both E L - and E H -clustering have the same time complexity. To the best of our knowledge, Hartigan’s method was not previously considered together with kernels, as we are proposing in E H -clustering algorithm. In [18], Hartigan’s method was applied to k-means problem with any Bregman divergence. It was shown that the number of Hartigan’s local optima is upper bounded by O(1/k) [18, Proposition 5.1]. In addition, it was provided examples where any initial partition correspond to a local optima of Lloyd’s method, while the number of local optima in Hartigan’s method is small and correspond to true partitions of the data. Empirically, the number of Hartigan’s local optima was considerably smaller than the number of Lloyd’s local optima. The above results indicate that Hartigan’s method provides several advantages over Lloyd’s method, a fact that will also be supported by our numerical experiments in the next section where E H outperforms of E L (kernel k-means) in several settings, specially in high dimensions. 23 VII. NUMERICAL EXPERIMENTS The main goal of this section is threefold. First, we want to compare E H -clustering in Euclidean space to k-means and GMM. Second, we want to compare E H -clustering, based on Hartigan’s method, to E L -clustering or kernel k-means, based on Lloyd’s method, and also to spectral clustering, when they all operate on the same kernel. Third, we want to illustrate the flexibility provided by energy clustering, which is able to cluster accurately in different settings while keeping the same kernel. The following experimental setup holds unless specified otherwise. We consider E H - clustering, E L -clustering and spectral clustering with the following semimetrics and corresponding generating kernels: ρα (x, y) = kx − ykα , Kα (x, y) = 21 (kxkα + kykα − kx − ykα ) , (34) ρeσ (x, y) = 2 − 2e− e σ (x, y) = e− K kx−yk 2σ , (35) kx−yk2 2σ 2 . (36) ρbσ (x, y) = 2 − 2e− kx−yk 2σ , kx−yk2 2σ 2 , b σ (x, y) = e− K The relation between kernel and semimetric is given by formula (6) where we fix x0 = 0. The standard ρ1 , from the original energy distance (1), will always be present in the experiments as a reference, being the implied choice unless explicitly mentioned. For k-means, GMM and spectral clustering we use the robust implementations of scikit-learn library [27], where k-means is initialized with k-means++ [28], and GMM with the output of k-means, making it more robust and preventing it from breaking in high dimensions. Spectral clustering implementation is based on [22]. We implemented E L -clustering as described in Algorithm 2, and E H -clustering as described in Algorithm 3. Both will also be initialized with k-means++. We run the algorithms 5 times with different initializations, picking the result with best objective function value. We evaluate clustering quality by the accuracy (28) based on the true labels. For each setting we show the average accuracy over 100 Monte Carlo trials, with error bars indicating standard error. We briefly mention that we compared E H -clustering, as described in Algorithm 3, to E 1D -clustering, described in Algorithm 1, for several univariate distributions. Both perform very closely. However, we omit these results since we will analyse more interesting scenarios in high dimensions. From the results of [17], summarized in the end of the previous section, we expect the 24 x1 −3 5 4 3 2 1 0 −1 −2 −3 x2 4 3 2 1 0 −1 −2 −3 −4 x2 8 6 4 2 0 −2 −4 −6 −8 x3 4 4 3 2 1 0 −1 −2 −3 −4 3 x1 2 1 0 −1 6 4 x3 2 −4 −6 4 10 3 8 2 6 1 4 0 −1 2 0 −2 −2 4 4 3 3 2 2 −3 −4 1 x5 x5 0 −2 x4 x4 −2 0 1 0 −1 −1 −3 −3 −2 −1 0 −3 −2 −1 −2 −2 1 x1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 x2 −3 −2 −1 0 x3 1 2 3 −3 −2 −1 0 1 x4 2 3 4 −3 −2 −1 0 1 x5 2 3 4 0 1 x1 2 3 4 −6 −4 −2 0 x2 2 4 (a) 6 −4−3−2−1 0 1 2 3 4 5 x3 −4 −2 0 2 x4 4 6 8 −3 −2 −1 0 1 x5 2 3 4 (a) FIG. 2. Pair plots for the first 5 dimensions. (a) Data normally distributed as in (37). (b) Data normally distributed as in (38). We sample 200 points for both cases. We can see that there is a considerable overlap between the clusters. improvement of Hartigan’s over Lloyd’s method to be more accentuated in high dimensions. Thus, we analyze how the algorithms degrade as the number of dimensions increase while keeping the number of points in each cluster fixed. Consider data from the Gaussian mixture iid x ∼ 21 N (µ1 , Σ1 ) + 12 N (µ2 , Σ2 ), Σ1 = Σ2 = ID , µ1 = (0, . . . , 0)> , | {z } ×D µ2 = 0.7(1, . . . , 1, 0, . . . , 0)> . | {z } | {z } ×10 (37) ×(D−10) To get some intuition about how separated data points from each class are, we show scatter plots between the first 5 dimensions in Fig. 2a. Note that the Bayes error is fixed as D increases, yielding an optimal accuracy of ≈ 0.86. We sample 200 points on each trial. The results are shown in Fig. 3a. We can see that E H and spectral clustering have practically the same performance, which is higher than E L -clustering (kernel k-means). Moreover, E H outperforms k-means and GMM, where the improvement is noticeable specially in high dimensions. Note that in this setting k-means and GMM are consistent models to the data, however, energy clustering degrades much less as dimension increases. Still for a two-class Gaussian mixture, we now allow the diagonal entries of one of the 25 0.95 0.85 0.90 0.75 0.70 0.65 0.60 0.55 0.85 accuracy accuracy 0.80 EH EL spectral k-means GMM 0.80 0.75 0.70 0.65 0.60 50 100 150 200 0.55 EH EL spectral k-means GMM 100 # dimensions 200 300 400 500 600 700 # dimensions (a) (b) FIG. 3. Comparison of E H -clustering, E L -clustering (kernel k-means), spectral clustering, k-means and GMM in high dimensional Gaussian settings. We plot the mean accuracy versus the number of dimensions, with error bars indicating standard error from 100 Monte Carlo runs. (a) Data normally distributed as in (37), with Bayes accuracy ≈ 0.86, over the range D ∈ [10, 200]. (b) Data normally distributed as in (38), with Bayes accuracy ≈ 0.95, over the range D ∈ [10, 700]. covariances to have different values by choosing iid x ∼ 12 N (µ1 , Σ1 ) + 12 N (µ2 , Σ2 ), µ1 = (0, . . . , 0)> , | {z } ×D µ2 = (1, . . . , 1, 0, . . . , 0)> , | {z } | {z } ×10 ×(D−10) Σ1 = ID ,  Σ2 =  e 10 Σ 0 0 ID−10 e 10 = diag(1.367, 3.175, 3.247, 4.403, 1.249, 1.969, 4.035, 4.237, 2.813, 3.637). Σ  , (38) We simply chose a fixed set of 10 numbers uniformly at random on the interval [1, 5] for the e 10 , and any other choice would give analogous results. We show pair plots of diagonal of Σ this data in Fig. 2b. We sample a total of 200 points from (38) on each trial. The Bayes error is kept fixed when increasing D yielding an optimal accuracy ≈ 0.95. In Fig. 3b we see that GMM performs better in low dimensions, but it quickly degenerates as D increases. The same is true for k-means and E L -clustering. However, E H and spectral clustering remains much more stable in high dimensions. Notice that a naive implementation of GMM should not be able to estimate the covariances when D & 100, however, scikit-learn library uses k-means output as initialization, therefore the output of GMM in this implementation is at least as good as k-means and the algorithm is more robust in high dimensions. 26 Consider sampling data from the following Gaussian mixture in R20 : iid x ∼ 21 N (µ1 , Σ1 ) + 12 N (µ2 , Σ2 ), µ1 = (0, . . . , 0)> , | {z } ×20 µ2 = 12 (1, . . . , 1, 0, . . . , 0)> , | {z } | {z } 5 Σ1 = 12 I20 , Σ2 = I20 . (39) 15 The optimal accuracy based on Bayes classification error is ≈ 0.90. We increase the sample size n ∈ [10, 400] and show the accuracy versus n for the different kernels (34) and (35) within E H -clustering algorithm, which are compared to k-means and GMM. The results are in Fig. 4a. Note that for small n all methods are superior than GMM, which slowly catches up and tend to optimal Bayes, as expected since it is a consistent model to the data. Note e 1 is as accurate as GMM for large number of points, also that E H -clustering with kernel K however, it is superior for small number of points. Still for the same setting, in Fig. 4b we show the difference in accuracy provided by E H minus E L and E H minus spectral clustering, e 1 . Note that E H was always superior than kernel k-means and when using the kernel K spectral clustering, otherwise there would be points negative values on the y-axis. Consider the same experiment but now with a lognormal mixture, iid 1 2 x∼ exp {N (µ1 , Σ1 )} + 12 exp {N (µ2 , Σ2 )} , µ1 = (0, . . . , 0)> , | {z } ×20 µ2 = 12 (1, . . . , 1, 0, . . . , 0)> , | {z } | {z } 5 Σ1 = 12 I20 , Σ2 = I20 . (40) 15 The results are in Fig. 4c. Energy clustering still performs accurately, with any of the utilized kernels, providing better results than k-means and GMM on this non-normal data. e 1 still provides the best results for small number of points, but its performance is The kernel K eventually achieved by K1/2 , indicating that α ≈ 1/2 in the standard energy distance should be more appropriate for skewed distributions. In Fig. 4d we show the difference between E H e 1 . Again, the accuracy clustering to kernel k-means and spectral clustering, with the kernel K provided by E H is higher than the other methods, although not much higher than spectral clustering in this example. The two experiments of Fig. 4 illustrate how energy clustering is more flexible, performing well in different settings with the same kernel, contrary to k-means and GMM. In Fig. 5a–c we have complex two dimensional datasets. The two parallel cigars in (a) have 200 points each. The concentric circles in (b) and (c) have 400 points for each class. We apply E H -clustering with the kernels (34), (35) and (36). We also consider the best 27 0.90 0.85 0.85 0.80 E H , ρ1 0.75 E H , ρ1/2 E H , ρe1 0.70 50 100 150 200 250 300 350 k-means GMM 0.70 50 100 150 200 250 # points # points (a) (c) 300 350 400 0.25 difference in accuracy difference in accuracy E H , ρe1 0.75 0.55 400 0.25 0.20 0.15 0.10 0.00 E H , ρ1/2 0.80 0.60 0.30 0.05 E H , ρ1 0.65 k-means GMM 0.65 0.60 accuracy accuracy 0.90 EH − EL E H − spectral 50 100 150 200 250 300 350 400 EH − EL 0.20 E H − spectral 0.15 0.10 0.05 0.00 50 100 150 200 250 # points # points (b) (d) 300 350 400 FIG. 4. E H -clustering with kernels (34) and (35) versus k-means and GMM. In both settings Bayes accuracy is ≈ 0.9. We show average accuracy (error bars are standard error) versus number of points for 100 Monte Carlo trials. (a,b) Gaussian mixture (39). (c,d) Lognormal mixture (40). The plots in (c) and (d) consider the difference in accuracy between E H versus E L (kernel k-means) e1. and spectral clustering, with the kernel K (a) (b) (c) (d) FIG. 5. (a) Parallel cigars. (b) Two concentric circles with noise. (c) Three concentric circles with noise. (d) MNIST handwritten digits. Clustering results are in Table I and Table II. 28 TABLE I. Clustering data from Fig. 5a–c. Fig. 5a ρ1 E H -clustering spectral-clustering k-means GMM 0.705 ± 0.065 ρ1/2 0.952 ± 0.048 ρ1 Fig. 5b Fig. 5c 0.521 ± 0.005 ρ1 0.393 ± 0.020 ρ1/2 0.522 ± 0.004 ρ1/2 0.486 ± 0.040 ρe2 0.9987 ± 0.0008 ρe1 0.778 ± 0.075 ρe2 0.666 ± 0.007 ρe2 0.557 ± 0.014 ρb1 0.732 ± 0.002 ρb2 0.364 ± 0.004 0.522 ± 0.004 7 0.368 ± 0.005 7 0.595 ± 0.011 7 0.465 ± 0.030 ρb2 0.956 ± 0.020 7 0.550 ± 0.011 7 0.903 ± 0.064 ρb1 7 1.0 ± 0.0 ρb2 0.676 ± 0.002 TABLE II. Clustering MNIST data from Fig. 5d. {0, 1, . . . , 4} {0, 1, . . . , 6} {0, 1, . . . , 8} {0, 1, . . . , 9} σ 10.41 10.41 10.37 10.19 ρ1 0.873 ± 0.025 0.731 ± 0.016 0.687 ± 0.016 0.581 ± 0.011 ρ1/2 0.874 ± 0.027 0.722 ± 0.017 0.647 ± 0.017 0.600 ± 0.009 0.847 ± 0.031 0.695 ± 0.023 0.657 ± 0.014 0.584 ± 0.013 Class Subset parameter E H -clustering ρeσ ρbσ 0.891 ± 0.009 0.759 ± 0.011 0.704 ± 0.011 0.591 ± 0.012 0.769 ± 0.012 0.678 ± 0.014 0.649 ± 0.018 0.565 ± 0.009 k-means 7 0.878 ± 0.010 0.744 ± 0.008 0.695 ± 0.012 0.557 ± 0.012 GMM 7 0.839 ± 0.015 0.694 ± 0.010 0.621 ± 0.009 0.540 ± 0.009 spectral-clustering ρbσ kernel choice for each example for spectral clustering. Moreover, we consider k-means and GMM. We perform 10 Monte Carlo runs for each example. The results are in Table I. For (a) we initialize all algorithms with k-means++, and for (b) and (c) we initialize at random. E H has superior performance in every example, and in particular better than the spectral clustering. In (a) the standard kernel from energy statistics in Euclidean space, K1 and K1/2 , are able to provide accurate results, however, for the examples in (b) and (c) the b 1 and K b 2 provide a significant improvement. kernel choice is more sensitive, where K Next, we consider the infamous MNIST handwritten digits as illustrated in Fig. 5d. Each 29 data point is an 8-bit gray scale image forming a 784-dimensional vector corresponding to the digits {0, 1, . . . , 9}. We compute the parameter n 1 X kxi − xj k2 , σ = 2 n i,j=1 2 from a separate training set, to be used in the kernels (35) and (36). We consider subsets of {0, 1, . . . , 9}, sampling 100 points for each class. The results are shown in Table II, where kernels and parameters are indicated. E H -clustering performs slightly better than k-means and GMM, however the difference is not considerable. Unsupervised clustering on MNIST without any feature extraction is not trivial. For instance, the same experiment was performed in [29] where a low-rank transformation is learned then subsequently used in subspace clustering, providing very accurate results. It would be interesting to explore analogous methods for learning a better representation of the data and subsequently apply E H -clustering. VIII. DISCUSSION We proposed clustering from the perspective of generalized energy statistics, valid for arbitrary spaces of negative type. Our mathematical formulation of energy clustering reduces to a QCQP in the associated RKHS, as demonstrated in Proposition 2. We showed that the optimization problem is equivalent to kernel k-means, once the kernel is fixed; see Proposition 3. Energy statistics, however, fixes a family of standard kernels in Euclidean space, and more general kernels on spaces of negative type can also be obtained. We also considered a weighted version of energy statistics, whose clustering formulation establishes connections with graph partitioning. We proposed the iterative E H -clustering algorithm based on Hartigan’s method, which was compared to kernel k-means algorithm based on Lloyd’s heuristic. Both have the same time complexity, however, numerical and theoretical results provide compelling evidence that E H -clustering is more robust with a superior performance, specially in high dimensions. Furthermore, energy clustering, with standard kernels from energy statistics, outperformed k-means and GMM on several settings, illustrating the flexibility of the proposed method which is model-free. In many settings, the iterative E H -clustering also surpassed spectral clustering, which is solves a relaxation of the original QCQP, and in other settings performed closely but never worse. Note that spec30 accuracy 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 EH EL spectral k-means GMM 0 50 100 150 200 # unbalanced points FIG. 6. Comparison of energy clustering algorithms to k-means and GMM on unbalanced clusters. The data is normally distributed as (41), where we vary m ∈ [0, 240], and in each case we do 100 Monte Carlo runs showing the average accuracy with standard error. tral clustering is more expensive than our iterative method, going up to O(n3 ), and finding eigenvectors of very large matrices is problematic. A limitation of the proposed methods for energy clustering is that it cannot handle accurately highly unbalanced clusters. As an illustration, consider the following Gaussian mixture: n1 n1 N (µ1 , Σ1 ) + N (µ1 , Σ1 ), µ1 = (0, 0, 0, 0)> , µ2 = 1.5 × (1, 1, 0, 0)> , 2N 2N   1 I2 0  , n1 = N − m, n2 = N + m, N = 300. Σ1 = I4 , Σ2 =  2 0 I2 iid x∼ (41) We then increase m ∈ [0, 240] making the clusters progressively more unbalanced. We plot the average accuracy over 100 Monte Carlo runs for each m, with error bars indicating standard error. The results are shown in Fig. 6. For highly unbalanced clusters we see that GMM performs better than the other methods, which have basically similar performance. Based on this experiment, an interesting problem would be to extend E H -clustering algorithm to account for highly unbalanced clusters. Moreover, it would be interesting to formally demonstrate cases where energy clustering is a consistent in the large n limit. A soft version of energy clustering is also an interesting extension. Finally, kernel methods can benefit from sparsity and fixed-rank approximations of the Gram matrix, and there is plenty of room to make E H -clustering algorithm more scalable. 31 ACKNOWLEDGMENTS We would like to thank Carey Priebe for discussions. We would like to acknowledge the support of the Transformative Research Award (NIH #R01NS092474) and the Defense Advanced Research Projects Agencys (DARPA) SIMPLEX program through SPAWAR contract N66001-15-C-4041. [1] G. J. Székely and M. L. Rizzo. 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Bayesian Probabilistic Numerical Methods Jon Cockayne∗ Chris Oates† Tim Sullivan‡ Mark Girolami§ arXiv:1702.03673v2 [stat.ME] 7 Jul 2017 July 10, 2017 The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with a challenging industrial application presented. 1. Introduction Numerical computation underpins almost all of modern scientific and industrial research and development. The impact of a finite computational budget is that problems whose solutions are high- or infinite-dimensional, such as the solution of differential equations, must be discretised in order to be solved. The result is an approximation to the object of interest. The declining rate of processor improvement as physical limits are reached is in contrast to the surge in complexity of modern inference problems, and as a result the error incurred by discretisation is attracting increased interest (e.g. Capistrán et al., 2016). The situation is epitomised in modern climate models, where use of single-precision arithmetic has been explored to permit finer temporal resolution. However, when computing in singleprecision, a detailed time discretisation can increase total error, due to the increased number of single precision computations, and in practice some form of ad-hoc trade-off is sought (Harvey and Verseghy, 2015). It has been argued that statistical considerations can permit more principled error control strategies for such models (Hennig et al., 2015). Numerical methods are designed to mitigate discretisation errors of all forms (Press et al., 2007). Nonetheless, the introduction of error is unavoidable and it is the role of the numerical analyst to provide control of this error (Oberkampf and Roy, 2013). The central theoretical results of numerical analysis have in general not been obtained through statistical considerations. ∗ University of Warwick, j.cockayne@warwick.ac.uk Newcastle University and Alan Turing Institute, chris.oates@ncl.ac.uk ‡ Free University of Berlin and Zuse Institute Berlin, sullivan@zib.de § Imperial College London and Alan Turing Institute, m.girolami@imperial.ac.uk † 1 More recently, the connection of discretisation error to statistics was noted as far back as Henrici (1963); Hull and Swenson (1966), who argued that discretisation error can be modelled using a series of independent random perturbations to standard numerical methods. However, numerical analysts have cast doubt on this approach, since discretisation error can be highly structured; see Kahan (1996) and Higham (2002, Section 2.8). To address these objections, the field of probabilistic numerics has emerged with the aim to properly quantify the uncertainty introduced through discretisation in numerical methods. The foundations of probabilistic numerics were laid in the 1970s and 1980s, where an important shift in emphasis occurred from the descriptive statistical models of the 1960s to the use of formal inference modalities that generalise across classes of numerical tasks. In a remarkable series of papers, Larkin (1969, 1970, 1972); Kuelbs et al. (1972); Larkin (1974, 1979a,b), Mike Larkin presented now classical results in probabilistic numerics, in particular establishing the correspondence between Gaussian measures on Hilbert spaces and optimal numerical methods. Re-discovered and re-emphasised on a number of occasions, the role for statisticians in this new outlook was clearly captured in Kadane and Wasilkowski (1985): Statistics can be thought of as a set of tools used in making decisions and inferences in the face of uncertainty. Algorithms typically operate in such an environment. Perhaps then, statisticians might join the teams of scholars addressing algorithmic issues. The 1980s culminated in development of Bayesian optimisation methods (Mockus, 1989; Törn and Žilinskas, 1989), as well as the relation of smoothing splines to Bayesian estimation (Kimeldorf and Wahba, 1970b; Diaconis and Freedman, 1983). The modern notion of a probabilistic numerical method (henceforth PNM) was described in Hennig et al. (2015); these are algorithms whose output is a distribution over an unknown, deterministic quantity of interest, such as the numerical value of an integral. Recent research in this field includes PNMs for numerical linear algebra (Hennig, 2015; Bartels and Hennig, 2016), numerical solution of ordinary differential equations (ODEs; Schober et al., 2014; Kersting and Hennig, 2016; Schober et al., 2016; Conrad et al., 2016; Chkrebtii et al., 2016), numerical solution of partial differential equations (PDEs; Owhadi, 2015; Cockayne et al., 2016; Conrad et al., 2016) and numerical integration (O’Hagan, 1991; Briol et al., 2016). Open Problems Despite numerous recent successes and achievements, there is currently no general statistical foundation for PNMs, due to the infinite-dimensional nature of the problems being solved. For instance, at present it is not clear under what conditions a PNM is welldefined, except for in the standard conjugate Gaussian framework considered in (Larkin, 1972). This limits the extent to which domain-specific knowledge, such as boundedness of an integrand or monotonicity of a solution to a differential equation, can be encoded in PNMs. In contrast, classical numerical methods often exploit such information to achieve substantial reduction in discretisation error. For instance, finite element methods for solution of PDEs proceed based on a mesh that is designed to be more refined in areas of the domain where greater variation of the solution is anticipated (Strang and Fix, 1973). Furthermore, although PNMs have been proposed for many standard numerical tasks (see Section 2.6.1), the lack of common theoretical foundations makes comparison of these methods difficult. Again taking PDEs as an example, Cockayne et al. (2016) placed a probability distribution on the unknown solution of the PDE, whereas Conrad et al. (2016) placed a probability distribution on the unknown discretisation error of a numerical method. The uncertainty modelled in each case is fundamentally different, but at present there is no framework in which to 2 articulate the relationship between the two approaches. Furthermore, though PNMs are often reported as being “Bayesian” there is no clear definition of what this ought to entail. A more profound consequence of the lack of common foundation occurs when we seek to compose multiple PNMs. For example, multi-physics cardiac models involve coupled ODEs and PDEs which must each be discretised and approximately solved to estimate a clinical quantity of interest (Niederer et al., 2011). The composition of successive discretisations leads to non-trivial error propagation and accumulation that could be quantified, in a statistical sense, with PNMs. However, proper composition of multiple PNMs for solutions of ODEs and PDEs requires that these PNMs share common statistical foundations that ensure coherence of the overall statistical output. These foundations remain to be established. Contributions The main contribution of this paper is to establish rigorous foundations for PNMs: The first contribution is to argue for an explicit definition of a “Bayesian” PNM. Our framework generalises the seminal work of Larkin (1972) and builds on the modern and popular mathematical framework of Stuart (2010). This illuminates subtle distinctions among existing methods and clarifies the sense in which non-Bayesian methods are approximations to Bayesian PNMs. The second contribution is to establish when PNMs are well-defined outside of the conjugate Gaussian context. For exploration of non-linear, non-Gaussian models, a numerical approximation scheme is developed and shown to asymptotically approach the posterior distribution of interest. Our aim here is not to develop new or more computationally efficient PNMs, but to understand when such development can be well-defined. The third contribution is to discuss pipelines of composed PNMs. This is a critical area of development for probabilistic numerics; in isolation, the error of a numerical method can often be studied and understood, but when composed into a pipeline the resulting error structure may be non-trivial and its analysis becomes more difficult. The real power of probabilistic numerics lies in its application to pipelines of numerical methods, where the probabilistic formulation permits analysis of variance (ANOVA) to understand the contribution of each discretisation to the overall numerical error. This paper introduces conditions under which a composition of PNMs can be considered to provide meaningful output, so that ANOVA can be justified. Structure of the Paper In Section 2 we argue for an explicit definition of Bayesian PNM and establish when such methods are well-defined. Section 3 establishes connections to other related fields, in particular with relation to evaluating the performance of PNMs. In Section 4 we develop useful numerical approximations to the output of Bayesian PNMs. Section 5 develops the theory of composition for multiple PNMs. Finally, in Section 6 we present applications of the techniques discussed in this paper. All proofs can be found in either the Appendix or the Electronic Supplement. 2. Probabilistic Numerical Methods The aim of this section is to provide rigorous statistical foundations for PNMs. 2.1. Notation For a measurable space (X , ΣX ), the shorthand PX will be used to denote the set of all distributions on (X , ΣX ). For µ, µ0 ∈ PX we write µ  µ0 when µ is absolutely continuous with respect 3 to µ. The notation δ(x) will be used to denote a Dirac measure on x ∈ X , so that δ(x) ∈ PX . Let 1[S] denote the indicator function of an event S ∈ ΣX . For a measurable function f :X →R R and a distribution µ ∈ PX , we will on occasion use the notation µ(f ) = f (x)µ(dx) and kf k∞ = supx∈X |f (x)|. The point-wise product of two functions f and g is denoted f · g. For a function or operator T , T# denotes the associated push-forward operator1 that acts on measures on the domain of T . Let ⊥ ⊥ denote conditional independence. The subset `p ⊂ R∞ is defined P∞ to consist of sequences (ui ) for which i=1 |ui |p is convergent. C(0, 1) will be used to denote the set of continuous functions on (0, 1). 2.2. Definition of a PNM To first build intuition, consider numerical approximation of the Lebesgue integral Z x(t)ν(dt) for some integrable function x : D → R, with respect to a measure ν on D. Here we may directly interrogate the integrand x(t) at any t ∈ D, but unless D is finite we cannot evaluate x at all t ∈ D with a finite computational budget. Nonetheless, there are many algorithms for approximation of this integral based on information {x(ti )}ni=1 at some collection of locations {ti }ni=1 . To see the abstract structure of this problem, assume the state variable x exists in a measurable space (X , ΣX ). Information about x is provided through an information operator A : X → A whose range is a measurable space (A, ΣA ). Thus, for the Lebesgue integration problem, the information operator is   x(t1 )   A(x) =  ...  = a ∈ A. (2.1) x(tn ) The space X , in this case a space of functions, can be high- or infinite-dimensional, but the space A of information is assumed to be finite-dimensional in accordance with our finite computational budget. In this paper we make explicit a quantity of interest (QoI) Q(x), defined by a map Q : X → Q into a measurable space (Q, ΣQ ). This captures that x itself may not be the object of interest for the numerical problem; for the Lebesgue integration illustration, the QoI is not R x itself but Q(x) = x(t)ν(dt). The standard approach to such computational problems is to construct an algorithm which, when applied, produces some approximation q̂(a) of Q(x) based on the information a, whose theoretical convergence order can be studied. A successful algorithm will often tailor the information operator A to the QoI Q. For example, classical Gaussian cubature specifies sigma points {t∗i }ni=1 at which the integrand must be evaluated, based on exact integration of certain polynomial test functions. The probabilistic numerical approach, instead, begins with the introduction of a random variable X on (X , ΣX ). The true state X = x is fixed but unknown; the randomness is used an abstract device used to represent epistemic uncertainty about x prior to evolution of the information operator (Hennig et al., 2015). This is now formalised: 1 Recall that, for measurable T : X → A, the pushforward T# µ of a distribution µ ∈ PX is defined as T# µ(A) = µ(T −1 (A)) for all A ∈ ΣA . 4 Definition 2.1 (Belief Distribution). An element µ ∈ PX is a belief distribution 2 for x if it carries the formal semantics of belief about the true, unknown state variable x. Thus we may consider µ to be the law of X. The construction of an appropriate belief distribution µ for a specific numerical task is not the focus of this research and has been considered in detail in previous work; see the Electronic Supplement for an overview of this material. Rather we consider the problem of how one updates the belief distribution µ in response to the information A(x) = a obtained about the unknown x. Generic approaches to update belief distributions, which generalise Bayesian inference beyond the unique update demanded in Bayes theorem, were formalised in Bissiri et al. (2016); de Carvalho et al. (2017). Definition 2.2 (Probabilistic Numerical Method). Let (X , ΣX ), (A, ΣA ) and (Q, ΣQ ) be measurable spaces and let A : X → A, Q : X → Q and B : PX × A → PQ where A and Q are measurable functions. The pair M = (A, B) is called a probabilistic numerical method for estimation of a quantity of interest Q. The map A is called an information operator, and the map B is called a belief update operator. The output of a PNM is a distribution B(µ, a) ∈ PQ . This holds the formal status of a belief distribution for the value of Q(x), based on both the initial belief µ about the value of x and the information a that are input to the PNM. An objection sometimes raised to this construction is that x itself is not random. We emphasise that this work does not propose that x should be considered as such; the random variable X is a formal statistical device used to represent epistemic uncertainty (Kadane, 2011; Lindley, 2014). Thus, there is no distinction from traditional statistics, in which x represents a fixed but unknown parameter and X encodes epistemic uncertainty about this parameter. Before presenting specific instances of this general framework, we comment on the potential analogy between A and the likelihood function, and between B and Bayes’ theorem. Whilst intuitively correct, the mathematical developments in this paper are not well-suited to these terms; in Section 2.5 we show that Bayes formula is not well-defined, as the posterior distribution is not absolutely continuous with respect to the prior. To strengthen intuition we now give specific examples of established PNMs: Example 2.3 (Probabilistic Integration). Consider the numerical integration problem earlier discussed. Take D ⊆ Rd , X a separable Banach space of real-valued functions on D, and ΣX the Borel σ-algebra for X . The space (X , ΣX ) is endowed with a Gaussian belief distribution µ ∈ PX . Given information A(x) = a, define µa to be the restriction of µ to those functions which interpolate x at the points {ti }ni=1 ; that µa is again Gaussian follows from linearity of the R information operator (see Bogachev, 1998, for details). The QoI Q remains Q(x) = x(t)ν(dt). This problem was first considered by Larkin (1972). The belief update operator proposed therein, and later considered in Diaconis (1988); O’Hagan (1991) and others, was B(µ, a) = Q# µa . Since Gaussians are closed under linear projection, the PNM output B(µ, a) is a univariate Gaussian whose mean and variance can be expressed in closed-form for certain choices of Gaussian covariance function and reference measure ν on D. Specifically, if µ has mean function m : X → R and covariance function k : X × X → R, then B(µ, a) = N(z > K−1 (a − m̄), z0 − z > K−1 z) 2 (2.2) Two remarks are in order: First, we have avoided the use of “prior” as this abstract framework encompasses both Bayesian and non-Bayesian PNMs (to be defined). Second, the use of “belief” differs to the set-valued belief functions in Dempster–Shafer theory, which do not require that µ(E) + µ(E c ) = 1 (Shafer, 1976). 5 R where m̄, z ∈ Rn are defined as m̄i = m(ti ), zi = k(t, ti )ν(dt), K ∈ Rn×n is defined as RR Ki,j = k(ti , tj ) and z0 = k(t, t0 )(ν × ν)d(t × t0 ) ∈ R. This method was extensively studied in Briol et al. (2016), who provided a listing of (ν, k) combinations for which z and z0 possess a closed-form. An interesting fact is that the mean of B(µ, a) coincides with classical cubature rules for different choices of µ and A (Diaconis, 1988; Särkkä et al., 2016). In Section 3 we will show that this is a typical feature of PNMs. The crucial distinction between PNMs and classical numerical methods is the distributional nature of B(µ, a), which carries the formal semantics of belief about the QoI. The full distribution B(µ, a) was examined in Briol et al. (2016), who established contraction to the exact value of the integral under certain smoothness conditions on the Gaussian covariance function and on the integrand. See also Kanagawa et al. (2016); Karvonen and Särkkä (2017). Example 2.4 (Probabilistic Meshless Method). As a canonical example of a PDE, take the following elliptic problem with Dirichlet boundary conditions −∇ · (κ∇x) = f in D x=g on ∂D (2.3) where we assume D ⊂ Rd and κ : D → Rd×d is a known coefficient. Let X be a separable Banach space of appropriately differentiable real-valued functions and take ΣX to be the Borel σ-algebra for X . In contrast to the first illustration, the QoI here is Q(x) = x, as the goal is to make inferences about the solution of the PDE itself. Such problems were considered in Cockayne et al. (2016) wherein µ was restricted to be a Gaussian distribution on X . The information operator was constructed by choosing finite sets of locations T1 = {t1,1 , . . . , t1,n1 } ⊂ D and T2 = {t2,1 , . . . , t2,n2 } ⊂ ∂D at which the system defined in Eq. (2.3) was evaluated, so that     −∇ · (κ(t1,1 )∇x(t1,1 )) f (t1,1 )    ..  ..    .  .     −∇ · (κ(t1,n1 )∇x(t1,n1 )) f (t1,n1 )     . A(x) =  a=   x(t2,1 )    g(t2,1 )     ..  ..    .  . x(t2,n2 ) g(t2,n2 ) The belief update operator was chosen to be B(µ, a) = µa , where µa is the restriction of µ to those functions for which A(x) = a is satisfied. In the setting of a linear system of PDEs such as that in Eq. (2.3), the distribution B(µ, a) is again Gaussian (Bogachev, 1998). Full details are provided in Cockayne et al. (2016). As in the previous example, we note that the mean of B(µ, a) coincides with the numerical solution to the PDE provided by a classical method (the symmetric collocation method; Fasshauer, 1999). The full distribution B(µ, a) provides uncertainty quantification for the unknown exact solution and can again be shown to contract to the exact solution under certain smoothness conditions (Cockayne et al., 2016). This method was further analysed for a specific choice of covariance operator in the belief distribution µ, in an impressive contribution from Owhadi (2017). 6 2.2.1. Classical Numerical Methods Standard numerical methods fit into the above framework, as can be seen by taking B(µ, a) = δ ◦ b(a) (2.4) independent of the distribution µ, where a function b : A → Q gives the output of some classical numerical method for solving the problem of interest. Here δ : Q → PQ maps b(a) ∈ Q to a Dirac measure centred on b(a). Thus, information in a ∈ A is used to construct a point estimate b(a) ∈ Q for the QoI. The formal language of probabilities is not used in classical numerical analysis to describe numerical error. However, in many cases the classical and probabilistic analyses are mathematically equivalent. For instance, there is an equivalence between the standard deviation of B(µ, a) for probabilistic integration and the worst-case error for numerical cubature rules from numerical analysis (Novak and Woźniakowski, 2010). The explanation for this phenomenon will be given in Section 3. 2.3. Bayesian PNMs Having defined a PNM, we now state the central definition of this paper, that is of a Bayesian PNM. Define µa to be the conditional distribution of the random variable X, given the event A(X) = a. For now we assume that this can be defined without ambiguity and reserve a more technical treatment of conditional probabilities for Section 2.5. In this work we followed Larkin (1972) and cast the problem of determining x in Eq. (2.1) as a problem of Bayesian inversion, a framework now popular in applied mathematics and uncertainty quantification research (Stuart, 2010). However, in a standard Bayesian inverse problem the observed quantity a is assumed to be corrupted with measurement error, which is described by a “likelihood”. This leads, under mild assumptions, to general versions of Bayes’ theorem (see Stuart, 2010, Section 2.2) For PNM, however, the information is not corrupted with measurement error. As a result, the support of the likelihood is a null set under the prior, making the standard approaches to such problems, including Bayes’ theorem, ill-defined outside of the conjugate Gaussian case when unknowns are infinite-dimensional. This necessitates a new definition: Definition 2.5 (Bayesian Probabilistic Numerical Method). A probabilistic numerical method M = (A, B) is said to be Bayesian 3 for a quantity of interest Q if, for all µ ∈ PX , the output B(µ, a) = Q# µa , for A# µ-almost-all a ∈ A. That is, a PNM is Bayesian if the output of the PNM is the push-forward of the conditional distribution µa through Q. This definition is familiar from the examples in Section 2.2, which are both examples of Bayesian PNMs. For Bayesian PNMs we adopt the traditional terminology in which µ is the prior for x and the output Q# µa the posterior for Q(x). Note that, for fixed A and µ, the Bayesian choice of belief update operator B (if it exists) is uniquely defined. It is emphasised that the class of Bayesian PNMs is a subclass of all PNMs; examples of nonBayesian PNMs are provided in Section 2.6.1. Our analysis is focussed on Bayesian PNMs due to 3 The use of “Bayesian” contrasts with Bissiri et al. (2016), for whom all belief update operators represent Bayesian learning algorithms to some greater or lesser extent. An alternative term could be “lossless”, since all the information in a is conditioned upon in µa . 7 their appealing Bayesian interpretation and ease of generalisation to pipelines of computation in Section 5. For non-Bayesian PNMs, careful definition and analysis of the belief update operator is necessary to enable proper interpretation of the uncertainty quantification being provided. In particular, the analysis of non-Bayesian PNMs may present considerable challenges in the context of computational pipelines, whereas for Bayesian PNMs this is shown in Section 5 to be straight-forward. 2.4. Model Evidence A cornerstone of the Bayesian framework is the model evidence, or marginal likelihood (MacKay, 1992). Let A ⊆ Rn be equipped with the Lebesgue reference measure λ, such that A# µ admits a density pA = dA# µ/dλ. Then the model evidence pA (a), based on the information that A(x) = a, can be used as the basis for Bayesian model comparison. In particular, two prior distributions µ, µ̃, can be compared through the Bayes factor BF := dA# µ̃ p̃A (a) = (a), pA (a) dA# µ (2.5) where p̃A = dA# µ̃/dλ. Here the second expression is independent of the choice of reference measure λ and is thus valid for general A. The model evidence has been explored in connection with the design of Bayesian PNM. For the integration and PDE examples 2.3 and 2.4, the model evidence has a closed form and was investigated in Briol et al. (2016); Cockayne et al. (2016). In Section 6 we investigate the model evidence in the context of non-linear ODEs and PDEs for which it must be approximated. 2.5. The Disintegration Theorem The purpose of this section is to formalise µa and to determine conditions under which µa exists and is well-defined. From Definition 2.5, the output of a Bayesian PNM is B(µ, a) = Q# µa . If µa exists, the pushforward Q# µa exists as Q is assumed to be measurable; thus, in this section, we focus on the rigorous definition of µa . Unlike many problems of Bayesian inversion, proceeding by an analogue of Bayes’ theorem is not possible. Let X a = {x ∈ X : A(x) = a}. Then we observe that, if it is measurable, X a may be a set of zero measure under µ. Standard techniques for infinite-dimensional Bayesian inversion rely on constructing a posterior distribution based on its Radon–Nikodým derivative with respect to the prior (Stuart, 2010). However, when µa 6 µ no Radon–Nikodým derivative exists and we must turn to other approaches to establish when a Bayesian PNM is well-defined. Conditioning on null sets is technical and was formalised in the celebrated construction of measure-theoretic probability by Kolmogorov (1933). The central challenge is to establish uniqueness of conditional probabilities. For this work we exploit the disintegration theorem to ensure our constructions are well-defined. The definition below is due to Dellacherie and Meyer (1978, p.78), and a statistical introduction to disintegration can be found in Chang and Pollard (1997). Definition 2.6 (Disintegration). For µ ∈ PX , a collection {µa }a∈A ⊂ PX is a disintegration of µ with respect to the (measurable) map A : X → A if: 1 (Concentration:) µa (X \ X a ) = 0 for A# µ-almost all a ∈ A; and for each measurable f : X → [0, ∞) it holds that 8 2 (Measurability:) a 7→ µa (f ) is measurable; R 3 (Conditioning:) µ(f ) = µa (f )A# µ(da). The concept of disintegration extends the usual concept of conditioning of random variables to the case where X a is a null set, in a way closely related to regular conditional distributions (Kolmogorov, 1933). Existence of disintegrations is guaranteed under general weak conditions: Theorem 2.7 (Disintegration Theorem; Thm. 1 of Chang and Pollard (1997)). Let X be a metric space, ΣX be the Borel σ-algebra and µ ∈ PX be Radon. Let ΣA be countably generated and contain all singletons {a} for a ∈ A. Then there exists a disintegration {µa }a∈A of µ with respect to A. Moreover, if {ν a }a∈A is another such disintegration, then {a ∈ A : µa 6= ν a } is a A# µ null set. The requirement that µ is Radon is weak and is implied when X is a Radon space, which encompasses, for example, separable complete metric spaces. The requirement that ΣA is countably generated is also weak and includes the standard case where A = Rn with the Borel σ-algebra. From Theorem 2.7 it follows that {µa }a∈A exists and is essentially unique for all of the examples considered in this paper. Thus, under mild conditions, we have established that Bayesian PNMs are well-defined, in that an essentially unique disintegration {µa }a∈A exists. It is noted that a variational definition of µa has been posited as an alternative approach, for when the existence of a disintegration is difficult to establish (p3 of Garcia Trillos and Sanz-Alonso, 2017). 2.6. Prior Construction The Gaussian distribution is popular as a prior in the PNM literature for its tractability, both in the fact that finite-dimensional distributions take a closed-form and that an explicit conditioning formula exists. More general priors, such as Besov priors (Dashti et al., 2012) and Cauchy priors (Sullivan, 2016) are less easily accessed. In this section we summarise a common construction for these prior distributions, designed to ensure that a disintegration will exist. Let {φi }∞ i=0 denote an orthogonal Schauder basis for X , assumed to be a separable Banach space in this section. Then any x ∈ X can be represented through an expansion x = x0 + ∞ X ui φi (2.6) i=0 for some fixed element x0 ∈ X and a sequence u ∈ R∞ . Construction of measures µ ∈ PX is then reduced to construction of almost-surely convergent measures on R∞ and studying the pushforward of such measures into X . In particular, this will ensure that µ ∈ PX is Radon (as X is a separable complete metric space), a key requirement for existence of a disintegration {µa }a∈A . To this end it is common to split u into a stochastic and deterministic component; let ξ ∈ R∞ represent an i.i.d sequence of random variables, and γ ∈ `p for some p ∈ (1, ∞). Then with ui = γi ξi , for the prior distribution to be well-posed we require that almost-surely u ∈ `1 . Different choices of (ξ, γ) give rise to different distributions on X . For instance, ξi ∼ Uniform(−1, 1), γ ∈ `1 is termed a uniform prior and ξi ∼ N (0, 1) gives a Gaussian prior, where γ determines the regularity of the covariance operator C (Bogachev, 1998). The choice of ξi ∼ Cauchy(0, 1) gives a Cauchy prior in the sense of Sullivan (2016); here we require γ ∈ `1 ∩ ` log ` for X a separable Banach space, or γ ∈ `2 for when X is a Hilbert space. A range of prior specifications will be explored in Section 6, including non-Gaussian prior distributions for numerical solution of nonlinear ODEs. 9 2.6.1. Dichotomy of Existing PNMs This section concludes with an overview of existing PNMs with respect to our definition of a Bayesian PNM. This serves to clarify some subtle distinctions in existing literature, as well as to highlight the generality of our framework. To maintain brevity we have summarised our findings in Table 1. 3. Decision-Theoretic Treatment Next we assess the performance of PNMs from a decision-theoretic perspective (Berger, 1985) and explore connections to average-case analysis of classical numerical methods (Ritter, 2000). Note that the treatment here is agnostic to whether the PNM in question is Bayesian, and also encompasses classical numerical methods. Throughout, the existence of a disintegration {µa }a∈A will be assumed. 3.1. Loss and Risk Consider a generic loss function L : Q × Q → R where L(q † , q) describes the loss incurred when the true QoI q † = Q(x) is estimated with q ∈ Q. Integrability of L is assumed. The belief update operator B returns a distribution over Q which can be cast as a randomised decision rule for estimation of q † . For randomised decision rules, the risk function r : Q×PQ → R is defined as Z † r(q , ν) = L(q † , q)ν(dq) . The average risk of the PNM M = (A, B) with respect to µ ∈ PX is defined as Z R(µ, M ) = r(Q(x), B(µ, A(x)))µ(dx). (3.1) Here a state x ∼ µ is drawn at random and the risk of the PNM output B(µ, A(x)) is computed. We follow the convention of terming R(µ, M ) the Bayes risk of the PNM, though the usual objection that a frequentist expectation enters into the definition of the Bayes risk could be raised. Next, we consider a sequence A(n) of information operators indexed such that A(n) (x) is n-dimensional (i.e. n pieces of information are provided about x). Definition 3.1 (Contraction). A sequence M (n) = (A(n) , B (n) ) of PNMs is said to contract at a rate rn under a belief distribution µ if R(µ, M (n) ) = O(rn ). This definition allows for comparison of classical and probabilistic numerical methods (Kadane and Wasilkowski, 1983; Diaconis, 1988). In each case an important goal is to determine methods M (n) that contract as quickly as possible for a given distribution µ that defines the Bayes risk. This is the approach taken in average-case analysis (ACA; Ritter, 2000) and will be discussed in Section 3.4. For Examples 2.3 and 2.4 of Bayesian PNMs, Briol et al. (2016) and Cockayne et al. (2016) established rates of contraction for particular prior distributions µ; we refer the reader to those papers for details. 10 Method Integrator QoI Q(x) R x(t)ν(dt) Information A(x) {x(ti )}n i=1 Non- (or Approximate) Bayesian PNMs Osborne et al. (2012b,a); Gunter et al. (2014) {ti }n i=1 s.t. ti ∼ x {(ti , x1 (ti ))}n i=1 s.t. ti ∼ x2 {x(ti )}n i=1 {∇x(ti )}n i=1 {(x(ti ), ∇x(ti )}n i=1 Kong et al. (2003); Tan (2004); Kong et al. (2007) Optimiser R R f (t)x(dt) x1 (t)x2 (dt) arg min x(t) Oates et al. (2016a) Bayesian Optimisation (Mockus, 1989) Hennig and Kiefel (2013) Probabilistic Line Search (Mahsereci and Hennig, 2015) Probabilistic Bisection Algorithm (Horstein, 1963) {I[tmin < ti ]}n i=1 Linear Solver x ODE Solver x −1 b {I[tmin < ti ] + error}n i=1 {xti }n i=1 Waeber et al. (2013) {∇x(ti )}n i=1 (Skilling, 1992) Filtering Methods for IVPs (Schober et al., 2014; Chkrebtii et al., 2016; Kersting and Hennig, 2016; Teymur et al., 2016; Schober et al., 2016) Finite Difference Methods (John and Wu, 2017) Hull and Swenson (1966); Mosbach and Turner (2009) Stochastic Euler (Krebs, 2016) Chkrebtii et al. (2016); Raissi et al. (2017) 11 ∇x + rounding error PDE Solver x(tend ) x {∇x(ti )}n i=1 {Dx(ti )}n i=1 Dx + discretisation error Bayesian PNMs Bayesian Quadrature (Larkin, 1974; Diaconis, 1988; O’Hagan, 1991) Probabilistic Linear Solvers (Hennig, 2015; Bartels and Hennig, 2016) Probabilistic Meshless Methods (Owhadi, 2015, 2017; Cockayne et al., 2016; Raissi et al., 2016) Conrad et al. (2016) Table 1: Comparison of several existing Probabilistic Numerical Methods (PNMs). 3.2. Bayes Decision Rules A (possibly randomised) decision rule is said to be a Bayes rule if it achieves the minimum Bayes risk among all decision rules. In the context of (not necessarily Bayesian) PNMs, let M = (A, B) and let   0 B(A) = B : R(µ, (A, B)) = inf0 R(µ, (A, B )) . B That is, for fixed A, B(A) is the set of all belief update operators that achieve minimum Bayes risk. This raises the natural question of which belief update operators yield Bayes rules. Although the definition of a Bayes rule applies generically to both probabilistic and deterministic numerical methods, it can be shown4 that if B(A) is non-empty, then there exists a B ∈ B(A) which takes the form of a classical numerical method, as expressed in Eq. (2.4). Thus in general, Bayesian PNMs do not constitute Bayes rules, as the extra uncertainty inflates the Bayes risk, so that such methods are not optimal. Nonetheless, there is a natural connection between Bayesian PNMs and Bayes rules, as exposed in Kadane and Wasilkowski (1983): Theorem 3.2. Let M = (A, B) be a Bayesian probabilistic numerical method for the QoI Q. Let (Q, h·, ·iQ ) be an inner-product space and let the loss function L have the form L(q † , q) = kq † − qk2Q , where k · kQ is the norm induced by the inner product. Then the decision rule that returns the mean of the distribution B(µ, a) is a Bayes rule for estimation of q † . This well-known fact from Bayesian decision theory5 is interesting in light of recent research in constructing PNMs whose mean functions correspond to classical numerical methods (Schober et al., 2014; Hennig, 2015; Särkkä et al., 2016; Teymur et al., 2016; Schober et al., 2016). Theorem 3.2 explains the results in Examples 2.3 and 2.4, in which both instances of Bayesian PNMs were demonstrated to be centred on an established classical method. 3.3. Optimal Information The previous section considered selection of the belief update operator B, but not of the information operator A. The choice of A determines the Bayes risk for a PNM, which leads to a problem of experimental design to minimise that risk. The theoretical study of optimal information is the focus of the information complexity literature (Traub et al., 1988; Novak and Woźniakowski, 2010), while other fields such as quasi-Monte Carlo (QMC, Dick and Pillichshammer, 2010) attempt to develop asymptotically optimal information operators for specific numerical tasks, such as the choice of evaluation points for numerical approximation of integrals in the case of QMC. Here we characterise optimal information for Bayesian PNMs. Consider the choice of A from a fixed subset Λ of the set of all possible information operators. To build intuition, for the task of numerical integration, Λ could represent all possible choices of locations {ti }ni=1 where the integrand is evaluated. For Bayesian PNM, one can ask for optimal information:  Aµ ∈ arg inf R(µ, M ) s.t. M = (A, B), B = Q# µA A∈Λ 4 5 The proof is included in the Electronic Supplement. This is the fact that the Bayes act is the posterior mean under squared-error loss (Berger, 1985). 12 where we have made explicit the fact that the optimal information depends on the choice of prior µ. Next we characterise Aµ , while an explicit example of optimal information for a Bayesian PNM is detailed in Example 3.4. 3.4. Connection to Average Case Analysis The decision theoretic framework in Section 3.1 is closely related to average-case analysis (ACA) of classical numerical methods (Ritter, 2000). In ACA the performance of a classical numerical method b : A → Q is studied in terms of the Bayes risk R(µ, M ) given in Eq. (3.1), for the PNM M = (A, B) with belief operator B(µ, a) = δ ◦ b(a) as in Eq. (2.4). ACA is concerned with the study of optimal information:   A∗µ ∈ arg inf inf R(µ, M ) s.t. M = (A, B), B = δ ◦ b . A∈Λ b In general there is no reason to expect Aµ and A∗µ to coincide, since Bayesian PNM are not Bayes rules6 . Indeed, an explicit example where Aµ 6= A∗µ is presented in Appendix S3. However, we can establish sufficient conditions under which optimal information for a Bayesian PNM is the same as optimal information for ACA: Theorem 3.3. Let (Q, h·, ·iQ ) be an inner product space and the loss function L have the form L(q † , q) = kq † − qk2Q where k · kQ is the norm induced by the inner product. Then the optimal information Aµ for a Bayesian PNM and A∗µ for ACA are identical. It is emphasised that this result is not a trivial consequence of the correspondance between Bayes rules and worst case optimal methods, as exposed in Kadane and Wasilkowski (1983). To the best of our knowledge, information-based complexity research has studied A∗µ but not Aµ . Theorem 3.3 establishes that, for the squared norm loss, we can extract results on optimal average case information from the ACA literature and use them to construct optimal Bayesian PNMs. An example is provided next. Example 3.4 (Optimal Information for Probabilistic Integration). To illustrate optimal information for Bayesian PNMs, we revisit the first worked example of ACA, due to Sul0 din (1959, 1960). Set X = {x ∈ C(0, 1) : x(0) = 0} and take the belief distribution µ to be induced from the Weiner process on X , i.e. a Gaussian process with mean 0 and covariance function R1 k(t, t0 ) = min(t, t0 ). Our QoI is Q(x) = 0 x(t)dt and the loss function is L(q, q 0 ) = (q − q 0 )2 . Consider standard information A(x) = (x(t1 ), . . . , x(tn )) for n fixed knots 0 ≤ t1 < · · · < tn ≤ 1. Our aim is to determine knots ti that represent optimal information for a Bayesian PNM with respect to µ and L. Motivated by Theorem 3.3 we first solve the optimal information problem for ACA and then derive Pn the associated PNM. It will be sufficient to restrict attention to linear methods b(a) = i=1 wi x(ti ) with wi ∈ R. This allows a closed-form expression for the average error:   X n n X 1 1 2 R(µ, (A, δ ◦ b)) = − 2 wi ti − ti + wi wj min(ti , tj ). 3 2 i=1 (3.2) i,j=1 Standard calculus can be used to minimise Eq. (3.2) over both the weights {wi }ni=1 and the locations {ti }ni=1 ; the full calculation can be found in Chapter 2, Section 3.3 of Ritter (2000). 6 The distribution Q# µa will in general not be supported on the set of Bayes acts. 13 The result is an ACA optimal method n b(A(x)) = 2 X ∗ x(ti ), 2n + 1 t∗i = i=1 2i 2n + 1 which is recognised as the trapezium rule with equally spaced knots. The associated contraction rate rn is n−1 (Lee and Wasilkowski, 1986). From Theorem 3.3 we have that ACA optimal information is also optimal information for the Bayesian PNM. Thus the optimal Bayesian PNM M = (A, B) for the belief distribution µ is uniquely determined:  ∗  ! x(t1 ) n 2 X 1  ..  A(x) =  .  , B(µ, a) = N ai , . 2n + 1 3(2n + 1)2 ∗ i=1 x(tn ) Note how the PNM is centred on the ACA optimal method. However the PNM itself is not a Bayes rule; it in fact carries twice the Bayes risk as the ACA method. This illustration can be generalised. It is known that for µ induced from the Weiner process on ∂ s x, Q a linear functional and φ a loss function that is convex and symmetric, equi-spaced evaluation points are essentially optimal information, the Bayes rule is the natural spline of degree 2s + 1, and the contraction rate rn is essentially n−(s+1) ; see Lee and Wasilkowski (1986) for a complete treatment. This completes our performance assessment for PNMs; next we turn to computational matters. 4. Numerical Disintegration In this section we discuss algorithms to access the output from a Bayesian PNM. The approach considered in this paper is to form an explicit approximation to µa that can be sampled. The construction of a sampling scheme can exploit sophisticated Monte Carlo methods and allow probing B(µ, a) at a computational cost that is de-coupled from the potentially substantial cost of obtaining the information a itself. The construction of an approximation to µa is non-trivial on a technical level. As shown in Section 2.5, under weak conditions on the space X and the operator A, the disintegration µa is well-defined for A# µ-almost all a ∈ A. The approach considered in this work is based on sampling from an approximate distribution µaδ which converges in an appropriate sense to µa in the δ ↓ 0 limit. This follows in a similar spirit to Ackerman et al. (2017). 4.1. Sequential Approximation of a Disintegration Suppose that A is an open subset of Rn and that the distribution A# µ ∈ PA , admits a continuous and positive density pA with respect to Lebesgue measure on A. Further endow A with the structure of a Hilbert space, with norm k · kA . Let φ : R+ → R+ denote a decreasing function, to be specified, that is continuous at 0, with φ(0) = 1 and limr→∞ φ(r) = 0. Consider   1 kA(x) − akA µaδ (dx) := a φ µ(dx) Zδ δ 14 where the normalisation constant Zδa := Z φ  kã − akA δ  pA (dã) is non-zero since pA is bounded away from 0 on a neighbourhood of a ∈ A and φ is bounded away from 0 on a sufficiently small interval [0, γ]. Our aim is to approximate µa with µaδ for small bandwidth parameter δ. The construction, which can be considered a mathematical generalisation of approximate Bayesian computation (Del Moral et al., 2012), ensures that µaδ  µ. The role of φ is to admit states x ∈ X for which A(x) is close to a but not necessarily equal. It is assumed to be sufficiently regular: R Assumption 4.1. There exists α > 0 such that Cφα := rα+n−1 φ(r)dr < ∞. To discuss the convergence of µaδ to µa we must first select a metric on PX . Let F be a normed space of (measurable) functions f : X → R with norm k·kF . For measures ν, ν 0 ∈ PX , define dF (ν, ν 0 ) = sup |ν(f ) − ν 0 (f )|. kf kF ≤1 This formulation encompasses many common probability metrics such as the total variation distance and Wasserstein distance (Müller, 1997). However, not all spaces of functions F lead 0 to useful theory. In particular the total variation distance between µa and µa for a 6= a0 will be one in general. Furthermore depending on the choice of F, dF may be merely a pseudometric7 . Sufficient conditions for weak convergence with respect to F are now established: Assumption 4.2. The map a 7→ µa is almost everywhere α-Hölder continuous in dF , i.e. 0 dF (µa , µa ) ≤ Cµα ka − a0 kαA for some constant Cµα > 0 and for A# µ almost all a, a0 ∈ A. Sufficient conditions for Assumption 4.2 are discussed in Ackerman et al. (2017), but are somewhat technical. Theorem 4.3. Let C̄φα := Cφα /Cφ0 . Then, for δ > 0 sufficiently small, dF (µaδ , µa ) ≤ Cµα (1 + C̄φα )δ α for A# µ almost all a ∈ A. This result justifies the approximation of µa by µaδ when the QoI can be well-approximated by integrals with respect to F. This result is stronger than that of earlier work, such as Pfanzagl (1979), in that it holds for infinite-dimensional X , though it also relies upon the stronger Hölder continuity assumption. The specific form for φ is not fundamental, but can impact upon rate constants. For the n choice φ(r) = 1[r < 1] we have C̄φα = α+n , which can be bounded independent of the dimension n of A. On the other hand, for φ(r) = exp(− 12 r2 ) it can be shown that, for α ∈ N, C̄φα = (α + n − 1)!! (n − 1)!! (4.1) so that the constant C̄φα might not be bounded. In general this necessitates effective Monte Carlo methods that are able to sample from the regime where δ can be extremely small, in order to control the overall approximation error. 7 For a pseudometric, dF (x, y) = 0 =⇒ x = y need not hold. 15 4.2. Computation for Series Priors The series representation of µ in Eq. (2.6) of Section 2.6 is infinite-dimensional and thus cannot, in general, be instantiated. To this end, define XN = x0 + span{φ0 , . . . , φN } and define the associated projection operator PN : X → XN as ! ∞ N X X PN x0 + ui φi := x0 + ui φi . i=0 i=0 A natural approach is to compute with the modified information operator A ◦ PN instead of A. This has the effect of updating the distribution of the first N + 1 coefficients and leaving the tail unchanged, to produce an output µaδ,N . Then computation performed in the Bayesian update step is finite-dimensional, whilst instantiation of the posterior itself remains infinitedimensional. A “likelihood-informed” choice of basis {φi } in such problems was considered in Cui et al. (2016). Inspired by this approach, we next considered convergence of the output µaδ,N to µaδ in the limit N → ∞. In this section it is additionally required that φ be everywhere continuous with φ > 0. Let ϕ = − log φ, so that ϕ is a continuous bijection of R+ to itself. The following are also assumed: Assumption 4.4. For each R > 0, it holds that |ϕ(r) − ϕ(r0 )| ≤ CR |r − r0 | for some constant CR and all r, r0 < R. Assumption 4.5. kA(x) − A ◦ PN (x)kA ≤ exp(m(kxkX ))Ψ(N ) for all x ∈ X , where m is measurable and satisfies EX∼µ [exp(2m(kXkX ))] < ∞ and Ψ(N ) vanishes as N is increased. Assumption 4.6. supx∈X kA(x)kA < ∞. Assumption 4.7. kf k∞ ≤ CF kf kF for some constant CF and all f ∈ F. Assumption 4.4 holds for the case ϕ(r) = 12 r2 with constant CR = R. Assumption 4.5 is standard in the inverse problem literature; for instance it is shown to hold for certain series priors in Theorem 3.4 of Cotter et al. (2010). Assumption 4.6 is, in essence, a compactness assumption, in that it is implied by compactness of the state space X when A is linear. In this sense it is a strong assumption; however it can be enforced in our experiments, where X is unbounded, through a threshold map ( A(x) if kA(x)kA ≤ λmax , Ã(x) := A(x) λmax kA(x)kA if kA(x)kA > λmax , where λmax is a large pre-defined constant. Assumption 4.7 places a restriction on the probability metric dF in which our result is stated. The following theorem has its proof in the Electronic Supplement: Theorem 4.8. For some constant Cδ , dependent on δ, it holds that dF (µaδ,N , µaδ ) ≤ Cδ Ψ(N ). An immediate consequence of Theorems 4.3 and 4.8 is that the total approximation error can be bounded by applying the triangle inequality: dF (µa , µaδ,N ) ≤ Cµα (1 + C̄φα )δ α + Cδ Ψ(N ). In particular, we have convergence of µaδ,N to µa in the δ ↓ 0 limit provided that the number of basis functions satisfies Cδ Ψ(N ) = o(1). 16 The approximate posterior µaδ,N analysed above can be sampledP when µ is Gaussian, since the first N + 1 coefficients can be handled with MCMC and the tail ∞ i=N +1 ui φi , being Gaussian, can be sampled. However, when µ is non-Gaussian the tail is not recognised in a form that can be sampled. For the experiments in Section 6, in which both Gaussian and non-Gaussian priors µ are considered, the series in Eq. (2.6) was truncated at level N + 1, with the resultant prior denoted µN . The associated posterior was then entirely supported on the finite-dimensional subspace XN ; this is mathematically equivalent to working with the projected output PN µaδ,N . Analysis of prior truncation, as opposed to modification of the information operator just reported, is known to be difficult. Indeed, while µN converges to µ weakly, it does not do so in total variation, and this deficiency generally transfers to the associated posteriors. In general the impact of prior perturbation is a subtle topic — see e.g. Owhadi et al. (2015) and the references therein — and we therefore defer theoretical analysis of this approximation to future work. 4.3. Monte Carlo Methods for Numerical Disintegration The previous sections established a sequence of well-defined distributions µaδ (or µaδ,N for nonGaussian models) which converge (in a specific weak sense) to the exact disintegration µa . From construction, µaδ  µa and this is sufficient to allow standard Monte Carlo methods to be used. The construction of Monte Carlo methods is de-coupled from the core material in the main text and the main methodological considerations are well-documented (e.g. Girolami and Calderhead, 2011). For the experiments reported in subsequent sections two approaches were explored; a Sequential Monte Carlo (SMC) method (Doucet et al., 2001) and a parallel tempering method (Geyer, 1991). This provided a transparent sampling scheme, whose non-asymptotic approximation error can be theoretically understood. In particular, they provide robust estimators of model evidence that can be used for Bayesian model comparison. Full details of the Monte-Carlo methods used for this work, along with associated theoretical analysis for the SMC method, are contained in Section S4.1 of the Electronic Supplement. 5. Computational Pipelines and PNM The last theoretical development in this paper concerns composition of several PNMs. Most analysis of numerical methods focuses on the error incurred by an individual method. However, real-world computational procedures typically rely on the composition of several numerical methods. The manner in which accumulated discretisation error affects computational output may be highly non-trivial (Roy, 2010; Anderson, 2011; Babuška and Söderlind, 2016). An extreme example occurs when one of the numerical methods in a pipeline is charged with integration of a chaotic dynamical system (Strogatz, 2014). In recent work, Chkrebtii et al. (2016), Conrad et al. (2016) and Cockayne et al. (2016) each used PNMs within a broader statistical procedure to estimate unknown parameters in systems of ODEs and PDEs. The probabilistic description of discretisation error was incorporated into the data-likelihood, resulting in posterior distributions for parameters with inflated uncertainty to properly account for the inferential impact of discretisation error. However, beyond these limited works, no examination of the composition of PNMs has been performed. In particular, the question of which PNMs can be composed, and when the output of such a composition is meaningful, has not been addressed. This is important; for instance, if the output of a 17 composition of PNMs is to be used for analysis of variance to elucidate the main sources of discretisation error, then it is important that such output is meaningful. This section defines a pipeline as an abstract graphical object that may be combined with a collection of compatible PNMs. It is proven that when compatible Bayesian PNMs are employed in the pipeline, the distributional output of the pipeline carries a Bayesian interpretation under an explicit conditional independence condition on the prior µ. To build intuition, for the simple case where two Bayesian PNMs are composed in series, our results provide conditions for when, informally, the output B2 (B1 (µ, a1 ), a2 ) corresponds to a single Bayesian procedure B(µ, (a1 , a2 )). To reduce the notational and technical burden, in this section we will not provide rigorous measure theoretic details; however we note that those details broadly follow the same pattern as in Section 2.5. 5.1. Computational Pipelines To analyse pipelines of PNMs, we consider n such methods M1 , . . . , Mn , where each method Mi = (Ai , Bi ) is defined on a common8 state space X and targets a QoI Qi ∈ Qi . A pipeline will be represented as a directed graphical model, wherein the QoIs Qi from parent methods constitute information operators for child methods. It may be that a method will take quantities from multiple parents as input. To allow for this, we suppose that the information operator Ai : X → Ai can be decomposed into components Ai,j : X → Ai,j such that Ai = (Ai,1 , . . . , Ai,m(i) ) and Ai = Ai,1 × · · · × Ai,m(i) . Thus, each component Ai,j can be thought of as the QoI output by one of the parents of the method Mi . Without loss of generality we designate the nth QoI Qn to be the principal QoI. That is, the purpose of the computational pipeline is to estimate Qn . The case of multiple principal QoI is a simple extension not described herein. Nodes with no immediate children are called terminal nodes, while nodes with no immediate parents are called source nodes. We denote by A the set of all source nodes. Definition 5.1 (Pipeline). A pipeline P is a directed acyclic graph defined as follows: • Nodes are of two kinds: Information nodes are depicted by , and method nodes are depicted by . • The graph is bipartite, so that edges connect a method node to an information node or vice-versa. That is, edges are of the form  →  or  → . • There are n method nodes, each with a unique label in {1, . . . , n}. • The method node labelled i has m(i) parents and one child. Its in-edges are assigned a unique label in {1, . . . , m(i)}. • There is a unique terminal node and it is the child of method node n. This represents the principal QoI Qn . Example 5.2 (Distributed Integration). Recall the numerical integration problem of Example 3.4 and, as a thought experiment, consider partitioning the domain of integration in order to distribute computation: Z Z Z 1 8 | 0 0.5 x(t)dt = {z } | 0 (c) 1 x(t)dt + x(t)dt 0.5 {z } | {z } (a) (5.1) (b) This is without loss of generality, since X can be taken as the union of all state spaces required by the individual methods. 18 B1 (µ, ·) x(t1 ), . . . , x(tm−1 ) x(tm ) x(tm+1 ), . . . , x(t2m ) B2 (µ, ·) R 0.5 0 x(t)dt B3 (µ, ·) R1 0.5 x(t)dt R1 0 x(t)dt Figure 1: An intuitive representation of Example 5.2. 1 1 1 2 1 2 3 2 2 Figure 2: The pipeline P corresponding to Figure 1. To keep presentation simple we consider an integral over [0, 1] with 2m + 1 equidistant knots ti = i/2m. Let M1 be a Bayesian PNM for estimating Q1 (x) = (a) and M2 be a Bayesian PNM for estimating Q2 (x) = (b). In terms of our notational convention, we divide the information operator into four components; Ai,j , for i, j ∈ {1, 2}. A1,1 and A2,2 contain the information unique to M1 and M2 . Specifically     x(t1 ) x(tm+1 )     .. .. A1,1 (x) =  A2,2 (x) =  , . . . x(t2m ) x(tm−1 ) A1,2 and A2,1 contain the information that is shared between the two methods; that is A1,2 = A2,1 = {x(tm )}. To complete the specification we need a third PNM for estimation of Q3 (x) = (c) which we denote M3 and which combines the outputs of M1 and M2 by simply adding them together. Formally this has information operator A3 (x) = (A3,1 (x), A3,2 (x)) where A3,1 (x) = (a) and A3,2 (x) = (b). Its belief update operator is given by: B3 (µ, (a3,1 , a3,2 )) = δ(a3,1 + a3,2 ) An intuitive graphical representation of this set-up is shown in Figure 1. The pipeline P itself, which is identical to Figure 1 but with additional node and edge labels, is shown in Figure 2. In general, the method node labelled i is taken to represent the method Mi . The in-edge to this node labelled j is taken to represent the information provided by the relationship Ai,j (x) = ai,j . Here ai,j can either be deterministic information provided to the pipeline, or statistical information derived from the output of another PNM. To make this formal and to “match the input-output spaces” we next define what it means for the collection of methods Mi to be compatible with the pipeline P . Informally, this describes the conditions that must be satisfied for method nodes in a pipeline to be able to connect to each other. Definition 5.3 (Compatible). The collection (M1 , . . . , Mn ) of PNMs is compatible with the pipeline P if the following two requirements are satisfied: 19 (i) (Method nodes which share an information node must have consistent information spaces and information operators.) For a motif i j0 i0 j we have that Ai,i0 = Aj,j 0 and Ai,i0 = Aj,j 0 . (ii) (The space Qi for the output of a previous method must be consistent with the information space of the next method.) For a motif j0 i j we have that Qi = Aj,j 0 . Note that we do not require the converse of (i) at this stage; that is, the same information can be represented by more than one node in the pipeline. This permits redundancy in the pipeline, in that information is not recycled. It will transpire that pipelines with such redundancy are non-Bayesian. The role of the pipeline P is to specify the order in which information, either deterministic of statistical, is propagated through the collection of PNMs. This is illustrated next: Example 5.4 (Propagation of Information). For the pipeline in Figure 2, the propagation of information proceeds as follows:: 1. The source nodes, representing A(x) = {A1,1 (x), A1,2 (x) = A2,1 (x), A2,2 (x)} are evaluated as {a1,1 , a1,2 = a2,1 , a2,2 }. This represents all the information on x that is provided. 2. The distributions µ(1) := B1 (µ, (a1,1 , a1,2 )) µ(2) := B2 (µ, (a2,1 , a2,2 )) are computed. 3. The push-forward distribution µ(3) := (B3 )# (µ, µ(1) × µ(2) ) is computed. Here µ(1) × µ(2) is defined on the Cartesian product ΣA3,1 × ΣA3,2 with independent components µ(1) and µ(2) . The notation (B3 )# refers to the push-forward of the function B3 (µ, ·) over its second argument. The distribution µ(3) is the output of the pipeline and is a distribution over the principal QoI Q3 (x). The procedure in Example 5.4 can be formalised, but to keep the presentation and notation succinct, we leave this implicit: 20 1 4 2 3 6 5 Figure 3: Dependence graph G(P ) corresponding to the pipeline P in Figure 2. The nodes are indexed with a topological ordering (shown). Definition 5.5 (Computation). For a collection (M1 , . . . , Mn ) of PNMs that are compatible with a pipeline P , the computation P (M1 , . . . , Mn ) is defined as the PNM with information operator A and belief update operator B that takes µ and A(x) = a as input and returns the distribution µ(n) as its output B(µ, a), obtained through the procedure outlined in Example 5.4. That is, the computation P (M1 , . . . , Mn ) is a PNM for the principal QoI Qn . Note that this definition includes a classical numerical work-flow just as a PNM encompasses a standard numerical method. 5.2. Bayesian Computational Pipelines Noting that P (M1 , . . . , Mn ) is itself a PNM, there is a natural definition for when such a computation can be called Bayesian: Definition 5.6 (Bayesian Computation). Denote by (A, B) the information and belief operators associated with the computation P (M1 , . . . , Mn ) and let {µa }a∈A be a disintegration of µ with respect to the information operator A. The computation P (M1 , . . . , Mn ) is said to be Bayesian for the QoI Qn if B(µ, a) = (Qn )# µa for A# µ-almost-all a ∈ A. This is clearly an appealing property; the output of a Bayesian computation can be interpreted as a posterior distribution over the QoI Qn (x) given the prior µ and the information A(x). Or, more informally, the “pipeline is lossless with information”. However, at face value it seems difficult to verify whether a given computation P (M1 , . . . , Mn ) is Bayesian, since it depends on both the individual PNMs Mi and the pipeline P that combines them. Our next aim is to establish verifiable sufficient conditions, for which we require another definition: Definition 5.7 (Dependence Graph). The dependence graph of a pipeline P is the directed acyclic graph G(P ) obtained by taking the pipeline P , removing the method nodes and replacing all  →  →  motifs with direct edges  → . The dependency graph for Example 5.2 is shown in Figure 3. For a computation P (M1 , . . . , Mn ), each of the J distinct nodes in G(P ) can be associated with a random variable Yj where either Yj = Ak,l (X) for some k, l, when the node is a source, or otherwise Yj = Qk (X), for some k. Randomness here is understood to be due to X ∼ µ, so that the distribution of the {Yj }Jj=1 is a function of µ. The convention used here is that the Yj are indexed according to a topological ordering on G(P ), which has the properties that (i) the source nodes correspond to indices 1, . . . , I, and (ii) the final random variable is YJ = Qn (X). 21 Definition 5.8 (Coherence). Consider a computation P (M1 , . . . , Mn ). Denote by π(j) ⊆ {1, . . . , j − 1} the parent set of node j in the dependence graph G(P ). Then we say that µ ∈ PX is coherent for the computation P (M1 , . . . , Mn ) if the implied joint distribution of the random variables Y1 , . . . , YJ satisfies: Yj ⊥ ⊥ Y{1,...,j−1}\π(j) | Yπ(j) for all j = I + 1, . . . , J. Note that this is weaker than the Markov condition for directed acyclic graphs (see Lauritzen, 1991), since we do not insist that the variables represented by the source nodes are independent. It is emphasised that, for a given µ ∈ PX , the coherence condition can in general be checked and verified. The following result provides sufficient and verifiable conditions which ensure that a computation composed of individual Bayesian PNMs is a Bayesian computation: Theorem 5.9. Let M1 , . . . , Mn be Bayesian PNMs and let µ ∈ PX be coherent for the computation P (M1 , . . . , Mn ). Then it holds that the computation P (M1 , . . . , Mn ) is Bayesian for the QoI Qn . Conversely, if non-Bayesian PNM are combined then the computation P (M1 , . . . , Mn ) need not be Bayesian in general. Example 5.10 (Example 5.2, continued). The random variables Yi in this example are: Z 0.5 Z 1 m−1 2m Y1 = {X(ti )}i=1 , Y2 = X(tm ), Y3 = {X(ti )}i=m+1 , Y4 = X(t)dt, Y5 = X(t)dt. 0 0.5 From G(P ) in Figure 3, coherence condition in Definition 5.8 requires that the non-trivial conditional independences Y4 ⊥ ⊥ Y3 | {Y1 , Y2 } and Y5 ⊥⊥ Y1 | {Y2 , Y3 } hold. Thus the distribution µ is coherent for the computation if and only if, for X ∼R µ, the assoR 0.5 P (M1 , M2 , M3 ) 2m 1 ⊥ ciated information variables satisfy 0 X(t)dt ⊥⊥ {X(ti )}i=m+1 |{X(ti )}m i=1 and 0.5 X(t)dt ⊥ 2m . {X(ti )}m−1 |{X(t )} i i=m i=1 The distribution µ induced by the Weiner process on x in Example 3.4 satisfies these conditions. Indeed, under µ the stochastic process {x(t) : t > tm } is conditionally independent of its history {x(t) : t < tm } given the current state x(tm ). Thus for this choice of µ, from Theorem 5.9 we have that P (M1 , M2 , M3 ) is Bayesian and parallel computation of (a) and (b) in Eq. (5.1) can be justified from a Bayesian statistical standpoint. However, for the alternative of belief distributions induced by the Weiner process on ∂ s x, this condition is not satisfied and the computation P (M1 , M2 , M3 ) is not Bayesian. To turn this into a Bayesian procedure for these alternative belief distributions it would be required that A1,2 (x) provides information about the derivatives ∂ k x(tm ) for all orders k ≤ s. 5.3. Monte Carlo Methods for Probabilistic Computation The most direct approach to access µ(n) is to sample from each Bayesian PNM and treat the output samples as inputs to subsequent PNM. This is sometimes known as ancestral sampling in the Bayesian network literature (e.g. Paige and Wood, 2015), and is illustrated in the following example: Example 5.11 (Ancestral Sampling for PNM). For Example 5.2, ancestral sampling proceeds as follows: 22 1. Draw initial samples q1 ∼ B1 (µ, (a1,1 , a1,2 )) q2 ∼ B2 (µ, (a2,1 , a2,2 )) 2. Draw a final sample q3 ∼ B3 (µ, (q1 , q2 )) Then q3 is a draw from µ(3) . Ancestral sampling requires that PNM outputs can be sampled. Such sampling methods were discussed in Section 4.3. For a more general approach, sequential Monte Carlo methods can be used to propagate a collection of particles through the pipeline P , similar to work on SMC for general graphical models (Briers et al., 2005; Ihler and McAllester, 2009; Lienart et al., 2015; Lindsten et al., 2017; Paige and Wood, 2015). 6. Numerical Experiments In this final section of the paper we present three numerical experiments. The first is a linear PDE, the second is a nonlinear ODE and the third is an application to a problem in industrial process monitoring, described by a pipeline of PNM. In each case we experiment with nonGaussian belief distributions and, in doing so, go beyond previous work. 6.1. Poisson Equation Our first illustration is an instance of the Poisson equation, a linear PDE with mixed DirichletNeumann boundary conditions: −∇2 x(t) = 0 x(t) = t1 x(t) = 1 − t1 ∂x/∂t2 = 0 t ∈ (0, 1)2 (6.1) t1 ∈ [0, 1] t2 = 0 (6.2) t2 = 1 (6.3) t2 ∈ (0, 1) t1 = 0, 1 (6.4) t1 ∈ [0, 1] A model solution to this system, generated with a finite-element method on a fine mesh, is shown in Figure 4. As the spatial domain for this problem is two-dimensional, the basis used for specification of the belief distribution is more complex. Here tensor products of orthogonal polynomials have been used: φi (t) = Cj (2t1 − 1)Ck (2t2 − 1), i + j ≤ Nc . The polynomials Ci were chosen to be normalised Chebyshev polynomials of the first kind. Prior specification then follows the formulation given in Section 2.6, where the remaining parameters were chosen to be x0 ≡ 1, and γi = α(i + 1)−2 . The random variables ξ were taken to be either Gaussian or Cauchy and the polynomial basis was truncated to N = 45 terms, corresponding to a maximum polynomial degree of NC = 8. For both priors the parameter α was set to α = 1. Note that closed-form expressions are available for analysis under the Gaussian prior (Cockayne et al., 2016) but, to simplify interpretation of empirical results, were not exploited. Mathematical background on Cauchy priors can be found in Sullivan (2016). The information operator was defined by a set of locations ti ∈ [0, 1]2 , i = 0, . . . , Nt , where  either the interior condition or one of the boundary conditions was enforced. Denote by tI,i 23 1.0 0.8 t2 0.6 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Figure 4: Model solution x(t), t = (t1 , t2 ), generated by application of a finite element method based on a triangular mesh of 50 × 50 elements.   the set of interior points, tD,j the set of Dirichlet boundary points and tN,k the set of Neumann boundary points, where i = 1, . . . , NI , j = 1, . . . , ND and k = 1, . . . , NN , with n = NI + ND + NN . Then, the information operator is given by the concatenation of the conditions defined above: A(x) = [AI (x)> , AD (x)> , AN (x)> ]> ,       ∂ N,1 ) −∇2 x(tI,1 ) x(tD,1 ) ∂t1 x(t       .. .. .. D N AI (x) =   , A (x) =   , A (x) =   . . . 2 I,N D,N ∂ N,N I D N −∇ x(t ) x(t ) ) ∂t1 x(t The Bayesian PNM output was approximated by numerical disintegration and sampled with a Monte Carlo method whose description is reserved for the Electronic Supplement. In Figure 5 the mean and pointwise standard-deviations of the posterior distributions are plotted for Gaussian and Cauchy priors with n = 16. There is little qualitative difference between the posterior distributions for the Gaussian and Cauchy priors. The mean functions match closely to the mean function from the model solution, as given in Figure 4. The posterior variance is lowest near to the Dirichlet boundaries where the solution is known, and peaks where the Neumann condition is imposed. This is to be expected, as evaluations of the Neumann boundary condition provide less information about the solution itself. Next, the posterior distribution of the spectrum {ui } was investigated. In Figure 6 the posterior distribution over these coefficients is plotted and it is seen that the correlation structure between coefficients is non-trivial, c.f. the joint distribution between u0 and u3 . Last, in Figure 7 convergence of the posterior distribution is plotted as the number of design points is varied, for n = 16, 25, 36. In each case a Gaussian prior was used. As expected, the standard deviation in the posterior distribution is seen to decrease as the number of design points is increased. At n = 36, the shape of the region of highest uncertainty changes markedly, with the most uncertain region lying between the Dirichlet boundary and the first evaluation points on the Neumann boundary. This is likely due to the fact that the number of evaluation points is approaching the size of the polynomial basis; when the number of points equals the size of the basis the system is completely determined for a linear model. Thus, we need N  n in order for discretisation error to be quantified. 24 0.8 t2 0.6 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.8 0.6 t2 1.0 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 0.0250 0.0225 0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 (a) Gaussian prior 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.8 t2 0.6 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 0.8 0.6 t2 1.0 0.4 0.2 0.0 0.0 1.0 0.2 0.4 t1 0.6 0.8 0.0250 0.0225 0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 (b) Cauchy prior Figure 5: Posterior distributions for the solution x of the Poisson equation, with n = 16 and different choices of prior distribution. Left: Posterior mean. Design points for the interior, Dirichlet and Neumann boundary conditions are indicated by green dots, green squares and green crosses, respectively. Right: Posterior standard deviation. 25 u0 8.5 6.2 3.8 1.5 u1 0.0 0.0 -0.0 -0.0 u2 0.0 0.0 -0.0 -0.0 u3 0.0 0.0 -0.0 -0.0 u4 -0.8 -0.8 -0.8 -0.8 u5 0.0 0.0 -0.0 -0.0 1.5 1.6 1.7 -0.0 0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 0.1 -0.8 -0.8 -0.8 -0.0 0.0 0.0 Figure 6: Posterior distributions for the first six coefficients of the spectrum for the solution x of the Poisson equation, obtained with Monte Carlo methods and numerical disintegration, based on δ = 0.0008, n = 16. (NB: The posterior is Gaussian and can be obtained in closed-form, but we opted to additionally illustrate the Monte Carlo method.) 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 (a) n = 16 0.6 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 (b) n = 25 0.0250 0.0225 0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 0.8 0.6 t2 t2 0.6 0.8 t2 0.8 0.0250 0.0225 0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 0.4 0.2 0.0 0.0 0.2 0.4 t1 0.6 0.8 0.0250 0.0225 0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 (c) n = 36 Figure 7: Heat map of the point-wise standard deviation for the solution x to the Poisson equation as the number n of design points is varied. In each case a Gaussian prior has been used. 26 4 10 -2 10 -4 1 un x(t) 2 0 10 -6 10 -8 1 10 -10 2 10 -12 3 Positive Negative 10 0 3 0 2 4 t 6 8 10 10 -14 0 10 20 30 40 i 50 60 70 80 Figure 8: Two distinct solutions for the Painlevé ODE. The spectral plot on the right shows the true coefficients {ui }, as determined by a model solver (the MatLab package chebfun). 6.2. The Painlevé ODE In this section a Bayesian PNM is developed to solve a nonlinear ODE based on Painlevé’s first transcendental x00 = x2 − t, t−1/2 x(t) → 1 t ∈ [0, ∞)x(0) =0 as t → ∞ . To permit computation, the right-boundary condition was relaxed by truncating the domain to √ [0, 10] and using the modified condition x(10) = 10. Two distinct solutions are known, illustrated in Figure 8 (left). These model solutions were obtained using the deflation technique described in Farrell et al. (2015). The spectrum plot in Figure 8 (right) represents the coefficients {ui } obtained when each solution is represented over a basis of normalised Chebyshev polynomials. As those polynomials are orthonormal with respect to the L2 -inner-product, the slower decay for the negative solution compared to the positive solution is equivalent to the negative solution having a larger L2 -norm. This explains the preference that optimisation-based numerical solvers have for returning the positive solution in general, and also explains some of the results now presented. Such systems for which multiple solutions exist have been studied before in the context of PNM, both in Chkrebtii et al. (2016) and in Cockayne et al. (2016). It was noted in both papers that existence of multiple solutions can present a substantial challenge to classical numerical methods. To build a Bayesian PNM, a prior µ for this problem was defined by using a series expansion as in Eq. (2.6). The basis functions were φi (t) = Ci ( 21 (t − 5)) where the Ci were normalised Chebyshev polynomials of the first kind. Both Gaussian and Cauchy priors were considered by taking ui := γi ξi , where ξi were taken to be either standard Gaussian or standard Cauchy and in in each case x0 (t) ≡ 0. In accordance with the exponential convergence rate for spectral methods when the solution to the system is a smooth function, the sequence of scale parameters was set to γi = αβ −i , where α = 8 and β = 1.5. These values were chosen by inspection of the true spectra (obtained with Matlab’s “chebfun” package) to ensure that both solutions were in the support of the prior. 27 The information operator A was defined by the choice of locations {tj }, j = 1, . . . , m, which determine the locations at which the posterior will be constrained. Analysis for several values of m was performed. In each case t1 = 0, tm = 10 and the remaining tj were equally spaced on [0, 10]. To be explicit, the information operator was  00  x (t1 ) − (x(t1 ))2   ..   .  00  A(x) = x (tm ) − (x(tm ))2      x(0) x(10) with the last two√elements enforcing the boundary conditions. Thus our information was a = [−t1 , . . . , −tm , 0, 10], which is n = m + 2 dimensional. The Bayesian PNM output B(µ, a) was approximated via numerical disintegration with the first N = 40 terms of the series representation used. This was sampled with Monte Carlo methods, the details of which are reserved for the Electronic Supplement. Results for a selection of bandwidths δ, with n = 17, are shown in Figure 9. Note that a strong preference for the positive solution is expressed at the smallest δ, with mass around both solutions at larger δ. For the Gaussian prior, some mass remained around the negative solution at the smallest δ, while this was not so for the Cauchy prior. This reflects the fact that, for a collection of independent univariate Cauchy random variables, one element is likely to be significantly larger in magnitude than the others, which favours faster decay for the remaining elements. Using the calculation described in Section S4.4, model evidence was computed for both the Gaussian and the Cauchy prior at n = 15. The Bayes factor for the Cauchy, compared to the Gaussian prior, was found to be 20.26, which constitutes strong evidence in favour of a Cauchy prior for this problem at the given level of discretisation. In Figure 10 the posterior distributions for first six coefficients ui at n = 17 and δ = 1 are plotted. Strong multimodality is clear, as well as skewed correlation structure between the coefficients. Illustration of such posteriors for smaller δ is difficult as the posteriors become extremely peaked. Figure 11 displays convergence of the posterior distributions as n is increased. Of particular interest is that for n = 12, the posterior distribution based on a Gaussian prior becomes trimodal. For each prior, the posterior mass settles on the positive solution to the system at n = 22. This is in accordance with the fact that this solution has smaller L2 -norm. This perhaps reflects the fact that, while in the limiting case both solutions should have an equal likelihood, the curvature of the likelihood at each mode may differ. Prior truncation may also be influential; in Figure 12 the log-likelihood of the negative solution increases at a slower rate than that of the positive solution. Thus, while in the setting of an infinite prior series neither solution should be preferred, in practice truncation might bias one solution over the other. Lastly, it is clear that the parameters α and β may also have a significant effect on which solution is preferred. Further theoretical work will be required to understand many of the phenomena that we have just described. Of particular interest is how a preference for the negative solution could be encoded into a PNM. Owing to the flexible specification the information operator, there is considerable choice in this matter. An elegant approach is the introduction of additional, inequality-based information x0 (0) ≤ 0 . 28 (6.5) (a) Gaussian Prior (b) Cauchy Prior. Figure 9: Posterior samples for the Painlevé system for n = 17. Blue and green dashed lines represent the positive and negative solutions determined with chebfun. Grey lines are samples from an approximation to the posterior provided by numerical disintegration (bandwidth parameter δ). 29 0.8 0.6 0.4 0.1 3.5 2.8 2.1 1.4 -0.3 -0.8 -1.2 -1.7 u0 u1 u2 u3 0.7 0.2 -0.2 -0.7 u4 1.6 0.9 0.2 -0.6 u5 0.7 0.2 -0.2 -0.7 2.3 3.0 3.7 1.4 2.5 3.5 -1.7 -1.0 -0.3 -0.7 0.0 0.7 -0.6 0.5 1.6 -0.7 0.0 0.7 Figure 10: Posterior distributions for the first six coefficients obtained with numerical disintegration (bandwidth parameter δ = 1), at n = 17. Vertical dashed lines on the diagonal plots indicate the value of the coefficients for the positive (blue) and negative (green) solutions determined with chebfun. 30 Figure 11: Convergence for the numerical disintegration scheme as n is increased. Left: Gaussian prior. Right: Cauchy prior. In all cases δ = 10−4 . 31 Positive Negative 108 106 − log φ  kA(x)−ak δ  1010 104 102 0 10 20 30 40 50 60 70 N Figure 12: Negative-log-likelihoods for the point-estimates of coefficients for the postive and negative solutions given by chebfun, as the truncation level N is varied. The fact that the likelihood for the positive solution decreases more rapidly than that of the negative solution suggests indicates that the posterior may have a preference for that solution over the other, though the level N = 40 has been selected in an attempt to minimise the impact. Such information can be difficult to incorporate in standard numerical algorithms, but is of interest in many physical problems (Kinderlehrer and Stampacchia, 2000). For Bayesian PNM we can extend the information operator to include 1[x0 (0) ≤ 0]. Posterior distributions for the Gaussian prior at n = 17 are shown in Figure 13. Note that posterior mass has settled close to the negative solution. This highlights the simplicity with which Bayesian PNMs can encode a preference for a particular solution when a multiplicity of solutions exist. 6.3. Application to Industrial Process Monitoring This final application illustrates how statistical models for discretisation error can be propagated through a pipeline of computation to model how these errors are accumulated. Hydrocyclones are machines used to separate solid particles from a liquid in which they are suspended, or two liquids of different densities, using centrifugal forces. High pressure fluid is injected into the top of a tank to create a vortex. The induced centrifugal force causes denser material to move to the wall of the tank while lighter material concentrates in the centre, where it can be extracted. They have widespread applications, including in areas such as environmental engineering and the petrochemical industry (Sripriya et al., 2007). An illustration of the operation is given in Figure 14. To ensure the materials are well-separated the hydrocyclone must be moitored to allow adjustment of the input flow-rate. This is also important for safe operation, owing to the high pressures involved (Bradley, 2013). However, direct monitoring is impossible owing to the opaque walls of the equipment and the high interior pressure. For this purpose electrical impedance tomography (EIT) has been proposed to allow monitoring of the contents (Gutierrez et al., 2000). EIT is a technique which allows recovery of an interior conductivity field based upon measurements of voltage obtained from applying a stimulating current on the boundary. It is suited to this problem, as the two materials in the hydrocyclone will generally be of different conductivities. In its simplified form due to Calderón (1980), EIT is described by a linear partial 32 Figure 13: Posterior distribution at n = 17, based on a Gaussian prior, with the negative boundary condition given by Eqn. (6.5) enforced. Left: δ = 0.99. Right: δ = 0.0001. overflow input flow less dense more dense underflow (a) Hydrocyclone tank schematic (b) Cross-section (top of tank) Figure 14: A schematic description of hydrocyclone equipment. (a) The tank is cone-shaped with overflow and underflow pipes positioned to extract the separated contents. (b) Fluid, a mixture to be separated, is injected at high pressure at the top of the tank to create a vortex. Under correct operation, denser materials are directed toward the centre of the tank and less-dense materials are forced to the peripheries of the tank. 33 differential equation similar to that in Section 6.1, but with modified boundary conditions to incorporate the stimulating currents and measured voltages: −∇ · (a(t)∇x(t)) = 0  ∂x ce a(t) (t) = 0 ∂n t∈D t = te e t ∈ ∂D \ {te }N e=1 (6.6) where D denotes the domain, modelling the hydrocyclone tank, e indexes the stimulating electrodes, te ∈ ∂D are the corresponding locations of the electrodes on ∂D, a is the unknown ∂ conductivity field to be determined and ∂n denotes the derivative with respect to the outward 1 pointing normal vector. The electrode t is referred to as the reference electrode. The vector c = (c1 , . . . , cNe ) denotes the stimulation current pattern. Several stimulation patterns were considered, denoted cj , j = 1, . . . , Nj . The experimental data described in West et al. (2005) were considered. In the experiment, a cylindrical perspex tank was used with a single ring of eight electrodes. Translation invariance in the vertical direction means that the contents are effectively a single 2D region and electrical conductivity can be modelled as a 2D field. At the start of the experiment, a mixing impeller was used to create a rotational flow. This was then removed and, after a few seconds, concentrated potassium chloride solution was carefully injected into the tap water initially filling the tank. Data, denoted yτ , were collected at regular time intervals by application of several stimulation patterns c1 , . . . , cM . To formulate the statistical problem, consider parameterising the conductivity field as a(τ, t), where τ ∈ [0, T ] is a temporal index while t ∈ D is the spatial coordinate and D is the circular domain representing the perspex tank in the experiment. A log-Gaussian prior was placed over the conductivity field so that log ais a Gaussian process with separable covariance function  kt−t0 k2 0 0 0 ka ((τ, t), (τ , t )) := λ min(τ, τ ) exp − 2`2 where ` is a length-scale parameter representing the anticipated spatial variation of the conductivity field and λ is a parameter controlling the amplitude of the field. Here ` was fixed to ` = 0.3, while λ = 10−3 . The problem of estimating a based on data can be well-posed in the Bayesian framework (Dunlop and Stuart, 2016). Full details of this experiment can be found in the accompanying report Oates et al. (2017). Our aim is to use a PNM to account for the effect of discretisation on inferences that are made on the conductivity  field. For fixed τ , a Gaussian prior was posited for x, with covariance kt−t0 k2 0 kx (t, t ) := exp − 2`2 where `x was fixed to `x = 0.3. The associated Bayesian PNM, a x probabilistic meshless method (PMM), was described in Example 2.4. The statistical inference procedure is formulated in a pipeline of computations in Figure 15. It is assumed that the desired outcome is to monitor the contents of the tank while the current contents are being mixed. This suggests a particle filter approach where a PMM Mτ is employed to handle the intractable likelihood p(yτ |aτ ) that involves the exact solution of a PDE. The distribution of aτ given y1 , . . . , yτ is denoted πτ an the computation P (M1 , . . . , Mτ ) is Bayesian only if the particle approximation error due to the use of a particle filter is overlooked. To briefly illustrate the method, Figure 16 presents posterior means for the field a(τ, ·), for each post-injection time point τ = 1, . . . , 8. These are based on a particle approximation of size P = 500, with method nodes based upon a Bayesian PNM, as in Example 2.4, with n = 119 design points. The high conductivity region representing the potassium chloride solution can be seen rotating through the domain in the frames after injection, with its conductivity reducing as it mixes with the water. The full posterior distribution over the conductivity field is inflated as a result of explicitly modelling the discretisation error; an extensive analysis of these results will be reported in the upcoming Oates et al. (2017). 34 yτ 2 1 ... ... τ dπτ −1 dπ0 dπτ dπ0 Figure 15: Pipeline for hydrocyclone application: The method node (black) represents the use of PMM solvers, which are incorporated into the likelihood for evolving the particles according to a Markov transition kernel. τ =1 1.0 τ =2 1.0 0.5 τ =3 1.0 0.5 τ =4 1.0 0.5 0.5 138.1 0.0 0.0 0.0 0.0 123.3 −0.5 −0.5 −0.5 −0.5 108.4 93.6 −1.0 −1.0 −0.5 0.0 0.5 1.0 τ =5 1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 τ =6 1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 τ =7 1.0 −1.0 −1.0 −0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.0 −0.5 −0.5 −0.5 −0.5 0.5 1.0 τ =8 1.0 0.5 0.0 78.7 63.9 49.0 34.1 19.3 −1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −1.0 4.4 −0.5 0.0 0.5 1.0 Figure 16: Mean conductivity fields recovered in the hydrocyclone experiment, for the first 8 frames post-injection. 35 9 Pipeline Static 30 8 Pipeline - Static 28 24 22 R D σ(t) dt 26 20 18 7 6 5 4 3 16 2 14 1 2 3 4 5 6 7 8 1 τ 2 3 4 5 6 7 8 τ Figure 17: Left: Integrated standard-deviation over the domain, for the first 8 frames postinjection, for both the pipeline and the static approaches described in the text. Right: The difference between these two quantities. R In Figure 17, the integrated standard-deviation D σ(t) dt is shown for τ = 1, . . . , 8 for both the “pipeline”, as described above, and a “static” approach in which no uncertainty was propagated. In this static approach a symmetric collocation PDE solver9 was used to solve the forward problem, and a separate Bayesian inversion problem was solved at each time point. The parameters of the symmetric collocation solver were identical to those used in the PMM. In the left panel we observe some structural periodicity, present in both the pipeline and the static approach. We speculate that this may be due to the rotation of the medium causing the area of high conductivity to periodically reach an area of the domain, relative to the 8 sensors, in which it is particularly easy to recover. With this periodicity subtracted in the right panel, there was a clear increase in posterior uncertainty in the pipeline compared to the static approach, which is depicted. Temporal regularisation would usually be expected to reduce uncertainty; thus, the fact that the overall uncertainty increased with τ , relative to the static formulation, demonstrates that we have quantified and propagated uncertainty due to successive discretisation of the PDE at each time point. 7. Discussion This paper has established statistical foundations for PNMs and investigated the Bayesian case in detail. Through connection to Bayesian inverse problems (Stuart, 2010), we have established when Bayesian PNM can be well-defined and when the output can be considered meaningful. The presentation touched on several important issues and a brief discussion of the most salient points is now provided. Bayesian vs Non-Bayesian PNMs The decision to focus on Bayesian PNMs was motivated by the observation that the output of a pipeline of PNMs can only be guaranteed to admit a valid Bayesian interpretation if the constituent PNMs are each Bayesian and the prior distribution is coherent. Indeed, Theorem 5.9 demonstrated that prior coherence can be established at a local level, essentially via a local Markov condition, so that Bayesian PNMs provide a extensible modelling framework as required to solve more challenging numerical tasks. These results support a research strategy that focuses on Bayesian PNMs, so that error can be propagated in a manner that is meaningful. 9 Recall that the PMM has a corresponding symmetric collocation solution to the PDE as its mean function. 36 On the other hand, there are pragmatic reasons why either approximations to Bayesian PNMs, or indeed, non-Bayesian PNMs might be useful. The predominant reason would be to circumvent the off-line computational costs that can be associated with Bayesian PNMs, such as the use of numerical disintegration developed in this research. Recent research efforts, such as Schober et al. (2014, 2016) and Kersting and Hennig (2016) for the solution of ODEs, have aimed for computational costs that are competitive with classical methods, at the expense of fully Bayesian estimation for the solution of the ODE. Such methods are of interest as nonBayesian PNMs, but their role in pipelines of PNMs is unclear. Our contribution serves to make this explicit. Computational Cost The present research focused on the more fundamental cost of access to the information A(x), rather than the additional CPU time required to obtain the PNM output. Indeed, numerical disintegration constituted the predominant computational cost in the applications that were reported. However, we stress that in many challenging applications gated by discretisation error, such as occur with climate models, the fundamental cost of the information A(x) will be dominant. Furthermore, the Monte Carlo methods that were employed for numerical disintegration admit substantial improvements (e.g. in a similar vein to Botev and Kroese, 2012; Koskela et al., 2016). The objective of this paper was to establish statistical foundations that will permit the development of more sophisticated and efficient Bayesian PNMs. Prior Elicitation Throughout this work we assumed that a belief distribution µ was provided. The question of whose belief is represented in µ has been discussed by several authors and a chronology is included in the Electronic Supplement. Of these perspectives we mention in particular Hennig et al. (2015), wherein µ is the belief of an agent that “we get to design”. This offers a connection to frequentist statistics, in that an agent can be designed to ensure favourable frequentist properties hold. A robust statistics perspective is also relevant and one such approach would be to consider a generalised Bayes risk (Eq. (3.1)) wherein the state variable X used for assessment is assumed to be drawn from a distribution µ̃ 6= µ. This offers an opportunity to derive Bayesian PNMs that are robust to certain forms of prior mis-specification. This direction was not considered in the present paper, but has been pursued in the ACA literature for classical numerical methods (see Chapter IV, Section 4 of Ritter, 2000). In general, the specification of prior distributions for robust inference on an infinite-dimensional state space can be difficult. The consistency and robustness of Bayesian inference procedures — particularly with respect to perturbations of the prior such as those arising from numerical approximations — in such settings is a subtle topic, with both positive (Castillo and Nickl, 2014; Doob, 1949; Kleijn and van der Vaart, 2012; Le Cam, 1953) and negative (Diaconis and Freedman, 1986; Freedman, 1963; Owhadi et al., 2015) results depending upon fine topological and geometric details. In the context of computational pipelines, the challenge of eliciting a coherent prior is closely connected to the challenge of eliciting a single unified prior based on the conflicting input of multiple experts (French, 2011; Albert et al., 2012). Consistent Estimation The present paper focused on foundations. Further methodological work will be required to establish sufficient conditions for when B(µ, An (x† )) collapses to an atom on a single element q † = Q(x† ) representing the data-generating QoI in the limit as the amount of information, n, is increased. There are two questions here; (i) when is q † identifiable from the given information, and (ii) at what rate does B(µ, An (x† )) concentrate on q † . 37 Generalisation and Extensions Two more directions are highlighted for extension of this work. First, note that in this paper the information operator A : X → A was treated as a deterministic object. However, in some applications there is auxiliary randomness in the acquisition of information. For our integration example, nodes ti might arise as random samples from a reference distribution on [0, 1]. Or, observations x(ti ) themselves might occur with measurement error, for example due to finite precision arithmetic. Then a more elaborate model A : X × Ω → A would be required, where Ω is a probability space that injects randomness into the information operator. This is the setting of, for instance, randomised quasi-Monte Carlo methods. Future work will extend the framework of PNMs to include randomised information operators of this kind. As a second direction, recall that in an adaptive algorithm the choice of the information is made in an iterative procedure that is informed by the information observed up to that point. For the canonical illustration in Example 3.4 and its generalisations discussed there, it can be proven that adaptive algorithms do not out-perform non-adaptive algorithms in average case error (Lee and Wasilkowski, 1986). However, outside this setting adaptation can be beneficial and should be investigated in the context of Bayesian PNM. Connection with Probabilistic Programming The central goal of probabilistic programming (PP) is to automate statistical computation, through symbolic representation of statistical objects and operations on those objects. The formalism of pipelines as graphical models presented in this work can be compared to similar efforts to establish PP languages (Goodman et al., 2012). For instance, a method node in a pipeline can be related to a monad aggregating several distributions into a single output distribution (Ścibior et al., 2015). An important challenge in PP is the automation of computing conditional distributions (Shan and Ramsey, 2017). Numerical disintegration and extensions thereof might be of independent interest to this field (e.g. extending Wood et al., 2014). Acknowledgements CJO was supported by the Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers. TJS was supported by the Excellence Initiative of the German Research Foundation (DFG) through the Free University of Berlin. MG was supported by the Engineering and Physical Sciences (EPSRC) grants EP/J016934/1, EP/K034154/1, an EPSRC Mathematical Sciences Established Career Research Fellowship and a Lloyds Register Foundation grant for Programme on Data-Centric Engineering. This material was based upon work partially supported by the National Science Foundation under Grant DMS1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The authors are grateful to Amazon for the provision of AWS credits and to the authors of the Eigen and Eigency libraries in Python. Appendices A. Proofs Proof of Theorem 3.3. The following observation will be required; the joint density of X and A = A(X) can be expressed in two ways: δ(A(x))(da)µ(dx) = µa (dx)A# µ(da) 38 (A.1) which holds almost everywhere from the definition of a disintegration {µa }a∈A . Note that our integrability assumption justifies the interchange of integrals from Fubini’s theorem. The Bayes risk for a Bayesian PNM MBPNM = (A, BBPNM ), BBPNM (µ, a) = Q# µa , can be expressed as: Z R(µ, MBPNM ) = r(x, B(µ, A(x)))µ(dx) ZZ = L(Q(x), q)Q# µA(x) (dq)µ(dx) (since M Bayesian) ZZZ = L(Q(x), q)Q# µa (dq)δ(A(x))(da)µ(dx) ZZZ = L(Q(x), Q(x0 ))µa (dx0 )µa (dx)A# µ(da) (from Eq. (A.1)) On the other hand, let b(a) ∈ arg min q∈Q Z L(Q(x), q)µa (dx) be a Bayes act. Then the Bayes risk associated with such a method MBR = (A, BBR ), BBR (µ, a) = δ(b(a)), can be expressed as: Z R(µ, MBR ) = L(Q(x), b(A(x)))µ(dx) ZZ = L(Q(x), b(a))δ(A(x))(da)µ(dx) ZZ = L(Q(x), b(a))µa (dx)A# µ(da) (from Eq. (A.1)) Next we use the inner product structure on Q and the form of the loss function as L(q, q 0 ) = kq − q 0 k2Q to argue that R(µ, MBPNM ) = 2R(µ, MBR ), which in turn implies that the optimal information Aµ for Bayesian PNM and A∗µ for ACA are identical. For this final step, fix a ∈ A and denote the random variables Qa (X) = Q(X) − b(a) that are induced according to X ∼ µa . Denote by Q̃a an independent copy of Qa generated from X̃ ∼ µa . The notation E will be used to refer to the expectation taken over X, X̃. Then we have Q(X) − Q(X̃) = (Q(X) − b(a)) − (Q(X̃) − b(a)) = Qa (X) − Q̃a (X̃) and moreover, from Theorem 3.2 the posterior mean of Q(X) is b(a) and thus E[Qa ] = E[Q̃a ] = 0. Then Z R(µ, MBPNM ) = E[kQa − Q̃a k2Q ]A# µ(da) Z = E[kQa k2Q − 2hQa , Q̃a iQ + kQ̃aA k2Q ]A# µ(da) Z = 2 E[kQa k2Q ]A# µ(da) (since E[Qa ] = 0 and Qa ⊥⊥ Q̃a ) = 2R(µ, MBR ) as required. 39 Proof of Theorem 4.3. Fix f ∈ F and a ∈ A. Then:   Z kA(x) − akA 1 a µ(dx) µδ (f ) = a f (x)φ Zδ δ   ZZ 1 kã − akA = a f (x)φ µã (dx)A# µ(dã) Zδ δ  Z  1 kã − akA = a φ µã (f )A# µ(dã) Zδ δ Z = µã (f )A# µaδ (dã). (from Eq. (A.1)) Thus |µaδ (f ) a Z [µã (f ) − µa (f )]A# µaδ (dã) Z α ≤ Cµ kf kF kã − akαA A# µaδ (dã) − µ (f )| = (Assumption 4.2). (A.2) Now consider the random variable R := kA(X) − akA δ (A.3) induced from X ∼ µ. The existence of a continuous and positive density pA implies that R also admits a density on [0, ∞), denoted pR,δ . The fact that pA is uniform on an infinitesimal neighbourhood of a implies that pR,δ (r) is proportional to the surface area of a hypersphere of radius δr centred on a ∈ A: pR.δ (r) = 2π n/2 (δr)n−1 (pA (a) + o(1)) Γ( n2 ) (A.4) This is valid since A is open and the hypersphere will be contained in A for r sufficiently small. Eq. (A.2) can then be evaluated:   R A Z kã − akαA φ kã−ak A# µ(dã) δ kã − akαA A# µaδ (dã) = R  kã−akA  φ A# µ(dã) δ R α r φ(r)pR,δ (r)dr = δα R (change of variables; Eq. A.3). (A.5) φ(r)pR,δ (r)dr R α+n−1 r φ(r)dr −−→ R n−1 (from Eq. A.4) δ↓0 r φ(r)dr Cφα = 0 (< ∞ from Assumption 4.1). Cφ Thus, for δ sufficiently small, Eq. (A.5) can be bounded above by δ α (1+ C̄φα ) where C̄φα := Cφα /Cφ0 and “1” is in this case an arbitrary positive constant. This establishes the upper bound |µaδ (f ) − µa (f )| ≤ Cµα (1 + C̄φα )kf kF δ α for δ sufficiently small and completes the proof. 40 Proof of Theorem 5.9. To reduce the notation, suppose that the random variables Y1 , . . . , YJ admit a joint density p(y1 , . . . , yJ ), However, we emphasise that existence of a density is not required for the proof to hold. To further reduce notation, denote ya:b = (ya , . . . , yb ). The output of the computation P (M1 , . . . , Mn ) was defined algorithmically in Definition 5.5 and illustrated in Example 5.4. Our aim is to show that this algorithmic output coincides with the distribution (Qn )# µa on Qn , which is identified in the present notation with p(yJ |y1:I ). For j ∈ {I + 1, . . . , J}, the coherence condition on Y1 , . . . , YJ translates into the present notation as p(yj |y1:j−1 ) = p(yj |yπ(j) ). This allows us to deduce that: Z Z p(yJ |y1:I ) = · · · p(yI+1:J |y1:I )dyI+1:J−1 = = Z Z ··· ··· Z Z J Y j=I+1 J Y j=I+1 p(yj |y1:j−1 )dyI+1:J−1 p(yj |yπ(j) )dyI+1:J−1 . The right hand side is recognised as the output of the computation P (M1 , . . . , Mn ), as defined in Definition 5.5. This completes the proof. References N. L. Ackerman, C. E. Freer, and D. M. Roy. On computability and disintegration. Mathematical Structures in Computer Science, 2017. To appear. I. Albert, S. Donnet, C. Guihenneuc-Jouyaux, S. Low-Choy, K. Mengersen, and J. Rousseau. 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In Proceedings of Artificial Intelligence and Statistics (AISTATS), 2014. 47 ∗ University of Warwick, j.cockayne@warwick.ac.uk Newcastle University and Alan Turing Institute, chris.oates@ncl.ac.uk ‡ Free University of Berlin and Zuse Institute Berlin, sullivan@zib.de § Imperial College London and Alan Turing Institute, m.girolami@imperial.ac.uk † 48 Electronic Supplement to the paper Bayesian Probabilistic Numerical Methods S1. Philosophical Status of the Belief Distribution The aim of this section is to discuss in detail the semantic status of the belief distribution µ in a probabilistic numerical method (PNM). In Section S1.1 we survey historical work on this topic, while in Section S1.2 more recent literature is covered. Then in Section S1.3 we highlight some philosophical objections and their counter-arguments. S1.1. Historical Precedent The use of probabilistic and statistical methods to model a deterministic mathematical object can be traced back to Poincaré (1912), who used a stochastic model to construct interpolation formulae. In brief, Poincaré formulated a polynomial f (x) = a0 + a1 x + · · · + am xm whose coefficients ai were modelled as independent Gaussian random variables. Thus Poincaré in effect constructed a Gaussian measure over the Hilbert space with basis {1, x, . . . , xm }. This pre-empted Kimeldorf and Wahba (1970a,b) and others, which associated spline interpolation formulae to the means of Gaussian measures over Hilbert spaces. The first explicit statistical model for numerical error (of which we are aware) was in the literature on rounding error in the numerical solution of ordinary differential equations (ODE), as summarised in Hull and Swenson (1966). Therein it was supposed that rounding, by which we mean representation of a real number x = 0.a1 a2 a3 a4 . . . ∈ [0, 1] in a truncated form x̂ = 0.a1 a2 a3 a4 . . . an , is such that the error e = x − x̂ can be reasonably modelled by a uniform random variable on [−5 × 10−(n+1) , 5 × 10−(n+1) ]. This implies a distribution µ over the unknown value of x given x̂. The contribution of Hull and Swenson (1966) and others was to replace the last digit an , in each stored number that arises in the numerical solution of an ODE, with a uniformly chosen element of {0, . . . , 9}. This performs approximate propagation of the numerical uncertainty due to rounding error through further computation and, in their case, induces a distribution over the solution space of the ODE. Note that this work focused on rounding error, rather than the (time) discretisation error that is intrinsic to numerical ODE solvers; this could reflect the limited precision arithmetic that was available from the computer hardware of the period. Larkin (1972) was an important historical paper for PNMs, being the first to set out the modern statistical agenda for PNMs: 49 In any particular problem situation we are given certain specific properties of the solution, e.g. a finite number of ordinate or derivative values at fixed abscissae. If we can assume no more than this basic information we can conclude only that our required solution is a member of that class of functions which possesses the given properties - a tautology which is unlikely to appeal to an experimental scientist! Clearly, we need to be given, or to assume, extra information in order to make more definite statements about the required function. Typically, we shall assume general properties, such as continuity or non-negativity of the solution and/or its derivatives, and use the given specific properties in order to assist in making a selection from the class K of all functions possessing the assumed general properties. We shall choose K either to be a Hilbert space or to be simply related to one. This description defines a set K of permissible functions, rather than an explicit distribution over K, but it is clear that Larkin envisaged numerical analysis as an instance of statistical estimation: In the present approach, an a priori localisation is achieved effectively by making an assumption about the relative likelihoods of elements of the Hilbert space of possible candidates for the solution to the original problem. Among other things, this permits, at least in principle, the derivation of joint probability density functions for functionals on the space and also allows us to evaluate confidence limits on the estimate of a required functional (in terms of given values of other functionals) without any extra information about the norm of the function in question. Later, Diaconis (1988) re-iterated this argument for the construction of K more explicitly, considering numerical integration of the function    x + x2 + cos(x) f (x) = exp cosh . 3 + sin(x3 ) over the unit interval. In particular, Diaconis asked: “What does it mean to ‘know’ a function?” The formula says some things (e.g. f is smooth, positive and bounded by 20 on [0, 1]) but there are many other facts about f that we don’t know (e.g. is f monotone, unimodal or convex?) This argument was provided as justification for belief distributions that encode certain basic features, such as the smoothness of the integrand. The belief distributions that were then considered in Diaconis’ paper were Gaussian distributions on K. Diaconis, as well as Larkin (1972); Kadane and Wasilkowski (1983), observed that some classical numerical methods are Bayes rules in this context. The arguments of these papers are intrinsic to modern PNMs. However, the associated theoretical analysis of computation under finite information has proceeded outside of statistics, in the applied mathematical literature, where it is usually presented without a statistical context. That research is reviewed next. S1.2. Contemporary Outlook The mathematical foundations of computation based on finite information are established in the field of information-based complexity (IBC). The monograph of Traub et al. (1988) presents the foundations of IBC. In brief, the starting point for IBC is the mantra that 50 To compute fast you need to compute with partial information (∼ Houman Owhadi, SIAM UQ 2016) This motivates the search for optimal approximations based on finite information, in either the worst-case or average-case sense of optimal. The particular development of PNMs that we presented in the main text is somewhat aligned to average-case analysis (ACA) and we focus on that literature in what follows. Among the earliest work on ACA, Sul0 din (1959, 1960) studied numerical integration and L2 function approximation in the setting where µ was induced from the Weiner process, with a focus on optimal linear methods. Later, Sacks and Ylvisaker (1970) moved from analysis with fixed µ to analysis over a class of µ defined by the smoothness properties of their covariance kernels. At the same time Kimeldorf and Wahba (1970a,b) established optimality properties of splines in reproducing kernel Hilbert spaces in the ACA context. Kadane and Wasilkowski (1985); Diaconis (1988) discussed the connection between ACA and Bayesian statistics. A general framework for ACA was formalised in the IBC monograph of Traub et al. (1988), while Ritter (2000) provides a more recent account. Game theoretic arguments have recently been explored in Owhadi (2015), who argued that the optimal prior for probabilistic meshless methods (Cockayne et al., 2016) is a particular Gaussian measure under a game theoretic framework where the energy norm is the loss function. This provides one route to the specification of default or objective priors for PNMs which deserves further exploration in general. The question of “whose” belief is captured in µ was addressed in Hennig et al. (2015), where it was argued that the prior information in µ represents that of a hypothetical agent (numerical analyst) which [. . . ] we are allowed to design (∼ Michael Osborne, personal correspondence, 2016). This represents a more pragmatic approach to the design of PNM. S1.3. Paradise Lost? Typical numerical algorithms contain several different sources of discretisation error. Consider the solution of the wave equation: A standard finite element method involves both spatial and temporal discretisations, a series of numerical quadrature problems, as well as the use of finite precision arithmetic for all numerical calculations. Yet, decades of numerical analysis have led to highly optimised computer codes such that these methods can be routinely used. To develop PNM for solution of the wave equation, which accounts for each separate source of discretisation error, is it required to unpick and reconstruct such established numerical algorithms? This would be an unattractive prospect that would detract from further research into PNMs. Our view is that there is a choice for which discretisation errors to model. In practice the PNMs implemented in this work were run on floating point precision machines, yet we did not model rounding error in their output. This was because, in our examples, floating point error is insignificant compared to discretisation error and so we chose not to model it. This is in line with the view that a model is a useful simplification of the real world. S2. Existence of Non-Randomised Bayes Rule In this section we recall an argument for the general existence of non-randomised Bayes rules, that was stated without proof in the main text. Sufficient conditions for Fubini’s theorem to hold are assumed. 51 Proposition S2.1. Let B(A) be non-empty. Then B(A) contains a classical numerical method of the form B(µ, a) = δ ◦ b(a) where b(a) is a Bayes act for each a ∈ A. Proof. Let C be the set of belief update operators of the classical form B(µ, a) = δ ◦ b(a). Suppose there exists a belief update operator B ∗ ∈ B(A) \ C. Then B ∗ can be characterised as a non-atomic distribution π over the elements of C. Its risk can be computed as: Z ∗ R(µ, (A, B )) = r(Q(x), B ∗ (µ, A(x)))µ(dx) ZZ = L(Q(x), b(A(x)))π(db)µ(dx) Z = R(µ, (A, δ ◦ b))π(db). If we had R(µ, (A, B ∗ )) < R(µ, (A, δ ◦ b)) for all δ ◦ b ∈ C we would have a contradiction, so it follows that B(A) ∩ C is non-empty. This completes the proof. S3. Optimal Information: A Counterexample In this section we demonstrate that the optimal information Aµ for Bayesian PNM and the optimal information A∗µ from average case analysis are different in general. Let X = {♠, ♦, ♥, ♣} be a discrete set, with quantity of interest Q(x) = 1[x = ♠] and information operator A(x) = 1[x ∈ S] so that Q = A = {0, 1}. In particular, Q is not a vector space and hence not an inner product space as specified in Theorem 3.3. Consider two possible choices, S = {♠, ♦} and S = {♠, ♦, ♥}. Assume a uniform prior over X . Consider the 0-1 loss function L(q, q 0 ) = 1[q 6= q 0 ]. It will be shown that ACA optimal information for this example can be based on either S = {♠, ♦} or S = {♠, ♦, ♥} whereas PNM optimal information must be based on S = {♠, ♦, ♥}. Thus Bayesian PNM optimal information Aµ and ACA optimal information A∗µ need not coincide in general. The classical case considers a method of the form MBR = (A, BBR ), BBR = δ ◦ b, where b(a) = 1[a = 0]c0 + 1[a = 1]c1 for some c0 , c1 ∈ {0, 1}. The Bayes risk is R(µ, MBR ) = Case of S = {♠, ♦}: 1 4 X x∈{♠,♦,♥,♣} 1[x ∈ / S] L(c0 , 1[x = ♠]) + 1[x ∈ S] L(c1 , 1[x = ♠]). We have 4 R(µ, MBR ) = L(c1 , 1) + L(c1 , 0) + L(c0 , 0) + L(c0 , 0) = 1[c1 = 0] + 1[c1 = 1] + 2 × 1[c0 = 1] which is minimised by c1 ∈ {0, 1} and c0 = 0 to obtain a minimum Bayes risk of 41 . Case of S = {♠, ♦, ♥}: We have 4 R(µ, MBR ) = L(c1 , 1) + L(c1 , 0) + L(c1 , 0) + L(c0 , 0) = 1[c1 = 0] + 2 × 1[c1 = 1] + 1[c0 = 1] 52 which is minimised by c0 = 0 and c1 = 0 to again obtain a minimum Bayes risk of 14 . Thus the ACA optimal information can be based on either S = {♠, ♦} or S = {♠, ♦, ♥}. On the other hand, for the Bayesian PNM we have that MBPNM = (A, BBPNM ), BBPNM = Q# µA and R(µ, MBPNM ) = 1 4 X x∈{♠,♦,♥,♣} 1[x ∈ / S]L(0, 0)   1 1 + 1[x ∈ S] (1 − )L(0, 1[x = ♠]) + L(1, 1[x = ♠]) . |S| |S| Case of S = {♠, ♦}: We have 4 R(µ, MBPNM ) = 1 1 + +0+0 2 2 = 1. 2 1 1 + + +0 3 3 3 = 4 . 3 Case of S = {♠, ♦, ♥}: We have 4 R(µ, MBPNM ) = Thus the PNM optimal information is S = {♠, ♦} and not S = {♠, ♦, ♥}. Hence, PNM and ACA optimal information differ in general. S4. Monte Carlo Methods for Numerical Disintegration In this section, Monte Carlo methods for sampling from the distribution µaδ (or µaδ,N ; the N subscript will be suppressed to reduce notation in the sequel) are considered. The Monte Carlo approximation of µaδ is, in effect, a problem in rare event simulation as most of the mass of µaδ will be confined to a set S such that µ(S) is small. Rare events pose some difficulties for classical Monte Carlo, as an enormous number of draws can be required to study the rare event of interest. In the literature there are two major solutions proposed. Importance sampling (Robert and Casella, 2013) samples from a modified process, under which the event of interest is more likely, then re-weights these samples to compensate for the adjustment. Conversely, in splitting (Botev and Kroese, 2012) trajectories of the process are constructed in a genetic fashion, by retaining and duplicating those which approach the events of interest and discarding others. Splitting is closely related to SMC (Cérou et al., 2012) and Feynman–Kac models (Del Moral, 2004). The splitting approach is described in the following section, while in Section S4.3 a parallel tempering (PT) algorithm is described. In spirit these approaches are similar in that they employ a tempering approach to ease sampling the relaxed posterior distribution for a small value of δ. The SMC method employs a particle approximation to accomplish this, while the PT algorithm uses coupled Markov chains. S4.1. Sequential Monte Carlo Algorithms for Numerical Disintegration Let {δi }m i=0 be such that δ0 = ∞, δm = δ and δi > δi+1 > 0 for all i < m − 1. Furthermore let a {Ki }m i=1 be some set of Markov transition kernels that leave µδi invariant, for which Ki (·, S) is measurable for all S ∈ ΣX and Ki (x, ·) is an element of PX for all x ∈ X . Then our SMC for numerical disintegration (SMC-ND) algorithm, based on P particles, is given in Algorithm 1. 53 Sample x0j ∼ µ for j = 1, . . . , P [Initialise] for i = 1, . . . , m do Sample xi−1 ∼ Ki (xi−1 j j , ·) for j = 1, . . . , P [Move] −1 i−1 φ δ kA(x )−akA ) ( i j for j = 1, . . . , P [Re-weight] Set wji ← −1 φ(δi−1 kA(xi−1 )−akA ) j P i P Sample xij ∼ Discrete({xi−1 j }j=1 ; {wj }j=1 ) for j = 1, . . . , P [Re-sample] end Algorithm 1: Sequential Monte Carlo for Numerical Disintegration (SMC-ND). Here we have used Discrete({xj }Pj=1 ; {wj }Pj=1 ) to denote the discrete distribution which puts mass proportional to wj on the state xj ∈ X . The output of the SMC-ND algorithm is an empirical approximation1 µaδm ,P P 1 X δ(xm = j ) P j=1 P to µaδm based on a population of P particles {xm j }j=1 . There is substantial room to extend and improve the SMC-ND algorithm based on the wide body of literature available on this subject (e.g. Doucet et al., 2001; Del Moral et al., 2006; Beskos et al., 2017; Ellam et al., 2016), but we defer all such improvements for future work. Our aim in the remainder is to establish the approximation properties of the SMC-ND output. This will be based on theoretical results in Del Moral et al. (2006). Assumption S4.1. φ > 0 on R+ . Assumption S4.2. For all i = 0, . . . , m − 1 and all x, y ∈ X , it holds that Ki+1 (x, ·)  Ki+1 (y, ·). Furthermore there exist constants i > 0 such that the Radon–Nikodým derivative dKi+1 (x, ·) ≥ i . dKi+1 (y, ·) Assumption S4.1 ensures that Algorithm 1 is well-defined, else it can happen that all particles are assigned zero weight and re-sampling will fail. However, the result that we obtain in Theorem S4.3 below can also be established in the special case of an indicator function φ(r) = 1[r < 1]. The details for this variation of the results are also included in the sequel. The interpretation of Assumption S4.2 is that, for fixed i, transition kernels do not allocate arbitrarily large or small amounts of mass to different areas of the state space, as a function of their first argument. This poses a constraint on the choice of Markov kernels for the SMC-ND algorithm. Theorem S4.3. For all δ ∈ {δi }m i=0 and fixed p ≥ 1 it holds that E  µaδ,P (f ) − µaδ (f ) p
1 p ≤ Cp kf kF √ P m−1 for some constant Cp independent of P but dependent on {δi }m i=0 , p and {i }i=0 . 1 The bandwidth parameter δ and the use of δ to denote an atomic distribution should not be confused. 54 The proof of Theorem S4.3 is presented next. Note that the established bound is independent of δ ∈ {δi }m i=0 ; this is therefore a uniform convergence result. The assumptions and the conclusion of Theorem S4.3 can be weakened in several directions, as discussed in detail in (Del Moral et al., 2006). Development of SMC methods in the context of high-dimensional and infinitedimensional state spaces has also been considered in Beskos et al. (2014, 2015). S4.2. Proof of Theorem S4.3 In this section we establish the uniform convergence of the SMC-ND algorithm as claimed in Theorem S4.3. This relies on a powerful technical result from Del Moral (2004), whose context is now established. S4.2.1. Feynman–Kac Models Let (Ei , Ei ) for i = 0, . . . , m be a collection of measurable spaces. Let η0 be a measure on E0 and let Γi index a collection of Markov transition kernels from Ei−1 to Ei . Let Gi : Ei → (0, 1] be a collection of functions, which are referred to as potentials. The triplets (η0 , Gi , Γi ) are associated with Feynman–Kac measures ηi on Ei defined as, for bounded and measurable functions fi on Ei ; ηi (fi ) = γi (fi ) γi (1)  γi (fi ) = Eη0 fi (X i ) i−1 Y j=0  Gj (X j ) where the expectation is taken with respect to the Markov process X i defined by X 0 ∼ η0 and X i |X i−1 ∼ Γi (X i−1 , ·). The Feynman–Kac measures can be associated with a (non-unique) McKean interpretation of the form ηi+1 = ηi Λi+1,ηi where the Λi+1,η are a collection of Markov transitions for which the following compatibility condition holds: ηΛi+1,η = Gi ηΓi+1 η(Gi ) Then the ηi can be interpreted as the ith step marginal distribution of the non-homogeneous Markov chain defined by X 0 ∼ η0 and X i+1 |X i ∼ Λi+1,ηi (X i , ·). The corresponding P -particle model is defined on EiP = Ei × · · · × Ei and has X0 ∼ η0P P(Xi ∈ dxi |Xi ) = P Y Λi,ηP (Xji−1 , dxij ) i−1 j=1 P where ηiP = P1 Pj=1 δ(Xji ) is an empirical (random) measure on Ei . The SMC-ND algorithm can be cast as an instance of such a P -particle model, as is made clear later. The result that we require from Del Moral (2004) is given next. Denote by Osc1 (Ei ) the set of measurable functions fi on Ei for which sup{|fi (xi ) − fi (y i )| : xi , y i ∈ Ei } ≤ 1. Theorem (Theorem 7.4.4 in Del Moral (2004)). Suppose that: 55 i G i i i (G) There exist G i ∈ (0, 1] such that Gi (x ) ≥ i Gi (y ) > 0 for all x , y ∈ Ei . (M1 ) There exist Γi ∈ (0, 1) such that Γi+1 (xi , ·) ≥ Γi Γi+1 (y i , ·) for all xi , y i ∈ Ei . Then for p ≥ 1 and any valid McKean interpretation Λi,η , the associated P -particle model ηiP satisfies the uniform (in i) bound √ sup sup P E[|ηiP (fi ) − ηi (fi )|p ]1/p ≤ Cp 0≤i≤m fi ∈Osc1 (Ei ) m Γ m−1 for some constant Cp independent of P but dependent on {G i }i=0 and {i }i=0 . The actual statement in Del Moral (2004) contains a more general version of (M1 ) and a more explicit decomposition of the constant Cp ; however the simpler version presented here is sufficient for the purposes of the present paper. S4.2.2. Case A: Positive Function φ(r) > 0 First we prove Theorem S4.3 as it is stated. Later the assumption of φ > 0 will be relaxed. SMC-ND as a Feynman–Kac Model The aim here is to demonstrate that the SMC-ND algorithm fits into the framework of Section S4.2.1 for a specific McKean interpretation. This connection will then be used to establish uniform convergence for the SMC-ND algorithm as a consequence of Theorem 7.4.4 in Del Moral (2004). For the state spaces we associate each Ei = X and Ei = ΣX . For the potentials we associate   1 kA(xi ) − akA φ δi+1   Gi (xi ) = φ δ1i kA(xi ) − akA which clearly does not vanish and takes values in (0, 1] since δi > δi+1 and φ is decreasing. For the Markov transitions we associate Γi+1 with Ki+1 . The Feynman–Kac measures associated with the SMC-ND algorithm can be cast as a nonhomogeneous Markov chain with transitions Λi+1,η . Here Λi+1,ηi acts on the current measure ηi on X by first propagating as ηi Ki+1 and then “warping” this measure with the potential Gi ; i.e. Gi ηΛi+1,η = ηΓi+1 . η(Gi ) This demonstrates that the SMC-ND algorithm is the P -particle model corresponding to the McKean interpretation Λi+1,η of the Feynman–Kac triplet (η0 , Gi , Γi ). Thus the SMC-ND algorithm can be studied in the context of Section S4.2.1, which we report next. Note that it is common in applications of SMC to perform the “Re-sample” step before the “Move” step - our choice of order was required for the McKean framework that is the basis of the theoretical results in Del Moral et al. (2006). It is known in the SMC “folk lore” that the order of these steps can be interchanged. Proof of Uniform Convergence Result for SMC-ND It remains to verify the hypotheses of Theorem 7.4.4 in Del Moral (2004). Condition (G) is satisfied if and only if   1 i φ kA(x ) − akA δi+1 56 is bounded below, since φ  1 kA(xi ) − akA δi  is bounded above by 1. Since φ is continuous, decreasing and satisfies φ > 0 (Assumption S4.1), 1 it suffices to show that its argument δi+1 kA(x) − akA is upper-bounded. This is the content of Assumption 4.6 in the main text, which shows that 1 1 kA(x) − akA ≤ sup kA(x)kA + kakA δi δi x∈X 1 =: G < ∞. i Condition (M1 ) requires that Γi+1 (xi , S) ≥ Γi Γi+1 (y i , S) for all xi , y i ∈ Ei and S ∈ Ei+1 . From construction this is equivalent to Ki+1 (xi , S) ≥ Γi Ki+1 (y i , S) for all xi , y i ∈ X and S ∈ ΣX . This is the content of Assumption S4.2. Thus we have established the hypotheses of Theorem 7.4.4 in Del Moral (2004) for the SMCND algorithm. Theorem S4.3 is a re-statement of this result. For the statement of the result we used the kf kF norm, based on the fact that (from Assumption 4.7) kfi kOsc(Ei ) ≤ 2kf k∞ ≤ 2CF kf kF . S4.2.3. Case B: Indicator Function φ(r) = 1[r < 1] The previous analysis required that φ > 0 on R+ . However, the most basic choice for φ is the indicator function φ(r) = 1[r < 1] which can take the value 0. The case of an indicator function demands special attention, since Algorithm 1 can fail in this case if all particles are assigned zero weight. If this occurs, then we just define µαδ,P (f ) = 0. To be specific, the SMCND algorithm associated to the indicator function φ for approximation of the integral µaδ (f ) is stated as Algorithm 2 next. Let Xδa = {x ∈ X : kA(x) − akA < δ}. If there is some iteration i at which, after applying the kernel Ki to each particle, no particle lies within Xδai , the algorithm fails. As a result it is critical to ensure that the distance between successive δi is small so that the probability of failure is controlled. This requirement is made formal next. To establish the approximation properties of the random measure µaδm ,P , two assumptions are required. These are intended to replace Assumptions S4.1, S4.2 and Assumption 4.6 from the main text: Assumption S4.4. For all i = 0, . . . , m − 1 and all xi ∈ Xδai , it holds that Ki+1 (xi , Xδai+1 ) > 0. Assumption S4.5. For all i = 0, . . . , m − 1 and all xi , y i ∈ Xδai , Ki+1 (xi , ·)  Ki+1 (y i , ·). Furthermore there exist constants i > 0 such that the Radon–Nikodým derivative dKi+1 (xi , ·) ≥ i . dKi+1 (y i , ·) Assumption S4.4 requires that the probability of reaching Xδai+1 when starting in Xδai and applying the transition kernel Ki+1 , is bounded away from zero. Assumption S4.5 ensures that, for fixed i, transition kernels do not allocate arbitrarily large or small amounts of mass to different areas of the state space, as a function of their first argument. 57 Sample x0j ∼ µ for j = 1, . . . , P [Initialise] for i = 1, . . . , n do Sample xij ∼ Ki (xi−1 j , ·) for j = 1, . . . , P [Sample] i i Ei ← {xj : xj ∈ Xδai } if Ei = ∅ then Return µaδ,P (f ) ← 0 end for j = 1, . . . , P do if xij ∈ / Ei then xij ∼ Uniform(Ei ) [Re-sample] end end end P Return µaδ,P (f ) ← P1 Pj=1 f (xnj ). Algorithm 2: Sequential Monte Carlo for Numerical Disintegration (SMC-ND), for the case where φ(r) = 1[r < 1]. Theorem S4.6. For the alternative situation of an indicator function, it holds that for all δ ∈ {δi }m i=0 and fixed p ≥ 1, E  µaδ,P (f ) − µaδ (f ) p
1 p ≤ Cp kf kF √ P for some constant Cp independent of P but dependent on p and {i }m−1 i=0 . Cérou et al. (2012) proposed an algorithm similar to the one herein but focussed on approximation of the probability of a rare event rather than sampling from the rare event itself. In particular the theoretical results provided are in terms of these probabilities rather than how well the measure restricted to the rare event is approximated. Furthermore, many of the results therein focused upon an idealised version of the problem, in which it was assumed that the intermediate restricted measures can be sampled directly; this avoids the issues with vanishing potentials indicated in Del Moral (2004). A similar algorithm was discussed in Ścibior et al. (2015) but was not shown to be theoretically sound. The remainder of this Section establishes Theorem S4.6. SMC-ND as a Feynman–Kac Model The aim here is to demonstrate that Algorithm 2 fits into the framework of Section S4.2.1 for a specific McKean interpretation. This is analogous to the proof of Theorem S4.3. A technical complication is that the potentials Gi must take values in (0, 1], which precludes the “obvious” choice of Ei = X and Gi (xi ) as indicator functions for the sets Xδai . Instead, we associate Ei = Xδai and Ei with the corresponding restriction of ΣX . For the potentials we then take Gi (xi ) = 1 for all xi ∈ Ei , which clearly does not vanish and takes values in (0, 1]. For the Markov transitions Γi+1 from Ei to Ei+1 we consider Γi+1 (xi , dxi+1 ) ∝ Ki+1 (xi , xi+1 ) which is the restriction of Ki+1 to Ei+1 . For the latter to be well-defined it is required that the 58 normalisation constant Z Ki+1 (xi , xi+1 )dxi+1 > 0 Ei+1 xi for all ∈ Ei , so that there is a positive probability of reaching Ei+1 from Ei . This is the content of Assumption S4.4. The Feynman–Kac measures associated with Algorithm 2 can be cast as a non-homogeneous Markov chain with transitions Λi+1,η . Here Λi+1,ηi acts on the current measure ηi on Ei by first propagating as ηi Ki+1 and then restricting this measure to Ei+1 . This procedure is seen to be identical to the Markov transition Γi+1 defined above and, since the potentials Gi ≡ 1, it follows that ηΛi+1,η = ηΓi+1 Gi = ηΓi+1 . η(Gi ) This demonstrates that Algorithm 2 is the P -particle model corresponding to the McKean interpretation Λi+1,η of the Feynman–Kac triplet (η0 , Gi , Γi ). Thus the SMC-ND algorithm can be studied in the context of Section S4.2.1, which we report next. Proof of Uniform Convergence Result for SMC-ND It remains to verify the hypotheses of Theorem 7.4.4 in Del Moral (2004). Condition (G) is satisfied with no further assumption, since Gi ≡ 1 and we can take G i = 1. Condition (M1 ) requires that Γi+1 (xi , S) ≥ Γi Γi+1 (y i , S) for all xi , y i ∈ Ei and S ∈ Ei+1 . From construction this is equivalent to Ki+1 (xi , S) ≥ Γi Ki+1 (y i , S) for all xi , y i ∈ Ei and S ∈ Ei+1 . This is the content of Assumption S4.5. Thus we have established the hypotheses of Theorem 7.4.4 in Del Moral (2004) for Algorithm 2 and in doing so have established Theorem S4.6. S4.3. Parallel Tempering for Numerical Disintegration a Let Ki , {δi }m i=1 be as in Section S4.1. The PT algorithm (Geyer, 1991) for sampling from µδm runs m Markov chains in parallel, one for each temperature, by alternately applying Ki , then randomly proposing to “swap” the current state of two of the chains. Commonly only swaps of adjacent chains are considered; to this end suppose at iteration j an index q ∈ {0, . . . , m − 1} has been selected. Denote by xq the state of the chain with µaδq as its invariant measure. Then to ensure the correct invariant distribution of all chains is maintained, the swap of state xq and xq+1 is accepted with probability α(xq , xq+1 ) = πq (xq+1 )πq+1 (xq ) πq (xq )πq+1 (xq+1 ) (S4.1) where πq denotes the density of the target distribution µaδq with respect to a suitable reference measure. The density notation can be justified since in our experiments the sampler was applied to the finite-dimensional distributions µaδq ,N and so the reference measure can be taken to be the Lebesgue measure on RN . 59 Given some initial xi0 for i = 1, . . . , m [Initialise] for j = 1, . . . , P do Sample x̂ij ∼ Ki (xij−1 , ·) for i = 1, . . . , m [Move] Sample q ∼ Uniform(0, m − 1) if U (0, 1) < α(xqj , xq+1 ) then j q q+1 Set xj = x̂j and xq+1 = x̂qj [Accept Swap] j else Set xqj = x̂qj and xq+1 = x̂q+1 [Reject Swap] j j i i For i 6= q, q + 1, set xj = x̂j [Update] end Algorithm 3: Parallel Tempering for Numerical Disintegration The PT algorithm for numerical disintegration is described in Algorithm 3. The samples P a {xm j }j=1 are approximate draws from the distribution µδm . Algorithms 1 and 3 are each valid for sampling from a target measure µaδ . The choice of which algorithm to use is problem dependent, and each algorithm has been applied in the experiments in Section 6. S4.4. Estimation of Model Evidence The model evidence pA (a) was estimated as a by-product of the numerical disintegration algorithm developed. Attention is restricted to the specific relaxation function φ(r) = exp(−r2 ). Then the thermodynamic integral identity (Gelman and Meng, 1998) can be exploited to calculate the model evidence: Z Z 1 1 log pA (a) = − lim 2 kA(x) − ak2A dµaδ/√t dt δ↓0 δ 0 √ where the parameterisation δ 7→ δ/ t is such that t = 0 corresponds to the prior, while t = 1 corresponds to the distribution µaδ . To approximate this integral, the outer integral is first discretised. To this end, fix a sequence ∞ = δ0 < δ1 < · · · < δm of relaxation parameters. For convenience this may be the same √ sequence as used to apply numerical disintegration. Then for δm small, and letting ti = δm /δi : Z m 1 X log pA (a) ≈ − 2 (ti − ti−1 ) kA(x) − ak2A dµaδm /√ti δm i=1 Thus we obtain a consistent approximation Z m  X 1 1 log pA (a) ≈ − kA(x) − ak2A dµaδi 2 − δ2 δ i i−1 | i=1 {z } (∗) The terms (∗) were estimated via Monte Carlo, based on samples from the distributions µaδi obtained through numerical disintegration. Higher-order quadrature rules and variance reduction techniques can be used, but were not implemented for this work (Oates et al., 2016b). 60 S4.5. Monte Carlo Details for Painlevé Transcendental Sampling of the posterior was performed for a temperature schedule of m = 1600 steps, equally spaced on a logarithmic scale from 10 to 10−4 , for an ensemble of P = 200 particles. Specification of appropriate transition kernels Ki for this problem was challenging due both to the high dimension and the empirical observation that, for small δ, mixing of the chains tends to be poor. This is likely due to the nonlinearity of the information operator which leads to highly a complex posterior structure. For this reason a gradient-based sampler was used to construct the transition kernel; the Metropolis-adjusted Langevin algorithm (MALA) (Roberts and Tweedie, 1996). Denote by uk the coefficients [ukj ]N j=1 at iteration k of MALA. Then, recall that MALA has proposals given by p uk+1 = uk + τi Γ∇ log πi (uk ) + 2τi ΓW where W is a standard Gaussian distribution and Γ ∈ RN ×N is a positive definite preconditioning matrix. The τi were taken to be fixed for each kernel Ki to a value found empirically to provide a reasonable acceptance rate. πi denotes the unnormalised target distribution for Ki , here given by ! N −a Ax πi (uk ) = φ q N (uk ) δi P N N where xN = N i=0 ui φi and q (·) denotes the prior density of the coefficients [uj ]j=1 . To ensure proposals were scaled to match the decay of the prior for the coefficients, we took Γ = diag(γ), the diagonal matrix which has the coefficients γi on its diagonal. Even with such a transition kernel, mixing is generally poor. To compensate k was taken to be large; for n = 12, 17 we took k = 10, 000, while for n = 22 we took k = 40, 000. We note that such a large number of temperature levels and transitions makes computation expensive, highlighting the importance of future work toward methods for approximating the Bayesian posterior in a more computationally efficient manner. S4.6. Monte Carlo Details for Poisson Equation The posterior distribution was obtained by use of the PT algorithm, for m = 20 temperatures equally spaced on a logarithmic scale between 10−2 and 10−4 . The transition kernels Ki were given by 10 iterations of a MALA sampler, with preconditioner as described earlier and parameter τ chosen to achieve a good acceptance rate. The number of iterations P was taken to be 106 when n = 25 and 107 when n = 25 or n = 36. S5. Truncation of the Prior Distribution (Proof of Theorem 4.8) In this section we present the proof of Theorem 4.8 in the main text. We use a general result on the well-posedness of Bayesian inverse problems: Theorem S5.1 (Theorem 4.6 in Sullivan (2016)). Let X and A be separable quasi-Banach spaces over R. Suppose that dµaδ exp(−Φδ (x; a)) = (S5.1) dµ Zδa where the potential function Φδ satisfies: 61 S0 Φδ (x; ·) is continuous for each x ∈ X , Φδ (·; a) is measurable for each a ∈ A, and for every r > 0, there exists M0,r,δ ∈ R such that, for all (x, a) ∈ X × A with kxkX < r and kakA < r, |Φδ (x; a)| ≤ M0,r,δ . S1 For every r > 0, there exists a measurable M1,r,δ : R+ → R such that, for all (x, a) ∈ X ×A with kakA < r,
Φδ (x; a) ≥ M1,r,δ kxkX . S2 For every r > 0, there exists a measurable M2,r,δ : R+ → R+ such that, for all (x, a, ã) ∈ X × A × A with kakA < r, kãkA < r,

|Φδ (x; a) − Φδ (x; ã)| ≤ exp M2,r,δ kxkX ka − ãkA . Let Φδ,N be an approximation to Φδ that satisfies (S1-S3) with Mi,r,δ independent of N , and such that S3 Ψ : N → R+ is such that, for every r > 0, there exists a measurable M3,r,δ : R+ → R+ , such that, for all (x, a) ∈ X × A with kakA < r,

|Φδ,N (x; a) − Φδ (x; a)| ≤ exp M3,r,δ kxkX Ψ(N ). S4 For some r > 0,   EX∼µ exp(2M3,r,δ (kXkX ) − M1,r,δ (kXkX )) < ∞. (S5.2) Let dH denote the Hellinger distance on PX . Then there exists a constant Cδ , independent of N , such that
dH µaδ,N , µaδ ≤ Cδ Ψ(N ) where µaδ,N is the posterior distribution based on the potential function Φδ,N instead of Φδ . This allows us to establish conditions on A and µ that guarantee stability under truncation of the prior: Proof of Theorem 4.8. Let ϕ be as in Section 4.1, and let   kA(x) − akA Φδ (x; a) = ϕ δ   kA ◦ PN (x) − akA Φδ,N (x; a) = ϕ . δ Our task is to check the conditions of Theorem S5.1 hold for Φδ and Φδ,N . S0 First, note that Φδ (x; ·) is continuous (since ϕ is continuous from Assumption 4.1 and Φδ (x; ·) is a composition of continuous functions) and that Φδ (·; a) is measurable (since φ is measurable and Φδ (·; a) is a composition of measurable functions). Second, note that ϕ is a continuous bijection from (0, ∞) to itself with ϕ(0) = 0. Thus ϕ−1 exists and we can consider δϕ−1 sup{|Φδ (x; a)| : kxkX , kakA < r} = sup{kA(x) − akA : kxkX , kakA < r} ≤ sup kA(x)kA + r x∈X ≤ ∞ (Assumption 4.6). Thus we can take M0,r,δ = ϕ( 1δ supx∈X kA(x)kA + rδ ). 62 S1 Since Φδ (x; a) ≥ 0 we can take M1,r,δ = 0. S2 Given r > 0 let R = bound 1 δ supx∈X kA(x)kA + rδ , which is finite by Assumption 4.6. The upper    kA(x) − ãkA kA(x) − akA −ϕ |Φδ (x; a) − Φδ (x; ã)| = ϕ δ δ kA(x) − akA kA(x) − ãkA ≤ CR − (Assumption 4.4) δ δ CR ≤ ka − ãkA (reverse triangle inequality) δ  demonstrates that we can take M2,r,δ = max{0, log( CδR )}. Minor variation on the above arguments show that S1-3 also hold for Φδ,N with the same constants Mi,r,δ . S3 Let CR be defined as in S2. The upper bound     kA ◦ PN (x) − akA kA(x) − akA |Φδ,N (x; a) − Φδ (x; a)| = ϕ −ϕ δ δ kA ◦ PN (x) − akA kA(x) − akA (Assumption 4.4) ≤ CR − δ δ CR ≤ kA ◦ PN (x) − A(x)kA (reverse triangle inequality) δ CR ≤ exp(m(kxkX ))Ψ(N ) (Assumption 4.5) δ demonstrates that we can take M3,r,δ (kxkX ) = max{0, log( CδR ) + m(kxkX )}. S4 Let CR be defined as in S2. The upper bound EX∼µ [exp(2M3,r,δ (kXkX ) − M1,r,δ (kXkX ))] = EX∼µ [exp(2 max{0, log(CR /δ) + m(kXkX )})] CR ≤1+ EX∼µ [exp(2m(kXkX ))] δ < ∞ (Assumption 4.5) establishes the last of the conditions for Theorem S5.1 to hold.
Thus from Theorem S5.1, dH µaδ,N , µaδ ≤ Cδ Ψ(N ). The proof is completed since Assumption 4.7 implies that dF ≤ CF−1 dTV where dTV is the total √ variation distance based on F = {f : kf k∞ ≤ 1}; in turn it is a standard fact that dTV ≤ 2dH . 63 

Inverse Stability Problem and Applications to Renewables Integration arXiv:1703.04491v5 [] 14 Oct 2017 Thanh Long Vu∗ , Hung Dinh Nguyen, Alexandre Megretski, Jean-Jacques Slotine and Konstantin Turitsyn Abstract—In modern power systems, the operating point, at which the demand and supply are balanced, may take different values due to changes in loads and renewable generation levels. Understanding the dynamics of stressed power systems with a range of operating points would be essential to assuring their reliable operation, and possibly allow higher integration of renewable resources. This letter introduces a non-traditional way to think about the stability assessment problem of power systems. Instead of estimating the set of initial states leading to a given operating condition, we characterize the set of operating conditions that a power grid converges to from a given initial state under changes in power injections and lines. We term this problem as “inverse stability”, a problem which is rarely addressed in the control and systems literature, and hence, poorly understood. Exploiting quadratic approximations of the system’s energy function, we introduce an estimate of the inverse stability region. Also, we briefly describe three important applications of the inverse stability notion: (i) robust stability assessment of power systems w.r.t. different renewable generation levels, (ii) stability-constrained optimal power flow (sOPF), and (iii) stability-guaranteed corrective action design. Index Terms—Power grids, renewables integration, transient stability, inverse stability, emergency control, energy function I. I NTRODUCTION R ENEWABLE generations, e.g., wind and solar, are increasingly installed into electric power grids to reduce CO2 emission from the electricity generation sector. Yet, their natural intermittency presents a major challenge to the delivery of consistent power that is necessary for today’s grid operation, in which generation must instantly meet load. Also, the inherently low inertia of renewable generators limits the grid’s controllability and makes it easy for the grid to lose its stability. The existing power grids and management tools were not designed to deal with these new challenges. Therefore, new stability assessment and control design tools are needed to adapt to the changes in architecture and dynamic behavior expected in the future power grids. Transient stability assessment of power system certifies that the system state converges to a stable operating condition after the system experiences large disturbances. Traditionally, this task is handled by using either time domain simulation (e.g., [1]), or by utilizing the energy method (e.g., [2], [3]) and the Lyapunov function method (e.g., [4]) to estimate the stability region of a given equilibrium point (EP), i.e., the set of initial states from which the system state will converge to that EP. In modern renewable power grids, the operating point may take different values under the real-time clearing of electricity ∗ Corresponding author. Email: longvu@mit.edu. All the authors are with the Massachusetts Institute of Technology, Cambridge, MA 02139, USA. markets, intermittent renewable generations, changing loads, and external disturbances. Dealing with the situation when the EP can change over a wide range makes the transient stability assessment even more technically difficult and computationally cumbersome. In this letter, rather than considering the classical stability assessment problem, we formulate the inverse stability assessment problem. This problem concerns with estimating the region around a given initial state δ0 , called “inverse stability region” A(δ0 ), so that whenever the power injections or power lines change and lead to an EP in A(δ0 ), the system state will converge from δ0 to that EP. Indeed, the convergence from δ0 to an EP is guaranteed when the system’s energy function is bounded under some threshold [2], [3]. In [5], we observed that if the EP is in the interior of the set P characterized by phasor angular differences smaller than π/2, then the nonlinear power flows can be strictly bounded by linear functions of angular differences. Exploiting this observation, we show that the energy function of power system can be approximated by quadratic functions of the EP and the system state, and from which we obtain an estimate of the inverse stability region. The remarkable advantage of the inverse stability certificate is making it possible to exploit the change in EP to achieve useful dynamical properties. We will briefly discuss three applications of this certificate, which are of importance to the integration of large-scale renewable resources: Robust stability assessment: For a typical power system composed of several components and integrated with different levels of renewable generations, there are many contingencies that need to be reassessed on a regular basis. Most of these contingencies correspond to failures of relatively small and insignificant components, so the post-fault dynamics is probably transiently stable. Therefore, most of the computational effort is spent on the analysis of non-critical scenarios. This computational burden could be greatly alleviated by a robust transient stability assessment toolbox that could certify the system’s stability w.r.t. a broad range of uncertainties. In this letter, we show that the inverse stability certificate can be employed to assess the transient stability of power systems for various levels of power injections. Stability-constrained OPF: Under large disturbances, a power system with an operating condition derived by solving the conventional OPF problem may not survive. It is therefore desirable to design operating conditions so that the system can withstand large disturbances. This can be carried out by incorporating the transient stability constraint into OPF together with the normal voltage and thermal constraints. Though this problem was discussed in the literature (e.g., [6]), there is no way to precisely formulate and solve the stability-constrained OPF problem because transient stability is a dynamic concept and differential equations are involved in the stability constraint. Fortunately, the inverse stability certificate allows for a natural incorporation of the stability constraint into the OPF problem as a static constraint of placing the EP in a given set. Stability-guaranteed corrective actions: Traditional protection strategies focus on the safety of individual components, and the level of coordination among component protection systems is far from perfect. Also, they do not take full advantage of the new flexible and fast electronics resources available in modern power systems, and largely rely on customer-harmful actions like load shedding. These considerations motivated us to coordinate widespread flexible electronics resources as a system-level customer-friendly corrective action with guaranteed stability [7]. This letter presents a unconventional control way in which we relocate the operating point, by appropriately redispatching power injections, to attract the emergency state and stabilize the power systems under emergency situations. II. I NVERSE STABILITY PROBLEM OF POWER SYSTEMS In this letter, we utilize the structure-preserving model to describe the power system dynamics [8]. This model naturally incorporates the dynamics of the generators’ rotor angle and the response of load power output to frequency deviation. Mathematically, the grid is described by an undirected graph A(N , E), where N = {1, 2, . . . , |N |} is the set of buses and E ⊆ N × N is the set of transmission lines {k, j}, k, j ∈ N . Here, |A| denotes the number of elements of set A. The sets of generator buses and load buses are denoted by G and L. We assume that the grid is lossless with constant voltage magnitudes Vk , k ∈ N , and the reactive powers are ignored. Then, the grid’s dynamics is described by [8]: X mk δ¨k + dk δ˙k + akj sin(δk − δj ) =Pk , k ∈ G, (1a) {k,j}∈E dk δ˙k + X akj sin(δk − δj ) =Pk , k ∈ L, (1b) {k,j}∈E where equation (1a) applies at the dynamics of generator buses and equation (1b) applies at the dynamics of load buses. Here akj = Vk Vj Bkj , where Bkj is the (normalized) susceptance of the transmission line {k, j} connecting the k th bus and j th bus. Nk is the set of neighboring buses of the k th bus (see [9] for more details). Let δ(t) = [δ1 (t) ... δ|N | (t) δ̇1 (t) ... δ̇|N | (t)]> be the state of the system (1) at time t (for simplicity, we will denote the system state by δ). Note that Eqs. (1) are invariant under any uniform shift of the angles δk → δk + c. However, the state δ can be unambiguously characterized by the angle differences δkj = δk − δj and the frequencies δ̇k . Normally, a power grid operates at an operating condition of the pre-fault dynamics. Under the fault, the system evolves according to the fault-on dynamics. After some time period, the fault is cleared or self-clears, and the system is at the socalled fault-cleared state δ0 (the fault-cleared state is usually estimated by simulating the fault-on dynamics, and hence, is assumed to be known). Then, the power system experiences the so-called post-fault dynamics. The transient stability assessment certifies whether the post-fault state converges from δ0 to a stable EP δ ∗ . Mathematically, the operating condition ∗ > δ ∗ = [δ1∗ ... δ|N | 0 ... 0] is a solution of the power-flow like equations: X akj sin δkj =Pk , k ∈ N . (2) {k,j}∈E With renewable generations or under power redispatching, the power injections Pk take different values. Also, the couplings akj can be changed by using the FACTS devices. Assume akj ≤ akj ≤ ākj . In those situations, the resulting EP δ ∗ also takes different values. Therefore, we want to characterize the region of EPs so that the post-fault state always converges from a given initial state δ0 to the EP whenever the EP is in this region. Though the EP can take different values, it is assumed to be fixed in each transient stability assessment because the power injections and couplings can be assumed to be unchanged in the very fast time scale of transient dynamics (i.e., 1 to 10 seconds). We consider the following problem: • Inverse (Asymptotic) Stability Problem: Consider a given initial state δ0 . Assume that power injections and the line susceptances can take different values. Estimate the region of stable EPs so that the state of the system (1) always converges from δ0 to the EP in this region. This problem will be addressed with the inverse stability certificate to be presented in the next section. III. E NERGY FUNCTION AND INVERSE STABILITY CERTIFICATE A. Stability assessment by using energy function Before introducing the inverse stability certificate addressing the inverse stability problem in the previous section, we present a normal stability certificate for system with the fixed power injections and line parameters. For the power system described by Eqs. (1), consider the energy function: X Z δkj X mk δ̇ 2 ∗ k E(δ, δ ∗ ) = + akj (sin ξ − sin δkj )dξ ∗ 2 δkj {k,j}∈E k∈G (3) Then, along every trajectory of (1), we have X X ∗ Ė(δ, δ ∗ ) = mk δ̇k δ̈k + akj (sin δkj − sin δkj )δ̇kj {k,j}∈E k∈G = X δ̇k (Pk − dk δ˙k − X δ̇k (Pk − dk δ˙k − X X akj sin(δk − δj )) {k,j}∈E k∈L + akj sin(δk − δj )) {k,j}∈E k∈G + X ∗ akj (sin δkj − sin δkj )(δ̇k − δ˙j ) {k,j}∈E =− X k∈N dk (δ˙k )2 ≤ 0, (4) in which the last equation is obtained from (2). Hence, E(δ, δ ∗ ) is always decreasing along every trajectory of (1). Consider the set P defined by |δkj | ≤ π/2, {k, j} ∈ E, and the set Φ = {δ ∈ P : E(δ, δ ∗ ) < Emin (δ ∗ )}, where Emin (δ ∗ ) = minδ∈∂P E(δ, δ ∗ ) and ∂P is the boundary of P. Φ is invariant w.r.t. (1), and bounded as the state δ is characterized by the angle differences and the frequencies. Though Φ is not closed, the decrease of E(δ, δ ∗ ) inside Φ assures the limit set to be inside Φ. As such, we can apply the LaSalle’s Invariance Principle and use a proof similar to that of Theorem 1 in [5] to show that, if δ0 is inside Φ then the system state will only evolve inside this set and eventually converge to δ ∗ . So, to check if the system state converges from δ0 ∈ P to δ ∗ , we only need to check if E(δ0 , δ ∗ ) < Emin (δ ∗ ). B. Inverse stability certificate 1 0.8 R(δ0) M 0.6 0.4 δ∗ y 0.2 δ0 B(δ0 ) 0 −0.2 −0.4 Λ −0.6 −0.8 P −1 −1 −0.5 0 0.5 1 1.5 2 2.5 Fig. 1. For a power system with a xgiven initial state δ0 , if the EP δ ∗ is inside the set A(δ0 ) = Λ ∩ B(δ0 ) surrounding δ0 then the system state will converge from δ0 to the EP δ ∗ since E(δ0 , δ ∗ ) < Emin . For a given initial state δ0 ∈ P, we will construct a region surrounding it so that whenever the operating condition δ ∗ is in this region then E(δ0 , δ ∗ ) < Emin (δ ∗ ). Hence, the grid state will converge from δ0 to δ ∗ according to the stability certificate in Section III-A. Indeed, we establish quadratic bounds of the energy function for every δ ∗ in the set Λ defined by inequali1 − sin λ ties |δkj | ≤ λ < π/2, ∀{k, j} ∈ E. Let g = > 0. In π/2 − λ ∗ [5], we observed that for δ ∈ Λ, ξ ∈ P, ∗ 2 X mk δ̇ 2 X (δkj − δkj ) k E(δ, δ ) ≥ +g akj , 2 2 ∗ (5) {k,j}∈E ∗ 2 X mk δ̇ 2 X (δkj − δkj ) k + akj . 2 2 (6) {k,j}∈E k∈G Define the following functions D(δ, δ ∗ ) = g X {k,j}∈E F (δ, δ ∗ ) = akj ∗ 2 (δkj − δkj ) , 2 X X mk δ̇ 2 (δkj − k + ākj 2 2 k∈G (9) The following is our main result regarding inverse stability of power system, as illustrated in Fig. 1. Theorem 1: Consider a given initial state δ0 inside the set P. Assume that the EP of the system takes different values in the set A(δ0 ) = Λ ∩ B(δ0 ), where the set B(δ0 ) is defined as in (9). Then, the system state always converges from the given initial state δ0 to the EP. Proof: See Appendix VI.  Remark 1: In this paper, we limit the grid to be described by the simplified model (1) which captures the dynamics of the generators’ rotor angle and the response of load power output to frequency deviation. More realistic models should take into account voltage variations, reactive powers, dynamics of the rotor flux, and controllers (e.g., droop controls and power system stabilizers). It should be noted that the results in this paper is extendable to more realistic models. The reason is that all the key results in this paper rely on the analysis of the system’s energy function, in which we combine the energy function-based transient stability analysis with the quadratic bounds of the energy function. In high-order models, the energy function is more complicated, yet it still can be bounded by quadratic functions. Hence, we can combine the energy function-based transient stability analysis of the more realistic models in [2]–[4] with the quadratic bounds of the energy function to extend the results in this paper to these higher-order models. This approach may also extend to other higher-order models in the port-Hamiltonian formulation (e.g., [10], [11]) by applying the appropriate approximation on the Lyapunov functions established for these models. A. Robust stability assessment Hence, for all δ ∗ ∈ Λ, δ ∈ P, we have E(δ, δ ∗ ) ≤ B(δ0 ) = {δ : F (δ0 , δ) ≤ R(δ0 )/4}. IV. A PPLICATIONS OF I NVERSE S TABILITY C ERTIFICATE ∗ 2 ∗ ∗ ∗ 2 g(ξ − δkj ) ≤ (ξ − δkj )(sin ξ − sin δkj ) ≤ (ξ − δkj ) . k∈G For a given initial state δ0 inside the set P, we calculate the “distance” from this initial state to the boundary of the set P : R(δ0 ) = minδ∈∂P D(δ0 , δ). Let B(δ0 ) be the neighborhood of δ0 defined by The robust transient stability problem that we consider involves situations where there is uncertainty in power injections Pk , e.g., due to intermittent renewable generations. Formally, for a given fault-cleared state δ0 , we need to certify the transient stability of the post-fault dynamics described by (1) with respect to fluctuations of the power injections, which consequently lead to different values of the post-fault EP δ ∗ as a solution of the power flow equations (2). Therefore, we consider the following robust stability problem [5]: • ∗ 2 δkj ) (7) . {k,j}∈E Using (5) and (6), we can bound the energy function as D(δ, δ ∗ ) ≤ E(δ, δ ∗ ) ≤ F (δ, δ ∗ ), ∀δ ∈ P, δ ∗ ∈ Λ. (8) Robust stability assessment: Given a fault-cleared state δ0 , certify the transient stability of (1) w.r.t. a set of stable EPs δ ∗ resulted from different levels of power injections. Utilizing the inverse stability certificate, we can assure robust stability of renewable power systems whenever the resulting EP is inside the set A(δ0 ) = Λ ∩ B(δ0 ). To check that the EP is in the set Λ, we can apply the criterion from [12], which states that the EP will be in the set Λ if the power injections p = [P1 , ..., P|N | ]T satisfy kL† pkE,∞ ≤ sin λ, (10) where L† is the pseudoinverse of the network Laplacian matrix and the norm kxkE,∞ is defined by kxkE,∞ = max{i,j}∈E |x(i) − x(j)|. On the other hand, some similar sufficient condition could be developed so that we can verify that the EP is in the set B(δ0 ) by checking the power injections. This will help us certify robust stability of the system by only checking the power injections. B. Stability-constrained OPF Stability-constrained OPF problem concerns with determining the optimal operating condition with respect to the voltage and thermal constraints, as well as the stability constraint. While the voltage and thermal constraints are well modeled via algebraic equations or inequalities, it is still an open question as to how to include the stability constraint into OPF formulation since stability is a dynamic concept and differential equations are involved [6]. Mathematically, a standard OPF problem is usually stated as follows (refer to [6] for more detailed formulation): s.t. min c(P ) (11) P (V, δ) = P (12) Q(V, δ) = Q (13) V ≤ V ≤ V̄ (14) S ≤ |S(V, δ)| ≤ S̄ (15) where c(P ) is a quadratic cost function, the decision variables P are typically the generator scheduled electrical power outputs, the equality constraints (12)-(13) stand for the power flow equations, and the inequality constraints (14)-(15) stand for the voltage and thermal limits of branch flows through transmission lines and transformers. Assume that the stability constraint is to make sure that the system state will converge from a given fault-cleared state δ0 to the designed operating condition, and that the reactive power is negligible. With the inverse stability certificate, the stability constraint can be relaxed and formulated as δ ∈ A(δ0 ). Basically, the inverse stability certificate transforms the dynamic problem of stability into a static problem of placing the prospective EP into a set. In summary, we obtain a relaxation of the stability-constrained OPF problem as follows: s.t. C. Emergency control design min c(P ) (16) P (V, δ) = P (17) V ≤ V ≤ V̄ (18) S ≤ |S(V, δ)| ≤ S̄ (19) δ ∈ A(δ0 ). (20) Solution of this optimization problem in an optimal operating condition at which the cost function is minimized and the voltage/thermal constraints are respected. Furthermore, the stability constraint is guaranteed by the inverse stability certificate, and the system state is ensured to converge from the fault-cleared δ0 to the operating condition. δ0 δ ∗1 δ ∗2 δ ∗N δ ∗desired Fig. 2. Power dispatching to relocate the stable EPs δi∗ so that the faultcleared state, which is possibly unstable if there is no controls, is driven ∗ through a sequence of EPs back to the desired EP δdesired . The placement of these EPs is determined by applying the inverse stability certificate. Another application of the inverse stability certificate, that will be detailed in this section, is designing stabilityguaranteed corrective actions that can drive the post-fault dynamics to a desired stability regime. As illustrated in Fig. 2, for a given fault-cleared state δ0 , by applying the inverse stability certificate, we can appropriately dispatch the power injections Pk to relocate the EP of the system so that the postfault dynamics can be attracted from the fault-cleared state ∗ δ0 through a sequence of EPs δ1∗ , ..., δN to the desired EP ∗ δdesired . In other words, we subsequently redispatch the power injections so that the system state converges from δ0 to δ1∗ , ∗ ∗ to δdesired . This and then, from δ1∗ to δ2∗ , and finally, from δN type of corrective actions reduces the need for prolonged load shedding and full state measurement. Also, this control method is unconventional where the operating point is relocated as desired, instead of being fixed as in the classical controls. Mathematically, we consider the following problem: • Emergency Control Design: Given a fault-cleared state ∗ δ0 and a desired stable EP δdesired , determine the feasible dispatching of power injections Pk to relocate the EPs so that the post-fault dynamics is driven from the fault-cleared state δ0 through the set of designed EPs ∗ to the desired EP δdesired . To solve this problem, we can design the first EP δ1∗ by minimizing kL† pkE,∞ over all possible power injections. The optimum power injection will result in an EP which is most far away from the stability margin |δkj | = π/2, and hence, the stability region of the first EP δ1∗ probably contains the given fault-cleared state. To design the sequence of EPs, in each step, we carry out the following tasks: ∗ ∗ • Calculate the distance R(δi−1 ) from δi−1 to the boundary ∗ ∗ of the set P, i.e., R(δi−1 ) = minδ∈∂P D(δ, δi−1 ). Noting ∗ that minimization of D(δ, δi−1 ) over the boundary of the set P is a convex problem with a quadratic objective function and linear constraints. Hence, we can quickly ∗ obtain R(δi−1 ). ∗ ∗ • Determine the set B(δi−1 ) and the set A(δi−1 ). ∗ • The next EP δi will be chosen as the intersection of ∗ the boundary of the set A(δi−1 ) and the line segment ∗ ∗ connecting δi−1 and δdesired . • The power injections Pk that we have to redispatch will P (i) be determined by Pk = j∈Nk akj sin δi∗kj for all k. 0.05 G δ1 (t) δ2 (t) δ3 (t) δ4 (t) δ5 (t) δ6 (t) δ7 (t) δ8 (t) δ9 (t) G 0 2 8 7 9 3 Angles (rad) −0.05 5 6 4 −0.1 −0.15 −0.2 −0.25 1 G −0.3 Fig. 3. 3-generator 9-bus system with frequency-dependent dynamic loads. 2 0 1 2 3 4 5 6 time (s) Fig. 5. Stable dynamics with power injection control: Convergence of buses angles from the fault-cleared state to δ1∗ in the post-fault dynamics 1 δ 1(t) δ 2(t) δ 3(t) δ 4(t) δ 5(t) δ 6(t) δ 7(t) δ 8(t) δ 9(t) −2 −3 −4 −5 −6 −7 0 0.5 1 1.5 2 2.5 3 3.5 7 D 2 (t) 6 Distance to δ2∗ Angles (rad) 0 −1 5 4 3 2 1 0 0 4 1 2 3 4 5 This power dispatch will place the new EP at δi∗ which is in the inverse stability region of the previous EP ∗ δi−1 . Therefore, the controlled post-fault dynamics will ∗ converge from δi−1 to δi∗ . This procedure strictly reduces the distance from EP to ∗ (it can be proved that there exists a constant d > 0 so δdesired that such distance reduces at least d in each step). Hence, after ∗ will be sufficiently near the desired EP some steps, the EP δN ∗ δdesired so that the convergence of the system state to the ∗ desired EP δdesired will be guaranteed. Node 1 2 3 4 5 6 7 8 9 V (p.u.) 1.0284 1.0085 0.9522 1.0627 1.0707 1.0749 1.0490 1.0579 1.0521 TABLE I Pk (p.u.) 3.6466 4.5735 3.8173 -3.4771 -3.5798 -3.3112 -0.5639 -0.5000 -0.6054 B US VOLTAGES , MECHANICAL INPUTS , AND STATIC LOADS . To illustrate that this control works well in stabilizing some possibly unstable fault-cleared state δ0 , we consider the 3machine 9-bus system with 3 generator buses and 6 frequencydependent load buses as in Fig. 3. The susceptances of the transmission lines are as follows: B14 = 17.3611p.u., B27 = 16.0000p.u., B39 = 17.0648p.u., B45 = 11.7647p.u., B57 = 6.2112p.u., B64 = 10.8696p.u., B78 = 13.8889p.u., B89 = 9.9206p.u., B96 = 5.8824p.u. The parameters for generators are: m1 = 0.1254, m2 = 0.034, m3 = 0.016, d1 = 0.0627, d2 = 0.017, d3 = 0.008. For simplicity, we take dk = 0.05, k = 4 . . . , 9. Assume that the fault trips the line between 7 8 9 10 Fig. 6. Effect of power dispatching control: the convergence of the distance D2 (t) to 0. Here, the Euclid distance qP D2 (t) between a state δ and the second 9 ∗ 2 EP δ2∗ is defined as D2 (t) = i=2 (δi1 (t) − δ2i1 ) . 0.5 Ddesired(t) 0.4 ∗ Distance to δdesired Fig. 4. Unstable post-fault dynamics when there is no controls: |δ45 | and |δ57 | evolve to 2π, triggering the tripping of lines {4, 5} and {5, 7}. 6 time (s) time (s) 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 time (s) Fig. 7. Autonomous dynamics when we switch the power injections to the desired values: the convergence of the distance Ddesired (t) to 0. Here, ∗ the distance Ddesired (t) between qP a state δ and the desired EP δdesired 9 ∗ 2 is defined as Ddesired (t) = i=2 (δi1 (t) − δdesired ) . i1 buses 5 and 7, and make the power injections to fluctuate. When the fault is cleared this line is re-closed. We also assume the fluctuation of the generation (probably due to renewables) and load so that the voltages Vk and power injections Pk of the ∗ post-fault dynamics are given in Tab. I. The stable EP δdesired is calculated as [−0.1629 0.4416 0.3623 −0.3563 −0.3608 − 0.3651 0.1680 0.1362 0.1371]> . However, the fault-cleared state, with angles [0.025 − 0.023 0.041 0.012 − 2.917 − 0.004 0.907 0.021 0.023]> and generators angular velocity [−0.016 − 0.021 0.014]> , is outside the set P. It can be seen from Fig. 4 that the uncontrolled post-fault dynamics is not stable since |δ45 | and |δ57 | quickly evolve from initial values to 2π, which will activate the protective devices to trip the lines. Using CVX software [13] to minimize kL† pkE,∞ , we obtain the new power injections at buses 1-6 as follows: P1 = 0.5890, P2 = 0.5930, P3 = 0.5989, P4 = −0.0333, P5 = −0.0617, and P6 = −0.0165. Accord- ingly, the minimum value of kL† pkE,∞ = 0.0350 < sin(π/89). Hence, the first EP obtained from equation (2) will be in the set defined by the inequalities |δkj | ≤ π/89, ∀{k, j} ∈ E, and can be approximated by δ1∗ ≈ L† p = [0.0581 0.0042 0.0070 0.0271 0.0042 0.0070 − 0.0308 − 0.0486 − 0.0281]> . The simulation results confirm that the post-fault dynamics is made stable by applying the optimum power injection control, as showed in Fig. 5. Using the above procedure, after one step, we can find that ∗ δ2∗ = 0.9259δdesired + 0.0741δ1∗ is the intersection of the set ∗ A(δ1∗ ) and the line segment connecting δ1∗ and δdesired . This EP is inside the inverse stability region of δ1∗ , and hence the system state will converge from δ1∗ to δ2∗ when we do the (2) power dispatching Pk corresponding to δ2∗ . On the other ∗ hand, δ2∗ is very near the desired EP δdesired and it is easy to ∗ check that δdesired is in the inverse stability region of δ2∗ , and thus the system state will converge from δ2∗ to the desired ∗ EP δdesired . Such convergence of the controlled post-fault dynamics is confirmed in Figs. 6-7. Note that D(δ0 , M ) ≥ R(δ0 ) as R(δ0 ) is the distance from δ0 to the boundary of the set P. This, together with (21), leads to E(M, δ ∗ ) + E(δ0 , δ ∗ ) ≥ R(δ0 )/2. From (8) and (9), we have E(δ0 , δ ∗ ) ≤ F (δ0 , δ ∗ ) < R(δ0 )/4. Hence, V. C ONCLUSIONS Electric power grids possess rich dynamical behaviours, e.g., nonlinear interaction, prohibition of global stability, and exhibition of significant uncertainties, that challenge the maintenance of their reliable operation and pose interesting questions to control and power communities. This letter characterized a surprising property termed as “inverse stability”, which was rarely investigated and poorly understood (though some related inverse problems were addressed in [14]). This new notion could change the way we think about the stability assessment problem. Instead of estimating the set of initial states leading to a given operating condition, we characterized the set of operating conditions that a power grid converges to from a given initial state under changes in power injections and lines. In addition, we briefly described three applications of the inverse stability certificate: (i) assessing the stability of renewable power systems, (ii) solving the stability-constrained OPF problem, and (iii) designing power dispatching remedial actions to recover the transient stability of power systems. Remarkably, we showed that robust stability due to the fluctuation of renewable generations can be effectively assessed, and that the stability constraint can be incorporated as a static constraint into the conventional OPF. We also illustrated a unconventional control method, in which we appropriately relocate the operating point to attract a given fault-cleared state that originally leads to an unstable dynamics. R EFERENCES VI. A PPENDIX : P ROOF OF T HEOREM 1 For each EP δ ∗ ∈ A(δ0 ), let M be the point on the boundary of the set P so that E(M, δ ∗ ) = Emin (δ ∗ ) = minδ∈∂P E(δ, δ ∗ ), as showed in Fig. 1. From (8), we have E(M, δ ∗ ) + E(δ0 , δ ∗ ) ≥ D(M, δ ∗ ) + D(δ0 , δ ∗ ) ∗ 2 ∗ 2 X (δMkj − δkj ) + (δ0kj − δkj ) =g akj 2 {k,j}∈E ≥g X {k,j}∈E akj (δMkj − δ0kj )2 D(δ0 , M ) = . 4 2 (21) E(M, δ ∗ ) > R(δ0 )/4 > E(δ0 , δ ∗ ). (22) Therefore, for any δ ∗ ∈ A(δ0 ), we have E(δ0 , δ ∗ ) < Emin (δ ∗ ). By applying the stability analysis in Section III-A, we conclude that the initial state δ0 must be inside the stability region of the EP δ ∗ and the system state will converge from the initial state δ0 to the EP δ ∗ . VII. ACKNOWLEDGEMENTS This work was supported by the MIT/Skoltech, Ministry of Education and Science of Russian Federation (Grant no. 14.615.21.0001.), and NSF under Contracts 1508666 and 1550015. [1] I. Nagel, L. Fabre, M. Pastre, F. Krummenacher, R. Cherkaoui, and M. Kayal, “High-Speed Power System Transient Stability Simulation Using Highly Dedicated Hardware,” Power Systems, IEEE Transactions on, vol. 28, no. 4, pp. 4218–4227, 2013. [2] A. Pai, Energy Function Analysis for Power System Stability, ser. Power Electronics and Power Systems. Springer US, 2012. [Online]. Available: https://books.google.com/books?id=1HDgBwAAQBAJ [3] H.-D. Chiang, Direct Methods for Stability Analysis of Electric Power Systems, ser. Theoretical Foundation, BCU Methodologies, and Applications. Hoboken, NJ, USA: John Wiley & Sons, Mar. 2011. [4] R. Davy and I. A. Hiskens, “Lyapunov functions for multi-machine power systems with dynamic loads,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 44, no. 9, pp. 796–812, 1997. [5] T. L. Vu and K. Turitsyn, “A framework for robust assessment of power grid stability and resiliency,” IEEE Transactions on Automatic Control, vol. 62, no. 3, pp. 1165–1177, March 2017. [6] A. Pizano-Martianez and C. R. Fuerte-Esquivel and D. Ruiz-Vega, “Global Transient Stability-Constrained Optimal Power Flow Using an OMIB Reference Trajectory,” IEEE Transactions on Power Systems, vol. 25, no. 1, pp. 392–403, Feb 2010. [7] T. L. Vu, S. Chatzivasileiadis, H. D. Chiang, and K. Turitsyn, “Structural emergency control paradigm,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 7, no. 3, pp. 371–382, Sept 2017. [8] A. R. Bergen and D. J. Hill, “A structure preserving model for power system stability analysis,” Power Apparatus and Systems, IEEE Transactions on, no. 1, pp. 25–35, 1981. [9] T. L. Vu, S. M. A. Araifi, M. S. E. Moursi, and K. Turitsyn, “Toward simulation-free estimation of critical clearing time,” IEEE Trans. on Power Systems, vol. 31, no. 6, pp. 4722–4731, Nov 2016. [10] S. Fiaz and D. Zonetti and R. Ortega and J.M.A. Scherpen and A.J. van der Schaft, “A port-Hamiltonian approach to power network modeling and analysis,” European Journal of Control, vol. 19, no. 6, pp. 477 – 485, 2013. [11] S. Y. Caliskan and P. Tabuada, “Compositional Transient Stability Analysis of Multimachine Power Networks,” IEEE Transactions on Control of Network Systems, vol. 1, no. 1, pp. 4–14, March 2014. [12] F. Dorfler, M. Chertkov, and F. Bullo, “Synchronization in complex oscillator networks and smart grids,” Proceedings of the National Academy of Sciences, vol. 110, no. 6, pp. 2005–2010, 2013. [13] Grant, Michael C. and Boyd, Stephen P. and Ye, Yinu, “CVX: Matlab software for disciplined convex programming (web page and software),” Available at http://cvxr.com/cvx. [14] I. A. Hiskens, “Power system modeling for inverse problems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 3, pp. 539–551, March 2004. 

arXiv:1201.3325v2 [] 14 Aug 2012 SIMPLICIAL COMPLEXES WITH RIGID DEPTH ADNAN ASLAM AND VIVIANA ENE Abstract. We extend a result of Minh and Trung [8] to get criteria for √ depth I = depth I where I is an unmixed monomial ideal of the polynomial ring S = K[x1 , . . . , xn ]. As an application we characterize all the pure simplicial complexes ∆ which have rigid depth, that is, which satisfy the con√ dition that for every unmixed monomial ideal I ⊂ S with I = I∆ one has depth(I) = depth(I∆ ). Introduction Let S = K[x1 , . . . , xn ] be the polynomial ring over a field K and I ⊂ S a monomial ideal. In [5], the authors compare the properties of I with the properties of its √ radical by using the inequality βi (I) ≥ βi ( I). In particular, from the inequality √ between the Betti numbers, one gets the inequality depth(S/I)√≤ depth(S/ I), which implies, for instance, that S/I is Cohen-Macaulay if S/ I is so. In [8], the authors presented criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. We extend their criteria to characterize the √ unmixed monomial ideals for which the equality depth(S/I) = depth(S/ I) holds. We recall that an ideal I ⊂ S is unmixed if the associated prime ideals of S/I are the minimal prime ideals of I. Let ∆ be a pure simplicial complex with the facet set denoted, as usual, by F (∆), T and let I∆ = F ∈F (∆) PF be its Stanley-Reisner ideal. For any subset F ⊂ [n], we denoted by PF the monomial prime ideal generated by √ the variables xi with i∈ / F . Let I ⊂ S be an unmixed monomial ideal such that I = I∆ and assume T that I = F ∈F (∆) IF where IF is the PF -primary component of I. Following [8], for every a ∈ Nn , a = (a1 , . . . , an ), we set xa = xa1 1 · · · xann and denote by ∆a the simplicial complex on the set [n] with the facet set F (∆a ) = {F ∈ F (∆) | xa ∈ / IF }. Moreover, for every simplicial complex Γ with F (Γ) ⊆ F (∆), we set [ \ IG }. IF \ LΓ (I) = {a ∈ Nn | xa ∈ F ∈F (∆)\F (Γ) G∈F (Γ) In Section 1, we prove the following theorem which is a natural extension of Theorem 1.6 in [8]. 1991 Mathematics Subject Classification. Primary 13C15, Secondary 13F55,13D45. Key words and phrases. Monomial ideals, Simplicial complexes, Stanley-Reisner rings, Depth. The second author was supported by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023. 1 2 ADNAN ASLAM AND VIVIANA ENE Theorem 1. Let ∆ be a pure simplicial √ complex with depth K[∆] = t. Let I ⊂ S be an unmixed monomial ideal with I = I∆ . Then the following conditions are equivalent: √ (a) depth(S/I) = depth(S/ I), (b) depth K[∆a ] ≥ t for all a ∈ Nn , (c) LΓ (I) = ∅ for every simplicial complex Γ with F (Γ) ⊆ F (∆) and depth K[Γ] < t. As a main application of the above theorem we study in Section 2 a special class of simplicial complexes. We say that a pure simplicial complex has √ rigid depth if for every unmixed monomial ideal I ⊂ S with I = I∆ one has depth(S/I) = depth(S/I∆ ). In Theorem 2.3 which generalizes [5, Theorem 3.2], we give necessary and sufficient conditions for ∆ to have rigid depth. In particular, from this characterization, it follows that if a pure simplicial complex has rigid depth over a field of characteristic 0, then it has rigid depth over any field. In the last part we discuss the behavior of rigid depth in connection to the skeletons of the simplicial complex. √ 1. Criteria for depth(S/I) = depth(S/ I) Let S = K[x1 , . . . , xn ] be the polynomial ring over a field K. Let I ⊂ S be an √ unmixed monomial ideal such that I = I∆ where ∆ is a pure simplicial complex T / F ) for with the facet set F (∆). Then I∆ = F ∈F (∆) PF , where PF = (xi | i ∈ T every F ∈ F (∆). Let I = F ∈F (∆) IF where IF is the PF -primary component of I. In order to prove the main result of this section we need to recall some facts from [8, Section 1]. For a = (a1 , . . . , an ) ∈ Zn , let Ga = {i | ai < 0}. We denote by ∆a the simplicial complex on [n] of all the sets of the form F \ Ga where Ga ⊂ F ⊂ [n] and such that F satisfies the condition xa ∈ / ISF where SF = S[x−1 | i ∈ F ]. It is i shown in [8, Section 1] that if ∆a is non-empty, then ∆a is a pure subcomplex of ∆ of dim ∆a = dim ∆ − |Ga |. For every simplicial subcomplex Γ of ∆ with F (Γ) ⊂ F (∆) we set [ \ IG }. IF \ LΓ (I) = {a ∈ Nn | xa ∈ F ∈F (∆)\F (Γ) G∈F (Γ) By [8, Lemma 1.5], we have (1) ∆a = Γ if and only if a ∈ LΓ (I). For the proof of the next theorem we also need to recall Takayama’s formula [9]. i For every degree a ∈ Zn we denote by Hm (S/I)a the a-component of the ith local cohomology module of S/I with respect to the homogeneous maximal ideal of S. For 1 ≤ j ≤ n, let ρj (I) = max{νj (u) | u is a minimal generator of I}, where by νj (u) we mean the exponent of the variable xj in u. If xj does not divide u, then we use the usual convention, νj (u) = 0. SIMPLICIAL COMPLEXES WITH RIGID DEPTH 3 Theorem 1.1 (Takayama’s formula).   dimK H̃i−|Ga |−1 (∆a , K), if Ga ∈ ∆ and i dimK Hm (S/I)a = aj < ρj (I) for 1 ≤ j ≤ n,  0, else. The next theorem is a natural extension of [8, Theorem 1.6]. Theorem 1.2. Let ∆ be a pure simplicial √ complex with depth K[∆] = t. Let I ⊂ S be an unmixed monomial ideal with I = I∆ . The following conditions are equivalent: (a) depth(S/I) = t, (b) depth K[∆a ] ≥ t for all a ∈ Nn with ∆a 6= ∅, (c) LΓ (I) = ∅ for every simplicial complex Γ with F (Γ) ⊆ F (∆) and depth K[Γ] < t. Proof. The proof of this theorem follows closely the ideas of the proof of [8, Theorem 1.6]. For the equivalence (a) ⇔ (b) we need to recall some known facts about local cohomology; see [2, Section A. 7]. For any finitely generated graded S-module i M we have depth M ≥ t if and only if Hm (M√ ) = 0 for all i < t. Therefore, in our hypothesis, and since depth(S/I) ≤ depth(S/ I) = t, we get (2) i depth(S/I) = t ⇔ Hm (S/I) = 0 for i < t. In addition, for every a ∈ Nn , we get (3) i depth(K[∆a ]) ≥ t ⇔ Hm (K[∆a ]) = 0 for i < t. For b ∈ Zn , we set Gb = {i | bi < 0} and Hb = {i | bi > 0}. By using [2, Theorem A.7.3], for every b ∈ Zn , we obtain i dimK Hm (K[∆a ])b = dimK H̃i−|Gb |−1 (linkstar Hb Gb ; K). Here we denoted by star Hb the star of Hb in ∆a , and by linkstar Hb Gb the link of Gb in the complex star Hb . We recall that if Γ is a simplicial complex and F is a face of Γ, then starΓ F = {G | F ∪G ∈ Γ} and linkΓ F = {G | F ∪G ∈ Γ and F ∩G = ∅}. Therefore, the equivalence (3) my be written depth K[∆a ] ≥ t (4) ⇔ H̃i−|Gb |−1 (linkstar Hb Gb ; K) = 0 for i < t and for every b ∈ Zn . Since linkstar Hb Gb is acyclic for Hb 6= ∅ and star Hb = ∆a if Hb = ∅, we get depth K[∆a ] ≥ t (5) ⇔ H̃i−|Gb |−1 (link∆a Gb ; K) = 0 for i < t and for every b ∈ Zn . By Takayama’s formula, the equivalence (2) may be rewritten depth(S/I) = t (6) ⇔ dimK H̃i−|Gb |−1 (∆b ; K) = 0 for i < t and for every b ∈ Zn . 4 ADNAN ASLAM AND VIVIANA ENE Now, the equivalence (a)⇔ (b) follows by relations (5) and (6) if we notice that, by the proof of (i) ⇒ (ii) in [8, Theorem 1.6], we have link∆a Gb = ∆b for any Gb ∈ ∆a . For the rest of the proof we only need to use (1). Indeed, for (b) ⇒ (c), let us assume that LΓ (I) 6= ∅ for some subcomplex Γ of ∆ with F (Γ) ⊂ F (∆) and such that depth(K[Γ]) < t. Then there exists a ∈ LΓ (I), hence Γ = ∆a . But this equality is impossible since depth(K[∆a ]) ≥ t. For (c) ⇒ (b), let us assume that there exists a ∈ Nn such that depth K[∆a ] < t. Then, for Γ = ∆a we get LΓ (I) 6= ∅, a contradiction.  Obviously, for t = dim K[∆] in the above theorem we recover Theorem 1.6 in [8]. The above theorem is especially useful in the situation when I is either an intersection of monomial prime ideal powers or an intersection of irreducible monomial ideals. The first class of ideals may be studied with completely similar arguments to those used in [8, Section 1]. In the sequel we discuss ideals which are intersections of irreducible monomial ideals. Tr Let F (∆) = {F1 , . . . , Fr } and I = i=1 IFi be an intersection of irreducible a monomial ideals, that is, for every 1 ≤ i ≤ r, IFi = (xj ij | j 6∈ Fi ) for some positive exponents aij . As a consequence √ of the above theorem, one may express the condition depth(S/I) = depth(S/ I) in terms of linear inequalities on the exponents aij . Proposition √ 1.3. The set of exponents (aij ) for which the equality depth(S/I) = depth(S/ I) holds consists of all points of positive integer coordinates in a finite union of rational cones in Rr(n−d) . Proof. Let Γ be a subcomplex of ∆ with depth(K[Γ]) < t and F (∆) \ F (Γ) = {Fi1 , . . . , Fis } where 1 ≤ i1 < · · · < is ≤ r. The condition LΓ (I) = ∅ gives s \ aiq j (xj q=1 :j∈ / Fiq ) ⊆ [ IFk . k∈{i / 1 ,...,is } This implies that the following conditions must hold ai j ai j a lcm(xj1 1 1 , xj2 2 2 , . . . , xjsis js ) ∈ [ IFk k∈{i / 1 ,...,is } for all s-tuples (j1 , j2 , . . . , js ), with jq ∈ / Fiq for 1 ≤ q ≤ s. This is equivalent to saying that for every s-tuple (j1 , j2 , . . . , js ), with jq ∈ / Fiq for 1 ≤ q ≤ s, there exists 1 ≤ q ≤ s such that aiq jq ≥ min{akjq : k 6= i1 , i2 , . . . , is , }.  In the following example we consider tetrahedral type ideals. SIMPLICIAL COMPLEXES WITH RIGID DEPTH 5 Example 1.4. Let ∆ be the 4-cycle, that is, I∆ = (x1 , x2 ) ∩ (x1 , x4 ) ∩ (x2 , x3 ) ∩ (x3 , x4 ). Note that S/I∆ is Cohen-Macaulay, hence depth(S/I∆ ) = 2. Let I = (xa1 1 , xa2 2 ) ∩ (xa1 3 , xa4 4 ) ∩ (xa2 5 , xa3 6 ) ∩ (xa3 7 , xa4 8 ). Then depth(S/I) = depth(S/I∆ ), that is, I is a Cohen-Macaulay ideal, if and only if one of the following condition holds: (1) (2) (3) (4) a3 a2 a5 a1 ≤ a1 , ≤ a5 , ≤ a2 , ≤ a3 , a2 a6 a1 a4 = a5 , = a7 , = a3 , = a8 , a7 a4 a8 a6 ≤ a6 . ≤ a8 . ≤ a4 . ≤ a7 . In order to prove the above claim, we first notice that any subcomplex Γ of ∆ which has depth(K[Γ]) < 2 corresponds to a disconnected subgraph of ∆. But ∆ has two disconnected subgraphs which correspond to the pair of disjoint edges    {1, 2}, {3, 4} and {1, 4}, {2, 3} . Let Γ be the subgraph {1, 2}, {3, 4} . Then the inequalities of the proof of Proposition 1.3 give (a1 ≤ a3 or a2 ≤ a5 ) and (a1 ≤ a3 or a7 ≤ a6 ) and (a8 ≤ a4 or a2 ≤ a5 ) and (a8 ≤ a4 or a7 ≤ a6 ), which is equivalent to (7) (a1 ≤ a3 and a8 ≤ a4 ) or (a2 ≤ a5 and a7 ≤ a6 ). Now we consider the other disconnected subgraph which corresponds to the pair  of disjoint edges {1, 4}, {2, 3} and get, similarly, (8) (a3 ≤ a1 and a5 ≤ a2 ) or (a6 ≤ a7 and a4 ≤ a8 ). By intersecting conditions (7) and (8), we get the desired relations. Note that in this example the union of the four rational cones defined by the set of the linear inequalities (1) − (4) is not a convex set. Indeed, if we take the exponent vectors a = (3, 5, 1, 3, 5, 9, 7, 9) and a′ = (1, 3, 1, 1, 7, 11, 11, 1), then the corresponding ideals are both Cohen-Macaulay. However, for the vector b = (a + a′ )/2 = (2, 4, 1, 2, 6, 10, 9, 5), the corresponding ideal is not Cohen-Macaulay. 2. Rigid depth Definition 2.1. Let ∆ be a pure simplicial complex. √ We say that ∆ has rigid depth if for every unmixed monomial ideal I ⊂ S with I = I∆ one has depth(S/I) = depth(S/I∆ ). For example, any pure simplicial complex ∆ with depth(K[∆]) = 1 has rigid depth. In this section we characterize all the pure simplicial complexes which have rigid depth. In the next theorem we will use the formula given in the following proposition for computing the depth of a Stanley-Reisner ring. We recall that the ith skeleton of a simplicial complex ∆ is defined as ∆(i) = {F ∈ ∆ | dim F ≤ i}. Proposition 2.2. [6] Let ∆ be a simplicial complex of dimension d − 1. Then: depth(K[∆]) = max{i | ∆(i) is Cohen-Macaulay} + 1. 6 ADNAN ASLAM AND VIVIANA ENE The following theorem generalizes [5, Theorem 3.2]. Theorem 2.3. Let ∆ be a pure simplicial complex with depth(K[∆]) = t and T I∆ = F ∈F (∆) PF . The following statements are equivalent: (a) ∆ has rigid depth. T (b) depth(S/I) = t for every ideal I = F ∈F (∆) IF where IF are irreducible √ monomial ideals with IF = PF for all F ∈ F (∆). T (c) depth(S/I) = t for every ideal I = F ∈F (∆) PFmF where mF are positive integers. (d) depth(K[Γ]) ≥ t for every subcomplex Γ of ∆ with F (Γ) ⊂ F (∆). (e) For every subcomplex Γ of ∆ with F (Γ) ⊂ F (∆), the skeleton Γ(t−1) is Cohen-Macaulay. (f) Let F (∆) = {F1 , . . . , Fr }. Then, for every 1 ≤ k ≤ min{r, t} and for any indices 1 ≤ i1 < · · · < ik ≤ r, we have |Fi1 ∩ · · · ∩ Fik | ≥ t − k + 1. Proof. (a) ⇒ (b) and (a) ⇒ (c) are trivial. (b) ⇒ (d): Let Γ be a subcompex of ∆ with F (Γ) ⊂ F (∆). We have to show that depth(K[Γ]) ≥ t. For every F ∈ F (Γ), let IF = (x2i | i ∈ / F ), and for every T F ∈ F (∆) \ F (Γ) let IF = PF = (xi | i ∈ / F ). Let I = F ∈F (∆) IF . By assumption, depth(S/I) = t. Let S ′ ⊂ K[x1 , . . . , xn , y1 , . . . , yn ] be the polynomial ring over K in all the variables which are needed for the polarization of I, and let I p ⊂ S ′ be T the polarization of I. We have I p = F ∈F (∆) IFp , where  (xi yi | i ∈ / F ), if F ∈ F (Γ), p IF = PF , if F ∈ F (∆) \ F (Γ). Then proj dim(S ′ /I p ) = proj dim(S/I). Let N be the multiplicative set generated T p / F ) and by all the variables xi . Then IN = F ∈F (Γ) (yi | i ∈ proj dim(S ′ /I p )N ≤ proj dim(S ′ /I p ) = proj dim(S/I). This inequality implies that depth(K[Γ]) ≥ depth(S/I) = t. (d) ⇔ (e) follows immediately by applying the criterion given in Proposition 2.2. (d) ⇒ (f): We proceed by induction on k. The initial inductive step is trivial. Let k > 1 and assume that |Fi1 ∩ · · · ∩ Fiℓ | ≥ t − ℓ + 1 for 1 ≤ ℓ < k and for any 1 ≤ i1 < · · · < iℓ ≤ r. Obviously, it is enough to show that |F1 ∩ · · ·∩ Fk | ≥ t − k + 1. By [3, Theorem 1.1], we have the following exact sequence of S-modules: (9) 0 → Tk S i=1 PFi → k M S → P Fi i=1 M 1≤i<j≤k S S → ··· → → 0. PFi + PFj PF1 + · · · + PFk Tk By assumption, depth(S/ i=1 PFi ) ≥ t. We decompose the above sequence in k − 1 short exact sequences as follows: 0 → Tk S i=1 0 → U1 → PFi M → 1≤i<j≤k k M S → U1 → 0, P i=1 Fi S → U2 → 0, PFi + PFj SIMPLICIAL COMPLEXES WITH RIGID DEPTH 7 .. . 0 → Uk−2 → M 1≤j1 <···<jk−1 ≤k PFj1 S S → 0. → + · · · PFjk−1 PF1 + · · · + PFk Note that, for all ℓ and any 1 ≤ j1 < · · · < jℓ ≤ k, we have PFj1 + · · · + PFjℓ = PFj1 ∩···∩Fjℓ . In particular, S/(PFj1 + · · · + PFjℓ ) is Cohen-Macaulay of depth equal to |Fj1 ∩ · · · ∩ Fjℓ |. Therefore, M depth( S/(PFj1 + · · · PFjℓ )) ≥ t − ℓ + 1 1≤j1 <···<jℓ ≤k for every 1 ≤ ℓ < k and any 1 ≤ j1 < · · · < jℓ ≤ k. Now, by using the inductive hypothesis and by applying Depth Lemma in the first k − 2 above short exact sequences from top to bottom, step by step, we obtain depth(U1 ) ≥ t−1, depth(U2 ) ≥ t − 2, . . . , depth(Uk−2 ) ≥ t − k + 2. Finally, by applying Depth Lemma in the last short exact sequence, since the depth of the middle term is ≥ t − k + 2, we get depth(S/(PF1 + · · · + PFk )) = |F1 ∩ · · · ∩ Fk | ≥ t − k + 1. (f)⇒(d): Let Γ be a subcomplex of ∆ with F (Γ) = {Fj1 , . . . , Fjk } ⊂ F (∆). We have to show that depth(K[Γ]) ≥ t. We may obviously assume that k < r and the facets of Γ are F1 , . . . , Fk . If k ≤ t, then we use the short exact sequences derived from (9) in the proof of (d) ⇒ (f) and, by applying successively Depth Lemma from bottom to the top, we get, step by step, depth(Uk−2 ) ≥ t − k + 2, . . . , depth(U2 ) ≥ t− 2, depth(U1 ) ≥ t− 1, and, finally, from the first exact sequence, depth(K[Γ]) ≥ t. If t < k, we use only the first t short exact sequences, that is, we stop at M S → Ut → 0. 0 → Ut−1 → PFj1 + · · · + PFjt 1≤j1 <···<jt ≤k Since the middle term in this short exact sequence has depth ≥ 1, we get depth(Ut−1 ) ≥ 1. Next, by using the same arguments as before, we get depth(Ut−2 ) ≥ 2, . . . , depth(U1 ) ≥ t − 1, and, finally, depth(K[∆]) ≥ t, as desired. The implication (d) ⇒ (a) follows by Theorem 1.2. Finally, the implication (c) ⇒ (e) follows similarly to the proof of Corollary 1.9 in [8].  In order to state the first consequence of the above theorem, we need to know the behavior of the depth of a Stanley-Reisner ring over a field when passing from characteristic 0 to characteristic p > 0. We show in the next lemma that the Betti numbers of the Stanley-Reisner ring can only go up when passing from characteristic 0 to a positive characteristic which, in particular, implies that the depth does not increase. This result is certainly known. However we include here its proof since we could not find any precise reference. The argument of the proof was communicated to the second author by Ezra Miller. Lemma 2.4. Let ∆ be a simplicial complex on the vertex set [n] and let K, L be two fields with char K = 0, char L = p > 0. Then βi (K[∆]) ≤ βi (L[∆]) for all i. 8 ADNAN ASLAM AND VIVIANA ENE Proof. Any field is flat over its prime field. Therefore, since char K = 0, we have βi (K[∆]) = βi (Q[∆]) for all i, and since char L = p, we have βi (K[∆]) = βi (Fp [∆]) for all i, where Fp is the prime field of characteristic p. In other words, the Betti numbers depend only on the characteristic of the base field. Let Zp be the local ring of the integers at the prime p. The ring Zp [X] is *local ([1, Section 1.5]) and the Stanley-Reisner ideal I∆ ⊂ Zp [X] is *homogeneous. Let F be a minimal free resolution of Zp [∆] over Zp [x1 , . . . , xn ]. Since p is a nonzerodivisor on Zp [∆], by [7, Lemma 8.27], the quotient F /pF is a minimal free resolution of Fp [∆] over Fp [x1 , . . . , xn ]. On the other hand, the localization F [p−1 ] by inverting p is a free resolution, not necessarily minimal, of Q[∆] over Q[x1 , . . . , xn ]. Since the modules in F /pF and F [p−1 ] have the same ranks, it follows that βi (Q[∆]) ≤ βi (Fp [∆]) for all i which leads to the desired inequalities.  Corollary 2.5. Let ∆ be a pure simplicial complex with rigid depth over a field of characteristic 0. Then ∆ has rigid depth over any field. Proof. Let K be a field of characteristic 0 and L a field of characteristic p > 0. The above lemma implies that proj dim K[∆] ≤ proj dim L[∆]. By AuslanderBuchsbaum formula, it follows that depth K[∆] ≥ depth L[∆]. Therefore, the desired statement follows by applying the combinatorial condition (f) of Theorem 2.3.  Example 2.6. Let ∆ be the six-vertex triangulation of the real projective plane; see [1, Section 5.3]. If char K 6= 2, then ∆ is Cohen-Macaulay over K, hence depth(K[∆]) = 2, and, by condition (f) of Theorem 2.3, it follows that ∆ does not have rigid depth over K. But if char K = 2, then depth(K[∆]) = 1, and, consequently, ∆ has rigid depth over K. The simplicial complexes with one or two facets have rigid depth. Lemma 2.7. Let ∆ be a pure simplicial complex with at most two facets. Then ∆ has rigid depth. Proof. We only need to consider the case of simplicial complexes with two facets since the other case is obvious. Let dim ∆ = d − 1 and F (∆) = {F, G}. We show that depth(K[∆]) = t if and only if |F ∩ G| = t − 1. Then the claim follows by condition (f) in Theorem 2.3. We consider the exact sequence 0 → K[∆] → (S/PF ) ⊕ (S/PG ) → S/(PF + PG ) ∼ = K[xi | i ∈ F ∩ G] → 0. As (S/PF ) ⊕ (S/PG ) and S/(PF + PG ) are Cohen-Macaulay of dimensions d and, respectively, |F ∩ G|, it follows that depth(K[∆]) = t if and only if |F ∩ G| = t − 1.  Example 2.8. Let ∆ and Γ be the simplicial complexes with F (∆) = {{1, 2, 3}, {1, 4, 5}} and F (Γ) = {{1, 2, 3}, {1, 3, 4}}. Obviously, by Lemma 2.7, ∆ is nonCohen-Macaulay of rigid depth 2, while Γ is Cohen-Macaulay of rigid depth. In the sequel we investigate whether the rigid depth property is preserved by the skeletons of the simplicial complexes with rigid depth. The next example shows that this is not the case. SIMPLICIAL COMPLEXES WITH RIGID DEPTH 9 Example 2.9. Let ∆ be the simplicial complex on the vertex set [8] with F (∆) = {F, G} where F = {1, 2, 3, 4, 5} and G = {1, 2, 6, 7, 8}. Then, by Lemma 2.7 and its proof, it follows that depth(K[∆]) = 3 and ∆ has rigid depth. Let ∆(3) be the 3-dimensional skeleton of ∆ and Γ the subcomplex of ∆(3) with the facets G1 = {1, 2, 3, 5} and G2 = {2, 6, 7, 8}. Then, again by the proof the above lemma, we get depth(K[Γ]) = 2. But depth K[∆(3) ] = 3, thus the skeleton ∆(3) of ∆ does not have rigid depth since it does not satisfy condition (d) in Theorem 2.3. However, as an application of Theorem 2.3, we prove the following Proposition 2.10. Let ∆ be a pure simplicial complex with rigid depth and let t = depth(K[∆]). If ∆(i) has rigid depth for some i ≥ t − 1, then ∆(j) has rigid depth for every j ≥ i. Proof. By [4], we know that depth(K[∆(i) ]) = t for i ≥ t − 1. It is enough to show that if ∆(i) has rigid depth for some i ≥ t − 1, then ∆(i+1) has the same property. Let Γ ⊂ ∆(i+1) be a subcomplex with F (Γ) ⊂ F (∆(i+1) ). Then Γ(i) is a subcomplex of ∆(i) and F (Γ(i) ) ⊂ F (∆(i) ). By our assumption and by using condition (e) in Theorem 2.3, it follows that Γ(t−1) is Cohen-Macaulay. Therefore, ∆(i+1) satisfies condition (e) in Theorem 2.3, which ends our proof.  Acknowledgment We thank Jürgen Herzog for helpful discussions on the subject of this paper and Ezra Miller for the proof of Lemma 2.4. We would also like to thank the referee for his valuable suggestions to improve our paper. References [1] W. Bruns, J. Herzog, Cohen-Macaulay rings, Revised Ed., Cambridge University Press, 1998. [2] J. Herzog, T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2010. [3] J. Herzog, D. Popescu, M. Vlădoiu, Stanley depth and size of a monomial ideal, to appear in Proceed. AMS. [4] J. Herzog, A. Soleyman Jahan, X. Zheng, Skeletons of monomial ideals, Math. Nachr. 283 (2010), 1403–1408. [5] J. Herzog, Y. Takayama, N. Terai, On the radical of a monomial ideal, Arch. Math. 85 (2005) 397-408. [6] T. Hibi, Quotient algebras of Stanley-Reisner rings and local cohomology, J. Algebra, 140, (1991), 336–343. [7] E. Miller, B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer, 2005. [8] N. C. Minh, N. V. Trung, Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals, Adv. Math. 226 (2011), 1285–1306. [9] Y. Takayama, Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48 (2005), 327–344. [10] R. H. Villarreal, Monomial algebras, Marcel Dekker, 2001. Abdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore, Pakistan. E-mail address: adnanaslam15@yahoo.com 10 ADNAN ASLAM AND VIVIANA ENE Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Buchaest, Romania E-mail address: vivian@univ-ovidius.ro 

Orthogonal Series Density Estimation for Complex Surveys Shangyuan Ye, Ye Liang and Ibrahim A. Ahmad arXiv:1709.06588v2 [stat.ME] 26 Sep 2017 Department of Statistics, Oklahoma State University Abstract We propose an orthogonal series density estimator for complex surveys, where samples are neither independent nor identically distributed. The proposed estimator is proved to be design-unbiased and asymptotically design-consistent. The asymptotic normality is proved under both design and combined spaces. Two data driven estimators are proposed based on the proposed oracle estimator. We show the efficiency of the proposed estimators in simulation studies. A real survey data example is provided for an illustration. Keywords: Nonparametric, asymptotic, survey sampling, orthogonal basis, HorvitzThompson estimator, mean integrated squared error. 1 Introduction Nonparametric methods are popular for density estimations. Most work in the area of nonparametric density estimation was for independent and identically distributed samples. However, both assumptions are violated if the samples are from a finite population using a complex sampling design. Bellhouse and Stafford (1999) and Buskirk (1999) proposed kernel density estimators (KDE) by incorporating sampling weights, and their asymptotic properties were studied by Buskirk and Lohr (2005). Kernel methods for clustered samples and stratified samples were studied in Breunig (2001) and Breunig (2008), respectively. One disadvantage of the KDE is that all samples are needed to evaluate the estimator. However, in some circumstances, there is a practical need to evaluate the estimator without using all samples for confidentiality or storage reasons. For example, many surveys are routinely conducted and sampling data are constantly collected. Data managers want to publish exact estimators without releasing all original data. In Section 6, we provide a real data example from Oklahoma M-SISNet, which is a routinely conducted survey on climate policies and public views. The orthogonal series estimators are useful alternatives to KDEs, without needing to release or store all samples. 1 The basic idea of the orthogonal series method is that any square integrable function f , in our case a density function, can be projected onto an orthogonal basis {ϕj }: f (x) = P∞ j=0 θj ϕj (x), where θj = Z ϕj (x)f (x)dx = E(ϕj (X)) (1) is called the jth Fourier coefficient. Some of the work in orthogonal series density estimation (OSDE) was covered in monographs by Efromovich (1999) and Tarter and Lock (1993), among others. Efromovich (2010) gave a brief introduction of this method. Walter (1994) discussed properties of different bases. Donoho et al. (1996) and Efromovich (1996) studied data driven estimators. Asymptotic properties were studied by Pinsker (1980) and Efromovich and Pinsker (1982). In this paper, we study the OSDE for samples from complex surveys. To the best of our knowledge, no previous work has been done on developing OSDE for finite populations. We propose a Horvitz-Thompson type of OSDE, incorporating sampling weights from the complex survey. We show that the proposed OSDE is design-unbiased and asymptotically design-consistent. We further prove the asymptotic normality of the proposed estimator. We compare the lower bound of minimax mean integrated squared error (MISE) with the I.I.D. case in Efromovich and Pinsker (1982). We propose two data driven estimators and show their efficiency in a simulation study. Finally, we analyze the M-SISNet survey data using the proposed estimation. All proofs to theorems and corollaries are given in the appendix. 2 Notations Consider a finite population labeled as U = {1, 2, ..., N}. A survey variable x is associated with each unit in the finite population. A subset s of size n is selected from U according to some fixed-size sampling design P̧(·). The first and second order inclusion probabilities from the sampling design P̧(·) are πi = Pr(i ∈ s) and πij = Pr(i, j ∈ s), respectively. The inverse of the first order inclusion probability defines the sampling weight di = πi−1 , ∀i ∈ s. The inference approach used in this paper for complex surveys is the combined designmodel-based approach originated in Hartley and Sielken (1975). This approach accounts for two sources of variability. The first one is from the fact that the finite population is a realization from a superpopulation, that is, the units xU = {x1 , x2 , ..., xN } are considered independent random variables with a common distribution function F , whose density function is f . The second one is from the complex sampling procedure which leads to a 2 sample x = {x1 , x2 , . . . , xn }. Denote w = {w1 , w2 , . . . , wn } design variables that determine the sampling weights. The sampling design P̧(·) is embedded within a probability space (S, J̧, PP̧ ). The expectation and variance operator with respect to the sampling design are denoted by EP̧ (·) = EP̧ (· | xU ) and VarP̧ (·) = VarP̧ (· | xU ), respectively. The superpopulation ξ, from which the finite population is realized, is embedded within a probability space (Ω, F̧, Pξ ). The sample x and the design variables w are ξ-measurable. The expectation and variance operator with respect to the model are denoted by Eξ (·) and Varξ (·), respectively. Assume that, given the design variables w, the product space, which couples the model and the design spaces, is (Ω × S, F̧ × J̧, Pξ × PP̧ ). The combined expectation and variance operators are denoted by EC (·) and VarC (·), where EC (·) = Eξ [EP̧ (· | xU )] and VarC (·) = Eξ [VarP̧ (· | xU )] + Varξ [EP̧ (· | xU )]. 3 Main Results Consider a sample s = {x1 , x2 , ..., xn } drawn from a finite population xU using some fixedsize sampling design P̧(·). Our goal is to estimate the hypothetical density function f of the superpopulation. Equation (1) implies that θj can be estimated using the Horvitz-Thompson (HT) estimator for the finite population mean θ̂j = N −1 n X di ϕj (xi ), (2) i=1 where N is the finite population size and di = πi−1 is the sampling weight for unit i. The HT estimator is a well known design unbiased estimator (Fuller, 2009). The basis {ϕj } can be Fourier, polynomial, spline, wavelet, or others. Properties of different bases are discussed in Efromovich (2010). We consider the cosine basis throughout the paper, which is defined √ as {ϕ0 = 1, ϕj = 2 cos(πjx)}, j = 1, 2, · · · , x ∈ [0, 1]. Regarding the compact support [0, 1] for the density, we adopt the argument in Wahba (1981):“it might be preferable to assume the true density has compact support and to scale the data to interior of [0, 1].” Analogous to Efromovich (1999), we propose an orthogonal series estimator in the form fˆ(x) = fˆ(x, {wj }) = 1 + ∞ X wj θ̂j ϕj (x), (3) j=1 where θ̂j is the HT estimator for the Fourier coefficient as in (2) and wj ∈ [0, 1] is a shrinking R1 coefficient. Note that θ0 = 0 f (x)dx = 1. If xU is known for all units in the finite population, 3 we can write the population estimator for f (x) as fU (x) = fU (x, {wj }) = 1 + where θU,j = N PN −1 i=1 ∞ X wj θU,j ϕj (x), (4) j=1 ϕj (xi ). The following theorems and a corollary show properties of our proposed estimator under both design and combined spaces. Theorem 1 considers unbiasedness and consistency under the design space. PP P∞ 2 πik Theorem 1 Suppose f ∈ L2 (R), δ = N −1 i=1 wi < ∞. i6=k πi πk − N < ∞ and Then, the estimator fˆ(x, {wj }) is design-unbiased and asymptotically design-consistent for fU (x, {wj }), i.e., h h i i ˆ ˆ EP̧ f (x, {wj }) = fU (x, {wj }) and ΓP̧ = VarP̧ f (x, {wj }) → 0 as N → ∞. Theorem 2 shows the asymptotic normality of the proposed estimator fˆ(x, {wj }) under the design space. Theorem 2 Suppose that all assumptions in Theorem 1 hold. As N → ∞, fˆ(x, {wj }) − fU (x, {wj }) Γ̂P̧ L P̧ −−→ N(0, 1), (5) where Γ̂P̧ = N −1 J X wj2 (1 + 2−1/2 θ̂2j + δ θ̂j2 )(1 + 2−1/2 ϕ2j (x)). j=1 We then show the asymptotic normality of the proposed estimator fˆ(x, {wj }) under the combined inference. Define a Sobolev Class of k-fold differentiable densities as F̧(k, Q) = P∞ P 2k 2 {f : f (x) = 1 + ∞ j=1 (πj) θj ≤ Q < ∞}, k ≥ 1. Note that for any j=1 θj ϕj (x), f ∈ F̧(k, Q), f is 1-periodic, f (k−1) is absolute differentiable and f (k) ∈ L2 (R). Theorem 3 Suppose that f ∈ F̧(k, Q) and all assumptions in Theorem 2 hold. Then, fˆ(x, {wj }) − f (x) LC h i −→ N(0, 1) as N → ∞, VarC fˆ(x, {wj }) (6) h i P where VarC fˆ(x, {wj }) = N −1 Jj=1 wj2 bj (1 + 2−1/2 ϕ2j (x)) and bj = 2 + 21/2 θ2j + (δ − 1)θj2 + oN (1). 4 The following corollary is a direct result of using Theorem 3 and Efromovich and Pinsker (1982). It shows the lower bound of the minimax MISE for the proposed estimator fˆ(x, {wj }) under the Sobolev class. Corollary 1 Let f ∈ F̧(k, Q) and fˆ(x, {wj }) be the estimator in Theorem 3. The lower bound of the minimax MISE, under the combined inference approach, is given by: h i R(F̧) = inf sup MISEC fˆ(x, {wj }) ≥ N −2k/(2k+1) P (k, Q, b)(1 + oN (1)), (7) {wj } f ∈F̧(k,Q) o2k/(2k+1) n k and b = 2. where P (k, Q, b) = Q1/(2k+1) π(k+1)b Remark that this lower bound is of the same form as the I.I.D. case in Efromovich and Pinsker (1982), but with b = 2 instead of b = 1. 4 Data Driven Estimators The choice of shrinking coefficients ŵj is not unique. To get a proper data driven estimator, we start with the oracle estimator (3), and then obtain ŵj by minimizing the MISE for the oracle estimator. Here, we propose two estimators: a truncated estimator and a smoothed truncated estimator, mimicking those in the I.I.D. case. The truncated estimator, denoted by fˆT , is an estimator with ŵj = 1 for j ≤ J, and ŵj = 0 for j > J. Alternatively we can write ŵj = Ij≤J . Then, only the truncation parameter J needs to be estimated. Notice that the MISE of this estimator is J h h i Z i X 2 ˆ MISEC f (x, {wj }) = VarC (θ̂j ) − θj − f 2 (x)dx. j=1 Since f 2 (x)dx is fixed and an unbiased estimator for θj2 is θ̂j2 −N −1 bj , a data-driven estimate for J can be obtained from R Jˆ = arg min J X (2N −1 b̂j − θ̂j2 ), j=1 where b̂j is the plug-in estimator of bj . In practice, the solution is obtained through a numerical search. Efromovich (1999) suggests to set the upper bound for Jˆ to be ⌊4+0.5 ln n⌋ for the search. Theoretically, the minimum of the MISE can be approximated in the following corollary. 5 Corollary 2 Let f ∈ F̧(k, Q), k > 1/2. The MISE of fˆT is minimized when J ≈ N 1/(2k+1) H1 (k, b, c), (8) and the minimum is approximately R(fˆT ) = MISEC (fˆT (x, {ŵj })) ≈ N −2k/(2k+1) H2 (k, b, c), −1/(2k+1) where H1 (k, b, c) = b and c is a constant.  2k+1 (2k+2)c −1/(2k+1) 2k/(2k+1) , H2 (k, b, c) = b (9)  2k+1 (2k+2)c −1/(2k+1) , One possible modification for fˆT is to shrink each Fourier coefficient toward zero. We call this estimator the smoothed truncated estimator, denoted by fˆS . It is constructed similarly as the truncated estimator, with the first J Fourier coefficients shrunk by multiplying the optimal smoothing coefficients wj∗, obtained from the proof of Corollary 1. Mathematically, ŵj = ŵj∗ Ij≤J , where ŵj∗ = (θ̂j2 − N −1 b̂j )/θ̂j2 is a direct plug-in estimator for wj∗ . A potential problem of the nonparametric density estimation is that the estimator may not be a valid density function. A simple modification is to define the L2 -projection of fˆT (or fˆS ) onto a class of non-negative densities, f˜T (x) = max{0, fˆT (x) − const.}, where the normalizing constant is to make f˜T integrate to 1. It has been proved that the constant always exists and is unique (Glad et al., 2003). 5 Simulation We compared our proposed estimators with the series estimator that ignores the finite population and sampling designs, through a Monte Carlo simulation study. We considered estimating density functions for three sampling designs: (1) the simple random sample without replacement (SRSWOR), (2) the stratified sampling and (3) the Poisson sampling. Note that the Poisson sampling has a random size with units independently sampled and hence violates our assumption of fixed size sampling. 1. For the SRSWOR, we considered two superpopulations: the standard normal distribution N(0, 1) and a mixture normal distribution 0.4N(−1, 0.5) + 0.6N(1, 1). 2. For the stratified sampling, we considered two superpopulations: a two-component mixture normal 0.4N(−1, 0.5) + 0.6N(1, 1) and a three-component mixture normal 6 0.3N(−1, 0.15) + 0.4N(0, 0.15) + 0.3N(1, 0.15). We designed two strata for the twocomponent mixture and three strata for the three-component mixture. A proportional stratified sampling is used. 3. For the Poisson sampling, we considered the same two superpopulations as in (1). We specified the expected sample size for the Poisson sampling to be n, and generated the first order inclusion probabilities for the Poisson sampling using the function “inclusionprobabilities” in the R package “sampling” (Till and Matei, 2016). For all cases, we considered a finite population of size N = 1, 000 drawn from each of the superpopulations. We repeated drawing the finite population for m1 = 100 times. For each of the finite population, we drew samples according to the sampling design, with increasing sample sizes: n = 20, 40, 60 and 80. The replication number for each finite population is m2 = 10, 000. The performance of estimators is measured by a Monte Carlo approximation of the MISE: Z m1 X m2 h i2 1 X f˜ij (x) − f (x) dx. MISEMC (f˜) = mm 1 2 i=1 j=1 The results of the simulation study are shown in Table 1. In general, the I.I.D. series estimator, which ignores the sampling design, performs the worst in all cases. However, it is not surprising to see that the improvement for the proposed estimators is much more in stratified sampling than in SRSWOR or Poisson sampling. It confirms the necessity of incorporating stratification sampling weights into the series estimator for a complex survey. Lastly, the smoothed truncated estimator performs better than the truncated estimator in most scenarios. 6 Oklahoma M-SISNet Survey The Oklahoma Weather, Society and Government Survey conducted by Meso-Scale Integrated Sociogeographic Network (M-SISNet) measures Oklahomans’ perceptions of weather in the state, their views on government policies and societal issues and their use of water and energy. The survey is routinely conduced at the end of each season. Until the end of 2016, 12 waves of survey data have been collected. It is desired that estimates can be obtained without constantly pulling out the original data. The sampling design has two separated phases. In Phase I, a simple random sample of size n = 1, 500 is selected from statewide households. In Phase II, a stratified oversample is selected from five special study areas: Payne County, 7 Oklahoma City County, Kiamichi County, Washita County and Canadian County. In each stratum, the sample size is fixed to be 200. The second phase can be viewed as a stratified sampling over the entire state with six strata: n1 = · · · = n5 = 200 and n6 = 0, where the sixth stratum contains households not in the five special study areas. This design with oversampling is not a typical fixed-size complex survey. The first-order inclusion probabilities are approximately πhi = nh /Nh + n/N, for i = 1, . . . , Nh and h = 1, . . . , 6. Note that for units not in the five areas, this inclusion probability is simply n/N. We presents OSDEs for two continuous variables for illustration: the monthly electricity bill and the monthly water bill. Figure 1 shows OSDEs of the two variables for all seasons in 2015. Acknowledgement This research is partially supported by National Science Foundation under Grant No. OIA1301789. Appendix Proof of Theorem 1 Proof. We first show that fˆ(x, {wj }) is design-unbiased:   ∞ h i X EP̧ fˆ(x, {wj }) = EP̧ 1 + wj θ̂j ϕj (x) j=1 = 1+ = 1+ ∞ X j=1 ∞ X wj EP̧ (θ̂j )ϕj (x) wj θU,j ϕj (x) j=1 = fU (x, {wj }). It remains to show that fˆ(x, {wj }) is asymptotically design-consistent, that is, the design-variance of fˆ(x, {wj }) approaches zero in the limit. We need the simple fact that √ ϕ2j (x) = [ 2 cos(πjx)]2 = 1 + cos(π2jx) = 1 + 2−1/2 ϕ2j (x). 8 Then, we have  ΓP̧ = VarP̧ 1 + = ∞ X ∞ X j=1  wj θ̂j ϕj (x) wj2 ϕ2j (x)VarP̧ (θ̂j ) j=1 = ∞ X wj2 j=1 h # " n i X di ϕj (xi ) , 1 + 2−1/2 ϕ2j (x) N −2 VarP̧ i=1 and VarP̧ " n X i=1 # di ϕj (xi ) = VarP̧ = EP̧ = N X i=1 − = " " N X i=1 Ii di ϕj (xi ) i=1 N X #2 Ii di ϕj (xi ) i=1 − ( EP̧   EP̧ (Ii2 )d2i EP̧ ϕ2j (xi ) + (N X N X # EP̧ (Ii )di EP̧ [ϕj (xi )] i=1 " N X #)2 Ii di ϕj (xi ) i=1 XX πik di dk EP̧ [ϕj (xi )] EP̧ [ϕk (xk )] i6=k )2 h i XX π ik 2 2 EP̧ 1 + 2−1/2 ϕ2j (xi ) + θ − N 2 θU,j πi πk U,j i6=k 2 = N (1 + 2−1/2 θU,2j + δθU,j ) ≤ N M, 2 ≤ M < ∞ for every j. where 1 + 2−1/2 θU,2j + δθU,j P 2 Hence, ΓP̧ ≤ N −1 M ∞ j=1 wj → 0 as N → ∞. Proof of Theorem 2 Proof. By the definition of θ̂j and θU,j , we have fˆ(x, {wj }) = 1 + = 1+ ∞ X wj θ̂j ϕj (x) j=1 N X i=1 9 Ii di ∞ X j=1 wj ϕj (x)ϕj (xi ), and  E Ii di ∞ X j=1  wj ϕj (x)ϕj (xi ) = = ∞ X wj ϕj (x)E [ϕj (xi )] j=1 ∞ X wj θU,j ϕj (x). j=1 Also, from the proof of Theorem 1, we have   ∞ ∞ X X 2 wj2 (1 + 2−1/2 θU,2j + δθU,j ) wj ϕj (x)ϕj (xi ) = Var Ii di j=1 j=1 ≤ B ∞ X j=1 wj2 < ∞ by assumption. Therefore, by the Lindeberg-Lévy central limit theorem, we have fˆ(x, {wj }) − fU (x, {wj }) LP̧ −−→ N (0, 1). ΓP̧ (10) It remains to show that Γ̂P̧ is consistent for ΓP̧ under design, or equivalently, P P̧ |Γ̂P̧ − ΓP̧ | −−→ 0, as n → N. (11) Condition (11) can be proved by using the facts that θ̂j is design unbiased and E(θ̂j2 ) = θj2 + Var(θ̂j ) → θj2 as n → N . Then, Theorem 2 is proved by using the equations (10) and (11) in conjunction with Slutsky’s theorem. Proof of Theorem 3 Proof. Since fU (x, {wj }) is the standard OSDE from an I.I.D. sample which is the finite population, then fU (x, {wj }) − f (x) Lξ −→ N (0, 1). Varξ [fU (x, {wj })] 10 (12) The asymptotic distribution of the I.I.D. OSDE under Sobolev class is obtained from Efromovich (1999), Chapter 7. Also, J h i h i X VarC wj θ̂j ϕj (x) VarC fˆ(x, {wj }) = j=1 = J X wj2 (1 + 2−1/2 ϕ2j (x))VarC (θˆj ) (13) j=1 Next, we calculate the variance of θ̂j by using Theorem 1: h i h i VarC (θ̂j ) = Eξ VarP̧ (θ̂j ) + Varξ EP̧ (θ̂j ) h i 2 ) + Varξ (θU,j ) = Eξ N −1 (1 + 2−1/2 θU,2j + δθU,j h i 2 ) + Varξ (θU,j ) = N −1 1 + 2−1/2 θ2j + δEξ (θU,j (14) 2 ) and Var (θ Then, we evaluate Eξ (θU,j ξ U,j ) separately. Based on a standard result in the I.I.D. case, we have Varξ (θU,j ) = N −1 (1 + 2−1/2 θ2j − θj2 ) (15) and 2 Eξ (θU,j ) = E2ξ (θU,j ) + Varξ (θU,j ) = N −1 (1 + 2−1/2 θ2j − θj2 ) + θj2 . Then, plug equations (15) and (16) into (14), we have h i VarC (θ̂j ) = N −1 2 + 21/2 θ2j + (δ − 1)θj2 + oN (1) = N −1 bj . (16) (17) Hence, plug (17) into (13) we can get the variance of fˆ under the combined inference approach. Finally, apply Theorem 5.1 in Bleuer and Kratina (1999), Theorem 3 is proved. Proof of Corollary 1 Proof. The proof is similar to Efromovich and Pinsker (1982). We sketch the steps as follows. We first evaluate the linear minimax MISE for the functions in the Sobolev class defined above. That is, we optimize wj∗ ’s that minimize MISEC (fˆ). Notice that EC (θ̂j ) = Eξ [EP̧ (θ̂j )] = Eξ (θU,j ) = θj 11 implying that θ̂j is an unbiased estimator of θj . Therefore, Z  h i MISEC fˆ(x, {wj }) = EC (f − fˆ)2 ∞ n o i h X wj2 VarC (θ̂j ) + θj2 − 2wj θj2 + θj2 . = (18) j=1 A straightforward calculation yields that wj∗ = θj2 θj2 + VarC (θ̂j ) . (19) Plug equation (19) into (18), h i sup MISEC fˆ(x, {wj }) {wj } f ∈F̧(k,Q) ∞ X θj2 VarC (θ̂j ) ≥ sup , 2 f ∈F̧(k,Q) j=1 θj + VarC (θ̂j ) RL (F̧) = inf (20) where VarC (θ̂j ) is of the form (17). Plug (17) into (20), and use the Lagrange multiplier to show that the maximum of (6) is attained at θj2 = N −1 (µ/(πj)k − bj )+ , where µ is determined by the constraint obtain P∞ 2k 2 j=1 (πj) θj (21) ≤ Q. Plug equation (21) back to (20), we RL (F̧) ≥ N −2k/(2k+1) P (k, Q, b). Pinsker (1980) shows that for Sobolev ball F̧, the linear minimax risk is asymptotically equal to the minimax risk, that is, R(F̧) = RL (F̧)(1 + oN (1)). Therefore Corollary 1 is proved. Proof of Corollary 2 Proof. Let ŵj = Ij≤J . Plug equation (17) into (18), we have R(fˆT ) = N −1 J X j=1 bj + ∞ X j=J+1 θj2 ≈ N −1 bJ + 12 ∞ X j=J+1 θj2 . (22) Notice that for f ∈ F̧(k, Q). By a straightforward calculation, we have θj2 = cj −2(k+1) (Efromovich, 1999). Therefore, ∞ X j=J+1 θj2 ≈c Z ∞ j −2(k+1) dj = J c J −2k−1 . 2k + 1 (23) Plug (23) into (22) and optimize J, Corollary 2 is proved. References Bellhouse, D. and Stafford, J. (1999), ‘Density estimation from complex surveys’, Statistica Sinica 9, 407–424. Bleuer, S. and Kratina, I. (1999), ‘On the two-phase framework for joint model and designbased inference’, The Annals of Statistics 33, 2789–2810. Breunig, R. (2001), ‘Density estimation for clustered data’, Econometric Reviews 20, 353– 367. Breunig, R. (2008), ‘Nonparametric density estimation for stratified samples’, Statistics and Probability Letters 78, 2194–2200. Buskirk, T. (1999), Using nonparametric methods for density estimation with complex survey data, Technical report, PhD thesis, Department of Mathematics, Arizona State University. Buskirk, T. and Lohr, S. (2005), ‘Asymptotic properties of kernel density estimation with complex survey data’, Journal of Statistical Planning and Inference 128, 165–190. Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1996), ‘Density estimation by wavelet thresholding’, Annals of Statistics 24, 508–539. Efromovich, S. (1996), ‘Adaptive orthogonal series density estimation for small samples’, Computational Statistics and Data Analysis 22, 599–617. Efromovich, S. (1999), Nonparametric Curve Estimation: Methods, Theorey and Applications, New York: Springer. Efromovich, S. (2010), ‘Orthogonal series density estimation’, WIREs Comp Stat 2, 467–476. 13 Efromovich, S. and Pinsker, M. (1982), ‘Estimation of square-integrable probability density of a random variable’, Problems of Information Transmission 18, 19–38. Fuller, W. (2009), Sampling Statistics, Wiley, New York. Glad, I., Hjort, N. and Ushakov, N. (2003), ‘Correction of density estimators that are not densities’, Scandinavian Journal of Statistics 30, 415–427. Hartley, H. and Sielken, R. (1975), ‘A super-population viewpoint for finite population sampling’, Biometrics 31, 411–422. Pinsker, M. (1980), ‘Optimal filtration of square-integrable signals in Gaussian noise’, Problems Inform. Transmission 16, 53–68. Tarter, M. and Lock, M. (1993), Model-Free Curve Estimation, New York: Chapman and Hall. Till, Y. and Matei, A. (2016), sampling: Survey Sampling. R package version 2.8. URL: https://CRAN.R-project.org/package=sampling Wahba, G. (1981), ‘Data-based optimal smoothing of orthogonal series density estimates’, The Annals of Statistics 9, 146–156. Walter, G. (1994), Wavelets and other Orthogonal Systems with Applications, London: CRC Press. 14 Table 1: Monte Carlo approximation of MISE for three sampling designs and two superpopulations. The finite population size is N = 1, 000. The replication size of the finite population is m1 = 100, and the replication size of the sample is m2 = 10, 000. Three estimators are compared: the truncated estimator, the smoothed estimator and the series estimator ignoring finite population and sampling design (I.I.D.). SRSWOR n 20 40 60 80 Standard Normal Mixture Normal Truncated Smoothed I.I.D. Truncated Smoothed 0.0232 0.0220 0.0290 0.0498 0.0480 0.0150 0.0140 0.0157 0.0311 0.0318 0.0116 0.0109 0.0121 0.0226 0.0234 0.0094 0.0089 0.0100 0.0173 0.0180 I.I.D. 0.0535 0.0388 0.0335 0.0219 n 20 40 60 80 Poisson Sampling Standard Normal Mixture Normal Truncated Smoothed I.I.D. Truncated Smoothed 0.0497 0.0481 0.0527 0.0580 0.0442 0.0281 0.0270 0.0392 0.0344 0.0294 0.0241 0.0229 0.0237 0.0283 0.0280 0.0201 0.0190 0.0211 0.0235 0.0234 I.I.D. 0.0705 0.0399 0.0322 0.0285 n 20 40 60 80 Stratified Sampling Two Strata Three Strata Truncated Smoothed I.I.D. Truncated Smoothed 0.0415 0.0409 0.0739 0.2847 0.2826 0.0231 0.0230 0.0688 0.2731 0.2718 0.0181 0.0180 0.0672 0.0426 0.0419 0.0142 0.0142 0.0675 0.0412 0.0406 I.I.D. 0.3106 0.3309 0.1132 0.1175 15 0.020 (a) Water Bill 0.000 0.005 Density 0.010 0.015 Winter Spring Summer Fall 0 50 100 150 200 Amount (in dollars) 250 300 (b) Electricity Bill 0.000 Density 0.004 0.008 Winter Spring Summer Fall 0 50 100 150 200 Amount (in dollars) 250 300 Figure 1: OSDEs of the electricity bill and the water bill for seasonal waves in 2015. 16 

A Parametric MPC Approach to Balancing the Cost of Abstraction for Differential-Drive Mobile Robots arXiv:1802.07199v1 [cs.RO] 20 Feb 2018 Paul Glotfelter and Magnus Egerstedt Abstract— When designing control strategies for differentialdrive mobile robots, one standard tool is the consideration of a point at a fixed distance along a line orthogonal to the wheel axis instead of the full pose of the vehicle. This abstraction supports replacing the non-holonomic, three-state unicycle model with a much simpler two-state single-integrator model (i.e., a velocitycontrolled point). Yet this transformation comes at a performance cost, through the robot’s precision and maneuverability. This work contains derivations for expressions of these precision and maneuverability costs in terms of the transformation’s parameters. Furthermore, these costs show that only selecting the parameter once over the course of an application may cause an undue loss of precision. Model Predictive Control (MPC) represents one such method to ameliorate this condition. However, MPC typically realizes a control signal, rather than a parameter, so this work also proposes a Parametric Model Predictive Control (PMPC) method for parameter and sampling horizon optimization. Experimental results are presented that demonstrate the effects of the parameterization on the deployment of algorithms developed for the single-integrator model on actual differential-drive mobile robots. I. I NTRODUCTION Models are always abstractions in that they capture some pertinent aspects of the system under consideration whereas they neglect others. But models only have value inasmuch as they allow for valid predictions or as generators of design strategies. For example, in a significant portion of the many recent, multi-agent robotics algorithms for achieving coordinated objectives, single-integrator models are employed (e.g., [1], [2], [3], [4]). Arguably, such simple models have enabled complex control strategies to be developed, yet, at the end of the day, they have to be deployed on actual physical robots. This paper formally investigates how to strike a balance between performance and maneuverability when mapping single-integrator controllers onto differentialdrive mobile robots. Due to the single-integrator model’s prevalence as a design tool, a number of methods have been developed for mapping from single-integrator models to more complex, non-holonomic models. For example, the authors of [5] achieve a map from single integrator to unicycle by leveraging a control structure introduced in [6]. However, this map does not come with formal guarantees about the degree to which the unicycle system approximates the singleintegrator system. One effective solution to this problem is to utilize a so-called Near-Identity Diffeomorphism (NID) This research was sponsored by Grants No. 1531195 from the U.S. National Science Foundation. The authors are with the Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA 30332, USA, {paul.glotfelter,magnus}@gatech.edu. between single-integrator and unicycle systems, as in [7], [8], where the basic idea is to perturb the original system ever-so-slightly (the near-identity part) and then show that there exists a diffeomorphism between a lower-dimensional version of the perturbed system’s dynamics and the singleintegrator dynamics. As the size of the perturbation is given as a design parameter, a bound on how far the original system may deviate from the single-integrator system follows automatically. A concept similar to NIDs from single-integrator to unicycle dynamics appears in the literature in different formats. For example, [9] utilizes this technique from a kinematics viewpoint to stabilize a differential-drive-like system. This "look-ahead" technique also arises in feedback linearization methods as a mathematical tool to ensure that the differentialdrive system is feedback linearizable (e.g., [10], [11]). This paper utilizes the ideas in [7], [8] to show that the NID incurs an abstraction cost, in terms of precision and maneuverability, that is based on the physical geometry of the differential-drive robots; in particular, the precision cost focuses on increasing the degree to which the singleintegrator system matches the unicycle-modeled system, and the maneuverability cost utilizes physical properties of the differential-drive systems to limit the maneuverability requirements imposed by the transformation. By striking a balance between these two costs, a one-parameter family of abstractions arises. However, the maneuverability cost shows that only selecting the parameter once over the course of an experiment may cause a loss of precision. A potential solution to this issue is to repeatedly optimize the parameter based on the system’s model and a suitable cost metric. Model Predictive Control (MPC) represents one such method. In particular, MPC approaches solve an optimal control problem over a time interval, utilize a portion of the controller, and re-solve the problem over the next time interval, effectively producing a state- and time-based controller. The authors of [12], [13] produce such a Parametric Model Predictive Control (PMPC) formulation. However, this formulation does not permit the cost metric to influence the time interval, which has practical performance implications. Using the formulated precision and maneuverability costs, this work formulates an appropriate PMPC cost metric and extends the work in [12], [13] to integrate a sampling horizon cost directly into the PMPC program. This paper is organized as follows: Sec. II presents the system of interest and introduces the inherent trade-off contained in the NID. Sec. III discusses the PMPC formulation. Sec. IV formulates the cost functions that allow a balanced selection of the NID’s parameters, with respect to the generated cost functions. To demonstrate and verify the main results of this work, Sec. V shows data from simulations and physical experiments, with Sec. VI concluding the paper. II. F ROM U NICYCLES TO S INGLE I NTEGRATORS This article uses the following mathematical notation. The expression k · k is the usual Euclidean norm. The symbol ∂x f (x) represents the partial derivative of the function f : Rn → Rm with respect to the variable x, assuming the convention that ∂x f (x) ∈ Rm×n . The symbol R≥0 refers to the real numbers that are greater than or equal to zero. As the focus of the paper is effective abstractions for controlling differential-drive robots, this section establishes the Near-Identity Diffeomorphism (NID) that provides a relationship between single-integrator and unicycle models. That is, systems whose pose is given by planar positions T x̄ = [x1 x2 ] and orientations θ, with the full state given  T T T = [x1 x2 θ] . The associated unicycle by x = x̄ θ dynamics are given by (dropping the dependence on time t)    R(θ)e1 0 v ẋ = , (1) 0 1 ω where the control inputs v, ω ∈ R are the linear and rotational velocities, respectively, 0 is a zero-vector of the appropriate dimension, and    T cos(θ) − sin(θ) e1 = 1 0 , R(θ) = . sin(θ) cos(θ) Letting  ux = v ω T be the collective control input to the unicycle-modeled agent, the objective becomes to turn this model into a singleintegrator model. To this end, we here recall the developments in [7]. Let xsi ∈ R2 be given by xsi = Φ(x, l) = x̄ + lR(θ)e1 , (2) where l ∈ (0, ∞) is a constant. The map Φ(x, l) is, in fact, the NID, as defined in [7]. Geometrically, the point xsi is simply given by a point at a distance l directly in front of the unicycle with pose x. Now, assume that the dynamics of xsi are given by a controller ẋsi = usi , where usi ∈ R2 is continuously differentiable, and compare this system to the time-derivative of (2), which yields   cos(θ) −l sin(θ) ẋsi = usi = ux = Rl (θ)ux . (3) sin(θ) l cos(θ) Note that the NID maps from three degrees of freedom to two degrees of freedom. As a consequence, the resulting unicycle controller cannot explicitly affect the orientation θ of the unicycle model. By [7], Rl (θ) is invertible, yielding a relationship between usi and ux . Consequently, (3) allows the transformation of linear, single-integrator algorithms into algorithms in terms of the non-linear, unicycle dynamics. Note that in this paper, which is different from [7], we let l˙ = 0 over the PMPC time intervals (i.e., l is a constant value). The unicycle model in (1) is not directly realizable on a differential-drive mobile robot. However, the relationship between the control inputs to the unicycle model and the differential-drive model is given by rw rw (ωr + ωl ), ω = (ωr − ωl ), (4) v= 2 lw where ωr and ωl are the right and left wheel velocities, respectively. The wheel radius rw and base length lw encode the geometric properties of the robot. In the discussion above, the parameter l (i.e., the distance off the wheel axis to the new point) is not canonical. Moreover, it plays an important role since kx̄ − xsi k = l. (5) The above equation seems to indicate that one should simply choose l ∈ (0, ∞) to be as small as possible. However, the following sections show that small values of l induce high maneuverability costs. In order to strike a balance between precision and maneuverability, we will, for the remainder of this paper, assume that the control input to the unicycle model is given by ux = Rl (θ)−1 usi , where usi is the control input supplied by a single-integrator algorithm. Sec. IV contains the further investigation of the effects of the parameter l on the precision and maneuverability implications of the transformation in (2). III. A PARAMETRIC MPC F ORMULATION Having introduced the system of interest, this section contains a derivation of a Parametric Model Predictive Control (PMPC) method with a variable sampling interval for general, nonlinear systems. Later, Sec. V utilizes a specific case of these results. In general, MPC methods solve an optimal control problem over a time interval and use only a portion of the obtained controller (for a small amount of time) before resolving the problem, producing a timeand state-based controller. In this case, PMPC optimizes the parameters of a system. That is, this method finds the optimal, constant parameters of a system, rather than a timevarying control input, over a time interval. For clarity, this section specifies dependencies on time t. Let ẋ(t) = f (x(t), p, t), xt0 = x(t0 ), where x(t) ∈ Rn , p ∈ Rm , and f (·) is continuously differentiable in x, measurable in t. The program t0Z+∆t arg min J(p, ∆t) = p∈Rm ,∆t∈R≥0 s.t. ẋ(t) = f (x(t), p, t) x(t0 ) = xt0 , L(x(s), p, s)ds + C(∆t) t0 expresses the PMPC problem of interest, where L(·) is continuously differentiable in x and p. Note that, in this case, both ∆t and p are decision variables determined by the PMPC program. A. Optimality Conditions This section contains the derivation of the necessary, first-order optimality conditions for the PMPC formulation, realizing gradients for the proposed cost. In particular, the derivation proceeds by calculus of variations. ˜ ∆t), Proposition 1. The augmented cost derivatives ∂p J(p, ˜ ∂∆t J(p, ∆t) are ∆t) = 0. Applying the mean value theorem and taking the limit as  → 0 shows that ˜ ˜ ∆t) J(p+γ, ∆t+τ ) − J(p, = lim →0    t +∆t 0Z  ∂p L(x(s), p, s)+λ(s)T ∂p f (x(s), p, s)ds γ t0 +[∂∆t C(∆t)+L(x(t0 +∆t), p, t0 +∆t)] τ, which is linear in τ and γ, and provides the final expressions t0Z+∆t ˜ ∆t) = ∂p J(p, t0Z+∆t ˜ ∆t) = ∂p J(p, ∂p L(x(s), p, s) + λ(s)T ∂p f (x(s), p, s)ds t0 ˜ ∆t) = L(x(t0 + ∆t), p, t0 + ∆t) + ∂∆t C(∆t), ∂∆t J(p, ˜ ∆t) (i.e., J(p, ∆t) augmented where the augmented cost J(p, with the dynamics constraint) is given by ˜ ∆t) = J(p, t0Z+∆t L(x(s), p, s)+λ(s)T (f (x(s), p, s)− ẋ(s))ds+C(∆t). t0 Proof. The proof proceeds by calculus of variations. Perturb p and ∆t as p 7→ p + γ and ∆t 7→ ∆t + τ , where γ ∈ Rm , τ ∈ R. The perturbed augmented cost is ˜ J(p+γ, ∆t+τ ) = t0 +∆t+τ Z L(x(s)+η(s), p+γ, s) t0 T λ(s) (f (x(s)+η(s), p+γ, s)− ẋ(s)−η̇(s))ds+ C(∆t+τ )+o(). Performing a Taylor expansion yields that ˜ J(p+γ, ∆t+τ ) = t0 +∆t+τ Z L(x(s), p, s)+∂x L(x(s), p, s)η(s)+∂p L(x(s), p, s)γ t0 +λT (f (x(s), p, s)+∂x f (x(s), p, s)η(s) +∂p f (x(s), p, s)γ − ẋ(s)−η̇(s))ds +C(∆t) + ∂∆t C(∆t)τ +o(). The proof now proceeds with multiple steps. First, the application of integration by parts to the quantity λ(t)T η̇(t). ˜ + γ, ∆t + τ ) − Second, the subtraction of the costs J(p ˜ ∆t). Note that, to subtract the costs properly, the inteJ(p, ˜ gral in J(p+γ, t+τ ) must be broken up into two intervals: [t, t + ∆t] and [t + ∆t, t + ∆t + τ ]. Furthermore, the costate assumes the usual definition: λ̇(t) = −∂x L(x(t), p, t)T − ∂x f (x(t), p, t)T λ(t) with the boundary condition λ(t0 + ∂p L(x(s), p, s)+λ(t)T ∂p f (x(s), p, s)ds t0 ˜ ∆t) = ∂∆t C(∆t)+L(x(t0 + ∆t), p, t0 +∆t), ∂∆t J(p, completing the proof. Interestingly, both of the usual conditions for free parameters and final time still hold, and the first-order, necessary optimality conditions for candidate solutions p∗ and ∆t∗ are that ˜ ∗ , ∆t∗ ) = 0, ∂∆t J(p ˜ ∗ , ∆t∗ ) = 0. ∂p J(p Furthermore, this formulation becomes amenable to solution by numerical methods for the optimal parameters p∗ and ˜ ∆t) can also ∆t∗ . In such cases, the expression for ∂p J(p, be expressed as a costate-like variable ξ : [t0 , t0 +∆t] → Rm with dynamics ˙ = −∂p L(x(t), p, t)T − ∂p f (x(t), p, t)T λ(t) ξ(t) ξ(t0 + ∆t) = 0, where ξ(·) is defined as t0Z+∆t ∂p L(x(s), p, s)T +∂p f (x(s), p, s)T λ(s)ds. ξ(t) = t In this case, the necessary optimality condition is that ξ(t0 ) = 0. B. Numerical Methods The above expressions allow for applications of typical gradient descent methods. Many such methods could apply, and this article presents one simple method in Alg. 1. Note that this algorithm procures the decision variables over one sampling interval [t0 , t0 + ∆t]. In practice, one typically applies this algorithm repeatedly. For example, the experiments in Sec. V-C consecutively apply this algorithm to solve the PMPC problem. IV. P RECISION VS . M ANEUVERABILITY As already noted in Sec. II, the parameter l is a design parameter. This section discusses the importance and effects of selecting l and proposes precision and maneuverability costs that elucidate the selection of this parameter and its impact on the differential-drive system. These derivations influence the PMPC cost metric in Sec. V and, for comparison, an optimal, static parameterization. Algorithm 1 Gradient Descent Algorithm for PMPC 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: k←0 pk ← initial guess ∆tk ← initial guess ˜ k , ∆t) >  do ˜ k , ∆t) + ∂∆t J(p while ∂p J(p Solve forward for x(·) from xt using pk and ∆tk Solve backward for λ(·), ξ(·) using x(·) ˜ k , ∆t) and ∂∆t J(p ˜ k , ∆t) Compute gradients ∂p J(p ˜ k , ∆t)T pk+1 ← pk − γ1 ∂p J(p ˜ k , ∆t) ∆tk+1 ← ∆tk − γ2 ∂∆t J(p k ←k+1 and let θsi be the angle of the vector usi . From (3),(4) we can retrieve the magnitude of the difference in angular velocities, |ωr − ωl |, as |ωr − ωl | = = = = = A. Precision Cost Seeking to select l, we initially present a cost that incorporates the degree to which the transformed system in (2) represents the original system xsi over an arbitrary time duration T ≥ 0. As such, we model the precision cost by the averaged tracking error Z 1 T D1 (x̄, xsi ) = kx̄ − xsi k dt. (6) T 0 = = ≤ lw ω rw l T e Rl (θ)−1 usi r 2 lw T −1 e T R(−θ)R(θsi )ūsi rw 2 lw T −1 e T R(θsi − θ)ūsi rw 2     lw 1 cos(θsi − θ) − sin(θsi − θ) kusi k 0 sin(θsi − θ) cos(θsi − θ) 0 rw l    lw sin(θsi − θ) cos(θsi − θ) kusi k 0 rw l l lw kusi k sin(θsi − θ) rw l lw v̄ . rw l It immediately follows from (5) that D1 (x̄, p) can be directly written as a function of l, given by Z 1 T D1 (x̄, xsi ) = kx̄ − xsi k dt T 0 Z 1 T = l dt = l. (7) T 0 Thus, Prop. 2 yields an upper bound on the magnitude of the wheel-velocity difference This immediate result states that the smaller l is, the better the unicycle model tracks the single-integrator model. Proposition 3. Given that the control-input magnitude kusi k is upper-bounded by v̄, the magnitude of the forward velocity, |ωr + ωl |, is upper bounded by B. Maneuverability Cost In this section, we derive a geometrically-influenced maneuverability cost that models the degree to which the selection of l influences the maneuverability requirements of the unicycle-modeled system, with respect to the map defined in (2). That is, we wish to elucidate how the parameter l affects the expressions for the differential-drive agent’s forward velocity, wheel difference, and exerted control effort. To this end, we utilize the differential-drive model in (4). Initially, note that the magnitude of the wheel-velocity difference |ωr − ωl | represents a measure of the complexity of a maneuver that the differential-drive system performs. Using this definition as guidance, we state the following proposition. Proposition 2. Given that the control-input magnitude kusi k is upper-bounded by v̄, the magnitude of the wheel-velocity difference, |ωr − ωl |, is upper bounded by |ωr − ωl | ≤ |ωr + ωl | ≤ Proof. Let  e1 = 1 " 1 T −1 1 , T = 0 # 0  1 , ūsi = kusi k l T 0 2v̄ . rw T 0 and T −1 , ūsi , θsi be defined as in the proof of Prop. 2. Then, we have, through (4), that |ωr + ωl | = = = Proof. Let  e2 = 0 Prop. 3 shows a similar result for the forward velocity of the differential-drive agent. = lw v̄ . rw λ lw v̄ . rw l |ωr − ωl | ≤ = 2 v rw 2 T e Rl (θ)−1 usi rw 1 2 T −1 e T R(−θ)R(θsi )ūsi rw 1 2 [1 0] R(θsi − θ)ūsi rw   2 kusi k [cos(θsi − θ) − sin(θsi − θ)] 0 rw 2 kusi k cos(θsi − θ) rw 2v̄ ≤ . rw where α ∈ (0, 1). Now, we seek the optimal l such that (10) is minimized. That is, = l∗ = arg min D(l). (11) l So Prop. 3 reveals that the forward velocity of the differential-drive system remains independent of the selection of the parameter l. To elucidate an appropriate maneuverability cost in terms of l, define the average control effort exerted by the differential-drive system over an arbitrary time duration T >= 0 as Z 1 T |ωr − ωl | + |ωr + ωl | dt. T 0 Directly applying Props. 2,3 reveals that the above expression is bounded above by lw v̄ 2v̄ + . (8) rw l r The expression in (8) demonstrates an interesting quality of the system. As l grows large, the forward velocity dominates the control effort exerted by the differential-drive system. However, if l becomes small, then the choice of l affects the potentially exerted control effort. Thus, (8) reveals how l affects the maneuverability requirements imposed by the abstraction. The fact that we always pay the forward-velocity price, regardless of the selection of l, naturally excludes the forward velocity from the soon-to-be-formulated cost, because any selection of l results in the same cost bound; but the choice of l directly affects the cost associated with the wheel difference. Accordingly, the wheel difference must play a role in the final PMPC cost metric. With this conclusion in mind, we define the static maneuverability cost as lw v̄ D2 (l) = , (9) rw l which the static parameterization in the following section utilizes. C. An Optimal, One-Time Selection Sec. V utilizes the results in Sec. IV to formulate an appropriate cost metric for a PMPC program. To have a baseline comparison, this section formulates an optimal, one-time selection for the parameter l. That is, the selection occurs once over the experiment’s duration. This selection should strike a balance between precision and maneuverability. Eqns. (7) and (9) represent each of these facets, respectively, and introduce an inherent trade-off in selecting l. Making l smaller directly reduces the cost in (7). However, consider the relationship in (9); as l decreases, the differential-drive system accumulates a higher maneuverability cost. As such, the convex combination of (6) and (9) yields a precision and maneuverability cost in terms of l as D(l) = αD1 (l) + (1 − α)D2 (l) lw v̄ , = αl + (1 − α) rw l (10) (11) leads to Prop. 4. Proposition 4. The optimal l∗ is given by r 1 − α lw v̄ . l∗ = α rw Proof. We have that ∂ lw v̄ D(l) = α − (1 − α) ∂l rw l2 lw v̄ = α − (1 − α) . rw l2 Setting this equation equal to zero directly yields the minimizer r 1 − α lw v̄ ∗ l = . (12) α rw Note that the above result utilizes (9), which is an upper bound on the wheel velocity difference. Thus, the PMPC method should outperform this static selection, a suspicion that Sec. V investigates. D. PMPC Cost This section formulates a PMPC cost based on the analysis in Sec. IV. To increase precision, the parameter l must be minimized. However, (9) in Sec. IV-B indicates that the wheel velocity difference must be managed. Thus, precision and maneuverability are balanced with the cost L(x, l, t) = (1 − β)(ωr − ωl )2 + βl2 = (1 − β)((lw /rw )e2 Rl (θ)−1 (usi ))2 + βl2 , where β ∈ (0, 1). With this cost metric, the PMPC program becomes t0Z+∆t (1 − β)((lw /rw )e2 Rl (θ)−1 (usi ))2 + βl2 arg min l∈R,∆t∈R≥0 t0 + C(∆t)   R(θ)e1 0 s.t. ẋ = R (θ)−1 usi 0 1 l (13) x(t0 ) = xt0 , Note that the sampling cost C(∆t) and single-integrator control input usi have yet to be specified. Angular Velocity ( ) 0.1 0 -0.1 -0.2 Static PMPC -0.3 0 20 40 60 80 100 Time (s) Fig. 3: Angular velocity (ω) during the simulation. The simulation shows that the static selection (solid line) and PMPC method (dashed line) both generate similar angular velocity values. Fig. 1: The GRITSbot, which is a small, differential-drive mobile robot used in the Robotarium. This figure displays the base length and wheel radius of the GRITSbots. V. N UMERICAL R ESULTS To demonstrate the findings in Sec. IV, we conduct two separate tests: in simulation and on real hardware. The simulation portion shows the effects of a one-time parameter selection on the angular velocity versus the PMPC method. The experimental section contains the same implementation on a real, physical system: the Robotarium (www. robotarium.org). In particular, the experimental results highlight the practical differences between using a PMPC approach and a one-time selection. A. Experiment Setup This section proposes cost functions based on the results in Sec. IV and expresses the PMPC problem to be solved in simulation and on the Robotarium. Furthermore, this section also statically parameterizes the NID to provide a baseline comparison to the PMPC strategy. In this case, the particular setup involves a mobile robot tracking an ellipsoidal reference signal Parameter (l) 0.06 0.04 r(t) =   0.4 cos((1/10)t) . 0.2 sin((1/10)t) (14) For a single-integrator system, the controller usi = xsi − r + ṙ drives the single-integrator system to the reference exponentially quickly. Utilizing the transformation in Sec. IV yields the controller ux = Rl (θ)−1 (r − xsi + ṙ) = Rl (θ)−1 (r − (x̄ + lR(θ)e1 ) + ṙ). The GRITSbots of the Robotarium (shown in Fig. 1) have a wheel radius and base length of rw = 0.005 m, lw = 0.03 m. Furthermore, their maximum forward velocity is v̄ = 0.1 m/s. For this problem, we also consider the sampling cost 1 , ∆t which prevents the time horizon from becoming too small (i.e., the cost penalizes small time horizons). Substituting these values into (13), the particular PMPC problem to be solved is C(∆t) = 0.02 0 20 40 60 80 t0Z+∆t 100 Time (s) Fig. 2: Parameter (left) and sampling horizon (right) from PMPC simulation, which oscillate because of the ellipsoidal reference trajectory in (14). Due to the sharp maneuvers required, the time horizon shortens and the parameter increases on the left and right sides of the ellipse. On flatter regions, the PMPC reduces the parameter and increases the sampling time. The zoomed portion displays the discrete nature of the PMPC solution. (β − 1)((lw /rw )e2 Rl (θ)−1 (r − xsi +ṙ))2 arg min l∈R,∆t∈R≥0 t0 + βl2 ds + (1/∆t)  R(θ)e1 0 s.t. ẋ = R (θ)−1 (r − xsi + ṙ) 0 1 l  x(t0 ) = xt0 , where l and ∆t are the decision variables and β = 0.01. Both simulation and experimental results utilize Alg. 1 to l =0.030937 t =3.3119 l =0.027048 t =3.4262 l =0.043572 t =3.2757 Angular Velocity ( ) Fig. 4: Robot during the PMPC experiment. This figure shows that the parameter grows and sampling horizon shrinks when the robot must perform more complex maneuvers (i.e., on the left and right sides of the ellipse). Over the flatter portions of the ellipse, the parameter increases and sampling horizon (solid line) reduces, allowing the robot to track the reference (solid circle) more closely. with α = 0.99. This assignment to α in (12) implies that 1 l∗ = 0.078. 0.5 Note that this value of l∗ is only for the one-time selection. The PMPC method induces different parameter values every 0.033 s. 0 -0.5 Static PMPC -1 0 20 40 60 80 100 Time (s) Fig. 5: Angular velocity of robots for PMPC method (dashed line) versus static parameterization (solid line). In this case, both methods generate similar angular velocities, but the PMPC method produces better tracking. solve for the optimal parameters and time horizon online with the step-size values γ1 = 0.001, γ2 = 0.01. Each experiment initially executes Alg. 1 to termination; then, steps are performed each iteration to ensure that the current values stays close to the locally optimal solution realized by Alg. 1. In particular, each iteration takes 0.033 s, which is the Robotarium’s sampling interval. For comparison, the one-time selection method stems directly from the abstraction cost formulated in Sec. IV-C Parameter (l) 0.06 0.05 0.04 0.03 0.02 0 20 40 60 80 100 Time (s) Fig. 6: Parameter (left) and sampling horizon (right) from PMPC experiment on the Robotarium. The PMPC program reduces the sampling horizon and increases the parameter to cope with the sharp maneuvers required at the left and right sides of the ellipse. On flatter regions, the PMPC decreases the parameter and increases the time horizon, providing better reference tracking. The zoomed portion illustrates the discrete nature of the PMPC solution. B. Simulation Results This section contains the simulation results for the method described in Sec. V-A. In particular, the simulation compares the proposed PMPC method to the one-time selection process in Sec. IV-C, showing that the PMPC method can outperform the one-time selection. Fig. 3 shows the simulated angular velocities, and Fig. 2 shows the parameter and sampling horizon evolution. Both methods generate similar control inputs. However, Fig. 2 demonstrates that the PMPC method selects smaller parameter values, implying that this method provides better reference tracking. Additionally, Fig. 2 also shows that the sampling horizon shortens and the parameter increases around the left and right portions of the ellipse, because these regions require sharper maneuvers and incur a higher maneuverability cost. Furthermore, the ellipsoidal reference trajectory induces the oscillations in Fig. 2. Overall, these simulated results show that the PMPC method can outperform a static parameterization. C. Experimental Comparison This section contains the experimental results of the implementation described in Sec. V-A. The physical experiments for this paper were deployed on the Robotarium and serve to highlight the efficacy and validity of applying the PMPC approach on a real system. Additionally, the experiments display the propriety of the maneuverability cost outlined in Sec. IV-B. Figs. 5-6 display the angular velocity of the mobile robot, the sampling horizon, and the parameter selection, respectively. As in the simulated results, Fig. 5 shows that the static parameterization and PMPC method produce similar angular velocities, and Fig. 6 shows that the PMPC method is able to adaptively adjust the parameter and sampling horizon to handle variations in the reference signal. Moreover, on a physical system, the PMPC method still adjusts the time horizon and parameter to account for maneuverability requirements. For example, on the left and right sides of the ellipse, the maneuverability cost rises, because the reference turns sharply. Thus, the parameter increases and the sampling horizon decreases. Over flat portions of the ellipse, the maneuverability cost decreases, permitting the extension of the time horizon and reduction of the parameter (i.e., better tracking). That is, reductions of the maneuverability cost permit decreasing the parameter l, allowing the PMPC strategy to outperform the static parameterization. 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Online Model Estimation for Predictive Thermal Control of Buildings Peter Radecki, Member, IEEE, and Brandon Hencey, Member, IEEE  Ideally, a Building Automation System (BAS) would automatically modify set-points and load shedding based on weather, occupancy, and utility pricing predictions [8]. Every building has unique and time-varying thermal dynamics, occupancy, and heat loads which must be characterized accurately if a BAS is to apply model predictive controllers (MPC) to realize energy and monetary savings [3]. Additional considerations include: measured building data often contains low information content; engineering models contain designer’s intent instead of actual construction; and building’s usage evolves over time [9]. Unfortunately, in practice there has yet to be demonstrated a scalable, low-cost method to readily acquire these much needed accurate models of individual buildings’ unique thermal envelopes. For continuous commissioning and lifetime adaptability a low-cost scalable method to acquire control-oriented building models must: learn both the dynamics and the disturbance patterns quickly, provide stable extrapolation, be adaptable to future changes in building structure or use, and use existing available data. White-box, first-principles, forward modeling approaches are often inaccurate, not robust to changes, and take extensive engineering or research effort to build [2], [10]. Black-box approaches take up to 6-months to train and cannot be safely extrapolated [5], [6]. Recently gray-box methods have begun to show potential as a scalable option for learning control-oriented building models in some limited specific studies [11], [12], [13], [14]. We propose a multi-mode Unscented Kalman Filter (UKF) as a generalizable on-line gray-box data-driven method to learn the building’s multi-zone thermal dynamics and detect unknown time varying thermal loads. By coupling known building information and simple physics models with existing measurable building data we demonstrate how a probabilistic estimation framework can overcome shortcomings of many previously attempted specialized solutions. Our method adapts over time to continually learn both dynamics and disturbances while providing stable prediction performance. Continuing our work in [11], this paper aims to generalize our findings and method with the following contributions:  literature survey on control-oriented thermal modeling for buildings,  development of minimal parameterization for dynamics estimation,  generalized thermal disturbance pattern estimation,  multi-mode heuristic for simultaneous parameter and disturbance estimation. Abstract—This study proposes a general, scalable method to learn control-oriented thermal models of buildings that could enable wide-scale deployment of cost-effective predictive controls. An Unscented Kalman Filter augmented for parameter and disturbance estimation is shown to accurately learn and predict a building’s thermal response. Recent studies of heating, ventilating, and air conditioning (HVAC) systems have shown significant energy savings with advanced model predictive control (MPC). A scalable cost-effective method to readily acquire accurate, robust models of individual buildings’ unique thermal envelopes has historically been elusive and hindered the widespread deployment of prediction-based control systems. Continuous commissioning and lifetime performance of these thermal models requires deployment of on-line data-driven system identification and parameter estimation routines. We propose a novel gray-box approach using an Unscented Kalman Filter based on a multi-zone thermal network and validate it with EnergyPlus simulation data. The filter quickly learns parameters of a thermal network during periods of known or constrained loads and then characterizes unknown loads in order to provide accurate 24+ hour energy predictions. This study extends our initial investigation by formalizing parameter and disturbance estimation routines and demonstrating results across a year-long study. I. INTRODUCTION A. Overview Significant energy savings in buildings’ heating, ventilating, and air-conditioning (HVAC) systems could be realized with advanced control systems [1], but deployment of these control systems requires a method to readily acquire low cost models of buildings’ unique thermal envelopes [2], [3]. Previous studies have investigated several methods but generally arrived at non-scalable specialized solutions [4], [5], [6], [7]. This work was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. P. Radecki and B. Hencey were with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, 14850 USA. Currently Radecki is a Transmission Development Engineer at General Motors Powertrain in Detroit, Michigan, email: ppr27@cornell.edu. Currently Hencey is with the Air Force Research Laboratory in Dayton, Ohio, email: bhencey@gmail.com. Copyright © 2015 Peter Radecki. All Rights Reserved. 1 A comparison of UKF and EKF applied estimation techniques is included for the benefit of practicing engineers. The robustness of the UKF estimation technique learning both parameters and disturbances is demonstrated against a multi-zone high-fidelity EnergyPlus simulation in a yearlong study. A short explanation of our proposed method follows. Using a simple first order heat transfer model with multiple zones, the UKF estimates model parameters of the thermal dynamics during periods which have small or wellcharacterized thermal loads. After learning the dynamics during low disturbance periods, such as nighttime, the UKF is augmented to track unknown disturbances while continuing to improve its dynamics model. The UKF is simpler than an adaptive control technique to implement because it internally maintains a covariance quality metric which only adjusts parameter estimates if the incoming data provides new thermal information. Fig. 1 shows how the proposed UKF enables rapid deployment of advanced predictive controllers for BAS. The paper is organized as follows. First, a background section initially examines the scope of the problem and previous approaches before proposing and analyzing an extensible on-line data-driven approach. After deriving the thermal model and parameterization, we formulate and compare performance of the EKF and UKF. True utility of the UKF is then demonstrated across a year-long study. Based on data generated from our simple passive 5-zone thermal model plus a more complex passive 5-zone EnergyPlus simulated model, less than 2 weeks of training data is shown to make reliable 24-hour predictions. Based on testing and performance, a discussion of how the UKF fits into the building thermal modeling problem and an identification of areas of future research conclude the paper. Despite these significant advances, most modern buildings have realized only minimal energy savings [3]. BAS modelbased predictive controllers are rarely implemented and typically underperform [15]. Larger buildings’ successes in realizing energy savings has been generally limited to automated lighting control, changing nighttime temperature set-points, and load shedding during peak demand [15]. The data available from BAS is underutilized—generating trend plots instead of enabling intelligent energy management decision and control tasks [4]. It is not uncommon for buildings to simultaneously run cooling and heating components throughout all 12 months of the year [9]. Load shedding, reducing peak usage by duty cycling off portions of a system, is often based on some arbitrary component order and can significantly impact occupant comfort—if one portion of the system that is already at capacity sheds, it may not recover until nighttime [16]. Certain sites, such as Drury University and UC Merced have a human operator regularly check weather forecasts and vary temperature set points based on personal experience and intuition [16], [17]. These techniques are labor and expertise intensive. Usually, however, the cost of the human operator doesn’t justify the energy savings and limits widespread deployment. Optimal building thermal control is a multi-objective optimization problem involving user comfort, air quality, energy cost, smart grid demand response, and thermal dynamics, which generally cannot be performed optimally by a human operator without expertise. Realistic widespread improvement in building controls requires a scalable method to accurately learn unique thermal models. C. Building Controls and Modeling Survey Many researchers have evaluated and implemented custom one-off BAS demonstrating energy savings with MPC [18], but widespread implementation has been illusive because a scalable method to learn thermal models has not been available. Both forward and inverse modeling efforts have traditionally taken months of a researcher’s or engineer’s time to accurately generate. Given that the majority of buildings which will be in use by 2030 are currently over 10 years old, the retrofit problem is significant. A scalable method must handle new construction, existing structures, and remodeling projects and continue to function if a building is repurposed. White-box, or forward models can be generated by analyzing blueprints, building materials, and expected use patterns to generate a sophisticated simulation, but are generally not cost effective for retrofit or existing buildings. Furthermore, some studies have shown that predicted and actual energy consumption can differ by up to 40% using forward models from the design stage [10]. MPC can perform adequately given simple, accurate first or second order heat transfer models, so there is viability for inverse models [9], which use building data recorded over time to infer the dynamics. Today’s networked sensors and available computation power make data-driven models feasible. The inverse modeling paradigm can use either pure numerical methods in a black-box environment or physics, first-principles based methods in a gray-box environment. B. Building Automation Systems Background Buildings use 39% of the total US energy supply, a significant fraction of which is used to provide people with a comfortable indoor working and living environment by operating HVAC systems. It is estimated that 25% to 30% of building energy usage or around 10% of the total US energy consumption could potentially be reduced with component and controls upgrades [1], [2]. Reducing this energy consumption would also help significantly reduce CO2 generation [9]. BAS and their connected components advanced tremendously over the past few decades and have started to be widely deployed in modern buildings and retrofit into remodeled buildings. Utilizing technologies such as wireless sensing, LonTalk, and BACnet, BAS provide networked infrastructure making it easier for sensor information and control signals to be distributed throughout and between buildings. Component advancements include sensors that can detect occupants, CO2 level and light in addition to traditional temperature and humidity measurements. Furnaces, water heaters and air conditioners have all started to approach their maximum theoretical efficiency. The computational power necessary to run demand-responsive and predictive control algorithms is cheaply available. 2 Current Methodology: Up to 1 year of time per building Proposed Methodology: Weeks of time per building Analyze building plans/geometry Discover/Utilize networked sensors in building Manually forward model and simulate or perform custom study of building using recorded data. Recursive online gray‐box learning algorithms determine dynamics of building Apply advanced controls Apply advanced controls Fig. 1. Comparison of existing typical thermal modeling process left and proposed method right . Picture Credits: EnergyPlus: DOE, R. L. Smith Building center : ThermoAnalytics, and floor plan: 3 . gain wide acceptance because zone interactions affect BAS control systems [18]. In [11] we demonstrated the first published study of a scalable modeling and online estimation framework for multi-zone building states, parameters, and unmodeled dynamics. Since our study in 2012 several other researchers have validated our initial claims and highlighted new challenges. Studies using real and simulated data demonstrated individual aspects of the proposed scalable modeling and estimation framework. Massoumy in 2013 [13] demonstrated the applicability of gray-box estimation with real data collected from Michigan Tech’s new Lakeshore Center building–he used an off-line batch parameter estimation routine with online state estimates, validating the results of our EKF versus UKF comparison. Fux [12] demonstrated the relevance of 1R1C models to individual rooms or entire zones by generating accurate predictions from a simple single-zone model of an entire building using real data in an EKF. The study by Fux [12] also validated the concept of multi-mode learning and the importance of characterizing disturbances. Martincevic [14] used simulated data from IDA-ICE in a year-long study to demonstrate a 50-zone model that learned parameters without disturbances from a constrained UKF. The model had a prediction error of less than 1oC RMS error if no unmodeled disturbances were present. Other researchers demonstrated the extensibility of the KF as an online estimation framework and presented important insight. Studies showed ground coupling was not necessary in RC models for certain scenarios [8], extended an EKF for fault diagnosis and monitoring with real-data from a supermarket [20], used an Ensemble KF to constrain parameters to physically realistic values [21], and showed the applicability of RC models as a design tool using offline parameter estimation [22]. Lin [23] showed the need for meaningful inter-zone excitation in order to guarantee satisfactory information content is present in measured data. Lin’s result corresponded with our follow-up study using an UKF learned model for MPC [24]. Lin further noted that one simple plot demonstrating prediction accuracy versus Many researchers from present day to those who participated in the 1990’s ASHRAE Energy Predictor Shootouts have taken data-driven approaches, but their generated methods and models have typically been too specialized to scale to other buildings [2], [4], [5], [6]. A brief comparison of previously tried data-driven methods is provided before proposing our solution. Black-box models such as Artificial Neural Networks (ANN) are often impractical for modeling thermal dynamics in building systems because of the large amount of training data (6 months to a year) that must be analyzed before getting an accurate model [6]. Because ANN create an arbitrary representation of the system, they are sensitive to the quality of the data collected, may meld together effects from loads and dynamics, are not robust to component failures, and are not adaptable when building use and configuration patterns change [2], [18]. Pure numerical methods may have better applications where simple heuristic and physical models are impractical such as pattern recognition of occupancy, lighting, or thermal component loads. Gray-box methods, which use pre-existing knowledge of the dynamical structure, have been used off-line with genetic algorithms (GA) [5] and recursively on-line (in real-time). Off-line methods may be good for initial model acquisition but on-line methods are desirable—buildings age and change physically over time due to deterioration or reconfiguration and change temporally as occupant usage evolves. On-line methods can also be dual-purposed as process monitoring, analysis, and fault detection devices. Examination of traditional online gray-box techniques in buildings with adaptive control [2] and Extended Kalman Filters (EKF) [19] shows the need for further development. Historically, adaptive control techniques have fallen short because they require autonomous tuning of a complex forgetting factor, which requires actively monitoring the excitation level of incoming data as an information criterion used to enable or disable thermal model learning [2]. Many on-line methods [2], [4], and [5] have used single zone models for demonstrating utility, but in reality multi-zone simulations are necessary before any modeling method will 3 measured data is meaningless in that it doesn’t say anything about the model’s robustness for control. In summary, recent studies have shown RC models are applicable for thermal modeling of buildings and Kalman filtering can learn parameters for models of buildings using both simulated and real data. Our paper builds upon recent advancements with an explanation of unique parameterization and formalizes the multi-mode estimation technique with a generalized algorithm for learning any disturbance pattern. We conclude with a year-long study of simultaneous state, parameter, and disturbance estimation showing robust, meaningful prediction accuracy. Thermal radiation does play a significant role in the heating and cooling of many buildings. For the purposes of evaluating UKF parameter estimation and thermal load detection for a passive building, the radiation between surfaces and zones is linearly approximated and lumped with convection and conduction [25]. The proposed framework could readily be augmented for nonlinear radiation effects at the cost of increasing the number of associated parameters to learn. Thermal disturbances significantly affect most buildings but are often overly complex to model requiring information about building geometry and neighboring foliage [26]. Solar gain is treated as an unmodeled external disturbance. This simplification removes complexities of modeling diffuse and direct sunlight, shading, and night sky radiation temperature, and allows for simple disturbance generation in EnergyPlus by turning on or off environmental radiation transfer. The solar gain provides us with a specific periodic disturbance to estimate with patterns. In practice this technique could estimate any number of disturbances if one has some information about the disturbance frequency, intensity, or timing such as dusk and dawn times or building occupancy times. Common examples amenable to disturbance pattern estimation include occupant body-heat, equipment, computers, electrical loads, lighting, and HVAC. Using the 2-node example, a state space representation can be derived where is a vector of temperatures; A is a matrix of RC values; and is a vector of additive, independent, time-varying disturbances such as solar radiation. II. PARAMETER ESTIMATION FORMULATION A. Thermal Model A standard thermal network captures the dominant convection and conduction heat transfer modes and mass transfer occurring between zones inside and outside the building, while solar gain is treated as an unmodeled disturbance. Internal zone radiation is linearly approximated and lumped with convection and conduction [25]. The thermal network matches that commonly used in the community and provides a simple mechanism to explore simultaneous model and disturbance estimation. Subsequently developed filters use the thermal model but are not restricted to it. One could select a non-linear model incorporating HVAC dynamics and use it in the proposed estimation framework. Convection, conduction, and mass transfer heat flux (watts) into zone is contributed from the temperature differential to connected adjacent node(s) divided by the (degree/watt) plus an additive term thermal resistance (watts) representing disturbances. (Note: unless otherwise mentioned, subscripts denote zones.) 1 1 1 1 2 / and thermal capacity (joule/degree) The heat flux affects the time-based temperature rate of change . Based on [27], an n-node thermal network can be formalized by defining a simple undirected weighted graph with: nodes and temperatures ≔ 1,2, … , assigned capacitances ; edges ⊂ that connect adjacent nodes with weights ∀ , ∈ such that ∀ , ∈ that are assigned resistances. For a general thermal network with n nodes, the A matrix is 0 if , , ∉ Substituting for , the temperature rate of change of node i due to connection(s) with node(s) j and disturbance becomes / / . 1 The derived representation for temperature change due to heat transfer is mathematically analogous to voltage change due to current flow in a resistor-capacitor network. For visualization a simple 2-node example with two capacitances and one resistance is shown in Fig. 2. if ∑ , , ∈ . 3 if B. Parameterization A minimal set of independent parameters must be specified for filters to enforce the system dynamics during parameter estimation [28]. Over-parameterization causes unidentifiable parameter manifolds or extra degrees of Fig. 2. Two node example thermal network. 4 freedom and can result in violation of dynamics constraints and physics laws such as conservation of energy. In machine learning and system identification, indeterminate degrees of freedom can cause overfitting where the model learns the noise instead of the dynamics of interest. In estimation theory, parameter observability requires that the Fisher information matrix is invertible—redundant parameters or over parameterization breaks this observability criterion resulting in an unobservable subspace [29]. Efficient and reliable parameter estimation requires estimating a minimal number of parameters [28]. From Equation 2 there are only two unique parameters required to describe the A matrix despite it containing three variables—two resistances and one capacitance. The extra parameter acts as a scaling factor and can be quantified only if the heat flux q is provided in addition to the temperature histories. Without the scaling factor only a time-constant can be inferred. Suppose the time-constant for our system was 1. Then 1 and we get the plot in Fig. 3 of possible values for and , many of which are violations of physics first principles such as a negative thermal capacitance and resistance. Unfortunately, removing the negative-valued parameter space does not resolve the ambiguity in selecting the true resistance and capacitance values for the provided time constant. This ambiguity generally makes the estimation problem numerically unstable, theoretically unobservable, or practically unreliable. Rectifying the ambiguity could be done with actual heat flux information which is generally unavailable in practice, so for this study, selecting a minimal set of parameters mitigates the problem. Now we present methods based on a careful graph study to obtain a minimal parameter set for thermal network estimation. Because diagonal terms in A are linear combinations of the off-diagonal terms, parameter estimation is only performed for off-diagonals. RC products are estimated together in order to reduce the non-linearity of the estimation problem. Parameterization of trees, graphs with no cycles of which Fig. 2 is an example, with combined RC products automatically guarantees a minimal representation of the system. Unfortunately this minimal guarantee does not extend to graphs containing closed cycles. Fig. 4 is an example of a graph containing a cycle whose state space A matrix is shown. 4 We arbitrarily selected R13C1 to show that one of the six RC products is redundant and can be eliminated by multiplying and dividing the other RijCi parameters by each other around the cycle: In a graph, each cycle which uses at least one unique edge and passes through no nodes with infinite capacitance may be used to eliminate one redundant RC product from the estimation problem by multiplying and dividing around the loop. For any thermal network the total number of unique parameters should be one less than the sum of the number of resistances and capacitances. In general unique edges should be selected for elimination. Eliminating a shared edge between two cycles joins the two cycles mathematically through multiplication in the estimation routine which can negatively impact numerical stability. Selecting multiple redundant parameters to prune from a graph estimation problem should be done such that each redundant RC parameter lies on a globally unique edge for its respective cycle, and the shortest available cycle should be chosen for calculation in order to guarantee minimal parameter cross-sensitivity. Two nodes that have no shared conduction or convection are considered independent, and any edge directly connecting them is pruned from the graph to give the simplest representation. Independent ambient nodes such as external temperatures have infinite capacitance in the thermal network. External nodes may have unique update functions depending on the simulation and weather desired for the modeling exercise. As an example look back at Fig. 2, if were an external temperature, setting ∞ would give the following state space representation. 1 1 Indeterminate Parameterization: Rij *Ci =1 4 Ci 2 0 -2 -4 -4 -2 0 Rij 2 4 Fig. 3. Non‐minimal parameterization gives rise to estimation ambiguity. The curve satisfying 1 demonstrates the unobservable subspace which includes physically meaningless and cannot be uniquely identified. negative quantities. C1 R12 R13 C2 R23 C3 0 Fig. 4. Three‐node graph with one loop. 5 0 5 drawn from distribution . The measurement noise terms ̅ are drawn from distribution . The measurement function of the thermal network is linear so H is simply an n row identity matrix (provided all temperatures are measured) padded with columns of zeros for the parameters. The update portion of the filter therefore can be written as a simple linear Kalman Filter. However in the prediction step, calculation of the Jacobian matrix F depends whether one chooses to estimate RC or 1/RC parameters. A comparison of both cases demonstrates that RC parameters would be a poor choice, especially during the initial acquisition stage, because poor parameter estimates will be squared and could easily cause the filter to diverge and blowup. C. Extended Kalman Filter (EKF) For all tests the state space system is integrated at one minute interval time-steps with Euler integration to allow discrete-time filter implementation. Parameter estimation with the Kalman Filter is achieved by augmenting the temperature states … with unique parameters … 1/ and disturbances ̅ 1/ / … / together in the state representation , ̅ , . For the purposes of estimation, the full discrete-time stochastic system is 1 1 1 ̅ ̅ 6 ̅ ̅ Case 1: represents process noise, represents where represents estimation uncertainty in RC parameters, process noise for disturbances, and ̅ represents measurement noise. All noise terms are assumed zero mean, Gaussian, white, and stationary. Note that in 6 , the matrix is actually a vector of functions composed from ̅ as defined by the parameterization of the RC terms. This representation results in multiplication and division of estimated parameters through the dynamics function. Specifically, temperature is being multiplied by RC parameters necessitating non-linear estimation techniques. For baseline comparisons, an EKF and UKF are formulated. In order to define notation, the prediction and update steps of the discrete EKF are shown in 7 , but for proper treatment of the derivation and background of the Kalman Filter, please consult [29], [30], [31]. (Note: For brevity in the following Kalman Filter formulations, notation deviates from the modeling section: subscripts denote time rather than node indices.) Case 2: | | | , State Estimate State Covariance Update: | | | | | | | Innovation Innov. Covariance Optimal Kalman Gain State Estimate State Covariance f x xˆ k 1|k 1 , u k 1 , Hk  8 D. Unscented Kalman Filter (UKF) Unlike the EKF, which uses a Jacobian first order linearization evaluated at the current estimate to propagate a probability distribution through a non-linear transform, the UKF uses the Unscented Transform to pass a distribution through a nonlinear transform. Specifically the UKF samples (2n+1) points in the distribution, evaluates each point through the non-linear transform and then recombines these points to generate a transformed mean and covariance which is oftentimes more accurate and stable than that obtained from the single point EKF linearizations [32]. The samples, called sigma points, are evenly spaced to capture at least the first and second order moments of the distribution and are weighted such that the covariance and mean of the samples matches that of the original distribution. After being mapped through the non-linear transform the resulting points are multiplied by their assigned weights to determine the transformed mean and covariance. For linear systems both 7 Jacobians: Fk  1 1 Thus, for numerical stability the EKF must follow Case 2 and estimate 1/RC parameters. Measurement noise is specified based on the accuracy of the temperature sensors. Process noise is specified for the temperature states based on the level of zone aggregation used while the RC and disturbance process noise is set to an artificial value greater than zero in order to allow the filter to vary its estimate of these parameters through time. Increasing process noise level for any parameter indicates that the model isn’t confident of its ability to describe the process evolution of that parameter. Disturbances, which by their nature the model is not explicitly capturing, are biased and vary with time. In order to estimate the disturbances over time; their noise level is set to be non-zero. Because the RC values should be fairly constant while the disturbance bias may change throughout the course of a day, the noise level for RC parameters should be much smaller than disturbances. Predict: | 1 h x xˆ k |k 1 For the filter dynamics function , temperature state dynamics follow the previously derived thermal model while the RC and disturbance parameters are modeled as constants. Artificial process noise for the constant parameters, denoted , allows the filter to change its estimate of these values through time and allows the filter to track the true time , varying disturbance. The set of process noise terms , and are stacked as defined by the state and 6 the EKF and UKF perform identically to a traditional Kalman Filter but for certain non-linear systems a UKF can provide higher accuracy with the same order of calculation complexity. The UKF was implemented with the same augmented state vector and used the same measurement and process noise values as the EKF. Removing the requirement to design and calculate a Jacobian, the UKF is amenable to either RC or 1/RC parameter representations—both are evaluated in the results section. The standard values of 10 , 0, 2, typical for a Gaussian distribution, were used to generate the following samples and weights . , , | For: | 1, … , | | For: , A 10,000 run Monte Carlo simulation was conducted to compare the EKF and UKF performance for the simple twoparameter search problem. Table 1 shows the distributions for all parameter and filter values which were randomly sampled at the beginning of each test run. The distributions of parameter and temperature values were chosen such that some runs will have high levels of excitation while others will have no excitation—effectively T(0) equal to Text. Results of the simulation, shown in Table 2, demonstrated that the UKF outperformed the EKF for non-linear estimation of parameter p1 but had statistically similar performance for estimation of linear parameter p2. The UKF showed filter stability—resilience to exponential tracking divergence—while 2% of the EKF runs went unstable and did not complete execution. These results agree with published studies comparing the filters’ general performance in other application areas [33], [34]. The EKF could estimate linear additive disturbances, which is in agreement with [19], but when estimating coefficient parameters such as weights in a thermal network, the UKF is a more viable solution. Further testing showed that increasing the number of temperature zones or trying to directly estimate RC instead of its reciprocal, 1/RC, further degraded the EKF stability and performance. However for the UKF, comparison of estimating RC products and their reciprocals showed that direct RC parameters are more robust. For zones with little thermal connection, the filter estimating RC parameters will continue increasing estimates but will be finitely decreasing the covariance, so the estimate will eventually converge. When 1/RC parameters are utilized for the same zones, the estimate will be driven to zero, but the covariance will not | 1, … ,2 | | | 1 For: 1, … ,2 1 2∗ The samples were then recombined to give the a priori state and covariance estimates. | | | f | | TABLE I MONTE CARLO SAMPLED PARAMETER DISTRIBUTIONS | | | Variable | Range 0,1 0,20 ̂ 0 0,1 ̂ 0 0,20 0 0,100 0 0,10 0 | .5, .5 | | .5, .5 | 1,0,0 1, 0,10 , … Est. Process Variance 0,10 1, 0,5 10 , … Initial Est. Variance 0 0,5 10 = Uniform Distribution, = Normal Distribution. Truth Truth I.C. Estimation I.C. Estimation Truth I.C. Estimation True Meas. Error Est. Meas. Error True Process Variance Note that λ and W can be reused so only the χ terms need to be recalculated each iteration. Because the measurement function is linear, the unscented transform is only used in the prediction step; the measurement step is simply calculated as a linear Kalman filter update. E. EKF vs. UKF In order to compare an EKF and UKF the 2-node thermal network from Fig. 2 was chosen. Looking back to 5 , the system can be parameterized as Ap. 0 Quantity 0 TABLE 2 10,000 RUN MONTE CARLO RESULTS In order to compare both the parameter estimation and bias detection capabilities only T was sensed—Text was treated as was used a constant latent variable. A second parameter to estimate Text as a linear additive disturbance as shown. Test EKF EKF EKF UKF UKF UKF 9 7 Statistic Numerical Instability Count p1 Average Estimation Error p2 Average Estimation Error Numerical Instability Count p1 Average Estimation Error p2 Average Estimation Error Result 175 3.16 60.5 0 0.18 58.0 decrease as quickly, which can result in a zero crossing that violates conservation of energy by estimating a negative parameter. As a result, RC parameters were estimated by the UKF for all remaining tests. R.C. Disturbance Solar Gain 1000 0.1 0.05 0 -0.05 F. Disturbance Estimation and Pattern Recognition Direct sensing of disturbance heat flux is rarely practical in building systems. However, timing information for disturbances is typically available, so a practical method is presented to learn disturbances given only timing information with no prior disturbance quantification. This method could readily be augmented if additional disturbance heat flux data was available. Looking at thermal model 1 , a change in zone temperature may be explained away using either connected zones with their respective temperatures or additive disturbances. In order to get satisfactory estimation performance this ambiguity must be considered in the filter design lest it manifest problems akin to non-minimal parameterization. The engineering solution selected to rectify the problem, splits the estimation problem based on the presence of disturbances and manipulates the process covariance for estimated disturbance parameters based on timing of expected unquantified disturbances. From a control theory perspective the system does not have timeinvariant observability. However, buildings are a timevarying system that have some periodicity. Looking over a horizon, for example one day for solar disturbances, we can learn constant parameters when no disturbances are present and learn disturbances after having estimated the constant parameters. This partitioning enables time-varying observability. The presented method is not claimed to be optimal, rather it is a practical solution based on engineering judgment of typical scenarios common in buildings. Typically, a system can sense if people are using a building but cannot measure their heat flux or the equipment they use, likewise it can sense if the HVAC is on but not the exact heat flow delivered to a specific room. The approach attempts to use commonly available timing knowledge to quantify and infer disturbances that are not directly measurable and only partially predictable. Learning of disturbances was done in a Markov fashion: the estimator assumed no knowledge of previous historical disturbance patterns and estimated a new disturbance value at each time step based on the previous time step’s estimate, the dynamic model, and the current measurement. The disturbance states in the UKF were modeled as constants which have zero-mean Gaussian additive noise. A characteristic change in the disturbance such as a heater turning on or the sun coming up at dawn violates the zeromean assumption causing a bias in the disturbance. In order to track these sudden bias changes using a simple UKF, the variance(s) correlating to those specific zone(s) disturbances were inflated to allow the filter to acquire and track the new value. This artificial tuning of the covariance is similar to tuning a forgetting factor in adaptive control frameworks. In the Matlab-based simulation, heating or cooling was arbitrarily added to individual zones from 10am to noon, so 500 0 6 12 18 24 0 -6 1 0 0 -1 -1 6 6 12 18 24 12 18 Hour 24 x 10 Variance 1 0 0 -4 x 10 Variance 12 18 Hour 24 0 6 Fig. 5 Disturbance Parameter Process Variance Tuning the covariance was increased around those times. In the EnergyPlus simulation, the primary unmodeled disturbance was solar radiation, so the covariance was increased around dusk and dawn. This variance tuning is visually depicted in Fig. 5. Because the UKF can explain temperature swings by either tuning RC or disturbance values, a multi-mode approach was chosen as a uniform method to split the estimation problem. In order to acquire good estimates, the UKF was operated in two modes: A) Acquisition Mode: initially only RC products were estimated by running the filter at night when solar gains were at a minimum, and B) Monitoring Mode: both RC products and disturbances were estimated simultaneously after RC product estimates had started to converge to constant values. This splitting proved critical to obtaining good estimates from EnergyPlus data but unnecessary for the simple Matlab based simulation due to its consistently high level of external temperature excitation. Disturbance estimates from the entire multi-day learning period were heuristically combined in order to generate a 24hour pattern. This disturbance pattern was then used when predicting the building’s thermal response. The heuristic pattern recognition algorithm was a simple weighted average. For thermal network simulated data the true disturbances were identical every day. Thus the bias values learned at each time step were equally averaged to generate one 24-hour pattern for each bias term. Mathematically this can be written as summation over days of learning with minute time steps to generate a disturbance pattern correlating to zone based on the as shown, where the value inside UKF estimated bias brackets notates the minute-based time step index. , ∀ ∈ 1 24 60 10 The predominant unmodeled disturbance from EnergyPlus data was radiation from the sun, which is dependent on the cloud cover, time of day, season of year and other factors. For the purpose of engineering a robust simple solution we made a realistic assumption that we had a measurement of the average solar intensity for morning and afternoon and used this for both pattern recognition and simulation 8 predictions. The solar intensity reading was calculated as the summation of the Environment Direct Solar and Environment Diffuse Solar variables from EnergyPlus. Algorithm 1 contains the weighted average and prediction steps used for EnergyPlus simulations. ALGORITHM 1 ♦ INITIALIZATION -Max solar radiation: -Indices: zone , day , time of day -Dawn, midday, and dusk times: , -Morning & afternoon avg solar radiation: , -Estimated bias per zone : -Measured solar radiation intensity: , ♦ WEIGHTED DISTURBANCE PATTERN -Disturbance Pattern: for 1 to if /2 0.35 for 1 to 24 60 , if then elseif then , elseif then , else end end end / ♦ PREDICTED DISTURBANCE -Bias for Prediction: for 1 to … for 1 to 24 60 if then , , elseif then elseif then , , else end end III. SIMULATION RESULTS A. UKF 5-room Simulated Performance A six-node thermal network, corresponding to five internal zones and one external temperature shown in Fig. 6, was used to evaluate the UKF parameter estimation and thermal disturbance detection. For this first evaluation, measurement data was generated from a model whose dynamics were structurally identical to the dynamics used in the UKF. The 5-room models shown here are based on that in [11], but feature extended explanations, derivations, and simulation results. Given five finite capacitances and thirteen resistances, there were a total of seventeen unique RC products for the UKF to estimate in this model. A first test was run in acquisition mode, so disturbance estimation was disabled. The external temperature forcing function was composed from the sum of a 40 degree peak-to-peak sinusoid with period of one day, a 10 degree peak-to-peak sinusoid with period of 4 hours, and random noise which would allow the temperature to drift from day to day. Resistance and capacitance values were chosen such that thermal lag in the simulation would be similar order as thermal lag in a small to medium sized building. Temperature states were initialized with less than a degree of error while RC estimates were all arbitrarily initialized to 1000 with a standard deviation of 500. Using four days of recorded data, RC parameters were learned by the UKF. At the end of the four day simulation, a 48 hour prediction of the five zones’ temperatures was made using the acquired RC parameters. The true 48 hour external temperature profile was provided to both the true dynamics and UKF dynamics in order to establish a fair comparison baseline for evaluation of the UKF. In a real system inaccuracies in the weather forecast would degrade the prediction quality. , , then , / / By averaging the solar intensity before midday and after midday, two weights were determined for each day. In order to ensure sufficient signal to noise ratio for disturbance pattern estimation, a day’s disturbance estimates were discarded if the day’s total solar intensity averaged below 35% of the maximum solar intensity possible for that location. The remaining days had their bias estimation values multiplied by the ratio of maximum solar radiation to respective half day average measured intensity in order to normalize the bias values. Then the normalized bias values were averaged using the same equation for to estimate the repeated daily disturbance profile. When used for predictions, the disturbance profile was scaled by the predicted solar intensity weights to predict each zone’s unique solar disturbance quantity at each timestep throughout the day. Thus predictions utilized the final estimated RC parameter, 24-hr disturbance patterns, predicted external temperature profile, and half-day average predicted solar intensity. The pattern recognition shown in Algorithm 1 could be applied to any cyclic disturbance. The variables , , , and would be adapted to whatever period and timing knowledge a designer had of other loads or disturbances present in the environment, and the arrays and would be modified to contain the pattern array. Fig. 6. Left: Five‐room building used for simulations, Right: Top view of node labeled internal thermal network representation. White lines represent building internal walls. A sixth unlabeled node acts as an external temperature 4000 4000 3000 3000 Truth Estimate 2000 2000 3  variance 1000 1000 0 0 -1000 0 0.5 Time (Day) -1000 1 0 0.5 Time (Day) Fig. 7. Example estimation of two parameters from 5‐room building using RC‐thermal model data. 9 1 Prediction Performance Bias Estimates Temperature (deg/min) Temperature 80 70 60 Truth Prediction 50 40 0 0.5 1 Time (Day) 1.5 0.1 0.05 0 0 6 12 18 24 12 Time (hrs) 18 24 0 -0.05 -0.1 2 0 Truth 6 Estimate Fig. 9. The actual disturbances introduced into system for the two zones shown in Fig. 10 are labeled above as “Truth” and were repeated on a 24 hour cycle. Dashed lines represent the UKF 4‐day average estimate of the disturbances which was used for predictions in Fig. 10. Deg F Fig. 8. Example 48‐hour prediction window in a disturbance free environment based on RC estimates. External temperature forcing function was composed from two sinusoids plus white noise. Deg F Running the described simulation showed that some parameters are estimated very well while others are not; two characteristic examples of this are shown in Fig. 7. The external temperature does not fully excite all of the node-tonode thermal connections which limits the filter’s ability to precisely determine all of the parameters. Despite this numerical estimation error, the 48-hour prediction demonstrated excellent matching between the UKF estimated model and the true model. From a poorly performing filter one would expect predictions to have increasing divergence or lag as the prediction window is increased. Characteristic temperature predictions of the highest and lowest capacitance rooms are shown in Fig. 8 for visualization. In our chosen model the poorly estimated parameters correlate with higher order dynamics that neednot be accurately estimated for good model fitting. For practical applications this is analogous to having multiple zones that are always excited together such that their relative interaction need not be known for useful predictions. For a second evaluation daily repeating disturbances were introduced uniquely into each zone of the house. The disturbance states in the UKF were modeled as constants which have zero-mean Gaussian additive noise; in order to track a sudden bias change using a simple UKF, the variance correlating to that specific bias must be inflated to allow the filter to acquire and track the new value. This inflation must occur anytime the disturbance changes significantly enough that the zero mean Gaussian assumption is severely violated, such as when an HVAC system turns on or at dusk and dawn due to solar radiation. For the combined RC and disturbance estimation, 4 days of data were again used for training. Disturbances turned on at 10:00 and off at 12:00 each day, so the estimate covariance P was boosted at those times and the process variance was also temporarily increased when the disturbance was cycled on and off as previously shown in Fig. 5. With this variance tuning method, the filter had excellent disturbance tracking and similar RC estimation accuracy. Example tracking of two disturbances is shown in Fig. 9. Using the described simple heuristic pattern recognition with a 24 hour period shown in Equation 10 , the estimated biases from all four days were averaged to generate a single day estimate of the repeating disturbance pattern. A 48-hour prediction was then made using the average disturbance and final RC estimates along with the Prediction Performance 80 Temperature 75 70 65 Truth RC Only Pred RC + Dist Pred 60 55 0 0.5 1 Time (Day) 1.5 2 Fig. 10. Example 48‐hour prediction derived from RC estimates and estimated daily cyclic disturbances. Gray boxes denote times where external disturbances were present in the true system. An external temperature forcing function was composed from two sinusoids plus low amplitude white noise. Deg F exact external temperature profile. Again, excellent predictions were made with the estimated model. Fig. 10 plots temperatures of two zones comparing predictions from the RC only estimation model and the RC plus disturbance estimation model to the truth model. This evaluation provides good indication of the applicability of the UKF to thermal network parameter and disturbance estimation. B. UKF EnergyPlus Performance Given the excellent performance of the UKF on data generated by the thermal network model, the UKF was tested on data generated from an EnergyPlus simulation. Individual data traces show typical estimation performance which is validated in an aggregated year-long demonstration. This more realistic EnergyPlus model, shown in Fig. 6, had five rooms correlating to the five zones; realistic data for the floor, wall, and ceiling composition; and windows on the four exterior walls, which pointed in the cardinal compass directions. The structure was simulated with weather and solar radiation for Elmira, NY. In order to acquire good estimates, the UKF was operated in two modes: A) Acquisition Mode: night-only estimation of RC products, 10 Prediction Performance 40 Temperature 35 30 25 20 15 Fig. 11 Screenshot from Sustain showing afternoon sun on West Wall 10 Truth 0 RC Only Pred 0.5 RC + Dist Pred and B) Monitoring Mode: simultaneous estimation of both RC products and disturbances. Distinguishing learning modes was less of a concern for the thermal network simulated data because the disturbances were only on for short periods of time and the external temperature had a consistently high level of variation to excite the system. Because the primary unmodeled thermal disturbance was solar radiation, covariance tuning was done for the Monitoring Mode by increasing the bias states’ process variance for two hours starting at dawn and another two hours ending 20 minutes after dusk as previously shown in Fig. 5. Fig. 11 shows the path of the sun on March 9th and the variability of the sun path over the course of the year that will be simulated by EnergyPlus. The graphic was generated by Sustain, a front-end for EnergyPlus developed by researchers at Cornell University Program of Computer Graphics [35]. Due to the axis-inclination of the Earth, the sun’s path and solar gain varies over the year. Fig. 13 shows the resulting average disturbances from a four day test where biases were only estimated for the last two days and then combined into a 24-hour pattern. Notice how the East room is heated in the morning, the West room is heated in the afternoon, and the South room is heated all day, which correlates nicely with expected heat from solar radiation. Plots of the 48-hour predictions are shown in Fig. 12. Predictions which utilize solar disturbances have much higher accuracy than the RC only predictions. The accuracy of these predictions ground assumptions made in the thermal network formulation and more importantly demonstrate the utility of the UKF for system identification of buildings’ thermal envelope. 1 1.5 Time (Day) 2 Fig. 12. Top South Zone, Bottom West Zone of 48‐hour EnergyPlus predictions Deg C Temperature (deg/min) -4 2 1 0 -1 2 1 0 -1 2 1 0 -1 2 1 0 -1 x 10 0x 10-4 East 5 10 North 15 20 5 10 West 15 20 0x 10 5 10 South 15 20 0 5 10 -4 0x 10 -4 15 Time (hrs) 20 Fig. 13. Estimated 24‐hour cyclic disturbances from EnergyPlus data Deg C . calculation failure. From further analysis, simple estimation monitoring by a human or addition of heuristic rules to the existing framework would fix all 43 estimation routine failures. For example over 10 of the failures occurred because 4 consecutive days had less than 35% solar radiation causing no disturbance pattern to be learned. Fixing these sorts of numerical issues to guarantee 100% reliability in an automated algorithm is outside the scope of the current study. Results suggested the algorithm could easily be matured for practical application by adding a number of heuristics. Using the 314 successful estimation runs, we compared the 24-hour prediction simulations against the buildings’ truth simulation from EnergyPlus and found good accuracy—the models have enough fidelity to be used for control. Over a 12 hour prediction horizon the root mean square (RMS) temperature prediction error was 1.16oC and over 24 hours the RMS temperature prediction error was 1.48oC. ASHRAE standards mandates that vertical temperature stratification in an occupied zone should be less than 5.4oF (3oC) [36]. Home and office thermostats often use a dead-band of 4oF (2.2oC) to 8oF (4.4oC). The model’s prediction errors are well within these design bounds for the 24 hour prediction horizon. In Fig. 14 the RMS error for the prediction is shown over time C. UKF EnergyPlus Year Study Further analysis of the EnergyPlus generated data was conducted by analyzing a total year of data. A unique UKF instance was initialized each day on the first 357 days of the year and run for 7 days, 3 days in acquisition mode and 4 days in monitoring mode. Then the UKF was used to predict the building’s response on the eighth day and compared against the buildings actual response from EnergyPlus. This generated 357 sets of learned parameters, bias estimates, and 24-hour prediction simulations. Of the total set, 43 simulations resulted in estimation routine errors such as negative parameter estimates or covariance shrinking to zero causing an UKF matrix inverse 11 Standard Deviation of Prediction Error East Zone Prediction 2.5 Temperature (deg C) Temperature Deg. 20 2 1.5 1 7 day 14 day 28 day 0.5 0 0 0.2 0.4 0.6 Time (Day) 0.8 10 0 5 0 5 10 15 20 West Zone Prediction 25 30 25 30 20 10 1 10 15 20 Time (days) Fig. 15 Month prediction using a model learned from 1 week of data. Fig. 14. Root Mean Square error of temperature predictions of all 5 zones over the year for different lengths of estimation learning time. Deg C depending on the target application and available computation and sensing hardware. For example, Dobbs [37] compared accuracy of thermal models across different levels of RC zone aggregation—leveraging such a tool may aid control-oriented model creation for the UKF. Extensions to the UKF may offer new opportunities for fault detection and monitoring [38]. One outstanding challenge remaining with this online estimation technique is a demonstration of the learned models’ performance with model predictive controllers in practice. Some studies [13], [24], [23] have begun investigating how the quality of the learned model affects the performance of predictive controllers that use the model. The consensus to date is that intra-zone excitation is necessary in order to learn a building’s internal coupling. Before controlling the building in a novel way to maximize energy savings, a buildings internal thermal coupling must be known. Deriving methods to monitor the quality of the measured data and better learn the building’s thermal dynamics on-demand, by experimentally exciting the building, are the subject of a future paper by the authors. demonstrating good performance. Increased learning periods of 14 and 28 days further reduced prediction error but did not drive it to zero—likely because the simple RC model could not capture the entire fidelity of the EnergyPlus truth simulation. Additionally Fig. 15 shows a month-long stable prediction of the East and West zones based on a model learned from 7 days of data. This long prediction used the same data types as that in Fig. 12: correct zone initial temperature conditions, correct external temperatures over the horizon and half day average solar intensity values over the horizon. The longhorizon prediction accuracy demonstrates the learned model is unbiased, stable, and robust. To the author’s knowledge, this is the first year-long study of an online UKF estimating disturbances with parameters and states for a building. IV. DISCUSSION Results of the UKF estimation and model prediction capabilities have demonstrated the method as a powerful tool for thermal modeling of building systems. The simplicity with which a thermal network can be described combined with the numerical stability and robustness of the UKF are important factors which could enable its deployment as a scalable system identification routine for buildings thermal envelopes. No physics constraints were applied to ensure RC parameters were positive, or to inform the bias and disturbance estimation. Realistic estimation of values was solely dependent on the quality of the chosen thermal model representation, estimation technique, and measured data. The authors expect that good results obtained from this paper’s simulations would reflect realistic expectations of good perfomance in real world applications. Accurate bias tracking was achieved though covariance tuning, but this might not be scalable to certain buildings where disturbances occur on erratic schedules, so multiple hypothesis estimation or constraints may be augmented with a UKF to provide a more powerful solution. Further investigation into model selection and fidelity could lead to performance improvements for the UKF V. CONCLUSION A multi-mode implementation of a multi-zone UKF was presented as a scalable and rapidly deployable system identification routine for building thermal dynamics. Using a passive 5-room model, the UKF demonstrated the ability to learn both dynamics parameters for a thermal network and unknown disturbances. 24-Hour predictions from UKF estimated parameters yielded accurate results which were validated with EnergyPlus simulations using a full year of data. The UKF, a data-driven, model-based approach, amenable to augmentation with numerical methods, provides a promising step towards a scalable framework to realize advanced BAS predictive controllers. ACKNOWLEDGMENT We would like to thank Dave Bosworth for his help generating an EnergyPlus 5-room building dataset for evaluation of parameter estimation methods. 12 REFERENCES Europe, Istanbul, 2014. [15] S. Kiliccote, M. 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1 Chance-Constrained Day-Ahead Hourly Scheduling in Distribution System Operation Yi Gu1 , Student Member, IEEE, Huaiguang Jiang2, Member, IEEE, Jun Jason Zhang1 , Senior Member, IEEE, Yingchen Zhang2 , Senior Member, IEEE, Eduard Muljadi2, Fellow, IEEE, and Francisco J. Solis3 , Senior Member 1 arXiv:1711.10687v1 [] 29 Nov 2017 Dept. of Electrical and Computer Engineering, University of Denver, Denver, CO, 80210 2 National Renewable Energy Laboratory, Golden, CO, 80401 2 School of Mathematical and Natural Sciences, Arizona State University, Glendale, AZ, 85306 Abstract—This paper aims to propose a two-step approach for day-ahead hourly scheduling in a distribution system operation, which contains two operation costs, the operation cost at substation level and feeder level. In the first step, the objective is to minimize the electric power purchase from the day-ahead market with the stochastic optimization. The historical data of dayahead hourly electric power consumption is used to provide the forecast results with the forecasting error, which is presented by a chance constraint and formulated into a deterministic form by Gaussian mixture model (GMM). In the second step, the objective is to minimize the system loss. Considering the nonconvexity of the three-phase balanced AC optimal power flow problem in distribution systems, the second-order cone program (SOCP) is used to relax the problem. Then, a distributed optimization approach is built based on the alternating direction method of multiplier (ADMM). The results shows that the validity and effectiveness method. Index terms— Renewable energy integration, second-order cone program, Gaussian mixture model, optimal power flow, stochastic optimization, alternating direction method of multiplier, Gaussian mixture model N OMENCLATURE C f1 f2 β cDA , cRT , cP V cs GDA GRT GR λt GDL t Pij zij Gerr Vi n ǫ Total cost of the two steps. Operation cost at the substation level. System loss at feeder levels. The system loss weight. The unit price of the day-ahead market, realtime market and renewable generation. The price to resell the redundant power generated to the electric market. The electric power obtained from the dayahead market. The RT power consuming. The renewable generation. An occurrence probability helps to decide if the systems needs to buy the electric power at time t. The demand load at period t. The loss of branch lines from bus i to j. The impedance from bus i to j. Forecasting error model of the renewable energy. The voltage at bus i. The amount of the clusters in the model, n = 1, · · · , N . The weight of each cluster in GMM. x Ui , Ci pi , qi Vi , Ii si Ω Gf x = [x1 , x2 , · · · , xq ]T is the q-dimensional data vector. The parent node and the children nodes at node i, i ∈ NB . Active and reactive power at bus i. The voltage and the current. Injection power at bus i. Complex impedance. The forecasted renewable generation. I. BACKGROUND AND M OTIVATION In general power system operation, the day-ahead hourly scheduling [1]supplies the customers playing a more active role. They can be provided more benefits, such as lower system operation cost, advanced system reliability, and lower volatility in hourly [2]. The high penetration of the renewable energy [3] can lead to a lower net load power consumption from the bulk power system, however, it brings increasingly stochastic deviations in net load profiles [4]. Compared with the transmission systems, the distribution systems often work in an unbalanced polyphase state because of the asynchrony, asymmetry and the diversity of the load, which brings bigger challenge for the distribution system operation. In this paper, a stochastic optimization based two-step approach for dayahead scheduling is proposed to minimize the distribution system operation cost, which consists of the cost of net load consumption and the system loss. At the substation level, the high penetration of the renewable energy can reduce the electric power [5] purchasing from the day-ahead market for the distribution systems [6]. In [7]–[9], the stochastic programming optimization (SPO) can provide many potential benefits to the transmission systems. With the renewable energies, this paper focuses on minimizing the electric power purchasing from the day-ahead market at the substation level in the first step. The historical data of the day-ahead electric power purchasing is used to generate the forecast results and the forecasting errors, which can be formulated as a chance constraint for the SPO in distribution systems. According to the distribution of the forecasting errors, the chance constraint can be formulated into a deterministic form by Gaussian Mixture Model (GMM) [10], and the global minimum of the convex problem can be determined. In [2],the variability of the renewable energy [11] is managed by hourly demand response in day-ahead scheduling, without considering the stochastic net load deviation in an system, which equals to the operation cost at feeder level. β is used to limit the system loss weight in the optimization.The consymption of the load cost model at the substation level is simulated as: hour, which dramatically impacts the operation cost of system loss. However, the single line model is used compute the system loss, which ignores the three-phase balanced configuration in distribution systems. In the second step, this paper focuses on minimizing the cost of the system loss at the feeder level with the three-phase balanced model. The characteristic of the nonconvexity is a critical problem for the three-phase balanced AC optimal power flow. The heuristic methods are always applied to solve the problem, but which are hardly avoided falling into the local minimums [12]. Based on the alternating direction methods of multipliers (ADMM), a distributed method is provided to solve the AC optimal power flow problem. This paper is organized as follows: the proposed approach is described in Section II. In Section III, the operation cost in substation level is analyzed. In Section IV, the operation cost in feeder level is analyzed. In Section V, the numerical results are presented in IEEE standard distribution systems. The conclusion is Section VI. II. A RCHITECTURE S UMMARY A PPROACH AND OF THE f1 = t=1 AT THE where the time intervals is defined as t = 1, 2, , N T . cDA , cRT and cP V describe the unit price in the day-ahead, real-time market and solar generation, respectively. cs is the established price to sell the redundant power generated by distribution systems to the electric market, GDA and GRT are the electric power amount purchased from the day-ahead and real time market, GP V is the amount of the solar power generation in the distribution systems. λt is an occurrence probability which decides if the distribution systems needs to buy the electric from real-time market at time t. GDL is t V the demand load at time t. cDA GDA + cP V GP is the baset t case generation cost consists of the time-dependent power consumption by day-ahead scheduling and the renewable RT presents the deviation energy generation cost. λt · cRT t Gt power purchased from real-time market for compensation. And V − GDL + GP (1 − λt ) · cs (GDA t ) means the redundant energy t t can be resold to the market in a lower price. Several months of hourly power forecasting data in dayahead scheduling and actual net load power consumption are used to provide the forecast results [4], [17], [18]. The available hourly forecasting power in day-ahead market GDA (N T + 1) can be now described as following (Gerr is the day-ahead forecasting error) [19]: P ROPOSED GDA (N T + 1) = GDA (N T ) + Gerr (N T ) · GDA (N T ) (3) Subject to: S UBSTATION L EVEL GDA + λGRT + GP V = GDL (4a) cs < cDA < cRT t (4b) GRT,min ≤ GRT ≤ GRT,max GDA,min ≤ GDA ≤ GDA,max (4c) (4d) GP V,min ≤ GP V ≤ GP V,max (4e) λt = The total cost is calculated as (1), the sum of the operation cost at the substation level and feeder level: C = min(f1 + βf2 ) (2) o  V + (1 − λt ) · cs (GDA + GP − GDL t t t ) In Fig. 1, the proposed method consists two steps, the stochastic optimization of the electric power purchased from the day-ahead market [13] at the substation level and the system loss optimization for the three-phase balanced AC optimal power flow [14], [15] at the feeder level [16]. Fig. 1 shows the first step at the left, the historical data of the electric power purchased from the day-ahead market is employed to generate the forecast results and the distribution of the forecast errors. Then, combined with the day-ahead hourly scheduling model, an objective formulation is simulated with an chance constraint. A GMM based approach is used to convert the chance constraint into a deterministic problem. Finally, the day-ahead hourly optimal operation cost can be obtained at the first step. In the rest side of Fig. 1, a three-phase balanced AC optimal power flow is simulated to compute the distribution system at the feeder level. After the day-ahead hourly purchased electric power is determined, the three-phase balanced distribution system model is built to minimum the system loss. Considering the nonconvexity of the AC optimal power flow, an inequality constraint is built based on SOCP to relax the problem into a convex problem. After that, the objective function of system loss can be derived with ADMM. Finally, the three-phase balanced AC optimal power flow is used to minimize the system loss successfully. III. O PERATION C OST NT n X V RT RT (cDA GDA + cP V GP t t ) + (λt · ct Gt ) ( V 1 GDA + GP < GDL t t t V 0 GDA + GP ≥ GDL t t t P r(f (Gerr ) ≤ 0) > α (1) (4f) (4g) where (4g) indicates that day-ahead forecasting error [20] should be fulfilled with the probability α, and the chanceconstraint [21] can be converted in a deterministic formulation with the GMM. where C is the total cost, f1 is the purchased electric power from the day-ahead market, which equals to the operation cost at the substation level, f2 is the system loss of the distribution 2 Day-ahead Hourly Scheduling Model Forecast Results and Errors Second-order Cone Programming Based Relaxation Gaussian Mixed Model Objective Function of Threephase Unbalanced AC Optimal Power Flow Objective Function with Chance Constraint Objection Function Derivation with ADMM and Computation Objective Function with Deterministic Constraint Final Results Results of Day-ahead Hourly Scheduling Substation Level Stochastic Optimization at Hourly Level Convex Relaxation Three-phase Unbalanced Distribution System Model Alternating Direction Method of Multipliers System Data Collection Historical Purchased Power Data Feeder Level AC Optimal Power Flow at Minutes Level Fig. 1. The flowchart of proposed approach. 2) M-Step: In M-step, the parameters are recalculated and estimated to maximize the quantity of the expectation in (7). A. GMM with Expectation Maximization. Comparing with the regular GMM, a definition of expectation maximization EM based GMM is described. It is used to model the forecasting error model of the renewable generation. We have known that GMM is a particular form of the finite mixture model. For (5), more than one components is calculated as the sum with different weights (ǫ): p(x|Θ) = ǫp(x|θ) θ(t+1) = arg max R(θ|θt ) (7) The EM based GMM can decide the amount of clusters based on minimum description length (MDL) [24], [25]. The MDL criterion is frequently used on the field of selection in [26], because of the improved method is less sensitive to the initialization. (5) ǫ in (5) is calculated in [22] and has been described to be non-negative, that the sum equals to 1. Each component in the GMM model is a normal distribution and obeys to θn = (µ, Σ), which indicates the means vector and the covariance. According to [23], the parameters of the mixture model is computed with expectation maximization (EM), the expectation-step (called E-step) and maximization-step (called M-step) are described as below. 1) E-Step: The algorithm is ended when the function in (6) reaches the convergence. IV. O PERATION C OST AT THE F EEDER L EVEL When the substation level cost is determined, the objective function of the three-phase balanced AC optimal power flow is to minimum the system loss. As we know, the influence of the three-phase distribution system is small enough, the second order cone programming (SOCP) is used to calculate the system loss here [27]. which can be defined as follows [28], [29]: X F = Pij , (8) E R(θ|θt ) = Ex,θt [logL(θ; x)] where Pij is a branch loss from bus i to j, and Pij = |Iij |2 rij , Iij is the complex current from bus i to j, and zij is the complex impedance zij = rij + ixij . E is the branch set of the distribution system, which can be represented with the set of buses and branches: G = [V, E]. (6) Where L(θ; x) = p(x|θ). x is a set of the observed data from the given statistic model and the θ is unknown parameters along with the likelihood function in (6). 3 TABLE I T IME CONSUMING OF DIFFERENT ALGORITHMS BASED ON THREE 350 INDIVIDUAL SYSTEMS 300 IEEE 13-bus 1.78 s 189.43 s 7.20 s IEEE 34-bus 5.23 s 221.34 s 13.16 s IEEE 123-bus 14.66 s 459.21 s 25.58 s Total Cost ($) Method Proposed Method Simulated annealing Interior-Point Comparison with Different Alpha Alpha=95% Alpha=90% Alpha=80% 250 200 150 100 50 Based on the branch flow model, the SOCP relaxation inequalities can be represented as follows 0 0 (9) Vi,min ≤ Vi ≤ Vi,max , (10a) Iij ≤ Iij,max . (10b) min f (x) + g(z) (11a) s.t. x ∈ Kx , z ∈ Kz Ax + Bz = c (11b) x,z B. The performance of the proposed approach It is assumed that the renewable energy resources are located in bus 7, 23, 29, 35, 47, 49, 65, 76, 83, and 99 [30]. The result of the three-phase balanced AC optimal power flow is shown in Fit. 2(b). The errors of the primal residual and the dual residual are less than 0.5 × 10−3 after 5 iterations. After 30 iterations (less than 0.2 second), the curves of primal residual and the dual residual are coinciding and stable, which demonstrates the high speed and effectiveness of the convergency of the proposed approach. In Fig. 3, the comparison is made with different α in (4g) during 24 hours. The results shows that the total operation cost of the system is higher with a lower α, which indicates that a higher accuracy of error forecasting model can help to save more money. V. T HE N UMERICAL R ESULTS 250 251 30 28 26 46 108 45 27 43 44 23 21 19 20 38 36 35 18 135 37 9 10 7 149 1 8 12 150 152 13 5 52 53 54 55 94 6 95 93 195 610 56 85 79 77 78 76 80 84 76 88 90 92 16 74 72 17 4 75 73 57 96 34 15 3 71 70 69 67 160 60 58 59 61 2 99 98 197 97 62 39 68 14 11 451 450 100 101 66 40 22 114 104 103 102 63 41 113 107 106 64 65 105 42 24 112 110 300 151 109 47 25 48 350 111 51 50 49 25 As shown in Fig. 2(a), the IEEE 123-bus distribution system contains 118 basic branches, 85 unbalanced loads, 4 capacitors, and 11 three phase switches (6 initially closed and 5 initially opened), which is the topology taken in the preliminary results. The simulation platform is based on a computer server with a Xeon processor and 32 GB ram. The programming language are Python and Matlab. Then, a dual problem is build based on the objective function (8) and (11a) with ADMM and solved in parallel. 29 20 A. The test bench The standard optimization problem of ADMM is defined as follows: 31 15 Fig. 3. The total cost with different Alpha. where Sij indicate the complex power flow Sij = Pij + iQij , lij := |Iij |2 , and vi := |Vi |2 . The basic physical constraints can be defined as follows: 32 10 Time(Hours) |Sij |2 ≤ lij , vi 33 5 91 81 87 89 86 82 83 C. Performance comparison (a) As in Table I, the different algorithms are used on different individual power systems (IEEE trans 13, -34 and -123 Bus ). The proposed approach obtains a shortest time consuming on all of the three power systems, which demonstrates our method can work more efficiently than others. -3 2.5 x 10 Primal Residual Dual Residual 2 Error 1.5 1 VI. C ONCLUSION 0.5 0 1 5 10 15 20 25 30 35 Iteration Number (b) Fig. 2. (a) The IEEE 123-bus distribution system. (b) The residual error curves of the three-phase balanced AC optimal power flow. 4 In this paper, a stochastic optimization based approach is proposed for the chance-constrained day-ahead hourly scheduling problem in distribution system operation. The operation cost is divided into two parts, the operation cost at substation level and feeder level. In the operation cost at substation level, the proposed approach minimizes the electric power purchase from the day-ahead market with a stochastic optimization. In the operation cost at feeder level, the system loss is presented with the three-phase balanced AC optimal power flow [31]. In our work, the detailed flowchart and the description of the stochastic optimization with the forecasting errors is improved, which helps to describe our approach detailedly. And the detailed description of the derivation from the chance constraint into the deterministic form with GMM is provided. 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POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS OF RIGHT-ANGLED ARTIN AND COXETER GROUPS TARAS PANOV AND YAKOV VERYOVKIN arXiv:1603.06902v2 [] 1 Sep 2016 To the memory of Rainer Vogt Abstract. We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a rightangled Coxeter group. 1. Introduction Right-angled Artin and Coxeter groups are familiar objects in geometric group theory [Da2]. From the abstract categorical viewpoint, they are particular cases of graph product groups, corresponding to a sequence of m groups G = (G1 , . . . , Gm ) and a graph Γ on m vertices. Informally, the graph product group G Γ consists of words with letters from G1 , . . . , Gm in which the elements of Gi and Gj with i 6= j commute whenever {i, j} is an edge of Γ. The graph product group G Γ interpolates between the free product G1 ⋆ · · · ⋆ Gm (corresponding to a graph consisting of m disjoint vertices) and the cartesian product G1 × · · · × Gm (corresponding to a complete graph). Right-angled Artin and Coxeter groups RAΓ and RC Γ correspond to the cases Gi = Z and Gi = Z2 , respectively. The polyhedral product is a functorial combinatorial-topological construction assigning a topological space (X , A)K to a sequence of m pairs of topological spaces (X , A) = {(X1 , A1 ), . . . , (Xm , Am )} and a simplicial complex K on m vertices [BP1, BBCG, BP2]. It generalises the notion of a moment-angle complex ZK = (D2 , S 1 )K , which is a key object of study in toric topology. Polyhedral products also provide a unifying framework for several constructions of classifying spaces for right-angled Artin and Coxeter groups, their commutator subgroups, as well as general graph products groups. The description of the classifying spaces of graph product groups and their commutator subgroups was implicit in [PRV], where the canonical homotopy fibration m Y (EG, G)K −→ (BG)K −→ BGk . k=1 of polyhedral products was introduced and studied. To each graph Γ without loops and double edges one can assign a flag simplicial complex K, whose simplices are the vertex sets of complete subgraphs (or cliques) of Γ. For any flag complex K the polyhedral product (BG)K is the classifying 2010 Mathematics Subject Classification. 20F65, 20F12, 57M07. Key words and phrases. Right-angled Artin group, right-angled Coxeter group, graph product, commutator subgroup, polyhedral product. The research of the first author was carried out at the IITP RAS and supported by the Russian Science Foundation grant no. 14-50-00150. The research of the second author was supported by the Russian Foundation for Basic Research grant no. 14-01-00537. 1 2 TARAS PANOV AND YAKOV VERYOVKIN space for the corresponding graph product group G K = G Γ , while (EG, G)K is the classifying space for the commutator subgroup of G K . In the case of right-angled Artin group RAΓ = RAK , each BGi = BZ is a circle, so we obtain as (BG)K the subcomplex (S 1 )K in an m-torus introduced by Kim and Roush in [KR]. In the case of right-angled Coxeter group RC K , each BGi = BZ2 is an infinite real projective space RP ∞ , so the classifying space for RC K is a similarly defined subcomplex (RP ∞ )K in the m-fold product of RP ∞ . The classifying space for the commutator subgroup of RC K is a finite cubic subcomplex RK in an m-dimensional cube, while the classifying space for the commutator subgroup of RAK is an infinite cubic subcomplex LK in the m-dimensional cubic lattice. All these facts are summarised in Theorem 3.2 and Corollaries 3.3 and 3.4. The emphasis of [PRV] was on properties of graph products of topological (rather than discrete) groups, as part of the homotopy-theoretical study of toric spaces and their loop spaces. In the present work we concentrate on the study of the commutator subgroups for discrete graph product groups. Apart from a purely algebraic interest, our motivation lies in the fact that the commutator subgroups of graph products are the fundamental groups of very interesting aspherical spaces. From this topological perspective, right-angled Coxeter groups RC K are the most interesting. The commutator subgroup RC ′K is π1 (RK ) for a finite-dimensional aspherical complex RK , which turns out to be a manifold when K is a simplicial subdivision of sphere. When K is a cycle (the boundary of a polygon) or a triangulated 2-sphere, one obtains as RC ′K a surface group or a 3-manifold group respectively. These groups have attracted much attention recently in geometric group theory and lowdimensional topology. The manifolds RK corresponding to (the dual complexes of) higher-dimensional permutahedra and graph-associahedra also feature as the universal realisators in the works of Gaifullin [Ga1], [Ga2] on the problem of realisation of homology classes by manifolds. In Theorem 4.3 we give a simple criterion for the commutator subgroup of a graph product group to be free. In the case of right-angled Artin groups this result was obtained by Servatius, Droms and Servatius in [SDS]. In Theorem 4.5 we provide an explicit minimal generator set for the finitely generated commutator subgroup of a right-angled Coxeter group RC K . This generator set consists of nested iterated commutators of the canonical generators of RC K which appear in a special order determined by the combinatorics of K. Theorems 4.3 and Theorem 4.5 parallel the corresponding results obtained in [GPTW] for the loop homology algebras and rational homotopy Lie algebras of moment-angle complexes. Algebraically, these results of [GPTW] can be interpreted as a description of the commutator subalgebra in a special graph product graded Lie algebra (see Theorem 4.6). The results of Section 4 in the current paper constitute a group-theoretic analogue of the results of [GPTW] for graded associative and Lie algebras. We dedicate this article to the memory of Rainer Vogt, who shared his great knowledge and ideas with T. P. during insightful collaboration in the 2000s. The authors are grateful to Alexander Gaifullin for his invaluable comments and suggestions, which much helped in making the text more accessible. 2. Preliminaries We consider a finite ordered set [m] = {1, 2, . . . , m} and its subsets I = {i1 , . . . , ik } ⊂ [m], where I can be empty of the whole of [m]. Let K be an (abstract) simplicial complex on [m], i. e. K is a collection of subsets of [m] such that for any I ∈ K all subsets of I also belong to K. We always assume that the empty set ∅ and all one-element subsets {i} ⊂ [m] belong to K. We POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 3 refer to I ∈ K as a simplex (or a face) of K. One-element faces are vertices, and two-element faces are edges. Every abstract simplicial complex K has a geometric realisation |K|, which is a polyhedron in a Euclidean space (a union of convex geometric simplices). In all subsequent constructions it will be useful to keep in mind the geometric object |K| alongside with the abstract collection K. We recall the construction of the polyhedral product (see [BP1, BBCG, BP2]). Construction 2.1 (polyhedral product). Let K be a simplicial complex on [m] and let (X , A) = {(X1 , A1 ), . . . , (Xm , Am )} be a sequence of m pairs of pointed topological spaces, pt ∈ Ai ⊂ Xi , where pt denotes the basepoint. For each subset I ⊂ [m] we set (2.1) m Y  (X , A)I = (x1 , . . . , xm ) ∈ Xk : xk ∈ Ak for k ∈ /I k=1 and define the polyhedral product of (X , A) corresponding to K as [ [ Y Y  (X , A)K = (X , A)I = Ai . Xi × I∈K I∈K i∈I i∈I / In the case when all pairs (Xi , Ai ) are the same, i. e. Xi = X and Ai = A for i = 1, . . . , m, we use the notation (X, A)K for (X , A)K . Also, if each Ai = pt , then we use the abbreviated notation X K for (X , pt )K , and X K for (X, pt )K . This construction of the polyhedral product has the following categorical interpretation. Consider the face category cat(K), whose objects are simplices I ∈ K and morphisms are inclusions I ⊂ J. Let top denote the category of topological spaces. Define a cat(K)-diagram (a covariant functor from the small category cat(K) to the “large” category top) DK (X , A) : cat(K) −→ top, (2.2) I 7−→ (X , A)I , which maps the morphism I ⊂ J of cat(K) to the inclusion of spaces (X , A)I ⊂ (X , A)J . Then we have (2.3) (X , A)K = colim DK (X , A) = colim(X , A)I . I∈K Here colim denotes the colimit functor (also known as the direct limit functor ) from the category of cat(K)-diagrams of topological spaces to the category top. By definition, colim is the left adjoint to the constant diagram functor. The details of these constructions can be found, e. g., in [BP2, Appendix C]. Given a subset J ⊂ [m], consider the restriction of K to J: KJ = {I ∈ K : I ⊂ J}, which is also known as a full subcomplex of K. Recall that a subspace Y ⊂ X is called a retract of X if there exists a continuous map r : X → Y such that the r composition Y ֒→ X −→ Y is the identity. We record the following simple property of the polyhedral product. Proposition 2.2. (X, A)KJ is a retract of (X, A)K whenever KJ ⊂ K is a full subcomplex. Proof. We have  [ Y Y Ai , (X , A)K = Xi × I∈K i∈I i∈[m]\I (X , A)KJ = [ Y I∈K, I⊂J i∈I Xi × Y i∈J\I  Ai . 4 TARAS PANOV AND YAKOV VERYOVKIN Since each Ai is a pointed space, there is a canonical inclusion (X , A)KJ ֒→ (X , A)K . Furthermore, for each I ∈ K there is a projection Y Y Y Y Ai . Ai −→ Xi × rI : Xi × i∈I i∈I∩J i∈[m]\I i∈J\I Since KJ is a full subcomplex, the image of S rI belongs to (X , A)KJ . The projections rI patch together to give a retraction r = I∈K rI : (X , A)K → (X , A)KJ .  The following examples of polyhedral products feature throughout the paper. Example 2.3. 1. Let (X, A) = (S 1 , pt ), where S 1 is a circle. The corresponding polyhedral product (S 1 )K is a subcomplex in the m-torus (S 1 )m : [ (2.4) (S 1 )K = (S 1 )I ⊂ (S 1 )m . I∈K In particular, when K = {∅, {1}, . . . , {m}} (which is m disjoint points geometrically), the polyhedral product (S 1 )K is the wedge (S 1 )∨m of m circles. When K consists of all proper subsets of [m] (which geometrically corresponds to the boundary ∂∆m−1 of an (m − 1)-dimensional simplex), (S 1 )K is known as the fat wedge of m circles; it is obtained by removing the top-dimensional cell from the standard cell decomposition of an m-torus (S 1 )m . For a general K on m vertices, (S 1 )K sits between the m-fold wedge (S 1 )∨m and the m-fold cartesian product (S 1 )m . 2. Let (X, A) = (R, Z), where Z is the set of integer points on a real line R. We denote the corresponding polyhedral product by LK : [ (2.5) LK = (R, Z)K = (R, Z)I ⊂ Rm . I∈K When K consists of m disjoint points, LK is a grid in m-dimensional space Rm consisting of all lines parallel to one of the coordinate axis and passing though integer points. When K = ∂∆m−1 , the complex LK is the union of all integer hyperplanes parallel to coordinate hyperplanes. 3. Let (X, A) = (RP ∞ , pt ), where RP ∞ is an infinite-dimensional real projective space, which is also the classifying space BZ2 for the 2-element cyclic group Z2 . Consider the polyhedral product [ (2.6) (RP ∞ )K = (RP ∞ )I ⊂ (RP ∞ )m . I∈K Similarly to the first example above, (RP ∞ )K sits between the m-fold wedge (RP ∞ )∨m (corresponding to K consisting of m points) and the m-fold cartesian product (RP ∞ )m (corresponding to K = ∆m−1 ). 4. Let (X, A) = (D1 , S 0 ), where D1 is a closed interval (a convenient model is the segment [−1, 1]) and S 0 is its boundary, consisting of two points. The polyhedral product (D1 , S 0 )K is known as the real moment-angle complex [BP1, §3.5], [BP2] and is denoted by RK : [ (2.7) RK = (D1 , S 0 )K = (D1 , S 0 )I . I∈K It is a cubic subcomplex in the m-cube (D1 )m = [−1, 1]m . When K consists of m disjoint points, RK is the 1-dimensional skeleton of the cube [−1, 1]m . When K = ∂∆m−1 , RK is the boundary of the cube [−1, 1]m . In general, if {i1 , . . . , ik } is a face of K, then RK contains 2m−k cubic faces of dimension k which lie in the k-dimensional planes parallel to the {i1 , . . . , ik }th coordinate plane. POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 5 The space RK was introduced and studied in the works of Davis [Da1] and Davis–Januszkiewicz [DJ], although their construction was different. When |K| is homeomorphic to a sphere, RK is a topological manifold (this follows from the results of [Da1], see also [BP2] and [Ca, Theorem 2.3]). Furthermore, the manifold RK has a smooth structure when |K| is the boundary of a convex polytope. In this case RK is the universal abelian cover of the dual simple polytope P [DJ, §4.1]. The four polyhedral products above are related by the two homotopy fibrations [PRV], [BP2, §4.3] (2.8) LK −→ (S 1 )K −→ (S 1 )m , (2.9) RK −→ (RP ∞ )K −→ (RP ∞ )m . Construction 2.4 (right-angled Artin and Coxeter group). Let Γ be a graph on m vertices. We write {i, j} ∈ Γ when {i, j} is an edge. Denote by F (g1 , . . . , gm ) a free group with m generators corresponding to the vertices of Γ. The right-angled Artin group RAΓ corresponding to Γ is defined by generators and relations as follows:  (2.10) RAΓ = F (g1 , . . . , gm ) (gi gj = gj gi for {i, j} ∈ Γ). When Γ is a complete graph we have RAΓ = Zm , while when Γ has no edges we obtain the free group. The right-angled Coxeter group RC Γ is defined as  (2.11) RC Γ = F (g1 , . . . , gm ) (gi2 = 1, gi gj = gj gi for {i, j} ∈ Γ). Both right-angled Artin and Coxeter groups have a categorical interpretation similar to that of polyhedral products (see (2.3)). Namely, consider the following cat(K)-diagrams, this time in the category grp of groups: DK (Z) : cat(K) −→ grp, I 7−→ ZI , DK (Z2 ) : cat(K) −→ grp, I 7−→ ZI2 , Q Q where ZI = i∈I Z and ZI2 = i∈I Z2 . A morphism I ⊂ J of cat(K) is mapped to the monomorphism of groups ZI → ZJ and ZI2 → ZJ2 respectively. Then (2.12) I RAK1 = colimgrp DK (Z) = colimgrp I∈K Z , I RC K1 = colimgrp DK (Z2 ) = colimgrp I∈K Z2 , where K1 denotes the 1-skeleton of K, which is a graph. Here colimgrp denotes the colimit functor in grp. A missing face (or a minimal non-face) of K is a subset I ⊂ [m] such that I is not a simplex of K, but every proper subset of I is a simplex of K. A simplicial complex K is called a flag complex if each of its missing faces consists of two vertices. Equivalently, K is flag if any set of vertices of K which are pairwise connected by edges spans a simplex. A clique (or a complete subgraph) of a graph Γ is a subset I of vertices such that every two vertices in I are connected by an edge. Each flag complex K is the clique complex of its one-skeleton Γ = K1 , that is, the simplicial complex formed by filling in each clique of Γ by a face. Note that the colimits in (2.12), being the corresponding right-angled groups, depend only on the 1-skeleton of K and do not depend on missing faces with more than 2 vertices. For example, the colimits of the diagrams of groups D∆2 (Z) and D∂∆2 (Z) are both Z3 . This reflects the lack of “higher” commutativity in the category of groups: when generators gi commute pairwise, they commute altogether. This phenomenon is studied in more detail in [PRV] and [PR]. 6 TARAS PANOV AND YAKOV VERYOVKIN For these reasons we denote the right-angled Artin and Coxeter groups corresponding to the 1-skeleton of K simply by RAK and RC K respectively. By analogy with the polyhedral product of spaces X K = colimI∈K X I , we may consider the following more general construction of a discrete group. Construction 2.5 (graph product). Let K be a simplicial complex on [m] and let G = (G1 , . . . , Gm ) be a sequence of m groups, which we think of as discrete topological groups. We also assume that none of Gi is trivial, i. e. Gi 6= {1}. For each subset I ⊂ [m] we set m Y  G I = (g1 , . . . , gm ) ∈ Gk : gk = 1 for k ∈ /I . k=1 Then consider the following cat(K)-diagram of groups: DK (G) : cat(K) −→ grp, I 7−→ G I , which maps a morphism I ⊂ J to the canonical monomorphism of groups G I → G J . Define the group I G K = colimgrp DK (G) = colimgrp I∈K G . (2.13) The group G K depends only on the graph K1 and is called the graph product of the groups G1 , . . . , Gm . We have canonical homomorphisms G I → G K , I ∈ K, which can be shown to be injective. As in the case of right-angled Artin and Coxeter groups (corresponding to Gi = Z and Gi = Z2 respectively), one readily deduces the following more explicit description from the universal property of the colimit: Proposition 2.6. The is an isomorphism of groups m  Gk (gi gj = gj gi for gi ∈ Gi , gj ∈ Gj , {i, j} ∈ K), GK ∼ = ⋆ k=1 where ⋆ m k=1 Gk denotes the free product of the groups Gk . Remark. We use the symbol ⋆ to denote the free product of groups, instead of the more common ∗; the latter is reserved for the join of topological spaces. 3. Classifying spaces Here we collect the information about the classifying spaces for graph product groups. The results of this section are not new, but as they are spread across the literature we find it convenient to collect everything in one place. The corresponding references are given below. Recall that a path-connected space X is aspherical if πi (X) = 0 for i > 2. An aspherical space X is an Eilenberg–Mac Lane space K(π, 1) with π = π1 (X). Given a (discrete) group G, there is a universal G-covering EG → BG whose total space EG is contractible and the base BG, known as the classifying space for G, has the homotopy type K(G, 1) (i. e. π1 (BG) = G and πi (BG) = 0 for i > 2). We shall therefore switch between the notation BG and K(G, 1) freely. Note that BZ ≃ S 1 and BZ2 ≃ RP ∞ , with the universal coverings R → S 1 and ∞ S → RP ∞ respectively. Now we use the notation from Construction 2.5. The classifying space BG I is the product of BGi over i ∈ I. We therefore have the polyhedral product (BG)K corresponding to the sequence of pairs (BG, pt ) = {(BG1 , pt ), . . . , (BGm , pt )}. Similarly, we have the polyhedral product (EG, G)K corresponding to the sequence of pairs (EG, G) = {(EG1 , G1 ), . . . , (EGm , Gm )}. Here each Gi is included in EGi as the fibre of the covering EGi → BGi over the basepoint. POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 7 The homotopy fibrations (2.8) and (2.9) can be generalised as follows. Proposition 3.1. The sequence of canonical maps (EG, G)K −→ (BG)K −→ m Y BGk k=1 is a homotopy fibration. When each Gk is Z, we obtain the fibration (2.8), as the pair (EZ, Z) is homotopy equivalent to (R, Z). Similarly, when each Gk is Z2 , we obtain (2.9), as the pair (EZ2 , Z2 ) is homotopy equivalent to (D1 , S 0 ). Qm Proof of Proposition 3.1. We denote k=1 BGk by BG [m] ; this is compatible with the notation BG I . According to [PRV, Proposition 5.1], the homotopy fibre of the inclusion (BG)K → BG [m] can be identified with the homotopy colimit hocolimI∈K G [m] /G I of the cat(K)-diagram in top given on the objects by I 7→ G [m] /G I (where the latter is the quotient group, viewed as a discrete space) and sending a morphism I ⊂ J to the canonical projection G [m] /G I → G [m] /G J of the quotients. This diagram is not Reedy cofibrant, e. g. because G [m] /G I → G [m] /G J is not a cofibration of spaces. The latter map is homotopy equivalent to the closed cofibration (EG, G)I → (EG, G)J , which is a morphism in the cat(K)-diagram DK (EG, G), see (2.2). The diagram DK (EG, G) is Reedy cofibrant, see [BP2, Proposition 8.1.1]. Therefore, the homotopy fibre of the inclusion (BG)K → BG [m] is given by hocolimI∈K G [m] /G I ≃ colimI∈K (EG, G)I = (EG, G)K .  Now we state the following group-theoretic consequence of the homotopy fibration in Proposition 3.1. Theorem 3.2. Let K be a simplicial complex on m vertices, and let GK be a graph product group given by (2.13). (a) π1 ((BG)K ) ∼ = GK . (b) Both spaces (BG)K and (EG, G)K are aspherical if and only if K is flag. Hence, B(GK ) = (BG)K whenever K is flag. (c) πi ((BG)K ) ∼ = πi ((EG, G)K ) for i > 2. (d) Q π1 ((EG, G)K ) is isomorphic to the kernel of the canonical projection GK → m k=1 Gk . Proof. To prove (a) we proceed inductively by adding simplices to K one by one and use van Kampen’s Theorem. The base of the induction is K consisting of m disjoint points. Then (BG)K is the wedge BG1 ∨ · · · ∨ BGm , and π1 ((BG)K ) is the free product G1 ⋆ · · · ⋆ Gm . This is precisely G K , so (a) holds. Assume now that K′ is obtained from K by adding a single 1-dimensional simplex {i, j}. Then, by the definition of the polyhedral product, ′ (BG)K = (BG)K ∪ (BGi × BGj ), where the two pieces are glued along BGi ∨ BGj . By van Kampen’s Theorem, ′ π1 ((BG)K ) is the amalgamated free product π1 ((BG)K ) ⋆(Gi ⋆Gj ) (Gi × Gj ). The latter group is obtained from π1 ((BG)K ) by adding all relations of the form gi gj = ′ gj gi for gi ∈ Gi , gj ∈ Gj . By the inductive assumption, this is precisely G K . Adding simplices of dimension > 2 to K does not change G K and results in adding cells of dimension > 3 to (BG)K , which does not change the fundamental group π1 ((BG)K ). The inductive step is therefore complete, proving (a). 8 TARAS PANOV AND YAKOV VERYOVKIN Now we prove (b). The canonical homomorphisms G I → G K give rise to the maps of classifying spaces BG I → B(G K ). These define a morphism from the cat(K)-diagram DK (BG, pt ) to the constant diagram B(G K ), and hence a map colimI∈K BG I = (BG)K → B(G K ). (3.1) According to [PRV, Proposition 5.1], the homotopy fibre of the map (3.1) can be identified with the homotopy colimit hocolimI∈K G K /G I of the cat(K)-diagram in top given on the objects by I 7→ G K /G I (where the latter is the right coset, viewed as a discrete space) and sending a morphism I ⊂ J to the canonical projection G K /G I → G K /G J of cosets. By [PRV, Corollary 5.4], the homotopy colimit hocolimI∈K G K /G I is homeomorphic to the identification space
 (3.2) Bcat(K) × G K ∼ . Here Bcat(K) is the classifying space of cat(K), which is homeomorphic to the cone on |K|. The equivalence relation ∼ is defined as follows: (x, gh) ∼(x, g) whenever h ∈ G I and x ∈ B(I ↓ cat(K)), where I ↓ cat(K) is the undercategory, whose objects are J ∈ K such that I ⊂ J, and B(I ↓ cat(K)) is homeomorphic to the star of I in K. When K is a flag complex, the identification space (3.2) is contractible by [PRV, Proposition 6.1]. Therefore, the map (3.1) is a homotopy equivalence, which implies that (BG)K is aspherical when K is flag. Assume now that K is not flag. Choose a missing face J = {j1 , . . . , jk } ⊂ [m] with k > 3 vertices and consider the corresponding full subcomplex KJ . Then (BG)KJ is the fat wedge of the spaces {BGj , j ∈ J} (see Example 2.3.1), and it is a retract of (BG)K by Proposition 2.2. Hence, in order to see that (BG)K is not aspherical, it is enough to check that (BG)KJ is not aspherical. Let FW (X1 , . . . , Xk ) denote the fat wedge of spaces X1 , . . . , Xk . According to a result of Porter [Po], the homotopy fibre of the inclusion k Y Xi FW (X1 , . . . , Xk ) ֒→ i=1 k−1 is Σ ΩX1 ∧ · · · ∧ ΩXk , where Σ denotes the suspension and Ω the loop space functor. Q In our case we obtain that the homotopy fibre of the inclusion (BG)KJ → j∈J BGj is Σ k−1 Gj1 ∧ · · · ∧ Gjk . Since each Gj is a discrete space, the latter suspension is a wedge Q of (k − 1)-dimensional spheres. It has nontrivial homotopy group πk−1 . Since j∈J BGj is a K(π, 1)-space, the homotopy exact sequence implies that πk−1 ((BG)KJ ) 6= 0 for some k > 3. Hence, (BG)KJ and (BG)K are non-aspherical. Asphericity of (EG, G)K and statements (c) and (d)Q follow from the homotopy m exact sequence of the fibration in Proposition 3.1, as πi ( k=1 BGk ) = 0, i > 2.  Specialising to the cases Gk = Z and Gk = Z2 respectively we obtain the following results about right-angled Artin and Coxeter groups. Note that in these two Q cases the groups Gk are abelian, so G K → m G k=1 k is the abelianisation homo′ morphism, and its kernel is the commutator subgroup (G K ) . Corollary 3.3. Let K be a simplicial complex on m vertices, let (S 1 )K and LK be the polyhedral products given by (2.4) and (2.5) respectively, and let RAK be the corresponding right-angled Artin group. (a) π1 ((S 1 )K ) ∼ = RAK . (b) Both (S 1 )K and LK are aspherical if and only if K is flag. (c) πi ((S 1 )K ) ∼ = πi (LK ) for i > 2. (d) π1 (LK ) is isomorphic to the commutator subgroup RA′K . POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 9 Corollary 3.4. Let K be a simplicial complex on m vertices, let (RP ∞ )K and RK be the polyhedral products given by (2.6) and (2.7) respectively, and let RC K be the corresponding right-angled Coxeter group. (a) π1 ((RP ∞ )K ) ∼ = RC K . (b) Both (RP ∞ )K and RK are aspherical if and only if K is flag. (c) πi ((RP ∞ )K ) ∼ = πi (RK ) for i > 2. (d) π1 (RK ) is isomorphic to the commutator subgroup RC ′K . Remark. All ingredients in the proof of Theorem 3.2 are contained in [PRV]. The fact that the polyhedral product (BG)K is the classifying space for the graph product group G K whenever K is a flag complex implies that the classifying space functor converts the colimit of groups (defining the graph product) to the colimit of topological spaces (defining the polyhedral product). This is not the case when K is not flag because of the presence of higher Whitehead and Samelson products (see [PRV, PR, GT]), but the situation can be remedied by replacing colimits with homotopy colimits. All these facts were proved in [PRV] for arbitrary well-pointed topological groups. Statements (a) and (b) of Corollary 3.3, implying a homotopy equivalence (S 1 )K ≃ K(RAK , 1) for flag K, were obtained by Kim and Roush [KR, Theorem 10]. Statements (a) and (b) of Corollary 3.4, implying a homotopy equivalence (RP ∞ )K ≃ K(RC K , 1) for flag K, are implicit in the works of Davis [Da1] and Davis–Januszkiewicz [DJ, p. 437]. In particular, contractibility of the space (3.2) (which is the crucial step in the proof of Theorem 3.2 (b)) in the case of right-angled Coxeter group RC K follows from [Da1, Theorem 13.5]. The isomorphism between π1 (RK ) and the commutator subgroup RC ′K was also obtained in the work of Droms [Dr] (his cubic complex is the 2-dimensional skeleton of our complex RK , and therefore has the same fundamental group). In the case of a general graph product G K , the result that both spaces (BG)K and (EG, G)K are aspherical if and only if K is flag appeared in the work of Stafa [St, Theorem 1.1]. Example 3.5. Let K be an m-cycle (the boundary of an m-gon). A simple argument with Euler characteristic shows that RK is homeomorphic to a closed orientable surface of genus (m − 4)2m−3 + 1 (this observation goes back to a 1938 work of Coxeter, see [BP2, Proposition 4.1.8]). Therefore, the commutator subgroup of the corresponding right-angled Coxeter group RC K is a surface group. This example was studied in [SDS] and [Dr]. Similarly, when |K| ∼ = S 2 (which is equivalent to K being the boundary of a 3-dimensional simplicial polytope), RK is a 3-dimensional manifold. Therefore, the commutator subgroup of the corresponding RC K is a 3-manifold group. The fact that 3-manifold groups appear as subgroups in right-angled Artin and Coxeter groups has attracted much attention in the recent literature. All homology groups are considered with integer coefficients. The homology of RK is described by the following result. For the particular case of flag K it gives a description of the homology of the commutator subgroup RC ′K . Theorem 3.6 ([BP1], [BP2, §4.5]). For any k > 0, there is an isomorphism M e k−1 (KJ ), H Hk (RK ) ∼ = J⊂[m] e k−1 (KJ ) is the reduced simplicial homology group of KJ . where H The cohomology ring structure of H ∗ (RK ) is described in [Ca]. 10 TARAS PANOV AND YAKOV VERYOVKIN 4. The structure of the commutator subgroups By Theorem 3.2, m   Y Ker G K → Gk = π1 ((EG, G)K ). k=1 In the case of right-angled Artin or Coxeter groups (or, more generally, when each ′ Gk is abelian), the group above is the commutator subgroup (G K ) . We want to study the group π1 ((EG, G)K ), identify the class of simplicial complexes K for which this group is free, and describe a minimal generator set. We shall need the following modification of a result of Grbić and Theriault [GT]: Proposition 4.1. Let K = K1 ∪I K2 be a simplicial complex obtained by gluing K1 and K2 along a common face I, which may be empty. If the polyhedral products (EG, G)K1 and (EG, G)K2 are homotopy equivalent to wedges of circles, then (EG, G)K is also homotopy equivalent to a wedge of circles. Proof. We may assume that K has the vertex set [m] = {1, . . . , m}, K1 is the full subcomplex of K on the first m1 vertices {1, . . . , m1 }, K2 is the full subcomplex of K on the last m2 vertices {m − m2 + 1, . . . , m}, and the common face I is on the k vertices {m1 − k + 1, . . . , m1 }, where m1 < m, m2 < m and m = m1 + m2 − k. Consider the polyhedral product (CX , X )K corresponding to a sequence of pairs (CX , X ) = {(CX1 , X1 ), . . . , (CXm , Xm )}, where CXi denotes the cone on Xi . According to [GT, Theorem 6.12], (CX , X )K ≃ (M1 ∗ M2 ) ∨ ((CX , X )K1 ⋊ M2 ) ∨ (M1 ⋉ (CX , X )K2 ), Q 1 Qm where M1 = m i=1 Xi , M2 = i=m−m2 +1 Xi , M1 ∗ M2 denotes the join of M1 and M2 , X ⋊ Y denotes the right half-smash X × Y /pt × Y of two pointed spaces X, Y , and X ⋉ Y denotes their left half-smash X × Y /X × pt . In our case, each Xi = Gi is a discrete space, the pair (EGi , Gi ) is homotopy equivalent to (CGi , Gi ), and each of M1 , M2 in (4.1) is a discrete space. Hence, each of the three wedge summands in (4.1) is a wedge of circles, and so is (EG, G)K .  (4.1) A graph Γ is called chordal (or triangulated ) if each of its cycles with > 4 vertices has a chord (an edge joining two vertices that are not adjacent in the cycle). The following result gives an alternative characterisation of chordal graphs. Theorem 4.2 (Fulkerson–Gross [FG]). A graph is chordal if and only if its vertices can be ordered in such a way that, for each vertex i, the lesser neighbours of i form a clique. Such an ordering of vertices is called a perfect elimination ordering. Theorem 4.3. Let K be a flag simplicial complex on m vertices, let G = (G1 , . . . , Gm ) be a sequence of m nontrivial groups, and let GK be the graph product group given by (2.13). The following conditions are equivalent: Qm (a) Ker(GK → k=1 Gk ) is a free group; (b) (EG, G)K is homotopy equivalent to a wedge of circles; (c) Γ = K1 is a chordal graph. Proof. (b)⇒(a) This follows from Theorem 3.2 (d) and the fact that the fundamental group of a wedge of circles is free. (c)⇒(b) Here we use the argument from [GPTW, Theorem 4.6]. However, that argument contained an inaccuracy, which was pointed out by A. Gaifullin and corrected in the argument below. POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 11 Assume that the vertices of K are in perfect elimination order. We assign to each vertex i the clique Ii consisting of i and the lesser neighbours of i. Since K is a flag complex, Sm each clique Ii is a face. All maximal faces are among I1 , . . . , Im , so we have i=1 Ii = K. Furthermore, for each k = 1, . . . , m the perfect elimination ordering on K induces such an ordering on the full subcomplex K{1,...,k−1} , so we Sk−1 Sk−1 have i=1 Ii = K{1,...,k−1} . In particular, the simplicial complex i=1 Ii is flag as Sk−1 a full subcomplex in a flag complex. The intersection Ik ∩ i=1 Ii is a clique, so it Sk−1 is a face of i=1 Ii . An inductive argument using Proposition 4.1 then shows that (EG, G)K is a wedge of circles. Qm (a)⇒(c) Let Ker(G K → k=1 Gk ) be a free group. Suppose that the graph Γ = K1 is not chordal, and choose a chordless cycle J with |J| > 4. Then the full subcomplex KJ is the same cycle (the boundary of a |J|-gon). We first consider the case when each Gk is Z2 , so that (EG, G)K is RK . Then RKJ is homeomorphic to a closed orientable surface of genus (|J| − 4)2|J|−3 + 1 by [BP2, Proposition 4.1.8]. In particular, the fundamental group π1 (RKJ ) is not free. On the other hand, RKJ is a retract of RK by Proposition 2.2, so π1 (RKJ ) is a subgroup of the free group π1 (RK ) = Ker(RC K → (Z2 )m ). A contradiction. Now consider the general case. Note that the pair (EGk , Gk ) is homotopy equivalent to (CGk , Gk ), so we can consider (CG, G)K instead of (EG, G)K . Since each Gk is discrete and nontrivial, we may fix an inclusion of a pair of points S 0 ֒→ Gk ; then there is a retraction Gk → S 0 (it does not have to be a homomorphism of groups). It extends to a retraction of cones, so we have a retraction of pairs (CGk , Gk ) → (D1 , S 0 ). These retractions give rise to a retraction of polyhedral products (CG, G)K → (D1 , S 0 )K = RK . Hence, we have a composite retraction (CG, G)K → RK → RKJ , so π1 (RKJ ) includes as a subgroup in the free group Q 1 π1 (EG, G)K = Ker(G K → m k=1 Gk ). On the other hand, if K contains a chordless cycle J with |J| > 4, then π1 (RKJ ) is the fundamental group of a surface of positive genus, so it is not free. A contradiction.  Corollary 4.4. Let RAK and RC K be the right-angled Artin and Coxeter groups corresponding to a simplicial complex K. (a) The commutator subgroup RA′K is free if and only if K1 is a chordal graph. (b) The commutator subgroup RC ′K is free if and only if K1 is a chordal graph. Part (a) of Corollary 4.4 is the result of Servatius, Droms and Servatius [SDS]. The difference between parts (a) and (b) is that the commutator subgroup RA′K is infinitely generated, unless RAK = Zm , while the commutator subgroup RC ′K is finitely generated. We elaborate on this in the next theorem. Let (g, h) = g −1 h−1 gh denote the group commutator of two elements g, h. Theorem 4.5. Let RC K be the right-angled Coxeter group corresponding to a simplicial complex K on m vertices. The commutator subgroup RC ′K has a finite minP e 0 (KJ ) iterated commutators imal generator set consisting of J⊂[m] rank H (4.2) (gj , gi ), (gk1 , (gj , gi )), ..., (gk1 , (gk2 , · · · (gkm−2 , (gj , gi )) · · · )), where k1 < k2 < · · · < kℓ−2 < j > i, ks 6= i for any s, and i is the smallest vertex in a connected component not containing j of the subcomplex K{k1 ,...,kℓ−2 ,j,i} . Theorem 4.5 is similar to a result of [GPTW] describing the commutator subalgebra of the graded Lie algebra given by 
(4.3) LK = FLhu1 , . . . , um i [ui , ui ] = 0, [ui , uj ] = 0 for {i, j} ∈ K , where FLhu1 , . . . , um i is the free graded Lie algebra on generators ui of degree one, and [a, b] = −(−1)|a||b|[b, a] denotes the graded Lie bracket. The commutator 12 TARAS PANOV AND YAKOV VERYOVKIN subalgebra is the kernel of the Lie algebra homomorphism LK → CLhu1 , . . . , um i to the commutative (trivial) Lie algebra. The graded Lie algebra (4.3) is a graph product similar to the right-angled Coxeter group RC K . It has a colimit decomposition similar to (2.13), with each Gi replaced by the trivial Lie algebra CLhui = FLhui/([u, u] = 0) and the colimit taken in the category of graded Lie algebras. Theorem 4.6 ([GPTW, Theorem 4.3]). The commutator subalgebra of the graded P e 0 (KJ ) Lie algebra LK has a finite minimal generator set consisting of J⊂[m] rank H iterated commutators [uj , ui ], [uk1 , [uj , ui ]], ..., [uk1 , [uk2 , · · · [ukm−2 , [uj , ui ]] · · · ]], where k1 < k2 < · · · < kℓ−2 < j > i, ks 6= i for any s, and i is the smallest vertex in a connected component not containing j of the subcomplex K{k1 ,...,kℓ−2 ,j,i} . Although the scheme of the proof of Theorem 4.5 is similar to that for Theorem 4.6, more specific techniques are required to work with group commutators, as opposed to Lie algebra brackets. Nevertheless, most of these techniques are quite standard, and can be extracted from the classical texts like [MKS]. Proof of Theorem 4.5. The first part is a standard argument applicable to the commutator subgroup of an arbitrary group. An element of RC ′K is a product of commutators (a, b) with a, b ∈ RC K . Writing each of a, b as a word in the generators g1 , . . . , gm and using the Hall identities (a, bc) = (a, c)(a, b)((a, b), c), (4.4) (ab, c) = (a, c)((a, c), b)(b, c), n ni we express each element of RC ′K in terms of iterated commutators (gi1i1 , . . . , giℓ ℓ ) with nik ∈ Z and arbitrary bracketing. Since we have relations gi2 = 1 in RC K , we may assume that each nik is 1. We refer to ℓ > 2 as the length of an iterated commutator. If an iterated commutator (gi1 , . . . , giℓ ) contains a commutator (a, b) where each of a, b is itself a commutator, then we can remove such (gi1 , . . . , giℓ ) from the list of generators by writing (a, b) as a word in shorter commutators a, b and using (4.4) iteratively. We therefore obtain a generators set for RC ′K consisting only of nested iterated commutators, i. e. those not containing (a, b) where both a, b are commutators. The next step is to use the identity ((a, b), c) = (b, a)(c, (b, a))(a, b) and the identities (4.4) to express each nested commutator in terms of canonical nested commutators (gi1 , (gi2 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · )). The most important part of the proof is to express each canonical nested commutator in terms of canonical nested commutators in which the generators gi appear in a specific order. This will be done by a combination of algebraic and topological arguments and use the specifics of the group RC K . We first prove a particular case of the statement, corresponding to K consisting of m disjoint points. The group RC K is then a free product of m copies of Z2 . Lemma 4.7. Let G be a free product of m copies of Z2 , given by the presentation  G = F (g1 , . . . , gm ) (gi2 = 1, i = 1, . . . , m). Then the commutator subgroup G′ is a free group freely generated by the iterated commutators of the form (gj , gi ), (gk1 , (gj , gi )), ..., (gk1 , (gk2 , · · · (gkm−2 , (gj , gi )) · · · )), where k1 < k2 < · · · < kℓ−2 < j > i and ks
6= i for any s. Here, the number of commutators of length ℓ is equal to (ℓ − 1) m ℓ . POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS 13 Proof. The statement is clear for m = 1 (then G = Z2 ) and for m = 2 (then G = Z2 ⋆ Z2 and G′ ∼ = Z with generator (g2 , g1 )). For m = 3, the lemma says that the commutator subgroup of G = Z2 ⋆ Z2 ⋆ Z2 is freely generated by (g2 , g1 ), (g3 , g1 ), (g3 , g2 ), (g1 , (g3 , g2 )), (g2 , (g3 , g1 )). This is easy to see geometrically, by identifying RC ′K with π1 (RK ). In our case RK is the 1-skeleton of the 3-cube (see Example 2.3.4). We have (g1 , (g3 , g2 )) = g1 (g2 , g3 )g1 (g3 , g2 ), (g2 , (g3 , g1 )) = g2 (g1 , g3 )g2 (g3 , g1 ), and the elements (g2 , g1 ), (g3 , g1 ), (g3 , g2 ), g1 (g2 , g3 )g1 , g2 (g1 , g3 )g2 correspond to the loops around five different faces of RK , which freely generate its fundamental group. The general statement for arbitrary m can be proved by a similar topological argument, by identifying G′ with the fundamental group of the 1-skeleton of the m-dimensional cube (see [St, Proposition 3.6]). However, we record an algebraic argument for subsequent use. We have the commutator identity (4.5) (gq , (gp , x)) = (gq , x)(x, (gp , gq ))(gq , gp )(x, gp )(gp , (gq , x))(x, gq )(gp , gq )(gp , x), which can be deduced from the Hall–Witt identity, or checked directly. Note that if x is a canonical nested commutator, then the factor (x, (gp , gq )) can be expressed via nested commutators as in the beginning of the proof of Theorem 4.5. In this case we can use (4.5) to swap gp and gq in the commutator (gq , (gp , x)), by expressing it through (gp , (gq , x)) and canonical nested commutators of lesser length. In the subsequent arguments, we shall swap elements in an iterated commutator. Such a swap will change the element of the group represented by the commutator, but the two elements will always differ by a product of commutators of lesser length, as in the case of (gp , (gq , x)) and (gq , (gp , x)) in the argument above. If we want to swap elements gp and gq inside a canonical nested commutator of the form (· · · , (gp , (gq , x)) · · · ), where x is a smaller canonical nested commutator, then we need to use the first identity of (4.4) alongside with (4.5). Note that if both b and c in the identity (a, bc) = (a, c)(a, b)((a, b), c) are canonical nested commutators, then (a, c) and (a, b) are also canonical nested commutators, while ((a, b), c) = (b, a)c−1 (a, b)c is a product of nested commutators of lesser length. Therefore, using (4.5) together with the identities (4.4) we can change the order of any two generators in a commutator (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) within the positions i1 to iℓ−2 . We first use this observation to eliminate commutators (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) which contain a pair of repeating generators gi (i. e. have ip = iq for some p 6= q). Namely, if the repeating pair occurs within the positions i1 to iℓ−2 , then we use (4.5) to reduce the commutator to the form (· · · (gi , (gi , x)) · · · ), where x = (giℓ−1 , giℓ ), and use the relation (gi , (gi , x)) = (gi , x)(gi , x) to reduce the commutator to a product of commutators of lesser length. (Note that here we use the relation gi2 = 1 in G.) If one of the repeating generators is on the position iℓ−1 or iℓ , then we use (4.5) to reduce the commutator to the form (· · · (gi , (gi , gj )) · · · ) and use the relation (gi , (gi , gj )) = (gi , gj )(gi , gj ). As a result, we obtain a generator set for G′ consisting of commutators (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) with all different gi . This already shows that G′ is finitely generated. Now we use (4.5) to put the generators gi in (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) in an order. Choose the generator gik with the largest index ik . If it is not within the last three positions, then we use (4.5) to move it to the third-to-last position. The case m = 3 considered above shows that the commutator (gj , (gi , gk )) can be expressed through (gi , (gj , gk ), (gk , (gj , gi )) and commutators of lesser length. This allows us to move the generator gik with the largest index ik in (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) to the second-to-last position, and set j = ik . Then we use (4.5) and (4.4) to put the first ℓ − 2 generators in the commutator in an increasing order, and redefine 14 TARAS PANOV AND YAKOV VERYOVKIN their indices as k1 < · · · < kℓ−2 . As a result, we obtain a generator set for G′ consisting of commutators of the required form (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) where k1 < k2 < · · · < kℓ−2 < j > i and ks 6= i for any s. It remains to show that the of G′ is free. This gen
generating set Pmconstructed m m−1 erating set consists of N = ℓ=2 (ℓ − 1) ℓ = (m − 2)2 + 1 commutators. On the other hand, G′ ∼ = π1 (RK ), where RK the 1-skeleton of the m-cube. Then RK is homotopy equivalent to a wedge of N circles (as easy to see inductively or by computing the Euler characteristic), and π1 (RK ) is a free group of rank N . We therefore have a system of N generators in a free group of rank N . This system must be free, by the classical result that a free group of finite rank cannot be isomorphic to its proper quotient group, see [MKS, Theorem 2.13].  Now we resume the proof of Theorem 4.5. We shall exclude some commutators (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) from the generating set using the new commutativity relations. First assume that j and i are in the same connected component of the complex K{k1 ,...,kℓ−2 ,j,i} . We shall show that the corresponding commutator (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) can be excluded from the generating set. We choose a path from i to j, i. e. choose vertices i1 , . . . , in from k1 , . . . , kℓ−2 with the property that the edges {i, i1 }, {i1, i2 }, . . . , {in−1 , in }, {in , j} are all in K. We proceed by induction on the length of the path. The induction starts from the commutator (gj , gi ) = 1 corresponding to a one-edge path {i, j} ∈ K. Now assume that the path consists of n + 1 edges. Using the relation (4.5) we can move the elements gi1 , gi2 , . . . , gin in (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) to the right and restrict ourselves to the commutator (gi1 , (gi2 , · · · (gin , (gj , gi )) · · · )). Observe that in the presence of the commutation relation (gp , gq ) = 1 the identity (4.5) does not contain the factor (x, (gp , gq )) and therefore it allows us to change the order of gp and gq without assuming x to be a commutator. We therefore can convert the commutator (gi1 , (gi2 , · · · (gin , (gj , gi )) · · · )) (with {in , j} ∈ K) to the commutator (gj , (gi1 , · · · (gin−1 , (gin , gi )) · · · )). The latter contains a commutator (gi1 , · · · (gin−1 , (gin , gi )) · · · ) corresponding to a shorter path {i, i1 , . . . , in }. By inductive hypothesis, it can be expressed through commutators of lesser length, and therefore excluded from the set of generators. We therefore obtain a generator set for RC ′K consisting of nested commutators (gk1 , · · · (gkℓ−2 , (gj , gi )) · · · ) with j and i in different connected components of the complex K{k1 ,...,kℓ−2 ,j,i} . Consider commutators (gk1 , · · · (gkℓ−2 , (gj , gi1 )) · · · ) and ′ ′ , j, i2 } (gk1′ , · · · (gkℓ−2 , (gj , gi2 )) · · · ) such that {k1 , . . . , kℓ−2 , j, i1 } = {k1′ , . . . , kℓ−2 and i1 , i2 lie in the same connected component of K{k1 ,...,kℓ−2 ,j,i1 } which is different from the connected component containing j. We claim that one of these commutators can be expressed through the other and commutators of lesser length. To see this, we argue as in the previous paragraph, by considering a path in K{k1 ,...,kℓ−2 ,j,i1 } between i1 and i2 , and then reducing it inductively to a one-edge path. This leaves us with the pair of commutators (gi2 , (gj , gi1 )) and (gi1 , (gj , gi2 )) where {i1 , i2 } ∈ K, {i1 , j} ∈ / K, {i2 , j} ∈ / K. The claim then follows easily from the relation (gi1 , gi2 ) = 1 (compare the case m = 3 of Lemma 4.7). Thus, to enumerate independent commutators, we use the convention of writing (gk1 , · · · (gkℓ−2 , (gj , gi )) · · · ) where i is the smallest vertex in its connected component within K{k1 ,...,kℓ−2 ,j,i} . This leaves us with precisely the set of commutators (4.2). It remains to show that this generating set is minimal. For this we once again recall that RC ′K = π1 (RK ). The first homology group H1 (RK ) is isomorphic to RC ′K /RC ′′K , where RC ′′K is the commutator subgroup of RC ′K . On the other POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS hand, we have H1 (RK ) ∼ = M J⊂[m] 15 e 0 (KJ ) H by Theorem 3.6. Hence, the number of generators in the abelian group H1 (RK ) ∼ = P e 0 (KJ ). The latter number agrees with the number of RC ′K /RC ′′K is J⊂[m] rank H e 0 (K) is one less the number of iterated commutators in the set (4.2), as rank H connected components of K.  r3 Example 4.8. 1. Let K be the simplicial complex 1 r r 2 r 4 on four vertices. Then the commutator subgroup RC ′K is free, and Theorem 4.5 gives the following free generators: (g3 , g1 ), (g4 , g1 ), (g4 , g2 ), (g4 , g3 ), (g2 , (g4 , g1 )), (g3 , (g4 , g1 )), (g1 , (g4 , g3 )), (g3 , (g4 , g2 )), (g2 , (g3 , (g4 , g1 ))). 2. Let K be an m-cycle with m > 4 vertices. Then K1 is not a chordal graph, so the group RC ′K is not free. One can see that RK is an orientable surface of genus (m − 4)2m−3 + 1 (see Example 3.5), so RC ′K ∼ = π1 (RK ) is a one-relator group. When m = 4, we get a 2-torus as RK , and Theorem 4.5 gives the generators a1 = (g3 , g1 ) and b1 = (g4 , g2 ). The single relation is obviously (a1 , b1 ) = 1. For m > 5 we do not know the explicit form of the single relation in the surface group RC ′K ∼ = π1 (RK ) in terms of the generators provided by Theorem 4.5. Compare [Ve], where the corresponding problem is studied for the commutator subalgebra of the graded Lie algebra from Theorem 4.6. References [BBCG] Anthony Bahri, Martin Bendersky, Frederic R. Cohen, and Samuel Gitler. The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces. Adv. Math. 225 (2010), no. 3, 1634–1668. [BP1] Victor Buchstaber and Taras Panov. Torus actions, combinatorial topology and homological algebra. Uspekhi Mat. Nauk 55 (2000), no. 5, 3–106 (Russian). Russian Math. Surveys 55 (2000), no. 5, 825–921 (English). [BP2] Victor Buchstaber and Taras Panov. Toric Topology. Math. Surv. and Monogr., 204, Amer. Math. Soc., Providence, RI, 2015. [Ca] Li Cai. On products in a real moment-angle manifold. Journal of the Mathematical Society of Japan, to appear; arXiv:1410.5543. [Da1] Michael W. Davis. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2) 117 (1983), no. 2, 293–324. [Da2] Michael W. Davis. The geometry and topology of Coxeter groups. London Math. Soc. Monographs Series, 32. Princeton Univ. Press, Princeton, NJ, 2008. [DJ] Michael W. Davis and Tadeusz Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62 (1991), no. 2, 417–451. [Dr] Carl Droms. A complex for right-angled Coxeter groups. Proc. Amer. Math. Soc. 131 (2003), no. 8, 2305–2311. [FG] D. R. Fulkerson and O. A. Gross. Incidence matrices and interval graphs. Pacific J. Math 15 (1965), 835–855. [Ga1] Alexander Gaifullin. Universal realisators for homology classes. Geom. Topol. 17 (2013), no. 3, 1745–1772. [Ga2] Alexander Gaifullin. Small covers of graph–asociahedra and realisation of cycles. Mat. Sbornik 207 (2016) (Russian); Sbornik Math. 207 (2016) (English translation, in this volume). [GPTW] Jelena Grbić, Taras Panov, Stephen Theriault and Jie Wu. Homotopy types of momentangle complexes for flag complexes. Trans. Amer. Math. Soc. 368 (2016), no. 9, 6663– 6682. [GT] Jelena Grbić and Stephen Theriault. Homotopy theory in toric topology. Russian Math. Surveys 71 (2016), no. 2. 16 [KR] [MKS] [PR] [PRV] [Po] [SDS] [St] [Ve] TARAS PANOV AND YAKOV VERYOVKIN Ki Hang Kim and Fred W. Roush. Homology of certain algebras defined by graphs. J. Pure Appl. Algebra 17 (1980), no. 2, 179–186. Wilhelm Magnus, Abraham Karrass and Donald Solitar. Combinatorial group theory. Presentations of groups in terms of generators and relations. Second revised edition. Dover Publications, Inc., New York, 1976. Taras Panov and Nigel Ray. Categorical aspects of toric topology. In: Toric Topology, M. Harada et al., eds. Contemp. Math., 460. Amer. Math. Soc., Providence, RI, 2008, pp. 293–322. Taras Panov, Nigel Ray and Rainer Vogt. Colimits, Stanley–Reiner algebras, and loop spaces. In: Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001). Progress in Math., 215, Birkhäuser, Basel, 2004, pp. 261–291. Gerald J. Porter. The homotopy groups of wedges of suspensions. Amer. J. Math. 88 (1966), 655–663. Herman Servatius, Carl Droms and Brigitte Servatius. Surface subgroups of graph groups. Proc. Amer. Math. Soc. 106 (1989), no. 3, 573–578. Mentor Stafa. On the fundamental group of certain polyhedral products. J. Pure Appl. Algebra 219 (2015), no. 6, 2279–2299. Yakov Veryovkin. Pontryagin algebras of some moment-angle-complexes. Dal’nevost. Mat. Zh. (2016), no. 1, 9–23 (in Russian); arXiv:1512.00283. Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory, 119991 Moscow, Russia, Institute for Theoretical and Experimental Physics, Moscow, Russia and Institute for Information Transmission Problems, Russian Academy of Sciences E-mail address: tpanov@mech.math.msu.su URL: http://higeom.math.msu.su/people/taras/ Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory, 119991 Moscow, Russia, Steklov Mathematical Institute, Russian Academy of Sciences E-mail address: verevkin j.a@mail.ru 

IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) arXiv:1609.09246v1 [] 29 Sep 2016 KAZUHIKO KURANO AND KAZUMA SHIMOMOTO Abstract. In this paper, we give a detailed proof to a result of Gabber (unpublished) on the ideal-adic completion of quasi-excellent rings, extending the previous work on Nishimura-Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent ring is excellent. 1. Introduction Throughout this paper, we assume that all rings are commutative and possess an identity. The aim of this article is to give a detailed proof to the following theorem (see Theorem 5.1). Main Theorem 1 (Nishimura-Nishimura, Gabber). Let A be a Noetherian ring, and I an ideal of A. Assume that A is I-adically complete. Then, if A/I is quasi-excellent, so is A. This result was proved in characteristic 0 by Nishimura-Nishimura in [12], using the resolution of singularities. More recently, the general case was settled by Gabber, using his theorem of local uniformization of quasi-excellent schemes. The idea of his proof is sketched in a letter [16] from Gabber to Laszlo. The above theorem is a special and difficult case of the Lifting Problem, which was formulated by Grothendieck [5, Remarque (7.4.8)]. For the precise definition of lifting problem as well as its variations, we refer the reader to Definition 2.4. As an important corollary, we obtain the following theorem (see Corollary 5.5). Main Theorem 2. Let A be an excellent ring with an ideal I ⊂ A. Then the I-adic completion of A is an excellent ring. In particular, if A is excellent, then the formal power series ring A[[x1 , . . . , xn ]] is excellent. Here is an outline of the paper. Key words and phrases. Ideal-adic completion, lifting problem, local uniformization, Nagata ring, quasiexcellent ring. 2010 Mathematics Subject Classification: 13B35, 13F25, 13F40. 1 2 K. KURANO AND K. SHIMOMOTO In § 2, we fix notation and definitions concerning excellent rings for which we refer the reader to [9] and [10]. Then we introduce terminology related to the class of ring theoretic properties preserved under ideal-adic completion and make a table on the known results. In § 3, we begin with the notions of quasi-excellent schemes and alteration coverings. Then we recall the recent result of Gabber on the existence of local unifromization of a quasi-excellent Noetherian scheme as an alteration covering. In § 4, using Gabber’s local uniformization theorem, we give a proof to the classical theorem of Brodmann and Rotthaus in the full generality. This result is an important step to prove the main result of this paper. In § 5, we finish the proof of the main result after proving a number of intermediate lemmas based on the ideas explained in [12] and [16] 2. Notation and conventions We use the following notation. Let I be an ideal of a ring A. Then denote by V (I) the set of all prime ideals of A containing I. Let D(I) := Spec A \ V (I). Let A be a ring with bI the I-adic completion lim A/I n . If (A, m) is a local ring, an ideal I. We denote by A ←−n b or A∧ the m-adic completion lim A/mn . Let A be an integral we simply denote by A ←−n domain. Denote by Q(A) the field of fractions of A. Now let P be a ring theoretic property of local Noetherian rings (e.g., regular, normal, reduced,. . .). Let A be a Noetherian algebra over a field K. We say that A has geometrically P over K, if every local ring of A ⊗K L is P for every finite field extension L/K. Let k(p) denote the residue field of A at p ∈ Spec A. For the following definition, we refer the reader to [5, (7.3.1)]. Definition 2.1. Let P be a ring theoretic property of local Noetherian rings. (1) A homomorphism of Noetherian rings ψ : A → B is said to be a P-homomorphism, if ψ is flat and the fiber B ⊗A k(p) is geometrically P over k(p) for any p ∈ Spec A. A homomorphism of Noetherian rings is said to be a regular homomorphism, if it is a P-homomorphism with P being regular. (2) A Noetherian ring A is said to be a P-ring, if for any p ∈ Spec A, the natural cp is a P-homomorphism. homomorphism Ap → A For the following definition, we refer the reader to [9] and [10]. Definition 2.2. Let A be a Noetherian ring. (1) We say that A is a G-ring (resp. Z-ring), if A is a P-ring with P being regular (resp. normal). (2) We say that A is a J2 -ring, if the regular locus of every finitely generated A-algebra is Zariski open. IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 3 (3) We say that A is catenary, if for any pair of prime ideals p ⊂ q of R, let p = p0 ⊂ p1 ⊂ · · · ⊂ pm = q and p = p′0 ⊂ p′1 ⊂ · · · ⊂ p′n = q be saturated strictly increasing chains of prime ideals between p and q, then we have m = n. We say that A is universally catenary, if every finitely generated A-algebra is catenary. A Noetherian ring is called quasi-excellent, if it is a G-ring and a J2 -ring. A Noetherian ring is called excellent, if it is a G-ring, a J2 -ring and universally catenary. Any field, the ring of integers and complete local rings are excellent rings. Polynomial rings over a G-ring are also G-rings in [9, Theorem 77]. One can prove the same for Z-rings in the same way. Therefore, the category of G-rings (that of Z-rings, that of J2 -rings or that of universally catenary rings) is closed under polynomial extensions, homomorphic images and localizations. It is known that any semi-local G-ring is a J2 -ring (cf. [9, Theorem 76]). Hence semilocal G-rings are quasi-excellent. Definition 2.3. Let A be a Noetherian ring. (1) We say that A is an N2 -ring (or Japanese ring), if A is an integral domain and the integral closure of A in any finite field extension of Q(A) is module-finite over A, where Q(A) is the field of fractions of A. (2) We say that A is a Nagata ring (or universally Japanese ring), if for any P ∈ Spec A the integral domain A/P is an N2 -ring. It is known that quasi-excellent rings are Nagata rings (cf. [9, Theorem 78]). The category of Nagata rings is closed under polynomial extensions, homomorphic images and localizations (cf. [9, Theorem 72]). Definition 2.4. Let P′ be a ring theoretic property of Noetherian rings. (1) We say that the Lifting Property (LP for short) holds for P′ , if the following condition holds. For any Noetherian ring A with an ideal I such that A is Iadically complete, if A/I is P′ , then A is P′ . (2) We say that the Local Lifting Property (LLP for short) holds for P′ , if the lifting property holds for any semi-local ring A. (3) We say that the Power Series Extension Property (P SEP for short) holds for P′ , if the following condition holds. For any Noetherian ring A, if A is P′ , then the formal power series ring A[[x]] is P′ . (4) We say that the Ideal-adic Completion Property (ICP for short) holds for P′ , if the following condition holds. For any Noetherian ring A with an ideal I, if A is P′ , then the I-adic completion of A is P′ . 4 K. KURANO AND K. SHIMOMOTO For any property P′ , it is easy to see the following implications (see Corollary 5.5 and its proof): LLP ⇐= LP =⇒ P SEP. If the category of Notherian rings having the property P′ is closed under homomorphic images, then the implication P SEP =⇒ ICP holds. If the category of Notherian rings having the property P′ is closed under polynomial extension, then the implication P SEP ⇐= ICP holds. In this article, we consider P′ as one of the following properties: G-ring, quasi-excellent, excellent, universally catenary, Nagata, Nagata Z-ring. Let us remark that the category of Noetherian rings with P′ as above is closed under polynomial extensions and homomorphic images. In 1970, Seydi [15] proved that PSEP holds for P′ = universally catenary. In 1975, Marot [8] proved that LP holds for P′ = Nagata. In 1979, Rotthaus [13] proved that LLP holds for P′ = G-ring. Hence, LLP holds for P′ = quasi-excellent. In 1981, Nishimura [11] found an example, and proved that ICP does not hold for P′ = G-ring. In 1982, Greco [4] found an example, and proved that LLP does not hold for both P′ = universally catenary and P′ = excellent. In 1987, Nishimura-Nishimura ([12, Theorem A]) proved that LP holds for P′ = Nagata Z-ring. They also proved that LP holds for P′ = quasi-excellent, if the ring contains a field of characteristic 0 ([12, Theorem B]). Recently, Gabber [16] proved that LP holds for P′ = quasi-excellent in general. The aim of this paper is to complete the following table by giving the details of Gabber’s theorem. LLP G-ring quasi-excellent excellent universally catenary Nagata × × LP PSEP ICP × × × × × Nagata Z-ring Here, remark that PSEP and ICP hold for P′ = excellent, since they hold for both P′ = quasi-excellent and P′ = universally catenary. IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 5 3. Gabber’s local uniformization theorem for quasi-excellent schemes In this section, let us recall a recent result on the existence of local uniformizations for quasi-excellent schemes, due to Gabber. Definition 3.1. A Noetherian scheme X is quasi-excellent (resp. excellent), if X admits an open affine covering, each of which is the spectrum of a quasi-excellent ring (resp. an excellent ring). Once this condition holds, then any other open affine covering has the same property. We introduce the notion of alteration covering of schemes. Definition 3.2. In this definition, we assume that all schemes are Notherian, and all morphisms are dominant, generically finite morphisms of finite type between Notherian schemes. Let Y be a Noetherian integral scheme. We say that a finite family of scheme maps {φi : Xi → Y }i=1,...,m is an alteration covering of Y , if there exist a proper morphism S f : V → Y , a Zariski open covering V = m i=1 Vi , together with a family of scheme maps {ψi : Vi → Xi }i=1,...,m such that the following diagram commutes for each i = 1, . . . , m Vi (3.1) ↓ ψi Xi −→ φi −→ V ↓f Y where Vi → V is the natural open immersion. If Xi is a regular integral scheme for each i = 1, . . . , m, we say that {φi : Xi → Y }i=1,...,m is a regular alteration covering of Y . We refer the reader to [7] for alteration coverings or the alteration topology in the general situation. The definiton adopted in [7] looks slightly different from the above one. However, we may resort to [7, Théorèm 3.2.1; EXPOSÉ II]. For the convenience of readers, notice that the alteration topology is the same as Voevodsky’s h-topology in the Noetherian case (see [2] for the proof of this fact). Let us state Gabber’s local uniformization theorem for which we refer the reader to [7]. Theorem 3.3 (Gabber). Assume that Y is a quasi-excellent Noetherian integral scheme. Then there exists a regular alteration covering of Y . Using the valuative criterion for proper maps and Gabber’s theorem, we obtain the following corollary. That is the reason why the above theorem is called “local uniformization theorem” for quasi-excellent schemes. 6 K. KURANO AND K. SHIMOMOTO Corollary 3.4. Assume that A is a quasi-excellent domain. Then there exists a finite field extension K/Q(A) such that if R is a valuation domain satisfying Q(R) = K and A ⊂ R, then there exists a regular domain B such that A ⊂ B ⊂ R and B is a finitely generated A-algebra. 4. A generalization of a theorem of Brodmann and Rotthaus The purpose of this section is to prove the following theorem. It was proved by Brodmann and Rotthaus [1] for rings containing a field of characteristic zero, using the resolution of singularities by Hironaka. Let rad(A) be the Jacobson radical of a ring A. Theorem 4.1. Let A be a Noetherian ring with an ideal I ⊂ rad(A). Assume that A/I is quasi-excellent and A is a G-ring. Then A is a J2 -ring. In other words, A is quasiexcellent. We need to prove a number of lemmas before proving this theorem. Let X be a Noetherian scheme. We denote by Reg(X) the regular locus of X, and put Sing(X) = X \Reg(X). Let us recall that a subset of a Noetherian scheme is open if and only if it is constructible and stable under generalization of points [6, II. Ex. 3.17, 3.18]. Lemma 4.2. Let A be a Noetherian ring with an ideal I ⊂ A and let π : X → Spec A be
a scheme map of finite type. Assume that A/I is a J2 -ring. Then Reg(X) ∩ π −1 V (I) is
open in π −1 V (I) . Proof. For the proof, we may assume that X is an affine scheme. Let B be an A-algebra of finite type and put X = Spec B. First, let us prove the following lemma: Claim 4.3. Assume that Z ⊂ V (IB) is a closed subset. Then Reg(X) ∩ Z is constructible in Z. Proof of the claim. We prove it by Noetherian induction. So let us suppose that any proper closed subset of Z satisfies the conclusion of the claim. We may assume that Z = V (q) for some prime ideal q ⊃ IB. Since A/I is a J2 -ring, we have
(4.1) Reg Spec(B/q) is a non-empty open set in Spec(B/q). Assume that q ∈ / Reg(X). Then we have Reg(X) ∩ Z = ∅ and this is evidently con- structible. Next, assume that q ∈ Reg(X). Then since Bq is a regular local ring, the maximal ideal qBq is generated by a regular sequence. This together with (4.1) implies the following: - There exists an element f ∈ B \ q such that qB[f −1 ] is generated by a B[f −1 ]regular sequence and B[f −1 ]/qB[f −1 ] is a regular ring. IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 7 Hence if p ∈ Spec B is taken such that p ∈ V (q) and p ∈ / V (q + f B), then Bp is regular. We have the decomposition:   
 Reg(X) ∩ V (q) = Reg(X) ∩ V (q + f B) ∪ Reg(X) ∩ V (q) \ V (q + f B) By Noetherian induction hypothesis, we see that Reg(X) ∩ V (q + f B) is constructible.
On the other hand, we have Reg(X) ∩ V (q) \ V (q + f B) = V (q) \ V (q + f B) and this is clearly open in V (q). We conclude that Reg(X) ∩ V (q) is constructible, which finishes the proof of the claim.  We can finish the proof of the lemma in the following way. It is easy to show that

Reg(X) ∩ π −1 V (I) is closed under taking generalizations of points inside π −1 V (I) .

 Combining Claim 4.3, we conclude that Reg(X) ∩ π −1 V (I) is open in π −1 V (I) . Using this lemma, we can prove the following crucial fact. Lemma 4.4. Let B be a Noetherian domain and let us choose q ∈ Spec B and an ideal J ⊂ B. Assume that Bq is a G-ring and B/J is a J2 -ring. Then there exists b ∈ B \ q together with an alteration covering {φb,i : Xb,i → Spec B[b−1 ]}i such that   B[b−1 ] (4.2) φ−1 Spec ⊂ Reg(Xb,i ) b,i JB[b−1 ] for all i. Proof. Since Bq is a quasi-excellent local domain by assumption, there exists a regular alteration covering {φi : Xi → Spec Bq }i=1,...,m , together with a proper map f : V → S Spec Bq with V := m i=1 Vi and {ψi : Vi → X}i by Theorem 3.3. By Chow’s lemma, we may assume that f : V → Spec Bq is projective. Then we can find an element b̃ ∈ B \q and an alteration covering {φb̃,i : Xb̃,i → Spec B[b̃−1 ]}i=1,...,m such that φi = φb̃,i ⊗B[b̃−1 ] Bq . Remark that, if φb̃,i (s) is a generalization of qB[b̃−1 ], then we have s ∈ Xi . In particular, OXb̃,i ,s is regular. (4.3) Let us put   B[b̃−1 ] Zb̃ := Spec , which is a closed subset of Spec B[b̃−1 ]. JB[b̃−1 ] Then we find that Wb̃,i := φ−1 (Zb̃ ) \ Reg(Xb̃,i ) is closed in φ−1 (Zb̃ ) b̃,i b̃,i by Lemma 4.2. Hence Wb̃,i is a closed subset of Xb̃,i and thus, φb̃,i (Wb̃,i ) is a constructible subset of Spec B[b̃−1 ] by Chevalley’s theorem (cf. [9, Theorem 6]). From this, it follows that the Zariski closure φb̃,i (Wb̃,i ) of φb̃,i (Wb̃,i ) is the set of all points of Spec B[b̃−1 ] that are obtained as a specialization of a point of φb̃,i (Wb̃,i ). 8 K. KURANO AND K. SHIMOMOTO Assume that we have q ∈ φb̃,i (Wb̃,i ). Then there is a point s ∈ Wb̃,i such that q ∈ {φb̃,i (s)}. By (4.3), the local ring OXb̃,i ,s is regular. Hence s ∈ Reg(Xb̃,i ), which is a contradiction to s ∈ Wb̃,i . Thus, we must get q ∈ / φb̃,i (Wb̃,i ). Let us choose an element 0 6= b ∈ B such that q ∈ Spec B[b−1 ] ⊂ Spec B[b̃−1 ] and Spec B[b−1 ] ∩ φb̃,i (Wb̃,i ) = ∅ for all i. Consider the fiber square: φb,i −→ Spec B[b−1 ] Xb,i ↓ ↓ φb̃,i −→ Spec B[b̃−1 ] Xb̃,i Then we have Wb̃,i ∩ Xb,i = ∅. Let us put   B[b−1 ] and Wb,i := φ−1 Zb := Spec b,i (Zb ) \ Reg(Xb,i ). JB[b−1 ] Since Zb = Zb̃ ∩ Spec B[b−1 ], we get

−1 −1 (Zb̃ ) ∩ Xb,i \ Reg(Xb̃,i ) = Wb̃,i ∩ Xb,i = ∅. Wb,i = φb̃,i (Zb̃ ) ∩ Xb,i \ Reg(Xb,i ) = φb̃,i This proves the assertion (4.2).  Lemma 4.5. Let {φi : Xi → Y }i=1,...,m be an alteration covering and let y1 , . . . , yl be a sequence of points in Y such that yj+1 ∈ {yj } for j = 1, . . . , l − 1, where {yj } denotes the Zariski closure of {yj } in Y . Then there exist i and a sequence of points x1 , . . . , xl in Xi such that φi (xj ) = yj for j = 1, . . . , l and xj+1 ∈ {xj } for j = 1, . . . , l − 1. Proof. By assumption, for i = 1, . . . , m, there is a commutative diagram: Vi ↓ ψi Xi where V = Sm i=1 Vi −→ φi −→ V ↓f Y f is a Zariski open covering and V − → Y is a proper surjective map. Let us find a sequence v1 , . . . , vl in V such that vj maps to yj for j = 1, . . . , l and vj+1 ∈ {vj } for j = 1, . . . , l − 1. First, lift y1 to a point v1 ∈ V via f . Suppose that a sequence v1 , . . . , vt in V has been found such that f (vj ) = yj for j = 1, . . . , t and vj+1 ∈ {vj } for j = 1, . . . , t − 1. So let us find vt+1 ∈ V with the required condition. Since f is a proper map, f ({vt }) is equal to {yt }. Hence yt+1 ∈ f ({vt }) and there is a lift vt+1 of yt+1 such that vt+1 ∈ {vt }. Here, we have vl ∈ Vi for some i. Since Vi is closed under generalizations, v1 , . . . , vl are contained in Vi . Then we see that the sequence x1 := ψi (v1 ), . . . , xl := ψi (vl ) in Xi satisfies the required conditions.  IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 9 Proof of Theorem 4.1. In order to show that A is a J2 -ring, it is enough to prove that the regular locus of any finite A-algebra is open (cf. [9, Theorem73]). Let B be a Noetherian domain that is finitely generated as an A-module. It suffices to prove that Reg(B) contains a non-empty open subset of Spec B by Nagata’s topological criterion (cf. [10, Theorem 24.4]). Since A → B is module-finite, we have IB ⊂ rad(B) and B/IB is quasi-excellent. Since B is a G-ring, Bq is quasi-excellent for any q ∈ Spec B by [9, Theorem 76]. By Lemma 4.4, there exists bq ∈ B \ q and an alteration covering {φbq ,i : Xbq ,i → Spec B[b−1 q ]}i such that (4.4) φ−1 bq ,i  Spec B[b−1 q ] IB[b−1 q ]  ⊂ Reg(Xbq ,i ) for all i. There exists a family of elements b1 , . . . , bs ∈ {bq | q ∈ Spec B} such that (4.5) −1 Spec B = Spec B[b−1 1 ] ∪ · · · ∪ Spec B[bs ]. There exists a finite family of alteration coverings {φbj ,i : Xbj ,i → Spec B[b−1 j ]}i with the same property as (4.4) by letting bj = bq . By the lemma of generic flatness (cf. [9, (22.A)]), we can find an element 0 6= c ∈ B such that the induced map
−1 −1 −1 −1 φ−1 bj ,i Spec B[bj c ] → Spec B[bj c ] is flat for all i and j. If we can show that B[c−1 ] is regular, then the proof is finished. Let us pick p ∈ Spec B such that c ∈ / p. Then we want to prove that Bp is regular. Let us choose a maximal ideal m ∈ Spec B such that p ⊂ m. Since IB ⊂ rad(B), it follows that IB ⊂ m. By (4.5), there exists j such that p, m ∈ Spec B[b−1 j ]. By Lemma 4.5, there exist x1 , x2 ∈ Xbj ,i for some i such that x2 ∈ {x1 }, φbj ,i (x1 ) = p and φbj ,i (x2 ) = m. By (4.4) together with the fact IB ⊂ m, the local ring OXbj ,i ,x2 is regular. Since x1 is a generalization of x2 , OXbj ,i ,x1 is also a regular local ring. Here, Bp → OXbj ,i ,x1 is flat, as c∈ / p. Therefore, Bp is regular, as desired.  5. Lifting problem for quasi-excellent rings In this section, we shall prove the main theorem: Theorem 5.1 (Nishimura-Nishimura, Gabber). Let A be a Noetherian ring, and I an ideal of A. Assume that A is I-adically complete. Then, if A/I is quasi-excellent, so is A. Proof. Assume the contrary. Let A be a Noetherian ring, and I an ideal of A. Suppose that A is I-adically complete, A/I is quasi-excellent, but A is not quasi-excellent. Step 1. We shall reduce this problem to a simpler case as long as possible. (1-1) The following are well-known facts. 10 K. KURANO AND K. SHIMOMOTO • Let R be a Noetherian ring and I1 , I2 be ideals of R such that I1 ⊃ I2 . If R is I1 -adically complete, then R is I2 -adically complete. • Let R be a Noetherian ring and J1 , J2 be ideals of R. If R is J1 -adically complete, then R/J2 is ((J1 + J2 )/J2 )-adically complete. We shall use these facts without proving them. Suppose I = (a1 , . . . , at ). Put Ii = (a1 , . . . , ai ) for i = 1, . . . , t and I0 = (0). Remark that A/Ii is (Ii+1 /Ii )-adically complete, and (A/Ii )/(Ii+1 /Ii ) = A/Ii+1 for i = 0, 1, . . . , t − 1. Here, Ii+1 /Ii is a principal ideal of A/Ii . Remember that A/It is quasi-excellent, but A/I0 is not so. Therefore, there exists i such that A/Ii+1 is quasiexcellent, but A/Ii is not so. Replacing A/Ii and Ii+1 /Ii with A and I respectively, we may assume that (A1) I is the principal ideal generated by some x 6= 0, that is, I = (x). (1-2) We put F = {J | A/J is not quasi-excellent}. Since F contains (0), the set F is not empty and there exists a maximal element J0 in F. Replacing A/J0 by A, we may assume that (A2) if J 6= (0), then A/J is quasi-excellent. (1-3) By Theorem 4.1, A is not a G-ring. There exist prime ideals P and Q of A such that P ⊃ Q, and the generic fiber of AP /QAP −→ (AP /QAP )∧ is not geometrically regular. On the other hand, if Q 6= (0), the above map is a regular homomorphism by (A2). Therefore we know Q = (0), that is, A is an integral domain. Since quasi-excellent rings are Nagata [9, Theorem 78], we know that A/xA is a Nagata ring. Since the lifting property holds for Nagata rings by Marot [8], A is a Nagata domain. Let A be the integral closure of A in Q(A). Then A is module-finite over A. By Greco’s theorem [3, Theorem 3.1], A is not quasi-excellent, since A is not so. Here A is xA-adically complete and A satisfies (A2). Replacing A with A, we further assume that (A3) A is a Nagata normal domain. (1-4) Since A/xA is quasi-excellent, A/xA is a Nagata Z-ring. Since the lifting property holds for Nagata Z-rings by Nishimura-Nishimura [12, Theorem A], A is also a Nagata cP is a local normal domain for any prime ideal P of Z-ring. Since A is a normal Z-ring, A A. Then by [10, Theorem 31.6], A is universally catenary. Thus we know IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 11 (A4) A is a Z-ring and universally catenary. Step 2. We assume (A1), (A2), (A3) and (A4). By Theorem 4.1, A is not a G-ring. By [9, Theorem 75], there exists a maximal ideal cm is not regular. Consider the fibers of m of A such that the homomorphism Am → A cm . By (A2), the fibers except for the generic fiber are geometrically regular. So Am → A we concentrate on the generic fiber. Let L be a finite algebraic extension of Q(A), where Q(A) is the field of fractions of A. Let BL be the integral closure of A in L. Remark that BL is a finite A-module by (A3). Consider the following fiber squares: L = L ↑ ↑ ↑ ↑ −→ cm L ⊗A A ↑ = cm = BL −→ BL ⊗A Am −→ BL ⊗A A A −→ Am −→ ↑ cm A Q nL \ ⊗BL (B L )n Q \ n (BL )n Here n runs over all the maximal ideals of BL lying over m. \ We want to discuss whether L ⊗BL (B L )n is regular or not for L and n. Recall that (BL )n /x(BL )n is a G-ring since A/xA is quasi-excellent. Let (BL )∗n be the x(BL )n -adic completion of (BL )n . By the local lifting property for G-rings (Rotthaus [13]), (BL )∗n is a G-ring. (5.1)
∗ \ ∗ \ Hence, the homomorphism (BL )∗n → (B L )n = (BL )n is regular and Sing (BL )n is a closed \ subset of Spec(BL )∗ . Therefore, L ⊗B (BL )∗ is regular if and only if L ⊗B (B L )n is n L n L regular. For a finite algebraic extension L of Q(A), we put ) (
∗ is a minimal prime ideal of Sing (B )∗ such that q∗ ∩ B = (0), q L L n n n SL = q∗n where n is a maximal ideal of BL . Here, L ⊗BL (BL )∗n is not regular for some L and n, since A is not a G-ring. Therefore, SL is not empty for some L. We put (5.2) h0 = min{ht Q | Q ∈ SL for some L}. Remember that A is a Z-ring by (A4). Since BL is a finite A-module, BL is also a ∗ \ Z-ring. It is easy to see that (B L )n is the completion of both (BL )n and (BL )n . Since (BL )n and (BL )∗n are Z-rings (cf. (5.1)), (5.3) \ (BL )n , (BL )∗n and (B L )n are local normal domains. Therefore we know (5.4) h0 ≥ 2. 12 K. KURANO AND K. SHIMOMOTO We shall prove the following claim in the rest of Step 2. Claim 5.2. Let p be a prime ideal of A. If ht p ≤ h0 , then Ap is excellent. Let p be a prime ideal of A such that 0 < ht p ≤ h0 . By (A4), A is universally catenary. It is enough to prove that Ap is a G-ring. By (A2) and [9, Theorem 75], it is enough to cp is geometrically regular. Let L be a finite algebraic show that the generic fiber of Ap → A extension of Q(A), and BL be the integral closure of A in L. Consider the following fiber squares. L = L ↑ ↑ ↑ ↑ −→ cp L ⊗A A = ↑ cp = BL −→ BL ⊗A Ap −→ BL ⊗A A A −→ Ap −→ Q qL ↑ c Ap \ ⊗BL (B L )q Q \ q (BL )q Here q runs over all the prime ideals of BL lying over p. Since A is normal, we have 0 < ht q = ht p ≤ h0 . (5.5) \ It is enough to show that L ⊗BL (B L )q is regular. Let n be a maximal ideal of BL such that n ⊃ q. Let q∗ be a minimal prime ideal of q(BL )∗n . Since (BL )n → (BL )∗n is flat, we have q∗ ∩ BL = q. Consider the commutative diagram: (BL )∗n −→ ((BL )∗n )q∗ ↑ BL −→ (BL )n −→ ↑α (BL )q β −→ (((BL )∗n )q∗ )∧ −→ ↑ α̂ \ (BL )q The map α as above is a flat local homomorphism. Since the closed fiber of α is of dimension 0, we have dim(BL )q = dim((BL )∗n )q∗ . In particular ht q = ht q∗ . (5.6)
By (5.1), β is a regular homomorphism and Sing (BL )∗n is a closed subset of Spec(BL )∗n .
√ ∗ Let c∗n be the defining ideal of Sing (BL )∗n satisfying c∗n = cn . We put T = (((BL )∗n )q∗ )∧ . Since β is a regular homomorphism, c∗n T is a defining ideal of Sing(T ). Suppose c∗n = q∗n,1 ∩ · · · ∩ q∗n,s , where q∗n,i ’s are prime ideals of (BL )∗n such that q∗n,i 6⊃ q∗n,j if i 6= j. One of the following three cases occurs: Case 1. If none of q∗n,i ’s is contained in q∗ , then c∗n T = T . Case 2. If q∗n,i = q∗ for some i, then c∗n T = q∗ T . Case 3. Suppose that q∗n,1 , . . . , q∗n,t are properly contained in q∗ , and q∗n,t+1 , . . . , q∗n,s are not contained in q∗ for some t satisfying 1 ≤ t ≤ s. Then, c∗n T = q∗n,1 T ∩ · · · ∩ q∗n,t T . IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 13 In any case as above, we can verify c∗n T ∩ BL 6= (0) as follows. In Case 1, we have
c∗n T ∩ BL = BL 6= (0). In Case 2, we have q∗ T ∩ BL = q∗ T ∩ (BL )∗n ∩ BL = q∗ ∩ BL = q 6= (0) by (5.5). In Case 3, suppose that q∗n,i is properly contained in q∗ . Then by (5.5)
and (5.6), we have ht q∗n,i < h0 . Since q∗n,i is in Sing (BL )∗n , we have q∗n,i ∩ BL 6= (0) by the minimality of h0 . Take 0 6= b ∈ c∗n T ∩ BL . Since c∗n T is the defining ideal of Sing(T ), T ⊗BL BL [b−1 ] is regular. Since −1 α̂⊗1 −1 \ (B L )q ⊗BL BL [b ] −→ T ⊗BL BL [b ] −1 \ \ is faithfully flat, (B L )q ⊗BL BL [b ] is regular. Hence, (BL )q ⊗BL L is regular. We have completed the proof of Claim 5.2. Step 3. Here, we shall complete the proof of Theorem 5.1. First of all, remember the following Rotthaus’ Hilfssatz (cf. [12, Theorem 1.9 and Proposition 1.18] or originally [14]). Theorem 5.3 (Rotthaus’ Hilfssatz). Let B be a Noetherian ring and n ∈ Max B. Assume that B is xB-adically complete Nagata ring. We put Γ(n) = {γ | n ∈ γ ⊂ Max B, # γ < ∞}. For γ ∈ Γ(n), we put Sγ = B \ ∪a∈γ a and Bγ = Sγ−1 B. Consider the homomorphism Bγ∗ → Bn∗ induced by Bγ → Bn , where ( )∗ denotes the (x)-adic completion. Let q∗n be a minimal prime ideal of Sing(Bn∗ ). For each γ ∈ Γ(n), we put q∗γ := q∗n ∩ Bγ∗ . We define n [
o ∆γ (x) := Q ∩ B | Q ∈ Min(B ∗ /q∗ ) (Bγ∗ /q∗γ )/(x) , ∆(x) := ∆γ (x) γ γ γ∈Γ(n) where (Bγ∗ /q∗γ ) is the normalization of Bγ∗ /q∗γ in the field of fractions. Assume the following two conditions: (i) for each γ ∈ Γ(n), ht q∗γ > 0, (ii) # △(x) < ∞. Then q∗n ∩ B 6= (0) is satisfied. We refer the reader to [12] for the proof of this theorem. We deeply use it in our proof. Now, we start to prove Theorem 5.1. Let L be a finite algebraic extension of Q(A). Let B be the integral closure of A in L. Let n be a maximal ideal of B. Suppose that q∗n is a minimal prime ideal of Sing(Bn∗ ) such that (5.7) q∗n ∩ B = (0) 14 K. KURANO AND K. SHIMOMOTO and ht q∗n = h0 . We remark that such L, B, n, q∗n certainly exist by the definition of h0 (see (5.2)). We define Bn∗ , Bγ , Bγ∗ , q∗γ , ∆γ (x), ∆(x) as in Theorem 5.3. Recall that Bγ is a semi-local ring satisfying Max(Bγ ) = {aBγ | a ∈ γ}. Since B is xB-adically complete, any maximal ideal of B contains x. Since Bγ /xBγ is isomorphic to Bγ∗ /xBγ∗ , there exists one-to-one correspondence between γ and the set of maximal ideals of Bγ∗ . Let n∗ be the maximal ideal of Bγ∗ corresponding to n, that is, n∗ = nBγ∗ . In the rest of this proof, we shall prove the conditions (i) and (ii) in Theorem 5.3. Then, it contradicts (5.7) and this completes the proof of Theorem 5.1. Put Cγ = Bγ∗ /q∗γ . Since A is a Nagata ring, Cγ is a Nagata ring, too. Therefore, the normalization Cγ is a finite Cγ -module. Note that x 6∈ q∗γ since q∗n ∩ B = (0). Take Q ∈ MinCγ (Cγ /xCγ ). Since Cγ is a universally catenary Nagata Z-ring *, we find that Q∩Cγ is a minimal prime ideal of xCγ . Therefore, △γ (x) defined in Theorem 5.3 coincides with
{Q̃ ∩ B | Q̃ ∈ MinBγ∗ Bγ∗ /q∗γ + xBγ∗ }. Here, we shall prove the following claim. Claim 5.4. (1) For each γ ∈ Γ(n), ht q∗γ = h0 . (2) For any Q ∈ △(x), ht Q = h0 + 1. First, we shall prove (1). Consider the following homomorphisms. (5.8) f g cn , (Bγ∗ )n∗ −→ Bn∗ −→ B where n∗ = nBγ∗ . By the local lifting property for G-rings, Bγ∗ is a G-ring. Since \ ∗ cn = (B B γ )n∗ , gf is a regular homomorphism. Since g is faithfully flat, f is a regular homomorphism by [9, (33.B)] or [10, Theorem 32.1]. Since q∗γ (Bγ∗ )n∗ = q∗n ∩ (Bγ∗ )n∗ , (5.9) q∗γ (Bγ∗ )n∗ is a minimal prime ideal of Sing((Bγ∗ )n∗ ). Furthermore, q∗n is a minimal prime ideal of q∗γ Bn∗ . Then, we have ht q∗γ = ht q∗γ (Bγ∗ )n∗ = ht q∗n = h0 . The assertion (1) has thus been proved. Since h0 ≥ 2 as in (5.4), the condition (i) in Theorem 5.3 follows from the above assertion (1). Next, we prove (2). Take Q ∈ △γ (x) for some γ ∈ Γ(n). We shall prove that Bγ∗ is normal. For a ∈ γ, let a∗ denote the maximal ideal aBγ∗ of ca is a normal local domain. By the Bγ∗ . Here B is a normal Z-ring, since A is. Therefore, B * Assume that C is a universally catenary Nagata Z-domain. Let C be the normalization of C. Let Q (resp. P ) be a prime ideal of C (resp. C). It is easy to see, if Q ∩ C = P , then ht Q = ht P . IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 15 \ ∗ c local lifting property for G-rings, (Bγ∗ )a∗ is a G-ring. Therefore, (Bγ∗ )a∗ → (B γ )a∗ = Ba is a regular homomorphism. Thus, (Bγ∗ )a∗ is normal. Hence, we know that Bγ∗ is normal.
By definition, there exists Q̃ ∈ MinBγ∗ Bγ∗ /q∗γ + xBγ∗ such that Q = Q̃ ∩ B. Since Bγ → Bγ∗ is flat and QBγ∗ = Q̃, we have (5.10) ht Q̃ = dim(Bγ∗ )Q̃ = dim(Bγ )Q = ht Q. Since Bγ∗ is a Noetherian normal ring, Bγ∗ is the direct product of finitely many integrally closed domains. Remember that Bγ∗ is universally catenary. Then we have ht Q̃ = ht q∗γ + 1. (5.11) By (5.10), (5.11) and the assertion (1) together, we obtain ht Q = h0 + 1. We have completed the proof of Claim 5.4. Let q be a prime ideal of B such that ht q ≤ h0 . Since A is normal, ht(q∩A) = ht q ≤ h0 . Hence, A(q∩A) is excellent by Claim 5.2. Therefore, Bq is excellent. Let us remember that B/xB is quasi-excellent. Then by Lemma 4.4, there exists bq ∈ B \q such that there exists an alteration covering (5.12) φbq ,i {Xbq ,i −→ Spec(B[b−1 q ])}i such that (5.13) φ−1 bq ,i  for each i. We put Ω= Spec [ q B[b−1 q ] xB[b−1 q ]  ⊂ Reg(Xbq ,i ) Spec(B[b−1 q ]) ⊂ Spec(B). By definition, Ω is an open set that contains all the prime ideals of B of height less than or equal to h0 . Hence, the complement Ωc contains only finitely many prime ideals of height h0 + 1. If △(x) is contained in Ωc , then △(x) must be a finite set by Claim 5.4 (2). Thus, it suffices to prove △(x) ⊂ Ωc . Assume the contrary. Take Q ∈ △(x) ∩ Ω. Since Q ∈ △(x), there exists Q̃ ∈
MinBγ∗ Bγ∗ /q∗γ + xBγ∗ such that Q̃ ∩ B = Q for some γ ∈ Γ(n). Since Q ∈ Ω, we find that Q ∈ Spec(B[b−1 q ]) for some q with ht q ≤ h0 . Since (5.12) is an alteration covering, we have a proper surjective generically finite S map π : V → Spec(B[b−1 q ]), together with an open covering V = i Vi and a morphism ψi : Vi → Xbq ,i for each i with commutative diagrams as in (3.1). Consider the following 16 K. KURANO AND K. SHIMOMOTO diagram: Vi′ ↓ ψi′ Xb′ q ,i ↓g Xbq ,i V′ ⊂ ↓ f′ φ′bq ,i ∗ −→ Spec(Bγ∗ [b−1 q ]) −→ Spec(Bγ ) φbq ,i −→ ↓h ↓ Spec(B[b−1 q ]) −→ Spec(B) We put ( )′ = ( ) ×Spec B Spec Bγ∗ . Then, both Q̃ and q∗γ are contained in Spec(Bγ∗ [b−1 q ]). ′ Since f ′ : V ′ → Spec(Bγ∗ [b−1 q ]) is proper surjective, there exist ξ1 , ξ2 ∈ V such that S f ′ (ξ1 ) = Q̃, f ′ (ξ2 ) = q∗γ , and ξ1 is a specialization of ξ2 . Since V ′ = i Vi′ is a Zariski open covering, we have ξ1 ∈ Vi′ for some i. Since Vi′ is closed under generalization, both ξ1 and ξ2 are contained in Vi′ . Here, we put η1 = ψi′ (ξ1 ), η2 = ψi′ (ξ2 ) and ζ1 = g(ψi′ (ξ1 )). Since x ∈ Q and φbq ,i (ζ1 ) = Q, we know that OXbq ,i ,ζ1 is a regular local ring by (5.13). Since B → Bγ∗ is flat, OXbq ,i ,ζ1 → OXb′ q ,i ,η1 is a flat local homomorphism. Its closed fiber is the identity since the maximal ideal of OXbq ,i ,ζ1 contains x. Thus, OXb′ q ,i local ring. Since η2 is a generalization of η1 , OXb′ q ,i (Bγ∗ )q∗γ → OXb′ q ,i ,η2 is flat, since h(q∗γ ) ,η2 ,η1 is a regular is also a regular local ring. Here, = (0) by (5.7). Therefore, (Bγ∗ )q∗γ is a regular local ring. It contradicts (5.9). The condition (ii) in Theorem 5.3 has been proved. We have completed the proof of Theorem 5.1.  Now we obtain the following corollary. Corollary 5.5. Let A be an excellent ring with an ideal I ⊂ A. Then the I-adic completion of A is an excellent ring. In particular, if A is excellent, then the formal power series ring A[[x1 , . . . , xn ]] is excellent. bI denote the I-adic completion of A. We want to prove that A bI is excellent. Proof. Let A bI /I A bI is also excellent. In particular, it is As A is excellent by assumption, A/I ∼ = A bI is quasi-excellent. So it suffices to prove that A bI is quasi-excellent. By Theorem 5.1, A universally catenary. For this, let I = (t1 , . . . , tm ) with ti ∈ A. By [15, Théorèm 1.12], the formal power series ring A[[T1 , . . . , Tm ]] is universally catenary. As there is an isomorphism [10, Theorem 8.12] bI ∼ A = A[[T1 , . . . , Tm ]]/(T1 − t1 , . . . , Tm − tm ), bI is universally catenary and hence excellent. we see that A Finally, assume that A is excellent. Then the polynomial algebra A[x1 , . . . , xn ] is excel- lent, and the (x1 , . . . , xn )-adic completion of A[x1 , . . . , xn ] is A[[x1 , . . . , xn ]]. Hence it is excellent. This proves the corollary.  IDEAL-ADIC COMPLETION OF QUASI-EXCELLENT RINGS (AFTER GABBER) 17 Acknowledgement . The authors are grateful to Professor O. Gabber for permitting us to write this paper. The authors are also grateful to Professor J. Nishimura for listening to the proof of the main theorem and providing us with useful suggestions kindly. References [1] M. Brodmann and C. Rotthaus, Ü ber den regulären Ort in Ausgezeichneten Ringen, Math. Z. 175 (1980) 81–85. [2] T. G. Goodwillie and S. Lichtenbaum, A cohomological bound for the h-topology, Amer. J. Math. 123 (2001) 425–443. [3] S. Greco, Two theorems on excellent rings, Nagoya Math. J. 60 (1976), 139–149. [4] S. Greco, A note on universally catenary rings, Nagoya Math. J. 87 (1982), 95–100. [5] A. Grothendieck, Élements de Géométrie Algébrique IV, Publications Math. I.H.E.S. 24 (1965). [6] R. Hartshorne, Algebraic Geometry, Springer-Verlag (1977). [7] L. Illusie, Y. Laszlo, and F. Orgogozo, Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque 363-364 (2014). [8] J. Marot, Sur les anneaux universellement japonais, Bull. Soc. Math. France 103 (1975), 103–111. [9] H. Matsumura, Commutative algebra. Second edition, . Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. [10] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge (1986). [11] J. Nishimura, Ideal-adic completion of Noetherian rings, J. Math. Kyoto Univ. 21 (1981), 153–169. [12] J. Nishimura and T. Nishimura, Ideal-adic completion of Noetherian rings II, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (1987) 453–467. [13] C. Rotthaus, Komplettierung semilokaler quasiausgezeichneter Ringe, Nagoya Math. J. 76 (1979), 173–180. [14] C. Rotthaus, Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253 (1980), 213–226. [15] H. Seydi, Anneaux henséliens et conditions de chaı̂nes, Bull. Soc. Math. France 98 (1970) 9–31. [16] A letter from O. Gabber to Y. Laszlo (2007). Department of Mathematics, School of Science and Technology, Meiji University, Higashimata 1-1-1, Tama-ku, Kawasaki 214-8571, Japan E-mail address: kurano@isc.meiji.ac.jp Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku, Tokyo 156-8550, Japan E-mail address: shimomotokazuma@gmail.com 

Automated Speed and Lane Change Decision Making using Deep Reinforcement Learning Carl-Johan Hoel∗† , Krister Wolff∗ , Leo Laine∗† ∗ Chalmers University of Technology, 412 96 Göteborg, Sweden Group Trucks Technology, 405 08 Göteborg, Sweden Email: {carl-johan.hoel, krister.wolff, leo.laine}@chalmers.se arXiv:1803.10056v1 [cs.RO] 14 Mar 2018 † Volvo Abstract—This paper introduces a method, based on deep reinforcement learning, for automatically generating a general purpose decision making function. A Deep Q-Network agent was trained in a simulated environment to handle speed and lane change decisions for a truck-trailer combination. In a highway driving case, it is shown that the method produced an agent that matched or surpassed the performance of a commonly used reference model. To demonstrate the generality of the method, the exact same algorithm was also tested by training it for an overtaking case on a road with oncoming traffic. Furthermore, a novel way of applying a convolutional neural network to high level input that represents interchangeable objects is also introduced. I. I NTRODUCTION By automating heavy vehicles, there is potential for a significant productivity increase, see e.g. [1]. One of the challenges in developing autonomous vehicles is that they need to make decisions in complex environments, ranging from highway driving to less structured areas inside cities. To predict all possible traffic situations, and code how to handle them, would be a time consuming and error prone work, if at all feasible. Therefore, a method that can learn a suitable behavior from its own experiences would be desirable. Ideally, such a method should be applicable to all possible environments. This paper introduces how a specific machine learning algorithm can be applied to automated driving, here tested on a highway driving case and an overtaking case. Traditionally, rule based gap acceptance models are common to make lane changing decisions, see for example [2] or [3]. More recent methods often consider the utility of a potential lane change. Either the utility of changing to a specific lane is estimated, see [4] or [5], or the total utility (also called the expected return) over a time horizon is maximized by solving a partially observable Markov decisions process (POMDP), see [6] or [7]. Two commonly used models for speed control and to decide when to change lanes are the Intelligent driver model (IDM) [8] and the Minimize overall braking induced by lane changes (MOBIL) model [9]. The combination of these two models was used as a baseline when evaluating the method presented in this paper. A common problem with most existing methods for autonomous driving is that they target one specific driving case. For example, the ones mentioned above are designed for highway driving, but if a different case is considered, such as driving on a road with oncoming traffic, a completely different method is required. In an attempt to overcome this issue, we introduced a more general approach in [10]. This method is based on a genetic algorithm, which is used to automatically train a general-purpose driver model that can handle different cases. However, the method still requires some features to be defined manually, in order to adapt its rules and actions to different driving cases. During the last years, the field of deep learning has made revolutionary progress in many areas, see e.g. [11] or [12]. By combining deep neural networks with reinforcement learning, artificial intelligence has evolved in different domains, from playing Atari games [13], to continuous control [14], reaching a super human performance in the game of Go [15] and beating the best chess computers [16]. Deep reinforcement learning has also successfully been used for some special applications in the field of autonomous driving, see e.g. [17] and [18]. This paper introduces a method based on a Deep Q-Network (DQN) agent [13] that, from training in a simulated environment, automatically generates a decision making function. To the extent of the authors’ knowledge, this method has not previously been applied to this problem. The main benefit of the presented method is that it is general, i.e. not limited to a specific driving case. For highway driving, it is shown that it can generate an agent that performs better than the combination of the IDM and MOBIL model. Furthermore, with no tuning, the same method can be applied to a different setting, in this case driving on a road with oncoming traffic. Two important differences compared to our previous approach in [10] is that the method presented in this paper does not need any hand crafted features and that the training is significantly faster. Moreover, this paper introduces a novel way of using a convolutional neural network architecture by applying it to high level sensor data, representing interchangeable objects, which improves and speeds up the learning process. This paper is organized as follows: The DQN algorithm and how it was implemented is described in Sect. II. Next, Sect. III gives an overview of the IDM and the MOBIL model, and describes how the simulations were set up. In Sect. IV, the results are presented, followed by a discussion in Sect. V. Finally the conclusions are given in Sect. VI. II. S PEED AND LANE CHANGE DECISION MAKING In this paper, the task of deciding when to change lanes and to control the speed of the vehicle under consideration (henceforth referred to as the ego vehicle) is viewed as a reinforcement learning problem. A Deep Q-Network (DQN) agent [13] is used to learn the Q-function, which describes how beneficial different actions are in a given state. The state of the surrounding vehicles and the available lanes are known to the agent, and its objective is to choose which action to take, which for example could be to change lanes, brake or accelerate. The details of the procedure are described in this section. A. Reinforcement learning Reinforcement learning is a branch of machine learning, where an agent acts in an environment and tries to learn a policy, π, that maximizes a cumulative reward function. The policy defines which action, a, to take, given a state, s. The state of the environment will then change to a new state, s0 , and return a reward, r. The reinforcement learning problem is often modeled as a Markov Decision Process (MDP), which is defined as the tuple hS, A, T, R, γi, where S is the set of states, A is the set of actions, T : S × A → S is the state transition probability function, R : S × A × S → R is the reward function and γ ∈ [0, 1] is a discount factor. An MDP satisfies the Markov property, which means that the probability distribution of the future states depends only on the current state and action, and not on the history of previous states. At every time step, t, the goal of the agent is to maximize the future discounted return, defined as ∞ X γ k rt+k , (1) Rt = k=0 where rt+k is the reward given at step t + k. See [19] for a comprehensive introduction to reinforcement learning and MDPs. B. Deep Q-Network In the reinforcement learning algorithm called Q-learning [20], the agent tries to learn the optimal action value function, Q∗ (s, a). This function is defined as the maximum expected return when being in a state, s, taking some action, a, and then following the optimal policy, π ∗ . This is described by Q∗ (s, a) = max E [Rt |st = s, at = a, π] . π (2) The optimal action value function follows the Bellman equation, see [20], h i ∗ 0 0 Q∗ (s, a) = E r + γ max Q (s , a )|s, a , (3) 0 a which is based on the intuition that if the values of Q∗ (s0 , a0 ) are known, the optimal policy is to select an action, a0 , that maximizes the expected value of Q∗ (s0 , a0 ). In the DQN algorithm [13], Q-learning is combined with deep learning. A deep neural network with weights θ is used as a function approximator of the optimal value function, i.e. Q(s, a; θ) ≈ Q∗ (s, a). The network is then trained by adjusting its parameters, θi , at every iteration, i, to minimize the error in the Bellman equation. This is typically done with stochastic gradient descent, where mini-batches with size M of experiences, described by the tuple et = (st , at , rt , st+1 ), are drawn from an experience replay memory. The loss function at iteration i is defined as h i 0 0 − 2 Li (θi ) = EM (r + γ max Q(s , a ; θ ) − Q(s, a; θ )) . (4) i i 0 a Here, θi− are the network parameters used to calculate the target at iteration i. In order to make the learning process more stable, these parameters are held fixed for a number of iterations and then periodically updated with the latest version of the trained parameters, θi . The trade off between exploration and exploitation is handled by following an -greedy policy. This means that a random action is selected with probability , and otherwise the action with the highest value is chosen. For further details on the DQN algorithm, see [13]. Q-learning and the DQN algorithm are known to overestimate the action value function under some conditions. A further development is the Double DQN algorithm [21], which aims to decouple the action selection and action evaluation. This is done by updating Eq. 4 to  Li (θi ) = EM r + γQ(s0 , arg max Q(s0 , a; θi ); θi− ) a
2  − Q(s, a; θi ) . (5) C. Implementation The Double DQN algorithm, outlined above, was applied to control a vehicle in the two test cases, described in Sect. III-B. The details of the implementation are presented below. 1) MDP formulation: Since the intention of other road users cannot be observed, the speed and lane change decision making problem can be modeled as a partially observable Markov decision process (POMDP) [22]. To address the partial observability, the POMDP can be approximated by an MDP with a k-Markov approximation, where the state consists of the last k observations, st = (ot−k+1 , ot−k+2 , . . . , ot ) [13]. However, for the method presented in this paper, it proved sufficient to set k = 1, i.e. to simply use the last observation. Two different agents were investigated in this study. They both used the same state input, s, defined as a vector with 27 elements, which contained information on the ego vehicle’s speed, available lanes and states of the 8 surrounding vehicles. Table I shows the configuration of the state (see Sect. III for details on how the traffic environment was simulated). The first agent only controlled the lane changing decisions, whereas the speed was automatically controlled by the IDM. This gave a direct comparison to the lane change decisions taken by the MOBIL model, in which the speed also was controlled by the IDM (see Sect. III-A for details). The other agent controlled both the lane changing decisions and the speed. Here, the speed was changed by choosing between four different acceleration options: full brake (−9 m/s2 ), medium brake (−2 m/s2 ), maintain speed (0 m/s2 ) and accelerate (+2 m/s2 ). The action spaces of the two agents are given in Table II. When a decision to change lanes was taken, the intended lane of the lateral control model, described in Sect. III-B, was changed. Both agents took decisions at an interval of ∆t = 1 s. A simple reward function was used. Normally, at every time step, a positive reward was given, equal to the normalized distance driven during that time step. This reward was calculated ego ego as ∆d/∆dmax , where ∆dmax = ∆tvmax , and vmax was the maximum possible speed of the ego vehicle. However, if a collision occurred, or the ego vehicle drove out of the road (it TABLE I S TATE INPUT VECTOR USED BY THE AGENTS . s1 , s2 AND s3 DESCRIBE THE STATE OF THE EGO VEHICLE AND THE AVAILABLE LANES , WHEREAS s3i+1 , s3i+2 AND s3i+3 , FOR i = 1,2,...8, REPRESENT THE STATE OF THE SURROUNDING VEHICLES . s1 s2 s3 s3i+1 s3i+2 s3i+3 max vego /vego Normalized ego vehicle speed,  1, if there is a lane to the left 0, otherwise  1, if there is a lane to the right 0, otherwise Normalized relative position of vehicle i, ∆si /∆smax Normalized relative speed of vehicle i, ∆vi /vmax   −1, if vehicle i is two lanes to the right of the ego vehicle     −0.5, if vehicle i is one lane to the right of the ego vehicle 0, if vehicle i is in the same lane as the ego vehicle    0.5, if vehicle i is one lane to the left of the ego vehicle    1, if vehicle i is two lanes to the left of the ego vehicle TABLE II ACTION SPACES OF THE TWO AGENTS . Agent1 a1 a2 a3 Stay in current lane Change lanes to the left Change lanes to the right Agent2 a1 a2 a3 a4 a5 a6 Stay in current lane, keep current speed Stay in current lane, accelerate with -2 m/s2 Stay in current lane, accelerate with -9 m/s2 Stay in current lane, accelerate with 2 m/s2 Change lanes to the left, keep current speed Change lanes to the right, keep current speed could choose to change lanes to one that did not exist), a penalizing reward of −10 was given and the episode was terminated. If the ego vehicle ended up in a near collision, defined as being one vehicle length (4.8 m) from another vehicle, a reward of −10 was also given, but the episode was not terminated. Finally, to limit the number of lane changes, a reward of −1 was given when a lane changing action was chosen. 2) Neural network design: Two different neural network architectures were investigated in this study. Both had 27 input neurons, for the state described above. The final output layer had 3 output neurons for Agent 1 and 6 output neurons for Agent 2, where the value of neuron ni represented the value function when choosing action ai , i.e. Q(s, ai ). The first architecture was a standard fully connected neural network (FCNN), with two hidden layers. Each layer consisted of nhidden neurons, set to 512, and rectified linear units (ReLUs) were used as activation functions [23]. The final output layer used a linear activation function. The second architecture introduces a new way of applying temporal convolutional neural networks (CNNs). CNNs are inspired by the structure of the visual cortex in animals. By their architecture and weight sharing properties, they create a space and shift invariance, and reduce the number of parameters to be optimized. This has made them successful in the field of computer vision, where they have been applied directly to low level input, consisting of pixel values. For further details on 27x1 35x1 64x1 3 3x1 or 6x1 merge 8x32 8x32 1x32 24 32 filters size 3x1 stride 3 32 filters size 1x32 stride 1 maxpool merge fully connected fully connected output input Fig. 1. The second network architecture, which used convolutional neural networks and max pooling to create translational invariance between the input from different surrounding vehicles. See the main text for further explanations. CNNs, see e.g. [12]. In this study, a CNN architecture was applied to a high level input, which described the state of identical, interchangeable objects, see Fig. 1. Two convolutional layers were applied to the part of the state vector that represented the relative position, speed and lane of the surrounding vehicles. The first layer had nconv1 filters, set to 32, with filter size 3, stride 3 and ReLU activation functions. This structure created an output of 8 × 32 signals. Since there were 3 neighbouring input neurons that described the properties of each of the 8 surrounding vehicles, by setting the filter size and stride to 3, each row of the output only depended on one vehicle. The second layer had nconv2 filters, set to 32, with filter size 1, stride 1 and ReLU activation functions. This further aggregated knowledge about each vehicle in every row of the 8 × 32 output signal. After the second convolutional layer, a max pooling layer was added. This structure created a translational invariance of the input that described the relative state of the different vehicles, i.e. the result would be the same if e.g. the input describing vehicle 3 and vehicle 4 switched position in the input vector. This translational invariance, in combination with the reduced number of optimizable parameters, simplified and sped up the training of the network. See Sect. V for a further discussion on why a CNN architecture was beneficial in this setting. The output of the max pooling layer was then concatenated with the rest of the input vector. A fully connected layer with nfull units, here set to 64, and ReLu activation functions followed. Finally, the output layer had 3 or 6 neurons, both with linear activation functions. 3) Training details: The network was trained by using the Double DQN algorithm, described in Sect. II-B. During training, the policy followed an -greedy behavior, where  decreased linearly from start to end over N−end iterations. A discount factor, γ, was used for future rewards. The target network was updated every Nupdate iterations by cloning the online parameters, i.e. setting θi− = θi , at the updating step. Learning started after Nstart iterations and a replay memory of size Mreplay was used. Mini-batches of training samples with size Mmini were uniformly drawn from the replay memory and the network was updated using the RMSProp algorithm [24], with a learning rate of η. In order to improve the stability, error clipping was used by limiting the error term r + γQ(s0 , arg maxa Q(s0 , a; θi ); θi− ) − Q(s, a; θi ) to [−1, 1]. TABLE III H YPERPARAMETERS USED TO TRAIN THE DQN AGENTS . Discount factor, γ Learning start iteration, Nstart Replay memory size, Mreplay Initial exploration constant, start Final exploration constant, end Final exploration iteration, N-end Learning rate, η Mini-batch size, Mmini Target network update frequency, Nupdate 0.99 50,000 500,000 1 0.1 500,000 0.00025 32 30,000 The hyperparameters of the training are summarized in Table III. Due to the computational complexity, a systematic grid search was not performed. Instead, the hyperparameter values were selected from an informal search, based upon the values given in [13] and [21]. The state space, described above, did not provide any information on where in an episode the agent was at a given time step, e.g. if it was in the beginning or close to the end (Sect. III-B describes how an episode was defined). The reason for this choice was that the goal was to train an agent that performed well in highway driving of infinite length. Therefore, the longitudinal position was irrelevant. However, at the end of a successful episode, the future discounted return, Rend , was 0. To avoid that the agent learned this, the last experience eend was not stored in the experience replay memory. Thereby, the agent was tricked to believe that the episode continued forever. III. S IMULATION SETUP A highway case was used as the main way to test the algorithm outlined above. To evaluate the performance of the agent, a reference model, consisting of the IDM and MOBIL model, was used. This section briefly summarizes the reference model, describes how the simulations were set up and how the performance was measured. Moreover, in order to show the versatility of the proposed method, it was further tested in a secondary overtaking case with oncoming traffic, which is also described here. A. Reference model The IDM [8] is widely used in transportation research to model the longitudinal dynamics of a vehicle. With this model, the speed of the ego vehicle, v, varies according to   δ  ∗ 2  v d (v, ∆v) v̇ = a 1 − − , (6) v0 d √ d∗ (v, ∆v) = d0 + vT + v∆v/(2 ab). (7) The vehicle’s speed depends on the distance to the vehicle in front, d, and the speed difference (approach rate), ∆v. Table IV shows the parameters that are used to tune the model. The values were taken from the original paper [8]. The MOBIL model [9] makes decisions on when to change lanes by maximizing the acceleration of the vehicle in consideration and the surrounding vehicles. For a lane change to be allowed, the induced acceleration of the following car in the new lane, an , must fulfill a safety criterion, an > −bsafe . To TABLE IV IDM AND MOBIL MODEL PARAMETERS . Minimum gap distance, s0 Safe time headway, T Maximal acceleration, a Desired deceleration, b Acceleration exponent, δ 2m 1.6 s 0.7 m/s2 1.7 m/s2 4 Politeness factor, p Changing threshold, ath Maximum safe deceleration, bsafe 0 0.1 m/s2 4 m/s2 predict the acceleration of the ego and surrounding vehicles, the IDM model is used. If the safety criterion is met, MOBIL changes lanes if ãe − ae + p ((ãn − an ) + (ão − ao )) > ath , (8) where ae , an and ao are the accelerations of the ego vehicle, the trailing vehicle in the target lane, and the trailing vehicle in the current lane, respectively, assuming that the ego vehicle stays in its lane. Furthermore, ãe , ãn and ão are the corresponding accelerations if the lane change is carried out. The politeness factor, p, controls how the effect on other vehicles is valued. To perform a lane change, the collective acceleration gain must be higher than a threshold, ∆ath . If there are lanes available both to the left and to the right, the same criterion is applied to both options. If both criteria are fulfilled, the option with the highest acceleration gain is chosen. The parameter values of the MOBIL model are shown in Table IV. They were taken from the original paper [9], except for the politeness factor, here set to 0. This setting provided a more fair comparison to the DQN agent, since then neither method considered possible acceleration losses of the surrounding vehicles. B. Traffic simulation 1) Highway case: A highway case was used as the main way to test the method presented in this paper. This case was similar to the one used in the previous study [10]. For completeness, it is summarized below. A three-lane highway was used, where the ego vehicle to be controlled was surrounded by 8 other vehicles. The ego vehicle consisted of a 16.5 m long truck-semitrailer combination and the surrounding vehicles were normal 4.8 m long passenger cars. These surrounding vehicles stayed in their initial lanes and followed the IDM model longitudinally. Overtaking was allowed both on the left and the right side of another vehicle. An example of an initial traffic situation is shown in Fig. 2a. Although normal highway driving mostly consists of traffic with rather constant speeds and small accelerations, occasionally vehicles brake hard, or even at the maximum of their capability to avoid collisions. Drivers can also decide to suddenly increase their speed rapidly. Therefore, in order for the agent to learn to keep a safe inter-vehicle distance, such quick speed changes need to be included in the training process. The surrounding vehicles in the simulations were assigned different desired speed trajectories. To speed up the training of the agent, these trajectories contain frequent speed changes, (a) (b) Fig. 2. (a) Example of an initial traffic situation for the highway case, which was used as the main way to test the algorithm. (b) Example of a traffic situation for a secondary overtaking case with oncoming traffic, showing the situation 10 seconds from the initial state. In both cases, the ego vehicle (truck-trailer combination) is shown in green and black. The arrows represent the velocities of the vehicles. Speed (m/s) 40 30 20 10 0 0 200 400 600 800 Position (m) Fig. 3. Example of six different randomly generated speed trajectories, defined for different positions along the highway. The solid lines are fast trajectories, applied to vehicles starting behind the ego vehicle, whereas the dashed lines are slow trajectories, applied to vehicles starting in front of the ego vehicle. TABLE V PARAMETERS OF THE SIMULATED HIGHWAY CASE . Maximum initial vehicle spread, dlong Minimum initial inter-vehicle distance, d∆ + Front vehicle minimum speed, vmin + Front vehicle maximum speed, vmax − Rear vehicle minimum speed, vmin − Rear vehicle maximum speed, vmax ego Initial ego vehicle speed, vinit ego Maximum ego vehicle speed, vmax Episode length, dmax 200 m 25 m 16.7 m/s (60 km/h) 23.6 m/s (85 km/h) 26.4 m/s (95 km/h) 33.3 m/s (120 km/h) 25 m/s (90 km/h) 25 m/s (90 km/h) 800 m which occurred more often than during normal highway driving. Some examples are shown in Fig. 3. The ego vehicle initially started in the middle lane, surrounded by 8 other vehicles. These were randomly positioned in the lanes, within dlong longitudinally and with a minimum inter-vehicle distance d∆ . The initial and maximum ego ego ego vehicle speed was vinit and vmax respectively. Vehicles that were positioned in front of the ego vehicle were assigned + + slower speed trajectories, in the range [vmin , vmax ], whereas vehicles placed behind the ego vehicle were assigned faster − − speed trajectories, in the range [vmin , vmax ]. Episodes where two vehicles were placed too close together with a large speed difference, thus causing an unavoidable collision, were deleted. Each episode was dmax long. The values of the mentioned parameters are presented in Table V. Further details on the setup of the simulations, and how the speed trajectories were generated, are described in [10]. 2) Overtaking case: In order to illustrate the generality of the method presented in this paper, a secondary overtaking case, including two-way traffic, was also tested. Fig. 2b shows an example of this case. The ego vehicle started in the right ego lane, with an initial speed of vinit , set to 25 m/s. Another vehicle, which followed a random slow speed profile (defined above), was placed 50 m in front of the ego vehicle. Two oncoming vehicles, also following slow speed profiles, were placed in the left, oncoming lane, at a random distance between 300 and 1100 m in front of the ego vehicle. 3) Vehicle dynamics and lateral control: In both the highway and the overtaking case, the dynamics of the vehicles were simulated by using simple kinematic models. A lane following two-point visual control model [25] was used to control the vehicles laterally. As mentioned in Sect. II-C, when the agent decided to change lanes, the setpoint of this model was changed to the new desired lane. The same procedure was used if the MOBIL model decided to change lanes. C. Performance index In order to evaluate how the DQN agent performed compared to the reference driver model (presented in Sect. III-A) in a specific episode of the highway case, a performance index p̃ was defined as p̃ = (d/dmax )(v̄/v̄ref ). (9) Here, d is the distance driven by the ego vehicle (limited by a collision or the episode length), dmax is the episode length, v̄ is the average speed of the ego vehicle and v̄ref is the average speed when the reference model controlled the ego vehicle through the episode. With this definition, the distance driven by the ego vehicle was the dominant limiting factor when a collision occurred. However, if the agent managed to complete the episode without collisions, the average speed determined the performance index. For the overtaking case, the reference model described above cannot be used. Instead, the performance index was simply defines as p̃o = (d/dmax )(v̄/v̄refIDM ). Here, v̄refIDM was the mean speed of the ego vehicle when it was controlled by the IDM through the same episode, i.e. when it did not overtake the preceding vehicle. IV. R ESULTS This section focuses on the results that were obtained for the highway case, described in Sect. III-B, which was the main way of testing the presented method. It also briefly explains and discusses some characteristics of the results, whereas a TABLE VI S UMMARY OF THE RESULTS OF THE DIFFERENT AGENTS FOR THE HIGHWAY CASE AND THE OVERTAKING CASE . Highway case Agent1CNN Agent2CNN Agent1FCNN Agent2FCNN 1 Overtaking case Collision free episodes Performance index, p̃ Collision free episodes Performance index, p̃o 100% 100% 98% 86% 1.01 1.10 0.98 0.96 100% 100% - 1.06 1.11 - more general discussion follows in Sect. V. The results regarding the overtaking case are collected in Sect. IV-C. As described in Sect. II, two agents with different action spaces were investigated. Agent1 only decided when to change lanes, whereas Agent2 decided both the speed and when to change lanes. Furthermore, two different neural network architectures were used. In summary, the four variants were Agent1FCNN , Agent1CNN , Agent2FCNN and Agent2CNN . Five different runs were carried out for the four agent variants, where each run had different random seeds for the DQN and the traffic simulation. The networks were trained for 2 million iterations (3 million for Agent2FCNN ), and at every 50,000 iterations, they were evaluated over 1,000 random episodes. Note that these evaluation episodes were randomly generated, and not presented to the agents during training. During the evaluation runs, the performance index described in Sect. III-C was used to compare the agents’ and the reference model’s behaviour. The results are shown in Fig. 4, which presents the average proportion, p̂, of successfully completed, i.e. collision free, evaluation episodes of the four agent variants, and in Fig. 5, which shows their average performance index, p̃. The final performance of the fully trained agents is summarized in Table VI. A. Agents using a CNN In Fig. 4, it can be seen that Agent1CNN solved all the episodes already after 100,000 iterations, which is the first evaluation after that the training started at 50,000 iterations. At this point it had learned to always stay in its lane, in order to avoid collisions. Since it often got blocked by slower vehicles, its average performance index was therefore lower than 1 at this point, see Fig. 5. However, after around 600,000 iterations, Agent1CNN had learned to carry out lane changes when necessary, and performed similar to the reference model. Fig. 4 shows that Agent2CNN quickly figured out how to change lanes and increase its speed to solve most of the episodes. Its performance index was on par with the reference model early on during the training, at around 250,000 iterations, see Fig. 5. Then, at 400,000 iterations, it solved all the evaluation episodes without collisions. With more training, there were still no collisions, but the performance index increased and stabilized at 1.1. Fig. 6 shows a histogram of the performance index for 1,000 evaluation episodes, which were run by the final trained version of Agent1CNN and Agent2CNN . Since all the episodes were completed without collisions, the performance index was Agent1CNN 0.5 Agent2CNN Agent1FCNN Agent2FCNN 0 0 0.5 1 1.5 2 2.5 3 106 Iteration Fig. 4. Proportion of episodes solved without collisions by the different agents during training. 1 Agent1CNN Agent2CNN 0.5 Agent1FCNN Agent2FCNN Reference model 0 0 0.5 1 1.5 2 2.5 3 106 Iteration Fig. 5. Performance index of the different agents during training. Mean: 1.01 Mean: 1.10 0.2 0.2 0.1 0.1 0 0.8 1 1.2 Performance index 1.4 0 0.8 1 1.2 1.4 Performance index Fig. 6. Histogram of the performance index at the end of the training for Agent1CNN (left) and Agent2CNN (right). simply the speed ratio v̄/v̄ref . In the figure, it can be seen that most often there was a small difference between the average speed of the agents and the reference model. There were also some outliers, which were both faster and slower than the reference model. The explanation for these is that the episodes were randomly generated, which meant that even a reasonable action could get the ego vehicle into a situation where it got locked in and could not overtake the surrounding vehicles. Therefore, a small difference in behaviour could lead to such situations for both the trained agents and the reference model, which explains the outliers. Furthermore, the peak at index 1 for Agent2CNN is explained by that there were some episodes when the lane in front of the ego vehicle was free from the start. Then both the reference model and the agents drove at the maximum speed through the whole episode. To further illustrate the properties of the agents, and how they developed during training, the percentage of chosen actions is shown in Fig. 7. For Agent1CNN , it can be seen that it quickly figured out that changing lanes can lead to collisions, and therefore it chose to stay in its lane almost 100% of the time in the beginning. This explains why it completed all the episodes already from the first evaluation point after its training started. However, as training proceeded, it figured out when it safely could change lanes, and thereby perform better. At the end of its training, it chose to change lanes 1 Action proportion 0.02 0.015 0.5 0.01 Agent1CNN a2 0.005 Agent2CNN a3 0 0 0 0.5 1 1.5 0 2 1 1.5 2 106 Iteration Fig. 8. Proportion of overtaking episodes solved without collisions by the different agents during training. 1 Action proportion 0.5 106 Iteration a1 a2 a3 0.5 a4 1 a5 a6 0 Agent1CNN 0.5 0 0.5 1 1.5 Iteration 2 Agent2CNN 6 10 Fig. 7. Top: proportion of actions chosen by Agent1CNN during training. Due to the scale difference, a1 , i.e. stay in the current lane, is here left out. Bottom: proportion of actions chosen by Agent2CNN during training. Both plots start at 100,000 iterations, since that is the first evaluation point after that training started at 50,000 iterations. around 1% of the time. Agent2CNN first learned a short sighted strategy, where it accelerated most of the time to obtain a high immediate reward. This naturally led to many rear end collisions. However, when its training proceeded, it learned to control its speed by braking or idling, and to change lanes when necessary. Reassuringly, both agents learned to change lanes to the left and right equally often. B. Agents using a FCNN Both Agent1FCNN and Agent2FCNN failed to complete all the evaluation episodes without collisions, see Fig. 4 and Table VI. Naturally, Agent1FCNN solved a significantly higher fraction of the episodes and performed better than Agent2FCNN , since it only needed to decide when to change lanes, and not control the speed. In the beginning, it learned to always stay in its lane, and thereby solved all episodes without collisions, but reached a lower performance index than the reference model, see Fig. 5. With more training, it started to change lanes and performed reasonably well, but sometimes caused collisions. Agent2FCNN performed significantly worse and collided in 14% of the episodes by the end of its training. A longer training run was carried out for Agent1FCNN and Agent2FCNN , but after 20 million iterations, the results were the same. C. Overtaking case In order to demonstrate the generality of the method presented in this paper, the same algorithm was applied to an overtaking situation, described in Sect. III-B. Fig. 8, Fig. 9 and Table VI show the proportion of successfully completed evaluation episodes, p̂, and the modified performance index, p̃o , of Agent1CNN and Agent2CNN . By the end of the training, both agents solved all episodes without collisions. Furthermore, in all the episodes, the ego vehicle overtook the slower vehicle, resulting in performance indexes above 1. IDM (no overtaking) 0 0 0.5 1 Iteration 1.5 2 106 Fig. 9. Performance index of the different agents during training on the overtaking case. V. D ISCUSSION In Table VI, it can be seen that both Agent1 and Agent2 with the convolutional neural network architecture solved all the episodes without collisions. The performance of Agent1CNN was on par with the reference model. Since they both used the IDM to control the speed, this result indicates that the trained agent and the MOBIL model took lane changing decisions with similar quality. However, when adding the possibility for the agent to also control its speed, as in Agent2CNN , the trained agent had the freedom to find better strategies and could therefore outperform the reference model. This result illustrates that for a better performance, lateral and longitudinal decisions should not be completely separated. As expected, using a CNN architecture resulted in a significantly better performance than a FCNN architecture, see e.g. Table VI. The reason for this is, as mentioned in Sect. II-C, that the CNN architecture creates a translational invariance of the input that describes the relative state of the different vehicles. This is reasonable, since it is desirable that the agent reacts the same way to other vehicles’ behaviour, independently of where they are positioned in the input vector. Furthermore, since CNNs share weights, the complexity of the network is reduced, which in itself speeds up the learning process. This way of using CNNs can be compared to how they previously were introduced and applied to low level input, often on pixels in an image, where they provide a spatial invariance when identifying features, see e.g. [26]. The results of this paper show that it can also be beneficial to apply CNNs to high level input of interchangeable objects, such as the state description shown in Sect. II-C. As mentioned in Sect. II-C, a simple reward function was used. Naturally, the choice of reward function strongly affects the resulting behaviour. For example, when no penalty was given for a lane change, the agent found solutions where it constantly demanded lane changes in opposite directions, which made the vehicle drive in between two lanes. In this study, a simple reward function worked well, but for other cases a more careful design may be required. One way to determine a reward function that mimics human preferences is to use inverse reinforcement learning [27]. In a previous paper, [10], we presented a different method, based on a genetic algorithm, that automatically can generate a driving model for similar cases as described here. That method is also general and it was shown that it is applicable to different cases, but it requires some hand crafted features when designing the structure of its rules. However, the method presented in this paper requires no such hand crafted features, and instead uses the measured state, described in Table I, directly as input. Furthermore, the method in [10] achieved a similar performance when it comes to safety and average speed, but the number of necessary training episodes was between one and two orders of magnitude higher than for the method that was investigated in this study. Therefore, the new method is clearly advantageous compared to the previous one. An important remark is that when training an agent by using the method presented in this paper, the agent will only be able to solve the type of situations that it is exposed to in the simulations. It is therefore important that the design of the simulated traffic environment covers the intended case. Furthermore, when using machine learning to produce a decision making function, it is hard to guarantee functional safety. Therefore, it is common to use an underlying safety layer, which verifies the safety of a planned trajectory before it is executed by the vehicle control system, see e.g. [28]. VI. C ONCLUSION AND FUTURE WORK The main results of this paper show that a Deep Q-Network agent can be trained to make decisions in autonomous driving, without the need of any hand crafted features. In a highway case, the DQN agents performed on par with, or better than, a reference model based on the IDM and MOBIL model. Furthermore, the generality of the method was demonstrated by applying it to a case with oncoming traffic. In both cases, the trained agents handled all episodes without collisions. Another important conclusion is that, for the presented method, applying a CNN to high level input that represents interchangeable objects can both speed up the learning process and increase the performance of the trained agent. Topics for future work include to further analyze the generality of this method by applying it to other cases, such as crossings and roundabouts, and to systematically investigate the impact of different parameters and network architectures. Moreover, it would be interesting to apply prioritized experience replay [29], which is a method where important experiences are repeated more frequently during the training process. This could potentially improve and speed up the learning process. ACKNOWLEDGMENT This work was partially supported by the Wallenberg Artificial Intelligence, Autonomous Systems and Software Program (WASP), funded by Knut and Alice Wallenberg Foundation, and partially by Vinnova FFI. R EFERENCES [1] D. J. Fagnant and K. Kockelman, “Preparing a nation for autonomous vehicles: opportunities, barriers and policy recommendations,” Transportation Research Part A: Policy and Practice, vol. 77, pp. 167 – 181, 2015. [2] P. Gipps, “A model for the structure of lane-changing decisions,” Transportation Research Part B: Methodological, vol. 20, no. 5, pp. 403 – 414, 1986. [3] K. I. Ahmed, “Modeling drivers’ acceleration and lane changing behavior,” Ph.D. dissertation, Massachusetts Institute of Technology, 1999. [4] J. Eggert and F. Damerow, “Complex lane change behavior in the foresighted driver model,” in 2015 IEEE 18th International Conference on Intelligent Transportation Systems, 2015, pp. 1747–1754. [5] J. 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arXiv:1712.08397v1 [math.RA] 22 Dec 2017 KÄHLER–POISSON ALGEBRAS JOAKIM ARNLIND AND AHMED AL-SHUJARY Abstract. We introduce Kähler–Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a Kähler–Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated Kähler–Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts. 1. Introduction Poisson manifolds and their geometry have been of great interest over the last decades. Besides from being important from a purely mathematical point of view, they are also fundamental to areas in mathematical and theoretical physics. Many authors have studied the geometric and algebraic properties of symplectic and Poisson manifolds together in relation to concepts such as connections, local structure and cohomology (see e.g. [Lic77, Wei83, Bry88, Hue90]). Moreover, there is a well developed field of deformations of Poisson structures, perhaps most famous through Kontsevich’s result on the existence of formal deformations [Kon03]. The ring of smooth functions on a Poisson manifold is a Poisson algebra and it seems quite natural to ask to what extent geometric properties and concepts may be introduced in an arbitrary Poisson algebra, without making reference to an underlying manifold. The methods of algebraic geometry can readily be extended to Poisson algebras (see e.g. [Ber79]); however, this will not be directly relevant to us as we shall start by focusing on metric aspects. Our work is mainly motivated by the results in [AHH12, AH14], where it is shown that one may reformulate the Riemannian geometry of an embedded Kähler manifold M entirely in terms of the Poisson structure on the algebra smooth functions of M . Let us also mention that the starting point of our approach is quite similar to that of [Hue90] (although metric aspects were not considered there). In this note, we show that any Poisson algebra, fulfilling an “almost Kähler condition”, enjoys many properties similar to those of the algebra of smooth functions on an almost Kähler manifold, opening up for a more metric treatment of Poisson algebras. Such algebras will be called “Kähler–Poisson algebras”, and we show that one may associate a Kähler–Poisson algebra to every algebra in a large class of Poisson algebras. In particular, we prove the existence of a unique Levi-Civita connection on the module generated by the inner derivations, and show that the curvature operator has all the classical symmetries. As our approach is quite close to the theory of Lie-Rinehart algebras, we start by introducing metric Lie-Rinehart 1 2 JOAKIM ARNLIND AND AHMED AL-SHUJARY algebras and recall a few results on the Levi-Civita connection and the corresponding curvature. In physics, the dynamics of quantum systems are found by using a correspondence between Poisson brackets of functions on the classical manifold, and the commutator of operators in the quantum system. Thus, understanding how properties of the underlying manifold may be expressed in Poisson algebraic terms enables both interpretation and definition of quantum mechanical quantities. For instance, this has been used in the context of matrix models to identify emergent geometry (cf. [BS10, AHH12]). Let us briefly outline the contents of the paper. In Section 2 we recall a few the results from [AH14], in order to motivate and understand the introduction of a Kähler type condition for Poisson algebras, and Section 3 explains how the theory of Lie-Rinehart algebras can be extended to include metric aspects. In Section 4, we define Kähler–Poisson algebras and investigate their basic properties as well as showing that one may associate a Kähler–Poisson algebra to an arbitrary Poisson algebra in a large class of algebras. In Section 5 we derive a compact formula for the Levi-Civita connection as well as introducing Ricci and scalar curvature. Section 6 presents a number of examples together with a few detailed computations. Remark 1.1. We have become aware of the fact that the terminology Kähler– Poisson structure (resp. Kähler–Poisson manifold ) is used for certain Poisson structures on a complex manifold where the Poisson bivector is of type (1, 1) (see e.g. [Kar02]), but we hope that this will not be a source of confusion for the reader. 2. Poisson algebraic formulation of almost Kähler manifolds In [AH14] it was shown that the geometry of embedded almost Kähler manifolds can be reformulated entirely in the Poisson algebra of smooth functions. As we shall develop an algebraic analogue of this fact, let us briefly recall the main construction. Let (Σ, ω) denote a n-dimensional symplectic manifold and let g be a metric on Σ. Furthermore, let us assume that x : (Σ, g) → (Rm , ḡ) is an isometric embedding of Σ into Rm (with the metric ḡ), and write
p → x(p) = x1 (p), x2 (p), . . . , xm (p) . The results in [AH14] state that the Riemannian geometry of Σ may be formulated in terms of the Poisson algebra generated by the embedding coordinates x1 , . . . , xm . These results hold true as long as there exists a non-zero function γ ∈ C ∞ (Σ) such that (2.1) γ 2 g ab = θap θbq gpq where θab and gab denote the components of the Poisson bivector and the metric in local coordinates {ua }na=1 , respectively. If (Σ, ω, g) is an almost Kähler manifold then it follows from the compatibility condition ω(X, Y ) = g(X, J(Y )) (where J denotes the almost complex structure on Σ) that relation (2.1) holds with γ = 1. In local coordinates, the isometric embedding is characterized by

gab = ḡ ij ∂a xi ∂b xj , and the Poisson bracket is computed as {f, h} = θab (∂a f )(∂b h). KÄHLER–POISSON ALGEBRAS 3 Note that in the above and following formulas, indices i, j, k, . . . run from 1 to m and indices a, b, c, . . . run from 1 to n. Defining D : Tp Rm → Tp Rm as D(X) ≡ Di j X j ∂i = 1 i k {x , x }ḡkl {xj , xl }ḡjm X m ∂i γ2 for X = X i ∂i ∈ Tp Rm , one computes 1 ab θ (∂a xi )(∂b xk )ḡkl θpq (∂p xj )(∂q xl )ḡ jm X m γ2 1 = 2 θab θpq gbq (∂a xi )(∂p xj )ḡ jm X m = g ap (∂a xi )(∂p xj )ḡ jm X m , γ D(X)i = by using (2.1). Hence, the map D is identified as the orthogonal projection onto Tp Σ, seen as a subspace of Tp Rm . Having the projection operator at hand, one may directly proceed to develop the theory of submanifolds. For instance, the Levi-Civita connection ∇ on Σ is given by
¯ XY ∇X Y = D ∇ ¯ is the Levi-Civita connection on (Rm , ḡ). In the where X, Y ∈ Γ(T Σ) and ∇ particular case (but generically applicable, by Nash’s theorem [Nas56]) when ḡ is the Euclidean metric, the above formula reduces to ∇X Y i = 1 γ4 m X {xi , xk }{xj , xk }X l {xl , xn }{Y j , xn }. i,j,k,l,n=1 As we intend to develop an analogous theory for Poisson algebras, without any reference to a manifold, we would like to reformulate (2.1) in terms of Poisson algebraic expressions. Using that gab = ḡij (∂a xi )(∂b xj ) and {xi , xk } = θab (∂a xi )(∂b xj ), one derives γ 2 g ab = θap θbq gpq ⇒ γ 2 δca = θap θbq gpq gbc ⇒ γ 2 θar = θap θbq gpq gbc θcr ⇒ γ 2 {xi , xj } = (∂a xi )(∂r xj )θap θbq θcr ḡ kl (∂p xk )(∂q xl )ḡ mn (∂b xm )(∂c xn ) ⇒ γ 2 {xi , xj } = −{xi , xk }ḡkl {xl , xn }ḡnm {xm , xj } which is equivalent to the statement that (2.2) γ 2 {f, h} = −{f, xi }ḡij {xj , xk }ḡkl {xl , h} for all f, h ∈ C ∞ (Σ). Given γ 2 , ḡij and x1 , . . . , xm , the above equation makes sense in an arbitrary Poisson algebra. The main purpose of this paper is to study algebras which satisfy such a relation. 3. Metric Lie-Rinehart algebras The idea of modeling the algebraic structures of differential geometry in a commutative algebra is quite old. We shall follow a pedestrian approach, were we assume that a (commutative) algebra A is given (corresponding to the algebra of functions), together with an A-module g (corresponding to the module of vector fields) which is also a Lie algebra and has an action on A as derivations. Under appropriate assumptions on the ingoing objects, such systems has been studied by many authors over the years, see e.g [Her53, Koz60, Pal61, Rin63, Nel67, Hue90]. Our starting 4 JOAKIM ARNLIND AND AHMED AL-SHUJARY point is the definition given by G. Rinehart [Rin63]. In the following, we let the field K denote either R or C. Definition 3.1 (Lie-Rinehart algebra). Let A be a commutative K-algebra and let g be an A-module which is also a Lie algebra over K. Given a map ω : g → Der(A), the pair (A, g) is called a Lie-Rinehart algebra if
(3.1) ω(aα)(b) = a ω(α)(b)
(3.2) [α, aβ] = a[α, β] + ω(α)(a) β, for α, β ∈ g and a, b ∈ A. (In most cases, we will leave out ω and write α(a) instead of ω(α)(a).) Let us point out some immediate examples of Lie-Rinehart algebras. Example 3.2. Let A be an algebra and let g = Der(A) be the A-module of derivations of A. It is easy to check that Der(A) is a Lie algebra with respect to composition of derivations, i.e. [α, β](a) = α(β(a)) − β(α(a)). The pair (A, Der(A)) is a Lie-Rinehart algebra with respect to the action of elements of Der(A) as derivations. Example 3.3. Let A = C ∞ (M ) be the algebra (over R) of smooth functions on a manifold M , and let g = X (A) be the A-module of vector fields on M . With respect to the standard action of a vector field as a derivation of C ∞ (M ), the pair (C ∞ (M ), X (A)) is a Lie-Rinehart algebra. Morphisms of Lie-Rinehart algebras are defined as follows. Definition 3.4. Let (A1 , g1 ) and (A2 , g2 ) be Lie-Rinehart algebras. A morphism of Lie-Rinehart algebras is a pair of maps (φ, ψ), with φ : A1 → A2 an algebra homomorphism and ψ : g1 → g2 a Lie algebra homomorphism, such that

ψ(aα) = φ(a)ψ(α) and φ α(a) = ψ(α) φ(a) , for all a ∈ A1 and α ∈ g1 . A lot of attention has been given to the cohomology of the Chevalley–Eilenberg complex consisting of alternating A-multilinear maps with values in a module M . Namely, defining C k (g, M ) to be the A-module of alternating maps from gk to an (A, g)-module M , on introduces the standard differential d : C k (g, M ) → C k+1 (g, M ) as dτ (α1 , . . . , αk+1 ) = (3.3) k+1 X i=1 +
(−1)i+1 αi τ (α1 , . . . , α̂i , . . . , αk+1 ) k+1 X i<j
(−1)i+j τ [αi , αj ], α1 , . . . , α̂i , . . . , α̂j , . . . , αk+1 , where α̂i indicates that αi is not present among the arguments. The fact that d ◦ d = 0 implies that one can construct the cohomology of this complex in analogy with de Rahm cohomology of smooth manifolds. However, as we shall be more interested in Riemannian aspects, it is natural to study the case when there exists a metric on the module g. More precisely, we make the following definition. KÄHLER–POISSON ALGEBRAS 5 Definition 3.5. Let (A, g) be a Lie-Rinehart algebra and let M be an A-module. An A-bilinear form g : M × M → A is called a metric on M if it holds that (1) g(m1 , m2 ) = g(m2 , m1 ) for all m1 , m2 ∈
M , (2) the map ĝ : M → M ∗ , given by ĝ(m1 ) (m2 ) = g(m1 , m2 ), is an A-module isomorphism, where M ∗ denotes the dual of M . We shall often refer to property (2) as the metric being non-degenerate. Definition 3.6. A metric Lie-Rinehart algebra (A, g, g) is a Lie-Rinehart algebra (A, g) together with a metric g : g × g → A. Let us introduce morphisms of metric Lie-Rinehart algebras as morphisms of LieRinehart algebras that preserve the metric. Definition 3.7. Let (A1 , g1 , g1 ) and (A2 , g2 , g2 ) be metric Lie-Rinehart algebras. A morphism of metric Lie-Rinehart algebras is a morphism of Lie-Rinehart algebras (φ, ψ) : (A1 , g1 ) → (A2 , g2 ) such that

φ g1 (α, β) = g2 ψ(α), ψ(β) for all α, β ∈ g1 . The theory of affine connections can readily be introduced, together with torsionfreeness and metric compatibility. Definition 3.8. Let (A, g) be a Lie-Rinehart algebra and let M be an A-module. A connection ∇ on M is a map ∇ : g → EndK (M ), written as α → ∇α , such that (1) ∇aα+β = a∇α + ∇β (2) ∇α (am) = a∇α m + α(a)m for all a ∈ A, α, β ∈ g and m ∈ M . Definition 3.9. Let (A, g) be a Lie-Rinehart algebra and let M be an A-module with connection ∇ and metric g. The connection is called metric if
(3.4) α g(m1 , m2 ) = g(∇α m1 , m2 ) + g(m1 , ∇α m2 ) for all α ∈ g and m1 , m2 ∈ M . Definition 3.10. Let (A, g) be a Lie-Rinehart algebra and let ∇ be a connection on g. The connection is called torsion-free if ∇α β − ∇β α − [α, β] = 0 for all α, β ∈ g. As in differential geometry, one can show that there exists a unique torsion-free and metric connection associated to the Riemannian metric. The first step involves proving Kozul’s formula. Proposition 3.11. Let (A, g, g) be a metric Lie-Rinehart algebra. If ∇ is a metric and torsion-free connection on g then it holds that



2g ∇α β, γ = α g(β, γ) + β g(γ, α) − γ g(α, β) (3.5) + g(β, [γ, α]) + g(γ, [α, β]) − g(α, [β, γ]) for all α, β, γ ∈ g. 6 JOAKIM ARNLIND AND AHMED AL-SHUJARY Proof. Starting from the right-hand-side of (3.5) and using the metric condition to rewrite the first three terms as
α g(β, γ) = g(∇α β, γ) + g(β, ∇α γ), together with the torsion-free condition to rewrite the last three terms as g(β, [γ, α]) = g(β, ∇γ α) − g(β, ∇α γ), immediately gives 2g(∇α β, γ).  By using Proposition 3.11 together with the fact that the metric is non-degenerate, one obtains the following result. Proposition 3.12. Let (A, g, g) be a metric Lie-Rinehart algebra. Then there exists a unique metric and torsion-free connection on g. Remark 3.13. The unique connection in Proposition 3.12 will be referred to as the Levi-Civita connection of a metric Lie-Rinehart algebra. Proof. For every α, β ∈ g, the right-hand-side of (3.5) defines a linear form ω ∈ g∗


ω(γ) = 12 α g(β, γ) + 12 β g(γ, α) − 21 γ g(α, β) + 21 g(β, [γ, α]) + 21 g(γ, [α, β]) − 21 g(α, [β, γ]). By assumption (see Definition 3.5), the metric induces an isomorphism map ĝ : g → g∗ , which implies that there exists an element ∇α β = ĝ −1 (ω) ∈ g such that g(∇α β, γ) = ω(γ). This shows that ∇α β exists for all α, β ∈ g such that relation (3.5) is satisfied. Next, let us show that ∇ defines a connection on g, which amounts to checking the four properties in Definition 3.8. This is a straight-forward computation using (3.5) and the fact that, for instance, g(∇aα β, γ) = g(a∇α β, γ) for all γ ∈ g implies that ∇aα β = a∇α β since the metric is non-degenerate. Let us illustrate the computation with the following example. From (3.5) it follows that


2g(∇aα β, γ) = aα g(β, γ) + β g(γ, aα) − γ g(aα, β) + g(β, [γ, aα]) + g(γ, [aα, β]) − g(aα, [β, γ])


= aα g(β, γ) + aβ g(γ, α) + β(a)g(γ, α) − aγ g(α, β) − γ(a)g(α, β) + g(β, γ(a)α + a[γ, α]) + g(γ, −β(a)α + a[α, β]) − ag(α, [β, γ]) = 2ag(∇α β, γ) + β(a)g(γ, α) − γ(a)g(α, β) + γ(a)g(β, α) − β(a)g(γ, α) = 2ag(∇α β, γ). The remaining properties of a connection is proved in an analogous way. To show that ∇ is metric, one again uses (3.5) to substitute g(∇α β, γ) and g(β, ∇α γ) and find that
α g(β, γ) − g(∇α β, γ) − g(β, ∇α γ) = 0. That the torsion-free condition holds follows from g(∇α β, γ) − g(∇β α, γ) − g([α, β], γ) = 0, which can be seen using (3.5). Hence, we conclude that there exists a metric and torsion-free affine connection satisfying (3.5). Moreover, since the metric is nondegenerate, such a connection is unique. Finally, as every metric and torsion-free KÄHLER–POISSON ALGEBRAS 7 connection on g satisfies (3.5) (by Proposition 3.11) we conclude that there exists a unique metric and torsion-free connection on g.  In what follows, we shall recall some of the properties satisfied by a metric and torsion-free connection. The differential geometric proofs goes through with only a change in notation needed, but we provide them here for easy reference, and to adapt the formulation to our particular situation. We refer to [Koz60, Nel67] for a nice overview of differential geometric constructions in modules over general commutative algebras. Following the usual definitions, we introduce the curvature as (3.6) R(α, β)γ = ∇α ∇β γ − ∇β ∇α γ − ∇[α,β] γ as well as R(α, β, γ) = R(α, β)γ R(α, β, γ, δ) = g(α, R(γ, δ)β). Let us also consider the extension of ∇ to multilinear maps T : gk → A k

X T α1 , . . . , ∇β αi , . . . , αk , (∇β T )(α1 , . . . , αk ) = β T (α1 , . . . , αk ) − i=1 k as well as to g-valued multilinear maps T : g → g k

X T α1 , . . . , ∇β αi , . . . , αk . (∇β T )(α1 , . . . , αk ) = ∇β T (α1 , . . . , αk ) − i=1 As in classical geometry, one proceeds to derive the Bianchi identities. Proposition 3.14. Let ∇ be the Levi-Civita connection of a metric Lie-Rinehart algbera (A, g, g) and let R denote corresponding curvature. Then it holds that (3.7) (3.8) R(α, β, γ) + R(γ, α, β) + R(β, γ, α) = 0,


∇α R (β, γ, δ) + ∇β R (γ, α, δ) + ∇γ R (α, β, δ) = 0, for all α, β, γ, δ ∈ g. Proof. The first Bianchi identity (3.7) is proven by acting with ∇γ on the torsion free condition ∇α β − ∇β α − [α, β] = 0, and then summing over cyclic permutations of α, β, γ. Since [[α, β], γ] + [[β, γ], α] + [[γ, α], β] = 0, the desired result follows. The second identity
is obtained by a cyclic permutation (of α, β, γ) in R ∇α β − ∇β α − [α, β], γ, δ = 0. One has
0 = R ∇α β − ∇β α − [α, β], γ, δ + cycl. = R(∇γ α, β, δ) + R(α, ∇γ β, δ) − R([α, β], γ, δ) + cycl. On the other hand, one has (∇γ R)(α, β, δ) = ∇γ R(α, β, δ) − R(∇γ α, β, δ) − R(α, ∇γ β, δ) − R(α, β, ∇γ δ), and substituting this into the previous equation yields
0 = ∇γ R(α, β, δ) − ∇γ R (α, β, δ) − R(α, β, ∇γ δ) − R([α, β], γ, δ) + cycl. After inserting the definition of R, and using that [[α, β], γ] + cycl. = 0, the second Bianchi identity follows.  8 JOAKIM ARNLIND AND AHMED AL-SHUJARY Finally, one is able to derive the classical symmetries of the curvature tensor. Proposition 3.15. Let ∇ be the Levi-Civita connection of a metric Lie-Rinehart algbera (A, g, g) and let R denote corresponding curvature. Then it holds that (3.9) R(α, β, γ, δ) = −R(β, α, γ, δ) = −R(α, β, δ, γ). (3.10) R(α, β, γ, δ) = R(δ, γ, α, β), for all α, β, γ, δ ∈ g. Proof. The identity R(α, β, γ, δ) = −R(α, β, δ, γ) follows immediately from the definition of R. Let us now prove that R(α, β, γ, δ) = −R(β, α, γ, δ). Starting from γ(δ(a)) − δ(γ(a)) − [γ, δ](a) = 0 and letting a = g(α, β) yields h i h i γ g(∇δ α, β) + g(α, ∇δ β) − δ g(∇γ α, β) + g(α, ∇γ β) − (∇[γ,δ] α, β) − (α, ∇[γ,δ] β) = 0. when using that ∇ is a metric connection; i.e τ (g(α, β)) = g(∇τ α, β) + g(α, ∇τ β) for τ = γ, δ, [γ, δ]. A further expansion using the metric property gives g(∇γ ∇δ α, β) + g(α, ∇γ ∇δ β) − g(∇δ ∇γ α, β) − g(α, ∇δ ∇γ β) − g(∇[γ,δ] α, β) − g(α, ∇[γ,δ] β) = 0, which is equivalent to g(R(γ, δ)α, β) = −g(R(γ, δ)β, α). Next, one can make use of equation (3.7) in Proposition 3.14, from which it follows that (3.11) R(α, β, γ, δ) + R(α, δ, β, γ) + R(α, γ, δ, β) = 0. It is a standard algebraic result that any quadri-linear map satisfying (3.9) and (3.11) also satisfies (3.10) (see e.g. [Hel01]).  4. Kähler–Poisson algebras In this section, we shall introduce a type of Poisson algebras, that resembles the smooth functions on an (isometrically) embedded almost Kähler manifold, in such a way that an analogue of Riemannian geometry may be developed. Namely, let us consider a unital Poisson algebra (A, {·, ·}) and let {x1 , . . . , xm } be a set of distinguished elements of A, corresponding to functions providing an embedding into Rm , in the geometrical case. One may also consider the setting of algebraic (Poisson) varieties where A is a finitely generated Poisson algebra and {x1 , . . . , xm } denotes a set of generators. Our aim is to introduce equation (2.2) in A and investigate just how far one may take the analogy with Riemannian geometry. After introducing Kähler–Poisson algebras below, we will show that they are, in a natural way, metric Lie-Rinehart algebras, which implies that the results of Section 3 can be applied; in particular, there exists a unique torsion-free metric connection on every Kähler–Poisson algebra. Note that Lie-Rinehart algebras related to Poisson algebras have extensively been studied by Huebschmann (see e.g. [Hue90, Hue99]). In Section 2 it was shown that the following identity holds on an almost Kähler manifold: (2.2) γ 2 {f, h} = −{f, xi }ḡij {xj , xk }ḡ kl {xl , h}. KÄHLER–POISSON ALGEBRAS 9 This equation is well-defined in a Poisson algebra, and we shall use it to define the main object of our investigation. Definition 4.1. Let A be a Poisson algebra over K and let {x1 , . . . , xm } ⊆ A. Given gij ∈ A, for i, j
= 1, . . . , m, such that gij = gji , we say that the triple K = A, {x1 , . . . , xm }, g is a Kähler–Poisson-algebra if there exists η ∈ A such that m X (4.1) η{a, xi }gij {xj , xk }gkl {xl , b} = −{a, b} i,j,k,l=1 for all a, b ∈ A. Remark 4.2. From now on, we shall use the differential geometric convention that repeated indices are summed over from 1 to m, and omit explicit summation symbols. Given a Kähler–Poisson-algebra K, we let g denote the A-module generated by all inner derivations, i.e. g = {a1 {c1 , ·} + · · · + aN {cN , ·} : ai , ci ∈ A and N ∈ N}. It is a standard fact that g is a Lie algebra over K with respect to

[α, β](a) = α β(a) − β α(a) . The matrix g induces a bilinear symmetric form on g, defined by (4.2) g(α, β) = α(xi )gij β(xj ), and we refer to g as the metric on g. To the metric g one may associate the map ĝ : g → g∗ defined as ĝ(α)(β) = g(α, β).
Proposition 4.3. If K = A, {x1 , . . . , xm }, g is a Kähler–Poisson-algebra then the metric g is non-degenerate; i.e. the map ĝ : g → g∗ is a module isomorphism. Proof. Let us first show that g is injective; i.e. we will show that ĝ(α)(β) = 0, for all β ∈ g, implies that α = 0. Thus, write α = αi {xi , ·}, and assume that g(α, β) = 0 for all β ∈ g. In particular, we can choose β = η{c, xk }gkm {·, xm }, for arbitrary c ∈ A, which implies that 0 = g(α, β) = ηαk {xk , xi }gij {c, xk }gkm {xj , xm } = −αk η{xk , xi }gij {xj , xm }gmk {xk , c}. Using the relation (4.1), one obtains αk {xk , c} = 0 for all c ∈ A, which is equivalent to α = 0. This shows that ĝ is injective. Let us now show that ĝ is surjective. Thus, let ω ∈ g∗ and set α = ηω({xi , ·})gij {xj , ·} ∈ g, which gives ĝ(α)(ak {bk , ·}) = ηω({xi , ·})gij {xj , xl }glm ak {bk , xm } = −ηak {bk , xm }gml {xl , xj }gji ω({xi , ·}). 10 JOAKIM ARNLIND AND AHMED AL-SHUJARY Since ω is a module homomorphism one obtains ĝ(α)(ak {bk , ·}) = ω(−ηak {bk , xm }gml {xl , xj }gji {xi , ·}) = ω(ak {bk , ·}), by using (4.1), which proves that every element of g∗ is in the image of ĝ. We conclude that ĝ is a module isomorphism.  Corollary 4.4. If (A, {x1 , . . . , xm }, g) is a Kähler–Poisson algebra then (A, g, g) is a metric Lie-Rinehart algebra. Proof. It is easy to check that (A, g) satisfies the conditions of a Lie-Rinehart algebra, and Proposition 4.3 implies that the metric is non-degenerate. Hence, (A, g, g) is a metric Lie-Rinehart algebra.  Let us now introduce some notation for Kähler–Poisson algebras. Thus, we set P ij = {xi , xj } P i (a) = {xi , a}, for a ∈ A, as well as Dij = ηP i k P jk = η{xi , xl }glk {xj , xk } Di (a) = ηP k (a)Pk i = η{xk , a}gkl {xl , xi }, and note that Dij = Dji . With respect to this notation, (4.1) can be stated as Di (a)Pi (b) = {a, b}. (4.3) The metric will be used to lower indices in analogy with differential geometry. E.g. P i j = P ik gkj Di j = Dik gkj . Furthermore, one immediately derives the following useful identities (4.4) Dij Pj (a) = P i (a), P ij Dj (a) = P i (a) and Di j Djk = Dik . by using (4.1). There is a natural embedding ι : g → Am , given by ι(ai {bi , ·}) = ai {bi , xk }ek , m where {ek }m k=1 denotes the canonical basis of the free module A . Moreover, g m defines a bilinear form on A via g(X, Y ) = X i gij Y j for X = X i ei ∈ Am and Y = Y i ei ∈ Am , and we introduce the map D : Am → Am by setting D(X) = Di j X j ei for X = X i ei ∈ Am . Proposition 4.5. The map D : Am → Am is an orthogonal projection; i.e. D2 (X) = D(X) and g(D(X), Y ) = g(X, D(Y )) for all X, Y ∈ Am . KÄHLER–POISSON ALGEBRAS 11 Proof. First, it is clear that D is an endomorphism of Am . It follows immediately from (4.4) that D2 (X) = Di j Dj k X k ei = Di j Djl glk X k ei = Dil glk X k ei = Di k X k ei = D(X). Furthermore, using that Dij = Dji one finds that
g D(X), Y = Di j X j gik Y k = X j Dil glj gik Y k = X j glj Dli gik Y k
= X j gjl Dl k Y k = g X, D(Y ) , which completes the proof.  From Proposition 4.5 we conclude that T A = im(D) is a finitely generated projective module. As a corollary, we prove that g is a finitely generated projective module by showing that g is isomorphic to T A. Proposition 4.6. The map ι : g → Am is an isomorphism from g to T A. Proof. First, it is clear from the definition that ι is a module homomorphism. Considered as a submodule of Am , elements of T A can be characterized by the fact that D(X) = X for all X ∈ T A. Thus, by showing that
D ι(ak {bk , ·}) = Di j ak {bk , xj }ei = −ak Di j P j (bk )ak ei = −ak P i (bk ) = ι(ak {bk , ·}) it follows that ι(ak {bk , ·}) ∈ T A. Let us now show that ι is injective; assume that ι(ak {bk , ·}) = 0, which implies that ak {bk , xi } = 0 for i = 1, . . . , m. Next, for arbitrary c ∈ A, we write ak {bk , c} = −ηak {bk , xi }gij P jl glm {xm , c}, by using (4.1). Since ak {bk , xi } = 0, one obtains ak {bk , c} = 0 for all c ∈ A. To prove that ι is surjective, we start from an arbitrary X = X i ei ∈ T A, and note that
ι X i gij Di (·) = X i gij Dik ek = D(X) = X by using that D(X) = X for all X ∈ T A. Hence, we may conclude that ι is an isomorphism from g to T A.  Corollary 4.7. g is a finitely generated projective module. Note that the above result is clearly not dependent on whether or not the underlying Poisson algebra has the structure of a Kähler–Poisson algebra, as the definition of g involves only inner derivations. Hence, as soon as the Poisson algebra admits the structure of a Kähler–Poisson algebra, it follows that the module of inner derivations is projective. Furthermore, the fact that g is a projective module has several implications for the underlying Lie-Rinehart algebra [Rin63, Hue90]. Next, let us show that the derivations Di generate g as an A-module. Proposition 4.8. The A-module g is generated by {D1 , . . . , Dm }. 12 JOAKIM ARNLIND AND AHMED AL-SHUJARY Proof. First of all, it is clear that every element in the module generated by Di , written as α(c) = αi Di (c) = ηαi {xi , xj }gjk {c, xk }, is an element of g. Conversely, let α ∈ g be an arbitrary element written as X α(c) = aN {bN , c}. N for c ∈ A. Using the Kähler–Poisson condition (4.1) one may write this as X X α(a) = aN {bN , c} = − ηaN {bN , xi }gij {xj , xk }gkl {xl , c} N = X N N i  aN {b , x }gij Dj (c), N which clearly lies in the module generated by {D1 , . . . , Dm }.  Thus, every α ∈ g may be written as α = αi Di for some αi ∈ A. It turns out that this is a very convenient way of writing elements of g, which shall be extensively used in the following. Note that if the Kähler–Poisson algebra comes from an almost Kähler manifold M , then Di is quite close to a partial derivative on M in the sense that (∂a xi )gik Dk (f ) = ∂a f , for f ∈ C ∞ (M ). 4.1. The trace of linear maps. As we shall be interested in both Ricci and scalar curvature, which are defined using traces of linear maps, we introduce
(4.5) tr(L) = g L(Di ), Dj Dij . for an A-linear map L : g → g. This trace coincides with the ordinary trace on g∗ ⊗A g; namely, consider X L= ωN ⊗A αN ∈ g∗ ⊗A g N as a linear map L : g → g in the standard way via X L(β) = ωN (β)αN , N together with tr(L) = X ωN (αN ). N i Writing αN = αN i D one finds that X X

k j kj g L(Di ), Dj Dij = g ωN (Di )αN ωN (Di )αN k D , D Dij = k D Dij N = X N k i ωN (αN k D iD ) = X N N k ωN (αN k D ) = X ωN (αN ). N In particular, this implies that the trace defined via (4.5) is independent of the Kähler–Poisson structure. KÄHLER–POISSON ALGEBRAS 13 4.2. Morphisms of Kähler–Poisson algebras. As Kähler–Poisson algebras are also metric Lie-Rinehart algebras, we shall require that a morphism of Kähler– Poisson algebras is also a morphism of metric Lie-Rinehart algebras (as defined in Section 3). However, as the definition of a Kähler–Poisson also involves the choice of a set of distinguished elements, we will require a morphism to respect the subalgebra generated by these elements. To this end, we start by making the following definition. Definition 4.9. Given a Kähler–Poisson algebra (A, {x1 , . . . , xm }, g), let Afin ⊆ A denote the subalgebra generated by {x1 , . . . , xm }. Equipped with this definition, we introduce morphisms of Kähler–Poisson algebras in the following way. ′ Definition 4.10. Let K = (A, {x1 , . . . , xm }, g) and K′ = (A′ , {y 1 , . . . , y m }, g ′ ) be Kähler–Poisson algebras together with their corresponding modules of derivations g and g′ , respectively. A morphism of Kähler–Poisson algebras is a pair of maps (φ, ψ), with φ : A → A′ and ψ : g → g′ , such that (φ, ψ) is a morphism of the metric Lie-Rinehart algebras (A, g, g) and (A, g′ , g ′ ) and φ is a Poisson algebra homomorphism such that φ(Afin ) ⊆ A′fin . Note that if the algebras are finitely generated such that A = Afin and A′ = A′fin (which is the case in many examples), the condition φ(Afin ) ⊆ A′fin is automatically satisfied. Although a morphism of Kähler–Poisson algebras is given by a choice of two maps φ and ψ, it is often the case that φ determines ψ in the following sense. ′ Proposition 4.11. Let (φ, ψ) : (A, {x1 , . . . , xm }, g) → (A′ , {y 1 , . . . , y m }, g ′ ) be a morphism of Kähler–Poisson algebras such that for all α′ ∈ g′
α′ φ(a) = 0 ∀ a ∈ A ⇒ α′ = 0 then Proof. Let
ψ a{b, ·}A = φ(a){φ(b), ·}A′ . ′ (φ, ψ) : (A, {x1 , . . . , xm }, g) → (A′ , {y 1 , . . . , y m }, g ′ ) be a morphism of Kähler–Poisson algebras fulfilling the assumption above. Since φ is a Poisson algebra homomorphism, one obtains for α = a{b, ·}A

φ α(c) = φ a{b, c}A = φ(a){φ(b), φ(c)}A′ for all a ∈ A. By the definition of a Lie-Rinehart morphism, this has to equal ψ(α)(φ(c)); i.e. ψ(α)(φ(c)) = φ(a){φ(b), φ(c)}A′ . Thus, ψ(α) agrees with φ(a){φ(b), ·}A′ on the image of φ, which implies that ψ(α) = φ(a){φ(b), ·}A′ since any derivation is determined by its action on the image of φ by assumption.  For instance, the requirements in Proposition 4.11 are clearly satisfied if φ is surjective. 14 JOAKIM ARNLIND AND AHMED AL-SHUJARY 4.3. Construction of Kähler–Poisson algebras. Given a Poisson algebra (A, {·, ·}) one may ask if there exist {x1 , . . . , xm } and gij such that (A, {x1 , . . . , xm }, g) is a Kähler–Poisson algebra? Let us consider the case when A is a finitely generated algebra, and let {x1 , . . . , xm } be an arbitrary set of generators. If we denote by P the matrix with entries {xi , xj } and by g the matrix with entries gij , the Kähler– Poisson condition (4.1) may be written in matrix notation as ηPgPgP = −P. Given an arbitrary antisymmetric matrix P, we shall find g by first writing P in a block diagonal form, with antisymmetric 2 × 2 matrices on the diagonal. This is a well known result in linear algebra, in which case the eigenvalues appear in the diagonal blocks. For an antisymmetric matrix with entries in a commutative ring, a similar result holds. Lemma 4.12. Let MN (R) denote the set of N × N matrices with entries in R. For N ≥ 2, let P ∈ MN (R) be an antisymmetric matric. Then there exists V ∈ MN (R), an antisymmetric Q ∈ MN −2 (R) and λ ∈ R such that   0 λ 0 . V T PV =  −λ 0 0 Q Proof. We shall construct the matrix V by using elementary row and column operations. Note that if a matrix E represents an elementary row operation, then E T PE is obtained by applying the elementary operation to both the row and the corresponding column. Denoting the matrix elements of P by pij , we start by constructing a matrix Vk such that (VkT PVk )k1 = (VkT PVk )k2 = 0 (which necessarily implies that also the (1k) and (2k) matrix elements are zero). To this end, let Vk1 denote the matrix representing the elementary row operation that multiplies the k’th row by p12 , and let Vk2 represent the operation that adds the first row, multiplied by −pk2 , to the k’th row. Furthermore, Vk3 represents the operation of adding the second row, multiplied by pk1 , to the k’th row. Setting Vk = Vk1 Vk2 Vk3 it is easy to see that VkT PVk is an antisymmetric matrix where the (1k), (2k), (k1) and (k2) matrix elements are zero. Consequently, we set V = V3 V4 · · · VN and conclude that V T PV is of the desired form.  Proposition 4.13. Let P ∈ MN (R) be an antisymmetric matric, and let N̂ denote the integer part of N/2. Then there exists V ∈ MN (R) and λ1 , . . . , λN̂ ∈ R such that V T PV = diag(Λ1 , . . . , ΛN̂ ) if N is even, T V PV = diag(Λ1 , . . . , ΛN̂ , 0) where Λk =  0 −λk if N is odd,  λk . 0 Proof. Let us prove the statement by using induction together with Lemma 4.12. Thus, assume that there exists V ∈ MN (R) such that V T PV = diag(Λ1 , . . . , Λk , Qk+1 ) where Qk+1 ∈ MN −2k is an antisymmetric matrix. Clearly, by Lemma 4.12, this holds true for k = 1. Next, assume that N − 2k ≥ 2. Applying Lemma 4.12 to KÄHLER–POISSON ALGEBRAS 15 T Qk+1 we conclude that there exists Vk+1 ∈ MN −2k (R) such that Vk+1 Qk+1 Vk+1 = diag(Λk+1 , Qk+2 ). Furthermore, defining Wk+1 ∈ MN (R) by Wk+1 = diag(12k , Vk+1 ) one finds that (V Wk+1 )T P(V Wk+1 ) = diag(Λ1 , . . . , Λk+1 , Qk+2 ). By induction, it follows that one may repeat this procedure until N − 2k < 2. If N is even, then N − 2k = 0 and the statement follows. If N is odd, then N − 2k = 1 and, since V T PV is antisymmetric, it follows that the (N N ) matrix element is zero, giving the stated result.  Returning to the case of a Poisson algebra generated by x1 , . . . , xm , assume for the moment that m = 2N for a positive integer N . By Proposition 4.13, there exists a matrix V V T PV = P0 where P0 is a block diagonal matrix of the form P0 = diag(Λ1 , . . . , ΛN ) with Λk =  0 −λk  λk . 0 In the same way, defining g0 = diag(g1 , . . . , gN ) with   λ 1 0 gk = λk 0 1 λ = λ1 · · · λN we set g = V g0 V T . Noting that P0 g0 P0 g0 P0 = −λ2 P0 one finds 0 = P0 g0 P0 g0 P0 + λ2 P0 = V T PV g0 V T P0 V gV T P0 V + λ2 V T PV
= V T PgPgP + λ2 P V It is a general fact that for an arbitrary matrix V there exists a matrix Ṽ such that Ṽ V = V Ṽ = (det V )1. Multiplying the above equation from the left by Ṽ T and from the right by Ṽ yields
(4.6) det(V )2 PgPgP + λ2 P = 0. As long as det(V ) is not a zero divisor, this implies that PgPgP = −λ2 P. Thus, given a finitely generated Poisson algebra A, the above procedure gives a rather general way to associate a localization A[λ−1 ] and a metric g to A, such that (A[λ−1 ], {x1 , . . . , xm }, g) is a Kähler–Poisson algebra. Note that the above argument, with only slight notational changes, also applies to the case when m is odd, in which case an extra block of 0 will appear in P0 . 16 JOAKIM ARNLIND AND AHMED AL-SHUJARY 5. The Levi-Civita connection Since every Kähler–Poisson algebra is also a metric Lie-Rinehart algebra, the results of Section 3 immediately applies. In particular, there exists a unique torsion-free and metric connection on the module g. In this section, we shall derive an explicit expression for the Levi-Civita connection of an arbitrary Kähler–Poisson algebra. It turns out to be convenient to formulate the results in terms of the generators {D1 , . . . , Dm }. Kozul’s formula gives the connection as


2g(∇Di Dj , Dk ) = Di g(Dj , Dk ) + Dj g(Dk , Di ) − Dk g(Di , Dj ) (5.1) − g([Dj , Dk ], Di ) + g([Dk , Di ], Dj ) + g([Di , Dj ], Dk ), and one notes that an element α = a{b, ·} ∈ g may be recovered from g(α, Di ) as g(α, Di )Di (f ) = a{b, xk }Di k Di (f ) = a{b, xk }Dk (f ) = a{b, f } = α(f ). Thus, one immediately obtains ∇Di Dj = g(∇Di Dj , Dk )Dk . However, it turns out that one can obtain a more compact formula for the connection. Let us start by proving the following result.

Lemma 5.1. g([Di , Dj ], Dk ) = Di Djk − Dj Dik . Proof. For convenience, let us introduce the notation P̂ ij = η{xi , xj } and, consequently, P̂ i j = P̂ ik gkj . In this notation, one finds Di (a) = P̂ i j {a, xj }. Thus, one obtains g([Di , Dj ], Dk ) = [Di , Dj ](xl )Dk l = P̂ i m {Djl , xm }Dk l − P̂ j n {Dil , xn }Dk l
= P̂ i m {P̂ j n {xl , xn }, xm } − P̂ j n {P̂ i m {xl , xm }, xn } Dk l
= P̂ i m P̂ j n − {{xn , xl }, xm } − {{xl , xm }, xn } Dk l
+ P̂ i m {P̂ j n , xm }{xl , xn } − P̂ j n {P̂ i m , xn }{xl , xm } Dk l = P̂ i m P̂ j n {{xm , xn }, xk } + P̂ i m {P̂ j n , xm }{xk , xn } − P̂ j n {P̂ i m , xn }{xk , xm }, by using the Jacobi identity together with {a, xi }Dk i = {a, xk }. Furthermore, in the second and third term, one uses Leibniz’s rule to obtain g([Di , Dj ], Dk ) = P̂ i m P̂ j n {{xm , xn }, xk } + P̂ i m {P̂ j n {xk , xn }, xm } − P̂ i m P̂ j n {{xk , xn }, xm } − P̂ j n {P̂ i m {xk , xm }, xn } + P̂ j n P̂ i m {{xk , xm }, xn }
= P̂ i m P̂ j n {{xm , xn }, xk } + {{xn , xk }, xm } + {{xk , xm }, xn }

+ Di Djk − Dj (Dik ) = Di Djk − Dj (Dik ), by again using the Jacobi identity.  The above result allows for the following formulation of the Levi-Civita connection for a Kähler–Poisson algebra. Proposition 5.2. If ∇ denotes the Levi-Civita connection of a Kähler–Poisson algebra K then 1 1 1 (5.2) ∇Di Dj = Di (Djk )Dk − Dj (Dik )Dk + Dk (Dij )Dk , 2 2 2 KÄHLER–POISSON ALGEBRAS 17 or, equivalently, ∇Di Dj = Γij k Dk where (5.3) Γij k = 1 1 1 i jl D (D )Dlk − Dj (Dil )Dlk + Dk (Dij ). 2 2 2 Proof. Since g(Di , Dj ) = Dij , Kozul’s formula (5.1) together with Lemma 5.1 gives 2g(∇Di Dj , Dk ) = Di (Djk ) + Dj (Dki ) − Dk (Dij ) − Dj (Dki ) + Dk (Dji ) + Dk (Dij ) − Di (Dkj ) + Di (Djk ) − Dj (Dik ) = Di (Djk ) − Dj (Dki ) + Dk (Dij ), which proves (5.2). The fact that one may write the connection as ∇Di Dj = Γij k Dk follows from Dij Dj = Di and Dk (a)Dk (b) = Dk (a)Dk (b).  Thus, for arbitrary elements of g, one obtains ∇α β = α(βi )Di + Γij k αi βj Dk (5.4) where α = αi Di and β = βi Di , and curvature is readily introduced as R(α, β)γ = ∇α ∇β γ − ∇β ∇α γ − ∇[α,β] γ. Ricci curvature is defined as Ric(α, β) = tr γ → R(γ, α)β and using the trace from Section 4.1, one obtains
Ric(α, β) = g(R(Di , α)β, Dj )Dij . To define the scalar curvature, one considers the Ricci curvature as a linear map Ric : g → g with Ric(α) = Ric(α, Di )Di , giving

S = tr α → Ric(α) = g R(Di , Dk )Dl , Dj Dij Dkl . Note that since the metric is nondegenerate, there exists a unique element ∇f ∈ g such that g(∇f, α) = α(f ) for all α ∈ g; we call ∇f the gradient of f . Now, it is easy to see that ∇f = Di (f )Di since g(Di (f )Di , αj Dj ) = Di (f )αj Dij = αj Dj (f ) = α(f ). The divergence of an element α ∈ g is defined as div(α) = tr(β → ∇β α), and, finally, the Laplacian ∆(f ) = div(∇f ). 18 JOAKIM ARNLIND AND AHMED AL-SHUJARY 6. Examples As shown in Section 2, the algebra of smooth functions on an almost Kähler manifold M becomes a Kähler–Poisson algebra when choosing x1 , . . . , xm to be embedding coordinates, providing an isometric embedding into Rm , endowed with the standard Euclidean metric. (Recall that, by Nash’s theorem [Nas56], such an embedding always exists.) In this section, we shall present examples of a more algebraic nature to illustrate the fact that algebras of smooth functions are not the only examples of Kähler–Poisson algebras. Keeping in mind the general construction procedure in Section 4.3, we consider finitely generated Poisson algebras with a low number of generators. 6.1. Poisson algebras generated by two elements. Let A be a unital Poisson algebra generated by the two elements x1 = x ∈ A and x2 = y ∈ A, and set   0 {x, y} P= −{x, y} 0 It is easy to check that for an arbitrary symmetric matrix g PgPgP = −{x, y}2 det(g)P. Thus, as long as {x, y}2 det(g) is not a zero-divisor, one may localize to obtain a Kähler–Poisson algebra K = (A[({x, y}2 det(g))−1 ], {x, y}, g). For the sake of illustrating the concepts and formulas we have developed so far, let us explicitly work out an example based on an algebra A0 , generated by two elements. Let us start by choosing an element λ ∈ A0 for which the localization A = A0 [p−1 , λ−1 ] exists, and then defining the metric as   1 1 0 g= λ 0 1 From the above considerations, we know that (A, {x, y}, g) is a Kähler–Poisson algebra with η = λ2 /p2 , where p = {x, y}. For convenience we also introduce γ = p/λ such that η = 1/γ 2. Let us start by computing the derivations Dx = D1 and Dy = D2 , which generate the module g: λ 1 {·, y} = − {y, ·} p γ λ 1 Dy = η{y, xi }gij {·, xj } = − {·, x} = {x, ·} p γ Dx = η{x, xi }gij {·, xj } = as well as 1 x 1 D and Dy = g2k Dk = Dy . λ λ Moreover, they provide an orthogonal set of generators since Dx = g1k Dk = 1 1 1 p2 {y, xi }gij {y, xj } = 2 =λ γ γ γ λ g(Dy , Dy ) = λ g(Dx , Dy ) = 0, g(Dx , Dx ) = KÄHLER–POISSON ALGEBRAS and one obtains
(Dij ) = g(Di , Dj ) = 19   λ 0 . 0 λ Note that g is a free module with basis {Dx , Dy } since ( ( x y −a γ1 {y, x} = 0 aD (x) + bD (x) = 0 aDx + bDy = 0 ⇒ ⇒ b γ1 {x, y} = 0 aDx (y) + bDy (y) = 0 ⇒ ( a=0 b=0 by using that λ is invertible. Let us introduce the derivation Dλ = γ −1 {λ, ·} and note that 1 x 1 D (λ)Dy − Dy (λ)Dx . λ λ From Proposition 5.2 one computes the connection: 1 1 1 ∇Dx Dx = D1 (D1k )Dk − D1 (D1k )Dk + Dk (D11 )Dk 2 2 2 1 y 1 i 1 x = D (λ)Dx + D (λ)Dy = D (λ)Di 2 2 2 and similarly 1 1 ∇Dy Dy = Dx (λ)Dx + Dy (λ)Dy = ∇Dx Dx 2 2 1 y 1 x y ∇Dx D = D (λ)Dy − D (λ)Dx = Dλ 2 2 1 1 y x ∇Dy D = D (λ)Dx − Dx (λ)Dy = −Dλ 2 2 Moreover, the curvature can readily be computed  
1
y 1 x y x 2 2 x y R(D , D )D = Dx (λ) + Dy (λ) − Dx D (λ) − Dy D (λ) D 2 2  

x 1 1 x y y 2 2 x y R(D , D )D = − Dx (λ) + Dy (λ) − Dx D (λ) − Dy D (λ) D , 2 2 Dλ = [Dx , Dy ] = as well as the scalar curvature 

1 S= Dx Dx (λ) + Dy Dy (λ) − 2Dx (λ)2 − 2Dy (λ)2 . λ Moreover, one finds that ∇f = Dx (f )Dx + Dy (f )Dy div(αx Dx + αy Dy ) = Dx (αx ) + Dy (αy )

∆(f ) = Dx Dx (f ) + Dy Dy (f )

= Dx Dx (f ) + Dy Dy (f ) − Dx (λ)Dx (f ) − Dy (λ)Dy (f ). 6.2. Poisson algebras generated by three elements. Let A be a unital Poisson algebra generated by x1 = x, x2 = y, x3 = z ∈ A. Writing {x, y} = a, {y, z} = b and {z, x} = c, i.e.   0 a −c b , P = −a 0 c −b 0 20 JOAKIM ARNLIND AND AHMED AL-SHUJARY one readily checks that for an arbitrary symmetric matrix g PgPgP = −τ P with τ = a2 |g|33 + b2 |g|11 + c2 |g|22 + 2ab|g|31 − 2ac|g|32 − 2bc|g|21 , where |g|ij denotes the determinant of the matrix obtained from g by deleting the i’th row and the j’th column. Thus, one may construct the Kähler–Poisson algebra K = {A[τ −1 ], {x, y, z}, g}. In particular, if g = diag(λ, λ, λ), then τ = λ2 (a2 + b2 + c2 ). Let us now construct a particular class of algebras with a natural geometric interpretation and a close connection to algebraic geometry. Let R[x, y, z] be the polynomial ring in three variables over the real numbers, and write x1 = x, x2 = y and x3 = z. For arbitrary C ∈ R[x, y, z], it is straight-forward to show that {xi , xj } = εijk ∂k C, where εijk denotes the totally antisymmetric symbol with ε123 = 1, defines a Poisson structure on R[x, y, z] which is well-defined on the quotient AC = R[x, y, z]/(C) since {xi , C} = {xi , xj }∂j C = εijk (∂k C)(∂j C) = 0. In the spirit of algebraic geometry, the algebra AC has a natural interpretation as the polynomial functions on the level set C(x, y, z) = 0 in R3 . Choosing the metric gij = δij 1 (corresponding to the Euclidean metric on R3 ) one obtains a Kähler–Poisson algebra (AbC , {x, y, z}, g) where
2
2
2 AbC = AC [τ −1 ] and τ = ∂x C + ∂y C + ∂z C , with η = τ −1 . Note that the points in R3 , for which τ (x, y, z) = 0, coincide with the singular points of C(x, y, z) = 0; i.e. points where ∂x C = ∂y C = ∂z C = 0. As an illustration, let us choose C = 21 (ax2 + by 2 + cz 2 − 1) for a, b, c ∈ R, giving {x, y} = cz, {y, z} = ax and and η = a2 x2 + b2 y 2 + c2 z 2 together with  b 2 y 2 + c2 z 2 ij  −abxy (D ) = η −acxz −abxy a2 x2 + c2 z 2 −bcyz {z, x} = by.
−1  −acxz −bcyz  . 2 2 a x + b2 y 2 A straight-forward, but somewhat lengthy, calculation gives  x  x  x  x D D D D b D y  b D y  R(Dy , Dz ) Dy  = ax R R(Dx , Dy ) Dy  = cz R Dz Dz Dz Dz    x  x 0 −cz by D D b = abcη 3  cz b Dy  where R 0 −ax , R(Dz , Dx ) Dy  = by R z z −by ax 0 D D KÄHLER–POISSON ALGEBRAS 21 and the scalar curvature becomes S = 2abcη 2 . 7. Summary In this note, we have introduced the concept of Kähler–Poisson algebras as a mean to study Poisson algebras from a metric point of view. As shown, the single relation (4.1) has consequences that allow for an identification of geometric objects in the algebra, which share crucial properties with their classical counterparts. The idea behind the construction was to identify a distinguished set of elements in the algebra that serve as “embedding coordinates”, and then construct the projection operator D that projects from the tangent space of the ambient manifold onto that of the embedded submanifold. It is somewhat surprising that (4.1) encodes the crucial elements that are needed for the algebra to resemble an algebra of functions on an almost Kähler manifold. As outlined in Section 4.3, a large class of Poisson algebras admit a Kähler– Poisson algebra as an associated localization, which shows a certain generality of our treatment. Thus, even if one is not interested in metric structures on a Poisson algebra, the tools we have developed might be of help. For instance, if a Poisson algebra can be given the structure of a Kähler–Poisson algebra, one immediately concludes that the module generated by the inner derivations is a finitely generated projective module. A statement which is clearly independent of any metric structure. A comparison with differential geometry is close at hand, where the structure of a Riemannian manifold can be used to prove results about the underlying manifold (or even the topological structure). Let us end with a brief outlook. After having studied the basic properties of Kähler–Poisson algebras in this paper, there are several natural questions that can be studied. For instance, what is the interplay between the cohomology (of Lie-Rinehart algebras) and the Levi-Civita connection? Can one perhaps use the connection to compute cohomology? Is there a natural way to study the moduli spaces of Poisson algebras; i.e. how many (non-isomorphic) Kähler–Poisson structures does there exist on a given Poisson algebra? We hope to return to these, and many other interesting questions, in the near future. Acknowledgments We would like to thank M. Izquierdo for ideas and discussions. Furthermore, J. A. is supported by the Swedish Research Council. References [AH14] J. Arnlind and G. Huisken. Pseudo-Riemannian geometry in terms of multi-linear brackets. Lett. Math. Phys., 104(12):1507–1521, 2014. [AHH12] J. Arnlind, J. Hoppe, and G. Huisken. Multi-linear formulation of differential geometry and matrix regularizations. J. Differential Geom., 91(1):1–39, 2012. [Ber79] R. Berger. Géométrie algébrique de Poisson. C. R. Acad. Sci. Paris Sér. A-B, 289(11):A583–A585, 1979. [Bry88] J.-L. Brylinski. A differential complex for Poisson manifolds. J. Differential Geom., 28(1):93–114, 1988. [BS10] D. N. Blaschke and H. Steinacker. Curvature and gravity actions for matrix models. Classical Quantum Gravity, 27(16):165010, 15, 2010. 22 JOAKIM ARNLIND AND AHMED AL-SHUJARY [Hel01] S. Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. [Her53] J.-C. Herz. Pseudo-algèbres de Lie. I. C. R. Acad. Sci. Paris, 236:1935–1937, 1953. [Hue90] J. Huebschmann. Poisson cohomology and quantization. J. Reine Angew. Math., 408:57– 113, 1990. [Hue99] J. Huebschmann. Extensions of Lie-Rinehart algebras and the Chern-Weil construction. In Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996), volume 227 of Contemp. Math., pages 145–176. Amer. Math. Soc., Providence, RI, 1999. [Kar02] A. Karabegov. A covariant Poisson deformation quantization with separation of variables up to the third order. Lett. Math. Phys., 61(3):255–261, 2002. [Kon03] M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66(3):157–216, 2003. [Koz60] J. L. Kozul. Lectures on fibre bundles and differential geometry. Tata Institute of Fundamental Research, Bombay, 1960. [Lic77] A. Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geometry, 12(2):253–300, 1977. [Nas56] J. Nash. The imbedding problem for Riemannian manifolds. Ann. of Math. (2), 63:20– 63, 1956. [Nel67] E. Nelson. Tensor Analysis. Princeton University Press, Princeton, New Jersey, 1967. [Pal61] R. S. Palais. The cohomology of Lie rings. In Proc. Sympos. Pure Math., Vol. III, pages 130–137. American Mathematical Society, Providence, R.I., 1961. [Rin63] G. S. Rinehart. Differential forms on general commutative algebras. Trans. Amer. Math. Soc., 108:195–222, 1963. [Wei83] A. Weinstein. The local structure of Poisson manifolds. J. Differential Geom., 18(3):523– 557, 1983. (Joakim Arnlind) Dept. of Math., Linköping University, 581 83 Linköping, Sweden E-mail address: joakim.arnlind@liu.se (Ahmed Al-Shujary) Dept. of Math., Linköping University, 581 83 Linköping, Sweden E-mail address: ahmed.al-shujary@liu.se 

Instruction Sequence Notations with Probabilistic Instructions arXiv:0906.3083v2 [] 1 Oct 2014 J.A. Bergstra and C.A. Middelburg Informatics Institute, Faculty of Science, University of Amsterdam, Science Park 107, 1098 XG Amsterdam, the Netherlands J.A.Bergstra@uva.nl,C.A.Middelburg@uva.nl Abstract. This paper concerns instruction sequences that contain probabilistic instructions, i.e. instructions that are themselves probabilistic by nature. We propose several kinds of probabilistic instructions, provide an informal operational meaning for each of them, and discuss related work. On purpose, we refrain from providing an ad hoc formal meaning for the proposed kinds of instructions. We also discuss the approach of projection semantics, which was introduced in earlier work on instruction sequences, in the light of probabilistic instruction sequences. Keywords: instruction sequence, probabilistic instruction, projection semantics. 1998 ACM Computing Classification: D.1.4, F.1.1, F.1.2. 1 Introduction In this paper, we take the first step on a new subject in a line of research whose working hypothesis is that the notion of an instruction sequence is relevant to diverse subjects in computer science (see e.g. [12,13,14,15]). In this line of research, an instruction sequence under execution is considered to produce a behaviour to be controlled by some execution environment: each step performed actuates the processing of an instruction by the execution environment and a reply returned at completion of the processing determines how the behaviour proceeds. The term service is used for a component of a system that provides an execution environment for instruction sequences, and a model of systems that provide execution environments for instruction sequences is called an execution architecture. This paper is concerned with probabilistic instruction sequences. We use the term probabilistic instruction sequence for an instruction sequence that contains probabilistic instructions, i.e. instructions that are themselves probabilistic by nature, rather than an instruction sequence of which the instructions are intended to be processed in a probabilistic way. We will propose several kinds of probabilistic instructions, provide an informal operational meaning for each of them, and discuss related work. We will refrain from a formal semantic analysis of the proposed kinds of probabilistic instructions. Moreover, we will not claim any form of completeness for the proposed kinds of probabilistic instructions. Other convincing kinds might be found in the future. We will leave unanalysed the topic of probabilistic instruction sequence processing, which includes all phenomena concerning services and execution architectures for which probabilistic analysis is necessary. Viewed from the perspective of machine-execution, execution of a probabilistic instruction sequence using an execution architecture without probabilistic features can only be a metaphor. Execution of a deterministic instruction sequence using an execution architecture with probabilistic features, i.e. an execution architecture that allows for probabilistic services, is far more plausible. Thus, it looks to be that probabilistic instruction sequences find their true meaning by translation into deterministic instruction sequences for execution architectures with probabilistic features. Indeed projection semantics, the approach to define the meaning of programs which was first presented in [9], need not be compromised when probabilistic instructions are taken into account. This paper is organized as follows. First, we set out the scope of the paper (Section 2) and review the special notation and terminology used in the paper (Section 3). Next, we propose several kinds of probabilistic instructions (Sections 4 and 5). Following this, we formulate a thesis on the behaviours produced by probabilistic instruction sequences under execution (Section 6) and discuss the approach of projection semantics in the light of probabilistic instruction sequences (Section 7). We also discuss related work (Section 8) and make some concluding remarks (Section 10). In the current version of this paper, we moreover mention some outcomes of a sequel to the work reported upon in this paper (Section 9). 2 On the Scope of this Paper We go into the scope of the paper and clarify its restrictions by giving the original motives. However, the first version of this paper triggered off work on the behaviour of probabilistic instruction sequences under execution and outcomes of that work invalidated some of the arguments used to motivate the restrictions. The relevant outcomes of the work concerned will be mentioned in Section 9. We will propose several kinds of probabilistic instructions, chosen because of their superficial similarity with kinds of deterministic instructions known from PGA (ProGram Algebra) [9], PGLD (ProGramming Language D) with indirect jumps [10], C (Code) [19], and other similar notations and not because any computational intuition about them is known or assumed. For each of these kinds, we will provide an informal operational meaning. Moreover, we will exemplify the possibility that the proposed unbounded probabilistic jump instructions are simulated by means of bounded probabilistic test instructions and bounded deterministic jump instructions. We will also refer to related work that introduces something similar to what we call a probabilistic instruction and connect the proposed kinds of probabilistic instructions with similar features found in related work. 2 We will refrain from a formal semantic analysis of the proposed kinds of probabilistic instructions. When we started with the work reported upon in this paper, the reasons for doing so were as follows: – In the non-probabilistic case, the subject reduces to the semantics of PGA. Although it seems obvious at first sight, different models, reflecting different levels of abstraction, can and have been distinguished (see e.g. [9]). Probabilities introduce a further ramification. – What we consider sensible is to analyse this double ramification fully. What we consider less useful is to provide one specific collection of design decisions and working out its details as a proof of concept. – We notice that for process algebra the ramification of semantic options after the incorporation of probabilistic features is remarkable, and even frustrating (see e.g. [24,27]). There is no reason to expect that the situation is much simpler here. – Once that a semantic strategy is mainly judged on its preparedness for a setting with multi-threading, the subject becomes intrinsically complex – like the preparedness for a setting with arbitrary interleaving complicates the semantic modeling of deterministic processes in process algebra. – We believe that a choice for a catalogue of kinds of probabilistic instructions can be made beforehand. Even if that choice will turn out to be wrong, because prolonged forthcoming semantic analysis may give rise to new, more natural, kinds of probabilistic instructions, it can at this stage best be driven by direct intuitions. In this paper, we will leave unanalysed the topic of probabilistic instruction sequence processing, i.e. probabilistic processing of instruction sequences, which includes all phenomena concerning services and concerning execution architectures for which probabilistic analysis is necessary. At the same time, we admit that probabilistic instruction sequence processing is a much more substantial topic than probabilistic instruction sequences, because of its machine-oriented scope. We take the line that a probabilistic instruction sequence finds its operational meaning by translation into a deterministic instruction sequence and execution using an execution architecture with probabilistic features. 3 Preliminaries In the remainder of this paper, we will use the notation and terminology regarding instructions and instruction sequences from PGA. The mathematical structure that we will use for quantities is a signed cancellation meadow. That is why we briefly review PGA and signed cancellation meadows in this section. In PGA, it is assumed that a fixed but arbitrary set of basic instructions has been given. The primitive instructions of PGA are the basic instructions and in addition: – for each basic instruction a, a positive test instruction +a; 3 – for each basic instruction a, a negative test instruction −a; – for each natural number l, a forward jump instruction #l; – a termination instruction !. The intuition is that the execution of a basic instruction a produces either T or F at its completion. In the case of a positive test instruction +a, a is executed and execution proceeds with the next primitive instruction if T is produced. Otherwise, the next primitive instruction is skipped and execution proceeds with the primitive instruction following the skipped one. If there is no next instruction to be executed, execution becomes inactive. In the case of a negative test instruction −a, the role of the value produced is reversed. In the case of a plain basic instruction a, execution always proceeds as if T is produced. The effect of a forward jump instruction #l is that execution proceeds with the l-th next instruction. If l equals 0 or the l-th next instruction does not exist, execution becomes inactive. The effect of the termination instruction ! is that execution terminates. The constants of PGA are the primitive instructions and the operators of PGA are: – the binary concatenation operator ; ; – the unary repetition operator ω . Terms are built as usual. We use infix notation for the concatenation operator and postfix notation for the repetition operator. A closed PGA term is considered to denote a non-empty, finite or periodic infinite sequence of primitive instructions.1 Closed PGA terms are considered equal if they denote the same instruction sequence. The axioms for instruction sequence equivalence are given in [9]. The unfolding equation X ω = X ; X ω is derivable from those equations. Moreover, each closed PGA term is derivably equal to one of the form P or P ; Qω , where P and Q are closed PGA terms in which the repetition operator does not occur. In [9], PGA is extended with a unit instruction operator u which turns sequences of instructions into single instructions. The result is called PGAu . In [35], the meaning of PGAu programs is described by a translation from PGAu programs into PGA programs. In the sequel, the following additional assumption is made: a fixed but arbitrary set of foci and a fixed but arbitrary set of methods have been given. Moreover, we will use f.m, where f is a focus and m is a method, as a general notation for basic instructions. In f.m, m is the instruction proper and f is the name of the service that is designated to process m. The signature of signed cancellation meadows consists of the following constants and operators: – the constants 0 and 1; – the binary addition operator + ; 1 A periodic infinite sequence is an infinite sequence with only finitely many distinct suffixes. 4 – – – – the the the the binary multiplication operator · ; unary additive inverse operator −; unary multiplicative inverse operator unary signum operator s. −1 ; Terms are build as usual. We use infix notation for the binary operators + and · , prefix notation for the unary operator −, and postfix notation for the unary operator −1 . We use the usual precedence convention to reduce the need for parentheses. We introduce subtraction and division as abbreviations: p − q abbreviates p + (−q) and p/q abbreviates p · (q −1 ). We use the notation n for numerals and the notation pn for exponentiation with a natural number as exponent. The term n is inductively defined as follows: 0 = 0 and n + 1 = n + 1. The term pn is inductively defined as follows: p0 = 1 and pn+1 = pn · p. The constants and operators from the signature of signed cancellation meadows are adopted from rational arithmetic, which gives an appropriate intuition about these constants and operators. The equational theories of signed cancellation meadows is given in [8]. In signed cancellation meadows, the functions min and max have a simple definition (see also [8]). A signed cancellation meadow is a cancellation meadow expanded with a signum operation. The prime example of cancellation meadows is the field of rational numbers with the multiplicative inverse operation made total by imposing that the multiplicative inverse of zero is zero, see e.g. [20]. 4 Probabilistic Basic and Test Instructions In this section, we propose several kinds of probabilistic basic and test instructions. It is assumed that a fixed but arbitrary signed cancellation meadow M has been given. Moreover, we write q̂, where q ∈ M, for max(0, min(1, q)). We propose the following probabilistic basic instructions: – %(), which produces T with probability 1/2 and F with probability 1/2; – %(q), which produces T with probability q̂ and F with probability 1 − q̂, for q ∈ M. The probabilistic basic instructions have no side-effect on a state. The basic instruction %() can be looked upon as a shorthand for %(1/2). We distinguish between %() and %(1/2) for reason of putting the emphasis on the fact that it is not necessary to bring in a notation for quantities ranging from 0 to 1 in order to design probabilistic instructions. Once that probabilistic basic instructions of the form %(q) are chosen, an unbounded ramification of options for the notation of quantities is opened up. We will assume that closed terms over the signature of signed √ cancellation meadows are used to denote quantities. Instructions such as %( 1 + 1) are implicit in the √ form %(q), assuming that it is known how to view as a notational extension of signed cancellation meadows (see e.g. [8]). Like all basic instructions, each probabilistic basic instruction give rise to two probabilistic test instructions: 5 – %() gives rise to +%() and −%(); – %(q) gives rise to +%(q) and −%(q). Probabilistic primitive instructions of the form +%(q) and −%(q) can be considered probabilistic branch instructions where q is the probability that the branch is not taken and taken, respectively, and likewise the probabilistic primitive instructions +%() and −%(). We find that the primitive instructions %() and %(q) can be replaced by #1 without loss of (intuitive) meaning. Of course, in a resource aware model, #1 may be much cheaper than %(q), especially if q is hard to compute. Suppose that %(q) is realized at a lower level by means of %(), which is possible, and suppose that q is a computable real number. The question arises whether the expectation of the time to execute %(q) is finite. To exemplify the possibility that %(q) is realized by means of %() in the case where q is a rational number, we look at the following probabilistic instruction sequences: −%(2/3) ; #3 ; a ; ! ; b ; ! , (+%() ; #3 ; a ; ! ; +%() ; #3 ; b ; !)ω . It is easy to see that these instruction sequences produce on execution the same behaviour: with probability 2/3, first a is performed and then termination follows; and with probability 1/3, first b is performed and then termination follows. In the case of computable real numbers other than rational numbers, use must be made of a service that does duty for the tape of a Turing machine (such a service is, for example, described in [18]). Let q ∈ M, and let random(q) be a service with a method get whose reply is T with probability q̂ and F with probability 1 − q̂. Then a reasonable view on the meaning of the probabilistic primitive instructions %(q), +%(q) and −%(q) is that they are translated into the deterministic primitive instructions random(q).get, +random(q).get and −random(q).get, respectively, and executed using an execution architecture that provides the probabilistic service random(q). Another option is possible here: instead of a different service random(q) for each q ∈ [0, 1] and a single method get, we could have a single service random with a different method get(q) for each q ∈ [0, 1]. In the latter case, %(q), +%(q) and −%(q) would be translated into the deterministic primitive instructions random.get(q), +random.get(q) and −random.get(q). 5 Probabilistic Jump Instructions In this section, we propose several kinds of probabilistic jump instructions. It is assumed that the signed cancellation meadow M has been expanded with an operation N such that, for all q ∈ M, N(q) = 0 iff q = n for some n ∈ N. We write l̄, where l ∈ M is such that N(l) = 0, for the unique n ∈ N such that l = n. We propose the following probabilistic jump instructions: 6 – #%U(k), having the same effect as #j with probability 1/k for j ∈ [1, k̄], for k ∈ M with N(k) = 0; – #%G(q)(k), having the same effect as #j with probability q̂ · (1 − q̂)j−1 for j ∈ [1, k̄], for q ∈ M and k ∈ M with N(k) = 0; – #%G(q)l, having the same effect as #l̄ · j with probability q̂ · (1 − q̂)j−1 for j ∈ [1, ∞), for q ∈ M and l ∈ M with N(l) = 0. The letter U in #%U(k) indicates a uniform probability distribution, and the letter G in #%G(q)(k) and #%G(q)l indicates a geometric probability distribution. Instructions of the forms #%U(k) and #%G(q)(k) are bounded probabilistic jump instructions, whereas instructions of the form #%G(q)l are unbounded probabilistic jump instructions. Like in the case of the probabilistic basic instructions, we propose in addition the following probabilistic jump instructions: – #%G()(k) as the special case of #%G(q)(k) where q = 1/2; – #%G()l as the special case of #%G(q)l where q = 1/2. We believe that it must be possible to eliminate all probabilistic jump instructions. In particular, we believe that it must be possible to eliminate all unbounded probabilistic jump instructions. This belief can be understood as the judgement that it is reasonable to expect from a semantic model of probabilistic instruction sequences that the following identity and similar ones hold: +a ; #%G()2 ; (+b ; ! ; c)ω = +a ; +%() ; #8 ; #10 ; (+b ; #5 ; #10 ; +%() ; #8 ; #10 ; ! ; #5 ; #10 ; +%() ; #8 ; #10 ; c ; #5 ; #10 ; +%() ; #8 ; #10)ω . Taking this identity and similar ones as our point of departure, the question arises what is the most simple model that justifies them. A more general question is whether instruction sequences with unbounded probabilistic jump instructions can be translated into ones without probabilistic jump instructions provided it does not bother us that the instruction sequences may become much longer (e.g. expectation of the length bounded, but worst case length unbounded). 6 The Probabilistic Process Algebra Thesis In the absence of probabilistic instructions, threads as considered in BTA (Basic Thread Algebra) [9] or its extension with thread-service interaction [17] can be used to model the behaviours produced by instruction sequences under execution.2 Processes as considered in general process algebras such as ACP [6], 2 In [9], BTA is introduced under the name BPPA (Basic Polarized Process Algebra). 7 CCS [32] and CSP [26] can be used as well, but they give rise to a more complicated modeling of the behaviours of instruction sequences under execution (see e.g. [13]). In the presence of probabilistic instructions, we would need a probabilistic thread algebra, i.e. a variant of BTA or its extension with thread-service interaction that covers probabilistic behaviours. When we started with the work reported upon in this paper, it appeared that any probabilistic thread algebra is inherently more complicated to such an extent that the advantage of not using a general process algebra evaporates. Moreover, it appeared that any probabilistic thread algebra requires justification by means of an appropriate probabilistic process algebra. This led us to the following thesis: Thesis. Modeling the behaviours produced by probabilistic instruction sequences under execution is a matter of using directly processes as considered in some probabilistic process algebra. A probabilistic thread algebra has to cover the interaction between instruction sequence behaviours and services. Two mechanisms are involved in that. They are called the use mechanism and the apply mechanism (see e.g. [17]). The difference between them is a matter of perspective: the former is concerned with the effect of services on behaviours of instruction sequences and therefore produces behaviours, whereas the latter is concerned with the effect of instruction sequence behaviours on services and therefore produces services. When we started with the work reported upon in this paper, it appeared that the use mechanism would make the development of a probabilistic thread algebra very intricate. The first version of this paper triggered off work on the behaviour of probabilistic instruction sequences under execution by which the thesis stated above is refuted. The ultimate point is that meanwhile an appropriate and relatively simple probabilistic thread algebra has been devised (see [16]). Moreover, our original expectations about probabilistic process algebras turned out to be too high. The first probabilistic process algebra is presented in [23] and the first probabilistic process algebra with an asynchronous parallel composition operator is presented in [5]. A recent overview of the work on probabilistic process algebras after that is given in [29]. This overview shows that the multitude of semantic ideas applied and the multitude of variants of certain operators devised have kept growing, and that convergence is far away. In other words, there is little well-established yet. In particular, for modeling the behaviours produced by probabilistic instruction sequences, we need operators for probabilistic choice, asynchronous parallel composition, and abstraction from internal actions. For this case, the attempts by one research group during about ten years to develop a satisfactory ACP-like process algebra (see e.g. [1,2,3]) have finally led to a promising process algebra. However, it is not yet clear whether the process algebra concerned will become well-established. 8 All this means that a justification of the above-mentioned probabilistic thread algebra by means of an appropriate probabilistic process algebra will be of a provisional nature for the time being. 7 Discussion of Projectionism Notice that once we move from deterministic instructions to probabilistic instructions, instruction sequence becomes an indispensable concept. Instruction sequences cannot be replaced by threads or processes without taking potentially premature design decisions. In preceding sections, however, we have outlined how instruction sequences with the different kinds of probabilistic instructions can be translated into instruction sequences without them. Therefore, it is a reasonable to claim that, like for deterministic instruction sequence notations, all probabilistic instruction sequence notations can be provided with a probabilistic semantics by translation of the instruction sequences concerned into appropriate single-pass instruction sequences. Thus, we have made it plausible that projectionism is feasible for probabilistic instruction sequences. Projectionism is the point of view that: – any instruction sequence P , and more general even any program P , first and for all represents a single-pass instruction sequence as considered in PGA; – this single-pass instruction sequence, found by a translation called a projection, represents in a natural and preferred way what is supposed to take place on execution of P ; – PGA provides the preferred notation for single-pass instruction sequences. In a rigid form, as in [9], projectionism provides a definition of what constitutes a program. The fact that projectionism is feasible for probabilistic instruction sequences, does not imply that it is uncomplicated. To give an idea of the complications that may arise, we will sketch below found challenges for projectionism. First, we introduce some special notation. Let N be a program notation. Then we write N2pga for the projection function that gives, for each program P in N, the closed PGA terms that denotes the single-pass instruction sequence that produces on execution the same behaviour as P . We have found the following challenges for projectionism: – Explosion of size. If N2pga(P ) is much longer than P , then the requirement that it represents in a natural way what is supposed to take place on execution of P is challenged. For example, if the primitive instructions of N include instructions to set and test up to n Boolean registers, then the projection to N2pga(P ) may give rise to a combinatorial explosion of size. In such cases, the usual compromise is to permit single-pass instruction sequences to make use of services (see e.g. [17]). – Degradation of performance. If N2pga(P )’s natural execution is much slower than P ’s execution, supposing a clear operational understanding of P , then 9 the requirement that it represents in a natural way what is supposed to take place on execution of P is challenged. For example, if the primitive instructions of N include indirect jump instructions, then the projection to N2pga(P ) may give rise to a degradation of performance (see e.g. [10]). – Incompatibility of services. If N2pga(P ) has to make use of services that are not deterministic, then the requirement that it represents in a natural way what is supposed to take place on execution of P is challenged. For example, if the primitive instructions of N include instructions of the form +%(q) or −%(q), then P cannot be projected to a single-pass instruction sequence without the use of probabilistic services. In this case, either probabilistic services must be permitted or probabilistic instruction sequences must not be considered programs. – Complexity of projection description. The description of N2pga may be so complex that it defeats N2pga(P )’s purpose of being a natural explanation of what is supposed to take place on execution of P . For example, the projection semantics given for recursion in [7] suffers from this kind of complexity when compared with the conventional denotational semantics. In such cases, projectionism may be maintained conceptually, but rejected pragmatically. – Aesthetic degradation. In N2pga(P ), something elegant may have been replaced by nasty details. For example, if N provides guarded commands, then N2pga(P ), which will be much more detailed, might be considered to exhibit signs of aesthetic degradation. This challenge is probably the most serious one, provided we accept that such elegant features belong to program notations. Of course, it may be decided to ignore aesthetic criteria altogether. However, more often than not, they have both conceptual and pragmatic importance. One might be of the opinion that conceptual projectionism can accept explosion of size and/or degradation of performance. We do not share this opinion: both challenges require a more drastic response than a mere shift from a pragmatic to a conceptual understanding of projectionism. This drastic response may include viewing certain mechanisms as intrinsically indispensable for either execution performance or program compactness. For example, it is reasonable to consider the basic instructions of the form %(q), where q is a computable real number, indispensable if the expectations of the times to execute their realizations by means of %() are not all finite. Nevertheless, projectionism looks to be reasonable for probabilistic programs: they can be projected adequately to deterministic single-pass instruction sequences for an execution architecture with probabilistic services. 8 Related Work In [38], a notation for probabilistic programs is introduced in which we can write, for example, random(p·δ0 +q·δ1 ). In general, random(λ) produces a value according to the probability distribution λ. In this case, δi is the probability distribution that gives probability 1 to i and probability 0 to other values. Thus, for p+q = 1, 10 p·δ0 +q·δ1 is the probability distribution that gives probability p to 0, probability q to 1, and probability 0 to other values. Clearly, random(p·δ0 +q·δ1 ) corresponds to %(p). Moreover, using this kind of notation, we could write #( k1 ·(δ1 +· · ·+δk̄ )) for #%U(k) and #(q̂ · δ1 + q̂ · (1 − q̂) · δ2 + · · · + q̂ · (1 − q̂)k−1 · δk ) for #%G(q)(k). In much work on probabilistic programming, see e.g. [25,30,33], we find the binary probabilistic choice operator p ⊕ (for p ∈ [0, 1]). This operator chooses between its operands, taking its left operand with probability p. Clearly, P p ⊕ Q can be taken as abbreviations for +%(p);u(P ;#2);u(Q). This kind of primitives dates back to [28] at least. Quite related, but from a different perspective, is the toss primitive introduced in [21]. The intuition is that toss(bm, p) assigns to the Boolean memory cell bm the value T with probability p̂ and the value F with probability 1− p̂. This means that toss(bm, p) has a side-effect on a state, which we understand as making use of a service. In other words, toss(bm, p) corresponds to a deterministic instruction intended to be processed by a probabilistic service. Common in probabilistic programming are assignments of values randomly chosen from some interval of natural numbers to program variables (see e.g. [37]). Clearly, such random assignments correspond also to deterministic instructions intended to be processed by probabilistic services. Suppose that x=i is a primitive instruction for assigning value i to program variable x. Then we can write: #%U(k) ; u(x=1 ; #k) ; u(x=2 ; #k−1) ; . . . ; u(x=k ; #1). This is a realistic representation of the assignment to x of a value randomly chosen from {1, . . . , k}. However, it is clear that this way of representing random assignments leads to an exponential blow up in the size of any concrete instruction sequence representation, provided the concrete representation of k is its decimal representation. The refinement oriented theory of programs uses demonic choice, usually written ⊓, as a primitive (see e.g. [30,31]). A demonic choice can be regarded as a probabilistic choice with unknown probabilities. Demonic choice could be written +⊓ in a PGA-like notation. However, a primitive instruction corresponding to demonic choice is not reasonable: no mechanism for the execution of +⊓ is conceivable. Demonic choice exists in the world of specifications, but not in the world of instruction sequences. This is definitely different with +%(p), because a mechanism for its execution is conceivable. Features similar to probabilistic jump instructions are not common in probabilistic programming. To our knowledge, the mention of probabilistic goto statements of the form pr goto {l1 , l2 } in [4] is the only mention of a similar feature in the computer science literature. The intuition is that pr goto {l1 , l2 }, where l1 and l2 are labels, has the same effect as goto l1 with probability 1/2 and has the same effect as goto l2 with probability 1/2. Clearly, this corresponds to a probabilistic jump instruction of the form #%U(1/2). It appears that quantum computing has something to offer that cannot be obtained by conventional computing: it makes a stateless generator of random bits available (see e.g. [22,34]). By that quantum computing indeed provides a justification of +%(1/2) as a probabilistic instruction. 11 9 On the Sequel to this Paper The first version of this paper triggered off work on the behaviour of probabilistic instruction sequences under execution and outcomes of that work invalidated some of the arguments used to motivate the restricted scope of this paper. In this section, we mention the relevant outcomes of that work. After the first version of this paper (i.e. [11]) appeared, it was found that: – The different levels of abstraction that have been distinguished in the nonprobabilistic case can be distinguished in the probabilistic case as well and only the models at the level of the behaviour of instruction sequences under execution are not essentially the same. – The semantic options at the behavioural level after the incorporation of probabilistic features are limited because of (a) the orientation towards behaviours of a special kind and (b) the semantic constraints induced by the informal explanations of the enumerated kinds of probabilistic instructions and the desired elimination property of all but one kind. – For the same reasons, preparedness for a setting with multi-threading does not really complicate matters. The current state of affairs is as follows: – BTA and its extension with thread-service interaction, which are used to describe the behaviour of instruction sequences under execution in the nonprobabilistic case, have been extended with probabilistic features in [16]. – In [16], we have added the probabilistic basic and test instructions proposed in Section 4 to PGLB (ProGramming Language B), an instruction sequence notation rooted in PGA and close to existing assembly languages, and have given a formal definition of the behaviours produced by the instruction sequences from the resulting instruction sequence notation in terms of non-probabilistic instructions and probabilistic services. – The bounded probabilistic jump instructions proposed in Section 5 can be given a behavioural semantics in the same vein. The unbounded probabilistic jump instructions fail to have a natural behavioural semantics in the setting of PGA because infinite instruction sequences are restricted to eventually periodic ones. – The extension of BTA with multi-threading, has been generalized to probabilistic multi-threading in [16] as well. 10 Conclusions We have made a notational proposal of probabilistic instructions with an informal semantics. By that we have contrasted probabilistic instructions in an execution architecture with deterministic services with deterministic instructions in an execution architecture with partly probabilistic services. The history of the proposed kinds of instructions can be traced. 12 We have refrained from an ad hoc formal semantic analysis of the proposed kinds of instructions. There are many solid semantic options, so many and so complex that another more distant analysis is necessary in advance to create a clear framework for the semantic analysis in question. The grounds of this work are our conceptions of what a theory of probabilistic instruction sequences and a complementary theory of probabilistic instruction sequence processing (i.e. execution architectures with probabilistic services) will lead to: – comprehensible explanations of relevant probabilistic algorithms, such as the Miller-Rabin probabilistic primality test [36], with precise descriptions of the kinds of instructions and services involved in them; – a solid account of pseudo-random Boolean values and pseudo-random numbers; – a thorough exposition of the different semantic options for probabilistic instruction sequences; – explanations of relevant quantum algorithms, such as Shor’s integer factorization algorithm [39], by first giving a clarifying analysis in terms of probabilistic instruction sequences or execution architectures with probabilistic services and only then showing how certain services in principle can be realized very efficiently with quantum computing. 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Formal verification of an interior point algorithm instanciation Guillaume Davy∗‡ , Eric Feron† , Pierre-Loic Garoche∗ , and Didier Henrion‡ arXiv:1801.03833v1 [cs.LO] 11 Jan 2018 ∗ Onera - The French Aerospace Lab, Toulouse, FRANCE ‡ CNRS LAAS, Toulouse, FRANCE † Georgia Institute of Technology, Atlanta GA, USA Abstract. With the increasing power of computers, real-time algorithms tends to become more complex and therefore require better guarantees of safety. Among algorithms sustaining autonomous embedded systems, model predictive control (MPC) is now used to compute online trajectories, for example in the SpaceX rocket landing. The core components of these algorithms, such as the convex optimization function, will then have to be certified at some point. This paper focuses specifically on that problem and presents a method to formally prove a primal linear programming implementation. We explain how to write and annotate the code with Hoare triples in a way that eases their automatic proof. The proof process itself is performed with the WP-plugin of Frama-C and only relies on SMT solvers. Combined with a framework producing all together both the embedded code and its annotations, this work would permit to certify advanced autonomous functions relying on online optimization. 1 Introduction The increasing power of computers, makes possible the use of complex numerical methods in real time within cyber-physical systems. These algorithms, despite having been studied for a long time, raise new issues when used online. Among these algorithms, we are concerned specifically with numerical optimization which is used in model predictive control (MPC) for example, by SpaceX to perform computation of trajectory for rocket landing [1]. These iterative algorithms perform complex math operations in a loop until they reach an -close optimal value. This implies some uncertainty on the number of iterations required but also on the reliability of the computed result. We address both these issues in this paper. As a first step, we focus on linear programming, with the long-term objective of proving more general convex optimization problems. We therefore chose to study an interior point algorithm (IPM, Interior Point Method) instead of the famous simplex methods. As a matter of fact simplex is bound to linear constraints while IPMs can address more general convex cones such as the Lorentz cone problems (quadratic programming QP and second-order cone programming SOCP), or Semi-Definite Programming (SDP). To give the best possible guarantee, we rely on formal methods to prove the algorithm soundness. More specifically we use Hoare Logic[2,3] to express what we expect from the algorithm and rely on Weakest Precondition[4] approach to prove that the code satisfy Fig. 1. Complete toolchain we are interested in, this article focus on writing C code and annotation them. Figure 1 sketches our fully automatic process which, when provided with a convex problem with some unknown values, generates the code, the associated annotation and prove it automatically. We are not going to present all the process in this paper but concentrate on how to write the embedded code, annotate it and automatize its proof. In this first work, we focus on the algorithm itself assuming a real semantics for float variables and leave the floating point problem for future work. However the algorithm itself is expressed in C with all the associated hassle and complexity. We proved the absence of runtime error, the functionality of the code and its termination. This paper is structured as follow. In Section 2 we present some key notions for both convex optimization and formal proof of our algorithm. In Section 3 we present the code structure supporting the later proof process. In Section 4 we introduce our code annotatations. Section 5 focuses on the proof process. Section 6 presents some experimental results. 2 Preliminaries In order to support the following analyses, we introduce the notions and notations used throughout the paper. First, we discuss Linear Programming (LP) problems and a primal IPM algorithm to solve it. Then, we introduce the reader to Hoare logic based reasoning. 2.1 Optimization Linear Programming is a class of optimization problems. A linear program is defined by a matrix A of size m × n, and two vectors b, c of respective size m, and n. Definition 1 (Linear program) Let us consider A ∈ Rm×n , b ∈ Rm and c ∈ Rn . We define P (A, b, c) as the linear program: min x∈Rn ,Ax≤b hc, xi with hc, xi = cT x (1) Definition 2 (Linear program solution) Let us consider the problem P (A, b, c) and assume that an optimal point x∗ exists and is unique. We have then the following definitions: Ef ={x ∈ Rn | Ax ≤ b} f (x) =hc, xi ∗ x = arg min f x∈Ef (feasible set of P ) (2) (cost function) (3) (optimal point) (4) Primal interior point algorithm. We decided to use interior point method (IPM) to enable the future extension of this work to more advanced convex programs. We chose a primal algorithm for its simplicity compared to primal/dual algorithms and we followed Nesterov’s book [5, chap. 4] for the theory. The following definitions present the key ingredients of IPM algorithms: barrier function, central path, Newton steps and approximate centering condition. Fig. 2. Barrier function Barrier function. Computing the extrema, i.e. minimum or maximum, of a function without additional constraints, could be done by analyzing the zeros of its gradiants. However this does not apply in presence of constraints. An approach that amounts to introducing a penalty function F : Ef → R to represent the feasible set, ie. the acceptable region. This function must tend towards infinity when x approaches the border of Ef , cf. Figure 2 for a logarithmic barrier function encoding a set of linear constraints. Definition 3 The adjusted cost function is a linear combination of the previous objective function f and the barrier function F . f˜(x, t) = t × f (x) + F (x) with t ∈ R (5) The variable t balances the impact of the barrier: when t = 0, f˜(x, t) is independent from the objective while when t → +∞, f˜(x, t) is equivalent to t × f (x). Central path. We are interested in the values of x minimizing f˜ when t varies from 0 to +∞. These values for x characterize a path, the central path: Definition 4 (Central path and analytic center) x ∗ : R+ t → Ef 7→ arg min f˜(x, t) (6) x∈Ef x∗ (0) is called the analytic center, it is independent from the cost function. The central path has an interesting property when t increases: Property 1 lim x∗ (t) = x∗ (7) t→+∞ The algorithm prerforms a sequence of iterations, updating a point X that follows the central path and eventually reaches the optimal point. At the beginning of an iteration, there exists a real t such that X = x∗ (t). Then t is increased by dt > 0 and x∗ (t+dt) is the new point X. This translation dX is performed by a Newton step as sketched in Fig. 3. One step along the central path Figure 3. Newton step. The Newton’s method computes an approximation of a root of a function G : Rk → Rl . It is a first order method, ie. it relies on the gradient of the function and, from a point in the domain of the neighbourhood of a root, performs a sequence of iterations, called Newton steps. Figure 4 illustrates one of such step. Definition 5 A Newton step transforms Yn into Yn+1 as follows: Fig. 4. Newton step for k =
−1 l=1 Yn+1 − Yn = − G0 (Yn ) G(Yn ) (8) We apply the Newton step to the gradient of f˜, computing its root which coincides with the minimum of f˜. We obtain dX = − F 00 (X)
−1 ((t + dt)c + F 0 (X)) (9) Self-concordant barrier. The convergence of the Newton method is guaranted only in the neighbourhood to the function root. This neighbourhood is called the region of quadratic convergence; this region evolves on each iteration since t varies. To guarantee that the iterate X remains in the region after each iteration, we require the barrier function to be self-concordant: Definition 6 (Self-concordant barrier) A closed convex function g is a νself-concordant barrier if 3 D3 g(x)[u, u, u] ≤ 2(uT g 00 (x)u) 4 (10) g 0 (x)T g 00 (x)g 0 (x) < ν (11) and From now on we assume that F is a self-concordant barrier. Thus F 00 is non-degenerate([5, Th4.1.3]) and we can define: Definition 7 (Local-norm) kyk∗x = q y T × F 00 (x)−1 × y (12) This local-norm allows to define the Approximate Centering Condition(ACC), the crucial property which guarantees that X remains in the region of quadratic convergence: Definition 8 (ACC) Let x ∈ Ef and t ∈ R+ , ACC(x, t, β) is a predicate defined by kf˜0 (x)k∗x = ktc + F 0 (x)k∗x ≤ β (13) In the following, we choose a specific value for β, as defined in (14). √ 3− 5 (14) β< 2 The only step remaining is the computation of the largest dt such that X remains in the region of quadratic convergence around x∗ (t + dt). dt = γ kck∗x (15) with γ a constant. This choice maintains the ACC at each iteration([5, Th4.2.8]): Theorem 1 (ACC preserved) If we have ACC(X, t, β) and γ ≤ then we also have ACC(X + dX, t + dt, β). √ β √ 1+ β −β For this work, we use the classic self-concordant barrier for linear program: m P F (x) = −log(bi − Ai × x) with A1 , An the columns of A. i=0 Importance of the analytic center. x∗F is required to initiate the algorithm. In case of offline use the value could be precomputed and validated. However in case of online use, its computation itself has to be proved. Fortunatly this can be done by a similiar algorithm with comparable proofs. 2.2 Formal methods For the same program, different semantics can be used to specify its behavior: (i) a denotational semantics, expressing the program as a mathematical function, (ii) an operational semantics, expressing it as a sequence of basic computations, or (iii) an axiomatic semantics. In the latter case, the semantics can be defined in an incomplete way, as a set of projective statements, i.e. observations. This idea was formalized by Floyd [3], Hoare [6] as a way to specify the expected behavior, the specification, of a program through pre- and post-condition, or assume-guarantee contracts. Definition 9 (Hoare Triple) Let C : M → M be a program with M the set of its possible memories. Let P and Q, two predicates on M. We say that the Hoare triple {P } C {Q} is valid when ∀m ∈ M, P (m) ⇒ Q(C(m)) (16) – P is called a precondition or requires and Q the postcondition or ensures. – If C is a function, {P } C {Q} is called a contract. These Hoare triples can be used to annotate programs written in C. In the following, we rely on the ANSI C Specification Language (ACSL)[7], the specification language of Frama-C, to annotate functions. The Frama-C tool processes the annotation language, identifying each Hoare Triple and converting them into logical formulas, using the Weakest Precondition strategy. Definition 10 (Weakest Precondition) The Weakest Precondition of a program c and a postcondition Q is a formula WP(C, Q) such that: 1. {WP(C, Q)} C {Q} is a valid Hoare triple 2. For all P , {P } C {Q} valid implies P ⇒ WP(C, Q) Theorem 2 (Proving Hoare Triple) WP(C, Q) = R P ⇒R {P } C {Q} The WP property can be computed mechanically, propagating back the postcondition along the program instructions. An example of such rules is given in Figure 5. The only exception is the while loop rule which requires to be provided with an invariant, this rule is presented later in the document in Figure. 12. Assignement Sequence Conditional WP(x = E, Q) = ∀y, y = E ⇒ Q[x ← y] WP(S2, Q) = O WP(S1, O) = R WP(S1;S2, Q) = R WP(S1, Q) = P1 WP(S2, Q) = P2 WP(if (E) S1 else S2, Q) = E ⇒ P1 ∧ ¬E ⇒ P2 Fig. 5. Examples of WP rules Automation The use of SMT solvers enables the automatic proof of some programs and their annotations. This is however only feasible for properties that could be solved without the need of proof assistant. This requires to write both programs (cf. Section 3) and annotations (cf. Section 4) with some considerations for the proof process. 3 Writing Provable Code In order to ease the proof process we write the algorithm code in a very specific manner. In the current section we present our modeling choices: we made all variables global, split the code into small meaningfull functions for matrix operations, and transform the while loop into a for loop to address the termination issue. Variables. One of the difficulties when analyzing C code are memory related issues. Two different pointers can reference the same part of the stack. A fine and complex modeling of the memory in the predicate encoding, such as separation logic[8] could address these issues. Another more pragmatic approach amounts to substitute all local variables and function arguments with global static variables. Two static arrays can’t overlap since their memory is allocated at compile time. Since we are targetting embedded system, static variables will also permit to compute and reduce memory footprints. However there are two major drawbacks: the code is a less readable and variables are accessible from any function. These two points usually lead the programmer to mistakes but could be accepted in case of code generation. We tag all variables with the function they belong to by prefixing each variable with its function name. This brings traceability. Function. Proving large functions is usually hard with SMT-based reasonR⇒Q {P } C {R} ing since the generated goals are too Function call WP(f(), Q) = P complex to be discharged automatically. A more efficient approach is to with void f() { C } associate small pieces of code with local contracts, These intermediate anFig. 6. WP rules used for function call notations act as cut-rules in the proof process. The Figure 6 presents the function call used in the WP algorithm. Let A = B[C] be a piece of code containing C. Replacing C by a call to f() { C } requires either to inline the call or to write a new contract {P } f() {Q}, characterizing two smaller goals instead of a larger one. Specifically in the proof of a B[f()], C has been replaced by P and Q which is simpler than a WP computation. Therefore instead of having one large function, our code is structured into several functions: one per basic operation. Each associated contract focuses on a really specific element of the proof without interference with the others. Thereby formulas sent to SMT solvers are smaller and the code is modular. Matrix operation. This is extremely useful for matrix operations. In C, a M × N Matrix operations is written as M × N scalar operation affecting an array representing the resulting matrix. With our methods these operations are gathered in a function annotated by the logic representation of the matrix operation, cf Figure 7. Contracts associated to this small function associate high-level matrix operation to the C low-level computation, acting as refinement contracts. The encoding hides low-level op-  erations to the rest of the code, lead- /*@ ensures MatVar(dX, 2, 1) ==\old(mat_scal(MatVar(cholesky, 2, 1), -1.0)); ing to two kinds of goals : @ assigns *(dX+(0..2)); */ – Low level operation (memory and basic matrix operation). – High level operation (mathematics on matrices). The structure of the final code after the split in small functions is shown in Figure 8. void set_dX() { dX[0] = -cholesky[0]; dX[1] = -cholesky[1]; } Fig. 7. Example of matrix operation: dx %= -cholesky encapsulated in a function for dx and cholesky of size 2 × 1 – compute fill X with the the analytic center and call pathfollowing. – pathfollowing contains the main loop which udpate dX and dt. – compute_pre compute Hessian and gradiant of F which are required for dt and dX. – udpate_dX and udpate_dt call the associated subfunction and update the corresponding value. – compute_dt performs (15), it requires to call Chowlesky to compute the local norm of c. – compute_dX performs (9), Chowlesky is used to inverse the hessian matrix. Fig. 8. Call tree of the implementation(rose boxes are matrix computation) While-loop. The interior point algorithm is iterative: it performs the same operation until reaching the stop condition. This stopping criteria depends on the desired precision : (1 + β)β 1 (1 + ) (17)  1−β Proving the termination amounts to find a suitable bound guaranting the obtention of a converged value. We present here the convergence proof of [5] and use it to ensure termination. It relies on the use of the following geometric progression: tk ≥ tstop = Definition 11 Lower(k) = γ k−1 γ(1 − 2β) (1 + ) (1 − β)kck∗x∗ 1+β F (18) This sequence minimizes t for each iteration k of the algorithm([5, Th4.2.9]): Theorem 3 For all k ∈ N∗ , tk ≥ Lower(k) (19) Combined with (17), a maximal number of iteration called klast can be computed: Theorem 4 (Required number of iterations) ln(1 + (β+1)∗β 1−β ) klast = 1 + γ∗(1−2β) ) − ln() − ln( (1−β)∗kck ∗ ∗ ln(1 + x γ β+1 ) F (20) Since we have a termination proof based on the number of iterations, the while-loop can be soundly replaced by a for-loop with klast iterations. As shown in Figure 9, the number of iterations is greater than the one obtained with original while-loop with the stopping criteria but it permits to have absolute guarantee on termination. Analytic center kck∗x∗ is required to Fig. 9. Evolution of t and Lower with the F compute klast , therefore a worst case algorithm, notice that t remains always execution time can be computed if greater than Lower and only if kck∗x∗ has a lower bound F at compilation time. 4 Annotate the code The code is prepared to ease its proof but the specification still remains to be formalized as function contracts, describing the computation of an -optimal solution. This requires to enrich ACSL with some new mathematics definitions. We introduce a set of axiomatic definition to specify optimization related properties. These definitions require, in turn, additional concepts related to matrices. Similar approaches were already proposed [9] but were too specfic to ellipsoid problems. We present here both the main annotation and the function local contracts, which ease the global proof. Matrix axiomatic. To write the mathematics property, we need to be able to express the notion of Matrix and operations over it. Therefore we defined a new ACSL axiomatic. An ACSL axiomatic permits the specifier to extend the ACSL language with new types and operators, acting as an algebraic specification.  axiomatic matrix { type LMat; First, we defined the new type: LMat standing for Logic Matrix. This type is abstract therefore it will be defined by its operators. // Getters logic integer getM(LMat A); logic integer getN(LMat A); logic real mat_get(LMat A, integer i, integer j); // Constructors logic LMat MatVar(double* ar, integer m, integer n) reads ar[0..(m*n)]; logic LMat MatCst_1_1(real x0); logic LMat MatCst_2_3(real x0, real x1, real x2, real x3, real x4, real x5); Getters allow to extract information from the type while constructors bind new LMat object. The first constructor is followed by a read clause stating which part of the memory affects the corresponding LMat object. The Constant constructor take directly the element of the matrix as argument, it can be replaced with an ACSL array for bigger matrix sizes. logic logic logic logic ... LMat LMat LMat LMat mat_add(LMat A, LMat B); mat_mult(LMat A, LMat B); transpose(LMat A); inv(LMat A); Then the theory defined the operations on the LMat type. These are defined axiomatically, with numerous axioms to cover their various behavior. We only give here a excerpt from that library. axiom getM_add: \forall LMat A, B; getM(mat_add(A, B))==getM(A); axiom mat_eq_def: \forall LMat A, B; (getM(A)==getM(B))==> (getN(A)==getN(A))==> (\forall integer i, j; 0<=i<getM(A) ==> 0<=j<getN(A) ==> mat_get(A,i,j)==(mat_get(B,i,j))==> A == B; ... } Matrix operations. As explained in previous Section, the matrix computation are encapsulated into smaller functions. Their contract states the equality between the resulting matrix and the operation computed. An extensionality axiom (mat_eq_def) is required to prove this kind of contract. Extensionality means that if two objects have the same external properties then they are equal. This axiom belong to the matrix axiomatix but is too general to be used therefore lemmas specific to the matrices size are added for each matrix affectation. This lemma can be proven with the previous axioms and therefore does not introduce more assumption. The proof remains difficult or hardly automatic for SMT solvers therefore we append additional assertions, as sketched in Figure 10, at then end of function stating all the hypothesis of the extensionality lemma. Proving these postconditions is straighfoward and smaller goals need now to be proven.  assert assert assert assert assert assert getM(MatVar(dX,2,1)) == 2; getN(MatVar(dX,2,1)) == 1; getM(MatVar(cholesky,2,1)) == 2; getN(MatVar(cholesky,2,1)) == 1; mat_get(MatVar(dX,2,1),0,0) == mat_get(\old(mat_scal(MatVar(cholesky,2,1),-1.0)),0,0); mat_get(MatVar(dX, 2, 1),1,0) == mat_get(\old(mat_scal(MatVar(cholesky,2,1),-1.0)),1,0); Fig. 10. Assertion appended to the function from figure 7 Assertions also act as cutWP(C, P ⇒ Q) = R WP(C, P ) = S rules in ACSL since it intro- Assert WP(C;assert P ;, Q) = R∧S duces the property in the set of hypothesis considered (see. Figure 11). Fig. 11. WP rules used for assert This works for small example, when scaling each C instruction is embedded inside a correctly annotated function. Optimization axiomatic. Beside generic matrix operators we also need some operators specifc to our algorithm.  axiomatic Optim { logic LMat hess(LMat x0, LMat x1, LMat x2); logic LMat grad(LMat x0, LMat x1, LMat x2); Hessian and gradiant are hard to define without real analysis which is well beyond the scope of this article. Therefore we decided to directly axiomatize some theorems relying on their definition like [5, Th4.1.14]. logic real sol(LMat x0, LMat x1, LMat x2); The sol operator represents x∗ , the exact solution which can be defined by Property 2 (Axiomatic characterization of Definition 2) s is a solution of 1 if and only if 1. For all y ∈ Ef , cT y ≥ s 2. For all y ∈ R, ∀x ∈ Ef , cT x ≥ y implies s ≥ y An ACSL equivalent definition is: logic real sol(LMat A, LMat b, LMat c); axiom sol_min: \forall LMat A, b, c; \forall LMat y; mat_gt(mat_mult(A, y), b) ==> dot(c, y) >= sol(A, b, c); axiom sol_greater: \forall LMat A, b, c; \forall Real y; (\forall LMat x; mat_gt(mat_mult(A, x), b) ==> dot(c, x) >= y) ==> sol(A, b, c) >= y; Then we defined some operators representing definitions 7, 8 and 11. logic real norm(LMat x0, LMat x1, LMat x2, LMat x3) = \sqrt(mat_get(mat_mult(transpose(x2), mat_mult(inv(hess(x0, x1, x3)), x2)), (0), (0))); logic boolean acc(LMat x0, LMat x1, LMat x2, real x3, LMat x4, real x5) = ((norm(x0, x1, mat_add(grad(x0, x1, x4), mat_scal(x2, x3)), x4))<=(x5)); ... } Contract on pathfollowing. A sound algorithm must produce a point in the feasible set such that its cost is -close to sol. This is asserted by two global post-conditions: ensures mat_gt(mat_mult(A, MatVar(X, N, 1)), b); ensures dot(MatVar(X, 2, 1), c) - sol(A, b, c) < EPSILON as well as two preconditions stating that X is feasible and close enough to the analytic center: requires mat_gt(mat_mult(A, MatVar(X, N, 1)), b); requires acc(A, b, c, 0, MatVar(X, N, 1), BETA); Thanks to our two new theories Matrix and Optim, writing and reading this contract is straighforward and can be checked by anyone familiar with linear programming. Main Loop. A loop needs to be annoted by an invariant to have its Weakest precondition computed (cf. Figure 12) For loop WP(E, I) = P (¬F ∧ I) ⇒ Q {F ∧ I} C;G {I} WP(for (E;F;G) inv I {C}, Q) = P Fig. 12. WP rules for a loop We need three invariants for our path following algorithms. The first one guarantees the feasibility of X while the second one states the conservation of the ACC (cf. Def. 8) The third invariant assert that t is increasing enough on each iteration, more specially that it is greater than a geometric progression(Definition 11).  /*@ @ @ for loop-invariant mat_gt(mat_mult(A, MatVar(X, N, 1)), b); loop-invariant acc(A, b, c, t, MatVar(X, N, 1), BETA); loop-invariant t > lower(l);*/ (int l = 0; l < NBR;l++) { ... } Proving the initialization is straighfoward, thanks to the main preconditions. The first invariant preservation is stated by [5, Th4.1.5] which was translated into an ACSL lemma, the second by Theorem 1 and the third one by Theorem 3. The last two loop invariants are combined to prove the second postcondition of pathfollowing thanks to Theorem 5 and NBR equals klast (20). Theorem 5 [5, Th4.2.7] Let t ≥ 0, and X such that ACC(X, t, β) then cT X − cT X ∗ < 1 (β + 1)β × (1 + ) t 1−β (21) Loop body. In the main loop there are three function calls: update_pre computing some common values, update_t and update_x(Figure 8). Therefore Theorem 1 is broken into several properties and the corresponding post-conditions. For example, the contract of update_t is:  /*@ requires MatVar(hess, N, @ requires acc(A, b, c, t, @ ensures acc(A, b, c, t, @ ensures t > \old(t)*(1 + void update_t(); N)==hess(A, b, MatVar(X, N, 1)); MatVar(X, N, 1), BETA); MatVar(X, N, 1), BETA + GAMMA); GAMMA/(1 + BETA));*/ The first postcondition is an intermediary results stating that: ACC(X, t + dt, β + γ) (22) This result is used as precondition for update_x. The second precondition corresponds to the product of t by the common ratio of the geometric progression Lower, cf. Definition 11 which will be used to prove the second invariant of the loop. The first precondition is a postcondition from update_pre and the second one is the first loop invariant. 5 Automatic proof with SMT solvers. For each annotated piece of code, the Frama-C WP plugin computes the Weakest precondition and generates all the first order formulas required to validate the Hoare triples. There are two main solutions to prove goals: proving them thanks to a proof assistant – this requires to be done by a human –, or proving them with a fully automatic SMT solver. We decided to rely only on SMT solvers in order to be able to completely automatize the process. Therefore it is better to have lots of small goals instead of several larger ones. We splited the code for this reason and we now split the proof of lemmas into several intermediate lemmas. For example, in order to prove (22) we wrote update_t_ensures1 where P1 is ACC(X, t, β) γ and P2 is dt = kck ∗ x ∀x, t, dt; P1 ⇒ P2 ⇒ ACC(X, t + dt, β) (update_t_ensures1) which itself need update_t_ensures1_l0 ∀x, t, dt; P1 ⇒ P2 ⇒ kF 0 (X) + c(t + dt)k∗x ≤ β + γ (update_t_ensures1_l0) Equation update_t_ensures1_l0 needs 3 lemmas to be proven: ∀x, t, dt; P2 ⇒ kc × dtk∗x = γ (update_t_ensures1_l3) ∀x, t, dt; P1 ⇒ kF 0 (X) + c × tk∗x ≤ β (update_t_ensures1_l2) ∀x, t, dt; P1 ⇒ P2 ⇒ kF 0 (X) + c(t + dt)k∗x ≤ kF 0 (X) + c × t)k∗x + kc × dtk∗x (update_t_ensures1_l1) The proof tree for the first ensures of update_t can be found in Figure 13. Fig. 13. Proof tree for (22)(In green proven goal, in white axioms) 6 Experimentations Frama-C is a powerful tool but not always built for our specific needs therefore we had to do some tricks to make it prove our goals. Using multiple files Frama-C automatically adds, for each goal, all the lemmas as hypotheses. This increases significantely the size of the goal. To avoid this issue that prevented some proofs, we wrote each function or lemma in a separate file. In this file we add as axioms all the lemmas required to prove the goal. This allows us to prove each goal independently with a minimal context. The impact of the separation into multiple function(Section 3) and the separation into multiple files is shown in Table 1. The annotated code can be retrieved from https://github.com/davyg/ proved_primal 7 Related work Related works include first activities related to the validation of numerical intensive control algorithms. This article is an extension of Wang et al [10] which was presenting annotations for a convex optimization algorithm, namely IMP, but the process was both manual and theoretical: the code annotated within Matlab and without machine checked proofs. An other work from the same authors [9] 2×5 Size of A 4 × 15 8 × 63 Experiences exp1 exp2 exp3 exp1 exp2 exp3 exp1 exp2 exp3 nb function 1 12 12 1 26 26 1 78 78 nb file 1 1 12 1 1 26 1 1 78 nb proven goal 21 48 48 43 97 97 12 257 264 nb goal 25 48 48 46 97 97 109 264 264 Table 1. Proof results for compute_dt with one function and one file(exp1), with multiple function(exp2) or with multiple file(exp3) and a Timeout of 30s for Alt-Ergo for random generated problem of specific sizes. presented a similar method than ours but limited to simple control algorithms, linear controllers. The required theories in ACSL were both different and less general than the ones we are proposing here. Concerning soundness of convex optimization, Cimini and Bemporad [11] presents a termination proof for a quadratic program but without any concerns for the proof of the implementation itself. A similar criticism applies to Tøndel, Johansen and Bemporad [12] where another possible approach to online linear programming is proposed, moreover it is unclear how this could scale and how to extend it to other convex programs. Roux et al [13,14] also presented a way to certify convex optimization algorithm, namely SDP and its sum-of-Squares (SOS) extension, but the certification is done a posteriori which is incompatible with online optimization. A last set of works, e.g. the work of Boldo et al [15], concerns the formal proof of complex numerical algorithms, relying only on theorem provers. Although code can be extracted from the proof, the code is usually not directly suitable for embedded system: too slow and require different compilation step which should also be proven to have the same guarantee than our method. 8 Conclusion In this article we presented a method to guarantee the safety of numerical algorithms in a critical embedded system. This allows to embed complex algorithms in critical real-time systems with formal guarantee on both their result and termination. This method was applied to a primal algorithm solving linear program. The implementation is first designed to be easier to prove. Then it is annotated so that in a third time Frama-C and SMT solver can prove the specification automatically. Combined to a code generator such as CVX [16] but with annotation generation it could lead to a tool taking an optimization problem and generating its code and proof automatically. We worked with real variables to concentrate on runtime errors, termination and functionality and left floating points errors for a future work. This proof relies on several point: the tools used, the axiomatics we wrote, the main ACSL contract and the theorems used as axioms. There is also some unchecked code which is independent from the core proof of the algorithm and remains for further work: the Chowlesky decomposition and the Hessian and gradiant computation. We also plan to extend the whole work to convex programming. References 1. Blackmore, L.: Autonomous precision landing of space rockets. National Academy of Engineering, Winter Bridge on Frontiers of Engineering 4(46) (December 2016) 2. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10) (1969) 576–580 3. Floyd, R.W.: Assigning meanings to programs. Proceedings of Symposium on Applied Mathematics 19 (1967) 19–32 4. Dijkstra, E.W.: Guarded commands, nondeterminacy and formal derivation of programs. Commun. ACM 18(8) (1975) 453–457 5. Nesterov, Y., Nemirovski, A.: Interior-point Polynomial Algorithms in Convex Programming. Volume 13 of Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (1994) 6. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12 (October 1969) 576–580 7. Baudin, P., Filliâtre, J.C., Marché, C., Monate, B., Moy, Y., Prevosto, V.: ACSL: ANSI/ISO C Specification Language. version 1.11. http://frama-c.com/ download/acsl.pdf 8. Reynolds, J.C.: Separation logic: a logic for shared mutable data structures. In: Proceedings 17th Annual IEEE Symposium on Logic in Computer Science. (2002) 55–74 9. Herencia-Zapana, H., Jobredeaux, R., Owre, S., Garoche, P.L., Feron, E., Perez, G., Ascariz, P.: Pvs linear algebra libraries for verification of control software algorithms in c/acsl. In Goodloe, A., Person, S., eds.: NASA Formal Methods Forth International Symposium, NFM 2012, Norfolk, VA USA, April 3-5, 2012. Proceedings. Volume 7226 of Lecture Notes in Computer Science., Springer (2012) 147–161 10. Wang, T., Jobredeaux, R., Pantel, M., Garoche, P.L., Feron, E., Henrion, D.: Credible autocoding of convex optimization algorithms. Optimization and Engineering 17(4) (Dec 2016) 781–812 11. Cimini, G., Bemporad, A.: Exact complexity certification of active-set methods for quadratic programming. IEEE Transactions on Automatic Control PP(99) (2017) 1–1 12. Tøndel, P., Johansen, T.A., Bemporad, A.: An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica 39(3) (2003) 489–497 13. Roux, P.: Formal proofs of rounding error bounds - with application to an automatic positive definiteness check. J. Autom. Reasoning 57(2) (2016) 135–156 14. Martin-Dorel, É., Roux, P.: A reflexive tactic for polynomial positivity using numerical solvers and floating-point computations. In Bertot, Y., Vafeiadis, V., eds.: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, CPP 2017, Paris, France, January 16-17, 2017, ACM (2017) 90–99 15. Boldo, S., Faissole, F., Chapoutot, A.: Round-off error analysis of explicit onestep numerical integration methods. In: 2017 IEEE 24th Symposium on Computer Arithmetic (ARITH). (July 2017) 82–89 16. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (March 2014) 

Adjacency Criterion For Gradient Flow With Multiple Local Maxima arXiv:1412.6731v3 [math.DS] 8 Sep 2017 Xudong Chen1 Abstract— In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of gradient flow which has as many as n! local maxima. In particular, we characterize the set of equilibria, compute the index of each critical manifold and moreover, find all pairs of adjacent neighbors. As an application of the adjacency criterion, we introduce a stochastic version of the double bracket flow and set up a Markov model to approximate the sample path behavior. The study of this specific prototype with its special structure provides insight into many other difficult problems involving simulated annealing. I. I NTRODUCTION Ascent/Descent equations often provide the most direct demonstration of the existence of a maximum/minima and can provide an easily implemented algorithm to find the maximum/minima (see, for example, [1]–[3]). Of course when the function being maximized/minimized has multiple local maxima/minima, steepest ascent/descent needs to be modified and for the last several decades the modification of choice has been some type of simulated annealing procedure. However, because simulated annealing is slow and subject to variable results because of its stochastic nature, there remains considerable interest in finding methods for improving its speed and, in general, learning more about its performance. But often this requires the knowledge of the sample path behavior. In this paper, we start with the development of a basic theorem about the geometry of a general class of gradient flows. In particular, we decompose the underlying space into disjoint regions of attraction associated with the gradient flow, and then establish the adjacency criterion. This theorem states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent, i.e, for them to share a boundary of co-dimension one. The adjacency criterion has a large potential impact on studying stochastic gradients. For example, to approximate the sample path behavior of a stochastic gradient, R. W. Brockett established a Markov model in [4] whose states consist of all stable equilibria. Each transition probability is evaluated by solving a related first-hitting time problem, the computational complexity is in a large-scale. The adjacency criterion then sheds light on the problem. Knowing adjacent neighbors reduces the computational amount as the transition probability between non-adjacent equilibria is 1 X. Chen is with the ECEE department, University of Colorado at Boulder. xudong.chen@colorado.edu negligible when compared with the transition probability between adjacent equilibria, especially under the case where the noise is moderate. The adjacency criterion also relates to a number of geometric factors. For example, the depth of the potential well associated with a stable equilibrium, the volume of a region of attraction, the area of the boundary shared by a pair of adjacent neighbors and etc, all these factors are important for explaining simulated annealing. As an demonstration of adjacency criterion, we consider a prototype system with the gradient flow of the type Ḣ = [H, [H, π(H)]], the map π projects a matrix onto its diagonal. This flow is a prototype for gradient systems with multiple stable equilibria, it has as many as n! stable equilibria, each is a diagonal matrix and one-to-one corresponds to an element in the permutation group Sn . In this paper, we will characterize all pairs of adjacent neighbors, the characterization is simple and clean. In particular, we will show that each pair of adjacent neighbors is related to a simple transposition, so then the n! regions of attraction are arranged in a way that each one has (n − 1) adjacent neighbors. A stochastic version of the gradient flow is studied at the end of this paper as a concrete example of the application of the adjacency criterion. In particular, we set up an optimal control problem as an approach to evaluate the transition probability. After this introduction, we proceed in steps: in section 2, we will introduce the adjacency criterion after a quick review of some basic notions about Morse-Bott gradient systems. In section 3, we will introduce the isospectral manifold, the normal metric and the double bracket flow under a special class of potentials functions. In section 4, we will focus on a specific quadratic potential function and show that the set of equilibria under the potential function is a mix of isolated points and continuum manifolds. In section 5, we will show that the quadratic potential is a MorseBott function by explicitly working out the Hessian of the potential. In particular, we will compute the index and the co-index of each critical manifold. In section 6, we will apply the adjacency criterion to characterize all pairs of adjacent neighbors. In the last section, we will work on a stochastic gradient as an application of the adjacency criterion. In particular, we will show how the adjacency criterion will simplify the evaluation of transition probability. II. A DJACENCY CRITERION In this section, we will introduce the adjacency criterion. Some mathematical backgrounds are needed here, in this paper we will only introduce terminologies that are necessary for establishing the criterion. A potential Ψ on a Riemannian manifold M is a MorseBott function if the set of equilibria under the gradient flow grad(Ψ) is a finite disjoint union of connected submanifolds {E1 , · · · , En }, and the Hessian of Ψ is nondegenerate when restricted at the normal space N p Ek at any point p ∈ Ek and for any Ek . (see, for example, [5] for more details about Morse-Bott functions) For convenience, we say each set Ek is a critical manifold. Assume Ψ is a Morse-Bott function, the index of each submanifold Ek is the number of negative eigenvalues of the Hessian restricted at N p Ek for some p ∈ Ek , this is welldefined because the index of Ek is independent of the choice of p. Similarly, the co-index of Ek is the number of positive eigenvalues of the Hessian restricted at N p Ek . An equation relating the index, co-index and the dimension of M is indEk + co-indEk + dim Ek = dim M So two stable manifolds S j and Sk are adjacent if and only if there is a Ki such that Ki ⊂ ∂W s (S j ) ∩ ∂W s (Sk ). (1) A stable critical manifold is then a critical manifold of coindex 0. Let Ek be a critical manifold, the stable manifold of Ek is defined by W s (Ek ) = {p ∈ M| lim ϕt (p) ∈ Ek } t→∞ (2) where ϕt (p) is the solution of the differential equation ṗ =grad(Ψ) parametrized by time t. The unstable manifold of Ek is defined in a similar way as W u (Ek ) = {p ∈ M| lim ϕt (p) ∈ Ek } t→−∞ and W u (Ek ) (3) The dimensions of W s (Ek ) are indEk + dim Ek and co-indEk + dim Ek respectively. A decomposition of M, with respect to stable/unstable manifolds of Ek ’s, is given by n [ M= W s (Ek ) (4) W u (Ek ) (5) k=1 or n [ M= k=1 If only stable critical manifolds {S1 , · · · , Sl } are concerned, then M= l [ (6) We are interested in how these cells are arranged in the underlying space. With terminologies above, we are now ready to state the adjacency criterion. Fact 1 (Adjacency Criterion): Suppose Ψ is a Morse-Bott function on a smooth manifold M. Let {S1 , · · · , Sl } be the collection of stable critical manifolds and let {K1 , · · · , Km } be the critical manifolds of co-index 1. If (W u (Ki ) − Ki ) ⊂ i=1 l [ W s (S j ) (7) j=1 then the boundary of each W s (S j ) is piece-wise smooth, each piece is a stable manifold of some Ki , i.e, ∂W s (S j ) = Before heading to its application on the isospectral manifold, we point out that the assumption described by equation (7) includes a broad range of gradient systems. For example, in a Morse-Smale gradient system, all equilibria are isolated points and the unstable manifold of each Ki is a one dimensional curve and its boundaries consists of stable equilibrium points as illustrated in Fig. 1. The class of Morse-Smale gradient system is a residual class in all gradient systems(see, for example, [6], [7] for more details about Morse-Smale systems). III. I SOSPECTRAL MANIFOLD AND A CLASS OF SYMMETRIC POTENTIALS W s (Si ) i=1 m [ Fig. 1. In this figure, each Si is a stable equilibrium, each Ki j is an equilibrium of co-index 1 and E is an equilibrium of co-index 2. The pair (S1 , S2 ), for example, consists of two adjacent neighbors because their regions of attraction share the stable manifold of K12 as a co-dimension one boundary. On the other hand, S1 and S3 are not adjacent because their regions of attraction intersect each other only at E. [ Ki ∈∂W s (S j ) W s (Ki ) (8) In this section, we will introduce the isospectral manifold with strongly disjoint eigenvalues, the normal metric and a special class of symmetric potential functions. Let Sym(Λ) denote the isospectral manifold, in case here, it refers to the collection of all n-by-n real symmetric matrices with a fixed set of eigenvalues Λ = {λ1 , · · · , λn }. We assume that the set Λ is strongly disjoint, i.e, • for any two nonempty disjoint subsets {λa1 , · · · , λa p } p and {λb1 , · · · , λbq } of Λ, the inequality 1p ∑k=1 λak 6= 1 q q ∑k=1 λbk holds. The assumption on Λ is a stronger version of the condition that all eigenvalues are distinct. Yet this condition is generic in the sense that if we randomly pick a vector (λ1 , · · · , λn ) in Rn , then Λ is strongly disjoint for almost sure because the p set defined by equations 1p ∑k=1 λak = 1q ∑qk=1 λbk is a finite n union of hyperplanes in R . On the other side, as we will see later this condition on Λ is necessary and sufficient for a class of potential functions on Sym(Λ) to be Morse-Bott functions which is a key assumption for application of the adjacency criterion. The so called normal metric g on Sym(Λ) is defined as follows, the tangent space TH Sym(Λ) consists of elements of the form [H, Ω] with Ω skew symmetric, since all eigenvalues are distinct, the adjoint map adH : Ω 7→ [H, Ω] is then an isomorphism. For any two tangent vectors [H, Ω1 ], [H, Ω2 ] in TH Sym(Λ), the normal metric g at H is then defined by g([H, Ω1 ], [H, Ω2 ]) := −tr(Ω1 Ω2 ) (9) It is routine to check that g is positive definite. Equipped with the normal metric, the gradient flow of a smooth function Ψ ∈ C∞ (Sym(Λ)) is then the double bracket flow Ḣ = [H, [H, Ψ0 (H)]] (10) where Ψ0 (H) denotes the derivative of Ψ with respect to H For convenience, we assume that eigenvalues are ordered as λ1 < · · · < λn and we denote by d1 , · · · , dn the diagonal entries of a matrix H in Sym(Λ). There is a special class of potentials functions Ψ on Sym(Λ), each is symmetric in diagonal entries and can be generated by a scalar function φ ∈ C∞ [λ1 , λn ] via the equation n Ψ(d1 , · · · , dn ) = ∑ Z di φ (x)dx (11) i=1 λ1 As the sum of the diagonal entries is constant for any matrix in Sym(Λ), there is actually a broad class of symmetric potentials can be expressed in this way. In fact, we have shown in [8] that if Ψ(d1 , · · · , dn ) can be expressed as a power series of symmetric polynomials, then there is a scalar function φ such that the equation (11) holds. Moreover, we have also shown in [8] that for almost all scalar functions φ , the potential Ψ generated by equation (11) is a Morse-Bott function. In this paper, we will work on a simple case where φ (x) = x. The corresponding potential is then in a diagonal form as Ψ= 1 n 2 ∑ di 2 i=1 (12) In the rest of paper, we will just call this specific quadratic potential the diagonal potential. In the next two sections, we will characterize the critical manifolds associated with the gradient flow and compute explicitly the Hessian at each equilibrium point. IV. T HE SET OF EQUILIBRIA ASSOCIATED WITH THE DIAGONAL POTENTIAL For convenience, we let π be a projection map sending each matrix H to the diagonal matrix diag(d1 , · · · , dn ). Then the gradient vector field f (H) with respect to the diagonal potential is simply f (H) = [H, [H, π(H)]] (13) A matrix H ∈ Sym(Λ) is an equilibrium if f (H) = 0 and it happens if and only if [H, π(H)] = 0 because tr(π(H) f (H)) = −tr([H, π(H)]2 ). This leads us to Lemma 2: If H is an equilibrium, then there exists a permutation matrix P such that PT HP is block-diagonal, i.e, PT HP = Diag(H1 , · · · , Hk ) (14) Suppose di1 , · · · , dini are diagonal entries of Hi , then di1 = · · · = dini (15) and this holds for each Hi . Proof: The commutator [H, π(H)] vanishes if and only if hi j (di − d j ) = 0 for each pair of (i, j). Let Λi be the set of eigenvalues of Hi , then (Λ1 , · · · , Λk ) is S a partition of Λ, i.e, Λ = ki=1 Λi and Λi ∩ Λ j = ∅ if i 6= j. For convenience, we let #(Λi ) denote the cardinality of Λi , let s(Λi ) denote the sum of its elements and we define µ(Λi ) := s(Λi )/#(Λi ) (16) di1 = · · · = dini = µ(Λi ) (17) then A symmetric matrix H is said to be irreducible if there isn’t a permutation matrix P such that PT HP is a nontrivial block-diagonal matrix. Lemma 3: Each Hi in equation (14) is irreducible, moreover µ(Λi ) 6= µ(Λ j ) if i 6= j. Proof: Suppose Hi is not irreducible, we may assume that Hi = Diag(Hi1 , Hi2 ), let Λi1 = Λi2 be the set of eigenvalues of Hi1 and Hi2 respectively, then µ(Λi1 ) = µ(Λi2 ) = µ(Λi ) which contradict the fact that Λ is strongly disjoint. By applying the same arguments, we conclude that µ(Λi ) and µ(Λ j ) are disjoint if i 6= j. An equilibrium H gives rise to a partition of Λ. Conversely a partition of Λ will correspond to a set of equilibria. Let A be the collection of all choices of partitions of Λ. Given a choice of partition α = (Λ1 , · · · , Λk ), we define a subset of Sym(Λ) as Eα := {Diag(H1 , · · · , Hk )|π(Hi ) = µ(Λi )} (18) clearly Eα consists exclusively of equilibria. Moreover it is a smooth manifold as a consequence of the next fact Fact 4: Let Λ0 = {λ10 , · · · , λn0 0 } be a subset of Λ and 0 let C(Λ0 ) ⊂ Rn be the convex hull of all vectors (λσ0 (1) , · · · , λσ0 (n0 ) ) where σ varies over all permutations of {1, · · · , n0 }, then the image of the projection 0 π : Sym(Λ0 ) −→ R#(Λ ) (19) C(Λ0 )(see, is the convex hull for example, [9]). Let ~e be a 0 vector of all ones in R#(Λ ) and define X(Λ0 ) := π −1 (µ(Λ0 )~e) then π is a submersion at each point H ∈ X(Λ0 ). (20) other hand, either PT Eα P = Eα or PT Eα P ∩ Eα = ∅, and P fixes Eα if and only if P is block-diagonal, i.e, P = Diag(P1 , · · · , Pk ) with each Pi a ni -by-ni permutation matrix. There are exactly Πki=1 ni ! many block-diagonal permutations matrices, so by Burnside’s counting formula, there are as many as n!/Πki=1 ni ! disjoint sets in the orbit of Eα , each is a smooth submanifold in Sym(Λ). Fig. 2. The convex hull C(Λ) under the case Λ = {λ1 , λ2 , λ3 }. Let ~e := (1, 1, 1)T , then the line R~e intersects C(Λ) perpendicularly at the point µ(Λ)~e. A complete proof can be found in [8]. Assuming the fact above, then an immediate consequence is that X(Λ0 ) is a smooth manifold of dimension 21 (#(Λ0 ) − 1)(#(Λ0 ) − 2). An identification is that Eα ' Πki=1 X(Λi ), so k dim Eα = ∑ (#(Λi ) − 1)(#(Λi ) − 2) (21) i=1 An upshot is that the set Eα is a set of discrete points if and only if each #(Λi ) is either one or two, in the case all Λi ’s are singletons, the set Eα consists only of a diagonal matrix. At this moment, we have characterized the set equilibria that are block-diagonal, equilibria that are off block-diagonal can be generated by letting the group of permutation matrices act on Eα ’s by conjugation as we will describe below. Lemma 5: If H is an equilibrium, then so is PT HP for any permutation matrix P. Proof: We check that π(PT HP) = PT π(H)P and hence conjugation by permutation matrices commutes with the commutator, i.e, PT [H, π(H)]P = [PT HP, π(PT HP)] So f (PT HP) = 0 (22) exactly when f (H) = 0. Lemma 5 can be regarded as a converse argument of lemma 2. Theorem 6: Suppose α = (Λ1 , · · · , Λk ) is a choice of partition and suppose #(Λi ) = ni . Let the group of permutation matrices act on Eα by conjugation, then the orbit of Eα contains as many as n!/Πki=1 ni ! disjoint smooth submanifolds in Sym(Λ), each consists exclusively of equilibria. The set of equilibria is then the union of orbits of Eα as α varies over A . Proof: For any permutation matrix P, the set PT Eα P consists exclusively of equilibria by lemma 5. On the In the rest of this paper, we will abuse the term critical manifold by referring it to an Eα or any set in its orbit as Eα may have multiple connected components. (see, for example, a discussion on the number of components in [8]). Notice that in the orbit of Eα , there exist multiple critical manifolds that are block-diagonal, for example, the orbit of a diagonal matrix is the set of all diagonal matrices. Actually given a choice of partition α = (Λ1 , · · · , Λn ), there are as many as k! critical manifolds that are block-diagonal leaving the others off block-diagonal. If we want to choose a canonical representative among the orbit, then permute Λ1 , · · · , Λk if necessary so that µ(Λ1 ) < · · · < µ(Λk ), this is possible because eigenvalues of Λ are strongly disjoint. V. T HE H ESSIAN OF THE DIAGONAL POTENTIAL In this section, we will show that the diagonal potential is a Morse-Bott function by explicitly working out the Hessian of the potential at each equilibrium point and compute its eigenvalues. Let H denote the Hessian of a potential Ψ. On a Riemannian manifold, it is defined by H = ∇2 Ψ (23) where ∇ is the Levi-Civita connection and if we evaluate the Hessian at an equilibrium point, then H (X,Y ) = X(g(grad(Ψ),Y )) (24) In this specific case, we have Fact 7: Suppose H is an equilibrium, then the Hessian H evaluated at H is given by H ([H, Ωi ], [H, Ω j ]) = −tr([H, Ωi ][π(H), Ω j ]) −hπ([H, Ωi ]), π([H, Ω j ])i (25) where h·, ·i is the normal inner-product in Rn . We omit the proof here as this is a direct computation following equation (24). Given a choice of partition α = (Λ1 , · · · , Λn ), we assume #(Λi ) = ni , then the dimension of the normal space at a point H ∈ Eα is given by k 1 dim NH Eα = {n(n − 1) − ∑ (ni − 1)(ni − 2)} 2 i=1 (26) For convenience, let nα := dim NH Eα . We will now construct an orthogonal basis N of NH Eα with respect to H , i.e, H (N, N) 6= 0, ∀N ∈ N (27) H (N, N 0 ) = 0, if N 6= N 0 (28) Suggested by the partition α, we divide a n-by-n matrix N into k-by-k blocks as   B11 · · · B1k  ..  .. (29) N =  ... . .  Bk1 · · · Bkk with the pq-th block of dimension n p -by-nq . The basis N consists of two parts: the block-diagonal part Nd and the off block-diagonal part No . 1. Constructing No : Write H = Diag(H1 , · · · , Hk ) with each Hi ∈ Sym(Λi ) and let ~vi1 , · · · ,~vini be the unitlength eigenvectors of Hi with respect to the eigenvalues λi1 , · · · , λini . Given integers p and q, define a set of symmetric matrices N pq in the way that for each matrix N pq,i j ∈ N pq , all blocks except B pq and Bqp are zeros while B pq (= BTqp ) = (λ pi − λq j )~v pi~vTq j (30) for some i = 1, · · · , n p and some j = 1, · · · , nq , so there are exactly n p nq symmetric matrices in the set N pq . Lemma 8: The set No := p<q N pq is contained in NH Eα . If N pq,i j ∈ N pq and N p0 q0 ,i0 j0 ∈ N p0 q0 , then 3. Orthogonality of No and Nd : First notice that by combining No and Nd , there are exactly nα matrices, i.e, k nα = ∑ 1≤i< j≤k ni n j + ∑ (ni − 1) (33) i=1 because of the equality ∑ki=1 ni = n. So it all remains to show that Lemma 11: Matrices in No are orthogonal to those in Nd with respect to the Hessian. Proof: Divide any n-by-n skew symmetric matrix Ω into k-by-k blocks as we did in equation (29). Define a skew symmetric matrix Ω pq,i j by setting all blocks but B pq and Bqp zeros while 1 B pq (= −BTqp ) = ~v pi~vTq j 2 then it is a straitforward computation that N pq,i j = [H, Ω pq,i j ] (34) (35) On the other side, if we define a skew symmetric matrix Ωs,t by setting all blocks zeros but leaving Bss = Ω̃s,t , then S Ns,t = [H, Ωs,t ] (36) A computation shows that H (N pq,i j , N p0 q0 ,i0 j0 ) = − 2(λ pi − λq j )(µ(Λ p ) − µ(Λq ))∆ pp0 ,qq0 ,ii0 , j j0 (31) with ∆ pp0 ,qq0 ,ii0 , j j0 := δ pp0 δqq0 δii0 δ j j0 and δi j is the Kronecker delta. This is a straitforward computation following the formula (25), more details can be found in [8]. The set No contains as many as ∑1≤i< j≤k ni n j off-block-diagonal matrices. 2. Constructing Nd : still assume H = Diag(H1 , · · · , Hk ) with each Hi ∈ Sym(Λi ). Fix i, we let ~e ∈ Rni be a vector with all entries ones and let ~e⊥ be the hyperplane perpendicular to ~e. An identification is that the tangent space of the convex hull C(Λ0 ) at any of its point is ~e⊥ . Lemma 9: Let {~u1 , · · · ,~uni −1 } be an orthonormal basis of ~e⊥ , then there exists a set of skew symmetric matrices {Ω̃i,1 , · · · , Ω̃i,ni −1 } such that π([Hi , Ω̃i, j ]) = u j . Proof: This follows fact 4 as the projection π is a submersion. Define a set of symmetric matrices Ni in the way that for each matrix Ni, j ∈ Ni , all blocks except Bii are zeros while Bii = [Hi , Ω̃i, j ] (32) for some j = 1, · · · , ni − 1, so there are exactly (ni − 1) symmetric matrices in the set Ni . Lemma 10: The set Nd := ki=1 Ni is contained in NH Eα . If Ni, j ∈ Ni and Ni0 , j0 ∈ Ni0 , then H (Ni, j , Ni0 , j0 ) = δii0 δ j j0 . S This again follows the formula (25). The set Nd contains as many as ∑ki=1 (ni − 1) block-diagonal matrices. [π(H), Ωs,t ] = 0 (37) π([H, Ω pq,i j ]) = 0 (38) These two equalities annihilate the right hand side of formula (25), so H (N pq,i j , Ns,t ) = 0. Theorem 12: The Hessian H is nondegenerate when restricted at the normal space NH Eα for any H ∈ Eα and any Eα . Moreover, H is invariant under conjugation of permutation matrices, so the diagonal potential by equation (12) is a Morse-Bott function. Proof: The invariance is an consequence of lemma 5 and formula (25). The rest is an outcome by combining lemma 8, lemma 10 and lemma 11. We end this section with a discussion on the basis N . For each symmetric matrix N ∈ NH Eα , there is a unique matrix N 0 ∈ NH Eα such that H (N, ·) = tr(N 0 ·) because H is nondegenerate. This induces a linear map LH on NH Eα by sending N to N 0 . Each linear subspace spanned by N pq or Ni is invariant under LH , moreover each matrix N pq,i j ∈ N pq is an eigenmatrix of LH with respect to the eigenvalue −(µ(Λ p ) − µ(Λq ))/(λ pi − λq j ). All eigenvalues from the linear subspace spanned by Nd are positive by lemma 10. So at this moment for each critical manifold, we have computed its index and co-index. VI. A PPLICATION OF ADJACENCY CRITERION ON THE ISOSPECTRAL MANIFOLD WITH THE DIAGONAL POTENTIAL In this section, we will work out all stable critical manifolds and characterize all pairs of adjacent neighbors. Lemma 13: A critical manifold is stable if and only if it is a singleton consisting of a diagonal matrix. Proof: Suppose M is a stable critical manifold, without loss of generality, we assume M = Eα because the Hessian is invariant under conjugation of permutation matrices, if Eα is stable, then so is any critical manifold in its orbit. Suppose α = (Λ1 , · · · , Λk ), then by lemma 10, each Λi is a singleton, otherwise there exist at least ∑ki=1 (#(Λi ) − 1) positive eigenvalues of LH . So M is necessarily a singleton consisting of a diagonal matrix. On the other hand, if H is a diagonal matrix, then it is a stable equilibrium because all eigenvalues of LH are equal −(λ pi − λq j )/(λ pi − λq j )=-1 as suggested by our discussion in the end of last section. A diagonal matrix as an stable equilibrium is worth having its own notation sσ , the subindex σ indicates the permutation on the set of indices {1, · · · , n}, i.e, sσ = (λσ (1) , · · · , λσ (n) ). We now characterize critical manifolds of co-index 1. Recall that λ1 < · · · < λn , we say that two eigenvalues λi < λ j are close in order if j = i + 1. Lemma 14: Suppose α = (Λ1 , · · · , Λk ) and Eα is a critical manifold is of co-index 1, then all but one Λ p are singletons, and Λ p consists of two eigenvalues and they are closed in order. Proof: By lemma 10 and lemma 13, we first conclude that there is exactly one Λ p among {Λ1 , · · · , Λk } such that it is not a singleton and #(Λ p ) = 2. The matrix set N p is then a singleton contains exactly one block-diagonal matrix N p . At this moment, there is at least one positive eigenvalue of LH with eigenmatrix N p because of lemma 10 and the fact that spanN p is an invariant subspace. Suppose Λ p = {λ p1 , λ p2 }, then to prevent from having extra positive eigenvalues of LH , two eigenvalues λ p1 and λ p2 are necessarily closed in order because if not, suppose λq is in between λ p1 and λ p2 . We consider the two eigenvalues of LH contributed by the two dimensional linear subspace spanned by N pq , these two eigenvalues are −(µ(Λ p ) − λq ))/(λ p1 − λq ) and −(µ(Λ p ) − λq )/(λ p2 − λq ). By assumption one is negative while the other is positive. On the other hand, suppose λ p1 < λ p2 and they are close in order, since λ p1 < µ(Λ p ) < λ p2 , it is clear that −(µ(Λ p ) − λq ))/(λ pi − λq ) < 0 for any i = 1, 2 and any λq ∈ Λ − {λi1 , λi2 }. Each critical manifold of co-index 1 is a discrete set of two elements. A typical set looks like   .. .     λi + λi+1 · · · ±(λi − λi+1 )   1  .. . . . .   (39) . . .  2   ±(λi − λi+1 ) · · · λi + λi+1   .. . To relate with diagonal matrices, we let Sn be the group of permutations on indices {1, · · · , n}, in convention a permutation is said to be a simple transposition if it is a 2-cycle (i, i + 1). We say two permutations σ1 and σ2 are related by a simple transposition σ̂ if σ1 · σ̂ = σ2 . Suppose sσ1 = diag(· · · , λi , · · · , λi+1 , · · · ) (40) sσ2 = diag(· · · , λi+1 , · · · , λi , · · · ) (41) we then denote by Kσ1 ,σ2 the two matrices in equation (39). The collection of equilibria of co-index 1 is then the union of Kσ1 ,σ2 where (σ1 , σ2 ) varies over all pairs of permutations that are related by a simple transposition. A closed set Z ⊂ Sym(Λ) is invariant if for any H ∈ Z, the solution ϕt (H) remains in Z for any t ∈ R. An observation is that Lemma 15: Let α = (Λ1 , · · · , Λk ) be a choice of partition. We define Zα := {Diag(H1 , · · · , Hk )|Hi ∈ Sym(Λi )} (42) then Zα is an invariant subset in Sym(Λ). Proof: If H ∈ Zα and we write H = Diag(H1 , · · · , Hk ), then the dynamical system is decoupled in the sense that Ḣi = [Hi , [Hi , π(Hi )]], ∀i = 1, · · · , k (43) and Ḣ = Diag(Ḣ1 , · · · , Ḣk ). Remark: Since the gradient flow f (H) commutes conjugation by permutation matrices, so PT Zα P is also an invariant subset of Sym(Λ) for any permutation matrix P. Theorem 16 (Adjacent neighbors of diagonal matrices): All stable critical manifolds are singletons, and they are diagonal matrices. Two diagonal matrices sσ1 and sσ2 are adjacent if and only if σ1 and σ2 are related by a simple transposition. Proof: Fix the pair (σ1 , σ2 ), and assume k+ and k− are the two matrices in Kσ1 ,σ2 . Let W u (k+ ) and W u (k− ) be the unstable manifolds of k+ and k− respectively, we will show that the boundary of either of unstable manifolds is {sσ1 , sσ2 }, i.e, ∂W u (k+ ) = ∂W u (k− ) = sσ1 ∪ sσ2 (44) But this is true by lemma 15 because if let Λ0 := {λ1 , λ2 } and consider the double bracket flow on Sym(Λ0 ), then there are exactly four isolated equilibria, two diagonal matrices as stable equilibria and the other two are   1 λi + λi+1 ±(λi − λi+1 ) 0 (45) k± = λi + λi+1 2 ±(λi − λi+1 ) The isospectral manifold Sym(Λ0 ) is diffeomorphic to the circle S1 , if we parametrize it by θ ∈ R, then the induced dynamical system of θ on the covering space R is then 1 θ̇ = − (λi − λi+1 )2 sin(2θ ) 2 (46) It is clear that Zπ are stable equilibria and (Z + 12 )π are the unstable ones, this is consistent with the earlier arguments. 0 ) Moreover this implies that the boundary of either W u (k+ 0 ) is the union of diag(λ , λ or W u (k− i i+1 ) and diag(λi+1 , λi ). This then completes the proof. where c is a normalization factor of the density. (see, for example, [10] the derivation of the formula.) This implies that the equiprobability surface of the stochastic flow coincides with the equipotential surface of Ψ. If ε is small enough, then the density function is highly peaked at diagonal matrices as they are the local maxima. So a typical trajectory will spend most of its time around stable equilibria, this suggests that we approximate the behavior of the sample path by setting up a Markov model whose states are the n! diagonal matrices, and the trajectory of a sample path is simplified by a chain T Tk−1 T 2 1 · · · −−→ sσk sσ2 −→ sσ1 −→ Fig. 3. Each vertex is the projection of a diagonal matrix and each edge is the projection of an unstable manifold W u (Kσi ,σ j ). For convenience, we use Cσi ,σ j to denote W u (Kσi ,σ j ) with emphasis that each W u (σi , σ j ) consists of two disjoint curves with the same image under π. There are as many as 21 (n − 1) · n! pairs of adjacent neighbors. To better understand the geometry behind the theorem, we consider the convex polytope C(Λ). Each vertex of the polytope, known as a 0-dim face, corresponds to a vector (λσ (1) , · · · , λσ (n) ). It is the image of a diagonal matrix under the projection map π. Each edge of the polytope, known as a 1-dim face, is then the image of an unstable manifold W u (Kσ ,σ 0 ). Notice that each W u (Kσ ,σ 0 ) has two disjoint curves, and both have the same image under π. It is clear then each edge corresponds to a pair of adjacent neighbors. VII. A PPLICATION OF ADJACENCY THEOREM ON STOCHASTIC GRADIENT FLOW There is a stochastic version of the double bracket flow by adding an isotropic noise into the equation dH =[H, [H, π(H)]]dt + ε ∑[Ωi j , H]dωi j + ε2 [Ωi j , [Ωi j , H]]dt 2 ∑ (47) The third term in the right hand side of equation above appears as a consequence of the Itô rule so that the solution still evolves on the isospectral manifold. In the literatures of NMR, the last two terms relate to the Lindblad terms modeling the heat bath. Each Ωi j is a skew symmetric matrix defined by Ωi j =~ei~eTj −~e j~eTi where {~e1 , · · · ,~en } is a standard basis in Rn . The significance of the stochastic effects is modeled by the scalar ε. It happens that there is an explicit formula for the steady state solution of the Fokker-Planck equation associated with our stochastic equation. It takes the form ρ(H) = c exp( 2Ψ(H) ) ε2 (48) (49) The rest of this section is to develop a method to evaluate the transition probabilities. There are two main challenging problems when coming to the computation: the scale and the model. For each state, there are as many as (n! − 1) transition probabilities we need to evaluate, so in general the amount of computation is about n!(n! − 1). To evaluate each transition probability, we need to investigate a corresponding first hitting time model: we ask for the probability P(T |sσi → sσ j ) of the event that the passage time is than T for a sample path to reach state sσ j from state sσi without visiting any other state. We now show how adjacency theorem enters to simplify the problems Reduce the computational complexity: Let Aσi be the set of adjacent neighbors of sσi . Suppose sσ j ∈ / Aσi , then we infer that under the case ε  1 P(T |sσi → sσ j )  ∑ P(T |sσi → sσ ) (50) σ ∈Aσi for any T > 0. This is reasonable because from geometric perspective, if a sample path escapes from the region of attraction W s (sσi ), there is a high percentage a path gets trapped by one of its neighbors. So our first step of approximation is to set P(T |sσi → sσ j ) = 0 for any σ j ∈ / Aσi . So then for each state, the amount of computation is reduced to (n − 1). Approximate the transition probability: For each state sσi , the sample space of transition probability is discrete in space Aσi but continuous in time. In general there is no exact formula for computing P(T |sσi → sσ j ). One approach to approximate the density function is to relate the first hitting problem to an optimal control problem. In quantum mechanics, it is known that the probability for a system to stay in a quantum state of energy E is proportional to exp(−E). This idea of Boltzmann sheds light on our problem. We consider the control problem Ḣ = [H, [H, π(H)]] + ε[H,U] (51) with U skew symmetric and the goal is to minimize the energy, i.e, 1 E(T |σi → σ j ) := − min U(t) 2 Z T tr(U 2 (t))dt (52) 0 under the assumption that H(0) = σi and H(T ) = σ j are fixed. We then approximate P(T |σi → σ j ) by a scalar proportion of exp(−E(T |σi → σ j )). R EFERENCES Fig. 4. Illustrating the idea behind equation (50): a typical sample path only connects adjacent neighbors while a nontypical sample path does not. By solving the Euler-Lagrange equation, we conclude that each optimal trajectory, or the energy minimizing path(EMP) has to satisfy Ḣ = [H, Ω] (53) Ω̇ = [H, [π(H), [H, π(H)]]] + [H, π([H, [H, π(H)]])] (54) It is hard to compute the MEP in general, however in the case H(0) and H(T ) are both diagonal matrices, each MEP coincides with a geodesic. In particular, there are two MEPs and they together form the unstable manifold of Kσ1 ,σ2 . This then simplifies the situation to a scalar problem: suppose the simple transposition relating σi and σ j is the 2-cycle (λ1 , λ2 ), then the control model is given by 1 θ̇ = − (λ1 − λ2 )2 sin(2θ ) + 2εu 2 and the goal is to minimize (55) RT 2 0 u dt. Before we ending this section, we point out that locating a MEP that connects two adjacent neighbors and computing the minimal consumption of energy is more than an ad-hoc plan for evaluating the transition probability. For example, consider the situation that the control is intermittent and impulse-like, and our goal is to steer the system from one stable equilibrium to the other. Then questions such as in what direction one can escape from a region of attraction by means of an impulse? How we can save most of energy and/or time to reach the target? These questions relate to the design of a path that concatenates pairs of adjacent neighbors, the analysis done in this section is then essential. ACKNOWLEDGEMENTS The author thanks Dr. Roger W. Brockett at Harvard University for his comments on an earlier draft. [1] R. W. Brockett. Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems. Linear algebra and its applications, 146:79–91, 1991. [2] A. M. Bloch. Steepest descent, linear programming and hamiltonian flows. Contemp. Math. AMS, 114:77–88, 1990. [3] R. W. Brockett and W. S. Wong. A gradient flow for the assignment problem. In New Trends in Systems Theory, pages 170–177. Springer, 1991. [4] R. W. Brockett. Modeling the transient behavior of stochastic gradient algorithms. In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 4461–4466. IEEE, 2011. [5] A. Banyaga and D. Hurtubise. Morse-bott homology. Transactions of the American Mathematical Society, 362(8):3997–4043, 2010. [6] A. Banyaga and D. Hurtubise. Lectures on Morse homology, volume 29. Springer Science & Business Media, 2013. [7] J. Palis. On morse-smale dynamical systems. Topology, 8(4):385–404, 1969. [8] X. Chen. Symmetric potential functions on the isospectral manifolds. in preparation. [9] A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620–630, 1954. [10] R. W. Brockett. Notes on stochastic processes on manifolds. In Systems and Control in the Twenty-first Century, pages 75–100. Springer, 1997. 

Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams Paweł Gawrychowski1 , Oleg Merkurev2 , Arseny M. Shur2 , and Przemysław Uznański3 1 arXiv:1610.03125v1 [] 10 Oct 2016 2 Institute of Informatics, University of Warsaw, Poland Institute of Mathematics and Computer Science, Ural Federal University, Ekaterinburg, Russia 3 Department of Computer Science, ETH Zürich, Switzerland January 19, 2018 Abstract We consider computing a longest palindrome in the streaming model, where the symbols arrive oneby-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of Ω(M log min{|Σ|, M }) bits of memory; here M = n/E for approximating the answer with additive error log n for approximating the answer with multiplicative error (1 + ε). Second, we design E, and M = log(1+ε) three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use O(M ) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled. 1 Introduction In the streaming model of computation, a very long input arrives sequentially in small portions and cannot be stored in full due to space limitation. While well-studied in general, this is a rather recent trend in algorithms on strings. The main goals are minimizing the space complexity, i.e., avoiding storing the already seen prefix of the string explicitly, and designing real-time algorithm, i.e., processing each symbol in worstcase constant time. However, the algorithms are usually randomized and return the correct answer with high probability. The prime example of a problem on string considered in the streaming model is pattern matching, where we want to detect an occurrence of a pattern in a given text. It is somewhat surprising that one can actually solve it using polylogarithmic space in the streaming model, as proved by Porat and Porat [15]. A simpler solution was later given by Ergün et al. [6], while Breslauer and Galil designed a real-time algorithm [3]. Similar questions studied in such setting include multiple-pattern matching [4], approximate pattern matching [5], and parametrized pattern matching [10]. We consider computing a longest palindrome in the streaming model, where a palindrome is a fragment which reads the same in both directions. This is one of the basic questions concerning regularities in texts and it has been extensively studied in the classical non-streaming setting, see [1, 8, 12, 14] and the references therein. The notion of palindromes, but with a slightly different meaning, is very important in computational 1 biology, where one considers strings over {A, T, C, G} and a palindrome is a sequence equal to its reverse complement (a reverse complement reverses the sequences and interchanges A with T and C with G); see [9] and the references therein for a discussion of their algorithmic aspects. Our results generalize to biological palindromes in a straightforward manner. We denote by LPS(S) the problem of finding the maximum length of a palindrome in a string S (and a starting position of a palindrome of such length in S). Solving LPS(S) in the streaming model was recently considered by Berenbrink et al. [2], who developed tradeoffs between the bound on the error and the space complexity for additive and multiplicative variants of the problem, that is, for approximating the length of the longest palindrome with either additive or multiplicative error. Their algorithms were Monte Carlo, i.e., returned the correct answer with high probability. They also proved that any Las Vegas algorithm achieving n log |Σ|) bits √ of memory, which matches the space complexity of additive error E must necessarily use Ω( E their solution up to a logarithmic factor in the E ∈ [1, n] range, but leaves a few questions. Firstly, does the lower bound √ still hold for Monte Carlo algorithms? Secondly, what is the best possible space complexity when E ∈ ( n, n] in the additive variant, and what about the multiplicative version? Finally, are there real-time algorithms achieving these optimal space bounds? We answer all these questions. Our main goal is to settle the space complexity of LPS. We start with the lower bounds in Sect. 2. First, we show that Las Vegas algorithms cannot achieve sublinear space complexity at all. Second, we prove a lower bound of Ω(M log min{|Σ|, M }) bits of memory for Monte Carlo algorithms; here M = n/E log n for approximating the answer with for approximating the answer with additive error E, and M = log(1+ε) multiplicative error (1+ε). Then, in Sect. 3 we design real-time Monte Carlo algorithms matching these lower bounds up to a logarithmic factor. More precisely, our algorithm for LPS with additive error E ∈ [1, n] uses O(n/E) words of space, while our algorithms for LPS with multiplicative error ε ∈ (0, 1] (resp., ε ∈ (1, n])
log n uses O log(nε) (resp., O( log(1+ε) )) words of space1 . Finally we present, for any m, a deterministic O(m)ε space real-time algorithm solving LPS exactly if the answer is less than m and detecting a palindrome of length ≥ m otherwise. The last result implies that if the input stream is fully random, then with high probability its longest palindrome can be found exactly by a real-time algorithm within logarithmic space. Notation and Definitions. Let S denote a string of length n over an alphabet Σ = {1, . . . , N }, where N is polynomial in n. We write S[i] for the ith symbol of S and S[i..j] for its substring (or factor ) S[i]S[i+1] · · · S[j]; thus, S[1..n] = S. A prefix (resp. suffix ) of S is a substring of the form S[1..j] (resp., S[j..n]). A string S is a palindrome if it equals its reversal S[n]S[n−1] · · · S[1]. By L(S) we denote the length of a longest palindrome which is a factor of S. The symbol log stands for the binary logarithm. We consider the streaming model of computation: the input string S[1..n] (called the stream) is read left to right, one symbol at a time, and cannot be stored, because the available space is sublinear in n. The space is counted as the number of O(log n)-bit machine words. An algorithm is real-time if the number of operations between two reads is bounded by a constant. An approximation algorithm for a maximization problem has additive error E (resp., multiplicative error ε) if it finds a solution with the cost at least OP T −E T (resp., OP 1+ε ), where OP T is the cost of optimal solution; here both E and ε can be functions of the size of the input. In the LPS(S) problem, OP T = L(S). A Las Vegas algorithm always returns a correct answer, but its working time and memory usage on the inputs of length n are random variables. A Monte Carlo algorithm gives a correct answer with high probability (greater than 1 − 1/n) and has deterministic working time and space. 2 Lower Bounds In this section we use Yao’s minimax principle [17] to prove lower bounds on the space complexity of the LPS problem in the streaming model, where the length n and the alphabet Σ of the input stream are specified. We denote this problem by LPSΣ [n]. 1 Note that (a) log(1 + ε) is equivalent to ε whenever ε < 1; (b) the space used by the algorithms is O(n) for any values of errors; (c) the multiplicative lower bound applies to ε > n−0.99 , and thus is not contradicting the algorithm space usage. 2 Theorem 2.1 (Yao’s minimax principle for randomized algorithms). Let X be the set of inputs for a problem and A be the set of all deterministic algorithms solving it. Then, for any x ∈ X and A ∈ A, the cost of running A on x is denoted by c(a, x) ≥ 0. Let p be the probability distribution over A, and let A be an algorithm chosen at random according to p. Let q be the probability distribution over X , and let X be an input chosen at random according to q. Then maxx∈X E[c(A, x)] ≥ mina∈A E[c(a, X)]. We use the above theorem for both Las Vegas and Monte Carlo algorithms. For Las Vegas algorithms, we consider only correct algorithms, and c(x, a) is the memory usage. For Monte Carlo algorithms, we consider all algorithms (not necessarily correct) with memory usage not exceeding a certain threshold, and c(x, a) is the correctness indicator function, i.e., c(x, a) = 0 if the algorithm is correct and c(x, a) = 1 otherwise. Our proofs will be based on appropriately chosen padding. The padding requires a constant number of fresh characters. If Σ is twice as large as the number of required fresh characters, we can still use half of it to construct a difficult input instance, which does not affect the asymptotics. Otherwise, we construct a difficult input instance over Σ, then add enough new fresh characters to facilitate the padding, and finally reduce the resulting larger alphabet to binary at the expense of increasing the size of the input by a constant factor. Lemma 2.2. For any alphabet Σ = {1, 2, . . . , σ} there exists a morphism h : Σ∗ → {0, 1}∗ such that, for any c ∈ Σ, |h(c)| = 2σ + 6 and, for any string w, w contains a palindrome of length ` if and only if h(w) contains a palindrome of length (2σ + 6) · `. Proof. We set: h(c) = 11s 01s−c 10011s−c 01c 1. Clearly |h(c)| = 2σ + 6 and, because every h(c) is a palindrome, if w contains a palindrome of length ` then h(w) contains a palindrome of length (2σ + 6) · `. Now assume that h(w) contains a palindrome of length (2σ + 6) · `, where ` ≥ 1. If ` = 1 then we obtain that w should contain a palindrome of length 1, which always holds. Otherwise, the palindrome contains 00 inside and we consider two cases. 1. The palindrome is centered inside 00. Then it corresponds to an odd palindrome of length ` in w. 2. The palindrome maps some 00 to another 00. Then it corresponds to an even palindrome of length ` in w. In either case, the claim holds. For the padding we will often use an infinite string ν = 01 11 02 12 03 13 . . ., or more precisely its prefixes of length d, denoted ν(d). Here 0 and 1 should be understood √ as two characters not belonging to the original alphabet. The longest palindrome in ν(d) has length O( d). Theorem 2.3 (Las Vegas approximation). Let A be a Las Vegas streaming algorithms solving LPSΣ [n] with additive error E ≤ 0.99n or multiplicative error (1 + ε) ≤ 100 using s(n) bits of memory. Then E[s(n)] = Ω(n log |Σ|). Proof. By Theorem 2.1, it is enough to construct a probability distribution P over Σn such that for any deterministic algorithm D, its expected memory usage on a string chosen according to P is Ω(n log |Σ|) in bits. Consider solving LPSΣ [n] with additive error E. We define P as the uniform distribution over ν( E2 )x$$yν( E2 )R , 0 where x, y ∈ Σn , n0 = n2 − E2 − 1, and $ is a special character not in Σ. Let us look at the memory usage of 0 D after having read ν( E2 )x. We say that x is “good” when the memory usage is at most n2 log |Σ| and “bad” 0 otherwise. Assume that 21 |Σ|n of all x’s are good, then there are two strings x 6= x0 such that the state of D after having read both ν( E2 )x and ν( E2 )x0 is exactly the same. Hence the behavior of D on ν( E2 )x$$xR ν( E2 )R and ν( E2 )x0 $$xR ν( E2 )R is exactly the same. The former is a palindrome of length n = 2n0 + E + 2, so D must answer at least 2n0 + 2, and consequently the latter also must contain a palindrome of length at least 3 2n0 + 2. A palindrome inside ν( E2 )x0 $$xR ν( E2 )R is either fully contained within ν( E2 ), x0 , xR or it is a middle √ palindrome. But the longest palindrome inside ν( E2 ) is of length O( E) < 2n0 + 2 (for n large enough) and the longest palindrome inside x or xR is of length n0 < 2n0 + 2, so since we have exluced other possibilities, ν( E2 )x0 $$xR ν( E2 )R contains a middle palindrome of length 2n0 + 2. This implies that x = x0 , which is a 0 contradiction. Therefore, at least 12 |Σ|n of all x’s are bad. But then the expected memory usage of D is at 0 least n4 log |Σ|, which for E ≤ 0.99n is Ω(n log |Σ|) as claimed. Now consider solving LPSΣ [n] with multiplicative error (1 + ε). An algorithm with multiplicative error ε , so if the expected memory usage of such (1 + ε) can also be considered as having additive error E = n · 1+ε an algorithm is o(n log |Σ|) and (1 + ε) ≤ 100 then we obtain an algorithm with additive error E ≤ 0.99n and expected memory usage o(n log |Σ|), which we already know to be impossible. Now we move to Monte Carlo algorithms. We first consider exact algorithms solving LPSΣ [n]; lower bounds on approximation algorithms will be then obtained by padding the input appropriately. We introduce an auxiliary problem midLPSΣ [n], which is to compute the length of the middle palindrome in a string of even length n over an alphabet Σ. Lemma 2.4. There exists a constant γ such that any randomized Monte Carlo streaming algorithm A solving midLPSΣ [n] or LPSΣ [n] exactly with probability 1 − n1 uses at least γ · n log min{|Σ|, n} bits of memory. Proof. First we prove that if A is a Monte Carlo streaming algorithm solving midLPSΣ [n] exactly using less 1 than b n2 log |Σ|c bits of memory, then its error probability is at least n|Σ| . By Theorem 2.1, it is enough to construct probability distribution P over Σn such that for any deterministic algorithm D using less than b n2 log |Σ|c bits of memory, the expected probability of error on a string 1 chosen according to P is at least n|Σ| . Let n0 = n 2. 0 For any x ∈ Σn , k ∈ {1, 2, . . . , n0 } and c ∈ Σ we define w(x, k, c) = x[1]x[2]x[3] . . . x[n0 ]x[n0 ]x[n0 − 1]x[n0 − 2] . . . x[k + 1]c0k−1 . Now P is the uniform distribution over all such w(x, k, c). 0 Choose an arbitrary maximal matching of strings from Σn into pairs (x, x0 ) such that D is in the same state after reading either x or x0 . At most one string per state of D is left unpaired, that is at most 0 0 n n 2b 2 log |Σ|c−1 strings in total. Since there are |Σ|n = 2n log |Σ| ≥ 2 · 2b 2 log |Σ|c−1 possible strings of length n0 , at least half of the strings are paired. Let s be longest common suffix of x and x0 , so x = vcs and x0 = v 0 c0 s, where c 6= c0 are single characters. Then D returns the same answer on w(x, n0 − |s|, c) and w(x0 , n0 − |s|, c), even though the length of the middle palindrome is exactly 2|s| in one of them, and at least 2|s| + 2 in the other one. Therefore, D errs on at least one of these two inputs. Similarly, it errs on either w(x, n0 − |s|, c0 ) 1 or w(x, n0 − |s|, c0 ). Thus the error probability is at least 2n01|Σ| = n|Σ| . Now we can prove the lemma for midLPSΣ [n] with a standard amplification trick. Say that we have a Monte Carlo streaming algorithm, which solves midLPSΣ [n] exactly with error probability ε using s(n) bits of memory. Then we can run its k instances simultaneously and return the most frequently reported answer. The new algorithm needs O(k · s(n)) bits of memory and its error probability εk satisfies: X k  εk ≤ (1 − ε)i εk−i ≤ 2k · εk/2 = (4ε)k/2 . i 2i<k log(4/n) 1−o(1) log n 1 = 61 1+log Let us choose κ = 61 log(1/(n|Σ|)) |Σ|/ log n = Θ( log n+log |Σ| ) = γ · log |Σ| log min{|Σ|, n}, for some constant γ. Now we can prove the Assume that A uses less than κ · n log |Σ| = γ · n log min{|Σ|, n}  1 theorem. 1 bits of memory. Then running 2κ ≥ 34 2κ (which holds since κ < 16 ) instances of A in parallel requires less than b n2 log |Σ|c bits of memory. But then the error probability of the new algorithm is bounded from above by 3   16κ   18 16 4 1 1 = ≤ n n|Σ| n|Σ| 4 which we have already shown to be impossible. The lower bound for midLPSΣ [n] can be translated into a lower bound for solving LPSΣ [n] exactly by padding the input so that the longest palindrome is centered in the middle. Let x = x[1]x[2] . . . x[n] be the input for midLPSΣ [n]. We define w(x) = x[1]x[2]x[3] . . . x[n/2]1 000 . . . 0 1x[n/2 + 1] . . . x[n]. n Now if the length of the middle palindrome in x is k, then w(x) contains a palindrome of length at least n + k + 2. In the other direction, any palindrome inside w(x) of length ≥ n must be centered somewhere in the middle block consisting of only zeroes and both ones are mapped to each other, so it must be the middle palindrome. Thus, the length of the longest palindrome inside w(x) is exactly n + k + 2, so we have reduced solving midLPSΣ [n] to solving LPSΣ [2n + 2]. We already know that solving midLPSΣ [n] with probability 1 ≥ 1 − n1 1 − n1 requires γ · n log min{|Σ|, n} bits of memory, so solving LPSΣ [2n + 2] with probability 1 − 2n+2 0 requires γ · n log{|Σ|, n} ≥ γ · (2n + 2) log min{|Σ|, 2n + 2} bits of memory. Notice that the reduction needs O(log n) additional bits of memory to count up to n, but for large n this is much smaller than the lower bound if we choose γ 0 < γ4 . To obtain a lower bound for Monte Carlo additive approximation, we observe that any algorithm solving E LPSΣ [n] with additive error E can be used to solve LPSΣ [ n−E E+1 ] exactly by inserting 2 zeroes between every two characters, in the very beginning, and in the very end. However, this reduction requires log( E2 ) ≤ log n additional bits of memory for counting up to E2 and cannot be used when the desired lower bound on the n n log min(|Σ|, E ) is significantly smaller than log n. Therefore, we need a separate required number of bits Ω( E technical lemma which implies that both additive and multiplicative approximation with error probability 1 n require Ω(log n) bits of space. Lemma 2.5. Let A be any randomized Monte Carlo streaming algorithm solving LPSΣ [n] with additive error at most 0.99n or multiplicative error at most n0.49 and error probability n1 . Then A uses Ω(log n) bits of memory. Proof. By Theorem 2.1, it is enough to construct a probability distribution P over Σn , such that for any deterministic algorithm D using at most s(n) = o(log n) bits of memory, the expected probability of error 1 on a string chosen according to P is 2s(n)+2 . 0 n0 Let n = s(n) + 1. For any x, y ∈ Σ , let w(x, y) = ν( n2 − n0 )R xy R ν( n2 − n0 ). Observe that if x = y then w(x, y) contains a palindrome of length n, and otherwise the longest palindrome there has length at most √ √ 2n0 + O( n) = O( n), thus any algorithm with additive error of at most 0.99n or with a multiplicative error at most n0.49 must be able to distinguish between these two cases (for n large enough). 0 Let S ⊆ Σn be an arbitrary family of strings of length n0 such that |S| = 2 · 2s(n) , and let P be the uniform distribution on all strings of the form w(x, y), where x and y are chosen uniformly and independently 0 from S. By a counting argument, we can create at least |S| 4 pairs (x, x ) of elements from S such that the state of D is the same after having read ν( n2 − n0 )R x and ν( n2 − n0 )R x0 . (If we create the pairs greedily, at most one such x per state of memory can be left unpaired, so at least |S| − 2s(n) = |S| 2 elements are paired.) Thus, D cannot distinguish between w(x, x0 ) and w(x, x), and between w(x0 , x0 ) and w(x0 , x), so its error 1 1 probability must be at least |S|/2 |S|2 = 4·2s(n) . Thus if s(n) = o(log n), the error rate is at least n for n large enough, a contradiction. Combining the reduction with the technical lemma and taking into account that we are reducing to a n problem with string length of Θ( E ), we obtain the following. Theorem 2.6 (Monte Carlo additive approximation). Let A be any randomized Monte Carlo streaming algorithm solving LPSΣ [n] with additive error E with probability 1 − n1 . If E ≤ 0.99n then A uses n n Ω( E log min{|Σ|, E }) bits of memory. 5 n Proof. Define σ = min{|Σ|, E }. n Because of Lemma 2.5 it is enough to prove that Ω( E log σ) is a lower bound when E≤ n γ · log σ. 2 log n (1) Assume that there is a Monte Carlo streaming algorithm A solving LPSΣ [n] with additive error E using n n o( E log σ) bits of memory and probability 1 − n1 . Let n0 = n−E/2 E/2+1 ≥ E (the last inequality, equivalent E to n ≥ E · E−2 holds because E ≤ 0.99n and because we can assume that E ≥ 200). Given a string x[1]x[2] . . . x[n0 ], we can simulate running A on 0E x[1]0E/2 x[2]0E/2jx[3] . . . k0E/2 x[n0 ]0E/2 to calculate R (using log(E/2) ≤ log n additional bits of memory), and then return R E/2+1 . We call this new Monte Carlo 0 streaming algorithm A . Recall that A reports the length of the longest palindrome with additive error E. Therefore, if the original string contains a palindrome of length r, the new string contains a palindrome of length E2 · (r + 1) + r, so R ≥ r(E/2 + 1) and A0 will return at least r. In the other direction, if A0 returns r, then the new string contains a palindrome of length r(E/2 + 1). If such palindrome is centered so that x[i] is matched with x[i + 1] for some i, then it clearly corresponds to a palindrome of length r in the original string. But otherwise every x[i] within the palindrome is matched with 0, so in fact the whole palindrome corresponds to a streak of consecutive zeroes in the new string and can be extended to the left and to the right to start and end with 0E , so again it corresponds to a palindrome of length r in 1 ≥ 1 − n10 and the original string. Therefore, A0 solves LPSΣ [n0 ] exactly with probability 1 − (n0 (E/2+1)+E/2) 0 uses o( n (E/2+1)+E/2 log σ) + log n = o(n0 log σ) + log n bits of memory. Observe that by Lemma 2.4 we get E/2 a lower bound γ γ n γ γ · n0 log min{|Σ|, n0 } ≥ · n0 log σ + · log σ ≥ · n0 log σ + log n 2 2 E 2 (where the last inequality holds because of Eq.(1)). Then, for large n we obtain contradiction as follows o(n0 log σ) + log n < γ 0 · n log σ + log n. 2 Finally, we consider multiplicative approximation. The proof follows the same basic idea as of Theorem 2.6, however is more technically involved. The main difference is that due to uneven padding, we are reducing to midLPSΣ [n0 ] instead of LPSΣ [n0 ]. Theorem 2.7 (Monte Carlo multiplicative approximation). Let A be any randomized Monte Carlo streaming algorithm solving LPSΣ [n] with multiplicative error (1 + ε) with probability 1 − n1 . If n−0.98 ≤ ε ≤ n0.49 then log n log n A uses Ω( log(1+ε) log min{|Σ|, log(1+ε) }) bits of memory. Proof. For ε ≥ n0.001 the claimed lower bound reduces to Ω(1) bits, which obviously holds. Thus we can assume that ε < n0.001 . Define 1 log n − 2}. σ = min{|Σ|, 50 log(1 + 2ε) log n First we argue that it is enough to prove that A uses Ω( log(1+ε) log σ) bits of memory. Since log(1 + 2ε) ≤ 0.001 log n + o(log n), we have that: 1 log n − 2 ≥ 18 − o(1) 50 log(1 + 2ε) (2) 1 log n log n − 2 = Θ( ). 50 log(1 + 2ε) log(1 + 2ε) (3) log(1 + 2ε) = Θ(log(1 + ε)) (4) and consequently: Finally, observe that: 6 because log 2(1 + ε) = Θ(log(1 + ε)) for ε ≥ 1, and log(1 + ε) = Θ(ε) for ε < 1. From (3) and (4) we conclude that: log n }). (5) log σ = Θ(log min{|Σ|, log(1 + ε) log n Because of Lemma 2.5 and equations (4) and (5), it is enough to prove that Ω( log(1+ε) log σ) is a lower bound when log σ log(1 + 2ε) ≤ γ · , (6) 100 log n log n as otherwise Ω( log(1+ε) log σ) = Ω( log(1+2ε) log σ) = Ω(log n). Assume that there is a Monte Carlo streaming algorithm A solving LPSΣ [n] with multiplicative error log n (1+ε) with probability 1− n1 using o( log(1+ε) log σ) bits of memory. Let x = x[1]x[2] . . . x[n0 ]x[n0 +1] . . . x[2n0 ] 0 be an input for midLPSΣ [2n0 ]. We choose n0 so that n = (1 + 2ε)n +1 · n0.99 . Then n0 = log(1+2ε) (n0.01 ) − 1 = log n 1 d+1 0 · n0.99 e for any 0 ≤ d ≤ n0 . 100 log(1+2ε) − 1. We choose i0 , i1 , i2 , i3 , . . . , in so that i0 + . . . + id = d(1 + 2ε) −0.98 0.99 0.01 (Observe that for ε = Ω(n ) we have i0 > n and i1 , . . . , id > 2n − 1.) Finally we define: w(x) = ν(in0 )R x[1]ν(in0 −1 )R . . . x[n0 ]ν(i0 )R ν(i0 )x[n0 + 1]ν(i1 ) . . . ν(in0 −1 )x[2n0 ]ν(in0 ). If x contains a middle palindrome of length exactly 2k, then w(x) contains a middle palindrome of length 2(1 + 2ε)k+1 · n0.99√ . Also, based on the properties of ν, any non-middle centered palindrome in w(x) has length at most O( n), which is less than n0.99 for n large enough. Since d2(1 + 2ε)k · n0.99 e · (1 + ε) < (2(1 + 2ε)k · n0.99 + 1) · (1 + ε) < 2(1 + 2ε)k+1 · n0.99 , value of k can be extracted from the answer of A. Thus, if A approximates the middle palindrome in w(x) with multiplicative error (1 + ε) with probability 1 − n1 log n using o( log(1+ε) log σ) bits of memory, we can construct a new algorithm A0 solving midLPSΣ [2n0 ] exactly 1 with probability 1 − n1 > 1 − 2n 0 using o( log n log σ) + log n log(1 + ε) (7) bits of memory. By Lemma 2.4 we get a lower bound γ · 2n0 log min{|Σ|, 2n0 } = ≥ γ log n · log σ − 2γ log σ 50 log(1 + 2ε) γ log n · log σ + log n − 2γ log σ 100 log(1 + 2ε) (8) (where the last inequality holds because of (6)). On the other hand, for large n   γ log n log n 1 · log σ − 2γ log σ + log n = − 2 γ log σ + log n 100 log(1+2ε) 100 log(1+2ε)   log n log σ + log n = Θ log(1+ε) so (8) exceeds (7), a contradiction. 3 Real-Time Algorithms In this section we design real-time Monte Carlo algorithms within the space bounds matching the lower bounds from Sect. 2 up to a factor bounded by log n. The algorithms make use of the hash function known as the Karp-Rabin fingerprint [11]. Let p be a fixed prime from the range [n3+α , n4+α ] for some α > 0, and r be a fixed integer randomly chosen from {1, . . . , p−1}. For a string S, its forward hash and reversed hash are defined, respectively, as ! ! n n X X F i R n−i+1 φ (S) = S[i] · r mod p and φ (S) = S[i] · r mod p . i=1 i=1 7 Clearly, the forward hash of a string coincides with the reversed hash of its reversal. Thus, if u is a palindrome, then φF (u) = φR (u). The converse is also true modulo the (improbable) collisions of hashes, because for two strings u 6= v of length m, the probability that φF (u) = φF (v) is at most m/p. This property allows one to detect palindromes with high probability by comparing hashes. (This approach is somewhat simpler than the one of [2]; in particular, we do not need “fingerprint pairs” used there.) In particular, a real-time algorithm makes O(n) comparisons and thus faces a collision with probability O(n−1−α ) by the choice of p. All further considerations assume that no collisions happen. For an input stream S, we denote F F (i, j) = φF (S[i..j]) and F R (i, j) = φR (S[i..j]). The next observation is quite important. Proposition 3.1 ( [3]). The following equalities hold:
F F (i, j) = r−(i−1) F F (1, j) − F F (1, i−1) mod p , F R (i, j) = F R (1, j) − rj−i+1 F R (1, i−1) mod p . Let I(i) denote the tuple (i, F F (1, i−1), F R (1, i−1), r−(i−1) mod p, ri mod p). The proposition below is immediate from definitions and Proposition 3.1. Proposition 3.2. 1) Given I(i) and S[i], the tuple I(i+1) can be computed in O(1) time. 2) Given I(i) and I(j+1), the string S[i..j] can be checked for being a palindrome in O(1) time. 3.1 Additive Error Theorem 3.3. There is a real-time Monte Carlo algorithm solving the problem LPS(S) with the additive error E = E(n) using O(n/E) space, where n = |S|. First we present a simple (and slow) algorithm which solves the posed problem, i.e., finds in S a palindrome of length `(S) ≥ L(S) − E, where L(S) is the length of the longest palindrome in S. Later this algorithm will be converted into a real-time one. We store the sets I(j) for some values of j in a doubly-linked list SP in the decreasing order of j’s. The longest palindrome currently found is stored as a pair answer = (pos, len), where pos is its initial position and len is its length. Let tE = b E2 c. In Algorithm ABasic we add I(j) to the list SP for each j divisible by tE . This allows us to check for palindromicity, at ith iteration, all factors of the form S[ktE ..i]. We assume throughout the section that at the beginning of ith iteration the value I(i) is stored in a variable I. Algorithm 1 : Algorithm ABasic, ith iteration 1: if i mod tE = 0 then 2: add I to the beginning of SP 3: read S[i]; compute I(i + 1) from I; I ← I(i + 1) 4: for all elements v of SP do 5: if S[v.i..i] is a palindrome and answer.len < i−v.i+1 then 6: answer ← (v.i, i−v.i+1) Proposition 3.4. Algorithm ABasic finds in S a palindrome of length `(S) ≥ L(S) − E using O(n/E) time per iteration and O(n/E) space. Proof. Both the time and space bounds arise from the size of the list SP , which is bounded by n/tE = O(n/E); the number of operations per iteration  isproportional to this size due to Proposition 3.2. Now let S[i..j] be a longest palindrome in S. Let k = tiE tE . Then i ≤ k < i + tE . At the kth iteration, I(k) was added to SP ; then the palindrome S[k..j−(k−i)] was found at the iteration j − (k − i). Its length is jE k j − (k − i) − k + 1 = j − i − 2(k − i) + 1 > (j − i + 1) − 2tE = L(S) − 2 ≥ L(S) − E, 2 as required. 8 The resource to speed up Algorithm ABasic stems from the following Lemma 3.5. During one iteration, the length answer.len increases by at most 2 · tE . Proof. Let S[j..i] be the longest palindrome found at the ith iteration. If i − j + 1 ≤ 2tE then the statement is obviously true. Otherwise the palindrome S[j+tE ..i−tE ] of length i − j + 1 − 2tE was found before (at the (i−tE )th iteration), and the statement holds again. Lemma 3.5 implies that at each iteration SP contains only two elements that can increase answer.len. Hence we get the following Algorithm A. Algorithm 2 : Algorithm A, ith iteration 1: if i mod tE = 0 then 2: add I to the beginning of SP 3: if i = tE then 4: sp ← f irst(SP ) 5: read S[i]; compute I(i + 1) from I; I ← I(i + 1) 6: sp ← previous(sp) 7: while i − sp.i + 1 ≤ answer.len and (sp 6= last(SP )) do 8: sp ← next(sp) 9: for all existing v in {sp, next(sp)} do 10: if S[v.i..i] is a palindrome and answer.len < i−v.i+1 then 11: answer ← (v.i, i−v.i+1) . if exists Due to Lemma 3.5, the cycle at lines 9–11 of Algorithm A computes the same sequence of values of answer as the cycle at lines 4–6 of Algorithm ABasic. Hence it finds a palindrome of required length by Proposition 3.4. Clearly, the space used by the two algorithms differs by a constant. To prove that an iteration of Algorithm A takes O(1) time, it suffices to note that the cycle in lines 7–8 performs at most two iterations. Theorem 3.3 is proved. 3.2 Multiplicative Error for ε ≤ 1 Theorem 3.6. There is a real-time Monte Carlo algorithm solving the problem LPS(S) with multiplicative
error ε = ε(n) ∈ (0, 1] using O log(nε) space, where n = |S|. ε As in the previous section, we first present a simpler algorithm MBasic with non-linear working time and then upgrade it to a real-time algorithm. The algorithm must find a palindrome of length `(S) ≥ L(S) 1+ε . The next lemma is straightforward. Lemma 3.7. If ε ∈ (0, 1], the condition `(S) ≥ L(S)(1 − ε/2) implies `(S) ≥ L(S) 1+ε .   We set qε = log 2ε . The main difference in the construction of algorithms with the multiplicative and additive error is that here all sets I(i) are added to the list SP , but then, after a certain number of steps, are deleted from it. The number of iterations the set I(i) is stored in SP is determined by the time-to-live function ttl (i) defined below. This function is responsible for both the correctness of the algorithm and the space bound. Let β(i) be the position of the rightmost 1 in the binary representation of i (the position 0 corresponds to the least significant bit). We define ttl (i) = 2qε +2+β(i) . (9) The definition is illustrated by Fig. 1. Next we state a few properties of the list SP . Lemma 3.8. For any integers a ≥ 1 and b ≥ 0, there exists a unique integer j ∈ [a, a + 2b ) such that ttl (j) ≥ 2qε +2+b . 9 Algorithm 3 : Algorithm MBasic, ith iteration 1: add I to the beginning of SP 2: for all v in SP do 3: if v.i + ttl (v.i) = i then 4: delete v from SP 5: read S[i]; compute I(i + 1) from I; I ← I(i + 1) 6: for all v in SP do 7: if S[v.i..i] is a palindrome and answer.len < i−v.i+1 then 8: answer ← (v.i, i−v.i+1) 1 8 16 21 24 28 32 36 38 40 42 44 46 53 Figure 1: The state of the list SP after the iteration i = 53 (qε = 1 is assumed). Black squares indicate the numbers j for which I(j) is currently stored. For example, (9) implies ttl (28) = 21+2+2 = 32, so I(28) will stay in SP until the iteration 28 + 32 = 60. Proof. By (9), ttl (j) ≥ 2qε +2+b if and only if β(j) ≥ b, i.e., j is divisible by 2b by the definition of β. Among any 2b consecutive integers, exactly one has this property. Figure 1 shows the partition of the range (0, i] into intervals having lengths that are powers of 2 (except for the leftmost interval). In general, this partition consists of the following intervals, right to left: l n m (i − 2qε +2 , i], (i − 2qε +3 , i − 2qε +2 ], . . . , (i − 2m , i − 2m−1 ], (0, i − 2m ], where m = log qε +2 − 1. (10) 2 Lemma 3.8 and (9) imply the following lemma on the distribution of the elements of SP . Lemma 3.9. After each iteration, the first interval (resp., the last interval; each of the remaining intervals) in (10) contains 2qε +2 (resp., at most 2qε +1 ; exactly 2qε +1 ) elements of the list SP . The number of the intervals in (10) is O(log(nε)), so from Lemma 3.9 and the definition of qε we have the following.
Lemma 3.10. After each iteration, the size of the list SP is O log(nε) . ε Proposition 3.11. Algorithm MBasic finds a palindrome of length `(S) ≥ L(S) 1+ε using O( log(nε) ) time per ε iteration and O( log(nε) ) space. ε Proof. Both the time per iteration and the space are dominated by the size of the list SP . Hence the required complexity bounds follow from Lemma 3.10. For the proof of correctness, let S[i..j] be a palindrome of length L(S). Further, let d = blog L(S)c. If d < qε + 2, the palindrome S[i..j] will be found exactly, because I(i) is in SP at the jth iteration: i + ttl (i) ≥ i + 2qε +2 ≥ i + 2d+1 > i + L(S) > j . Otherwise, by Lemma 3.8 there exists a unique k ∈ [i, i + 2d−qε −1 ) such that ttl (k) ≥ 2d+1 . Hence at the iteration j −(k −i) the palindrome S[i+(k−i)..j−(k−i)] will be found, because I(k) is in SP at this iteration: k + ttl (k) ≥ i + ttl (k) ≥ i + 2d+1 > j ≥ j − (k − i) . The length of this palindrome satisfies the requirement of the proposition: j − (k − i) − (i + (k − i)) + 1 = L(S) − 2(k − i) ≥ L(S) − 2d−qε ≥ L(S) − The reference to Lemma 3.7 finishes the proof. 10  L(S) ε ≥ L(S) 1 − . 2qε 2 Now we speed up Algorithm MBasic. It has two slow parts: deletions from the list SP and checks for palindromes. Lemmas 3.12 and 3.13 show that, similar to Sect. 3.1, O(1) checks are enough at each iteration. Lemma 3.12. Suppose that at some iteration the list SP contains consecutive elements I(d), I(c), I(b), I(a). Then b − a ≤ d − b. Proof. Let j be the number of the considered iteration. Note that a < b < c < d. Consider the interval in (10) containing a. If a ∈ (j − 2qε +2 , j], then b − a = 1 and d − b = 2, so the required inequality holds. Otherwise, let a ∈ (j −2qε +2+x , j −2qε +2+x−1 ]. Then by (9) β(a) ≥ x; moreover, any I(k) such that a < k ≤ j and β(k) ≥ x is in SP . Hence, b − a ≤ 2x . By Lemma 3.9 each interval, except for the leftmost one, contains at least 2qε +1 ≥ 4 elements. Thus each of the numbers b, c, d belongs either to the same interval as a or to the previous interval (j − 2qε +2+x−1 , j − 2qε +2+x−2 ]. Again by (9) we have β(b), β(c), β(d) ≥ x − 1. So c−b, d−c ≥ 2x−1 , whence the result. We call an element I(a) of SP valuable at ith iteration if i − a + 1 > answer.len and S[a..i] can be a palindrome. (That is, Algorithm MBasic does not store enough information to predict that the condition in its line 7 is false for v = I(a).) Lemma 3.13. At each iteration, SP contains at most three valuable elements. Moreover, if I(d0 ), I(d) are consecutive elements of SP and i − d0 < answer.len ≤ i − d, where i is the number of the current iteration, then the valuable elements are consecutive in SP , starting with I(d). Proof. Let d be as in the condition of the lemma. If I(d) is followed in SP by at most two elements, we are done. If it is not the case, let the next three elements be I(c), I(b), and I(a), respectively. If S[a..i] is a palindrome then S[a+(b−a)..i−(b−a)] is also a palindrome. At the iteration i−(b−a) the tuple I(b) was in SP , so this palindrome was found. Hence, at the ith iteration the value answer.len is at least the length of this palindrome, which is i − a + 1 − 2(b − a). By Lemma 3.12, b − a ≤ d − b, implying answer.len ≥ i − a + 1 − (b − a) − (d − b) = i − d + 1. This inequality contradicts the definition of d; hence, S[a..i] is not a palindrome. By the same argument, the elements following I(a) in SP do not produce palindromes as well. Thus, only the elements I(d), I(c), I(b) are valuable. Now we turn to deletions. The function ttl (x) has the following nice property. Lemma 3.14. The function x → x + ttl (x) is injective. Proof. Note that β(x + ttl (x)) = β(x) from the definition of ttl . Hence the equality x + ttl (x) = y + ttl (y) implies β(x) = β(y), then ttl (x) = ttl (y) by (9), and finally x = y. Lemma 3.14 implies that at most one element is deleted from SP at each iteration. To perform this deletion in O(1) time, we need an additional data structure. By BS(x) we denote a linked list of maximal segments of 1’s in the binary representation of x. For example, the binary representation of x = 12345 and BS(x) are as follows: 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 BS(12345) = {[0, 0], [3, 5], [10, 10], [12, 13]} Clearly, BS(x) uses O(log x) space. Lemma 3.15. Both β(x) and BS(x + 1) can be obtained from BS(x) in O(1) time. Proof. The first number in BS(x) is β(x). Let us construct BS(x + 1). Let [a, b] be the first segment in BS(x). If a > 2, then BS(x + 1) = [0, 0] ∪ BS(x). If a = 1, then BS(x + 1) = [0, b] ∪ (BS(x)\[1, b]). Now let a = 0. If BS(x) = {[0, b]} then BS(x + 1) = {[b+1, b+1]}. Otherwise let the second segment in BS(x) be [c, d]. If c > b + 2, then BS(x + 1) = [b+1, b+1] ∪ (BS(x)\[0, b]). Finally, if c = b + 2, then BS(x + 1) = [b+1, d] ∪ (BS(x)\{[0, b], [c, d]}). 11 Thus, if we support one list BS which is equal to BS(i) at the end of the ith iteration, we have β(i). If I(a) should be deleted from SP at this iteration, then β(a) = β(i) (see Lemma 3.14). The following lemma is trivial. Lemma 3.16. If a < b and ttl (a) = ttl (b), then I(a) is deleted from SP before I(b). By Lemma 3.16, the information about the positions with the same ttl (in other words, with the same β) are added to and deleted from SP in the same order. Hence it is possible to keep a queue QU (x) of the pointers to all elements of SP corresponding to the positions j with β(j) = x. These queues constitute the last ingredient of our real-time Algorithm M. Algorithm 4 : Algorithm M, ith iteration 1: add I to the beginning of SP 2: if i = 1 then 3: sp ← f irst(SP ) 4: compute BS[i] from BS; BS ← BS[i]; compute β(i) from BS 5: if QU (β(i)) is not empty then 6: v ← element of SP pointed by f irst(QU (β(i))) 7: if v = sp then 8: sp ← next(sp) 9: delete v; delete f irst(QU (β(i))) 10: add pointer to f irst(SP ) to QU (β(i)) 11: read S[i]; compute I(i + 1) from I; I ← I(i + 1) 12: sp ← previous(sp) 13: while i − sp.i + 1 ≤ answer.len and sp 6= last(SP ) do 14: sp ← next(sp) 15: for all existing v in {sp, next(sp), next(next(sp))} do 16: if S[v.i..i] is a palindrome and answer.len < i−v.i+1 then 17: answer ← (v.i, i−v.i+1) . if exists Proof of Theorem 3.6. After every iteration, Algorithm M has the same list SP (see Fig. 1) as Algorithm MBasic, because these algorithms add and delete the same elements. Due to Lemma 3.13, Algorithm M returns the same answer as Algorithm MBasic. Hence by Proposition 3.11 Algorithm M finds a palindrome of required length. Further, Algorithm M supports the list BS of size O(log n) and the array QU containing O(log n) queues of total size equal to the size of SP . Hence, it uses O( log(nε) ) space in total by Lemma 3.10. ε The cycle in lines 13–14 performs at most three iterations. Indeed, let z be the value of sp after the previous iteration. Then this cycle starts with sp = previous(z) (or with sp = z if z is the first element of SP ) and ends with sp = next(next(z)) at the latest. By Lemma 3.15, both BS(i) and β(i) can be computed in O(1) time. Therefore, each iteration takes O(1) time.

log n Remark Since for n−0.99 ≤ ε ≤ 1 the classes O log(1+ε) and O log(nε) coincide, Algorithm M uses space ε within a log n factor from the lower bound of Theorem 2.7. Furthermore, for an arbitrarily slowly growing function ϕ Algorithm M uses o(n) space whenever ε = ϕ(n) n . 3.3 Multiplicative Error for ε > 1 Theorem 3.17. There is a real-time Monte Carlo algorithm solving the problem LPS(S) with multiplicative log(n)
error ε = ε(n) ∈ (1, n] using O log(1+ε) space, where n = |S|. 12 We transform Algorithm
M into real-time Algorithm M’ which solves LPS(S) with the multiplicative log n error ε > 1 using O log(1+ε) space. The basic idea of transformation is to replace all binary representations with those in base proportional to 1 + ε, and thus shrink the size of the lists SP and BS. First, we assume without loss of generality that ε ≥ 7, as otherwise we can fix ε = 1 and apply Algorithm M. Fix k ≤ 21 (1 + ε) as the largest such even integer (in particular, k ≥ 4). Let β 0 (i) be the position of the rightmost non-zero digit in the k-ary representation of i. We define ( 0 9 · k β (i) if β 0 (i) > 0, 0 (11) ttl (i) = 2 4 otherwise. The list SP 0 is the analog of the list SP from Sect. 3.2. It contains, after ith iteration, the tuples I(j) for all positions j ≤ i such that j + ttl0 (j) > i. Similar to (10), we partition the range (0; i] into intervals and then count the elements of SP 0 in these intervals. The intervals are, right to left,     (i − 4, i], i − 92 k, i − 4 , i − 92 k 2 , i − 92 k , . . . , i − 92 k m , i − 92 k m−1 , 0, i − 92 k m . (12) For convenience, we enumerate the intervals starting with 0. Lemma 3.18. Each interval in (12) contains at most 5 elements of SP 0 . Each of the intervals 0, ..., m contains at least 3 elements of SP 0 . Proof. The 0th interval contains exactly 4 elements. For any j = 1, . . . , m+1, an element x of the jth interval is in SP 0 if and only if its position is divisible by k j ; see (11). The length of this interval is less  9 j 9 than 2 k , giving us upper bound of 2 = 5 elements. Similarly, if j 6= m+1, the jth interval has the length   9 j 9 j−1 and thus contains at least 92 k−1 elements of SP 0 . Since k ≥ 4, the claim follows. 2k − 2k k Next we modify Algorithm MBasic, replacing ttl by ttl0 and SP by SP 0 .
log n Proposition 3.19. Modified Algorithm MBasic finds a palindrome of length `(S) ≥ L(S) 1+ε using O log(1+ε) space.   Proof. Let S[i..j] be a palindrome of length L(S). Let d = logk L(S) . Without loss of generality we assume 4 0 d ≥ 0, as otherwise L(S) < 4 ≤ ttl (i) and the palindrome S[i..j] will be detected exactly. Since L(S) ≥ 4k d , let a1 < a2 < a3 < a4 < a5 be consecutive positions which are multiples of k d (i.e., β 0 (a1 ), . . . , β 0 (a5 ) ≥ d) such that a2 ≤ i+j 2 < a3 . Then in particular i < a1 , and there is a palindrome S[a1 ..(i + j − a1 )] such that a3 ≤ (i + j − a1 ) < a5 . Since a1 + ttl 0 (a1 ) ≥ a5 , this particular palindrome will be detected by the modified Algorithm MBasic; thus `(S) ≥ a3 − a1 = 2k d . However, we have L(S) < 4k d+1 , hence L(S) < 2k ≤ (1 + ε).
 log n `(S) log n 0 . Space complexity follows from bound on size of the list SP , which is at most 5 log k = O log(1+ε) To follow the framework from the case of ε ≤ 1, we provide analogous speedup to the checks for palindromes. We adopt the same notion of an element valuable at ith iteration as in Sect. 3.2. First we need the following property, which is a more general analog of Lemma 3.12; an analog of Lemma 3.13 is then proved with its help. Lemma 3.20. Suppose that at some iteration the list SP 0 contains consecutive elements I(d), I(c), and d ≤ i − answer.len, where i is the number of the current iteration. Further, let I(a) be another element of SP 0 at this iteration and a < c. If c, d belong to the same interval of (12), then I(a) is not valuable. Proof. Let c, d belong to the jth interval. Thus they are divisible by k j and d − c = k j . Since a < c, a is c+a j divisible by k j as well. One of the numbers d+a 2 , 2 is divisible by k ; take it as b. Let δ = b − a. If S[a..i] is a palindrome, then S[b..i − δ] is also a palindrome. Since at the ith iteration the left border of the jth interval was smaller than c, then at the (i − δ)th iteration this border was smaller than b; hence, I(b) was in SP 0 at that iteration, and the palindrome S[b..i − δ] was found. Its length is i − δ − b + 1 = i + 1 + a − 2b ≥ i + 1 + a − (d + a) ≥ i − d + 1 > answer.len, which is impossible by the definition of answer.len. So S[a..i] is not a palindrome, and the claim follows. 13 Lemma 3.21. At each iteration, SP 0 contains at most three valuable elements. Moreover, if I(d0 ), I(d) are consecutive elements of SP 0 and i − d0 < answer.len ≤ i − d, where i is the number of the current iteration, then the valuable elements are consecutive in SP 0 , starting with I(d). Proof. Let a < b < c < d be such that the elements I(d), I(c), I(b) are consecutive in SP 0 and I(a) belongs to SP 0 . Then either b, c or c, d are in the same interval of (12), and thus a is not valuable by Lemma 3.20. To complete the proof, we now turn to deletions, proving the following analog of Lemma 3.14. Lemma 3.22. The function h(x) = x + ttl 0 (x) maps at most two different values of x to the same value. Moreover, if h(x) = h(y) and β 0 (x) ≥ β 0 (y), then β 0 (x) = β 0 (h(x)) + 1 and β 0 (y) = 0. Proof. Let h(x) = h(y). If β 0 (x) = β 0 (y) then ttl0 (x) = ttl0 (y) by (11), implying x = y. Hence all preimages 0 of h(x) have different values of β 0 . Assume β 0 (x) > β 0 (y). Then we have, for some integer j, x = j · k β (x) 0 0 and h(x) = (j + 4)k β (x) + k2 · k β (x)−1 by (11). Since k is even, we get β 0 (h(x)) = β 0 (x) − 1. If β 0 (y) > 0, we repeat the same argument and obtain β 0 (x) = β 0 (y), contradicting our assumption. Thus β 0 (y) = 0. The claim now follows. We also define a list BS 0 (x), encoding of the k-ary representation of x. The
which maintains an RLE n 0 list BS 0 (x) has length O log , can be updated to BS (x+1) in O(1) time, and provides the value β 0 (x) log k in O(1) time also (cf. Lemma 3.15). Further, Lemma 3.16 holds for the function ttl0 , so we introduce the queues QU 0 (x) in the same way as the queues QU (x) in Sect. 3.2. Having all the ingredients, we present Algorithm M’ which speeds up the modified Algorithm MBasic and thus proves Theorem 3.17. The only significant difference between Algorithm M and Algorithm M’ is in the deletion of tuples from the list (compare lines 5–9 of Algorithm M against lines 5–15 of Algorithm M’). Algorithm 5 : Algorithm M’, ith iteration 1: add I to the beginning of SP 0 2: if i = 1 then 3: sp ← f irst(SP 0 ) 4: compute BS 0 [i] from BS 0 ; BS 0 ← BS 0 [i]; compute β(i) from BS 0 5: if QU 0 (β(i) + 1) is not empty then 6: v ← element of SP 0 pointed by f irst(QU 0 (β(i))) 7: if v.i + ttl0 (v.i) = i then 8: if v = sp then 9: sp ← next(sp) 10: delete v; delete f irst(QU 0 (β(i) + 1)) 11: v ← element of SP 0 pointed by f irst(QU 0 (0)) 12: if v.i + ttl0 (v.i) = i then 13: if v = sp then 14: sp ← next(sp) 15: delete v; delete f irst(QU 0 (0)) 16: add pointer to f irst(SP 0 ) to QU 0 (β(i)) 17: read S[i]; compute I(i + 1) from I; I ← I(i + 1) 18: sp ← previous(sp) 19: while i − sp.i + 1 ≤ answer.len and sp 6= last(SP 0 ) do 20: sp ← next(sp) 21: for all existing v in {sp, next(sp), next(next(sp))} do 22: if S[v.i..i] is a palindrome and answer.len < i−v.i+1 then 23: answer ← (v.i, i−v.i+1) 14 . if exists 3.4 The Case of Short Palindromes A typical string contains only short palindromes, as Lemma 3.23 below shows (for more on palindromes in random strings, see [16]). Knowing this, it is quite useful to have a deterministic real-time algorithm which finds a longest palindrome exactly if it is “short”, otherwise reporting that it is “long”. The aim of this section is to describe such an algorithm (Theorem 3.24). Lemma 3.23. If an input stream S ∈ Σ∗ is picked up uniformly at random among all strings of length n, where n ≥ |Σ|, then for any positive constant c the probability that S contains a palindrome of length greater log n is O(n−c ). than 2(c+1) log |Σ| Proof. A string S contains a palindrome of length greater than m if and only if S contains a palindrome of length m+1 or m+2. The probability P of containing such a palindrome is less than the expected number M of palindromes of length m+1 and m+2 in S. A factor of S of length l is a palindrome with probability 1/|Σ|bl/2c ; by linearity of expectation, we have M= Substituting m = 2(c+1) log n , log |Σ| n−m−1 n−m + . b(m+1)/2c |Σ| |Σ|b(m+2)/2c we get M = O(n−c ), as required. Theorem 3.24. Let m be a positive integer. There exists a deterministic real-time algorithm working in O(m) space, which - solves LPS(S) exactly if L(S) < m; - finds a palindrome of length m or m+1 as an approximated solution to LPS(S) if L(S) ≥ m. Proof. To prove Theorem 3.24, we present an algorithm based on the Manacher algorithm [14]. We add two features: work with a sliding window instead of the whole string to satisfy the space requirements and lazy computation to achieve real time. (The fact that the original Manacher algorithm admits a real-time version was shown by Galil [7]; we adjusted Galil’s approach to solve LPS.) The details follow. j−i We say that a palindromic factor S[i..j] has center i+j 2 and radius 2 . Thus, odd-length (even-length) palindromes have integer (resp., half integer) centers and radiuses. This looks a bit weird, but allows one to avoid separate processing of these two types of palindromes. Manacher’s algorithm computes, in an online fashion, an array of maximal radiuses of palindromes centered at every position of the input string S. A variation, which outputs the length L of the longest palindrome in a string S, is presented as Algorithm EBasic below. This variation is similar to the one of [13]. Here, n stays for the length of the input processed so far, c is the center of the longest suffix-palindrome of the processed string. The array of radiuses Rad has length 2n−1 and its elements are indexed by all integers and half integers from the interval [1, n]. Initially, Rad is filled with zeroes. The left endmarker is added to the string for convenience. After each iteration, the following invariant holds: the element Rad[i] has got its true value if i < c and equals zero if i > c; the value Rad[c] = n − c can increase at the next iteration. Note that the longest palindrome in S coincides with the longest suffix-palindrome of S[1..i] for some i. At the moment when the input stream ends, the algorithm has already found all such suffix-palindromes, so it can stop without filling the rest of the array Rad. Note that n calls to AddLetter perform at most 3n iterations of the cycle in lines 10–15 (each call performs the first iteration plus zero or more “additional” iterations; the value of c gets increased before each additional iteration and never decreases). So, Algorithm EBasic works in O(n) time but not in real time; for example, reading the last letter of the string an b requires n iterations of the cycle. By conditions of the theorem, we are not interested in palindromes of length > m+1. Thus, processing a suffix-palindrome of length m or m+1 we assume that the symbol comparison in line 12 fails. So the procedure AddLetter needs no access to S[i] or Rad[i] whenever i < n − m. Hence we store only recent values of S and Rad and use circular arrays CS and CRad of size O(m) for this purpose. For example, the symbol S[n−i] is stored in CS[(n−i) mod (m+1)] during m+1 successful iterations of the outer cycle (lines 3–6), and then is replaced by S[n−i+m+1]; the same scheme applies to the array Rad. In this way, all elements of S and Rad, needed by Algorithm EBasic, are accessible in constant time. Further, we define 15 Algorithm 6 : Algorithm EBasic, ith iteration 1: procedure Manacher 2: c ← 1; L ← 1; n ← 1; S[0] ←’#’ 3: while not (end of input) do 4: read(S[n + 1]); AddLetter(S[n + 1]) 5: if 2 ∗ Rad[c] + 1 > L then 6: L ← 2 ∗ Rad[c] + 1 7: return L 8: procedure AddLetter(a) 9: s←c 10: while c < n + 1 do 11: Rad[c] ← min(Rad[2 ∗ s − c], n − c) 12: if c + Rad[c] = n and S[c − Rad[c] − 1] = a then 13: Rad[c] ← Rad[c] + 1 14: break 15: c ← c + 0.5 16: n←n+1 . longest suf-pal of S[1..n + 1] is found . next candidate for the center a queue Q of size q for lazy computations; it contains symbols that are read from the input and await processing. Now we describe real-time Algorithm E. It reads input symbols to Q and stops when the end of the input is reached. After reading a symbol, it runs procedure Manacher, requiring the latter to pause after three “inner iterations” (of the cycle in lines 10–15). The procedure reads symbols from Q; if it tries to read from the empty queue, it pauses. When the procedure is called next time, it resumes from the moment it was stopped. Algorithm E returns the value of L after the last call to procedure Manacher. To analyze Algorithm E, consider the value X = q + n − c after some iteration (clearly, this iteration has number q+n) and look at the evolution of X over time. Let ∆f denote the variation of the quantity f at one iteration. Note that ∆(q+n) = 1. Let us describe ∆X. First assume that the iteration contains three inner iterations. Then ∆n = 0, 1, 2 or 3 and, respectively, ∆c = 1.5, 1, 0.5 or 0. Hence ∆X = 1 − ∆c = 1 + (∆n − 3)/2 = 1 − (1 − ∆q − 3)/2 = −(∆q)/2. If the number of inner iterations is one or two, then q becomes zero (and was 0 or 1 before this iteration); hence ∆n = 1 − ∆q ≥ 1. Then ∆c ≤ 0.5 and finally ∆X > 0. From these conditions on ∆X it follows that (∗) if the current value of q is positive, then the current value of X is less than the value of X at the moment when q was zero for the last time. Let X 0 be the previous value of X mentioned in (∗). Since the difference n − c does not exceed the radius of some palindrome, X 0 ≤ m/2. Since q ≤ X < X 0 , the queue Q uses O(m) space. Therefore the same space bound applies to Algorithm E. It remains to prove that Algorithm E returns the same number L as Algorithm EBasic with a sliding window, in spite of the fact that Algorithm E stops earlier (the content of Q remains unprocessed by procedure Manacher). Suppose that Algorithm E stops with q > 0 after n iterations. Then the longest palindrome that could be found by processing the symbols in Q has the radius X = n + q − c. Now consider the iteration mentioned in (∗) and let n0 and c0 be the values of n and c after it; so X 0 = n0 −c0 . Since q was zero after that iteration, procedure Manacher read the symbol S[n0 ] during it; hence, it tried to extend a suffix-palindrome of S[1..n0 −1] with the center c00 ≤ c0 . If this extension was successful, then a palindrome of radius at least X 0 was found. If it was unsuccessful, then c0 ≥ c00 + 1/2 and hence S[1..n0 −1] has a suffix-palindrome of length at least X 0 − 1/2. Thus, a palindrome of length X ≤ X 0 − 1/2 is not longer than a longest palindrome seen before, and processing the queue cannot change the value of L. Thus, Algorithm E is correct. 16 The center and the radius of the longest palindrome in S can be updated each time the inequality in line 6 of procedure Manacher holds. Theorem 3.24 is proved. Remark Lemma 3.23 and Theorem 3.24 show a practical way to solve LPS. Algorithm E is fast and lightweight (2m machine words for the array Rad, m symbols in the sliding window and at most m symbols in the queue; compare to 17 machine words per one tuple I(i) in the Monte Carlo algorithms). So it makes direct sense to run Algorithm M and Algorithm E, both in O(log n) space, in parallel. Then either Algorithm E will give an exact answer (which happens with high probability if the input stream is a “typical” string) or both algorithms will produce approximations: one of fixed length and one with an approximation guarantee (modulo the hash collision). References [1] A. Apostolico, D. Breslauer, and Z. Galil. Parallel detection of all palindromes in a string. Theoret. Comput. Sci., 141:163–173, 1995. [2] P. Berenbrink, F. Ergün, F. Mallmann-Trenn, and E. Sadeqi Azer. Palindrome recognition in the streaming model. In STACS 2014, volume 25 of LIPIcs, pages 149–161, Germany, 2014. Schloss DagstuhlLeibniz-Zentrum fuer Informatik, Dagstuhl Publishing. [3] D. Breslauer and Z. Galil. Real-time streaming string-matching. In Combinatorial Pattern Matching, volume 6661 of LNCS, pages 162–172, Berlin, 2011. Springer. [4] R. Clifford, A. Fontaine, E. Porat, B. Sach, and T. A. Starikovskaya. Dictionary matching in a stream. In ESA 2015, volume 9294 of LNCS, pages 361–372. Springer, 2015. [5] R. Clifford, A. Fontaine, E. Porat, B. Sach, and T. A. Starikovskaya. The k -mismatch problem revisited. In SODA 2016, pages 2039–2052. SIAM, 2016. [6] F. Ergün, H. Jowhari, and M. Saglam. Periodicity in streams. In RANDOM 2010, volume 6302 of LNCS, pages 545–559. Springer, 2010. [7] Z. Galil. Real-time algorithms for string-matching and palindrome recognition. In Proc. 8th annual ACM symposium on Theory of computing (STOC’76), pages 161–173, New York, USA, 1976. ACM. [8] Z. Galil and J. Seiferas. A linear-time on-line recognition algorithm for “Palstar”. J. ACM, 25:102–111, 1978. [9] P. Gawrychowski, F. Manea, and D. Nowotka. Testing generalised freeness of words. In STACS 2014, volume 25 of LIPIcs, pages 337–349. Dagstuhl Publishing, 2014. [10] M. Jalsenius, B. Porat, and B. Sach. Parameterized matching in the streaming model. In STACS 2013, volume 20 of LIPIcs, pages 400–411. Dagstuhl Publishing, 2013. [11] R. Karp and M. Rabin. Efficient randomized pattern matching algorithms. IBM Journal of Research and Development, 31:249–260, 1987. [12] D. E. Knuth, J. Morris, and V. Pratt. Fast pattern matching in strings. SIAM J. Comput., 6:323–350, 1977. [13] D. Kosolobov, M. Rubinchik, and A. M. Shur. Finding distinct subpalindromes online. In Proc. Prague Stringology Conference. PSC 2013, pages 63–69. Czech Technical University in Prague, 2013. [14] G. Manacher. A new linear-time on-line algorithm finding the smallest initial palindrome of a string. J. ACM, 22(3):346–351, 1975. 17 [15] B. Porat and E. Porat. Exact and approximate pattern matching in the streaming model. In FOCS 2009, pages 315–323. IEEE Computer Society, 2009. [16] M. Rubinchik and A. M. Shur. The number of distinct subpalindromes in random words. Fundamenta Informaticae, 145:371–384, 2016. [17] A. C.-C. Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In FOCS 1977, pages 222–227. IEEE Computer Society, 1977. 18 

On Hierarchical Communication Topologies in the π-calculus Emanuele D’Osualdo1 and C.-H. Luke Ong2 arXiv:1601.01725v2 [] 19 Apr 2016 1 TU Kaiserslautern dosualdo@cs.uni-kl.de 2 University of Oxford lo@cs.ox.ac.uk Abstract. This paper is concerned with the shape invariants satisfied by the communication topology of π-terms, and the automatic inference of these invariants. A π-term P is hierarchical if there is a finite forest T such that the communication topology of every term reachable from P satisfies a T-shaped invariant. We design a static analysis to prove a term hierarchical by means of a novel type system that enjoys decidable inference. The soundness proof of the type system employs a non-standard view of π-calculus reactions. The coverability problem for hierarchical terms is decidable. This is proved by showing that every hierarchical term is depth-bounded, an undecidable property known in the literature. We thus obtain an expressive static fragment of the π-calculus with decidable safety verification problems. 1 Introduction Concurrency is pervasive in computing. A standard approach is to organise concurrent software systems as a dynamic collection of processes that communicate by message passing. Because processes may be destroyed or created, the number of processes in the system changes in the course of the computation, and may be unbounded. Moreover the messages that are exchanged may contain process addresses. Consequently the communication topology of the system—the hypergraph [19, 18] connecting processes that can communicate directly—evolves over time. In particular, the connectivity of a process (i.e. its neighbourhood in this hypergraph) can change dynamically. The design and analysis of these systems is difficult: the dynamic reconfigurability alone renders verification problems undecidable. This paper is concerned with hierarchical systems, a new subclass of concurrent message-passing systems that enjoys decidability of safety verification problems, thanks to a shape constraint on the communication topology. The π-calculus of Milner, Parrow and Walker [19] is a process calculus designed to model systems with a dynamic communication topology. In the π-calculus, processes can be spawned dynamically, and they communicate by exchanging messages along synchronous channels. Furthermore channel names can themselves be created dynamically, and passed as messages, a salient feature known as mobility, as this enables processes to modify their neighbourhood at runtime. It is well known that the π-calculus is a Turing-complete model of computation. Verification problems on π-terms are therefore undecidable in general. There are 2 E. D’Osualdo and C.-H. L. Ong however useful fragments of the calculus that support automatic verification. The most expressive such fragment known to date is the depth-bounded π-calculus of Meyer [12]. Depth boundedness is a constraint on the shape of communication topologies. A π-term is depth-bounded if there is a number k such that every simple path3 in the communication topology of every reachable π-term has length bounded by k. Meyer [14] proved that termination and coverability (a class of safety properties) are decidable for depth-bounded terms. Unfortunately depth boundedness itself is an undecidable property [14], which is a serious impediment to the practical application of the depth-bounded fragment to verification. This paper offers a two-step approach to this problem. First we identify a (still undecidable) subclass of depth-bounded systems, called hierarchical, by a shape constraint on communication topologies (as opposed to numeric, as in the case of depth-boundedness). Secondly, by exploiting this richer structure, we define a type system, which in turn gives a static characterisation of an expressive and practically relevant fragment of the depth-bounded π-calculus. Example 1 (Client-server pattern). To illustrate our approach, consider a simple system implementing a client-server pattern. A server S is a process listening on a channel s which acts as its address. A client C knows the address of a server and has a private channel c that represents its identity. When the client wants to communicate with the server, it asynchronously sends c along the channel s. Upon receipt of the message, the server acquires knowledge of (the address of) the requesting client; and spawns a process A to answer the client’s request R asynchronously; the answer consists of a new piece of data, represented by a new name d, sent along the channel c. Then the server forgets the identity of the client and reverts to listening for new requests. Since only the requesting client knows c at this point, the server’s answer can only be received by the correct client. Figure 1a shows the communication topology of a server and a client, in the three phases of the protocol. The overall system is composed of an unbounded number of servers and clients, constructed according to the above protocol. The topology of a reachable configuration is depicted in Fig. 1b. While in general the topology of a mobile system can become arbitrarily complex, for such common patterns as client-server, the programmer often has a clear idea of the desired shape of the communication topology: there will be a number of servers, each with its cluster of clients; each client may in turn be waiting to receive a number of private replies. This suggests a hierarchical relationship between the names representing servers, clients and data, although the communication topology itself does not form a tree. T-compatibility and hierarchical terms Recall that in the π-calculus there is an important relation between terms, ≡, called structural congruence, which equates terms that differ only in irrelevant presentation details, but not in behaviour. For instance, the structural congruence laws for restriction tell us that the order of restrictions is irrelevant—νx.νy.P ≡ νy.νx.P —and that the scope 3 a simple path is a path with no repeating edges. On Hierarchical Communication Topologies in the π-calculus S s C d c s C cc c cc c c C C C R → S s S C c d d → S 3 A (a) the protocol d c C C C s A c ··· C d ··· d C C C R R R s A A A A C d ··· d A A c cc (b) a reachable configuration (c) forest representation Fig. 1. Evolution of the communication topology of a server interacting with a client. R represents a client’s pending request and A a server’s pending answer. of a restriction can be extended to processes that do not refer to the restricted name—i.e., (νx.P ) k Q ≡ νx.(P k Q) when x does not occur free in Q—without altering the meaning of the term. The former law is called exchange, the latter is called scope extrusion. Our first contribution is a formalisation in the π-calculus of the intuitive notion of hierarchy illustrated in Example 1. We shall often speak of the forest representation of a π-term P , forest(P ), which is a version of the abstract syntax tree of P that captures the nesting relationship between the active restrictions of the term. (A restriction of a π-term is active if it is not in the scope of a prefix.) Thus the internal nodes of a forest representation are labelled with (active) restriction names, and its leaf nodes are labelled with the sequential subterms. Given a π-term P , we are interested in not just forest(P ), but also forest(P 0 ) where P 0 ranges over the structural congruents of P , because these are all behaviourally equivalent representations. See Fig. 4 for an example of the respective forest representations of the structural congruents of a term. In our setting a hierarchy T is a finite forest of what we call base types. Given a finite forest T , we say that a term P is T-compatible if there is a term P 0 , which is structurally congruent to P , such that the parent relation of forest(P 0 ) is consistent with the partial order of T . In Example 1 we would introduce base types srv, cl and data associated with the restrictions νs, νc and νd respectively, and we would like the system to be compatible to the hierarchy T = srv  cl  data, where  is the is-parent-of relation. That is, we must be able to represent a configuration with a forest that, for instance, does not place a server name below a client name nor a client name below another client name. Such a representation is shown in Fig. 1c. In the Example, we want every reachable configuration of the system to be compatible with the hierarchy. We say that a π-term P is hierarchical if there is a hierarchy T such that every term reachable from P is T-compatible. Thus the hierarchy T is a shape invariant of the communication topology under reduction. 4 E. D’Osualdo and C.-H. L. Ong a b S a b ≡ R S a b → R S0 R0 Fig. 2. Standard view of π-calculus reactions It is instructive to express depth boundedness as a constraint on forest representation: a term P is depth-bounded if there is a constant k such that every term reachable from P has a structurally congruent P 0 whereby forest(P 0 ) has height bounded by k. It is straightforward to see that hierarchical terms are depth-bounded; the converse is however not true. A type system for hierarchical terms While membership of the hierarchical fragment is undecidable, by exploiting the forest structure, we have devised a novel type system that guarantees the invariance of T-compatibility under reduction. Furthermore type inference is decidable, so that the type system can be used to infer a hierarchy T with respect to which the input term is hierarchical. To the best of our knowledge, our type system is the first that can infer a shape invariant of the communication topology of a system. The typing rules that ensure invariance of T-compatibility under reduction arise from a new perspective of the π-calculus reaction, one that allows compatibility to a given hierarchy to be tracked more readily. Suppose we are presented with a T-compatible term P = C[S, R] where C[-, -] is the reaction context, and the two processes S = ahbi.S 0 and R = a(x).R0 are ready to communicate over a channel a. After sending the message b, S continues as the process S 0 , while upon receipt of b, R binds x to b and continues as R00 = R0 [ b/x ]. Schematically, the traditional understanding of this transaction is: first extrude the scope of b to include R, then let them react, as shown in Fig. 2. Instead, we seek to implement the reaction without scope extrusion: after the message is transmitted, the sender continues in-place as S 0 , while R00 is split in 0 0 two parts Rmig k R¬mig , one that uses the message (the migratable part) and one 0 that does not. As shown in Fig. 3, the migratable part of R00 , Rmig , is “installed” under b so that it can make use of the acquired name, while the non-migratable 0 one, R¬mig , can simply continue in-place. Crucially, the reaction context, C[-, -], is left unchanged. This means that if the starting term is T-compatible, the reaction context of the reactum is T-compatible as well. We can then focus on imposing constraints on the use of names of R0 0 so that the migration does not result in Rmig escaping the scope of previously bound names. By using these ideas, our type system is able to statically accept π-calculus encodings of such system as that discussed in Example 1. The type system can be used, not just to check that a given T is respected by the behaviour of a term, but also to infer a suitable T when it exists. Once typability of a term On Hierarchical Communication Topologies in the π-calculus a a → b S 5 R b R0 S 0 mig R0 ¬mig Fig. 3. T-compatibility preserving reaction is established, safety properties such as unreachability of error states, mutual exclusion or bounds on mailboxes, can be verified algorithmically. For instance, in Example 1, a coverability check can prove that each client can have at most one reply pending in its mailbox. To prove such a property, one needs to construct an argument that reasons about dynamically created names with a high degree of precision. This is something that counter abstraction and uniform abstractions based methods have great difficulty attaining. Our type system is (necessarily) incomplete in that there are depth-bounded, or even hierarchical, systems that cannot be typed. The class of π-terms that can be typed is non-trivial, and includes terms which generate an unbounded number of names and exhibit mobility. Outline. In Section 2 we review the π-calculus, depth-bounded terms, and related technical preliminaries. In Section 3 we introduce T-compatibility and the hierarchical terms. We present our type system in Section 4. Section 5 discusses soundness of the type system. In Section 6 we give a type inference algorithm; and in Section 7 we present results on expressivity and discuss applications. We conclude with related and future work in Sections 8 and 9. All missing definitions and proofs can be found in Appendix. 2 2.1 The π-calculus and the depth-bounded fragment Syntax and semantics We use a π-calculus with guarded replication to express recursion [16]. Fix a universe N of names representing channels and messages. The syntax is defined by the grammar: P 3 P, Q ::= 0 | νx.P | P1 k P2 | M | !M M ::= M + M | π.P π ::= a(x) | ahbi | τ process choice prefix Definition 1. Structural congruence, ≡, is the least relation that respects αconversion of bound names, and is associative and commutative with respect to + 6 E. D’Osualdo and C.-H. L. Ong (choice) and k (parallel composition) with 0 as the neutral element, and satisfies laws for restriction: νa.0 ≡ 0 and νa.νb.P ≡ νb.νa.P , and !P ≡ P k !P P k νa.Q ≡ νa.(P k Q) Replication (if a 6∈ fn(P )) Scope Extrusion In P = π.Q, we call Q the continuation of P and will often omit Q altogether when Q = 0. In a term νx.P we will occasionally refer to P as the scope of x. The name x is bound in both νx.P , and in a(x).P . We will write fn(P ), bn(P ) and bnν (P ) for the set of free, bound and restriction-bound names in P , respectively. A sub-term is active if it is not under a prefix. A name is active when it is bound by an active restriction. We write actν (P ) for the set of active names of P . Terms of the form M and !M are called sequential. We write S for the set of sequential terms, actS (P ) for the set of active sequential processes of P , and P i for the parallel composition of i copies of P . Intuitively, a sequential process acts like a thread running finite-control sequential code. A term τ .(P k Q) is the equivalent of spawning a process Q and continuing as P —although in this context the rôles of P and Q are interchangeable. Interaction is by synchronous communication over channels. An input prefix a(x) is a blocking receive on the channel a binding the variable x to the message. An output prefix ahbi is a blocking send of the message b along the channel a; here b is itself the name of a channel that can be used subsequently for further communication: an essential feature for mobility. A non-blocking send can be simulated by spawning a new process doing a blocking send. Restrictions are used to make a channel name private. A replication !(π.P ) can be understood as having a server that can spawn a new copy of P whenever a process tries to communicate with it. In other words it behaves like an infinite parallel composition (π.P k π.P k · · · ). For conciseness, we assume channels are unary (the extension to the polyadic case is straightforward). In contrast to process calculi without mobility, replication and systems of tail recursive equations are equivalent methods of defining recursive processes in the π-calculus [17, Section 3.1]. We rely on the following mild assumption, that the choice of names is unambiguous, especially when selecting a representative for a congruence class: Name Uniqueness Assumption. Each name in P is bound at most once and fn(P ) ∩ bn(P ) = ∅. Normal Form. The notion of hierarchy, which is central to this paper, and the associated type system depend heavily on structural congruence. These are criteria that, given a structure on names, require the existence of a specific representative of the structural congruence class exhibiting certain properties. However, we cannot assume the input term is presented as that representative; even worse, when the structure on names is not fixed (for example, when inferring types) we cannot fix a representative and be sure that it will witness the desired properties. Thus, in both the semantics and the type system, we manipulate On Hierarchical Communication Topologies in the π-calculus 7 a neutral type of representative called normal form, which is a variant of the standard form [19]. In this way we are not distracted by the particular syntactic representation we are presented with. We say that a term P is in normal form (P ∈ Pnf ) if it is in standard form and each of its inactive subterms is also in normal form. Formally, normal forms are defined by the grammar Pnf 3 N ::= νx1 . · · · νxn .(A1 k · · · k Am ) A ::= π1 .N1 + · · · + πn .Nn | !(π1 .N1 + · · · + πn .Nn ) where the sequences x1 . . . xn and A1 . . . Am may be empty; when they are both empty the normal form is the term 0. We further assume w.l.o.g. that normal forms Q satisfy Name Uniqueness. Given a finite set of indexes I = {i1 , P . . . , in } we write i∈I Ai for (Ai1 k · · · k Ain ), which is 0 when I is empty; and i∈I πi .Ni for (πi1 .Ni1 + · · · + πin .Nin ). This notation is justified by commutativity and associativity of the parallel and choice operators. Thanks to the structural laws of restriction, we also write νX.P where X = {x1 , . . . , xn }, or νx1 x2 · · · xn .P , for νx1 . · · · νxn .P ; or just P when X is empty. When X and Y are disjoint sets of names, we use juxtaposition for union. Every process P ∈ P is structurally congruent to a process in normal form. The function nf : P → Pnf , defined in Appendix, extracts, from a term, a structurally congruent normal form. Q Given a process P with normal form νX. i∈I Ai , the communication topology 4 of P , written GJP K, is defined as the labelled hypergraph with X as hyperedges and I as nodes, each labelled with the corresponding Ai . An hyperedge x ∈ X is connected with i just if x ∈ fn(Ai ). Semantics. We are interested in the reduction semantics of a π-term, which can be described using the following rule. Definition 2 (Semantics of π-calculus). The operational semantics of a term P0 ∈ P is defined by the (pointed) transition system (P, →, P0 ) on π-terms, where P0 is the initial term, and the transition relation, → ⊆ P 2 , is defined by P → Q if either (i) to (iv) hold, or (v) and (vi) hold, where (i) (ii) (iii) (iv) P ≡ νW.(S k R k C) ∈ Pnf , S = (ahbi.νYs .S 0 ) + Ms , R = (a(x).νYr .R0 ) + Mr , Q ≡ νW Ys Yr .(S 0 k R0 [ b/x ] k C), (v) P ≡ νW.(τ .νY.P 0 k C) ∈ Pnf , (vi) Q ≡ νW Y.(P 0 k C). We define the set of reachable terms from P as Reach(P ) := { Q | P →∗ Q }, writing →∗ to mean the reflexive, transitive closure of →. We refer to the restrictions, νYs , νYr and νY , as the restrictions activated by the transition P → Q. Notice that the use of structural congruence in the definition of → takes unfolding replication into account. 4 This definition arises from the “flow graphs” of [19]; see e.g. [14, p. 175] for a formal definition. 8 E. D’Osualdo and C.-H. L. Ong Example 2 (Client-server). We can model a variation of the client-server pattern sketched in the introduction, with the term νs c.P where P = !S k !C k !M , S = s(x).νd.xhdi, C = c(m).(shmi k m(y).chmi) and M = τ .νm.chmi. The term !S represents a server listening to a port s for a client’s requests. A request is a channel x that the client sends to the server for exchanging the response. After receiving x the server creates a new name d and sends it over x. The term !M creates unboundedly many clients, each with its own private mailbox m. A client on a mailbox m repeatedly sends requests to the server and concurrently waits for the answer on the mailbox before recursing. In the following examples, we use CCS-style nullary channels, which can be understood as a shorthand: c.P := c(x).P and c.P := νx.chxi.P where x 6∈ fn(P ). Example 3 (Resettable counter). A counter with reset is a process reacting to messages on three channels inc, dec and rst. An inc message increases the value of the counter, a dec message decreases it or causes a deadlock if the counter is zero, and a rst message resets the counter to zero. This behaviour is exhibited

by the process Ci = ! pi (t). inc i .(t k pi hti) + dec i .(t.pi hti) + rst i .(νt0i .pi ht0i i) . Here, the number of processes t in parallel with pi hti represents the current
value of the counter i. A system νp1 t1 .(C1 k p1 ht1 i) k νp2 t2 .(C2 k p2 ht2 i) can for instance simulate a two-counter machine when put in parallel with a finite control process sending signals along the channels inc i , dec i and rst i . Example 4 (Unbounded ring). Let R
= νm.νs0 .(M k mhs0 i k s0 ), S = !(s.n) and M = ! m(n).s0 .νs.(S k mhsi k s) . The term R implements an unboundedly growing ring. It initialises the ring with a single “master” node pointing at itself (s0 ) as the next in the ring. The term M , implementing the master node’s behaviour, waits on s0 and reacts to a signal by creating a new slave with address s connected with the previous next slave n. A slave S simply propagates the signals on its channel to the next in the ring. 2.2 Forest representation of terms In the technical developement of our ideas, we will manipulate the structure of terms in non-trivial ways. When reasoning about these manipulations, a term is best viewed as a forest representing (the relevant part of) its abstract syntax tree. Since we only aim to capture the active portion of the term, the active sequential subterms are the leaves of its forest view. Parallel composition corresponds to (unordered) branching, and names introduced by restriction are represented by internal (non-leaf) nodes. A forest is a simple, acyclic, directed graph, f = (Nf , f ), where the edge relation n1 f n2 means “n1 is the parent of n2 ”. We write ≤f and <f for the reflexive transitive and the transitive closure of f respectively. A path is a sequence of nodes, n1 . . . nk , such that for each i < k, ni f ni+1 . Henceforth we drop the subscript f from f , ≤f and <f (as there is no risk of confusion), and assume that all forests are finite. Thus every node has a unique path to a root (and that root is unique). On Hierarchical Communication Topologies in the π-calculus 1 2 a 3 c a b c A1 A1 A2 A3 A4 b A3 A2 A4 a 4 A1 b A1 A2 c A4 a c b A3 A2 A4 5 A2 9 b c a A3 A1 A4 A3 Fig. 4. Examples of forests in FJP K where P = νa b c.(A1 k A2 k A3 k A4 ), A1 = a(x), A2 = b(x), A3 = c(x) and A4 = ahbi. An L-labelled forest is a pair ϕ = (fϕ , `ϕ ) where fϕ is a forest and `ϕ : Nϕ → L is a labelling function on nodes. Given a path n1 . . . nk of fϕ , its trace is the induced sequence `ϕ (n1 ) . . . `ϕ (nk ). By abuse of language, a trace is an element of L∗ which is the trace of some path in the forest. We define L-labelled forests inductively from the empty forest (∅, ∅). We write ϕ1 ] ϕ2 for the disjoint union of forests ϕ1 and ϕ2 , and l[ϕ] for the forest with a single root, which is labelled with l ∈ L, and whose children are the respective roots of the forest ϕ. Since the choice of the set of nodes is irrelevant, we will always interpret equality between forests up to isomorphism (i.e. a bijection on nodes respecting parent and labeling). Definition 3 (Forest representation). We represent the structural congruence class of a term P ∈ P with the set of labelled forests FJP K := {forest(Q) | Q ≡ P } with labels in actν (P ) ] actS (P ) where forest(Q) is defined as  (∅, ∅)    Q[(∅, ∅)] forest(Q) :=  x[forest(Q0 )]    forest(Q1 ) ] forest(Q2 ) if if if if Q=0 Q is sequential Q = νx.Q0 Q = Q1 k Q2 Note that leaves (and only leaves) are labelled with sequential processes. The restriction height, heightν (forest(P )), is the length of the longest path formed of nodes labelled with names in forest(P ). In Fig. 4 we show some of the possible forest representations of an example term. 2.3 Depth-bounded terms Definition 4 (Depth-bounded term [12]). The nesting of restrictions of a term is given by the function nestν (M ) := nestν (!M ) := nestν (0) := 0 nestν (νx.P ) := 1 + nestν (P ) nestν (P k Q) := max(nestν (P ), nestν (Q)). 10 E. D’Osualdo and C.-H. L. Ong The depth of a term is defined as the minimal nesting of restrictions in its congruence class, depth(P ) := min {nestν (Q) | P ≡ Q}. A term P ∈ P is depthbounded if there exists k ∈ N such that for each Q ∈ Reach(P ), depth(Q) ≤ k. We write Pdb for the set of terms with bounded depth. Notice that nestν is not an invariant of structural congruence, whereas depth and depth-boundedness are. Example 5. Consider the congruent terms P and Q

P = νa.νb.νc. a(x) k bhci k c(y) ≡ νa.a(x) k νc. (νb.bhci) k c(y) = Q We have nestν (P ) = 3 and nestν (Q) = 2; but depth(P ) = depth(Q) = 2. It is straightforward to see that the nesting of restrictions of a term coincides with the height of its forest representation, i.e., for every P ∈ P, nestν (P ) = heightν (forest(P )). Example 6 (Depth-bounded term). The term in Example 2 is depth-bounded: all the reachable terms are congruent to terms of the form
Qijk = νs c. P k N i k Req j k Ans k for some i, j, k ∈ N where N = νm.chmi, Req = νm.(shmi k m(y).chmi) and Ans = νm.(νd.mhdi k m(y).chmi). For any i, j, k, nestν (Qijk ) ≤ 4. Example 7 (Depth-unbounded term). Consider the term in Example 4 and the following run: R →∗ νm s0 .(M k νs1 .(!(s1 .s0 ) k mhs1 i k s1 )) →∗ νm s0 .(M k νs1 .(!(s1 .s0 ) k νs2 .(!(s2 .s1 ) k mhs2 i k s2 ))) →∗ . . . The scopes of s0 , s1 , s2 and the rest of the instantiations of νs are inextricably nested, thus R has unbounded depth: for each n ≥ 1, a term with depth n is reachable. Depth boundedness is a semantic notion. Because the definition is a universal quantification over reachable terms, analysis of depth boundedness is difficult. Indeed the membership problem is undecidable [14]. In the communication topology interpretation, depth has a tight relationship with the maximum length of the simple paths. A path v1 e1 v2 . . . vn en vn+1 in GJP K is simple if it does not repeat hyper-edges, i.e., ei = 6 ej for all i 6= j. A term is depth-bounded if and only if there exists a bound on the length of the simple paths of the communication topology of each reachable term [12]. This allows terms to grow unboundedly in breadth, i.e., the degree of hyper-edges in the communication topology. Q A term P is Q embeddable in a term Q, written P  Q, if P ≡ νX. i∈I Ai ∈ Pnf and Q ≡ νXY.( i∈I Ai k R) ∈ Pnf for some term R. In [12] the term embedding ordering, , is shown to be both a simulation relation on π-terms, and an effective well-quasi ordering on depth-bounded terms. This makes the transition system (Reach(P )/≡ , →/≡ , P ) a well-structured transition system (WSTS) [7, 1] under the term embedding ordering. Consequently a number of verification problems are decidable for terms in Pdb . On Hierarchical Communication Topologies in the π-calculus 11 Theorem 1 (Decidability of termination [12]). The termination problem for depth-bounded terms, which asks, given a term P0 ∈ Pdb , if there is an infinite sequence P0 → P1 → . . ., is decidable. Theorem 2 (Decidability of coverability [12, 24]). The coverability problem for depth-bounded terms, which asks, given a term P ∈ Pdb and a query Q ∈ P, if there exists P 0 ∈ Reach(P ) such that Q  P 0 , is decidable. 3 T-compatibility and hierarchical terms A hierarchy is specified by a finite forest (T , ). In order to formally relate active restrictions in a term to nodes of the hierarchy T , we annotate restrictions with types. For the moment we view types abstractly as elements of a set T, equipped with a map base : T → T . An annotated restriction ν(x : τ ) where τ ∈ T will be associated with the node base(τ ) in the hierarchy T . Elements of T are called types, and those of T are called base types. In the simplest case and, especially for Section 3, we may assume T = T and base(t) = t. In Section 4 we will consider a set T of types generated from T , and a non-trivial base map. Definition 5 (Annotated term). A T-annotated π-term (or simply annotated π-term) P ∈ P T has the same syntax as ordinary π-terms except that restrictions take the form ν(x : τ ) where τ ∈ T. In the abbreviated form νX, X is a set of annotated names (x : τ ). Structural congruence, ≡, of annotated terms, is defined by Definition 1, with the proviso that the type annotations
are invariant under α-conversion
and replication. For example, ! π.ν(x : τ ).P ≡ π.ν(x : τ ).P k ! π.ν(x : τ ).P and ν(x : τ ).P ≡ ν(y : τ ).P [ y/x ]; observe that the annotated restrictions that occur in a replication unfolding are necessarily inactive. The forest representation of an annotated π-term is obtained from Definition 3 by replacing the case of Q = ν(x : τ ).Q0 by forest(ν(x : τ ).Q0 ) := (x, t)[forest(Q0 )] where base(τ ) = t. Thus the forests in FJP K have labels in (actν (P )×T )]actS (P ). We write FT for the set of forests with labels in (N × T ) ] S. We write PnfT for the set of T-annotated π-terms in normal form. The definition of the transition relation of annotated terms, P → Q, is obtained from Definition 2, where W, Ys , Yr and Y are now sets of annotated names, by replacing clauses (iv) and (vi) by (iv’) Q ≡ νW Ys0 Yr0 .(S 0 k R0 [ b/x ] k C) (vi’) Q ≡ νW Y 0 .(P 0 k C) respectively, such that Ys  N = Ys0  N , Yr  N = Yr0  N , and Y  N = Y  N , where X  N := {x ∈ N | ∃τ.(x : τ ) ∈ X}. I.e. the type annotation of the names that are activated by the transition (i.e. those from Ys , Yr and Y ) are not required to be preserved in Q. (By contrast, the annotation of every active restriction in P is preserved by the transition.) While in this context inactive annotations can 12 E. D’Osualdo and C.-H. L. Ong be ignored by the transitions, they will be used by the type system in Section 4, to establish invariance of T-compatible. Now we are ready to explain what it means for an annotated term P to be T-compatible: there is a forest in FJP K such that every trace of it projects to a chain in the partial order T . Definition 6 (T-compatibility). Let P ∈ P T be an annotated π-term. A forest ϕ ∈ FJP K is T-compatible if for every trace ((x1 , t1 ) . . . (xk , tk ) A) in ϕ it holds that t1 < t2 < . . . < tk . The π-term P is T-compatible if FJP K contains a T-compatible forest. A term is T-shaped if each of its subterms is T-compatible. As a property of annotated terms, T-compatibility is by definition invariant under structural congruence. A term P 0 ∈ P T is a type annotation (or simply annotation) of P ∈ P if its type-erasure, written pP q, coincides with P . (We omit the obvious definition of type-erasure.) A consistent annotation of a transition of terms, P → Q, is a choice function that, given an annotation P 0 of P , returns an annotation Q0 of Q such that P 0 → Q0 . Note that it follows from the definition that the annotation of every active restriction in P 0 is preserved in Q0 . The effect of the choice function is therefore to pick a possibly new annotation for each restriction in Q0 that is activated by the transition. Thus, given a semantics (P, →, P ) of a term P , and an annotation P 0 of P , and a consistent annotation for every transition of the semantics, there is a well-defined pointed transition system (P T , →0 , P 0 ) such that every transition sequence of the former lifts to a transition sequence of the latter. We call (P T , →0 , P 0 ) a consistent annotation of the semantics (P, →, P ). Definition 7 (Hierarchical term). A term P ∈ P is hierarchical if there exist a finite forest T = T and a consistent annotation (P T , →0 , P 0 ) of the semantics (P, →, P ) of P , such that all terms reachable from P 0 are T-compatible. Example 8. The term in Examples 2 and 6 is hierarchical: take the hierarchy T = s  c  m  d and annotate each name in Qijk as follows: s : s, c : c, m : m and d : d. The annotation is consistent, and forest(Qijk ) is T-compatible for all i,j and k. Example 4 gives an example of a term that is not hierarchical. The forest representation of the reachable terms shown in Example 7 does not have a bounded height, which means that if T has n base types, there is a reachable term with a representation of height bigger than n, which implies that there will be a path repeating a base type. Let us now study this fragment. First it is easy to see that invariance of T-compatibility under reduction →, for some finite T , puts a bound |T | on the height of the T-compatible reachable forests, and consequently a bound on depth. Theorem 3. Every hierarchical term is depth-bounded. The converse is false. Thanks to Theorem 2, an immediate corollary of Theorem 3 is that coverability and termination are decidable for hierarchical terms. On Hierarchical Communication Topologies in the π-calculus 13 Unfortunately, like the depth-bounded fragment, membership of the hierarchical fragment is undecidable. The proof is by adapting the argument for the undecidability of depth boundedness [14]. Lemma 1. Every terminating π-term is hierarchical. Proof. Since the transition system of a term, quotiented by structural congruence, is finitely branching, by König’s lemma the computation tree of a terminating term is finite, so it contains finitely many reachable processes and therefore finitely many names. Take the set of all (disambiguated) active names of the reachable terms and fix an arbitrary total order T on them. The consistent annotation with (x : x) for each name will prove the term hierarchical. Theorem 4. Determining whether an arbitrary π-term is hierarchical, is undecidable. Proof. The π-calculus is Turing-complete, so termination is undecidable. Suppose we had an algorithm to decide if a term is hierarchical. Then we could decide termination of an arbitrary π-term by first checking if the term is hierarchical; if the answer is yes, we can decide termination for it by Theorem 1, otherwise we know that it is not terminating by Lemma 1. Theorem 4—and the corresponding version for depth-bounded terms—is a serious impediment to any practical application of hierarchical terms to verifcation: when presented with a term to verify, one has to prove that it belongs to one of the two fragments, manually, before one can apply the relevant algorithms. While the two fragments have a lot in common, hierarchical systems have a richer structure, which we will exploit to define a type system that can prove a term hierarchical, in a feasible, sound but incomplete way. Thanks to the notion of hierarchy, we are thus able to statically capture an expressive fragment of the π-calculus that enjoys decidable coverability. 4 A type system for hierarchical topologies The purpose of this section is to devise a static check to determine if a term is hierarchical. To do so, we define a type system, parametrised over a forest T , which satisfies subject reduction. Furthermore we prove that if a term is typable then T-shapedness is preserved by reduction of the term. Typability together with T-shapedness of the initial term would then prove the term hierarchical. As we have seen in the introduction, the typing rules make use of a new perspective on π-calculus reactions. Take the term P = νa.(νb.ahbi.S k νc.a(x).R) = C[ahbi.S, a(x).R] where C[−1 , −2 ] = νa.(νb.[−1 ] k νc.[−2 ]) is the reaction context. Standardly the synchronisation of the two sequential processes over a is preceded by an extrusion 14 E. D’Osualdo and C.-H. L. Ong of the scope of b to include νc.a(x).R, followed by the actual reaction:

νa. νb.(ahbi.S) k νc.a(x).R ≡ νa.νb. ahbi.S k νc.a(x).R
→ νa.νb. S k νc.(R[ b/x ]) This dynamic reshuffling of scopes is problematic for establishing invariance of T-compatibility under reduction: notice how νc is brought into the scope of νb, possibly disrupting T-compatibility. (For example, the preceding reduction would break T-compatibility of the forest representations if the tree T is either a  c  b or b  a  c.) We therefore adopt a different view. After the message is transmitted, the sender continues in-place as S, while R is split into two parts Rmig k R¬mig , one that uses the message (the migratable one) and one that does not. The migratable portion Rmig is “installed” under νb so that it can make use of the acquired name, while the non-migratable one can simply continue in-place:

νa. νb.(ahbi.S) k νc.a(x).R → νa. νb.(S k Rmig [ b/x ]) k νc.R¬mig | {z } | {z } C[ahbi.S, a(x).R] C[SkRmig [ b/x ], R¬mig ] Crucially, the reaction context C is unchanged. This means that if the starting term is T-compatible, the context of the reactum is T-compatible as well. Naturally, this only makes sense if Rmig does not use c. Thus our typing rules impose constraints on the use of names of R so that the migration does not result in Rmig escaping the scope of bound names such as c. The formal definition of “migratable” is subtle. Consider the term
νf.a(x).νc d e. xhci k chdi k ahei.ehf i Upon synchronisation with νb.ahbi, surely xhci will need to be put under the scope of νb after substituting b for x, hence the first component of the continuation, xhci, is migratable. However this implies that the scope of νc will need to be placed under νb, which in turn implies that chdi needs to be considered migratable as well. On the other hand, νe.ahei.ehf i must be placed in the scope of f , which may not be known by the sender, so it is not considered migratable.The following definition makes these observations precise. Definition 8 (Linked to, tied to, migratable). Given a normal form P = Q νX. i∈I Ai we say that Ai is linked to Aj in P , written i ↔P j, if fn(Ai )∩fn(Aj )∩ X= 6 ∅. We define the tied-to relation as the transitive closure of ↔P . I.e. Ai is tied to Aj , written i aP j, if ∃k1 , . . . , kn ∈ I. i ↔P k1 ↔P k2 . . . ↔P kn ↔P j, for some n ≥ 0. Furthermore, we say that a name y is tied to Ai in P , written y /P i, if ∃j ∈ I. y ∈ Q fn(Aj ) ∧ j aP i. Given an input-prefixed normal form a(y).P where P = νX. i∈I Ai , we say that Ai is migratable in a(y).P , written Miga(y).P (i), if y /P i. These definitions have an intuitive meaning with respect to the communication topology of a normal form P : two sequential subterms are linked if they are connected by an hyperedge in the communication topology of P , and are tied to each other if there exists a path between them. On Hierarchical Communication Topologies in the π-calculus 15 The following lemma indicates how the tied-to relation fundamentally constrains the possible shape of the forest of a term. Q Lemma 2. Let P = νX. i∈I Ai ∈ Pnf , if i aP j then if a forest ϕ ∈ FJP K has leaves labelled with Ai and Aj respectively, they belong to the same tree in ϕ (i.e., have a common ancestor in ϕ). Example 9. Take the normal form P = νa b c.(A1 k A2 k A3 k A4 ) where A1 = a(x), A2 = b(x), A3 = c(x) and A4 = ahbi. We have 1 ↔P 4, 2 ↔P 4, therefore 1 aP 2 aP 4 and a /P 2. In Fig. 4 we show some of the forests in FJP K. Forest 1 represents forest(P ). The fact that A1 , A2 and A4 are tied is reflected by the fact that none of the forests place them in disjoint trees. Now suppose we select only the forests in FJP K that respect the hierarchy a  b: in all the forests in this set, the nodes labelled with A1 , A2 and A4 have a as common ancestor (as in forests 1, 2, 3 and 4). In particular, in these forests A2 is necessarily a descendent of a even if a is not one of its free names. In Section 3 we introduced annotations in a rather abstract way by means of a generic domain of types T. In Definition 7 we ask for the existence of an annotation for the semantics of a term. Specifically, one can decide an arbitrary annotation for each active name. A type system however will examine the term statically, which means that it needs to know what could be a possible annotation for a variable, i.e., the name bound in an input action. This information is directly related to the notion of data-flow, that is the set of names that are bound to a variable during runtime. Since a static method cannot capture this information precisely, we make use of sorts [17], also known as simple types, to approximate it. The annotation of a restriction will carry not only which base type should be associated with its instances, but also instructions on how to annotate the messages received or sent through those instances. Concretely, we define T 3 τ ::= t | t[τ ] where t ∈ T is a base type. A name with type t cannot be used as a channel but can be used as a message; a name with type t[τ ] can be used to transmit a name of type τ . We will write base(τ ) for t when τ = t[τ 0 ] or τ = t. By abuse of notation we write, for a set of types X, base(X) for the set of base types of the types in X. As is standard, we keep track of the types of free names by means of a typing environment. An environment Γ is a partial map from names to types, which we will write as a set of type assignments, x : τ . Given a set of names X and an environment Γ , we write Γ (X) for the set {Γ (x) | x ∈ X ∩ dom(Γ )}. Given two environments Γ and Γ 0 with dom(Γ ) ∩ dom(Γ 0 ) = ∅, we write Γ Γ 0 for their union. For a type environment Γ we define minT (Γ ) := {(x : τ ) ∈ Γ | ∀(y : τ 0 ) ∈ Γ. base(τ 0 ) 6< base(τ )}. A judgement Γ `T P means that P ∈ PnfT can be typed under assumptions Γ , over the hierarchy T ; we say that P is typable if Γ `T P is provable for some 16 E. D’Osualdo and C.-H. L. Ong ∀i ∈ I. Γ, X `T Ai ∀i ∈ I. ∀x : τx ∈ X. x /P i =⇒ base(Γ (fn(Ai ))) < base(τx ) Par Q Γ `T νX. i∈I Ai ∀i ∈ I. Γ `T πi .Pi Choice P Γ `T i∈I πi .Pi a : ta [τb ] ∈ Γ Γ `T A Γ `T !A b : τb ∈ Γ Γ `T ahbi.Q Repl Γ `T Q Γ `T P Γ `T τ .P Tau Out Q a : ta [τx ] ∈ Γ Γ, x : τx `T νX. i∈I Ai
base(τx ) < ta ∨ ∀i ∈ I. Miga(x).P (i) =⇒ base(Γ (fn(Ai ) \ {a})) < ta In Q Γ `T a(x).νX. i∈I Ai Fig. 5. A type system for hierarchical terms. The term P stands for νX. Q i∈I Ai . Γ and T . An arbitrary term P ∈ P T is said to be typable if its normal form is. The typing rules are presented in Fig. 5. The type system presents several non-standard features. First, it is defined on normal forms as opposed to general π-terms. This choice is motivated by the fact that different syntactic presentations of the same term may be misleading when trying to analyse the relation between the structure of the term and T . The rules need to guarantee that a reduction will not break T-compatibility, which is a property of the congruence class of the term. As justified by Lemma 2, the scope of names in a congruence class may vary, but the tied-to relation puts constraints on the structure that must be obeyed by all members of the class. Therefore the type system is designed around this basic concept, rather than the specific scoping of any representative of the structural congruence class. Second, no type information is associated with the typed term, only restricted names hold type annotations. Third, while the rules are compositional, the constraints on base types have a global flavour due to the fact that they involve the structure of T which is a global parameter of typing proofs. Let us illustrate intuitively how the constraints enforced by the rules guarantee preservation of T-compatibility. Consider the term 

 P = νe a. νb. ahbi.A0 k νd. a(x).Q with Q = νc.(A1 k A2 k A3 ), A0 = b(y), A1 = xhci, A2 = c(z).ahei and A3 = ahdi. Let T be the forest with te  ta  tb  tc and ta  td , where tx is the base type of the (omitted) annotation of the restriction νx, for x ∈ {a, b, c, d, e}. The reader can check that forest(P ) is T-compatible. In the traditional understanding of mobility, we would interpret the commu
nication of b over a as an application of scope extrusion to include νd. a(x).Q in the scope of b and then syncronisation over a with the application of the On Hierarchical Communication Topologies in the π-calculus 17 substitution [ b/x ] to Q; note that the substitution is only valid because the scope of b has been extended to include the receiver. Our key observation is that we can instead interpret this communication as a migration of the subcomponents of Q that do get their scopes changed by the reduction, from the scope of the receiver to the scope of the sender. For this operation to be sound, the subcomponents of Q migrating to the sender’s scope cannot use the names that are in the scope of the receiver but not of the sender. In our specific example, after the synchronisation between the prefixes ahbi and a(x), b is substituted to x in A1 resulting in the term A01 = bhci and A0 , A01 , A2 and A3 become active. The scope of A0 can remain unchanged as it cannot know more names than before as a result of the communication. By contrast, A1 now knows b as a result of the substitution [ b/x ]: A1 needs to migrate under the scope of b. Since A1 uses c as well, the scope of c needs to be moved under b; however A2 uses c so it needs to migrate under b with the scope of c. A3 instead does not use neither b nor c so it can avoid migration and its scope remains unaltered. This information can be formalised using the tied-to relation: on one hand, A1 and A2 need to be moved together because 1 aQ 2 and they need to be moved because x /Q 1, 2. On the other hand, A3 is not tied to neither A1 nor A2 in Q and does not know x, thus it is not migratable. After reduction, our view of the reactum is the term  
νa. νb. A0 k νc.(A01 k A2 ) k νd.A3 the forest of which is T-compatible. Rule Par, applied to A1 and A2 , ensures that c has a base type that can be nested under the one of b. Rule In does not impose constraints on the base types of A3 because A3 is not migratable. It does however check that the base type of e is an ancestor of the one of a, thus ensuring that both receiver and sender are already in the scope of e. The base type of a does not need to be further constrained since the fact that the synchronisation happened on it implies that both the receiver and the sender were already under its scope; this implies, by T-compatibility of P , that c can be nested under a. We now describe the purpose of the rules of the type system in more detail. Most of the rules just drive the derivation through the structure of the term. The crucial constraints are checked by Par, In and Out. The Out rule. The main purpose of rule Out is enforcing types to be consistent with the dataflow of the process: the type of the argument of a channel a must agree with the types of all the names that may be sent over a. This is a very coarse sound over-approximation of the dataflow; if necessary it could be refined using well-known techniques from the literature but a simple approach is sufficient here to type interesting processes. The Par rule. Rule Par is best understood imagining the normal form to be typed, P , as the continuation of a prefix π.P . In this context a reduction exposes each of the active sequential subterms of P which need to have a place in a 18 E. D’Osualdo and C.-H. L. Ong T-compatible forest for the reactum. The constraint in Par can be read as follows. A “new” leaf Ai may refer to names already present in the forests of the reaction context; these names are the ones mentioned in both fn(Ai ) and Γ . Then we must be able to insert Ai so that we can find these names in its path. However, Ai must belong to a tree containing all the names in X that are tied to it in P . So by requiring every name tied to Ai to have a base type greater than any name in the context that Ai may refer to, we make sure that we can insert the continuation in the forest of the context without violating T-compatibility. Note that Γ (fn(Ai )) contains only types that annotate names both in Γ and fn(Ai ), that is, names which are not restricted by X and are referenced by Ai (and therefore come from the context). The In rule. Rule In serves two purposes: on the one hand it requires the type of the messages that can be sent through a to be consistent with the use of the variable x which will be bound to the messages; on the other hand, it constrains the base types of a and x so that synchronisation can be performed without breaking T-compatibility. The second purpose is achieved by distinguishing two cases, represented by the two disjuncts of the condition on base types of the rule. In the first case, the base type of the message is an ancestor of the base type of a in T . This implies that in any T-compatible forest representing a(x).P , the name b sent as message over a is already in the scope of P . Under this circumstance, there is no real mobility, P does not know new names by the effect of the substitution [ b/x ], and the T-compatibility constraints to be satisfied are in essence unaltered. The second case is more complicated as it involves genuine mobility. This case also requires a slightly non-standard feature: not only do the premises predicate on the direct subcomponents of an input prefixed term, but also on the direct subcomponents of the continuation. This is needed to be able to separate the continuation in two parts: the one requiring migration and the one that does not. The situation during execution is depicted in Fig. 6. The non migratable sequential terms behave exactly as the case of the first disjunct: their scope is unaltered. The migratable ones instead are intended to be inserted as descendents of the node representing the message b in the forest of the reaction context. For this to be valid without rearrangement of the forest of the context, we need all the names in the context that are referenced in the migratable terms, to be also in the scope at b; we make sure this is the case by requiring the free names of any migratable Ai that are from the context (i.e. in Γ ) to have base types smaller than the base type of a. The set base(Γ (fn(Ai ) \ {a})) indeed represents the base types of the names in the reaction context referenced in a migratable continuation Ai . In fact a is a name that needs to be in the scope of both the sender and the receiver at the same time, so it needs to be a common ancestor of sender and receiver in any T-compatible forest. Any name in the reaction context and in the continuation of the receiver, with a base type smaller than the one of a, will be an ancestor of a—and hence of the sender, the receiver and the node representing the message—in any T-compatible forest. Clearly, remembering a On Hierarchical Communication Topologies in the π-calculus a a → b ahbi 19 a(x).P b P mig P ¬mig Fig. 6. Explanation of constraints imposed by rule In. The dashed lines represent references to names restricted in the reduction context. is not harmful as it must be already in the scope of receiver and sender, so we exclude it from the constraint. Example 10. Take the normal form in Example 2. Let us fix T to be the forest s  c  m  d and annotate the normal form with the following types: s : τs = s[τm ], c : τc = c[τm ], m : τm = m[d] and d : d. We want to prove ∅ `T νs c.P . We can apply rule Par: in this case there are no conditions on types because, being the environment empty, we have base(∅(fn(A))) = ∅ for every active sequential term A of P . Let Γ = {(s : τs ), (c : τc )}. The rule requires Γ `T !S, Γ `T !C and Γ `T !M , which can be proved by proving typability of S, C and M under Γ by rule Repl. To prove Γ `T S we apply rule In; we have s : s[τm ] ∈ Γ and we need to prove that Γ, x : τm `T νd.xhdi. No constraints on base types are generated at this step since the migratable sequential term νd.xhdi does not contain free variables typed by Γ making Γ (fn(νd.xhdi) \ {a}) = Γ ({x}) empty. Next, Γ, x : τm `T νd.xhdi can be proved by applying rule Par which amounts to checking Γ, x : τm , d : d `T xhdi.0 (by a simple application of Out and the axiom Γ, x : τm , d : d `T 0) and verifying the condition—true in T —base(τm ) < base(τd ): in fact d is tied to xhdi and, for Γ 0 = Γ ∪ {x : τm }, base(Γ 0 (fn(xhdi))) = base(Γ 0 ({x, d})) = base({τm }). The proof for Γ `T M is similar and requires c < m which is true in T . Finally, we can prove Γ `T C using rule In; both the two continuations A1 = shmi and A2 = m(y).chmi are migratable in C and since base(τm ) < base(τc ) is false we need the other disjunct of the condition to be true. This amounts to checking that base(Γ (fn(A1 ) \ {c})) = base(Γ ({s, m})) = base({τs }) = s < c (note m 6∈ dom(Γ )) and base(Γ (fn(Aa ) \ {c})) = base(Γ (∅)) < c (that holds trivially). To complete the typing we need to show Γ, m : τm `T A1 and Γ, m : τm `T A2 . The former can be proved by a simple application of Out which does not impose further constraints on T . The latter is proved by applying In which requires base(τc ) < m, which holds in T . Note how, at every step, there is only one rule that applies to each subproof. Example 11. The term Example 4 is not typable under any T . To see why, one can build the proof tree without assumptions on T by assuming that each 20 E. D’Osualdo and C.-H. L. Ong restriction νx has base type tx . When typing mhsi we deduce that ts = tn , which is in contradiction with the constraint that tn < ts required by rule Par when typing νs.(S k mhsi k s). 5 Soundness of the type system We now establish the soundness of the type system. Theorem 5 will show how typability is preserved by reduction. Theorem 6 establishes the main property of the type system: if a term is typable then T-shapedness is invariant under reduction. This allows us to conclude that if a term is T-shaped and typable, then every term reachable from it will be T-shaped. The subtitution lemma states that substituting names without altering the types preserves typability. Lemma 3 (Substitution). Let P ∈ PnfT and Γ be a typing environment such that Γ (a) = Γ (b). Then it holds that if Γ `T P then Γ `T P [ b/a ]. Before we state the main theorem, we define the notion of P -safe type environment, which is a simple restriction on the types that can be assigned to names that are free at the top-level of a term. Definition 9 (P -safe environment). A type environment Γ is said to be P safe if for each x ∈ fn(P ) and (y : τ ) ∈ bnν (P ), base(Γ (x)) < base(τ ). Theorem 5 (Subject Reduction). Let P and Q be two terms in PnfT and Γ be a P -safe type environment. If Γ `T P and P → Q, then Γ `T Q. The proof is by careful analysis of how the typing proof for P can be adapted to derive a proof for Q. The only difficulty comes from the fact that some of the subterms of P will appear in Q with a substitution applied. However, typability of P ensures that we are only substituting names for names with the same type, thus allowing us to apply Lemma 3. To establish that T-shapedness is invariant under reduction for typable terms, we will need to show that starting from a typable T-shaped term P , any step will reduce it to a (typable) T-shaped term. The hypothesis of T-compatibility of P can be used to extract a T-compatible forest ϕ from FJP K. While many forests in FJP K can be witnesses of the T-compatibility of P , we want to characterise the shape of a witness that must exist if P is T-compatible. The proof of invariance relies on selecting a ϕ that does not impose unnecessary hierarchical dependencies among names. Such forest is identified by ΦT (nf(P )): it is the shallowest among all the T-compatible forests in FJP K. Definition 10 (ΦT ). The function ΦT : PnfT → FT is defined inductively as ] Q ΦT ( i∈I Ai ) := {Ai []} i∈I o Q (x, base(τ ))[ΦT (νYx . j∈Ix Aj )] (x : τ ) ∈ minT (X) Q ] ΦT (νZ. r∈R Ar ) ΦT (P ) := ]n On Hierarchical Communication Topologies in the π-calculus 21 Q where X 6= ∅, P = νX. i∈I Ai Ix = {i ∈ I | x /P i} and Yx = {(y : τ ) ∈ X | ∃i ∈ Ix . y ∈ fn(Ai )} \ minT (X)
S Z = X\ (x : τ )∈minT (X) Yx ∪ {x : τ }
S R = I\ (x : τ )∈minT (X) Ix Forest 4 of Fig. 4 is ΦT (P ) when every restriction νx has base type x (for x ∈ {a, b, c}) and T is the forest with nodes a, b and c and a single edge a  b. Lemma 4. Let P ∈ PnfT . Then: a) ΦT (P ) is a T-compatible forest; b) ΦT (P ) ∈ FJP K if and only if P is T-compatible; c) if P ≡ Q ∈ P T then ΦT (P ) ∈ FJQK if and only if Q is T-compatible. Theorem 6 (Invariance of T-shapedness). Let P and Q be terms in PnfT such that P → Q and Γ be a P -safe environment such that Γ `T P . Then, if P is T-shaped then Q is T-shaped. The key of the proof is a) the use of ΦT (P ) to extract a specific T-compatible forest, b) the definition of a way to insert the subtrees of the continuations of the reacting processes in the forest of reaction context, in a way that preserves T-compatibility. Thanks to the constraints of the typing rules, we will always be able to find a valid place in the reaction context where to attach the trees representing the reactum. 6 Type inference In this section we will show that it is possible to take any non-annotated normal form P and derive a forest T and an annotated version of P that can be typed under T . Inference for simple types has already been proved decidable in [8, 23]. In our case, since our types are not recursive, the algorithm concerned purely with the constraints imposed by the type system of the form τx = t[τy ] is even simpler. The main difficulty is inferring the structure of T . Let us first be more specific on assigning simple types. The number of ways a term P can be annotated with types are infinite, simply from the fact that types allow an arbitrary nesting as in t, t[t], t[t[t]] and so on. We observe that, however, there is no use annotating a restriction with a type with nesting deeper than the size of the program: the type system cannot inspect more deeply nested types. Thanks to this observation we can restrict ourselves to annotations with bounded nesting in the type’s structure. This also gives a bound on the number of base types that need to appear in the annotated term. Therefore, there are only finitely many possible annotations and possible forests under which P can be proved typably hierarchical. A naı̈ve inference algorithm can then enumerate all of them and type check each. 22 E. D’Osualdo and C.-H. L. Ong Theorem 7 (Decidability of inference). Given a normal form P ∈ Pnf , it is decidable if there exists a finite forest T , a T-annotated version P 0 ∈ P T of P and a P 0 -safe environment Γ such that P 0 is T-shaped and Γ `T P 0 . While enumerating all the relevant forests, annotations and environments is impractical, more clever strategies for inference exist. We start by annotating the term with type variables: each name x gets typed with a type variable tx . Then we start the type derivation, collecting all the constraints on types along the way. If we can find a T and type expressions to associate to each type variable, so that these constraints are satisfied, the process can be typed under T . By inspecting rules Par and In we observe that all the “tied-to” and “migratable” predicates do not depend on T so for any given P , the type constraints can be expressed simply by conjuctions and disjuctions of two kinds of basic predicates: 1. data-flow constraints of the form tx = tx [ty ] where tx is a base type variable; 2. base type constraints of the form base(tx ) < base(ty ) which correspond to constraints over the corresponding base type variables, e.g. tx < ty . Note that the P -safety condition on Γ translates to constraints of the second kind. The first kind of constraint can be solved using unification in linear time. If no solution exists, the process cannot be typed. This is the case of processes that cannot be simply typed. If unification is successful we get a set of equations over base type variables. Any assignment of those variables to nodes in a suitable forest that satisfies the constraints of the second kind would be a witness of typability. An example of the type inference in action can be found in Appendix. First we note that if there exists a T which makes P typable and T-compatible, then there exists a T 0 which does the same but is a linear chain of base types (i.e. a single tree with no branching). To see how, simply take T 0 to be any topological sort of T . Now, suppose we are presented with a set C of constraints of the form t < t0 (no disjuctions). One approach for solving them could be based on reductions to SAT or CLP(FD). We instead outline a direct algorithm. If the constraints are acyclic, i.e. it is not possible to derive t < t by transitivity, then there exists a finite forest satisfying the constraints, having as nodes the base type variables. To construct such forest, we can first represent the constraints as a graph with the base type variables as vertices and an edge between t and t0 just when t < t0 ∈ C. Then we can check the graph for acyclicity. If the test fails, the constraints are unsatisfiable. Otherwise, any topological sort of the graph will represent a forest satisfying C. We can modify this simple procedure to support constraints including disjuctions by using backtracking on the disjuncts. Every time we arrive at an acyclic assigment, we can check for T-shapedness (which takes linear time) and in case the check fails we can backtrack again. To speed up the backtracking algorithm, one can merge the acyclicity test with the T-compatibility check. Acyclicity can be checked by constructing a On Hierarchical Communication Topologies in the π-calculus 23 topological sort of the constraints graph. Every time we produce the next node in the sorting, we take a step in the construction of Φ(P ) using the fact that the currently produced node is the minimal base type among the remaining ones. We can then backtrack as soon as a choice contradicts T-compatibility. The complexity of the type checking problem is easily seen to be linear in the size of the program. This proves, in conjuction with the finiteness of the candidate guesses for T and annotations, that the type inference problem is in NP. We conjecture that inference is also NP-hard. We implemented the above algorithm in a tool called ‘James Bound’ (jb), available at http://github.com/bordaigorl/jamesbound. 7 7.1 Expressivity and verification Expressivity Typably hierarchical terms form a rather expressive fragment. Apart from including common patterns as the client-server one, they generalise powerful models of computation with decidable properties. Relations with variants of CCS are the easiest to establish: CCS can be seen as a syntactic subset of π-calculus when including 0-arity channels, which are very easily dealt with by straightforward specialisations of the typing rules for actions. One very expressive, yet not Turing-powerful, variant is CCS! [9] which can be seen as our π-calculus without mobility. Indeed, every CCS! process is typably hierarchical [4, Section 11.4]. Reset nets can be simulated by using resettable counters as defined in Example 3. The full encoding can be found in Appendix. The encoding preserves coverability but not reachability. CCS! was recently proven to have decidable reachability [9] so it is reasonable to ask whether reachability is decidable for typably hierarchical terms. We show this is not the case by introducing a weak encoding of Minsky machines (in Appendix). The encoding is weak in the sense that not all of the runs represent real runs of the encoded Minsky machine; however with reachability one can distinguish between the reachable terms that are encodings of reachable configurations and those which are not. We therefore reduce reachability of Minsky machines to reachability of typably hierarchical terms. Theorem 8. The reachability problem is undecidable for (typably) hierarchical terms. Theorem 8 can be used to clearly separate the (typably) hierarchical fragment from other models of concurrent computation as Petri Nets, which have decidable reachability and are thus less expressive. 7.2 Applications Although reachability is not decidable, coverability is often quite enough to prove non-trivial safety properties. To illustrate this point, let us consider Example 2 24 E. D’Osualdo and C.-H. L. Ong again. In our example, each client waits for a reply reaching its mailbox before issuing another request; moreover the server replies to each request with a single message. Together, these observations suggest that the mailboxes of each client will contain at most one message at all times. To automatically verify this property we could use a coverability algorithm for depth-bounded systems: since the example is typable, it is depth-bounded and such algorithm is guaranteed to terminate with a correct answer. To formulate the property as a coverability problem, we can ask for coverability of the following query: νs m.(!S k m(y).chmi k νd.mhdi k νd0 .mhd0 i). This is equivalent to asking whether a term is reachable that embeds a server connected with a client with a mailbox containing two messages. The query is not coverable and therefore we proved our property.5 Other examples of coverability properties are variants of secrecy properties. For instance, the coverability query νs m m0 .(!S k m(y).chmi k m0 (y).chm0 i k νd.(mhdi k m0 hdi)) encodes the property “can two different clients receive the same message?”, which cannot happen in our example. It is worth noting that this level of accuracy for proving such properties automatically is uncommon. Many approaches based on counter abstraction [22, 6] or CFA-style abstractions [5] would collapse the identities of clients by not distinguishing between different mailbox addresses. Instead a single counter is typically used to record the number of processes in the same control state and of messages. In our case, abstracting the mailbox addresses away has the effect of making the bounds on the clients’ mailboxes unprovable in the abstract model. A natural question at this point is: how can we go about verifying terms which cannot be typed, as the ring example? Coverability algorithms can be applied to untypable terms and they yield sound results when they terminate. But termination is not guaranteed, as the term in question may be depth-unbounded. However, even a failed typing attempt may reveal interesting information about the structure of a term. For instance, in Example 11 one may easily see that the cyclic dependencies in the constraints are caused by the names representing the “next” process identities. In the general case heuristics can be employed to automatically identify a minimal set of problematic restrictions. Once such restrictions are found, a counter abstraction could be applied to those restrictions only yielding a term that simulates the original one but introducing some spurious behaviour. Type inference can be run again on the the abstracted term; on failure, the process can be repeated, until a hierarchical abstraction is obtained. This abstract model can then be model checked instead of the original term, yielding sound but possibly imprecise results. 8 Related work Depth boundedness in the π-calculus was first proposed in [12] where it is proved that depth-bounded systems are well-structured transition systems. In [24] it is further proved that (forward) coverability is decidable even when the depth 5 To fully prove a bound on the mailbox capacity one may need to also ask another coverability question for the case where the two messages bear the same data-value d. On Hierarchical Communication Topologies in the π-calculus 25 bound k is not known a priori. In [25] an approximate algorithm for computing the cover set—an over-approximation of the set of reachable terms—of a system of depth bounded by k is presented. All these analyses rely on the assumption of depth boundedness and may even require a known bound on the depth to terminate. Several other interesting fragments of the π-calculus have been proposed in the literature, such as name bounded [10], mixed bounded [15], and structurally stationary [13]. Typically defined by a non-trivial condition on the set of reachable terms – a semantic property, membership becomes undecidable. Links with Petri nets via encodings of proper subsets of depth-bounded systems have been explored in [15]. Our type system can prove depth boundedness for processes that are breadth and name unbounded, and which cannot be simulated by Petri nets. In [2], Amadio and Meyssonnier consider fragments of the asynchronous π-calculus and show that coverability is decidable for the fragment with no mobility and bounded number of active sequential processes, via an encoding to Petri nets. Typably hierarchical systems can be seen as an extension of the result for a synchronous π-calculus with unbounded sequential processes and a restricted form of mobility. Recently Hüchting et al. [11] proved several relative classification results between fragments of π-calculus. Using Karp-Miller trees, they presented an algorithm to decide if an arbitrary π-term is bounded in depth by a given k. The construction is based on an (accelerated) exploration of the state space of the π-term, with non primitive recursive complexity, which makes it impractical. By contrast, our type system uses a very different technique leading to a quicker algorithm, at the expense of precision. Our forest-structured types can also act as specifications, offering more intensional information to the user than just a bound k. Our types are based on Milner’s sorts for the π-calculus [17, 8], later refined into I/O types [20] and their variants [21]. Based on these types is a system for termination of π-terms [3] that uses a notion of levels, enabling the definition of a lexicographical ordering. Our type system can also be used to determine termination of π-terms in an approximate but conservative way, by using it in conjuction with Theorem 1. Because the respective orderings between types of the two approaches are different in conception, we expect the terminating fragments isolated by the respective systems to be incomparable. 9 Future directions The type system we presented in Section 4 is conservative: the use of simple types, for example, renders the analysis context-insensitive. Although we have kept the system simple so as to focus on the novel aspects, a number of improvements are possible. First, the extension to the polyadic case is straightforward. Second, the type system can be made more precise by using subtyping and polymorphism to refine the analysis of control and data flow. Third, the typing rule for replication introduces a very heavy approximation: when typing a subterm, we have no information about which other parts of the term (crucially, which restrictions) 26 E. D’Osualdo and C.-H. L. Ong may be replicated. By incorporating some information about which names can be instantiated unboundedly in the types, the precision of the analysis can be greatly improved. The formalisation and validation of these extensions is a topic of ongoing research. Another direction worth exploring is the application of this machinery to heap manipulating programs and security protocols verification. Acknowledgement. We would like to thank Damien Zufferey for helpful discussions on the nature of depth boundedness, and Roland Meyer for insightful feedback on a previous version of this paper. Bibliography [1] P. A. Abdulla, K. Cerans, B. Jonsson, and Y. Tsay. General decidability theorems for infinite-state systems. In Symposium on Logic in Computer Science, pages 313–321. IEEE Computer Society, 1996. [2] R. M. Amadio and C. Meyssonnier. On decidability of the control reachability problem in the asynchronous π-calculus. Nordic Journal of Computing, 9 (2):70–101, 2002. [3] Y. Deng and D. Sangiorgi. Ensuring termination by typability. Information and Computation, 204(7):1045–1082, 2006. [4] E. D’Osualdo. Verification of Message Passing Concurrent Systems. PhD thesis, University of Oxford, 2015. URL http://ora.ox.ac.uk/objects/ uuid:f669b95b-f760-4de9-a62a-374d41172879. [5] E. D’Osualdo, J. Kochems, and C.-H. L. Ong. Automatic verification of erlang-style concurrency. In F. Logozzo and M. 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In Foundations of Software Science and Computation Structures (FoSSaCS), pages 94–108, 2010. [25] D. Zufferey, T. Wies, and T. Henzinger. Ideal abstractions for well-structured transition systems. In Verification, Model Checking, and Abstract Interpretation (VMCAI), pages 445–460, 2012. Appendix A Supplementary Material for Section 2 A.1 Definition and properties of nf The function nf : P → Pnf , defined in Definition 11, extracts, from a term, a normal form structurally equivalent to it. Definition 11 (nf(P )). We define the function nf : P → Pnf as follows: nf(0) := 0 nf(π.P ) := π. nf(P ) nf(M + M 0 ) := nf(M ) + nf(M 0 )  nf(P )    nf(Q) nf(P k Q) :=  νXP XQ .(NP k NQ )    nf(νx.P ) := νx. nf(P ) nf(!M ) := !(nf(M )) if nf(Q) = 0 6= nf(P ) if nf(P ) = 0 if nf(Q) = νXQ .NQ , nf(P ) = νXP .NP and actν (NP ) = actν (NQ ) = ∅ Lemma 5. For each P ∈ P, P ≡ nf(P ) Proof. A straightforward induction on P . Lemma 6. Let ϕ be a forest with labels in N ] S. Then ϕ = forest(Q) with Q ≡ Qϕ where Q Qϕ := νXϕ . (n,A)∈I A Xϕ := {`ϕ (n) ∈ N | n ∈ Nϕ } I := {(n, A) | `ϕ (n) = A ∈ S} provided i) ∀n ∈ Nϕ , if `ϕ (n) ∈ S then n has no children in ϕ, and ii) ∀n, n0 ∈ Nϕ , if `ϕ (n) = `ϕ (n0 ) ∈ N then n = n0 , and iii) ∀n ∈ Nϕ , if `ϕ (n) = A ∈ S then for each x ∈ Xϕ ∩ fn(A) there exists n0 <ϕ n such that `ϕ (n0 ) = x. Proof. We proceed by induction on the structure of ϕ. The base case is when ϕ = (∅, ∅), for which we have Qϕ = 0 and ϕ = forest(0). When ϕ = ϕ0 ] ϕ1 we have that if conditions 8.i, 8.ii and 8.iii hold for ϕ, they must hold for ϕ0 and ϕ1 as well, hence we can apply the induction hypothesis to them obtaining ϕi forest(Qi ) with Qi ≡ Qϕi (i ∈ {0, 1}). We have ϕ = forest(Q0 k Q1 ) by definition of forest, and we want to prove that Q0 k Q1 ≡ Qϕ . By condition 8.ii on ϕ, Xϕ0 and Xϕ1 must be disjoint; furthermore, by condition 8.iii on both 30 E. D’Osualdo and C.-H. L. Ong ϕ0 and ϕ1 we can conclude that fn(Qϕi ) ∩ Xϕ1−i = ∅. We can therefore apply scope extrusion: Q0 k Q1 ≡ Qϕ0 k Qϕ1 ≡ νXϕ0 Xϕ1 .(Pϕ0 k Pϕ1 ) = Qϕ . The last case is when ϕ = l[ϕ0 ]. Suppose conditions 8.i, 8.ii and 8.iii hold for ϕ. We distinguish two cases. If l = A ∈ S, by 8.i we have ϕ0 = (∅, ∅), ϕ = forest(A) and A = Qϕ . If l = x ∈ N then we observe that conditions 8.i, 8.ii and 8.iii hold for ϕ0 under the assumption that they hold for ϕ. Therefore ϕ0 = forest(Q0 ) with Q0 ≡ Qϕ0 , and, by definition of forest, ϕ = forest(νx.Q0 ). By condition 8.ii we have x 6∈ Xϕ0 so νx.Q0 ≡ νx.Qϕ0 ≡ ν(X ∪ {x}).Pϕ0 = Qϕ . B B.1 Supplementary Material for Section 3 Proof of Theorem 3 First, it is immediate to see that every hierarchical term is depth-bounded. Any T-compatible forest cannot repeat a type in a path, which means that the number of base types in T bounds the height of T-compatible forests. This automatically gives a bound on the depth of any T-compatible term. We show the converse is not true by presenting a depth-bounded process which is not hierarchical. Take P = (!A k !B k !(C1 + C2 )) where A = τ .ν(a : a).phai C1 = p(x).!(q(y).D) B = τ .ν(b : b).qhbi C2 = q(x).!(p(y).D) D = xhyi then P is depth-bounded. However we can show there is no choice for consistent annotations and T that can prove it hierarchical. Let h be the height of T . From P we can reach, by reducing the τ actions of A and B, any of the terms Qi,j = P k (νa.phai)i k (νb.qhbi)j (omitting annotations) for i, j ∈ N. The choice for annotations can potentially assign a different type in T to each νa and νb. Let n, m ∈ N be naturals strictly greater than 2h and consider the reachable term Qn,nm ; from this term we can reach a term   
m
 n ab Q = P k νa. νb.D[ a/x, b/y ] k ! q(y).D[ a/x ] by never selecting C2 as part of a redex. Each occurrence of a and b will have an annotation: we assume type tia is assigned to each occurrence i ≤ n of νa in Qab i ab and a type ti,j b is assigned to each occurrence j of νb under ν(a : ta ) in Q . Each ab occurrence of νa in Q has in its scope more than h occurrences of νb. We cannot extrude more than h occurrences of νb because we would necessarily violate T-compatibility by obtaining a path of length greater than h in the forest of the i,h+1 extruded term. Therefore, w.l.o.g., we can assume that the types ti,1 b , . . . , tb i are all descendants of ta , for each i ≤ n. Pictorially, the parent relation in T entails the relations in Fig. 7 where the edges represent <T . The type associations of the restrictions in Qab are already fixed in Qn,nm . From Qn,nm we can however also reach any of the terms   
n
 Qib = P k · · · k ν(b : ti,1 ). νa.D[ a/x, b/y ] k ! p(y).D[ b/x ] b On Hierarchical Communication Topologies in the π-calculus t1a 31 tn a t2a ··· t1,1 b t1,2 b · · · t1,h+1 t2,1 b b t2,2 b · · · t2,h+1 b tn,1 b tbn,2 · · · tn,h+1 b Fig. 7. Structure of T in the counterexample. for i ≤ m, by making C2 and ν(b : ti,1 b ).qhbi react and then repeatedly making !q(y).D react with each ν(a : tja ).phai. Let us consider Q1b . As before, we cannot extrude more than h occurrences of a or we would break T-compatibility. We must however extrude (a : t1a ) to get T-compatibility since t1a <T t1,1 b . From these two facts we can infer that there must be a type associated to one of the a, let it be t2a , such that t1a <T t1,1 <T t2a . We can apply the same argument to Q2b b 2,1 obtaining t1a <T t2a <T tb <T t3a . Since m > 2h we can repeat this h + 1 times and get t1a <T t2a <T . . . <T th+1 which contradicts the assumption that the a height of T is h. The reason why the counterexample presented in the proof above fails to be hierarchical is that (unboundedly many) names are used in fundamentally different ways in different branches of the execution. C C.1 Supplementary Material for Section 4 Proof of Lemma 2 We show that the claim holds in the case where Ai is linked to Aj in P . From this, a simple induction over the length of linked-to steps required to prove i aP j, can prove the lemma. Suppose i ↔P j. Let Y = fn(Ai ) ∩ fn(Aj ) ∩ {x | (x : τ ) ∈ X}, we have Y = 6 ∅. Both Ai and Aj are in the scope of each of the restrictions bounding names y ∈ Y in any of the processes Q in the congruence class of P , hence, by definition of forest, the nodes labelled with Ai and Aj generated by forest(Q) will have nodes labelled with (y, base(X(y))) as common ancestors. C.2 Some auxiliary lemmas Lemma 7. If forest(P ) is T-compatible then for any term Q which is an αrenaming of P , forest(Q) is T-compatible. Proof. Straightforward from the fact that T-compatibility depends only on the type annotations. Q Lemma 8. Let P = νX. i∈I Ai be a T-compatible normal form, Y ⊆ X and Q J ⊆ I. Then P 0 = νY. j∈J Aj is T-compatible. 32 E. D’Osualdo and C.-H. L. Ong Proof. Take a T-compatible forest ϕ ∈ FJP K. By Lemma 7 we can assume without loss of generality that ϕ = forest(Q) where proving Q ≡ P does not require α-renaming. Clearly, removing the leaves that do not correspond to sequential terms indexed by Y does not affect the T-compatibility of ϕ. Similarly, if a restriction (x : τ ) ∈ X is not in Y , we can remove the node of ϕ labelled with (x, base(τ )) by making its parent the new parent of its children. This operation is unambiguous under Name Uniqueness and does not affect T-compatibility, by transitivity of <. We then obtain a forest ϕ0 which is T-compatible and that, by Lemma 6, is the forest of a term congruent to the desired normal form P 0 . D D.1 Supplementary Material for Section 5 Some Elementary Properties of the Type System Lemma 9. Let P ∈ PnfT and Γ , Γ 0 be type environments. a) if Γ `T P then fn(P ) ⊆ dom(Γ ); b) if dom(Γ 0 ) ∩ bn(P ) = ∅ and fn(P ) ⊆ dom(Γ ), then Γ `T P if and only if Γ Γ 0 `T P ; c) if P ≡ P 0 ∈ PnfT then, Γ `T P if and only if Γ `T P 0 . D.2 Proof of Lemma 4 Item a) is an easy induction on the cardinality of X. Item b) requires more work. By item a) Φ(P ) is T-compatible so Φ(P ) ∈ FJP K proves that P is T-compatible. Q To prove the ⇐-direction we assume that P = νX. i∈I Ai is T-compatible and proceed by induction on the cardinality of X U to show that Φ(P ) ∈ Q FJP K. The Q base case is when X = ∅: Φ(P ) = Φ( i∈I Ai ) = i∈I {Ai []} = forest( i∈I Ai ) = forest(P ) ∈ FJP K. For the induction step, we observe that X 6= ∅ implies minT (X) 6= ∅ so, Z ⊂ X and for each (x : τ ) ∈ minT (X), Yx ⊂ X since x 6∈ Yx . This, together with Lemma 8, allows us to Q apply the induction hyQ potesis on the terms Px = νYx . j∈Ix Aj and PR = νZ. r∈R Ar , obtaining that there exist terms Qx ≡ Px and QR ≡ PR such that forest(Qx ) = Φ(Px ) and forest(QR ) = Φ(PRQ ) where all the forests forest(Qx ) and forest(QR ) are T-com{ν(x : τ ).Qx | (x : τ ) ∈ minT (X)} k QR , then forest(Q) = patible. Let Q = Φ(P ). To prove the Q Q claim we only need to show that Q ≡ P . We have Q ≡ {ν(x : τ ).νYx . j∈Ix Aj | (x : τ ) ∈ minT (X)} k PR and we want to apply extru
Q U sion to get Q ≡ νYU {Ix | (x : τ ) ∈ minT (X)}, min . i∈Imin Ai k PR for Imin = Ymin = minT (X) ] {Yx | (x : τ ) ∈ minT (X)} which adds an obligation to prove that i) Ix are all pairwise disjoint so that Imin is well-defined, ii) Yx are all pairwise disjoint and all disjoint from minT (X) so that Ymin is well-defined, On Hierarchical Communication Topologies in the π-calculus 33 iii) Yx ∩fn(Aj ) = ∅ for every j ∈ Iz with z = 6 x so that we can apply the extrusion rule. To prove condition i), assume by contradiction that there exists an i ∈ I and names x, y ∈ minT (X) with x 6= y, such that both x and y are tied to Ai in P . By transitivity of the tied-to relation, we have Ix = Iy . By Lemma 2 all the Aj with j ∈ Ix need to be in the same tree in any forest ϕ ∈ FJP K. Since P is T-compatible there exist such a ϕ which is T-compatible and has every Aj as label of leaves of the same tree. This tree will include a node nx labelled with (x, base(X(x))) and a node ny labelled with (y, base(X(y))). By T-compatibility of ϕ and the existence of a path between nx and ny we infer base(X(x)) < base(X(y)) or base(X(y)) < base(X(x)) which contradicts the assumption that x, y ∈ minT (X). Condition ii) follows from condition i): suppose there exists a (z : τ ) ∈ X ∩ Yx ∩ Yy for x 6= y, then we would have that z ∈ fn(Ai ) ∩ fn(Aj ) for some i ∈ Ix and j ∈ Iy , but then i aP j, meaning that i ∈ Iy and j ∈ Ix violating condition i). The fact that Yx ∩ minT (X) = ∅ follows from the definition of Yx . The same reasoning proves condition iii).Q
Q Now we have Q ≡ νYmin . i∈IQ Ai k νZ. r∈R Ar and we want to apply min extrusion again to get Q ≡ νYmin Z. {Ai | i ∈ (Imin ] R)} which is sound under the following conditions: iv) Ymin ∩ Z = ∅, v) Imin ∩ R = ∅, vi) Z ∩ fn(Ai ) = ∅ for all i 6∈ R of which the first two hold trivially by construction, while the last follows from condition viii) below, as a name in the intersection of Z and a fn(Ai ) would need to be in X but not in Ymin . To be able to conclude that Q ≡ P it remains to prove that vii) I = Imin ] R and viii) X = Ymin ] Z which are also trivially valid by inspection of their definitions. This concludes the proof for item b). Finally, for every Q ∈ P T such that Q ≡ P , Φ(P ) ∈ FJQK if and only if Φ(P ) ∈ FJP K by definition of FJ−K; since Φ(P ) is T-compatible we can infer that Q is T-compatible if and only if Φ(P ) ∈ FJQK, which proves item c). In light of Lemma 4, we can turn the computation of ΦT (P ) into an algorithm to check T-compatibility of P : it is sufficient to compute ΦT (P ) and check at each step that the sets Ix , R form a partition of I and the sets Yx , Z form a partition of X. If the checks fail ΦT (P ) 6∈ FJP K and P is not T-compatible, otherwise the obtained forest is a witness of T-compatibility. D.3 Further Properties of ΦT (P ) Q Lemma 10. Let P = νX. i∈I Ai ∈ PnfT be a T-compatible normal form. Then for every trace ((x1 , t1 ) . . . (xk , tk ) Aj ) in the forest Φ(P ), for every i ∈ {1, . . . , k}, we have xi /P j (i.e. xi is tied to Aj in P ). 34 E. D’Osualdo and C.-H. L. Ong Proof. Straightforward from the definition of Ix in Φ: when a node labelled by (x, t) is introduced, its subtree is extracted from a recursive call on a term that contains all and only the sequential terms that are tied to x. Remark 1. Φ(P ) satisfies conditions 8.i, 8.ii and 8.iii of Lemma 6. D.4 Proof of Lemma 3 We prove the lemma by induction on the structure of P . The base case is when P ≡ 0, where the claim trivially holds.Q P For the induction step, let P ≡ νX. i∈I Ai with Ai = j∈J πij .Pij , for some finite sets of indexes I and J. Since the presence of replication does not affect the typing proof, we can safely ignore that case as it follows the same argument. Let us assume Γ `T P and prove that Γ `T P [ b/a ]. Let Γ 0 be Γ ∪ X. From Γ `T P we have Γ, X `T Ai x /P i =⇒ base(Γ (fn(Ai ))) < base(τx ) (1) (2) for each i ∈ I and x : τx ∈ X. To extract from this assumptions a proof for Γ `T P [ b/a ], we need to prove that (1) and (2) hold after the substitution. Since the substitution does not apply to names in X and the tied to relation is only concerned with names in X, the only relevant effect of the substitution is modifying the set fn(Ai ) to fn(Ai [ b/a ]) = fn(Ai ) \ {a} ∪ {b} when a ∈ fn(Ai ); But since Γ (a) = Γ (b) by hypothesis, we have base(Γ (fn(Ai [ b/a ]))) < base(τx ). It remains to prove (1) holds after the substitution as well. This amounts to prove for each j ∈ J that Γ 0 `T πij .Pij =⇒ Γ 0 `T πij .Pij [ b/a ]; we prove this by cases. Suppose πij = αhβi for two names α and β, then from Γ 0 `T πij .Pij we know the following α : tα [τβ ] ∈ Γ 0 0 β : τβ ∈ Γ 0 Γ `T Pij (3) (4) Condition (3) is preserved after the substitution because it involves only types so, even if α or β are a, their types will be left untouched after they get substituted with b from the hypothesis that Γ (a) = Γ (b). Condition (4) implies Γ 0 `T Pij [ b/a ] by inductive hypothesis. Q Suppose now that πij = α(x) and Pij ≡ νY. k∈K A0k for some finite set of indexes K; by hypothesis we have: α : tα [τx ] ∈ Γ 0 0 Γ , x : τx `T Pij base(τx ) < tα ∨ ∀k ∈ K. Migπij .Pij (k) =⇒ base(Γ 0 (fn(A0k ) \ {α})) < tα (5) (6) (7) Now x and Y are bound names so they are not altered by substitutions. The substitution [ b/a ] can therefore only be affecting the truth of these conditions On Hierarchical Communication Topologies in the π-calculus 35 when α = a or when a ∈ fn(A0k ) \ (Y ∪ {x}). Since we know a and b are assigned the same type by Γ and Γ ⊆ Γ 0 , condition (5) still holds when substituting a for b. Condition (6) holds by inductive hypotesis. The first disjunct of condition (7) depends only on types, which are not changed by the substitution, so it holds after applying it if and only if it holds before the application. To see that the second disjunct also holds after the substitution we observe that the migratable condition depends on x and fn(A0k ) ∩ Y which are preserved by the substitution; moreover, if a ∈ fn(A0k ) \ {α} then Γ 0 (fn(A0k ) \ {α}) = Γ 0 (fn(A0k [ b/a ]) \ {α}). This shows that the premises needed to derive Γ 0 , x : τx0 `T πij .Pij [ b/a ] are implied by our hypothesis, which completes the proof. D.5 Proof of Theorem 5 We will only prove the result for the case when P → Q is caused by a synchronising send and receive action since the τ action case is similar and simpler. From P → Q we know that P ≡ νW.(S k R k C) ∈ PnfT with S ≡ (ahbi.νYs .S 0 )+Ms and R ≡ (a(x).νYr .R0 ) + Mr the synchronising sender and receiver respectively; Q Q≡ k C). In what follows, let W 0 = W Ys Yr , C = h∈H Ch , νW YsQ Yr .(S 0 k R0 [ b/x ] Q S 0 = i∈I Si0 and R0 = j∈J Rj0 , all normal forms. For annotated terms, the type system is syntax directed: there can be only one proof derivation for each typable term. By Lemma 9.c, from the hypothesis Γ `T P we can deduce Γ `T νW.(S k R k C). The proof derivation for this typing judgment can only be of the following shape: Γ W `T S Γ W `T R ∀h ∈ H. Γ W `T Ch Γ `T νW.(S k R k C) Ψ (8) where Ψ represents the rest of the conditions of the Par rule.6 The fact that P is typable implies that each of these premises must be provable. The derivation proving Γ, W `T S must be of the form a : ta [τb ] ∈ Γ W b : τb ∈ Γ W Γ W `T νYs .S 0 Γ W `T ahbi.νYs .S 0 Γ `T ahbi.νYs .S 0 + Ms ΨMs (9) where Γ W `T νYs .S 0 is proved by an inference of the shape ∀i ∈ I. Γ W Ys `T Si0 ∀i ∈ I. ΨSi0 Γ W `T νYs .S 0 (10) Analogously, Γ W `T R must be proved by an inference with the following shape a : ta [τx ] ∈ Γ W Γ W, x : τx `T νYr .R0 ΨR0 Γ W `T a(x).νYr .R0 ΨMr (11) 0 Γ W `T a(x).νYr .R + Mr 6 Note that Ψ is trivially true by P -safety of Γ . 36 E. D’Osualdo and C.-H. L. Ong and to prove Γ W, x : τx `T νYr .R0 ∀j ∈ J. Γ W, x : τx , Yr `T Rj0 Γ W, x : τx `T νYr .R0 ∀j ∈ J. ΨRj0 (12) We have to show that from this hypothesis we can infer that Γ `T Q or, equivalently (by Lemma 9.c), that Γ `T Q0 where Q0 = νW Ys Yr .(S 0 k R0 [ b/x ] k C). The derivation of this judgment can only end with an application of Par: ∀i ∈ I. Γ W 0 `T Si0 ∀j ∈ J. Γ W 0 `T Rj0 [ b/x ] ∀h ∈ H. Γ W 0 `T Ch Ψ0 Γ `T νW 0 .(S 0 k R0 [ b/x ] k C) In what follows we show how we can infer these premises are provable as a consequence of the provability of the premises of the proof of Γ `T νW.(S k R k C). From Lemma 9.b and Name Uniqueness, Γ W Ys `T Si0 from (10) implies Γ W 0 `T Si0 for each i ∈ I. Let Γr = Γ W, x : τx . We observe that by (9) and (11), τx = τb . From (11) we know that Γr Yr `T Rj0 which, by Lemma 3, implies Γr Yr `T Rj0 [ b/x ]. By Lemma 9.b we can infer Γr Yr Ys `T Rj0 [ b/x ] and by applying the same lemma again using fn(Rj0 [ b/x ]) ⊆ dom(Γ W Yr Ys ) and Name Uniqueness we obtain Γ W 0 `T Rj0 [ b/x ]. Again applying Lemma 9.b and Name Uniqueness, we have that Γ W `T Ch implies Γ W 0 `T Ch for each h ∈ H. To complete the proof we only need to prove that for each A ∈ {Si0 | i ∈ I} ∪ {Rj0 | j ∈ J} ∪ {Ch | h ∈ H}, Ψ 0 = ∀(x : τx ) ∈ W 0 . x tied to A in Q0 =⇒ base(Γ (fn(A))) < base(τx ) holds. This is trivially true by the hypothesis that Γ is P -safe. D.6 Proof of Theorem 6 We will consider the input output synchronisation case as the τ action one is similar and simpler. We will further assume that the sending action ahbi is such that ν(a : τa ) and ν(b : τb ) are both active restrictions of P , i.e. (a : τa ) ∈ W , (b : τb ) ∈ W with P ≡ νW.(S k R k C). The case when any of these two names is a free name of P can be easily handled with the aid of the assumption that Γ is P -safe. As in the proof of Theorem 5, the derivation of Γ `T P must follow the shape of (8). From T-shapedness of P we can conclude that both νYs .S 0 and νYr .R0 are T-shaped. We note that substitutions do not affect T-compatibility since they do not alter the set of bound names and their type annotations. Therefore, we can infer that νYr .R0 [ b/a ] is T-shaped. By Lemma 4 we know that ϕ = Φ(νW.(S k R k C)) ∈ FJP K, ϕr = Φ(νYr .R0 [ b/a ]) ∈ FJνYr .R0 [ b/x ]K and ϕs = Φ(νYs .S 0 ) ∈ FJνYs .S 0 K. Let ϕr = ϕmig ] ϕ¬mig where only ϕmig contains a leaf labelled with a term with b as a free name. These leaves will correspond On Hierarchical Communication Topologies in the π-calculus 37 to the continuations Rj0 that migrate in a(x).νYr .R0 , after the application of the substitution [ b/x ]. By assumption, inside P both S and R are in the scope of the restriction bounding a and S must also be in the scope of the restriction bounding b. Let ta = base(τa ) and tb = base(τb ), ϕ will contain two leaves nS and nR labelled with S and R respectively, having a common ancestor na labelled with (a, ta ); nS will have an ancestor nb labelled with (b, tb ). Let pa , pS and pR be the paths in ϕ leading from a root to na , nS and nR respectively. By T-compatibility of ϕ, we are left with only two possible cases: either 1) ta < tb or 2) tb < ta . Let us consider case 1) first. The tree in ϕ to which the nodes nS and nR belong, would have the following shape: na nb nS nR Now, we want to transform ϕ, by manipulating this tree, into a forest ϕ0 that is T-compatible by construction and such that there exists a term Q0 ≡ Q with forest(Q0 ) = ϕ0 , so that we can conclude Q is T-shaped. To do so, we introduce the following function, taking a labelled forest ϕ, a path p in ϕ and a labelled forest ρ and returning a labelled forest: ins(ϕ, p, ρ) := (Nϕ ] Nρ , ϕ ] ρ ] ins , `ϕ ] `ρ ) where n ins n0 if n0 ∈ minρ (Nρ ) and if `ρ (n0 ) = (y, ty ) then n ∈ max {m ∈ p | `ϕ (m) = (x, tx ), tx < ty } ϕ or if `ρ (n0 ) = A then n ∈ max {m ∈ p | `ϕ (m) = (x, tx ), x ∈ fn(A)}. ϕ Note that for each n0 , since p is a path, there can be at most one n such that n ins n0 . To obtain the desired ϕ0 , we first need to remove the leaves nS and nR from ϕ, as they represent the sequential processes which reacted, obtaining a forest ϕC . We argue that the ϕ0 we need is indeed ϕ0 = ins(ϕ1 , pS , ϕmig ) ϕ1 = ins(ϕ2 , pR , ϕ¬mig ) ϕ2 = ins(ϕC , pS , ϕs ) 38 E. D’Osualdo and C.-H. L. Ong It is easy to see that, by definition of ins, ϕ0 is T-compatible: ϕC , ϕs , ϕ¬mig and ϕmig are T-compatible by hypothesis, ins adds parent-edges only when they do not break T-compatibility. 0 To prove the claim we need to show that forest of a term congruent Qϕ is the 0 0 0 to νW Ys Yr .(S k R [ b/x ] k C). Let R = j∈J Rj0 , Jmig = {j ∈ J | x /νYr .R0 j}, J¬mig = J \ Jmig and Yr0 = {(x : τ ) ∈ Yr | x ∈ fn(Rj0 ), j ∈ J¬mig }. We know that no Rj0 with j ∈ J¬mig can contain x as a free name so Rj0 [ b/x ] = Rj0 . Now suppose we are able to prove that conditions 8.i, 8.ii and 8.iii of Lemma 6 hold for ϕC , ϕ1 , ϕ2 and ϕ0 . Then we could use Lemma 6 to prove a) b) c) d) ϕC = forest(QC ), QC ≡ QϕC = νW.C, ϕ2 = forest(Q2 ), Q2 ≡ Qϕ2 = νW Ys .(S 0 k C),Q ϕ1 = forest(Q1 ), Q1 ≡ Qϕ1 = νW Ys Yr0 .(S 0 k j∈J¬mig Rj0 k C), ϕ0 = forest(Q0 ), Q0 ≡ Qϕ0 = νW Ys Yr .(S 0 k R0 [ b/x ] k C) ≡ Q (it is straightforward to check that ϕC , ϕ2 , ϕ1 and ϕ0 have the right sets of nodes and labels to give rise to the right terms). We then proceed to check for each of the forests above that they satisfy conditions 8.i, 8.ii and 8.iii, thus proving the theorem. Condition 8.i requires that only leafs are labelled with sequential processes, condition that is easily satisfied by all of the above forests since none of the operations involved in their definition alters this property and the forests ϕ, ϕs and ϕr satisfy it by construction. Similarly, since νW.(S k R k C) is a normal form it satisfies Name Uniqueness, 8.ii is satisfied as we never use the same name more than once. Condition 8.iii holds on ϕ and hence it holds on ϕC since the latter contains all the nodes of ϕ labelled with names. Now consider ϕs : in the proof of Theorem 5 we established that Γ `T P implies that the premises ΨSi0 from (10) hold, that is base(Γ W (fn(Si0 ))) < base(τx ) holds for all Si0 for i ∈ I and all (x : τx ) ∈ Ys such that x /νYs .S 0 i. Since fn(Si0 ) ∩ W ⊆ fn(S 0 ) we know that every name (w : τw ) ∈ W such that w ∈ fn(Si0 ) will appear as a label (w, base(τw )) of a node nw in pS . Therefore, by definition of ins, we have that for each n ∈ NϕC , nw <ϕ2 n; in other words, in ϕ2 , every leaf in Nϕs labelled with Si0 is a descendent of a node labelled with (w, base(τw )) for each (w : τw ) ∈ W with w ∈ fn(Si0 ). This verifies condition 8.iii on ϕ2 . Similarly, by (12) the following premise must hold: base(Γ W (fn(Rj0 ))) < base(τx ) for all Rj0 for j ∈ J and all (y : τy ) ∈ Yr such that y /νYr .R0 j. We can then apply the same argument we applied to ϕ2 to show that condition 8.iii holds on ϕ1 . From (11) and the assumption ta < tb , we can conclude that the following premise must hold: base(Γ W (fn(Rj0 ) \ {a})) < ta for each j ∈ J such that Rj0 is migratable in a(x).νYr .R0 , i.e j ∈ Jmig . From this we can conclude that for every name (w : τw ) ∈ W such that w ∈ fn(Rj0 [ b/x ]) with j ∈ Jmig there must be a node in pa (and hence in pS ) labelled with (w, base(τw )). Now, some of the leaves in ϕmig will be labelled with terms having b as a free name; we show that in fact every node in ϕmig labelled with a (y, ty ) is indeed such that ty < tb . From the On Hierarchical Communication Topologies in the π-calculus 39 proof of Theorem 5 and Lemma 3 we know that from the hypothesis we can infer that Γ W `T νYr .R0 [ b/x ] and hence that for each j ∈ Jmig and each (y : τy ) ∈ Yr , if y is tied to Rj0 [ b/x ] in νYr .R0 [ b/x ] then base(Γ W (Rj0 [ b/x ])) < base(τy ). By Lemma 10 we know that every root of ϕmig is labelled with a name (y, ty ) which is tied to each of the leaves in its tree. Therefore each such ty satisfies base(Γ W (Rj0 [ b/x ])) < ty . By construction, there exists at least one j ∈ Jmig such that x ∈ fn(Rj0 ) and consequently such that b ∈ fn(Rj0 [ b/x ]). From this and b ∈ W we can conclude tb < ty for ty labelling a root in ϕmig . We can then conclude that {nb } = maxϕ2 {m ∈ pS | `ϕ (m) = (z, tz ), tz < ty } for each ty labelling a root of ϕmig , which means that each tree of ϕmig is placed as a subtree of nb in ϕ0 . This verifies condition 8.iii for ϕ0 completing the proof. Pictorially, the tree containing nS and nR in ϕ is now transformed in the following tree in ϕ0 : na ∈ ϕs ∈ ϕmig ∈ ϕ¬mig nb pS pR Case 2) — where tb < ta — is simpler as the migrating continuations can be treated just as the non-migrating ones. D.7 Role of ϕmig , ϕ¬mig and ins To illustrate the role of ϕmig , ϕ¬mig and the ins operation in the above proof, we show an example that would not be typable if we choose a simpler “migration” transformation. Consider the normal form P = νa b c.(!A k ahci) where A = a(x).νd.(ahdi k bhxi). To make types consistent we need annotations satisfying a : ta [t], b : tb [t], c : t and d : t. Any T satisfying the constraints tb < ta < t would allow us to prove ∅ `T P ; let then T be the forest with b  a  t with ta = a, tb = b and t = t. Let P 0 = νa b c d.(!A k ahdi k bhci) be the (only) successor of P . The following picture shows Φ(P ) in the middle, on the left a forest in FJP 0 K extracted by just putting the continuation of A under the message, on the right the forest obtained by using ins on the non-migrating continuations of A: b a c d bhci ahdi !A ← b a !A c ahci → !A b a c d bhci ahdi Clearly, the tree on the left is not T-compatible since c and d have the same base type t. Instead, the tree on the right can be obtained because ins inserts the non-migrating continuation as close to the root as possible. 40 E E. D’Osualdo and C.-H. L. Ong Supplementary Material for Section 6 E.1 A type inference example
Take the term ! τ .νs c.P of Example 2. We start by annotating each restriction νa with a fresh type variable ν(a : ta ). Then we perform a type derivation as in Example 10, obtaining the following data-flow constraints: ts = ts [tz ] tc = tc [tx ] tx = tx [ty ] tz = tx = tm = tz [td ] from which we learn that: - td is unconstrained; we use the base type variable td for base(td ); - ty = td ; - tx = tz and base(tm ) = tx . We can therefore completely specify the types just by associating ts , tc , tx and td to nodes in a forest: all the types would be determined as a consequence of the data-flow constraints, apart from td to which we can safely assign the type td . During the type derivation we also collected the following base type constraints: base(tz ) < base(td ) base(tc ) < base(tm ) base(tc ) < base(tx ) base(tx ) < base(tc ) ∨ base(ts ) < base(tc ) These can be simplified and normalised using the equations on types seen above obtaining the set Cνs c.P = {tx < tc ∨ ts < tc , tc < tx , tx < td } Hence any choice of T ⊇ {tx , tc , ts , td } such that ts <T tc <T tx <T td would make the typing succeed. F F.1 Supplementary Material for Section 7 Encoding of Reset nets A reset net N with n places is a finite set of transitions of the form (u, R) where n u ∈ {−1, 0, +1} is the update vector and R ⊆ {1, . . . , n} is the reset set. A marking m is a vector in Nn ; a transition (u, R) is said to be enabled at m if m − u > 0. The semantics of a reset net N with initial marking m0 is the transition system (Nn , [i, m0 ) where m [i m0 if there exists a transition (u, R) in N that is enabled in m and such that ( mi + ui if i 6∈ R m0i = 0 if i ∈ R On Hierarchical Communication Topologies in the π-calculus 41 To simulate place i in a reset net we can construct a term that implements a counter with increment and reset:

Ci = ! pi (t). inc i .(t k pi hti) + dec i .(t.pi hti) + rst i .(νt0i .pi ht0i i) Here, the number of processes t in parallel with pi hti represent the current value of the marking in place
i. A transition (u, R) is encoded as a process Tu,R = ! valid .Du .Iu .ZR .valid where Du = dec j1 . · · · .dec jk with {j1 , . . . , jk } = {j | uj < 0}, Iu = inc i1 . · · · .inc il with {i1 , . . . , il } = {i | ui > 0}, and ZR = rst r1 . · · · .rst rm with R = {r1 , . . . , rl }. A marking m is encoded by a process Q Q m
PN,m = valid k 1≤i≤n pi hti i k Ci k ti i k (u,R)∈N Tu,R Actions on the name valid act as a global lock: a transition may need many steps to complete, but by acquiring and releasing valid it can ensure no other transition will fire in between. If a transition tries to decrement a counter below zero, the counter would deadlock causing valid to be never released again. Therefore, the encoding preserves coverability: m is coverable in N from m0 if and only if PN,m is coverable from PN,m0 . Reachability is not preserved because each reset would generate some ‘garbage’ term νt.(t k . . . k t) and thus, even when m is reachable, PN,m might not be reachable alone, but only in parallel with some garbage. The reader can verify that any encoding P can be typed under the hierarchy valid < inc 1 < dec 1 < rst 1 < t1 < p1 < · · · < inc n < dec n < rst n < tn < pn < t01 < · · · < t0n by annotating each restriction νt0i as ν(t0i : t0i ) and using the P -safe environment {(x : x) | x ∈ fn(P )}. F.2 A weak encoding of Minsky machines A k-counters Minsky machine is a finite list of instructions I1 , . . . , In each of which can be either an increase or a decrease command. An increase command inc i j increases counter i and jumps to instruction Ij . A decrease command dec i j1 j2 decreases counter i jumping to instruction Ij1 if the counter is greater than zero, or jumps to Ij2 otherwise. We implement a counter i with
the process Ci of Example 3. An increase Im = inc i j is encoded by ! im .inc i .ij . A decrease
Im = dec i j1 j2 is encoded by ! im .(dec i .ij1 + rst i .ij2 ) . A configuration of a Minsky machine is the vector of values of its registers r1 , . . . , rk and the current instruction j; its encoding is the term Q Q ri 1≤i≤k νti .(pi hti i k ti ) k ij k 1≤m≤n PIm where PIm is the encoding of the instruction Im . When a counter is zero, performing a decrease command on it in the encoding presents a non-deterministic choice between sending a decrease or a reset signal 42 E. D’Osualdo and C.-H. L. Ong to the counter. In the branch where the decrease signal is sent, the counter process will deadlock, ending up in a term that is clearly not an encoding of a configuration of the Minsky machine. If instead a reset signal is sent, the counter will refresh the name t with a new name, but the old one would be discarded as there is no sequential term which knows it. When a counter is not zero, the branch where the decrease signal is sent will simply succeed, while the resetting one will generate some ‘garbage’ term νt.(t k . . . k t) in parallel with the rest of the encoding of the Minsky machine’s configuration. A configuration of the machine is thus reachable if and only if its encoding (without garbage) is reachable from the encoding of the machine. This proves Theorem 8. 

Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation arXiv:1710.06261v1 [] 17 Oct 2017 Yin Tat Lee∗, Santosh S. Vempala† October 18, 2017 Abstract We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number 2 of steps needed is O∗ (mn 3 ), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds. Contents . . . . . 3 3 5 6 7 7 2 Basics of Hamiltonian Monte Carlo 2.1 Hamiltonian Monte Carlo on Riemannian manifolds . . . . . . . . . . . . . . . . . . . 8 10 3 Convergence of Riemannian Hamiltonian Monte Carlo 3.1 Basics of geometric Markov chains . . . . . . . . . . . . . 3.2 Overlap of one-step distributions . . . . . . . . . . . . . . 3.2.1 Proof Outline . . . . . . . . . . . . . . . . . . . . . 3.2.2 Variation of Hamiltonian curve . . . . . . . . . . . 3.2.3 Local Uniqueness of Hamiltonian Curves . . . . . . 3.2.4 Smoothness of one-step distributions . . . . . . . . 3.3 Convergence bound . . . . . . . . . . . . . . . . . . . . . . 13 13 15 15 16 19 21 23 1 Introduction 1.1 Results . . . . 1.2 Approach and 1.3 Practicality . 1.4 Notation . . 1.5 Organization . . . . . . . . contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Improved analysis of the convergence 24 4.1 Improved one-to-one correspondence for Hamiltonian curve . . . . . . . . . . . . . . 25 4.2 Improved smoothness of one-step distributions . . . . . . . . . . . . . . . . . . . . . . 26 ∗ † University of Washington and Microsoft Research, yintat@uw.edu Georgia Tech, vempala@gatech.edu 1 5 Gibbs sampling on manifolds 29 5.1 Isoperimetry for Hessian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Sampling with the log barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6 Polytope volume computation: Gaussian cooling on manifolds 6.1 Algorithm: cooling schedule . . . . . . . . . . . . . . . . . . . . . 6.2 Correctness of the algorithm . . . . . . . . . . . . . . . . . . . . . 6.2.1 Initial and terminal conditions . . . . . . . . . . . . . . . 6.2.2 Variance of Yx . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Main lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Volume computation with the log barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 32 32 35 37 38 7 Logarithmic barrier 7.1 Riemannian geometry on ML (G2 ) . . . . 7.2 Hamiltonian walk on ML . . . . . . . . . 7.3 Randomness of the Hamiltonian flow (ℓ0 ) 7.4 Parameters R1 , R2 and R3 . . . . . . . . . 7.5 Stability of L2 + L4 + L∞ norm (ℓ1 ) . . . 7.6 Mixing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 40 42 46 53 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Matrix ODE 59 B Concentration 61 C Calculus 61 D Basic definitions of Riemannian geometry 62 D.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 D.2 Hessian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2 1 Introduction Hamiltonian dynamics provide an elegant alternative to Newtonian mechanics. The Hamiltonian H, which captures jointly the potential and kinetic energy of a particle, is a function of its position and velocity. First-order differential equations describe the change in both. ∂H(x, v) dx = , dt ∂v dv ∂H(x, v) =− . dt ∂x As we review in Section 2, these equations preserve the Hamiltonian H. Riemannian Hamiltonian Monte Carlo (or RHMC) [26, 25][6, 7] is a Markov Chain Monte Carlo method for sampling from a desired distribution. The target distribution is encoded in the definition of the Hamiltonian. Each step of the method consists of the following: At a current point x, 1. Pick a random velocity y according to a local distribution defined by x (in the simplest setting, this is the standard Gaussian distribution for every x). 2. Move along the Hamiltonian curve defined by Hamiltonian dynamics at (x, y) for time (distance) δ. For a suitable choice of H, the marginal distribution of the current point x approaches the desired target distribution. Conceptually, the main advantage of RHMC is that it does not require a Metropolis filter (as in the Metropolis-Hastings method) and its step sizes are therefore not severely limited even in high dimension. Over the past two decades, RHMC has become very popular in Statistics and Machine Learning, being applied to Bayesian learning, to evaluate expectations and likelihood of large models by sampling from the appropriate Gibbs distribution, etc. It has been reported to significantly outperform other known methods [2, 25] and much effort has been made to make each step efficient by the use of numerical methods for solving ODEs. In spite of all these developments and the remarkable empirical popularity of RHMC, analyzing its rate of convergence and thus rigorously explaining its success has remained an open question. 1.1 Results In this paper, we analyze the mixing rate of Hamiltonian Monte Carlo for a general function f as a Gibbs sampler, i.e., to generate samples from the density proportional to e−f (x) . The corresponding Hamiltonian is H(x, v) = f (x) + 21 log((2π)n det g(x)) + 12 v T g(x)−1 v for some metric g. We show that for x in a compact manifold, the conductance of the Markov chain is bounded in terms of a few parameters of the metric g and the function f . The parameters and resulting bounds are given in Corollary 28 and Theorem 30. Roughly speaking, the guarantee says that Hamiltonian Monte Carlo mixes in polynomial time for smooth Hamiltonians. We note that these guarantees use only the smoothness and Cheeger constant (expansion) of the function, without any convexity type assumptions. Thus, they might provide insight in nonconvex settings where (R)HMC is often applied. We then focus on logconcave densities in Rn , i.e., f (x) is a convex function. This class of functions appears naturally in many contexts and is known to be sampleable in polynomial-time given access to a function value oracle. For logconcave densities, the current fastest sampling algorithms use n4 function calls, even for uniform sampling [20, 23], and n2.5 oracle calls given a 3 warm start after appropriate rounding (linear transformation) [14]. In the prototypical setting of uniform sampling from a polytope Ax ≥ b, with m inequalities, the general complexity is no better, with each function evaluation taking O(mn) arithmetic operations, for an overall complexity of n4 · mn = mn5 in the worst case and n2.5 · mn after rounding from a warm start. The work of Kannan and Narayanan [11] gives an algorithm of complexity mn2 · mnω−1 from an arbitrary start and mn · mnω−1 from a warm start (here ω is the matrix multiplication exponent), which is better than the general case when the number of facets m is not too large. This was recently improved to mn0.75 · mnω−1 from a warm start [13]; the subquadratic complexity for the number of steps is significant since all known general oracle methods cannot below a quadratic number of steps. The leading algorithms and their guarantees are summarized in Table 1. Year 1997 [10] 2003 [21] 2009 [11] 2016 [13] 2016 [15] This paper Algorithm Ball walk# Hit-and-run# Dikin walk Geodesic walk Ball walk# RHMC Steps n3 n3 mn 3 mn 4 n2.5 2 mn 3 Cost per step mn mn mnω−1 mnω−1 mn mnω−1 Table 1: The complexity of uniform polytope sampling from a warm start, where each step of e every algorithm uses O(n) bit of randomness. The entries marked # are for general convex bodies presented by oracles, while the rest are for polytopes. In this paper, using RHMC, we improve the complexity of sampling polytopes. In fact we do this for a general family of Gibbs distributions, of the form e−αφ(x) where φ(x) is a convex function over a polytope. When φ(x) is the standard logarithmic barrier function and g(x) is its Hessian, 1 1 we get a sampling method that mixes in only n 6 m 2 + 1 2 2 n5 m5 1 1 α 5 +m− 5 + n3 α+m−1 steps from a warm start! When α = 1/m, the resulting distribution is very close to uniform over the polytope. Theorem 1. Let φ be the logarithmic barrier for a polytope M with m constraints and n variables. Hamiltonian Monte Carlo applied to the function f = exp(−αφ(x)) and the metric given by ∇2 φ with appropriate step size mixes in ! 1 2 1 3 3n3 1 1 n m e + m2 n6 O + α + m−1 α 31 + m− 13 steps where each step is the solution of a Hamiltonian ODE. In recent independent work, Mangoubi and Smith [24] analyze Euclidean HMC in the oracle setting, i.e., assuming an oracle for evaluating φ. Their analysis formally gives a dimension-independent convergence rate based on certain regularity assumptions such as strong convexity and smoothness of the Hamiltonian H. Unfortunately, these assumptions do not hold for the polytope sampling problem. An important application of sampling is integration. The complexity of integration for general logconcave functions is also n4 oracle calls. For polytopes, the most natural questions is computing its volume. For this problem, the current best complexity is n4 · mn, where the factor of O(mn) is the complexity of checking membership in a polytope. Thus, even for explicitly specified polytopes, the complexity of estimating the volume from previous work is asymptotically the same as that 4 for a general convex body given by a membership oracle. Here we obtain a volume algorithm with 2 complexity mn 3 · mnω−1 , improving substantially on previous algorithms. The volume algorithm is based using Hamiltonian Monte Carlo for sampling from a sequence of Gibbs distributions over polytopes. We remark that in the case when m = O(n)1 , the final complexity is o(n4 ) arithmetic operations, improving by more than a quadratic factor in the dimension over the previous best complexity of Õ(n6 ) operations for arbitrary polytopes. These results and prior developments are given in Table 2. Year 1989 [5] 1989-93 [17, 4, 1, 18, 19] 1997 [10] 2003 [22] 2015 [3] This paper Algorithm DFK many improvements DFK, Speedy walk, isotropy Annealing, hit-and-run Gaussian Cooling* RHMC + Gaussian Cooling Steps n23 n7 n5 n4 n3 2 mn 3 Cost per step mn mn mn mn mn mnω−1 e Table 2: The complexity of volume estimation, each step uses O(n) bit of randomness, all except * the last for general convex bodies (the result marked is for well-rounded convex bodies). The current paper applies to general polytopes, and is the first improvement utilizing their structure. Theorem 2. For any polytope P = {x : Ax ≥ b} with m constraints and n variables, and any ε > 0, the Hamiltonian volume algorithm estimates the volume of P to within 1 ± ε multiplicative   2 −2 e 3 factor using O mn ε steps where each step consists of solving a first-order ODE and takes time   e mnω−1 LO(1) logO(1) 1 and L is the bit complexity2 of the polytope. O ε A key ingredient in the analysis of RHMC is a new isoperimetric inequality for Gibbs distributions over manifolds. This inequality can be seen as an evidence of a manifold version of the KLS hyperplane conjecture. For the family of Gibbs distributions induced by convex functions with convex Hessians, the expansion is within a constant factor of that of a hyperplane cut. This result might be of independent interest. 1.2 Approach and contributions Traditional methods to sample from distributions in Rn are based on random walks that take straight line steps (grid walk, ball walk, hit-and-run). While this leads to polynomial-time convergence for logconcave distributions, the length of each step has to be small due to boundary effects, and a Metropolis filter (rejection sampling) has to be applied to ensure the limiting  distribution is the desired one. These walks cannot afford a step of length greater than δ = O √1n for a distribution in isotropic position, and take a quadratic number of steps even for the hypercube. The Dikin walk for polytopes [11], which explicitly takes into account the boundary of polytope at each step, has a varying step size, but still runs into similar issues and the bound on its convergence rate is O(mn) for a polytope with m facets. 1 We suspect that the LS barrier [12] might be used to get a faster algorithm even in the regime even if m is sub-exponential. However, our proof requires a delicate estimate of the fourth derivative of the barrier functions. Therefore, such a result either requires a new proof or a unpleasantly long version of the current proof. 2 L = log(m + dmax + kbk∞ ) where dmax is the largest absolute value of the determinant of a square sub-matrix of A. 5 In a recent paper [13], we introduced the geodesic walk. Rather than using straight lines in Euclidean space, each step of the walk is along a geodesic (locally shortest path) of a Riemannian metric. More precisely, each step first makes a deterministic move depending on the current point (drift), then moves along a geodesic in a random initial direction and finally uses a Metropolis filter. Each step can be computed by solving a first-order ODE. Due to the combination of drift and geodesic, the local 1-step distributions are smoother than that of the Dikin walk and larger steps can be taken while keeping a bounded rejection probability for the filter. For sampling polytopes, 3 the manifold/metric defined by the standard log barrier gives a convergence rate of mn 4 , going below the quadratic (or higher) bound of all previous sampling methods. One major difficulty with geodesic walk is ensuring the stationary distribution is uniform. For high dimensional problems, this necessitates taking a sufficiently small step size and then rejecting some samples according to the desired transition probabilities according to Metropolis filter. Unfortunately, computing these transition probabilities can be very expensive. For the geodesic walk, it entails solving an n × n size matrix ODE. Hamiltonian Monte Carlo bears some similarity to the geodesic walk — each step is a random (non-linear) curve. But the Hamiltonian-preserving nature of the process obviates the most expensive ingredient, Metropolis filter. Due to this, the step size can be made longer, and as a result we 2 obtain a faster sampling algorithm for polytopes that mixes in mn 3 steps (the per-step complexity remains essentially the same, needing the solution of an ODE). To get a faster algorithm for volume computation, we extends the analysis to a general family of Gibbs distributions, including f (x) = e−αφ(x) where φ(x) is the standard log-barrier and α > 0. We show that the smoothness we need for the sampling corresponding to a variant of self-concordance defined in Definition 46. Furthermore, we establish an isoperimetric inequality for this class of functions. This can be viewed as an extension of the KLS hyperplane conjecture from Euclidean to Riemannian metrics (the analogous case in Euclidean space to what we prove here is the isoperimetry of the Gaussian density function multiplied by any logconcave function, a case for which the KLS conjecture holds). The mixing rate for this family of functions is sublinear for α = Ω(1). Finally, we study the Gaussian Cooling schedule of [3]. We show that in the manifold setting, 2 the Gaussian distribution e−kxk /2 can be replaced by e−αφ(x) . Moreover, the speed of Gaussian Cooling depends on the “thin-shell” constant of the manifold and classical self-concordance of φ. Combining all of these ideas, we obtain a faster algorithm for polytope volume computation. The resulting complexity of polytope volume computation is the same as that of sampling uniformly 2 from a warm start: mn 3 steps. To illustrate the improvement, for polytopes with m = O(n) facets, 5 the new bound is n 3 while the previous best bound was n4 . 1.3 Practicality From the experiments, the ball walk/hit-and-run seem to mix in n2 steps, the geodesic walk seems to mix in sublinear number of steps (due to the Metropolis filter bottleneck) and RHMC seems to mix in only polylogarithmic number of steps. One advantage of RHMC compared to the geodesic walk is that it does not require the expensive Metropolis filter that involves solving n × n matrix ODEs. In the future, we plan to do an empirical comparison study of different sampling algorithms. We are hopeful that using RHMC we might finally be able to sample from polytopes in millions of dimensions after more than three decades of research on this topic! 6 1.4 Notation Throughout the paper, we use lowercase letter for vectors and vector fields and uppercase letter for d matrices and tensors. We use ek to denote coordinate vectors. We use dt for the usual derivative, df (c(t)) ∂ is the derivative of some function f along a curve c parametrized by t, we use ∂v for e.g. dt k th the usual partial derivative. We use D f (x)[v1 , v2 , · · · , vk ] for the k directional derivative of f at x along v1 , v2 , · · · , vk . We use ∇ for the usual gradient and the connection (manifold derivative, defined in Section D which takes into account the local metric), Dv for the directional derivative of a vector with respect to the vector (or vector field) v (again, defined in Section D), and Dt if the curve v(t) is clear from the context. We use g for the local metric. Given a point x ∈ M , g is a matrix with entries gij . Its inverse has entries g ij . Also, n is the dimension, m the number of inequalities. We use dT V for the total variation (or L1 ) distance between two distributions. 1.5 Organization In Section 2, we define the Riemannian Hamiltonian Monte Carlo and study its basic properties such as time-reversibility. In Section 3, we give the the first convergence rate analysis of RHMC. However, the convergence rate is weak for the sampling applications (it is polynomial, but not better than previous methods). In Section 4, we introduce more parameters and use them to get a tighter analysis of RHMC. In Section 5, we study the isoperimetric constant of f (x) = e−αφ(x) under the metric ∇2 φ(x). In Section 6, we study the generalized Gaussian Cooling schedule and its relation to the thin-shell constant. Finally, in Section 7, we compute the parameters we need for the log barrier function. 7 2 Basics of Hamiltonian Monte Carlo In this section, we define the Hamiltonian Monte Carlo method for sampling from a general distribution e−H(x,y) . Hamiltonian Monte Carlo uses curves instead of straight lines and this makes the walk time-reversible even if the target distribution is not uniform, with no need for a rejection sampling step. In contrast, classical approaches such as the ball walk require an explicit rejection step to converge to a desired stationary distribution. Definition 3. Given a continuous, twice-differentiable function H : M×Rn ⊂ Rn ×Rn → R (called the Hamiltonian, which often corresponds to the total energy of a system) where M is the x domain of H, we say (x(t), y(t)) follows a Hamiltonian curve if it satisfies the Hamiltonian equations ∂H(x, y) dx = , dt ∂y dy ∂H(x, y) =− . dt ∂x (2.1) def We define the map Tδ (x, y) = (x(δ), y(δ)) where the (x(t), y(t)) follows the Hamiltonian curve with the initial condition (x(0), y(0)) = (x, y). Hamiltonian Monte Carlo is the result of a sequence of randomly generated Hamiltonian curves. Algorithm 1: Hamiltonian Monte Carlo Input: some initial point x(1) ∈ M. for i = 1, 2, · · · , T do R 1 (k) Sample y (k+ 2 ) according to e−H(x ,y) /π(x(k) ) where π(x) = Rn e−H(x,y) dy. 1 With probability 12 , set (x(k+1) , y (k+1) ) = Tδ (x(k) , y (k+ 2 ) ). 1 Otherwise, (x(k+1) , y (k+1) ) = T−δ (x(k) , y (k+ 2 ) ). end Output: (x(T +1) , y (T +1) ). Lemma 4 (Energy Conservation). For any Hamiltonian curve (x(t), y(t)), we have that d H(x(t), y(t)) = 0. dt Proof. Note that ∂H dx ∂H dy ∂H ∂H ∂H ∂H d H(x(t), y(t)) = + = − = 0. dt ∂x dt ∂y dt ∂x ∂y ∂y ∂x Lemma 5 (Measure Preservation). For any t ≥ 0, we have that det (DTt (x, y)) = 1 where DTt (x, y) is the Jacobian of the map Tt at the point (x, y). 8 Proof. Let (x(t, s), y(t, s)) be a family of Hamiltonian curves given by Tt (x + sdx , y + sdy ). We write u(t) = ∂ ∂ x(t, s)|s=0 , v(t) = y(t, s)|s=0 . ∂s ∂s By differentiating the Hamiltonian equations (2.1) w.r.t. s, we have that du ∂ 2 H(x, y) ∂ 2 H(x, y) = u+ v, dt ∂y∂x ∂y∂y dv ∂ 2 H(x, y) ∂ 2 H(x, y) =− u− v, dt ∂x∂x ∂x∂y (u(0), v(0)) = (dx , dy ). This can be captured by the following matrix ODE ∂ 2 H(x(t),y(t)) ∂y∂x 2 − ∂ H(x(t),y(t)) ∂x∂x dΦ = dt ∂ 2 H(x(t),y(t)) ∂y∂y 2 − ∂ H(x(t),y(t)) ∂x∂y ! Φ(t) dx dy  Φ(0) = I using the equation DTt (x, y)  dx dy  =  u(t) v(t)  = Φ(t)  . Therefore, DTt (x, y) = Φ(t). Next, we observe that   d −1 d log det Φ(t) = Tr Φ(t) Φ(t) = Tr dt dt ∂ 2 H(x(t),y(t)) ∂y∂x 2 − ∂ H(x(t),y(t)) ∂x∂x ∂ 2 H(x(t),y(t)) ∂y∂y 2 − ∂ H(x(t),y(t)) ∂x∂y ! = 0. Hence, det Φ(t) = det Φ(0) = 1. Using the previous two lemmas, we now show that Hamiltonian Monte Carlo indeed converges to the desired distribution. Lemma 6 (Time reversibility). Let px (x′ ) denote the probability density of one step of the Hamiltonian Monte Carlo starting at x. We have that π(x)px (x′ ) = π(x′ )px′ (x) R for almost everywhere in x and x′ where π(x) = Rn e−H(x,y) dy. Proof. Fix x and x′ . Let Fδx (y) be the x component of Tδ (x, y). Let V+ = {y : Fδx (y) = x′ } and x (x) = x′ )}. Then, V− = {y : F−δ π(x)px (x′ ) = 1 2 Z y∈V+ 1 e−H(x,y)
+ x 2 det DFδ (y) Z y∈V− e−H(x,y)
. x (y) det DF−δ We note that this formula assumed that DFδx is invertible. Sard’s theorem showed that Fδx (N ) is def measure 0 where N = {y : DFsx (y) is not invertible}. Therefore, the formula is correct except for a measure zero subset. 9 By reversing time for the Hamiltonian curve, we have that for the same V± , Z Z ′ ′ ′ ′ 1 e−H(x ,y ) e−H(x ,y ) 1

+ π(x′ )px′ (x) = x′ (y ′ ) 2 y∈V+ det DF−δ 2 y∈V− det DFδx′ (y ′ ) (2.2) where y ′ denotes the y component of Tδ (x, y) and T−δ (x, y) in the first  and second  sum respectively. A B We compare the first terms in both equations. Let DTδ (x, y) = . Since Tδ ◦ T−δ = I C D and Tδ (x, y) = (x′ , y ′ ), the inverse function theorem shows that DT−δ (x′ , y ′ ) is the inverse map of DTδ (x, y). Hence, we have that   −1  A B · · · −A−1 B(D − CA−1 B)−1 ′ ′ DT−δ (x , y ) = . = ··· ··· C D ′ x (y ′ ) = −A−1 B(D − CA−1 B)−1 . Hence, we have that Therefore, we have that Fδx (y) = B and F−δ  
−1 x′ = det DF−δ (y ′ ) = det A−1 det B det D − CA−1 B |det B|   . A B det C D   A B = 1 (Lemma 5), we have that C D   x′ det DF−δ (y ′ ) = |det (DFδx (y))| . Using that det (DTt (x, y)) = det Hence, we have that 1 2 Z y∈V+ 1 e−H(x,y)
= x 2 det DFδ (y) = 1 2 −H(x′ ,y ′ ) Z Z y∈V+ e−H(x,y)
x′ (y ′ ) det DF−δ ′ y∈V+ ′ e−H(x ,y )
x′ (y ′ ) det DF−δ where we used that e−H(x,y) = e (Lemma 4) at the end. For the second term in (2.2), by the same calculation, we have that Z Z ′ ′ 1 e−H(x,y) e−H(x ,y ) 1

= x (y) 2 y∈V− det DF−δ 2 y∈V+ det DFδx′ (y ′ ) Combining both terms we have the result. The main challenge in analyzing Hamiltonian Monte Carlo is to bound its mixing time. 2.1 Hamiltonian Monte Carlo on Riemannian manifolds Suppose we want to sample from the distribution e−f (x) . We define the following energy function H: 1 1 def (2.3) H(x, v) = f (x) + log((2π)n det g(x)) + v T g(x)−1 v. 2 2 One can view x as the location and v as the velocity. The following lemma shows that the first variable x(t) in the Hamiltonian curve satisfies a second-order differential equation. When we view the domain M as a manifold, this equation is simply Dt dx dt = µ(x), namely, x acts like a particle under the force field µ. (For relevant background on manifolds, we refer the reader to Appendix D). 10 Lemma 7. In Euclidean coordinates, The Hamiltonian equation for (2.3) can be rewritten as Dt dx =µ(x), dt dx (0) ∼N (0, g(x)−1 ) dt   where µ(x) = −g(x)−1 ∇f (x) − 12 g(x)−1 Tr g(x)−1 Dg(x) and Dt is the Levi-Civita connection on the manifold M with metric g. Proof. From the definition of the Hamiltonian curve, we have that dx = g(x)−1 v dt  1 dx T dv 1  dx = −∇f (x) − Tr g(x)−1 Dg(x) + Dg(x) . dt 2 2 dt dt Putting the two equations together, we have that dx dv d2 x = − g(x)−1 Dg(x)[ ]g(x)−1 v + g(x)−1 2 dt dt dt   1 dx dx 1 dx T dx = − g(x)−1 Dg(x)[ ] − g(x)−1 ∇f (x) − g(x)−1 Tr g(x)−1 Dg(x) + g(x)−1 Dg(x) . dt dt 2 2 dt dt Hence, T   dx dx 1 1 d2 x dx −1 −1 dx + g(x) Dg(x)[ ] − g(x) = − g(x)−1 ∇f (x) − g(x)−1 Tr g(x)−1 Dg(x) . Dg(x) 2 dt dt dt 2 dt dt 2 (2.4) Using the formula of Christoffel symbols Dt dx d2 x X dxi dxj k = 2 + Γ ek dt dt dt dt ij where Γkij = 1 X kl g (∂j gli + ∂i glj − ∂l gij ), 2 l ijk we have that Dt X dxi dxj d2 x 1 dx = 2 + g(x)−1 (∂j gli + ∂i glj − ∂l gij )el dt dt 2 dt dt ijl = T d2 x dx dx 1 dx −1 −1 dx + g(x) Dg(x)[ ] − g(x) Dg(x) . dt2 dt dt 2 dt dt Putting this into (2.4) gives Dt   dx 1 = − g(x)−1 ∇f − g(x)−1 Tr g(x)−1 Dg(x) . dt 2 Motivated by this, we define the Hamiltonian map as the first component of the Hamiltonian dynamics operator T defined earlier. For the reader familiar with Riemannian geometry, this is similar to the exponential map (for background, see Appendix D). 11 Definition 8. Let Hamx,δ (vx ) = γ(δ) where γ(t) be the solution of the Hamiltonian equation ′ Dt dγ dt = µ with initial conditions γ(0) = x and γ (0) = vx . We also denote Hamx,1 (vx ) by Hamx (vx ). We now give two examples of Hamiltonian Monte Carlo. Example 9. When g(x) = I, the Hamiltonian curve acts like stochastic gradient descent for the function f with each random perturbation drawn from a standard Gaussian. Dt dx = −∇f (x). dt When g(x) = ∇2 f (x), the Hamiltonian curve acts like a stochastic Newton curve for the function f + ψ:
−1 dx Dt = − ∇2 f (x) ∇(f (x) + ψ(x)) dt where the volumetric function ψ(x) = log det ∇2 f (x). Next we derive a formula for the transition probability in Euclidean coordinates. Lemma 10. For any x ∈ M ⊂ Rn and s > 0, the probability density of the 1-step distribution from x is given by s   X 1 det (g(y)) 2 −1 exp − kvx kx (2.5) px (y) = |det(DHamx,δ (vx ))| (2π)n 2 vx :Hamx,δ (vx )=y where DHamx,δ (vx ) is the Jacobian of the Hamiltonian map Hamx,δ . Proof. We prove the formula by separately considering each vx ∈ Tx M s.t. Hamx,δ (vx ) = y, then summing up. In the tangent space Tx M, the point vx follows a Gaussian step. Therefore, the probability density of vx in Tx M is as follows:   1 1 2 exp − kv k pTxx M (vx ) = x x . 2 (2π)n/2 Let y = Hamx,δ (vx ) and F : Tx M → Rn be defined by F (v) = idM→Rn ◦ Hamx,δ (v). Here Rn is the same set as M but endowed with the Euclidean metric. Hence, we have DF (vx ) = DidM→Rn (y)DHamx,δ (vx ). The result follows from px (y) = |det(DF (vx ))|−1 pxTx M (vx ) and det DF (vx ) = det (DidM→Rn (y)) det (DHamx,δ (vx )) = det(g(y))−1/2 det (DHamx,δ (vx )) . 12 3 Convergence of Riemannian Hamiltonian Monte Carlo Hamiltonian Monte Carlo is a Markov chain on a manifold whose stationary stationary distribution has density q(x) proportional to exp(−f (x)). We will bound the conductance of this Markov chain and thereby its mixing time to converge to the stationary distribution. Bounding conductance involves showing (a) the induced metric on the state space satisfies a strong isoperimetric inequality and (b) two points that are close in metric distance are also close in probabilistic distance, i.e., the one-step distributions from two nearby points have large overlap. In this section and the next, we present general conductance bounds using parameters determined by the associated manifold. In Section 7, we bound these parameters for the manifold corresponding to the logarithmic barrier in a polytope. 3.1 Basics of geometric Markov chains For completeness, we will discuss some standard techniques in geometric random walks in this subsection. For a Markov chain with state space M, stationary distribution q and next step distribution pu (·) for any u ∈ M, the conductance of the Markov chain is R def S pu (M \ S)dq(u) φ = inf . S⊂M min {q(S), q(M \ S)} The conductance of an ergodic Markov chain allows us to bound its mixing time, i.e., the rate of convergence to its stationary distribution, e.g., via the following theorem of Lovász and Simonovits. Theorem 11 ([19]). Let qt be the distribution of the current point after t steps of a Markov chain with stationary distribution q and conductance at least φ, starting from initial distribution q0 . For any ε > 0, s  t dq0 (x) φ2 1 . Ex∼q0 dT V (qt , q) ≤ ε + 1− ε dq(x) 2 Definition 12. The isoperimetry of a metric space M with target distribution q is R d(S,x)≤δ q(x)dx − q(S) def ψ = inf min δ>0 S⊂M δ min {q(S), q(M \ S)} where d is the shortest path distance in M. The proof of the following theorem follows the standard outline for geometric random walks (see e.g., [29]). Lemma 13. Given a metric space M and a time-reversible Markov chain p on M with stationary distribution q. Fix any r > 0. Suppose that for any x, y ∈ M with d(x, z) < r, we have that dT V (px , py ) ≤ 0.9. Then, the conductance of the Markov chain is Ω(rψ). Proof. Let S be any measurable subset of M. Then our goal is to bound the conductance of the Markov chain R S px (M \ S) dq(x) = Ω (rψ) . min {q(S), q(M \ S)} R R Since the Markov chain is time-reversible (For any two subsets A, B, A px (B) dq(x) = B px (A) dq(x)), we can write the numerator of the left hand side above as ! Z Z 1 px (S) dq(x) . px (M \ S) dq(x) + 2 M\S S 13 Define S1 = {x ∈ S : px (M \ S) < 0.05} S2 = {x ∈ M \ S : px (S) < 0.05} S3 = M \ S1 \ S2 . Without loss of generality, we can assume that q(S1 ) ≥ (1/2)q(S) and q(S2 ) ≥ (1/2)q(M \ S) (if R not, S px (M \ S) dq(x) = Ω(1) and hence the conductance is Ω(1).) Next, we note that for any two points x ∈ S1 and y ∈ S2 , dT V (px , py ) > 0.9. Therefore, by the assumption, we have that d(x, y) ≥ r. Therefore, by the definition of ψr , we have that Z q(x)dx − q(S1 ) q(S3 ) ≥ d(S1 ,x)≤r ≥ rψ min {q(S1 ), q(M \ S1 )} ≥ rψ min {q(S1 ), q(S2 )} . Going back to the conductance, 1 2 Z S px (M \ S) dq(x) + Z ! px (S) dq(x) M\S 1 ≥ 2 Z (0.05)dq(x) S3 = Ω (rψ) min{q(S1 ), q(S2 )} = Ω (rψ) min{q(S), q(M \ S)}. Therefore, the conductance of the Markov chain is Ω(rψ). Combining Theorem 13 and Lemma 13 gives the following result for bounding mixing time of general geometric random walk. Lemma 14. Given a metric space M and a time-reversible Markov chain p on M with stationary distribution q. Suppose that there exist r > 0 and ψ > 0 such that 1. For any x, y ∈ M with d(x, z) < r, we have that dT V (px , py ) ≤ 0.9. 2. For any S ⊂ M, we have that Z 0<d(S,x)≤r q(x)dx ≥ rψ min {q(S), q(M \ S)} . Let qt be the distribution of the current point after t steps of a Markov chain with stationary distribution q starting from initial distribution q0 . For any ε > 0, s
t 1 dq0 (x) Ex∼q0 1 − Ω(r 2 ψ 2 ) . dT V (qt , q) ≤ ε + ε dq(x) 14 3.2 Overlap of one-step distributions The mixing of the walk depends on smoothness parameters of the manifold and the functions f, g used to define the Hamiltonian. Since each step of our walk involves a Gaussian vector, many smoothness parameters depend on choices of the random vector. Formally, let γ be the Hamiltonian curve used in a step of Hamiltonian Monte Carlo. In the analysis, we need a large fraction of Hamiltonian curves from any point on the manifold to be well-behaved. A Hamiltonian curve can be problematic when its velocity or length is too large and this happens with non-zero probability. Rather than using supremum bounds for our smoothness parameters, it suffices to use large probability bounds, where the probability is over the random choice of Hamiltonian curve at any point x ∈ Ω. To capture the notion that “most Hamiltonian curves are well-behaved”, we use an auxiliary function ℓ(γ) ≥ 0 which assigns a real number to each Hamiltonian curve γ and measures how “good” the curve is. The smoothness parameters assume that this function ℓ is bounded and Lipshitz. One possible choice of such ℓ is ℓ(γ) = kγ ′ (0)kγ(0) which measures the initial velocity, but this will give us a weaker bound. Instead, we use the following which jointly bounds the change in position (first term) and change in velocity (second term). Definition 15. An auxiliary function ℓ is a non-negative real-valued function on the set of Hamiltonian curves, i.e., maps γ : [0, δ] → M , with bounded parameters ℓ0 , ℓ1 such that 1. For any variation γs of a Hamiltonian curve (see Definition 18) with ℓ(γs ) ≤ ℓ0 , we have ! d d ′ ℓ(γs ) ≤ ℓ1 γs (0) + δ Ds γs (0) γs (0) . ds ds γs (0)   1 min 1, ℓℓ10δ where γ ∼ x indicates a ran2. For any x ∈ M , Pγ∼x (ℓ(γ) ≤ 21 ℓ0 ) ≥ 1 − 100 dom Hamiltonian curve starting at x, chosen by picking a random Gaussian initial velocity according to the local metric at x. 3.2.1 Proof Outline To bound the conductance of HMC, we need to show that one-step distributions from nearby points have large overlap for reasonably large step size δ. To this end, recall that the probability density of going from x to y is given by the following formula s   X 1 det (g(y)) 2 −1 exp − kvx kx . px (y) = |det (DHamx,δ (vx ))| (2π)n 2 vx :Hamx,δ (vx )=y In Section 3.2.2, we introduce the concept of variations of Hamiltonian curves and use it to bound |det (DHamx,δ (vx ))|−1 . We can show that px (y) is in fact close to s   X 1 det (g(y)) 1 2 · exp − kvx kx . (3.1) pex (y) = δn (2π)n 2 vx :Hamx,δ (vx )=y To compare px (y) with pz (y), we need to relate vx and vz that map x and z to y respectively. In Section 3.2.3, we shows that if x and z are close enough, for every vx , there is a unique vz such that vx is close to vz and that Hamz,δ (vz ) = Hamx,δ (vx ). Combining these facts, we obtain our main theorem for this section, stated in Subsection 3.2.4. 15 In the analysis, we use three important operators from the tangent space to itself. The motivation for defining these operators comes directly from Lemma 19, which studies the variation in Hamiltonian curves as the solution of a Jacobi equation. In words, the operator R(.) below allows us to write the change in the Hamiltonian curve as an ODE. Definition 16. Given a Hamiltonian curve γ, let R(γ, t), M (γ, t) and Φ(γ, t) be the operators from T M to T M defined by R(t)u = R(u, γ ′ (t))γ ′ (t), M (t)u = Du µ(γ(t)), Φ(t)u = M (t)u − R(t)u. When γ is explicit from the context, we simply write them as R(t), M (t) and Φ(t). The key parameter R1 we use in this section is a bound on the Frobenius norm of Φ formally defined as follows. Definition 17. Given a manifold M with metric g and an auxiliary function ℓ with parameters ℓ0 , ℓ1 , we define the smoothness parameter R1 depending only on M and the step size δ such that kΦ(γ, t)kF,γ(t) ≤ R1 for any γ such that ℓ(γ) ≤ ℓ0 and any 0 ≤ t ≤ δ where the Frobenius norm kAkF,γ(t) is defined by kAk2F,γ(t) = Eα,β∼N (0,g(x)−1 ) (αT Aβ)2 . The above definitions are related to but different from our previous paper analyzing the geodesic walk [13]. 3.2.2 Variation of Hamiltonian curve To bound the determinant of the Jacobian of Hamx , we study variations of Hamiltonian curves. Definition 18. We call γs (t) a Hamiltonian variation if γs (·) satisfies the Hamiltonian equation for s every s. We call ∂γ ∂s a Jacobi field. The following lemma shows that a Jacobi field satisfies the following Jacobi equation. Lemma 19. Given a path c(s), let γs (t) = Hamc(s) (t(v + sw)) be a Hamiltonian variation. The def Jacobi field ψ(t) = ∂ ∂s γs (t)|s=0 satisfies the following Jacobi equation Dt2 ψ(t) = Φ(t)ψ(t) (3.2) Let Γt parallel transport from Tγ(t,0) M to Tγ(0,0) M and ψ(t) = Γt ψ(t). Then, ψ(t) satisfies the following ODE on the tangent space Tγ(0,0) M: ′′ ψ (t) = Γt Φ(t)Γ−1 t ψ(t) ′ ψ (0) = w, ψ(0) = Ds c(0). 16 ∀t ≥ 0, (3.3) Proof. Taking derivative Ds on both sides of Dt ∂γ ∂t = µ(γ), and using Fact 67, we get ∂γ ∂t ∂γ ∂γ ∂γ ∂γ + R( , ) = Dt Ds ∂t ∂s ∂t ∂t ∂γ ∂γ ∂γ ∂γ = Dt2 + R( , ) . ∂s ∂s ∂t ∂t Ds µ(γ) = Ds Dt In short, we have Dt2 ψ(t) = Φ(t)ψ(t). This shows (3.2). Equation (3.3) follows from the fact that Dt v(t) = Γt
d Γ−1 t v(t) dt ′ for any vector field on γ0 (t) (see Definition 11 in the appendix) applied to v(t) = ψ (t). We now proceed to estimate the determinant of the Jacobian of Hamx . For this we will use the following elementary lemmas describing the solution of the following second-order matrix ODE: d2 Ψ(t) = Φ(t)Ψ(t), dt2 d Ψ(0) = B, dt Ψ(0) = A. (3.4) Lemma 20. Consider the matrix ODE (3.4). Let λ = max0≤t≤ℓ kΦ(t)k2 . For any t ≥ 0, we have that √ √ kBk kΨ(t)k2 ≤ kAk2 cosh( λt) + √ 2 sinh( λt). λ Lemma 21. Consider the matrix ODE (3.4). Let λ = max0≤t≤ℓ kΦ(t)kF . For any 0 ≤ t ≤ have that   t3 2 kΨ(t) − A − BtkF ≤ λ t kAk2 + kBk2 . 5 √1 , λ we In particular, this shows that Ψ(t) = A + Bt +  with kE(s)kF ≤ λ s2 kAk2 + s3 5  kBk2 . Z t 0 (t − s)Φ(s)(A + Bs + E(s))ds The proofs of these lemmas are in Appendix A. We continue with the main proof here. Lemma 22. Let γ(t) = Hamx (tvx ) be a Hamiltonian curve and step size δ satisfy 0 < δ2 ≤ R11 where R1 = max0≤t≤h kΦ(t)kF,γ(t) . Then DHamx,δ is invertible with kDHamx,δ − δIkF,γ(δ) ≤ 5δ . Also, we have, log det 
2  Z δ δ 2 R1 t(δ − t) 1 DHamx,δ (vx ) − TrΦ(t)dt ≤ . δ δ 10 0 17 (3.5) Proof. We want to compute DHamx,δ (vx )[w] for some w ∈ Tx M . By definition, we have that DHamx,δ (vx )[w] = ∂ γ(t, s)|t=δ,s=0 ∂s (3.6) where γ(t, s) = Hamx (t(vx + sw)). Define ψ(t) as in Lemma 19 with c(s) = x. So, Ds c(0) = 0, i.e., ψ(0) = 0. Then, by the lemma, ′′ ψ (t) = Γt Φ(t)Γ−1 t ψ(t), ′ ψ (0) = w, ψ(0) = 0. Now, we define Ψ be the solution of the matrix ODE Ψ′′ (t) = Γt Φ(t)Γ−1 t Ψ(t) ∀t ≥ 0, Ψ′ (0) = I, Ψ(0) = 0. By the definition of Ψ, we see that ψ(t) = Ψ(t)w. Therefore, we have that ∂ −1 γ(t, s)|s=0 = Γ−1 t ψ(t) = Γt Ψ(t)w. ∂s Combining it with (3.6), we have that DHamx,δ (vx ) = Γ−1 t Ψ(δ). Since Γt is an orthonormal matrix, we have that log det (DHamx,δ (vx )) = log det Ψ(δ). (3.7) Note that Γt Φ(t)Γ−1 t that F,γ(t) = kΦ(t)kF,γ(t) ≤ R1 for all 0 ≤ t ≤ δ. Using this, Lemma 21 shows 1 Ψ(δ) − I δ F,x ≤ R1  δ2 kIk2 5  ≤ 1 5 (3.8) Hence, Ψ(δ) is invertible, and so is DHamx . By Lemma 64, we have that 1 log det( Ψ(δ)) − Tr δ  1 Ψ(δ) − I δ  ≤  1 2 δ R1 5 2 . (3.9) Now we need to estimate Tr(Ψ(δ) − δI). Lemma 21 shows that Z δ Z δ (δ − t)Φ(t)E(t)dt t(δ − t)Φ(t)dt + Ψ(δ) = δI + 0 0 3 with kE(t)kF,γ(t) ≤ t 5R1 for all 0 ≤ t ≤ δ. Since  Z δ Z δ δ5 (δ − t) kΦ(t)kF,x kE(t)kF,x dt ≤ R12 , (δ − t)Φ(t)E(t)dt ≤ Tr 20 0 0 we have that Tr  1 Ψ(δ) − I − δ Z δ 0 t(δ − t) Φ(t)dt δ  ≤ δ4 2 R . 20 1 Combining (3.9) and (3.10), we have
2  2 Z δ δ 2 R1 t(δ − t) 1 2 δ4 2 1 TrΦ(t)dt ≤ δ R1 + R1 ≤ . log det( Ψ(δ)) − δ δ 5 20 10 0 Applying (3.7), we have the result. 18 (3.10) 3.2.3 Local Uniqueness of Hamiltonian Curves Next, we study the local uniqueness of Hamiltonian curves. We know that for every pair x, y, there can be multiple Hamiltonian curves connecting x and y. Due to this, the probability density px at y in M is the sum over all possible Hamiltonian curves connecting x and y. The next lemma establishes a 1-1 map between Hamiltonian curves connecting x to y as we vary x. Lemma 23. Let γ(t) = Hamx (tvx ) be a Hamiltonian curve and let the step size δ satisfy 0 < δ2 ≤ 1 R1 , where R1 = max0≤t≤δ kΦ(t)kF,γ(t) . Let the end points be x = γ(0) and y = γ(δ). Then there is an unique smooth invertible function v : U ⊂ M → V ⊂ T M such that y = Hamz,δ (v(z)) for any z ∈ U where U is a neighborhood of x and V is a neighborhood of vx = v(x). Furthermore, 5 we have that k∇η v(x)kx ≤ 2δ kηkx and 1 η + ∇η v(x) δ x ≤ 3 R1 δ kηkx . 2 Let γs (t) = Hamc(s) (t · v(c(s))) where c(s) is any path with c(0) = x and c′ (0) = η. Then, for all 0 ≤ t ≤ δ, we have that ∂ ∂s γs (t) s=0 γ(t) ≤ 5 kηkx and Ds γs′ (t) s=0 γ(t) ≤ 10 kηkx . δ Proof. Consider the smooth function f (z, w) = Hamz,δ (w). From Lemma 22, the Jacobian of w at (x, vx ) in the w variables, i.e., DHamx,δ (vx ), is invertible. Hence, the implicit function theorem shows that there is a open neighborhood U of x and a unique function v on U such that f (z, v(z)) = f (x, vx ), i.e. Hamz,δ (v(z)) = Hamx,δ (vx ) = y. To bound ∇η v(x), we let γs (t) = Hamc(s) (t · v(c(s))) and c(s) be any path with c(0) = x and ′ c (0) = η. Let Γt be the parallel transport from Tγ(t) M to Tγ(0) M. Define ψ(t) = Γt ∂ ∂s γs (t). s=0 Lemma 19 shows that ψ(t) satisfies the following ODE ′′ ψ (t) = Γt Φ(t)Γ−1 t ψ(t) ∀t ≥ 0, ′ ψ (0) = ∇η v(x), ψ(0) = η. Moreover, we know that ψ(δ) = 0 because γs (δ) = Hamc(s) (δ · v(c(s))) = Hamc(s),δ (v(c(s))) = y for small enough s. To bound k∇η v(x)kx , we note that Lemma 21 shows that   t 2 (3.11) ψ(t) − η − t · ∇η v(x) x ≤ R1 t kηkx + k∇η v(x)kx . 5 Since ψ(δ) = 0 and δ2 ≤ 1 R1 , we have that kη + δ · ∇η v(x)kx ≤ kηkx + 19 δ k∇η v(x)kx 5 which implies that δ k∇η v(x)kx − kηkx ≤ kηkx + Therefore, k∇η v(x)kx ≤ 5 2δ δ k∇η v(x)kx . 5 kηkx . More precisely, from (3.11), we have that 1 η + ∇η v(x) δ x ≤ 3 R1 δ kηkx . 2 Putting this into (3.11), for t ≤ δ we get ψ(t) x   5 1 2 ≤ kηkx + kηkx + R1 t kηkx + kηkx ≤ 5 kηkx . 2 2 Now, apply the conclusion of Lemma 21 after taking a derivative, we have that Z t ′ Φ(s)(η + s · ∇η v(x) + E(s))ds ψ (t) = ∇η v(x) + 0  where kE(s)kx ≤ R1 s2 kηkx + that t ≤ δ, we have ′ ψ (t) x s3 5 k∇η v(x)kx  ≤ 3 2 kηkx . Hence, bounding each term and noting   5 10 5 3 ≤ kηkx + δR1 1 + + kηkx . kηkx ≤ 2δ 2 2 δ When we vary x, the Hamiltonian curve γ from x to y varies and we need to bound ℓ(γ) over the variation. Lemma 24. Given a Hamiltonian curve γ(t) = Hamx (t · vx ) with step size δ satisfying δ2 ≤ R11 , let c(s) be any geodesic starting at γ(0). Let x = c(0) = γ(0) and y = γ(δ). Suppose that the auxiliary ℓ0 dc function ℓ satisfies ds ≤ 7ℓ and ℓ(γ) ≤ 21 ℓ0 . Then, there is a unique vector field v on c such c(0) 1 that y = Hamc(s),δ (v(s)). Moreover, this vector field is uniquely determined by the geodesic c(s) and any v(s) on this vector field. Also, we have that ℓ(Hamc(s),δ (v(s))) ≤ ℓ0 for all s ≤ 1. Proof. Let smax be the supremum of s such that v(s) can be defined continuously such that y = Hamc(s),δ (v(s)) and ℓ(γs ) ≤ ℓ0 where γs (t) = Hamc(s) (t · v(s)). Lemma 23 shows that there is a neighborhood N at x and a vector field u on N such that for any z ∈ N , we have that y = Hamz,δ (u(z)). Also, this lemma shows that u(s) is smooth and hence the parameter ℓ1 shows that ℓ(γs ) is Lipschitz in s. Therefore, ℓ(γs ) ≤ ℓ0 in a small neighborhood of 0. Hence smax > 0. Now, we show smax > 1 by contradiction. By the definition of smax , we have that ℓ(γs ) ≤ ℓ0 for dc 5 5 any 0 ≤ s < smax . Hence, we can apply Lemma 23 to show that kDs v(s)kγ(s) ≤ 2δ ds γ(s) = 2δ L where L is the length of c up to s = 1 (since the speed is constant on any geodesic and the curve is defined over [0, 1]). Therefore, the function v is Lipschitz and hence v(smax ) is well-defined and ℓ(γsmax ) ≤ ℓ0 by continuity. Hence, we can apply Lemma 23 at ℓ(smax ) and extend the domain of ℓ(s) beyond smax . 20 To bound ℓ(γs ) beyond smax , we note that kDs γs′ kγ(s) = kDs v(s)kγ(s) ≤ d ds c γ(0) = L. Hence, d ds ℓ(γs ) 5 2δ L and ≤ (L + 25 L)ℓ1 by the definition of ℓ1 . Therefore, if L ≤ that ℓ(γs ) ≤ ℓ(γ) + 21 ℓ0 ≤ ℓ0 for all s ≤ 1.01 wherever v(s) is the assumption that smax is the supremum. Hence, smax > 1. d ds c γ(s) ℓ0 7ℓ1 , = we have defined. Therefore, this contradicts The uniqueness follows from Lemma 23. 3.2.4 Smoothness of one-step distributions Lemma 25. For δ2 ≤ px , pz from x, z satisfy 1 √ 100 nR1 and δ3 ≤ √ℓ0 , 100 nR1 ℓ1 the one-step Hamiltonian walk distributions   1 1 dTV (px , pz ) = O d(x, z) + . δ 25 ℓ0 . Let c(s) be a unit speed geodesic connecting x and Proof. We first consider the case d(x, y) < 7ℓ 1 ℓ0 z of length L < 7ℓ1 .   def Let ℓe = min 1, ℓℓ10δ . By the definition of ℓ0 , with probability at least 1 − ℓe in paths γ start at x, we have that ℓ(γ) ≤ 21 ℓ0 . Let Vx be the set of vx such that ℓ(Hamx (t · vx )) ≤ 21 ℓ0 . Since the ℓ0 and ℓ(γ) ≤ 12 ℓ0 , for those vx , Lemma 24 shows there is a family distance from x to z is less than 7ℓ 1 of Hamiltonian curves γs that connect c(s) to y, and ℓ(γs ) ≤ ℓ0 for each of them. For any vx ∈ V , we have that ℓ(γs ) ≤ ℓ0 . When ℓ(γs ) ≤ ℓ0 , by the definition of R1 , we indeed have that kΦ(t)k ≤ R1 and hence Lemma 22 shows that
2   Z δ δ 2 R1 t(δ − t) 1 DHamx,δ (vx ) ≤ TrΦ(t)dt + log det δ δ 10 0
2 2 2 δ R1 1 e δ √ nR1 + ≤ ℓ ≤ 6 10 600 where we used our assumption on δ. We use p(vx ) to denote the probability density of choosing vx and s   det (g(Hamx (δ · vx ))) 1 def 2 exp − kvx kx . pe(vx ) = (2πδ2 )n 2 Hence, we have that C −1 · p(vx ) ≤ pe(vx ) ≤ C · p(vx ) (3.12) 1 e ℓ. As we noted, for every vx , there is a corresponding vz such that Hamz (δ·vz ) = y where C = 1+ 600 and ℓ(Hamz (t · vz )) ≤ ℓ0 . Therefore, we have that def (1 − C 2 )p(vx ) + C(e p(vx ) − pe(vz )) ≤ p(vx ) − p(vz ) ≤(1 − C −2 )p(vx ) + C −1 (e p(vx ) − pe(vz )).   R 1 Since Vx p(vx )dvx ≥ 1 − 100 min 1, ℓℓ10δ , we have that Z ℓe |p(vx ) − p(vz )| dvx + 100 Vx Z Z ℓe ≤ |e p(vx ) − pe(vz )| dvx p(vx )dvx ) + 2 (1 + 100 Vx Vx Z Z d ℓe +2 pe(vc(s) ) dsdvx ≤ 50 Vx s ds dT V (px , pz ) ≤ 21 (3.13) Note that d pe(v ) = ds c(s)  −  1 d kv(s)k2c(s) pe(vc(s) ). 2 ds Using (3.12), we have that pe(vc(s) ) ≤ 2 · p(vc(s) ) and hence Z Vx d pe(v ) dy ≤ ds c(s) Z d kv(s)k2c(s) p(vc(s) )dy Vx ds d kv(s)k2c(s) ≤Eℓ(γs )≤ℓ0 ds (3.14) Using that k∂s c(s)k = 1, Lemma 23 shows that 1 ∂s c(s) + Ds v(s) δ x 3 ≤ R1 δ. 2 Therefore, we have that d kv(s)k2c(s) = 2 hv(s), Ds v(s)ic(s) ds 2 hv(s), ∂s c(s)ic(s) + 3R1 δ kv(s)kc(s) . ≤ δ Since v(s) is a random Gaussian vector from the local metric, we have that hv(s), ∂s c(s)ic(s) = O(1) √ and kv(s)kc(s) = O( n) with high probability. Putting it into (3.14), we have that Eℓ(γs )≤ℓ0 d kv(s)k2c(s) = O ds   √ 1 + δR1 n . δ Putting this into 3.14, we have that   Z √ d 1 pe(vc(s) ) dy = O + δR1 n . δ Vx ds Putting this into (3.13), we get dTV (px , pz ) = O   √ ℓe 1 + δR1 n L + δ 50 ℓ0 for any L < 7ℓ . By taking a minimal length geodesic, and summing over segment of length 1 any x and z, we have   √ 1 1 + δR1 n d(x, z) + dTV (px , pz ) = O δ 25   1 1 =O d(x, z) + . δ 25 22 ℓ0 8ℓ1 , for 3.3 Convergence bound Combining Lemma 25 and Lemma 14, we have the following result. Theorem 26. Given a manifold M. Let ℓ0 , ℓ1 , R1 be the parameters of the Hamiltonian Monte Carlo defined in Definition 15 and 17. Let qt be the distribution of the current point after t steps Hamiltonian Monte Carlo with step size δ satisfying δ2 ≤ 1 √ 100 nR1 δ3 ≤ and ℓ √0 , 100 nR1 ℓ1 starting from initial distribution q0 . Let q be the distribution proportional to e−f . For any ε > 0, we have that s  t dq0 (x) (δψ)2 1 Ex∼q0 1− dT V (qt , q) ≤ ε + ε dq(x) 2 where ψ is the conductance of the manifold defined in Definition 12. For ℓ(γ) = kγ ′ (0)kγ(0) , we can bound ℓ0 and ℓ1 as follows: √ Lemma 27. For the auxiliary function ℓ(γ) = kγ ′ (0)kγ(0) , we have that ℓ0 = 10 n and ℓ1 = O( 1δ ). Furthermore, we have that γ ′ (t) γ(t) ≤ γ ′ (0) γ(0) + R0 t where R0 = supx∈M kµ(x)kx . √ Proof. For ℓ0 , we note that γ ′ (0) ∼ N (0, g(x)−1 ) and hence kγ ′ (0)kγ(0) ≤ 5 n with probability √ e−O( n) . For ℓ1 , we note that d d γ ′ (0) ℓ(γs ) = ds ds s = γs (0) ≤ 1 2 hDs γs′ (0), γs′ (0)iγs (0) d ds kγs′ (0)k2γs (0) kγs′ (0)kγs (0) kγs′ (0)kγs (0) ≤ Ds γs′ (0) γs (0) . Hence, we have that ℓ1 = 1δ . Next, we note that d γ ′ (t) dt 2 γ(t) = 2 Dt γ ′ (t), γ ′ (t) γ(t) ≤ 2 kµ(γ(t))kγ(t) γ ′ (t) γ(t) . Therefore, we get the last result. Combining Lemma 27 and Theorem 26, we have the following result. This result might be more convenient to establish an upper bound on the rate of convergence, as it depends on only two worst-case smoothness parameters. In the next section, we will see a more refined bound that uses the randomness of Hamiltonian curves via additional parameters. 23 Corollary 28. Given a manifold M. Let qt be the distribution of the current point after t steps Hamiltonian Monte Carlo with step size δ and q be the distribution proportional to e−f . Let R0 and R1 be parameters such that 1. kµ(x)kx ≤ R0 for any x ∈ M where µ is defined in Lemma 7. 2. Eα,β∼N (0,g(x)−1 ) hDα µ(x), βi2x ≤ R12 for any x ∈ M. 3. Eα,β∼N (0,g(x)−1 ) hR(α, v)v, βi 2x ≤ R12 for any x ∈ M and any kvkx ≤ Riemann curvature tensor of M. Suppose that δ ≤ √ n R0 and δ2 ≤ 1 √ , 100 nR1 √ n where R is the then for any ε > 0, we have that dT V (qt , q) ≤ ε + s 1 dq0 (x) Ex∼q0 ε dq(x)  1− (δψ)2 2 t where ψ is the conductance of the manifold defined in Definition 12. In short, the mixing time of Hamiltonian Monte Carlo is    2 √ R −2 0 e ψ . nR1 + O n Proof. The statement is basically restating the√ definition of R1 and R0 used in Lemma 27 and Theorem 26. The only difference is that if δ ≤ Rn0 , then we know that γ ′ (t) γ(t) ≤ γ ′ (0) γ(0) √ + R0 t = O( n) for all 0 ≤ t ≤ δ. Therefore, we can relax the constraints ℓ(γ) ≤ ℓ0 in the definition of R1 to simply √ kvkx ≤ n. It allows us to use R1 without mentioning the auxiliary function ℓ. 4 Improved analysis of the convergence Corollary 28 gives a polynomial mixing time for the log barrier function. There are two bottlenecks to improving the bound. First, the auxiliary function ℓ it used does not capture the fact each curve γ in Hamiltonian Monte Carlo follows a random initial direction. Second, Lemma 23 also does not take full advantage of the random initial direction. In this section, we focus on improving Lemma 23. Our main theorem for convergence can be stated as follows in terms of ψ and additional parameters ℓ0 , ℓ1 , R1 , R2 , R3 (see Definitions 12, 15, 16, 17, 33 and 31). It uses the following key lemma. Theorem 29. For δ2 ≤ x, z satisfy 1 R1 and δ5 ≤ ℓ0 , R21 ℓ1 the one-step Hamiltonian walk distributions px , pz from  1 1 dTV (px , pz ) = O δ R2 + + δR3 d(x, z) + . δ 25  2 Remark. The constant term at the end is an artifact that comes from bounding the probability of some bad events of the Hamiltonian walk, and can be made to arbitrary small. 24 We now prove this theorem. In a later section, we specialize to sampling distributions over polytopes using the logarithmic barrier, by defining a suitable auxiliary function and bounding all the parameters. The key ingredient is Theorem 29 about the overlap of one-step distributions, which we prove in the next section. Here is the consequence of Theorem 29 and Theorem 13. Theorem 30. Given a manifold M. Let ℓ0 , ℓ1 , R1 , R2 , R3 be the parameters of the Hamiltonian Monte Carlo defined in Definition 15, 17, 33 and 31. Let qt be the distribution of the current point after t steps Hamiltonian Monte Carlo starting from initial distribution q0 . Let q be the distribution proportional to e−f . Suppose that the step size δ satisfies δ2 ≤ 1 ℓ0 , δ5 ≤ 2 and δ3 R2 + δ2 R3 ≤ 1. R1 R1 ℓ 1 For any ε > 0, we have that s dT V (qt , q) ≤ ε + dq0 (x) 1 Ex∼q0 ε dq(x)  (δψ)2 1− 2 t where ψ is the conductance of the manifold. 4.1 Improved one-to-one correspondence for Hamiltonian curve In the previous section, we only used R1 to analyze how much a Hamiltonian curves change as one end point varies. Here we derive a more refined analysis of Lemma 23 using an additional parameter R3 . Definition 31. For a manifold M and auxiliary function ℓ, R3 is a constant such that for any Hamiltonian curve γ(t) of step size δ with ℓ(γ0 ) ≤ ℓ0 , if ζ(t) is the parallel transport of the vector γ ′ (0) along γ(t), then we have sup kΦ(t)ζ(t)kγ(t) ≤ R3 . 0≤t≤δ Lemma 32. Under the same assumptions as Lemma 23, we have that δ ∇η kv(x)k2x ≤ |hvx , ηix | + 3δ2 R3 kηkx . 2 Proof. Let χ = ∇η v(x). Define ψ(t) as in the proof of Lemma 23. Using Lemma 21 we get that 0 = ψ(δ) = η + δχ + Z 0 i.e., −δχ = η + with Z δ 0 kE(s)kx ≤ R1 δ (δ − s)Γs Φ(s)Γ−1 s (η + sχ + e(s))ds (δ − s)Γs Φ(s)Γ−1 s (η + sχ + E(s))ds  δ3 kχkx δ kηkx + 5 2 for all 0 ≤ s ≤ δ where we used k∇η v(x)kx ≤ 5 2δ  ≤ 2R1 δ2 kηkx kηkx at the end (by Lemma 23). 25 Therefore, δ |hv(x), χix | ≤ |hv(x), ηix | + δ2 sup hv(x), Γs Φ(s)Γ−1 s (η + sχ + E(s))ix . 2 0≤s≤δ Noting that kη + sχ + E(s)kx ≤ kηkx + δ kχkx + kE(s)kx ≤ (1 + 5 + 2R1 δ2 ) kηkx ≤ 6 kηkx , 2 δ |hv(x), χix | ≤ |hv(x), ηix | + 3δ2 sup x we have 0≤s≤δ = |hv(x), ηix | + 3δ2 sup 0≤s≤δ γ ′ (0)T Γs Φ(s)Γ−1 s Φ(s)Γs γ ′ (0) γ(s) kηkx kηkx ≤ |hv(x), ηix | + 3δ2 R3 kηkx . Finally, we note that 4.2 δ ∇η kv(x)k2x = δ hv(x), ∇η v(x)ix . 2 Improved smoothness of one-step distributions The proof of Theorem 29 is pretty similar to Lemma 25. First, we show that px (y) is in fact close to s  Z δ  X t(δ − t) 1 det (g(y)) 2 exp − TrΦ(t)dt − kvx kx . (4.1) pex (y) = (2πδ2 )n δ 2 0 vx :Hamx,δ (vx )=y Note that this is a more refined estimate than (3.1). We use Lemma 32 to bound the change of kvx k2x . For the change of TrΦ(t), we defer the calculation until the end of this section. Theorem 29. For δ2 ≤ x, z satisfy 1 R1 and δ5 ≤ ℓ0 , R21 ℓ1 the one-step Hamiltonian walk distributions px , pz from  1 1 dTV (px , pz ) = O δ R2 + + δR3 d(x, z) + . δ 25  Proof. We first consider the case d(x, y) < δ2 ≤ R11 and δ5 ≤ Rℓ20ℓ , we have that 2 ℓ0 7ℓ1 . By a similar argument as Lemma 25, using that 1 1 Z Z d ℓe +2 pe(vc(s) ) dsdvx dT V (px , pz ) ≤ 50 Vx s ds   def where ℓe = min 1, ℓℓ10δ . By direct calculation, we have d pe(v ) = ds c(s)  Z − δ 0  t(δ − t) d 1 d 2 ′ TrΦ(γs (t))dt − kv(s)kc(s) pe(vc(s) ). δ ds 2 ds By similar argument as (3.12), we have that pe(vc(s) ) ≤ 2 · p(vc(s) ) and hence  Z δ t(δ − t) d 1 d d 2 ′ pe(v ) ≤ 2 TrΦ(γs (t))dt + kv(s)kc(s) p(vc(s) ). ds c(s) δ ds 2 ds 0 26 (4.2) Since ℓ(γs ) ≤ ℓ0 , we can use Lemma 34 to get Z δ 0 Hence,
t(δ − t) d TrΦ(γs′ (t))dt ≤ O δ2 R2 . δ ds Z Z Z d 2 pe(vc(s) ) dvx ≤O(δ R2 ) p(vc(s) )dvx + Vx ds Vx Vx R For the first term, we note that Vx p(vc(s) )dvx ≤ 1. For the second term, we have that Z d kv(s)k2c(s) p(vc(s) )dvx ≤Eℓ(γs )≤ℓ0 Vx ds d kv(s)k2c(s) p(vc(s) )dvx . ds (4.3) d kv(s)k2c(s) . ds (4.4) By Lemma 32, we have that δ d kv(s)k2c(s) ≤ |hv(s), ∂s cix | + 3δ2 R3 k∂s ckx . 2 ds Since k∂s ckx = 1 and |hv(s), ∂s cix | = O(1) with high probability since v(s) is a Gaussian vector from the local metric. We have δ d kv(s)k2c(s) ≤ O(1) + 3δ2 R3 . 2 ds Putting it into (4.4) and (4.3), we have that   Z d 1 2 pe(vc(s) ) dvx = O δ R2 + + δR3 . δ Vx ds Putting this into 4.2, we get   1 ℓe dTV (px , pz ) = O δ R2 + + δR3 L + δ 50 2 ℓ0 . By taking a minimal length geodesic, and summing over segment of length for any L < 7ℓ 1 any x and z, we have   1 1 2 dTV (px , pz ) = O δ R2 + + δR3 d(x, z) + . δ 25 ℓ0 8ℓ1 , for Definition 33. Given a Hamiltonian curve γ(t) with ℓ(γ0 ) ≤ ℓ0 . Let R2 be a constant depending on the manifold M and the step size δ such that for any 0 ≤ t ≤ δ, any curve c(s) starting from γ(t) and any vector field v(s) on c(s) with v(0) = γ ′ (t), we have that ! dc d + δ k Ds v|s=0 kγ(t) R2 . TrΦ(v(s))|s=0 ≤ ds ds s=0 γ(t) 27 Lemma 34. For δ2 ≤ 1 R1 and ℓ(γs ) ≤ ℓ0 , we have Z δ 0
t(δ − t) d TrΦ(γs′ (t))dt ≤ O δ2 R2 δ ds where γs is a family of Hamiltonian curve that connect c(s) to y defined in Lemma 24. Proof. By Definition 33, we have that d TrΦ(γs′ (t)) ≤ ds ∂ ∂s γs (t) s=0 +δ γ(t) Ds γs′ (t) s=0 γ(t) ! · R2 . (4.5) d d d γs (0) = ds Hamc(s) (0) = ds c(s) is a unit vector and hence By definition of γs , we have that ds d ds γs (0) γ(0) = 1. Since Hamc(s),δ (v(s)) = y, Lemma 23 shows that ∂ ∂s γs (t) s=0 γ(t) ≤ 5 and Ds γs′ (t) Therefore, we have that d TrΦ(γs′ (t)) ≤ 15R2 . ds 28 s=0 γ(t) ≤ 10 . δ 5 5.1 Gibbs sampling on manifolds Isoperimetry for Hessian manifolds Here we derive a general isoperimetry bound, assuming that the manifold is defined by the Hessian of a convex function, and that the directional fourth directive is non-negative, a property satisfied, e.g., by the standard logarithmic barrier. Lemma 35. Let φ : [a, b] → R be a convex function such that φ′′ is also convex. For any x ∈ [a, b], we have that  Z b Z x p −φ(t) −φ(t) −φ(x) ′′ e dt . e dt, e ≥ 0.372 φ (x) min a x Let f (x) be the logconcave density proportional to VarX∼f (x) X ≤ e−φ(x) . Then, we have that O(1) . minx∈[a,b] φ′′ (x) p Proof. Case 1)p|φ′ (x)| ≥ a φ′′ (x) (we will pick a at the end). Without loss of generality, we have that φ′ (x) ≥ a φ′′ (x). Since φ is convex, we have that φ(t) ≥ φ(x) + φ′ (x)(t − x). Therefore, we have that Z Case 2) |φ′ (x)| ≤ a we have that p b x e−φ(t) dt ≤ Z ∞ ′ e−φ(x)−φ (x)(t−x) dt x e−φ(x) e−φ(x) = ′ ≤ p . φ (x) a φ′′ (x) φ′′ (x). Without loss of generality, we have that φ′′′ (x) ≥ 0. Since φ′′ is convex, φ′′ (t) ≥ φ′′ (x) + φ′′′ (x)(t − x) ≥ φ′′ (x) for t ≥ x. Hence, we have that ′ φ(t) = φ(x) + φ (x)(t − x) + Z t x (t − s)φ′′ (t)ds 1 ≥ φ(x) + φ (x)(t − x) + φ′′ (x)(t − x)2 2 ′ for all t ≥ x. Therefore, we have that Z Z b −φ(t) e dt ≤ x ∞ ′ 1 ′′ 2 e−φ(x)−φ (x)(t−x)− 2 φ (x)(t−x) ds −∞ Z ∞ 1 ′′ 2 ′ = e−φ(x) e−φ (x)s− 2 φ (x)s ds −∞ s 2 2π −φ(x)+ 21 φφ′′′(x) (x) = e φ′′ (x) s 2π −φ(x)+ a2 2 . e ≤ ′′ φ (x) 29 Hence, we have Z −φ(x) √ a2 e . 2πe 2 p φ′′ (x) x  2  − a2 e√ Combining both cases, the isoperimetric ratio is min a, 2π . Setting a to be the solution of p a2 ae 2 = √12π , this minimum is achieved at W (1/2π) > 0.372 where W is the inverse Lambert b e−φ(t) dt ≤ 1 . This proves the first result. function, i.e., W (x)eW (x) = 2π The variance of f follows from the fact that f is logconcave. This generalizes to higher dimension with no dependence on the dimension using localization, which we review next. Define an exponential needle E = (a, b, γ) as a segment [a, b] ⊆ Rn and γ ∈ R corresponding to the weight function eγt applied the segment [a, b]. The integral of an n-dimensional function h : Rn → R over this one dimensional needle is Z |b−a| Z b−a . h(a + tu)eγt dt where u = h= |b − a| 0 E Theorem 36 (Theorem 2.7 in [9]). Let f1 , f2 , f3 , f3 be four nonnegative continuous functions defined on Rn , and c1 , c2 > 0. Then, the following are equivalent: 1. For every logconcave function F defined on Rn with compact support,  c 1 Z  c 2 Z  c 1 Z Z F (x)f3 (x) dt ≤ F (x)f2 (x) dx F (x)f1 (x) dx Rn Rn Rn Rn F (x)f4 (x) dx  c2 2. For every exponential needle E, c 2 c 1 Z c 2 Z c 1 Z Z f4 f3 ≤ f2 f1 E E E E Lemma 37. Let φ : K ⊂ Rn → R be a convex function defined over a convex body K such that D 4 φ(x)[h, h, h, h] ≥ 0 for all x ∈ K and h ∈ Rn . Given any partition S1 , S2 , S3 of K with d = minx∈S1 ,y∈S2 d(x, y), i.e., the minimum distance between S1 and S2 in the Riemannian metric induced by φ. For any α > 0, we have that R −αφ(x) dx √ S3 e o = Ω( α · d) nR R min S1 e−αφ(x) dx, S2 e−αφ(x) dx Proof. By rescaling φ, we can assume α = 1. We write the desired inequality as follows, for a constant C, with χS being the indicator of set S: Z Z Z Z −φ(x) −φ(x) −φ(x) e−φ(x) χS3 (x) dx. e dx · e χS2 (x) dx ≤ e χS1 (x) dx · Cd Rn Rn Rn Rn Using the localization lemma for exponential needles (Theorem 36), with fi (x) being Cde−φ(x) χS1 (x), e−φ(x) χS2 (x), e−φ(x) and e−φ(x) χS3 (x) with respectively, it suffices to prove the following onedimensional inequality for functions φ defined on an interval and shifted by a linear term: Z 1 Z 1 e−φ((1−t)a+tb) e−ct χS2 ((1 − t)a + tb) dt e−φ((1−t)a+tb) e−ct χS1 ((1 − t)a + tb) dt · Cd 0 0 Z 1 Z 1 e−φ((1−t)a+tb) e−ct χS3 ((1 − t)a + tb) dt. e−φ((1−t)a+tb) e−ct dt · ≤ 0 0 30 Each Ti = {t : (1 − t)a + tb ∈ Si } is a union of intervals. By a standard argument (see [19]), it suffices to consider the case when each Si is a single interval and add up over all intervals in S3 . Thus it suffices to prove the statement in one dimension for all convex φ with convex φ′′ . In one-dimension, we have Z yp d(x, y) = φ′′ (t) dt. x Taking T3 = is [a′ , b′ ] ⊂ [a, b], the inequality we need to prove is that for any convex φ with convex φ′′ R a′ −φ(t) R b −φ(t) −φ(t) dt e dt b′ e a′ e ≥ Ω(1) a Rb R b′ p −φ(t) dt φ′′ (t) dt a e a′ R b′ −φ(t) e dt e−φ(x) ≥ minx∈[a′ ,b′ ] √ and that R ba′′ √ ′′ φ′′ (x) φ (t) dt a′ R b′ which is implied by noting 5.2 dt applying Lemma 35. Sampling with the log barrier For any polytope M = {Ax > b}, the logarithmic barrier function φ(x) is defined as φ(x) = − m X i=1 log(aTi x − bi ). Theorem 1. Let φ be the logarithmic barrier for a polytope M with m constraints and n variables. Hamiltonian Monte Carlo applied to the function f = exp(−αφ(x)) and the metric given by ∇2 φ with appropriate step size mixes in ! 1 2 1 3 3n3 1 1 n m e + m2 n6 O + α + m−1 α 31 + m− 13 steps where each step is the solution of a Hamiltonian ODE. √ Proof. Lemma 37 shows that the isoperimetric coefficient ψ is Ω( α). Also, we know that isoperi1 metric coefficient ψ is at worst Ω(m− 2 ) [16]. Lemma 61 shows that the condition of Theorem 30 is satisfied, thus implying the bound   √ e max( α, m− 12 )−2 δ−2 O    √ e max( α, m− 12 )−2 max n 32 , α 23 m 13 n 31 , αm 12 n 61 =O ! 2 1 1 1 1 2 3 + α 3 m 3 n 3 + αm 2 n 6 n e =O α + m−1 ! 1 2 1 3 3n3 1 1 n m e + m2 n6 =O + α + m−1 α 13 + m− 13 To implement the walk, we solve this ODE using the collocation method as described in [13]. The key lemma used in that paper is that the slack of the geodesic does not change by more than 1 ). Similarly in Lemma 53, we proved a constant multiplicative factor up to the step size δ = O( n1/4 the the Hamiltonian flow does not change by more than a constant multiplicative factor up to −1/3 n−1/4 √ ). Since the step size we use is Θ( n √ ), we can apply the the step size δ = O( 11/4 ) = O( 1+ α 1+ α M1 31 collocation method as described in [13] and obtain an algorithm to compute the Hamiltonian flow ω−1 logO(1) 1 ) time with additive error η distance in the local norm. Due to the exponential e in O(mn η convergence, it suffices for the sampling purpose with only a polylogarithmic overhead in the total running time. 6 Polytope volume computation: Gaussian cooling on manifolds The volume algorithm is essentially the Gaussian cooling algorithm introduced in [3]. Here we apply it to a sequence of Gibbs distributions rather than a sequence of Gaussians. More precisely, for a convex body K and a convex barrier function φ : K → R, we define (
exp −σ −2 φ(x) if x ∈ K 2 f (σ , x) = 0 otherwise and 2 F (σ ) = x∗ Z f (σ 2 , x) dx Rn where is the minimizer of φ (the center of K). Let µi be the probability distribution proportional to f (σi2 , x) where σi is the tempature of the Gibbs distribution to be fixed. The algorithm estimates each ratio in the following telescoping product: e−σ −2 φ(x∗ ) vol(K) ≈ F (σk ) = F (σ0 ) k 2 ) Y F (σi+1 i=1 F (σi2 ) for some large enough σk . 2 , x)/f (σ 2 , x). Then, Let x be a random sample point from µi and let Yx = f (σi+1 i Ex∼µi (Yx ) = 6.1 Algorithm: cooling schedule 6.2 Correctness of the algorithm 2 ) F (σi+1 . F (σi2 ) In this subsection, we prove the correctness of the algorithm. We separate the proof into two parts. In the first part, we estimate how small σ0 we start with should be and how large σk we end with should be. In the second part, we estimate the variance of the estimator Yx . 6.2.1 Initial and terminal conditions First, we need the following lemmas about self-concordance functions and logconcave functions: Lemma 38 ([27, Thm 2.1.1]). Let φ be a self-concordant function and x∗ be its minimizer. For any x such that φ(x) ≤ φ(x∗ ) + r 2 with r ≤ 12 , we have that φ(x) = φ(x∗ ) + 1 ± Θ(r) (x − x∗ )T ∇2 φ(x∗ )(x − x∗ ). 2 32 Algorithm 2: Volume(M , ε) Let σ02 = Θ(ε2 n−3 log−3 (n/ε)), 2 σi+1    σ 2 1 + √1 if ϑ ≤ nσi2 i n   = . σ 2 1 + min( √σi , 1 ) otherwise i 2 ϑ and ki = ( √ Θ( ε2n log nε ) Θ(( √ ϑ σi if ϑ ≤ nσi2 + 1)ǫ−2 log nε ) otherwise . Set i = 0. Compute x∗ = arg min φ(x). Assume that φ(x∗ ) = 0 by shifting the barrier function. Sample k0 points {X1 , . . . , Xk0 } from the Gaussian distribution centered at the minimizer x∗
−1 . of φ with covariance σ02 ∇2 φ(x∗ ) nϑ ϑ 2 while σi ≤ Θ(1) ε log ε do Sample ki points {X1 , . . . , Xk } using Hamiltonian Monte Carlo with target density f (σi2 , X) and the previous {X1 , . . . , Xk } as warm start. Compute the ratio ki 2 ,X ) f (σi+1 1 X j Wi+1 = · 2, X ) . ki f (σ j i j=1 Increment i. end n 1 Output: (2πσ02 ) 2 det(∇2 φ(x∗ ))− 2 W1 . . . Wi as the volume estimate. 33 Lemma 39 ([27, Prop 2.3.2]). For any ϑ-self-concordance barrier function φ on convex K, for any interior point x and y in K, we have that φ(x) ≤ φ(y) + ϑ ln 1 1 − πy (x) where πy (x) = inf{t ≥ 0|y + t−1 (x − y) ∈ K}. Lemma 40 ([23, Lem 5.16]). For any logconcave distribution f on Rn and any β ≥ 2, we have Px∼f (f (x) ≤ e−βn max f (y)) ≤ e−O(βn) . y 3 3 Lemma 41 (Large and small σ 2 ). Let φ be a ϑ-self concordant barrier function for K. If σn 2 log 2 1 1 1 and σn 2 log 2 σ1 ≤ 1, then we have that 1 σ ≤   3 1 3 1 n −2 ∗ ln F (σ) − ln e−σ φ(x ) (2πσ 2 ) 2 det(∇2 φ(x∗ ))− 2 ≤ O(σn 2 log 2 ). σ If σ 2 ≥ ϑ, then   −2 ∗ ln F (σ) − ln e−σ φ(x ) vol(K) ≤ O(σ −2 ϑ ln(σ 2 n/ϑ)). Proof. We begin with the first inequality. Let S be the set of x such that φ(x) ≤ φ(x∗ ) + βσ 2 n for some β ≥ 2 to be determined. Therefore, f (σ 2 , x) ≥ e−βn f (σ 2 , x∗ ) for all x ∈ S. Since f (σ 2 , x) is a logconcave function, Lemma 40 shows that Px∼f (S) ≤ e−O(βn) . Therefore, Z Z Z f (σ 2 , x) dx ≤ f (σ 2 , x) dx ≤ (1 + e−O(βn) ) f (σ 2 , x) dx. S S In short, we have −O(βn) F (σ) = (1 ± e ) Z f (σ 2 , x) dx. S By the construction of S, Lemma 38 and the fact that φ is self-concordant, if βσ 2 n ≤ 14 , we have that √ 1 ± Θ(σ βn) φ(x) = φ(x∗ ) + (x − x∗ )T ∇2 φ(x∗ )(x − x∗ ) 2 for all x ∈ S. Hence, Z √ σ −2 −2 ∗ ∗ T 2 ∗ ∗ F (σ) = (1 ± e−O(βn) )e−σ φ(x ) e−(1±O(σ βn)) 2 (x−x ) ∇ φ(x )(x−x ) dx. S Now we note that Z Z σ −2 σ −2 ∗ T 2 ∗ ∗ ∗ T 2 ∗ ∗ e−Θ( 2 (x−x ) ∇ φ(x )(x−x )) dx = e−O(βn) e− 2 (x−x ) ∇ φ(x )(x−x ) dx Sc because S σ −2 (x − x∗ )T ∇2 φ(x∗ )(x − x∗ ) = Ω(βn) 2 34 outside S. Therefore, F (σ) = (1 ± e−O(βn) )e−σ −O(βn) = (1 ± e −2 φ(x∗ ) Z 3 2 e−(1±O(σ √ Rn −σ−2 φ(x∗ ) ± O(σ(βn) ))e 3 = (1 ± e−O(βn) ± O(σ(βn) 2 ))e−σ −2 φ(x∗ ) −2 βn)) σ 2 (x−x∗ )T ∇2 φ(x∗ )(x−x∗ ) Z e− dx σ −2 (x−x∗ )T ∇2 φ(x∗ )(x−x∗ ) 2 dx Rn n 1 (2πσ 2 ) 2 det(∇2 φ(x∗ ))− 2 1 3 where we used that σ(βn) 2 = O(1) in first sentence and σ(βn) 2 = O(1) in the second sentence. Setting β = Θ(log σ1 ), we get the first result. For the second inequality, for any 0 ≤ t < 1 and any x ∈ x∗ + t(K − x∗ ), we have that πx∗ (x) ≤ t (πx∗ is defined in Lemma 39). Therefore, Lemma 39 shows that φ(x) ≤ φ(x∗ ) + ϑ ln 1 . 1−t Note that Px∼µ (x ∈ x∗ + t(K − x∗ )) = tn where µ is the uniform distribution in K. Therefore, for any 0 < β < 1, we have that Px∼µ (φ(x) ≤ φ(x∗ ) + ϑ ln 1 }) ≥ (1 − β)n . β Hence, vol(K) · e−σ −2 φ(x∗ ) ≥ F (σ 2 ) n  ≥ vol(K) · (1 − β) exp −σ −2  1 (φ(x ) + ϑ ln ) . β ∗ Setting β = σ −2 n−1 ϑ, we get the second result. 6.2.2 Variance of Yx Our goal is to estimate Ex∼µi (Yx ) within a target relative error. The algorithm estimates the quantity Ex∼µi (Yx ) by taking random sample points x1 , . . . , xk and computing the empirical estimate for Ex∼µi (Yx ) from the corresponding Yx1 , . . . , Yxk . The variance of Yxi divided by its expectation squared will give a bound on how many independent samples xi are needed to estimate Ex∼µi (Yx ) within the target accuracy. We have   R σ2 σi2 2φ(x) φ(x) − dx ) exp F ( 2σ2i+1 2 2 2 K σ σ i i+1 i −σi+1   Ex∼µi (Yx2 ) = = R φ(x) F (σi2 ) dx K exp − σ2 i and σ2 σ2 i ) F (σi2 )F ( 2σ2i+1 2 Ex∼µi (Yx2 ) i −σi+1 = 2 )2 Ex∼µi (Yx )2 F (σi+1 2 If we let σ 2 = σi+1 and σi2 = σ 2 /(1 + r), then we can further simplify as  2   2  σ σ 2 F 1+r F 1−r Ex∼µi (Yx ) = . Ex∼µi (Yx )2 F (σ 2 )2 35 (6.1) Lemma 42. For any 1 > r ≥ 0, we have that   2   2  σ σ Z r Z 1+t F 1−r F 1+r 1   = 4 Varx∼µs φ(x)dsdt ln σ 0 1−t F (σ 2 )2 2 where µs be the probability distribution proportional to f ( σs , x). 2 Proof. Fix σ 2 . Let g(t) = ln F ( σt ). Then, we have that   2   2    2   2  σ σ σ σ Z r F 1−r F 1+r d  F 1+t F 1−t    = dt ln ln F (σ 2 )2 F (σ 2 )2 0 dt Z r d d = g(1 + t) − g(1 − t)dt dt 0 dt Z r Z 1+t 2 d = g(s)dsdt. 2 ds 0 1−t For d2 g(s), ds2 (6.2) we have that Z  s  exp − 2 φ(x) dx σ K R
1 d K φ(x) · exp − σs2 φ(x) dx R
=− 2 · s σ ds K exp − σ2 φ(x) dx


2 ! R  2 R s 2 (x) · exp − s φ(x) dx φ(x) dx φ(x) · exp − φ 1 2 2 K R σ
σ − KR =

2 s 2 s σ K exp − σ2 φ(x) dx K exp − σ2 φ(x) dx   1 1 = 4 Ex∼µs φ2 (x) − (Ex∼µs φ(x))2 = 4 Varx∼µs φ(x). σ σ d2 d2 g(s) = ln ds2 ds2 Putting it into (6.2), we have the result. Now, we bound Varx∼µs φ(x). This can be viewed as a manifold version of the thin shell or variance hypothesis estimate. Lemma 43 (Thin shell estimates). Let φ be a ϑ-self concordant barrier function for K. Then, we have that  2  σ Varx∼µs φ(x) = O ϑ . s def Proof. Let Kt = {x ∈ K such that φ(x) ≤ t} and m be the number such that µs (Km ) = 21 . Let Km,r = {x such that d(x, y) ≤ r and y ∈ Km }. By repeatedly applying Lemma 37, we have that µs (Km,r ) = 1 − e−Ω( √ s r) σ . √ By our assumption on φ, for any x and y, we have that |φ(x) − φ(y)| ≤ ϑd(x, y). Therefore,√for √ s any x ∈ Km,r , we have that φ(x) ≤ m + ϑr. Therefore, with probability at least 1 − e−Ω( √σ r) √ √ s φ(x) ≥ m − ϑr. Hence, with 1 − e−Ω( σ r) in µs , it follows that φ(x) ≤ m + ϑr. Similarly, √ probability in µs , we have that |φ(x) − m| ≤ ϑr. The bound on the variance follows. 36 Now we are ready to prove the key lemma. Lemma 44. Let φ be a ϑ-self concordant barrier function for K. For any
Ex∼µi (Yx2 ) = O r 2 min 2 Ex∼µi (Yx )  1 2  ϑ ,n . σi2 > r ≥ 0, we have that Proof. Using Lemma 43 and Lemma 42, we have that   2   2  σ σ  2  F 1−r F 1+r r ϑ   =O . ln 2 σ2 F (σ 2 ) (6.3) This bounds is useful when σ 2 is large. case σ 2 is small, we recall that for any logconcave function f , the
function a → RFor the n a a f (x) dx is logconcave (Lemma 3.2 in [8]). In particular, this shows that an F a1 is logconcave 1 1−r in a. Therefore, with a = 1+r σ2 , σ2 , σ2 ,we have 1 F (σ 2 )2 ≥ σ 4n  1+r σ2 n F Rearranging the term, we have that  2   2  σ σ F 1−r F 1+r F (σ 2 )2 Therefore, we have that  ln  F  σ2 1+r F  F  σ2 1+r ≤   σ2 1−r (σ 2 )2 1−r σ2 F  n . n 1 (1 + r)(1 − r) σ2 1−r  .  Combining (6.1), (6.3) and (6.4), we have the result. 6.2.3 
 = O nr 2 . (6.4) Main lemma Lemma 45. Given any ϑ-self-concordance barrier φ on a convex set K and 0 < ε < 12 , the algorithm Volume(M , ε) outputs the volume of K to within a 1 ± ε multiplicative factor. −2 ∗ n 1 Proof. By our choice of ε, Lemma 41 shows that e−σ0 φ(x ) (2πσ02 ) 2 det(∇2 φ(x∗ ))− 2 is an 1± 4ε mul−2 ∗ tiplicative approximation of F (σ0 ) and that e−σk φ(x ) vol(K) is a 1± 4ε multiplicative approximation of F (σk ). Note that we shifted the function φ such that φ(x∗ ) = 0. Therefore, k 2 n 1 Y F (σi+1 ) ε vol(K) = (1 ± )(2πσ02 ) 2 det(∇2 φ(x∗ ))− 2 2) . 2 F (σ i i=1 2 , X)/f (σ 2 , X) is upper In Lemma 44, we showed that the variance of the estimator Y = f (σi+1 i q √ 2 bounded by O(1)(EY ) . Note that the algorithm takes O( n) iterations to double σi if ϑn ≤ σi √ and O( ϑσi−1 ) iterations otherwise. By a simple analysis of variance, to have relative error ε, it e i ) samples in each phase. suffices to have O(k 37 6.3 Volume computation with the log barrier In this section, we prove the Theorem 2, restated below for convenience. Theorem 2. For any polytope P = {x : Ax ≥ b} with m constraints and n variables, and any ε > 0, the Hamiltonian volume algorithm estimates the volume of P to within 1 ± ε multiplicative   2 −2 e mn 3 ε factor using O steps where each step consists of solving a first-order ODE and takes time   e mnω−1 LO(1) logO(1) 1 and L is the bit complexity3 of the polytope. O ε 1 m e n− 3 ). Since the number n , the mixing time of HMC is O(m ·√ √ e 2n ), the total number of σ 2 is O( n) and since we samples O( ǫ Proof. In the first part, when σ 2 ≤ of sampling phases to double such steps of HMC is √    √
n − 31 e e e O mn =O ×O n ×O 2 ǫ In the second part, when σ 2 ≥ m n, 2 m · n3 ǫ2 ! . √ 1 2 e −2n 3 −1 + the mixing time of HMC is O( σ +m 1 m3 n3 σ −2 3 1 1 3 +m−√ 1 + m 2 n 6 ). m −2 e Since the number of sampling phases to double σ 2 is O(1+ σm ) and since we sample O(( σ +1)ε ) in each phase, the total number of steps of HMC is ! !  √   √  1 2 2 1 3 3 n3 3 1 1 m m · n n m m −2 e ( e e + m2 n6 × O 1 + + + 1)ε . ×O =O O σ −2 + m−1 σ − 32 + m− 13 σ σ ǫ2 Combining both parts, the total number of steps of HMC is ! 2 3 m · n e . O ǫ2 3 A. L = log(m + dmax + kbk∞ ) where dmax is the largest absolute value of the determinant of a square sub-matrix of 38 7 Logarithmic barrier For any polytope M = {Ax > b}, the logarithmic barrier function φ(x) is defined as φ(x) = − m X i=1 log(aTi x − bi ). We denote the manifold induced by the logarithmic barrier on M by ML . The goal of this section is to analyze Hamiltonian Monte Carlo on ML . In Section 7.1, we give explicit formulas for various Riemannian geometry concepts on ML . In Section 7.2, we describe the HMC specialized to ML . In Sections 7.3 to 7.5, we bound the parameters required by Theorem 30, resulting in Theorem 1. The following parameters that are associated with barrier functions will be convenient. Definition 46. For a convex function f , let M1 , M2 and M3 be the smallest numbers such that
−1 1. M1 ≥ maxx∈M (∇f (x))T ATx Ax ∇f (x) and M1 ≥ n. 2. ∇2 f  M2 · ATx Ax . 3. Tr((ATx Ax )−1 ∇3 f (x)[v]) ≤ M3 kvkx for all v. For the case f = φ are the standard logarithmic barrier, these parameters are n, 1, tively. 7.1 √ n respec- Riemannian geometry on ML (G2 ) We use the following definitions throughout this section. Definition 47. For any matrix A ∈ Rm×n and vectors b ∈ Rm and x ∈ Rn , define 1. sx = Ax − b, Sx = Diag(sx ), Ax = Sx−1 A. 2. sx,v = Ax v, Sx,v = Diag(Ax v).   (2) = (Px )2ij . 3. Px = Ax (ATx Ax )−1 ATx , σx = diag(Px ), Σx = Diag(P ), Px ij 4. Gradient of φ: φi = − P ℓ
eTℓ Ax ei .
−1


P T ej . eℓ Ax ei eTℓ Ax ej , gij = eTi ATx Ax 5. Hessian of φ and its inverse: gij = φij = ATx Ax ij =


P 6. Third derivatives of φ: φijk = −2 ℓ eTℓ Ax ei eTℓ Ax ej eTℓ Ax ek . 7. For brevity (overloading notation), we define sγ ′ = sγ,γ ′ , sγ ′′ = sγ,γ ′′ , Sγ ′ = Sγ,γ ′ and Sγ ′′ = Sγ,γ ′′ for a curve γ(t). In this section, we will frequently use the following identities derived from elementary calculus (using only the chain/product rules and the formula for derivative of the inverse of a matrix). For reference, we include proofs in Appendix C. Fact 48. For any matrix A and any curve γ(t), we have dAγ = −Sγ ′ Aγ , dt dPγ = −Sγ ′ Pγ − Pγ Sγ ′ + 2Pγ Sγ ′ Pγ , dt dSγ ′ = Diag(−Sγ ′ Aγ γ ′ + Aγ γ ′′ ) = −Sγ2′ + Sγ ′′ , dt 39 We also use these matrix inequalities: Tr(AB) = Tr(BA), Tr(P AP ) ≤ Tr(A) for any psd matrix A; Tr(ABAT ) ≤ Tr(AZAT ) for any B  Z; the Cauchy-Schwartz, namely, Tr(AB) ≤ 1 1 Tr(AAT ) 2 Tr(BB T ) 2 . We note Px2 = Px because Px is a projection matrix. Since the manifold ML is naturally embedded in Rn , we can identify Tx ML with Euclidean coordinates. We have that hu, vix = uT ∇2 φ(x)v = uT ATx Ax v. We will use the following two lemmas proved in [13]. Lemma 49. Let w(t) be a vector field defined on a curve z(t) in ML . Then, ∇z ′ w =
−1 T
−1 T dw dw Az Sz ′ sz,w = Az Sz,w sz ′ . − ATz Az − ATz Az dt dt In particular, the equation for parallel transport on a curve γ(t) is given by
−1 T d Aγ Sγ ′ Aγ v. v(t) = ATγ Aγ dt (7.1) Lemma 50. Given u, v, w, x ∈ Tx ML , the Riemann Curvature Tensor at x is given by R(u, v)w = ATx Ax def
−1 ATx (Sx,v Px Sx,w − Diag(Px sx,v sx,w )) Ax u and the Ricci curvature Ric(u) = TrR(u, u) is given by Ric(u) = sTx,u Px(2) sx,u − σxT Px s2x,u where R(u, u) is the operator defined above. 7.2 Hamiltonian walk on ML We often work in Euclidean coordinates. In this case, the Hamiltonian walk is given by the formula in the next lemma. To implement the walk, we solve this ODE using the collocation method as described in [13], after first reducing it to a first-order ODE. The resulting complexity is Õ(mnω−1 ) per step. Lemma 51. The Hamiltonian curve at a point x in Euclidean coordinates is given by the following equations γ ′′ (t) = ATγ Aγ ′ γ (0) = w,
−1 ATγ s2γ ′ + µ(γ(t)) ∀t ≥ 0 γ(0) = x. where µ(x) = ATx Ax
−1 ATx σx − (ATx Ax )−1 ∇f (x) and w ∼ N (0, (ATγ Aγ )−1 ). Proof. Recall from Lemma 7 that the Hamiltonian walk is given by Dt dγ =µ(γ(t)), dt dγ (0) ∼N (0, g(x)−1 ) dt 40   where µ(x) = −g(x)−1 ∇f (x)− 12 g(x)−1 Tr g(x)−1 Dg(x) . By Lemma 49, applied with w(t) = γ ′ (t), z(t) = γ(t), we have
−1 T 2 dγ Dt Aγ sγ ′ . = γ ′′ (t) − ATγ Aγ dt For the formula of µ, we note that
∂
 1  1X (ATx Ax )−1 ij Tr g(x)−1 Dg(x) k = ATx Ax ji 2 2 ∂xk ij X X



(ATx Ax )−1 ij eTℓ Ax ei eTℓ Ax ej eTℓ Ax ek by Defn.47(6) = − ij =− = X ℓ ATx (ATx Ax )−1 Ax ℓ −ATx σx .
ℓℓ eTℓ Ax ek Therefore, µ(x) = −(ATx Ax )−1 ∇f (x) + ATx Ax
−1
ATx σx . Many parameters for Hamiltonian walk depends on the operator Φ(t). Here, we give a formula of Φ(t) in Euclidean coordinates. Lemma 52. Given a curve γ(t), in Euclidean coordinates, we have that Φ(t) = M (t) − R(t) where
−1
ATγ Sγ ′ Pγ Sγ ′ Aγ − ATγ Diag(Pγ s2γ ′ )Aγ  
−1  T  Ax Sx,µ − 3Σx + 2Px(2) Ax − ∇2 f (x) . M (t) = ATγ Aγ R(t) = ATγ Aγ Proof. Lemma 50 with v = w = γ ′ , gives the formula for R(t). For M (t), Lemma 49 with w = µ(x), z ′ = u shows that Du µ(x) = ∇u µ(x) − (ATx Ax )−1 ATx Sx,µ Ax u. For the first term ∇u µ(x), we note that
−1
−1 T ATx Sx,u Ax ATx Ax Ax σx
−1 ATx Sx,u σx − 3 ATx Ax
−1 T + 2 ATx Ax Ax diag(Px Sx,u Px )
−1
−1 ∇f (x) ATx Sx,u Ax ATx Ax − 2 ATx Ax ∇u µ(x) =2 ATx Ax − (ATx Ax )−1 ∇2 f (x)u. 41 Therefore, we have that Du µ(x)
−1 T (2)
−1 T Ax Px Ax u Ax Σx Ax u + 2 ATx Ax ATx Diag(Px σx )Ax u − 3 ATx Ax  

−1 −1 T ∇f (x) Ax u − (ATx Ax )−1 ∇2 f (x)u − (ATx Ax )−1 ATx Sx,µ Ax u Ax Diag Ax ATx Ax − 2 ATx Ax    
−1
−1  T  ∇f (x) − Sx,µ − 3Σx + 2Px(2) Ax − ∇2 f (x) u Ax 2Diag(Px σx ) − 2Diag Aγ ATx Ax = ATx Ax  
−1  T  = ATx Ax Ax 2Sx,µ − Sx,µ − 3Σx + 2Px(2) Ax − ∇2 f (x) u.  
−1  T  = ATx Ax Ax Sx,µ − 3Σx + 2Px(2) Ax − ∇2 f (x) u. =2 ATx Ax
−1 where we used the facts that µ = ATx Ax
−1 ATx σx − (ATx Ax )−1 ∇f (x) Sx,µ = Ax µ = Diag(Px σx − Ax (ATx Ax )−1 ∇f (x)). Remark. Note that R(t) and M (t) is symmetric in h·, ·iγ , but not in h·, ·i2 . That is why the formula does not look symmetric. 7.3 Randomness of the Hamiltonian flow (ℓ0 ) Many parameters of a Hessian manifold relate to how fast a Hamiltonian curve approaches the boundary of the polytope. Since the initial velocity of the curve is drawn from a   Hamiltonian 1 Gaussian distribution, one can imagine that sγ ′ (0) ∞ = O √m sγ ′ (0) 2 (each coordinate of sγ ′ measures the relative rate at which the curve is approaching the corresponding facet). So the walk initial approaches/leaves every facet of the polytope at roughly the same slow pace. If this holds for the entire walk, it would allow us to get very tight bounds on various parameters. Although we are not able to prove that sγ ′ (t) ∞ is stable throughout 0 ≤ t ≤ δ, we will show that sγ ′ (t) 4 is stable and thereby obtain a good bound on sγ ′ (t) ∞ . Throughout this section, we only use the randomness of the walk to prove that both sγ ′ (t) 4 and sγ ′ (t) ∞ are small with high probability. Looking ahead, we will show that sγ ′ (t) √ √ and sγ ′ (t) ∞ = O( log n + M1 δ) (Lemma 55), we define sγ ′ (t) def ℓ(γ) = max 0≤t≤δ n1/2 + 2 1/4 M1 + sγ ′ (t) 1/4 M1 4 sγ ′ (t) √∞ +√ + log n + M1 δ sγ ′ (0) n1/2 2 + sγ ′ (0) n1/4 4 1/4 4 = O(M1 ) sγ ′ (0) ∞ + √ log n ! to capture this randomness involves in generating the geodesic walk. This allows us to perturb the geodesic (Lemma 24) without worrying about the dependence on randomness. We first prove the the walk is stable in the L4 norm and hence ℓ(γ) can be simply approximated by sγ ′ (0) 4 and sγ ′ (0) ∞ . Lemma 53. Let γ be a Hamiltonian flow in ML starting at x. Let v4 = 1 0≤t≤ 1/4 , we have that 12(v4 +M1 ) 42 sγ ′ (0) 4 . Then, for 1. sγ ′ (t) 1/4 4 ≤ 2v4 + M1 . 2. kγ ′′ (t)k2γ(t) ≤ 128v44 + 30M1 . Proof. Let u(t) = sγ ′ (t) 4 . Then, we have (using Holder’s inequality in the first step),
d du ≤ Aγ γ ′ dt dt ≤ Aγ γ ′′ 4 4 2 = Aγ γ ′′ − Aγ γ ′ + u (t).
2 4 (7.2) Under the Euclidean coordinates, by Lemma 51 the Hamiltonian flow is given by γ ′′ (t) = ATγ Aγ with µ(x) = ATx Ax
−1 γ ′′
−1 ATγ s2γ ′ + µ(γ(t)) ATx σx − (ATx Ax )−1 ∇f (x). Hence, we have that 2 γ
−1 T
T
−1 T 2 Aγ sγ ′ Aγ Aγ Aγ Aγ Aγ ATγ Aγ
−1 T
T
−1 T + 3σγT Aγ ATγ Aγ Aγ Aγ Aγ Aγ Aγ σγ


−1 −1 + 3(∇f (x))T ATγ Aγ ATγ Aγ ATγ Aγ ∇f (x) X X
−1 ≤3 (s4γ ′ )i + 3 (σγ2 )i + 3(∇f (x))T ATγ Aγ ∇f (x) ≤3 s2γ ′
T i 4 i ≤3u (t) + 3(n + M1 ) ≤ 3u4 (t) + 6M1 Therefore, we have Aγ γ ′′ 4 ≤ Aγ γ ′′ 2 Plugging it into (7.2), we have that 1/4 (7.3) p ≤ 2u2 (t) + 3 M1 . p du ≤ 3u2 (t) + 3 M1 . dt Note that when u ≤ 2v4 + M1 , we have that p du 1/4 ≤ 12v42 + 9 M1 ≤ 12(v4 + M1 )2 . dt Since u(0) = v4 , for 0 ≤ t ≤ 1 1/4 , 12(v4 +M1 ) 1/4 we have that u(t) ≤ 2v4 + M1 and this gives the first inequality. Using (7.3), we get the second inequality. We can now prove that ℓ(γ) is small with high probability. Lemma 54. Assume that δ ≤ 1 1/4 , 36M1 ℓ1 = Ω(n1/4 δ) and n is large enough, we have that   ℓ0 1 min 1, Pγ∼x (ℓ(γ) ≥ 128) ≤ . 100 ℓ1 δ Therefore, we have ℓ0 ≤ 256. 43 Proof. From the definition of the Hamiltonian curve (Lemma 7), we have that Aγ γ ′ (0) = Bz
−1/2 and z ∼ N (0, I). where B = Aγ ATγ Aγ First, we estimate kAγ γ ′ (t)k4 . Lemma 65 shows that   Pz∼N (0,I) kBzk44 ≤  3 Note that P eTi B i 4 2 = P 2 i (σγ )i Pγ ′ (0) def X eTi B i 4 2 !1/4 4  2 s + kBk2→4 s  ≤ 1 − exp(− ). 2 ≤ n and kBk2→4 ≤ kBk2→2 = 1. Putting s =  ′ Aγ γ (0) 4 4 n1/4 2 , we have that √ n ). ≤ 11n ≤ 1 − exp(− 8  Therefore, we have that v4 = kAγ γ ′ (0)k4 ≤ 2n1/4 with probability at least 1 − exp(− apply Lemma 53 to get that 1/4 1/4 sγ ′ (t) 4 ≤ 2v4 + M1 ≤ 5M1 √ n 8 ). Now, we 1 1/4 . 12(v4 +M1 ) for all 0 ≤ t ≤ Next, we estimate kAγ γ ′ (t)k∞ . Since eTi Aγ γ ′ (0) = eTi Bx ∼ N (0, σi ), we have  2 √
t T ′ . Pγ ′ (0) ei Aγ γ (0) ≥ σi t ≤ 2 exp − 2 Hence, we have that Pγ ′ (0)  ′ Aγ γ (0) ∞    X p 2 log n ≥ 2 log n ≤ 2 exp − σi i   P P n log n Since i exp − 2 log is concave in σ, the maximum of exp − on the feasible set {0 ≤ i σi σi P σ ≤ 1, σi = n} occurs on its vertices. Hence, we have that   p 2 Pγ ′ (0) Aγ γ ′ (0) ∞ ≥ 2 log n ≤ 2n exp (−2 log n) = . n √ √ Lemma 53 shows that kAγ γ ′′ k∞ ≤ kγ ′′ kγ(t) ≤ 46 n + 6 M1 . Hence, for any 0 ≤ t ≤ δ, we have that sγ ′ (t)   ∞ Z t Aγ(t) γ ′′ (r) ∞ dr ≤ Aγ(t) γ (0) ∞ + 0   Z δ sγ(t),i ′ ≤ max Aγ(r) γ ′′ (r) Aγ(0) γ (0) ∞ + i,0≤s≤t sγ(s),i 0    p sγ(t),i  p √ ≤ max 2 log n + (46 n + 6 M1 )δ i,0≤s≤t sγ(s),i    p sγ(t),i  p ≤ max 2 log n + 52 M1 δ . i,0≤s≤t sγ(s),i ′ 44 ∞ dr  (7.4) sγ(t),i sγ(s),i Let z(t) = maxi,0≤s≤t sγ(t),i = sγ(s),i exp Z . Note that  sγ ′ (α),i dα t r since (Aγ(t)−b)i = (Aγ(r)−b)i exp Z t r  ai · γ ′ (α) dα (Aγ(α) − b)i Hence, we have that z ′ (t) ≤ z(t) sγ ′ (t) ∞ Solving this, since z(0) = 1, we get z(t) ≤ Since t ≤ δ ≤ 1 1/4 , 36M1 1
. √ √ 1 − 2 log n + 52 M1 δ t we have that z(t) ≤ 1.05. Putting this into (7.4), we have that sγ ′ (t) Finally, we estimate sγ ′ (t) 2 X Pz∼N (0,I) kBzk22 ≤  P i eTi B 2 2 = P ∞ p p ≤ 3 log n + 55 M1 δ. = kAγ γ ′ (t)k2 . Lemma 65 shows that   Note that   p p ≤ z 2 (t) 2 log n + 52 M1 δ . eTi B i 2 2 !1/2 2  2 r + kBk2→2 r   ≤ 1 − exp(− ). 2 1/2 ≤ n and kBk2→2 ≤ 1. Putting s = n 3 , we have that   n 2 Pγ ′ (0) Aγ γ ′ (0) 2 ≤ 2n ≤ 1 − exp(− ). 18 i (σγ )i Therefore, kAγ γ ′ (0)k22 ≤ 2n with high probability. Next, we note that d Aγ γ ′ dt 2 2 = Aγ γ ′ , Aγ γ ′′ − s2γ ′ E D
−1 T 2 Aγ sγ ′ + Aγ µ(γ(t)) − Aγ s2γ ′ = Aγ γ ′ , Aγ ATγ Aγ E D
−1 T 2 Aγ sγ ′ − s2γ ′ + Aγ γ ′ , Aγ µ(γ(t)) = Aγ γ ′ , Aγ ATγ Aγ = Aγ γ ′ , Aγ µ(γ(t)) . Using that µ(x) = ATx Ax d Aγ γ ′ dt 2 2 =
−1 ATx σx − (ATx Ax )−1 ∇f (x), we have that X X (γ ′ )i (∇f )i (sγ ′ )i (σγ )i − i i sX sX qX q 2 2 ≤ (sγ ′ )i (σγ )i + (sγ ′ )2i (∇f )T (ATγ Aγ )−1 (∇f ) i ≤ 2 Aγ γ ′ 2 p i M1 . √ ≤ M1 . Since δ ≤ d 1 ′ Therefore, we have that dt kAγ γ ′ k2 1/4 , we have that kAγ γ (t)k2 ≤ 36M1 1/4 1/4 √ √ M1 M1 1/4 kAγ γ ′ (0)k2 + 36 ≤ 2n + 36 . Therefore, we have that kAγ γ ′ (t)k2 ≤ 2 n + M1 with n probability at least 1 − exp(− 18 ). 45 Combining our estimates on sγ ′ (t) we have that sγ ′ (t) 2 1/4 1/2 n + M1 max P 0≤t≤δ + sγ ′ (t) 4 1/4 M1 2 , sγ ′ (t) +√ √ n 2 n ≤ exp(− ) + exp(− )+ . 18 8 n sγ ′ (t) log n + 4 and sγ ′ (t) √∞ M1 δ + ∞ sγ ′ (0) n1/2 and using the assumption on δ, 2 + sγ ′ (0) n1/4 4 sγ ′ (0) ∞ + √ log n ! ! ≥ 128 In Lemma 59, we indeed have that ℓ1 = Ω(n1/4 δ) and hence ℓℓ10δ = Ω(n−1/4 δ−2 ) = Ω(1). Therefore,   1 min 1, ℓℓ10δ when n is large enough. the probability is less than 100 Here, we collect some simple consequences of small ℓ(γ) that we will use later. Lemma 55. Given a Hamiltonian flow γ on ML with ℓ(γ) ≤ ℓ0 ≤ 256. For any 0 ≤ t ≤ δ, 1/4 1. kAγ γ ′ (t)k2 ≤ 256(n1/2 + M1 ), kAγ γ ′ (0)k2 ≤ 256n1/2 . 1/4 2. kAγ γ ′ (t)k4 ≤ 256M1 , kAγ γ ′ (0)k4 ≤ 256n1/4 .
√ √ √ 3. kAγ γ ′ (t)k∞ ≤ 256 log n + M1 δ , kAγ γ ′ (0)k∞ ≤ 256 log n. 4. kγ ′′ (t)k2γ ≤ 1013 M1 . Proof. The first three inequalities simply follow from the definition of ℓ(γ). Since kAγ γ ′ (0)k4 ≤ 1/4 256M1 , Lemma 53 shows the last inequality. 7.4 Parameters R1 , R2 and R3 Lemma 56. For a Hamiltonian curve γ on ML with ℓ(γ) ≤ ℓ0 , we have that sup kΦ(t)kF,γ(t) ≤ R1 0≤t≤ℓ √ √ with R1 = O( M1 + M2 n). Proof. Note that Φ(t) = M (t) − R(t) where
ATγ Sγ ′ Pγ Sγ ′ Aγ − ATγ Diag(Pγ s2γ ′ )Aγ , 
−1  T M (t) = ATγ Aγ Aγ (Sγ,µ − 3Σγ + 2Pγ(2) )Aγ − ∇2 f . R(t) = ATγ Aγ
−1 We bound the Frobenius norm of Φ(t) separately. For kR(t)kF,γ , we note that kR(t)k2F,γ ≤2 (ATγ Aγ )−1/2 ATγ Sγ ′ Pγ Sγ ′ Aγ (ATγ Aγ )−1/2 =2TrPγ S Pγ S Pγ S Pγ S + γ′ ≤4 sγ ′ γ′ γ′ γ′ 2 F + 2 (ATγ Aγ )−1/2 ATγ Diag(Pγ s2γ ′ )Aγ (ATγ Aγ )−1/2 2TrPγ Diag(Pγ s2γ ′ )Pγ Diag(Pγ s2γ ′ ) 4 . 4 46 2 F For kM (t)kF,γ , we note that kM (t)k2F,γ ≤2 (ATγ Aγ )−1/2 ATγ (Sγ,µ − 3Σγ + 2Pγ(2) )Aγ (ATγ Aγ )−1/2 2 F + 2 (ATγ Aγ )−1/2 ∇2 f (ATγ Aγ )−1/2 2 F ≤6TrPγ Sγ,µ Pγ Sγ,µ + 54TrPγ Σγ Pγ Σγ + 24TrPγ Pγ(2) Pγ Pγ(2) + 2M22 n 2 ≤6TrSγ,µ + 54TrΣ2γ + 24Tr(Pγ(2) )2 + 2M22 n. (2) where we used that diag(Pγ ) 2 ≤ kσγ k2 ≤ n and TrPγ M Pγ M = TrPγ M Pγ M Pγ = TrPγ M M Pγ = TrM Pγ Pγ M ≤ TrM 2 . Note that 2 2 ksγ,µ k22 = kµ(γ)k2γ = Pγ σγ − Aγ (ATγ Aγ )−1 ∇f (γ) and Aγ ATγ Aγ Therefore, we have that
−1 ∇f (γ(t)) 2 2 = ∇f (γ(t))T ATγ Aγ ≤ 2n + 2M1 ≤ 4M1
−1 ∇f (γ(t)) = M1 . kM (t)k2F,γ ≤ 102M1 + 2M22 n. The claim follows from Lemma 55. Lemma 57. Let γ be a Hamiltonian curve on ML with ℓ(γ) ≤ ℓ0 . Assume that δ2 ≤ √1n . For any 0 ≤ t ≤ δ, any curve c(r) starting from γ(t) and any vector field v(r) on c(r) with v(0) = γ ′ (t), we have that ! dc d + δ k Dr v|r=0 kγ(t) . TrΦ(v(r))|r=0 ≤ R2 dr dr r=0 γ(t) where 1/4 p √ M R2 = O( nM1 + nM1 δ2 + 1 + δ √ n log n √ + nM2 + M3 ). δ Proof. We first bound TrR(t). By Lemma 50, we know that (2) T Ric(v(r)) = sTc(r),v(r) Pc(r) sc(r),v(r) − σc(r) Pc(r) s2c(r),v(r) = Tr(Sc(r),v(r) Pc(r) Sc(r),v(r) Pc(r) ) − Tr(Diag(Pc(r) s2c(r),v(r) )Pc(r) ). For simplicity, we suppress the parameter r and hence, we have Ric(v) = Tr(Sc,v Pc Sc,v Pc ) − Tr(Diag(Pc s2c,v )Pc ). 47 d d c = c′ and dr v = v ′ (in Euclidean coordinates). Since We write dr d and dr Sc,v = −Sc′ Sc,v + Sc,v′ , we have that d dr Pc = −Sc′ Pc − Pc Sc′ + 2Pc Sc′ Pc d Ric(v) dr = −2Tr(Sc,v Sc′ Pc Sc,v Pc ) − 2Tr(Sc,v Pc Sc′ Sc,v Pc ) + 4Tr(Sc,v Pc Sc′ Pc Sc,v Pc ) −2Tr(Sc′ Sc,v Pc Sc,v Pc ) + 2Tr(Sc,v′ Pc Sc,v Pc ) +Tr(Diag(Pc s2c,v )Sc′ Pc ) + Tr(Diag(Pc s2c,v )Pc Sc′ ) − 2Tr(Diag(Pc s2c,v )Pc Sc′ Pc ) +Tr(Diag(Pc Sc′ s2c,v )Pc ) + Tr(Diag(Sc′ Pc s2c,v )Pc ) − 2Tr(Diag(Pc Sc′ Pc s2c,v )Pc ) +2Tr(Diag(Pc Sc,v Sc′ sc,v )Pc ) − 2Tr(Diag(Pc Sc,v sc,v′ )Pc ) = −6Tr(Sc,v Sc′ Pc Sc,v Pc ) + 4Tr(Sc,v Pc Sc′ Pc Sc,v Pc ) + 2Tr(Sc,v′ Pc Sc,v Pc ) +3Tr(Diag(Pc s2c,v )Sc′ Pc ) − 2Tr(Diag(Pc s2c,v )Pc Sc′ Pc ) +3Tr(Diag(Pc Sc′ s2c,v )Pc ) − 2Tr(Diag(Pc Sc′ Pc s2c,v )Pc ) −2Tr(Diag(Pc Sc,v sc,v′ )Pc ). d Let dr Ric(v) = (1) + (2) where (1) is the sum of all terms not involving v ′ and (2) is the sum of other terms. For the first term (1), we have that |(1)| ≤ 6 |Tr(Sc,v Sc′ Pc Sc,v Pc )| + 4 |Tr(Sc,v Pc Sc′ Pc Sc,v Pc )| +3 Tr(Diag(Pc s2c,v )Sc′ Pc ) + 2 Tr(Diag(Pc s2c,v )Pc Sc′ Pc ) +3 Tr(Diag(Pc Sc′ s2c,v )Pc ) + 2 Tr(Diag(Pc Sc′ Pc s2c,v )Pc ) sX sX 2 (sc,v )i (sc,v )2i + 4 ksc′ k∞ |Tr(Pc Sc,v Pc Sc,v Pc )| ≤ 6 ksc′ k∞ i i sX sX sX 4 4 (sc,v )i kSc′ k2 + 2 (sc,v )i (Pc Sc′ Pc )2ii +3 i i i sX sX sX sX (sc,v )4i (Pc )2ii + 2 ksc′ k∞ (sc,v )4i (Pc )2ii +3 ksc′ k∞ i ≤ ≤ i i 10 ksc′ k∞ ksc,v k22 + 3 ksc,v k24 ksc′ k2 √ 20 ksc′ k2 ksc,v k24 n. + √ 7 ksc′ k∞ ksc,v k24 n i 1/2 Since sc,v = sγ ′ at r = 0, we have that ksc,v k24 = O(M1 ) and hence p  |(1)| = O nM1 ksc′ k2 . For the second term (2), we have that |(2)| ≤ 2 Tr(Sc,v′ Pc Sc,v Pc ) + 2 Tr(Diag(Pc Sc,v sc,v′ )Pc ) sX √ (sc,v′ sc,v )2i ≤ 2 sc,v′ 2 ksc,v k2 + 2 n i    p p 1/4 n log n + nM1 δ sc,v′ ≤ O n1/2 + M1 sc,v′ 2 + O   p p 1/4 = O M1 + n log n + nM1 δ sc,v′ 2 48 2 where we used ksc,v k∞ = sγ ′ 1/4 M1 ) ∞ =O √ log n + √
M1 δ and at r = 0, we have ksc,v k2 = Aγ(0) γ ′ (0) sγ ′ 2 = O(n1/2 + in the second-to-last line. Note that at r = 0, by Lemma 49, we have Dr v = Therefore,
−1 T dv − ATc Ac Ac Sc′ sc,v . dr sc,v′ = Ac v ′ = Ac (Dr v) − Ac ATc Ac and hence sc,v′ 2 ≤ kDr vk + Ac ATc Ac ≤ kDr vk + sγ ′
−1
−1 ATc Sc′ sc,v ATc Sc′ sc,v 2 ksc′ k2 ∞ Therefore,   p   p p p 1/4 |(2)| = O M1 + n log n + nM1 δ kDr vk + log n + M1 δ ksc′ k2 . Therefore, we have d Ric(v(r))|s=0 dr p    p p 1/4 =O nM1 ksc′ k2 + O M1 + n log n + nM1 δ kDr vk  p   p p p 1/4 log n + M1 δ ksc′ k2 + O M1 + n log n + nM1 δ p  p p √ √ 1/4 1/4 =O nM1 + M1 log n + M1 M1 δ + n log n + nM1 δ2 ksc′ k2 ! √ 1/4 M1 n log n p + + nM1 δ kDr vk +O δ δ ! √ 1/4 p √ n log n M1 2 =O (ksc′ k2 + δ kDr vk) . + nM1 + nM1 δ + δ δ √ √ 3/4 where we used M1 δ = O( nM1 δ2 + nM1 ) at the last line. Next, we bound TrM (t). Lemma 52 shows that TrM (r) =Tr((ATc Ac )−1 ATc (Sc,µ − 3Σc + 2Pc(2) )Ac − (ATc Ac )−1 ∇2 f ) =Tr(Pc (Sc,µ − 3Σc + 2Pc(2) )) − Tr((ATc Ac )−1 ∇2 f ) =σcT Pc σc − σcT Ac (ATc Ac )−1 ∇f − 3Tr(Σ2c ) + 2Tr(Pc(3) ) − Tr((ATc Ac )−1 ∇2 f ). where in the last step we used Sc,µ = Ac µ = Diag(Pc σc − Ac (ATc Ac )−1 ∇f ). 49 2 = Since d dr Pc = −Sc′ Pc − Pc Sc′ + 2Pc Sc′ Pc and d dr Ac = −Sc′ Ac , we have that d TrM (r) dr = − σcT Sc′ Pc σc − σcT Pc Sc′ σc + 2σcT Pc Sc′ Pc σc − 4σcT Pc Sc′ σc + 4σcT Pc diag(Pc Sc′ Pc ) + 3σcT Sc′ Ac (ATc Ac )−1 ∇f − 2diag(Pc Sc′ Pc )T Ac (ATc Ac )−1 ∇f − 2σcT Ac (ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇f − σcT Ac (ATc Ac )−1 ∇2 f · c′ + 12Tr(Sc′ Σ2c ) − 12Tr(Σc diag(Pc Sc′ Pc )) − 6Tr(Pc(2) Sc′ Pc + Pc(2) Pc Sc′ ) + 12Tr(Pc(2) Pc Sc′ Pc ) − 2Tr((ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇2 f ) − Tr((ATc Ac )−1 ∇3 f [c′ ]). Simplifying it, we have d TrM (r) dr = − 6σcT Sc′ Pc σc + 2σcT Pc Sc′ Pc σc + 4σcT Pc Pc(2) sc′ + 3σcT Sc′ Ac (ATc Ac )−1 ∇f − 2sTc′ Pc(2) Ac (ATc Ac )−1 ∇f − 2σcT Ac (ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇f − σcT Ac (ATc Ac )−1 ∇2 f · c′ + 12Tr(Sc′ Σ2c ) − 12σcT Pc(2) sc′ − 6Tr(Pc(2) Sc′ Pc + Pc(2) Pc Sc′ ) + 12Tr(Pc(2) Pc Sc′ Pc ) − 2Tr((ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇2 f ) − Tr((ATc Ac )−1 ∇3 f [c′ ]). d TrM (r) = (3) + (4) where (3) is the sum of all terms not involving f and (4) is the sum of Let dr other terms with f . For the first term (3), we have that |(3)| ≤6 σcT Sc′ Pc σc + 2 σcT Pc Sc′ Pc σc + 4 σcT Pc Pc(2) sc′ + 12 Tr(Sc′ Σ2c ) + 12 σcT Pc(2) sc′ + 6 Tr(Pc(2) Sc′ Pc + Pc(2) Pc Sc′ ) + 12 Tr(Pc(2) Pc Sc′ Pc ) q √ √ √ √ ≤6 σcT Sc2′ σc n + 2 ksc′ k∞ n + 4 n ksc′ k2 + 12 ksc′ k2 n + 12 kSc′ k2 n + 6 diag(Pc Pc(2) ) 2 ksc′ k2 + 6 diag(Pc(2) Pc ) ≤36n ksc′ k2 . 50 2 ksc′ k2 + 12 diag(Pc Pc(2) Pc ) 2 ksc′ k2 For the second term (4), we have that (4) ≤3 σcT Sc′ Ac (ATc Ac )−1 ∇f + 2 sTc′ Pc(2) Ac (ATc Ac )−1 ∇f + 2 σcT Ac (ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇f + σcT Ac (ATc Ac )−1 ∇2 f · c′ + 2 Tr((ATc Ac )−1 ATc Sc′ Ac (ATc Ac )−1 ∇2 f ) + Tr((ATc Ac )−1 ∇3 f [c′ ]) q q q q (2) (2) T T T T −1 ≤3 sc′ Σc sc′ ∇f (Ac Ac ) ∇f + sc′ Pc Pc Pc sc′ ∇f T (ATc Ac )−1 ∇f q q q q T T T −1 T + 2 σc Pc Sc′ Pc Sc′ Pc σc ∇f (Ac Ac ) ∇f + σc Pc σc c′ ∇2 f (ATc Ac )−1 ∇2 f · c′ + 2 diag(Ac (ATc Ac )−1 ∇2 f (ATc Ac )−1 ATc ) 2 ksc′ k2 + Tr((ATc Ac )−1 ∇3 f [c′ ]) p p √ ≤4 ksc′ k2 M1 + 2 ksc′ k∞ nM1 + 3 nM2 ksc′ k2 + M3 ksc′ k2  p  √ ≤ 6 M1 + 3 nM2 + M3 ksc′ k2 . Therefore,   p √ d TrM (s) ≤ O n + M1 + nM2 + M3 ksc′ k2 . dr Lemma 58. Let γ be a Hamiltonian curve on ML with ℓ(γ) ≤ ℓ0 . Assume that δ ≤ ζ(t) be the parallel transport of the vector γ ′ (0) on γ(t). Then, 1 1/4 . 36M1 sup kΦ(t)ζ(t)kγ(t) ≤ R3 0≤t≤δ where  1 2 R3 = O M1 p 3 4 1 4 log n + M1 n δ + M2 n 1 2  . Proof. By Lemma 52, we have that 
−1  T Aγ (Sγ,µ − 3Σγ + 2Pγ(2) − Sγ ′ Pγ Sγ ′ + Diag(Pγ s2γ ′ ))Aγ − ∇2 f Φ(t) = ATγ Aγ = (1) + (2) where (2) is the last term − ATγ Aγ For the first term, we have that
−1 ∇2 f . k(1)ζkγ = Pγ (Sγ,µ − 3Σγ + 2Pγ(2) − Sγ ′ Pγ Sγ ′ + Diag(Pγ s2γ ′ ))sγ,ζ ≤ (Sγ,µ − 3Σγ + 2Pγ(2) − Sγ ′ Pγ Sγ ′ + Diag(Pγ s2γ ′ ))sγ,ζ Pγ(2) sγ,ζ 2 2 + Sγ ′ Pγ Sγ ′ sγ,ζ 2 + Diag(Pγ s2γ ′ )sγ,ζ ≤ kSγ,µ sγ,ζ k2 + 3 kΣγ sγ,ζ k2 + 2 2   ≤ ksγ,ζ k∞ ksγ,µ k2 + kσγ k2 + Pγ s2γ ′ 2 + 2 Pγ(2) sγ,ζ + sγ ′ ∞ Sγ ′ sγ,ζ 2 2  p  √ 2 (2) ≤ ksγ,ζ k∞ 2 M1 + n + sγ ′ 4 + 2 Pγ sγ,ζ + sγ ′ ∞ sγ ′ 4 ksγ,ζ k4 2 51 2 Let where we used that ksγ,µ k22 = kµ(x)k2x ≤ 4M1 , kσγ k2 ≤ n. Now, we note that Pγ(2) sγ,ζ 2 2 = X i  2 X  (Pγ )2ij (sγ,ζ )j  j ≤ ksγ,ζ k2∞ = ksγ,ζ k2∞ X i X   X  (Pγ )2ij  j σγ2 i
i = n ksγ,ζ k2∞ . √ √ 1/4 ′ log n + M1 δ), we have that = O(M ) and s = O( γ 1 4 ∞  p  2 k(1)ζkγ ≤ ksγ,ζ k∞ 5 M1 + sγ ′ 4 + sγ ′ ∞ sγ ′ 4 ksγ,ζ k4   p 1/2 1/4 3/4 ≤ O M1 ksγ,ζ k∞ + O( log nM1 + M1 δ) ksγ,ζ k4 . Using also that sγ ′ For the second term, we have that k(1)ζkγ = Pγ ∇2 f ζ 2 2 ≤ ζ T ∇2 f · ζ ≤ M2 ksγ,ζ k2 . Combining both terms, we have that   p 1/4 3/4 1/2 kΦ(t)ζkγ = O M1 ksγ,ζ k∞ + O( log nM1 + M1 δ) ksγ,ζ k4 + M2 ksγ,ζ k2 . (7.5) Now, we bound ksγ,ζ k2 , ksγ,ζ k4 and ksγ,ζ k∞ . (7.1) shows that
−1 T d ζ(t) = ATγ Aγ Aγ Sγ ′ Aγ ζ. dt Let wp (t) = kAγ ζ(t)kp . Then, we have that d d wp (t) ≤ Aγ ζ(t) dt dt p d ζ(t) dt ≤ Sγ ′ Aγ ζ(t) p + Aγ ≤ Sγ ′ Aγ ζ(t) p + Pγ Sγ ′ Aγ ζ ≤ sγ ′ d w2 (t) ≤ 2 sγ ′ For p = 2, we have that dt 1 that t ≤ δ ≤ 1/4 , we have ∞ w (t) + Sγ ′ Aγ ζ ∞ p w2 (t) ≤ e512( = O(n where we used that ζ(0) = γ ′ (0) and t ≤ O(n1/2 ). √ √ log n+ M1 δ)t 1/2 p . 2 w2 (t). Using that sγ ′ 36M1 p ∞ ≤ 256 √ log n + √
M1 δ and w2 (0) ) 1 1/4 12(v4 +M1 ) 52 at the end. Therefore, we have that ksγ,ζ k2 = For p = 4, we note that d w4 (t) ≤ sγ ′ ∞ w4 (t) + sγ ′ 4 ksγ,ζ k4 dt 1/4 ≤ 2 sγ ′ 4 w4 (t) = O(M1 w4 (t)). 1 1/4 , we have again 36M1 O(n1/4 ), we have that that w4 (t) = O(w4 (0)). Since w4 (0) = kAγ ζ(0)k4 = Since t ≤ δ ≤ kAγ γ ′ (0)k4 = w4 (t) = O(n1/4 ). For p = ∞, we note that d w∞ (t) ≤ sγ ′ ∞ w∞ (t) + sγ ′ 4 ksγ,ζ k4 dt 1/4 1/4 ≤ O(M1 w∞ (t)) + O(M1 n1/4 ). Again using t ≤ δ ≤ 1 1/4 , 36M1 √ 1/4 we have that w∞ (t) ≤ O( log n + M1 n1/4 δ). Combining our bounds on w2 , w4 , w∞ to (7.5), we get   1   3 3 1 p p 1 1 1 1 2 4 kΦ(t)ζkγ = O M1 log n + M1 n 4 δ + O( log nM14 n 4 + M14 n 4 δ) + O M2 n 2  1  3 p 1 1 2 4 4 2 = O M1 log n + M1 n δ + M2 n . 7.5 Stability of L2 + L4 + L∞ norm (ℓ1 ) Lemma 59. Given a family of Hamiltonian curves γr (t) on ML with ℓ(γ0 ) ≤ ℓ0 , for δ2 ≤ 1 √ √ , we have that M +M n 1 2   d 1 1/4 ℓ(γr ) ≤ O M1 δ + √ dr δ log n  1/4 Hence, ℓ1 = O M1 δ + Proof. For brevity, all d dr √1 δ log n d γr (0) dr  . are evaluated at r = 0. Since d sγ ′ dr r 2 ≤ sγr′ ∞ Aγr d γr dr d ′ dr sγr + δ Dr γr′ (0) γr (0) γr (0) ! (7.6) . d ′ = −Sγr , d γr sγr′ + Aγr dr γr , we have dr + Aγr 2 d ′ γ dr r . 2 For the last term, we note that Dr γr′ = Hence, we have Aγr d ′ γ dr r 2 ≤ Dr γr′ γr
−1 T d ′ γr − ATγr Aγr Aγr Sγr , d γr sγr′ . dr dr + Sγr , d γr sγ ′ dr 53 2 ≤ Dr γr′ γr + sγr′ ∞ sγr , d γr dr 2 . Therefore, we have that d sγ ′ dr r 2 d + Dr γ ′ γr dr γr p  d p =O log n + M1 δ γr dr ≤ 2 sγr′ ∞ γ + Dr γr′ . (7.7) γr Since γr is a family of Hamiltonian curves, Lemma 19 shows that ′′ ψ (t) = Γt Φ(t)Γ−1 t ψ(t) where ψ(t) is the parallel transport of d ψ(t) = γr (0) + Dr γr′ (0)t + dr d dr γr (t) Z 0 t from γr (t) to γr (0). By Lemma 49, we have that (t − r)Γr Φ(r)Γ−1 r ( d γr (0) + Dr γr′ (0)r + E(r))dr dr def d with kE(r)kF ≤ O(1)∆ and ∆ = dr γr (0) γr (0) + δ kDr γr′ (0)kγr (0) where we used that kΦ(t)kF,γ = √ √ 1 √ O( M1 + M2 n) (Lemma 56) and that s2 ≤ δ2 ≤ √M +M . 1 2 n Therefore, we have that Z t d γr (t) = ψ(t) γr (0) ≤ ∆ + O(∆) (t − s) Γr Φ(s)Γ−1 dr r γr (0) dr 0 γr (t) ≤ O(∆) √ √ where we used again kΦ(s)kF,γ = O( M1 + M2 n) and s2 ≤ δ2 ≤ ′ Similarly, we have that kDr γr′ (t)kγr (t) = ψ (t) √1 , n Putting these into (7.7) and using h ≤ d sγ ′ dr r We write ℓ(γr ) = max 0≤t≤δ sγr′ (t) n1/2 + 2 1/4 M1 + sγr′ (t) 1 √ . M1 +M2 n ≤ O( ∆ δ ). we have  ∆ M1 δ ∆ + δ p  1 M1 δ + =O ∆. δ =O 2 γr (0) √ 4 1/4 M1 p log n + sγr′ (t) +√ log n + p √∞ M1 δ + sγr′ (0) (7.8) 2 n1/2 + According to same calculation as (7.7), we can improve the estimate on d sγ ′ (0) dr r 2 ≤ p log n d γr (0) dr ∆ ≤ . δ sγr′ (0) n1/4 d ′ dr sγr 2 4 sγ ′ (0) ∞ + √r log n ! . for t = 0 and get + Dr γr′ (0) γr (7.9) 54 Using (7.8) and (7.9), we have that d d ′ ′ dr sγr (t) 2 dr sγr (0) 2 √ max + 1/4 0≤t≤δ log n M1 ! √ M1 δ + 1δ 1 ∆ + √ 1/4 δ log n M1 d ℓ(γr ) = O dr =O =O 7.6  1/4 M1 δ 1 + √ δ log n  ! ∆. Mixing Time Lemma 60. If f (x) = α · φ(x) (logarithmic barrier), then, we have that M1 = n + α2 m, M2 = α √ and M3 = 2α · n.
−1 Proof. For M1 , we note that (∇f (x))T ATx Ax ∇f (x) = α2 1T Ax (ATx Ax )−1 ATx 1 ≤ α2 m. Hence, M1 = n + α2 m. For M2 , it directly follows from the definition. For M3 , we note that Tr((ATx Ax )−1 ∇3 f (x)[v]) = −2αTr((ATx Ax )−1 ATx Sx,v Ax ) X σx,i (sx,v )i . = −2α i Hence, we have Tr((ATx Ax )−1 ∇3 f (x)[v]) sX sX √ 2 σx,i (sx,v )2i ≤ 2α n kvkx . ≤ 2α i i Using Theorem 29, we have the following Lemma 61. There is a universal constant c > 0 such that if the step size  1  1 1 1 1 1 1 δ ≤ c · min n− 3 , α− 3 m− 6 n− 6 , α− 2 m− 4 n− 12 , then, all the δ conditions for Theorem 30 are satisfied. Proof. In the previous section, we proved that if δ ≤ 1 1/4 36M1 and n is large enough, 1. ℓ0 = 256 (Lemma 54)   1/4 1 (Lemma 59) 2. ℓ1 = O M1 δ + δ√log n √ √ 3. R1 = O( M1 + M2 n) (Lemma 56) √ √ 4. R2 = O( nM1 + nM1 δ2 + 1/4 M1 δ + √ n log n δ + 55 √ nM2 + M3 ) (Lemma 57) 3 1√ 1 1 5. R3 = O(M12 log n + M14 n 4 δ + M2 n 2 ) (Lemma 58) 1 1 1 1 1 Substituting the value of M1 , M2 and M3 and using that δ . n− 3 , δ . α− 2 n− 3 and δ . α− 2 m− 4 , we have that  1 1  1 1. ℓ1 = O α 2 m 4 δ + δ√log n √ √ 2. R1 = O( n + α m) 1 √ √ √ 4 3. R2 = O(n + α nm + αm +δ n log n ) √ √ 3 3 1 4. R3 = O( n log n + α m log n + nδ + α 2 m 4 n 4 δ) Now, we verify all the δ conditions for Theorem 30. 1 1 1 Using δ . n− 3 and δ . α− 2 m− 4 , we have that δ2 . Using δ . n − 13 − 12 and δ . α m − 41 1 R1 . , we have that 1 1 1 δ5 R12 ℓ1 . δ5 (n + α2 m) · (α 2 m 4 δ + √ ) ≤ ℓ0 . δ log n For the last condition, we note that p √ √ 1 δ3 R2 + δ2 R3 .nδ3 + α nmδ3 + αm 4 δ2 + n log nδ2 p p 3 1 3 + n log nδ2 + α m log nδ2 + nδ3 + α 2 m 4 n 4 δ3 p p √ √ 3 1 1 3 . αm 4 δ2 + n log nδ2 + α m log nδ2 + nδ3 + α nmδ3 + α 2 m 4 n 4 δ3 p p √ 3 3 1 . n log nδ2 + α m log nδ2 + nδ3 + α nmδ3 + α 2 m 4 n 4 δ3 where we used that Therefore, if √ √ 1 αm 4 . (1 + α m) at the end  1  1 1 1 1 1 1 δ ≤ c · min n− 3 , α− 3 m− 6 n− 6 , α− 2 m− 4 n− 12 for small enough constant, then all the δ conditions for Theorem 30 are satisfied. Acknowledgement. We thank Ben Cousins for helpful discussions. This work was supported in part by NSF awards CCF-1563838, CCF-1717349 and CCF-1740551. References [1] D. Applegate and R. Kannan. Sampling and integration of near log-concave functions. In STOC, pages 156–163, 1991. [2] M. Betancourt. A Conceptual Introduction to Hamiltonian Monte Carlo. ArXiv e-prints, January 2017. [3] B. Cousins and S. Vempala. Bypassing KLS: Gaussian cooling and an O ∗ (n3 ) volume algorithm. In STOC, pages 539–548, 2015. [4] M. E. Dyer and A. M. Frieze. Computing the volume of a convex body: a case where randomness provably helps. In Proc. of AMS Symposium on Probabilistic Combinatorics and Its Applications, pages 123–170, 1991. 56 [5] M. E. Dyer, A. M. Frieze, and R. Kannan. 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In STOC, pages 561–570, 2009. [12] Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in õ (vrank) iterations and faster algorithms for maximum flow. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 424–433. IEEE, 2014. [13] Yin Tat Lee and Santosh S. Vempala. Geodesic walks in polytopes. CoRR, abs/1606.04696, 2016. [14] Yin Tat Lee and Santosh Srinivas Vempala. Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion. CoRR, abs/1612.01507, 2016. [15] Yin Tat Lee and Santosh Srinivas Vempala. Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion. In Proc. of IEEE FOCS, 2017. [16] Yin Tat Lee and Santosh Srinivas Vempala. Geodesic walks in polytopes. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 927–940, 2017. [17] L. Lovász and M. Simonovits. Mixing rate of Markov chains, an isoperimetric inequality, and computing the volume. In ROCS, pages 482–491, 1990. [18] L. Lovász and M. Simonovits. On the randomized complexity of volume and diameter. In Proc. 33rd IEEE Annual Symp. on Found. of Comp. Sci., pages 482–491, 1992. [19] L. Lovász and M. Simonovits. Random walks in a convex body and an improved volume algorithm. In Random Structures and Alg., volume 4, pages 359–412, 1993. [20] L. Lovász and S. Vempala. Fast algorithms for logconcave functions: sampling, rounding, integration and optimization. In FOCS, pages 57–68, 2006. [21] L. Lovász and S. Vempala. Hit-and-run from a corner. SIAM J. Computing, 35:985–1005, 2006. 57 [22] L. Lovász and S. Vempala. Simulated annealing in convex bodies and an O ∗ (n4 ) volume algorithm. J. Comput. Syst. Sci., 72(2):392–417, 2006. [23] L. Lovász and S. Vempala. The geometry of logconcave functions and sampling algorithms. Random Struct. Algorithms, 30(3):307–358, 2007. [24] Oren Mangoubi and Aaron Smith. Rapid mixing of hamiltonian monte carlo on strongly logconcave distributions. arXiv preprint arXiv:1708.07114, 2017. [25] Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, 54:113–162, 2010. [26] R.M. Neal. Bayesian Learning for Neural Networks. Lecture Notes in Statistics. Springer New York, 1996. [27] Yurii Nesterov, Arkadii Nemirovskii, and Yinyu Ye. Interior-point polynomial algorithms in convex programming, volume 13. SIAM, 1994. [28] Burt Totaro. The curvature of a hessian metric. 15(04):369–391, 2004. International Journal of Mathematics, [29] S. Vempala. Geometric random walks: A survey. MSRI Combinatorial and Computational Geometry, 52:573–612, 2005. 58 A Matrix ODE In this section, we prove Lemmas (20) and (21) for the solution of the ODE (3.4), restated below for convenience. d2 Ψ(t) = Φ(t)Ψ(t), dt2 d Ψ(0) = B, dt Ψ(0) = A. Lemma 62. Consider the matrix ODE (3.4). Let λ = max0≤t≤ℓ kΦ(t)k2 . For any t ≥ 0, we have that √ √ kBk kΨ(t)k2 ≤ kAk2 cosh( λt) + √ 2 sinh( λt). λ Proof. Note that Z ′ t Ψ(t) = Ψ(0) + tΨ (0) + (t − s)Ψ′′ (s)ds 0 Z t = A + tB + (t − s)Φ(s)Ψ(s)ds. (A.1) 0 Let a(t) = kΨ(t)k2 , then we have that a(t) ≤ kAk2 + t kBk2 + λ Z t (t − s)a(s)ds. 0 Let a(t) be the solution of the integral equation a(t) = kAk2 + t kBk2 + λ Z t 0 (t − s)a(s)ds. By induction, we have that a(t) ≤ a(t) for all t ≥ 0. By taking derivatives on both sides, we have that a′′ (t) = λa(t), a(0) = kAk2 , a′ (0) = kBk2 . Solving these equations, we have √ √ kBk kΨ(t)k2 = a(t) ≤ a(t) = kAk2 cosh( λt) + √ 2 sinh( λt) λ for all t ≥ 0. Lemma 63. Consider the matrix ODE (3.4). Let λ = max0≤t≤ℓ kΦ(t)kF . For any 0 ≤ t ≤ have that   t3 2 kΨ(t) − A − BtkF ≤ λ t kAk2 + kBk2 . 5 In particular, this shows that Ψ(t) = A + Bt +  with kE(s)kF ≤ λ s2 kAk2 + s3 5  kBk2 . Z t 0 (t − s)Φ(s)(A + Bs + E(s))ds 59 √1 , λ we Proof. Recall from (A.1) that Ψ(t) = A + tB + Z t 0 (A.2) (t − s)Φ(s)Ψ(s)ds. Let E(t) = Ψ(t) − (A + tB). Using Lemma 20, we have that kE(t)kF = Z ≤λ t 0 Z Z (t − s)Φ(s)Ψ(s)ds F t (t − s) kΨ(s)k2 ds 0 √ √ kBk (t − s) kAk2 cosh( λs) + √ 2 sinh( λs)ds λ  0 √ √ √ kBk2 = λ kAk2 (cosh( λt) − 1) + √ (sinh( λt) − λt) . λ ≤λ t √ √ √ Since 0 ≤ t ≤ √1λ , we have that cosh( λt) − 1 ≤ λt2 and sinh( λt) − λt ≤ the result. The last equality follows again from (A.2) λ3/2 t3 5 . This gives Next, we have an elementary lemma about the determinant. Lemma 64. Suppose that E is a matrix (not necessarily symmetric) with kEk2 ≤ 41 , we have |log det(I + E) − TrE| ≤ kEk2F . Proof. Let f (t) = log det(I + tE). Then, by Jacobi’s formula, we have
f ′ (t) = Tr (I + tE)−1 E , f ′′ (t) = −Tr((I + tE)−1 E(I + tE)−1 E). Since kEk2 ≤ 14 , we have that (I + tE)−1 f ′′ (t) = ≤ 2 ≤ 4 3 and hence Tr((I + tE)−1 E(I + tE)−1 E)
T Tr(E T (I + tE)−1 (I + tE)−1 E) ≤ 2 Tr(E T E) = 2 kEk2F . The result follows from Z 1 (1 − s)f ′′ (s)ds f (1) = f (0) + f (0) + 0 Z 1 (1 − s)f ′′ (s)ds. = Tr(E) + ′ 0 60 B Concentration Lemma 65 ([13, Ver 3, Lemma 90]). For p ≥ 1, we have p    !1/p  2 p+1 X p/2 2 Γ( 2 ) t p p     √ + kAk2→p t Px∼N (0,I) kAxkp ≤ kai k2 . ≤ 1 − exp − π 2 i In particular, we have  and  Px∼N (0,I) kAxk44 ≤  3   Px∼N (0,I) kAxk22 ≤  C X i X i kai k42 kai k22 !1/4 !1/2 4   2 t   + kAk2→4 t ≤ 1 − exp − 2 2   2 t . + kAk2→2 t  ≤ 1 − exp − 2 Calculus Proof of Fact 48. Recall Definition 47 and write dSγ−1 dAγ = A dt dt dSγ −1 Sγ A = −Sγ−1 dt   d(Aγ − b) −1 = −Sγ Diag Aγ dt
= −Diag Sγ−1 Aγ ′ Aγ = −Diag(Aγ γ ′ )Aγ = −Sγ ′ Aγ . For the second, using the first, dAγ (ATγ Aγ )−1 ATγ dPγ = dt dt d(ATγ Aγ )−1 T dAγ T dAγ = (Aγ Aγ )−1 ATγ + Aγ (ATγ Aγ )−1 + Aγ Aγ dt dt dt d(ATγ Aγ ) T (Aγ Aγ )−1 ATγ = −Sγ ′ Pγ − Pγ Sγ ′ − Aγ (ATγ Aγ )−1 dt
= −Sγ ′ Pγ − Pγ Sγ ′ + 2Aγ (ATγ Aγ )−1 ATγ Sγ ′ Aγ (ATγ Aγ )−1 ATγ = −Sγ ′ Pγ − Pγ Sγ ′ + 2Pγ Sγ ′ Pγ . And for the last, dSγ ′ = Diag dt  dAγ ′ γ + Aγ γ ′′ dt  = Diag(−Sγ ′ Aγ γ ′ + Aγ γ ′′ ) = −Sγ2′ + Sγ ′′ . 61 D Basic definitions of Riemannian geometry Here we recall basic notions of Riemannian geometry. One can think of a manifold M as a ndimensional “surface” in Rk for some k ≥ n. 1. Tangent space Tp M : For any point p, the tangent space Tp M of M at point p is a linear subspace of Rk of dimension n. Intuitively, Tp M is the vector space of possible directions that are tangential to the manifold at x. Equivalently, it can be thought as the first-order linear d c(t) is tangent approximation of the manifold M at p. For any curve c on M , the direction dt ′ to M and hence lies in Tc(t) M . When it is clear from context, we define c (t) = dc dt (t). For any n n open subset M of R , we can identify Tp M with R because all directions can be realized by derivatives of some curves in Rn . 2. Riemannian metric: For any v, u ∈ Tp M , the inner product (Riemannianq metric) at p is given by hv, uip and this allows us to define the norm of a vector kvkp = hv, vip . We call a manifold a Riemannian manifold if it is equipped with a Riemannian metric. When it is clear from context, we define hv, ui = hv, uip . In Rn , hv, uip is the usual ℓ2 inner product. 3. Differential (Pushforward) d: Given a function f from a manifold M to a manifold N , we define df (x) as the linear map from Tx M to Tf (x) N such that df (x)(c′ (0)) = (f ◦ c)′ (0) for any curve c on M starting at x = c(0). When M and N are Euclidean spaces, df (x) is the Jacobian of f at x. We can think of pushforward as a manifold Jacobian, i.e., the first-order approximation of a map from a manifold to a manifold. 4. Hessian manifold: We call M a Hessian manifold (induced by φ) if M is an open subset of Rn with the Riemannian metric at any point p ∈ M defined by hv, uip = v T ∇2 φ(p)u where v, u ∈ Tp M and φ is a smooth convex function on M . 5. Length: For any curve c : [0, 1] → M , we define its length by L(c) = Z 1 0 d c(t) dt dt. c(t) 6. Distance: For any x, y ∈ M , we define d(x, y) be the infimum of the lengths of all paths connecting x and y. In Rn , d(x, y) = kx − yk2 . 7. Geodesic: We call a curve γ(t) : [a, b] → M a geodesic if it satisfies both of the following conditions: (a) The curve γ(t) is parameterized with constant speed. Namely, for t ∈ [a, b]. d dt γ(t) γ(t) is constant (b) The curve is the locally shortest length curve between γ(a) and γ(b). Namely, for any family of curve c(t, s) with c(t, 0) = γ(t) and c(a, s) = γ(a) and c(b, s) = γ(b), we have Rb d d c(t, s) c(t,s) dt = 0. that ds s=0 a dt 62 Note that, if γ(t) is a geodesic, then γ(αt) is a geodesic for any α. Intuitively, geodesics are local shortest paths. In Rn , geodesics are straight lines. 8. Exponential map: The map expp : Tp M → M is defined as expp (v) = γv (1) where γv is the unique geodesic starting at p with initial velocity γv′ (0) equal to v. The exponential map takes a straight line tv ∈ Tp M to a geodesic γtv (1) = γv (t) ∈ M . Note that expp maps v and tv to points on the same geodesic. Intuitively, the exponential map can be thought as point-vector addition in a manifold. In Rn , we have expp (v) = p + v. 9. Parallel transport: Given any geodesic c(t) and a vector v such that hv, c′ (0)ic(0) = 0, we define the parallel transport Γ of v along c(t) by the following process: Take h to be infinitesimally small and v0 = v. For i = 1, 2, · · · , 1/h, we let vih be the vector orthogonal to c′ (ih) that minimizes the distance on the manifold between expc(ih) (hvih ) and expc((i−1)h) (hv(i−1)h ). Intuitively, the parallel transport finds the vectors on the curve such that their end points are closest to the end points of v. For general vector v ∈ Tc′ (0) , we write v = αc′ (0) + w and we define the parallel transport of v along c(t) is the sum of αc′ (t) and the parallel transport of w along c(t). For a non-geodesic curve, see the definition in Fact 66. 10. Orthonormal frame: Given vector fields v1 , v2 , · · · , vn on a subset of M , we call {vi }ni=1 is an orthonormal frame if hvi , vj ix = δij for all x. Given a curve c(t) and an orthonormal frame at c(0), we can extend it on the whole curve by parallel transport and it remains orthonormal on the whole curve. 11. Directional derivatives and the Levi-Civita connection: For a vector v ∈ Tp M and a vector field u in a neighborhood of p, let γv be the unique geodesic starting at p with initial velocity γv′ (0) = v. Define u(h) − u(0) ∇v u = lim h→0 h where u(h) ∈ Tp M is the parallel transport of u(γ(h)) from γ(h) to γ(0). Intuitively, LeviCivita connection is the directional derivative of u along direction v, taking the metric into d u(x + tv). When u is defined on a curve account. In particular, for Rn , we have ∇v u(x) = dt d d n c, we define Dt u = ∇c′ (t) u. In R , we have Dt u(γ(t)) = dt u(γ(t)). We reserve dt for the usual derivative with Euclidean coordinates. We list some basic facts about the definitions introduced above that are useful for computation and intuition. Fact 66. Given a manifold M , a curve c(t) ∈ M , a vector v and vector fields u, w on M , we have the following: 1. (alternative definition of parallel transport) v(t) is the parallel transport of v along c(t) if and only if ∇c′ (t) v(t) = 0. 2. (alternative definition of geodesic) c is a geodesic if and only if ∇c′ (t) c′ (t) = 0. 3. (linearity) ∇v (u + w) = ∇v u + ∇v w. 4. (product rule) For any scalar-valued function f, ∇v (f · u) = 63 ∂f ∂v u + f · ∇v u. 5. (metric preserving) d dt hu, wic(t) = hDt u, wic(t) + hu, Dt wic(t) . ∂c 6. (torsion free-ness) For any map c(t, s) from a subset of R2 to M , we have that Ds ∂c ∂t = Dt ∂s where Ds = ∇ ∂c and Dt = ∇ ∂c . ∂s ∂t 7. (alternative definition of Levi-Civita connection) ∇v u is the unique linear mapping from the product of vector and vector field to vector field that satisfies (3), (4), (5) and (6). D.1 Curvature Roughly speaking, curvature measures the amount by which a manifold deviates from Euclidean space. Given vector u, v ∈ Tp M , in this section, we define uv be the point obtained from moving from p along direction u with distance kukp (using geodesic), then moving along direction “v” with distance kvkp where “v” is the parallel transport of v along the path u. In Rn , uv is exactly p + u + v and hence uv = vu, namely, parallelograms close up. For a manifold, parallelograms almost close up, namely, d(uv, vu) = o(kuk kvk). This property is called being torsion-free. 1. Riemann curvature tensor: Three-dimensional parallelepipeds might not close up, and the curvature tensor measures how far they are from closing up. Given vector u, v, w ∈ Tp M , we define uvw as the point obtained by moving from uv along direction “w” for distance kwkp where “w” is the parallel transport of w along the path uv. In a manifold, parallelepipeds do not close up and the Riemann curvature tensor how much uvw deviates from vuw. Formally, for vector fields v, w, we define τv w be the parallel transport of w along the vector field v for one unit of time. Given vector field v, w, u, we define the Riemann curvature tensor by R(u, v)w = d d −1 −1 τ τ τsu τtv w ds dt su tv . (D.1) t,s=0 Riemann curvature tensor is a tensor, namely, R(u, v)w at point p depends only on u(p), v(p) and w(p). 2. Ricci curvature: Given a vector v ∈ Tp M , the Ricci curvature Ric(v) measures if the geodesics starting around p in direction v converge together. Positive Ricci curvature indicates the geodesics converge while negative curvature indicates they diverge. Let S(0) be a small shape around p and S(t) be the set of point obtained by moving S(0) along geodesics in the direction v for t units of time. Then, volS(t) = volS(0)(1 − Formally, we define Ric(v) = t2 Ric(v) + smaller terms). 2 X ui (D.2) hR(v, ui )ui , vi where ui is an orthonormal basis of Tp M . Equivalently, we have Ric(v) = Eu∼N (0,I) hR(v, u)u, vi. kvk2 . For Rn , Ric(v) = 0. For a sphere in n + 1 dimension with radius r, Ric(v) = n−1 r2 Fact 67 (Alternative definition of Riemann curvature tensor). Given any M -valued function c(t, s), ∂c we have vector fields ∂c ∂t and ∂s on M . Then, for any vector field z, R( ∂c ∂c , )z = ∇ ∂c ∇ ∂c z − ∇ ∂c ∇ ∂c z. ∂t ∂s ∂s ∂t ∂t ∂s Equivalently, we write R(∂t c, ∂s c)z = Dt Ds z − Ds Dt z. 64 Fact 68. Given vector fields v, u, w, z on M , hR(v, u)w, zi = hR(w, z)v, ui = − hR(u, v)w, zi = − hR(v, u)z, wi . D.2 Hessian manifolds 2 Recall that a manifold is called Hessian if it is a subset of Rn and its metric is given by gij = ∂x∂i ∂xj φ for some smooth convex function φ. We let gij be entries of the inverse matrix of gij . For example, P 2 ∂3 we have j gij gjk = δik . We use φij to denote ∂x∂i ∂xj φ and φijk to denote ∂xi ∂x j ∂xk φ. Since a Hessian manifold is a subset of Euclidean space, we identify tangent spaces Tp M by Euclidean coordinates. The following lemma gives formulas for the Levi-Civita connection and curvature under Euclidean coordinates. Lemma 69 ([28]). Given a Hessian manifold M , vector fields v, u, w, z on M , we have the following: 1. (Levi-Civita connection) ∇v u = and the Christoffel symbol P ∂uk ik vi ∂xi ek Γkij = + P k ijk vi uj Γij ek where ek are coordinate vectors 1 X kl g φijl . 2 l 2. (Riemann curvature tensor) hR(u, v)w, zi = Rklij = 3. (Ricci curvature) Ric(v) = 1 4 P P ijlk Rklij ui vj wl zk where 1 X pq g (φjkp φilq − φikp φjlq ) . 4 pq ijlkpq g pq g jl (φjkpφilq − φikp φjlq ) vi vk . 65 

arXiv:1710.08585v1 [cs.LG] 24 Oct 2017 Max-Margin Invariant Features from Transformed Unlabeled Data Dipan K. Pal, Ashwin A. Kannan∗, Gautam Arakalgud∗, Marios Savvides Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA 15213 {dipanp,aalapakk,garakalgud,marioss}@cmu.edu Abstract The study of representations invariant to common transformations of the data is important to learning. Most techniques have focused on local approximate invariance implemented within expensive optimization frameworks lacking explicit theoretical guarantees. In this paper, we study kernels that are invariant to a unitary group while having theoretical guarantees in addressing the important practical issue of unavailability of transformed versions of labelled data. A problem we call the Unlabeled Transformation Problem which is a special form of semisupervised learning and one-shot learning. We present a theoretically motivated alternate approach to the invariant kernel SVM based on which we propose MaxMargin Invariant Features (MMIF) to solve this problem. As an illustration, we design an framework for face recognition and demonstrate the efficacy of our approach on a large scale semi-synthetic dataset with 153,000 images and a new challenging protocol on Labelled Faces in the Wild (LFW) while out-performing strong baselines. 1 Introduction It is becoming increasingly important to learn well generalizing representations that are invariant to many common nuisance transformations of the data. Indeed, being invariant to intra-class transformations while being discriminative to between-class transformations can be said to be one of the fundamental problems in pattern recognition. The nuisance transformations can give rise to many ‘degrees of freedom’ even in a constrained task such as face recognition (e.g. pose, age-variation, illumination etc.). Explicitly factoring them out leads to improvements in recognition performance as found in [11, 8, 6]. It has also been shown that that features that are explicitly invariant to intra-class transformations allow the sample complexity of the recognition problem to be reduced [2]. To this end, the study of invariant representations and machinery built on the concept of explicit invariance is important. Invariance through Data Augmentation. Many approaches in the past have enforced invariance by generating transformed labelled training samples in some form such as [14, 19, 21, 10, 17, 4]. Perhaps, one of the most popular method for incorporating invariances in SVMs is the virtual support method (VSV) in [20], which used sequential runs of SVMs in order to find and augment the support vectors with transformed versions of themselves. Indecipherable transformations in data leads to shortage of transformed labelled samples. The above approaches however, assume that one has explicit knowledge about the transformation. This is a strong assumption. Indeed, in most general machine learning applications, the transformation ∗ Authors contributed equally 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. present in the data is not clear and cannot be modelled easily, e.g. transformations between different views of a general 3D object and between different sentences articulated by the same person. Methods which work on generating invariance by explicitly transforming or augmenting labelled training data cannot be applied to these scenarios. Further, in cases where we do know the transformations that exist and we actually can model them, it is difficult to generate transformed versions of very large labelled datasets. Hence there arises an important problem: how do we train models to be invariant to transformations in test data, when we do not have access to transformed labelled training samples ? Transformed unlabeled data Non-transformed labeled data Train Train Availability of unlabeled transformed data. Although it is difficult to obtain or generate transformed labelled data (due to the reasons mentioned above), unlabeled transformed data is more readily available. For instance, if different views of specific objects of interest are not available, one can simply collect views of general objects. Also, if different sentences spoken by a specific group of people are not available, one can simply collect those spoken by members of the general population. In both these scenarios, no explicit knowledge or model of the transformation is needed, thereby bypassing the problem of indecipherable transformations. This situation is common in vision e.g. only unlabeled transformed images are observed, but has so far mostly been addressed by the community by intense efforts in large scale data collection. Note that the transformed data that is collected is not required to be labelled. We now are in a position to state the central problem that this paper addresses. Test image invariant to not invariant to Figure 1: Max-Margin Invariant Features (MMIF) can solve an important problem we call the Unlabeled Transformation Problem. In the figure, a traditional classifier F (x) "learns" invariance to nuisance transformations directly from the labeled dataset X . On the other hand, our approach (MMIF) can incorporate additional invariance learned from any unlabeled data that undergoes the nuisance transformation of interest. The Unlabeled Transformation (UT) Problem: Having access to transformed versions of the training unlabeled data but not of labelled data, how do we learn a discriminative model of the labelled data, while being invariant to transformations present in the unlabeled data ? Overall approach. The approach presented in this paper however (see Fig. 1), can solve this problem and learn invariance to transformations observed only through unlabeled samples and does not need labelled training data augmentation. We explicitly and simultaneously address both problems of generating invariance to intra-class transformation (through invariant kernels) and being discriminative to inter or between class transformations (through max-margin classifiers). Given a new test sample, the final extracted feature is invariant to the transformations observed in the unlabeled set, and thereby generalizes using just a single example. This is an example of one-shot learning. Prior Art: Invariant Kernels. Kernel methods in machine learning have long been studied to considerable depth. Nonetheless, the study of invariant kernels and techniques to extract invariant features has received much less attention. An invariant kernel allows the kernel product to remain invariant under transformations of the inputs. Most instances of incorporating invariances focused on local invariances through regularization and optimization such as [20, 21, 3, 23]. Some other techniques were jittering kernels [19, 3] and tangent-distance kernels [5], both of which sacrificed the positive semi-definite property of its kernels and were computationally expensive. Though these methods have had some success, most of them still lack explicit theoretical guarantees towards invariance. The proposed invariant kernel SVM formulation on the other hand, develops a valid PSD kernel that is guaranteed to be invariant. [4] used group integration to arrive at invariant kernels but did not address the Unlabeled Transformation problem which our proposed kernels do address. Further, our proposed kernels allow for the formulation of the invariant SVM and application to large scale problems. Recently, [16] presented some work with invariant kernels. However, unlike our non-parametric formulation, they do not learn the group transformations from the data itself and assume known parametric transformations (i.e. they assume that transformation is computable). Key ideas. The key ideas in this paper are twofold. 2 1. The first is to model transformations using unitary groups (or sub-groups) leading to unitarygroup invariant kernels. Unitary transforms allow the dot product to be preserved and allow for interesting generalization properties leading to low sample complexity and also allow learning transformation invariance from unlabeled examples (thereby solving the Unlabeled Transformation Problem). Classes of learning problems, such as vision, often have transformations belonging to a unitary-group, that one would like to be invariant towards (such as translation and rotation). In practice however, [9] found that invariance to much more general transformations not captured by this model can been achieved. 2. Secondly, we combine max-margin classifiers with invariant kernels leading to non-linear max-margin unitary-group invariant classifiers. These theoretically motivated invariant non-linear SVMs form the foundation upon which Max-Margin Invariant Features (MMIF) are based. MMIF features can effectively solve the important Unlabeled Transformation Problem. To the best of our knowledge, this is the first theoretically proven formulation of this nature. Contributions. In contrast to many previous studies on invariant kernels, we study non-linear positive semi-definite unitary-group invariant kernels guaranteeing invariance that can address the UT Problem. One of our central theoretical results to applies group integration in the RKHS. It builds on the observation that, under unitary restrictions on the kernel map, group action in the input space is reciprocated in the RKHS. Using the proposed invariant kernel, we present a theoretically motivated approach towards a non-linear invariant SVM that can solve the UT Problem with explicit invariance guarantees. As our main theoretical contribution, we showcase a result on the generalization of max-margin classifiers in group-invariant subspaces. We propose Max-Margin Invariant Features (MMIF) to learn highly discriminative non-linear features that also solve the UT problem. On the practical side, we propose an approach to face recognition to combine MMIFs with a pre-trained deep learning feature extractor (in our case VGG-Face [13]). MMIF features can be used with deep learning whenever there is a need to focus on a particular transformation in data (in our application pose in face recognition) and can further improve performance. 2 Unitary-Group Invariant Kernels Premise: Consider a dataset of normalized samples along with labels X = {xi }, Y = {yi } ∀i ∈ 1...N with x ∈ Rd and y ∈ {+1, −1}. We now introduce into the dataset a number of unitary transformations g part of a locally compact unitary-group G. We note again that the set of transformations under consideration need not be the entire unitary group. They could very well be a subgroup. Our augmented normalized dataset becomes {gxi , yi } ∀g ∈ G ∀i. For clarity, we denote by gx the action of group element g ∈ G on x, i.e. gx = g(x). We also define an orbit of x under G as the set XG = {gx} ∀g ∈ G. Clearly, X ⊆ XG . An invariant function is defined as follows. Definition 2.1 (G-Invariant Function). For any group G, we define a function f : X → Rn to be G-invariant if f (x) = f (gx) ∀x ∈ X ∀g ∈ G. One method of generating an invariant towards a group is through group integration. Group integration has stemmed from classical invariant theory and can be shown to be a projection onto a G-invariant subspace for vector spaces. In such a space x = gx ∀g ∈ G and thus the representation x is invariant under the transformation of any element from the group G. This is ideal for recognition problems where one would want to be discriminative to between-class transformations (for e.g. between distinct subjects in face recognition) but be invariant to within-class transformations (for e.g. different images of the same subject). The set of transformations we model as G are the within-class transformations that we would like to be invariant towards. An invariant to any group G can be generated through the following basic (previously) known property (Lemma 2.1) based on group integration. d Lemma 2.1. (Invariance Property) Given a vector ω ∈ R R , and any R affine group G, for any fixed 0 0 g ∈ G and a normalized Haar measure dg, we have g G gω dg = G gω dg The Haar measure (dg) exists for every locally compact group and is unique up to a positive multiplicative constant (hence normalized). A similar property holds for discrete groups. Lemma 2.1 R results in the quantity G gω dg enjoy global invariance (encompassing all elements) to group G. This property allows one to generate a G-invariant subspace in the inherent space Rd through group integration. In practice, the integral corresponds to a summation over transformed samples. The 3 following two lemmas (novel results, and part R of our contribution) (Lemma 2.2 and 2.3) showcase elementary properties of the operator Ψ = G g dg for a unitary-group G 2 . These properties would prove useful in the analysis of unitary-group invariant kernels and features. R Lemma 2.2. If Ψ = G g dg for unitary G, then ΨT = Ψ R Lemma 2.3. (Unitary Projection) If Ψ = G g dg for any affine G, then ΨΨ = Ψ, i.e. it is a projection operator. Further, if G is unitary, then hω, Ψω 0 i = hΨω, ω 0 i ∀ω, ω 0 ∈ Rd Sample Complexity and Generalization. On applying the operator Ψ to the dataset X , all points in the set {gx | g ∈ G} for any x ∈ X map to the same point Ψx in the G-invariant subspace thereby reducing the number of distinct points by a factor of |G| (the cardinality of G, if G is finite). Theoretically, this would drastically reduce sample complexity while preserving linear feasibility (separability). It is trivial to observe that a perfect linear separator learned in XΨ = {Ψx | x ∈ X } would also be a perfect separator for XG , thus in theory achieving perfect generalization. Generalization here refers to the ability to perform correct classification even in the presence of the set of transformations G. We prove a similar result for Reproducing Kernel Hilbert Spaces (RKHS) in Section 2.2. This property is theoretically powerful since cardinality of G can be large. A classifier can avoid having to observe transformed versions {gx} of any x and yet generalize perfectly. is The case of Face Recognition. As an illustration, if the group G of transformations considered R pose (it is hypothesized that small changes in pose can be modeled as unitary [11]), then Ψ = G g dg represents a pose invariant subspace. In theory, all poses of a subject will converge to the same point in that subspace leading to near perfect pose invariant recognition. We have not yet leveraged the power of the unitary structure of the groups which is also critical in generalization to test cases as we would see later. We now present our central result showcasing that unitary kernels allow the unitary group action to reciprocate in a Reproducing Kernel Hilbert Space. This is critical to set the foundation for our core method called Max-Margin Invariant Features. 2.1 Group Actions Reciprocate in a Reproducing Kernel Hilbert Space Group integration provides exact invariance as seen in the previous section. However, it requires the group structure to be preserved, i.e. if the group structure is destroyed, group integration does not provide an invariant function. In the context of kernels, it is imperative that the group relation between the samples in XG be preserved in the kernel Hilbert space H corresponding to some kernel k with a mapping φ. If the kernel k is unitary in the following sense, then this is possible. Definition 2.2 (Unitary Kernel). A kernel k(x, y) = hφ(x), φ(y)i is a unitary kernel if, for a unitary group G, the mapping φ(x) : X → H satisfies hφ(gx), φ(gy)i = hφ(x), φ(y)i ∀g ∈ G, ∀x, y ∈ X . The unitary condition is fairly general, a common class of unitary kernels is the RBF kernel. We now define a transformation within the RKHS itself as gH : φ(x) → φ(gx) ∀φ(x) ∈ H for any g ∈ G where G is a unitary group. We then have the following result of significance. Theorem 2.4. (Covariance in the RKHS) If k(x, y) = hφ(x), φ(y)i is a unitary kernel in the sense of Definition 2.2, then gH is a unitary transformation, and the set GH = {gH | gH : φ(x) → φ(gx) ∀g ∈ G} is a unitary-group in H. Theorem 2.4 shows that the unitary-group structure is preserved in the RKHS. This paves the way for new theoretically motivated approaches to achieve invariance to transformations in the RKHS. There have been a few studies on group invariant kernels [4, 11]. However, [4] does not examine whether the unitary group structure is actually preserved in the RKHS, which is critical. Also, DIKF was recently proposed as a method utilizing group structure under the unitary kernel [11]. Our result is a generalization of the theorems they present. Theorem 2.4 shows that since the unitary group structure is preserved in the RKHS, any method involving group integration would be invariant in the original space. The preservation of the group structure allows more direct group invariance results to be applied in the RKHS. It also directly allows one to formulate a non-linear SVM while guaranteeing invariance theoretically leading to Max-Margin Invariant Features. 2 All proofs are presented in the supplementary material 4 2.2 Invariant Non-linear SVM: An Alternate Approach Through Group Integration We now apply the group integration approach to the kernel SVM. The decision function of SVMs can be written in the general form as fθ (x) = ω T φ(x) + b for some bias b ∈ R (we agglomerate all parameters of f in θ) where φ is the kernel feature map, i.e. φ : X → H. Reviewing the SVM, a maximum margin separator is found by minimizing loss functions such as the hinge loss along with a regularizer. In order to invoke invariance, we can now utilize group integration in the the kernel space H using Theorem 2.4. All points in the set {gx ∈ XG } get mapped to φ(gx) = gH φ(x) for a given g ∈ G in R the input space X . Group integration then results in a G-invariant subspace within H through ΨH = GH gH dgH using Lemma 2.1. Introducing Lagrange multipliers α = (α1 , α2 ...αN ) ∈ RN , the dual formulation (utilizing Lemma 2.2 and Lemma 2.3) then becomes min − α X i αi + 1X yi yj αi αj hΨH φ(xi ), ΨH φ(xj )i 2 i,j (1) P under the constraints 0 ≤ αi ≤ N1 ∀i. The SVM separator is then given by i αi yi = 0, P ∗ ∗ ωH = ΨH ω = i yi αi ΨH φ(xi ) thereby existing in the GH -invariant (or equivalently G-invariant) subspace ΨH within H (since g → gH is a bijection). Effectively, the SVM observes samples from ∗ XΨH = {x | φ(x) = ΨH φ(u), ∀u ∈ XG } and therefore ωH enjoys exact global invariance to G. ∗ Further, ΨH ω is a maximum-margin separator of {φ(XG )} (i.e. the set of all transformed samples). This can be shown by the following result. For a unitary group G and unitary kernel k(x, y) = hφ(x), φ(y)i, Theorem 2.5. (Generalization) R ∗ if ωH = ΨH ω ∗ = ( GH gH dgH ) ω ∗ is a perfect separator for {ΨH φ(X )} = {ΨH φ(x) | ∀x ∈ X }, then ΨH ω ∗ is also a perfect separator for {φ(XG )} = {φ(x) | x ∈ XG } with the same margin. Further, a max-margin separator of {ΨH φ(X )} is also a max-margin separator of {φ(XG )}. The invariant non-linear SVM in objective 1, observes samples in the form of ΨH φ(x) and obtains a max-margin separator ΨH ω ∗ . This allows for the generalization properties of max-margin classifiers to be combined with those of group invariant classifiers. While being invariant to nuisance transformations, max-margin classifiers can lead to highly discriminative features (more robust than DIKF [11] as we find in our experiments) that are invariant to within-class transformations. Theorem 2.5 shows that the margins of φ(XG ) and {ΨH φ(XG )} are deeply related and implies that ΨH φ(x) is a max-margin separator for both datasets. Theoretically, the invariant non-linear SVM is able to generalize to XG on just observing X and utilizing prior information in the form of G for all unitary kernels k. This is true in practice for linear kernels. For non-linear kernels in practice, the invariant SVM still needs to observe and integrate over transformed training inputs. Leveraging unitary group properties. During test time to achieve invariance, the SVM would require to observe and integrate over all possible transformations of the test sample. This is a huge computational and design bottleneck. We would ideally want to achieve invariance and generalize by observing just a single test sample, in effect perform one shot learning. This would not only be computationally much cheaper but make the classifier powerful owing to generalization to full transformed orbits of test samples by observing just that single sample. This is where unitarity of g helps and we leverage it in the form of the following Lemma. R Lemma 2.6. (Invariant Projection) If Ψ = G g dg for any unitary group G, then for any fixed g 0 ∈ G (including the identity element) we have hΨx0 , Ψω 0 i = hg 0 x0 , Ψω 0 i ∀ω, ω 0 ∈ Rd Assuming Ψω 0 is the learned SVM classifier, Lemma 2.6 shows that for any test x0 , the invariant dot product hΨx0 , Ψω 0 i which involves observing all transformations of x0 is equivalent to the quantity hg 0 x0 , Ψω 0 i which involves observing only one transformation of x0 . Hence one can model the entire orbit of x0 under G by a single sample g 0 x0 where g 0 ∈ G can be any particular transformation including identity. This drastically reduces sample complexity and vastly increases generalization capabilities of the classifier since one only need to observe one test sample to achieve invariance Lemma 2.6 also helps us in saving computation, allowing us to apply the computationally expensive Ψ (group integration) operation only once on he classifier and not the test sample. Thus, the kernel in the Invariant SVM formulation can be replaced by the form kΨ (x, y) = hφ(x), ΨH φ(y)i. For kernels in general, the GH -invariant subspace cannot be explicitly R computed since it lies in the RKHS. It is only implicitly projected upon through ΨH φ(xi ) = G φ(gxi )dgH . It is important to 5 Integration over the group (pooling) Class 1 Test Image Class 2 Class 3 Kernel Invariant Feature Test Image Class 4 (a) Invariant kernel feature extraction (b) SVM feature extraction leading to MMIF features Figure 2: MMIF Feature Extraction. (a) l(x) denotes the invariant kernel feature of any x which is invariant to the transformation G. Invariance is generated by group integration (or pooling). The invariant kernel feature learns invariance form the unlabeled transformed template set TG . Also, the faces depicted are actual samples from the large-scale mugshots data (∼ 153, 000 images). (b) Once the invariant features have been extracted for the labelled non-transformed dataset X , then the SVMs learned act as feature extractors. Each binary class SVM (different color) was trained on the invariant kernel feature of a random subset of l(X ) with random class assignments. The final MMIF feature for x is the concatenation of all SVM inner-products with l(x). note that during testing however, the SVM formulation will be invariant to transformations of the test sample regardless of a linear or non-linear kernel. Positive R Semi-Definiteness. The G-invariant kernel map is now of the form kΨ (x, y) = hφ(x), G φ(gy)dgH i. This preserves the positive semi-definite property of the kernel k while guaranteeing global invariance to unitary transformations., unlike jittering kernels [19, 3] and tangent-distance kernels [5]. If we wish to include invariance to scaling however (in the sense of scaling an image), then we would lose positive-semi-definiteness (it is also not a unitary transform). Nonetheless, [22] show that conditionally positive definite kernels still exist for transformations including scaling, although we focus of unitary transformations in this paper. 3 Max-Margin Invariant Features The previous section utilized a group integration approach to arrive a theoretically invariant non-linear SVM. It however does not Transformation problem i.e. the kernel kΨ (x, y) = R address the Unlabeled R hΨH φ(x), ΨH φ(y)i = h G φ(gx)dgH , G φ(gy)dgH i still requires observing transformed versions of the labelled input sample namely {gx | gx ∈ XG } (or atleast one of the labelled samples if we utilize Lemma 2.6). We now present our core approach called Max-Margin Invariant Features (MMIF) that does not require the observation of any transformed labelled training sample whatsoever. Assume that we have access to an unlabeled set of M templates T = {ti }i={1,...M } . We assume that we can observe all transformations under a unitary-group G, i.e. we have access to TG = {gti | ∀g ∈ G}i={1,...M } . Also, assume we have access to a set X = {xj }i={1,...D} of labelled data with N classes which are not transformed. We can extract an M -dimensional invariant kernel feature for each xj ∈ X as follows. Let the invariant kernel feature be l(x) ∈ RM to explicitly show the dependence on x. Then the ith dimension of l for any particular x is computed as Z l(x)i = hφ(x), ΨH φ(ti )i = hφ(x), Z gH φ(ti )dgH i = hφ(x), G φ(gti )dgH i (2) G The first equality utilizes Lemma 2.6 and the third equality uses Theorem 2.4. This is equivalent to observing all transformations of x since hφ(x), ΨH φ(ti )i = hΨH φ(x), φ(ti )i using Lemma 2.3. Thereby we have constructed a feature l(x) which is invariant to G without ever needing to observe transformed versions of the labelled vector x. We now briefly the training of the MMIF feature extractor. The matching metrics we use for this study is normalized cosine distance. 6 Training MMIF SVMs. To learn a K-dimensional MMIF feature (potentially independent of N ), we learn K independent binary-class linear SVMs. Each SVM trains on the labelled dataset l(X ) = {l(xj ) | j = {1, ...D}} with each sample being label +1 for some subset of the N classes (potentially P just one class) and the rest being labelled −1. This leads us to a classifier in the form of ωk = j yj αj l(xj ). Here, yj is the label of xj for the k th SVM. It is important to note that the unlabeled data was only used to extract l(xj ). Having multiple classes randomly labelled as positive allows the SVM to extract some feature that is common between them. This increases generalization by forcing the extracted feature to be more general (shared between multiple classes) rather than being highly tuned to a single class. Any K-dimensional MMIF feature can be trained through this technique leading to a higher dimensional feature vector useful in case where one has limited labelled samples and classes (N is small). During feature extraction, the K inner products (scores) of the test sample x0 with the K distinct binary-class SVMs provides the K-dimensional MMIF feature vector. This feature vector is highly discriminative due to the max-margin nature of SVMs while being invariant to G due to the invariant kernels. 0 MMIF. Given TG and X , the MMIF feature is defined as MMIF(x ) ∈ RK for any test x0 with P 0 each dimension k being computed as hl(x ), ωk i for ωk = j yj αj l(xj ) ∀xj ∈ X . Further, l(x0 ) ∈ RM ∀x with each dimension i being l(x0 )i = hφ(x0 ), ΨH φ(ti )i. The process is illustrated in Fig. 2. Inheriting transformation invariance from transformed unlabeled data: A special case of semisupervised learning. MMIF features can learn to be invariant to transformations (G) by observing them only through TG . It can then transfer the invariance knowledge to new unseen samples from X thereby becoming invariant to XG despite never having observed any samples from XG . This is a special case of semi-supervised learning where we leverage on the specific transformations present in the unlabeled data. This is a very useful property of MMIFs allowing one to learn transformation invariance from one source and sample points from another source while having powerful discrimination and generalization properties. The property is can be formally stated as the following Theorem. Theorem 3.1. (MMIF is invariant to learnt transformations) MMIF(x0 ) = MMIF(gx0 ) ∀x0 ∀g ∈ G where G is observed only through TG = {gti | ∀g ∈ G}i={1,...M } . Thus we find that MMIF can solve the Unlabeled Transformation Problem. MMIFs have an invariant and a discriminative component. The invariant component of MMIF allows it to generalize to new transformations of the test sample whereas the discriminative component allows for robust classification due to max-margin classifiers. These two properties allow MMIFs to be very useful as we find in our experiments on face recognition. Max and Mean Pooling in MMIF. Group integration in practice directly results in mean pooling. Recent work however, showed that group integration can be treated as a subset of I-theory where one tries to measure moments (or a subset of) of the distribution hx, gωi g ∈ G since the distribution itself is also an invariant [1]. Group integration can be seen as measuring the mean or the first moment of the distribution. One can also characterize using the infinite moment or the max of the distribution. We find in our experiments that max pooling outperforms mean pooling in general. All results in this paper however, still hold under the I-theory framework. MMIF on external feature extractors (deep networks). MMIF does not make any assumptions regarding its input and hence one can apply it to features extracted from any feature extractor in general. The goal of any feature extractor is to (ideally) be invariant to within-class transformation while maximizing between-class discrimination. However, most feature extractors are not trained to explicitly factor out specific transformations. If we have access to even a small dataset with the transformation we would like to be invariant to, we can transfer the invariance using MMIFs (e.g. it is unlikely to observe all poses of a person in datasets, but pose is an important nuisance transformation). Modelling general non-unitary transformations. General non-linear transformations such as out-of-plane rotation or pose variation are challenging to model. Nonetheless, a small variation in these transformations can be approximated by some unitary G assuming piece wise linearity through transformation-dependent sub-manifold unfolding [12]. Further, it was found that in practice, integrating over general transformations produced approximate invariance [9]. 7 1 0.9 0.9 0.8 Verification Rate Verification Rate 1 0.95 0.85 ∞ -DIKF (0.74) 1 -DIKF (0.61) NDP-∞ (0.41) NDP-1 (0.32) MMIF (Ours) (0.78) VGG Features (0.55) MMIF-VGG (Ours) (0.61) 0.8 0.75 0.7 0.65 0.6 0.55 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 -8 0.5 0.1 0.2 0.3 0.4 0.5 0.6 MMIF VGG (Ours)(0.71) VGG (0.56) 0.7 False Accept Rate (a) Invariant kernel feature extraction 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 False Accept Rate (b) SVM feature extraction leading to MMIF features Figure 3: (a) Pose-invariant face recognition results on the semi-synthetic large-scale mugshot database (testing on 114,750 images). Operating on pixels: MMIF (Pixels) outperforms invariance based methods DIKF [11] and invariant NDP [9]. Operating on deep features: MMIF trained on VGG-Face features [13] (MMIF-VGG) produces a significant improvement in performance. The numbers in the brackets represent VR at 0.1% FAR. (b) Face recognition results on LFW with raw VGG-Face features and MMIF trained on VGG-Face features. The values in the bracket show VR at 0.1% FAR. 4 Experiments on Face Recognition As illustration, we apply MMIFs using two modalities overall 1) on raw pixels and 2) on deep features from the pre-trained VGG-Face network [13]. We provide more implementation details and results discussion in the supplementary. A. MMIF on a large-scale semi-synthetic mugshot database (Raw-pixels and deep features). We utilize a large-scale semi-synthetic face dataset to generate the sets TG and X for MMIF. In this dataset, only two major transformations exist, that of pose variation and subject variation. All other transformations such as illumination, translation, rotation etc are strictly and synthetically controlled. This provides a very good benchmark for face recognition. where we want to be invariant to pose variation and be discriminative for subject variation. The experiment follows the exact protocol and data as described in [11] 3 We test on 750 subjects identities with 153 pose varied real-textured gray-scale image each (a total of 114,750 images) against each other resulting in about 13 billion pair-wise comparisons (compared to 6,000 for the standard LFW protocol). Results are reported as ROC curves along with VR at 0.1% FAR. Fig. 3(a) shows the ROC curves for this experiment. We find that MMIF features out-performs all baselines including VGG-Face features (pre-trained), DIKF and NDP approaches thereby demonstrating superior discriminability while being able to effectively capture pose-invariance from the transformed template set TG . MMIF is able to solve the Unlabeled Transformation problem by extracting transformation information from unlabeled TG . B. MMIF on LFW (deep features): Unseen subject protocol. In order to be able to effectively train under the scenario of general transformations and to challenge our algorithms, we define a new much harder protocol on LFW. We choose the top 500 subjects with a total of 6,300 images for training MMIF on VGG-Face features and test on the remaining subjects with 7,000 images. We perform all versus all matching, totalling upto 49 million matches (4 orders more than the official protocol). The evaluation metric is defined to be the standard ROC curve with verification rate reported at 0.1% false accept rate. We split the 500 subjects into two sets of 250 and use as TG and X . We do not use any alignment for this experiment, and the faces were cropped according to [18]. Fig. 3(b) shows the results of this experiment. We see that MMIF on VGG features significantly outperforms raw VGG on this protocol, boosting the VR at 0.1% FAR from 0.56 to 0.71. This demonstrates that MMIF is able to generate invariance for highly non-linear transformations that are not well-defined rendering it useful in real-world scenarios where transformations are unknown but observable. 3 We provide more details in the supplementary. Also note that we do not need utilize identity information, all that is required is the fact that a set of pose varied images belong to the same subject. Such data can be obtained through temporal sampling. 8 5 Main Experiments: Detailed notes supplementing the main paper. A. MMIF on a large-scale semi-synthetic mugshot database (Raw-pixels and deep features). MMIF template set TG and X . We utilize a large-scale semi-synthetic face dataset to generate the sets TG and X for MMIF. The face textures are sampled from real-faces although the poses are rendered using 3D model fit to each face independently, hence the dataset is semi-synthetic. This semi-synthetic dataset helps us to evaluate our algorithm in a clean setting, where there exists only one challenging nuisance transformation (pose variation). Therefore G models pose variation in faces. We utilize the same pose variation dataset generation procedure as described in [11] in order for a fair comparison. The poses were rendered varying from −40◦ to 40◦ (yaw) and −20◦ to 20◦ (pitch) in steps of 5◦ using 3D-GEM [15]. The total number of images we generate is 153 × 1000 = 153, 000 images. We align all faces by the two eye-center locations in a 168 × 128 crop. Protocol. Our first experiment is a direct comparison with approaches similar in spirit to ours, namely `∞ -DIKF and `1 -DIKF [11] and NDP-`∞ and NDP-`1 [9, 1]. We train on 250 subjects (38,250 images) and test each method on the remaining 750 subjects (114,750 images), matching all pose-varied images of a subject to each other. DIKF follows the same protocol as in [11]. For MMIF, we utilize the first 125 × 153 images (125 subjects with 153 poses each) as TG and the next 125 × 153 images as X . A total of 500 SVMs were trained on subsets of X (10 randomly chosen subjects per SVM with all images of 3 of those 10 subjects, again randomly chosen, being +1 and the rest being −1). Note that although X in this case contains pose variation, we do not integrate over them to generate invariance. All explicit invariance properties are generated through integration over TG . For testing, we compare all 153 images of the remaining unseen 750 subjects against each other (114,750 images). The algorithms are therefore tested on about 13 billion pair wise comparisons. Results are reported as ROC curves along with VR at 0.1% FAR. For this experiment, we report results working on 1) raw pixels directly and 2) 4096 dimensional features from the pre-trained VGG-Face network [13]. As a baseline, we also report results on using the VGG-Face features directly. Results. Fig.3(a) shows the ROC curves for this experiment. We find that MMIF features out-perform both DIKF and NDP approaches thereby demonstrating superior discriminability while being able to effectively capture pose-invariance from the transformed template set TG . We find that VGG-Face features suffer a handicap due to the images being grayscale. Nonetheless, MMIF is able to transfer pose-invariance from TG onto the VGG features. This significantly boosts performance owing to the fact that the main nuisance transformation is pose. MMIF being explicitly pose invariant along with solving the Unlabeled Transformation Problem is able to help VGG features while preserving the discriminability of the VGG features. In fact, the max-margin SVMs further add discriminability. This illustrates in a clean setting (dataset only contains synthetically generated pose variation as nuisance transformation), that MMIF is able to work well in conjunction with deep learning features, thereby rendering itself immediately usable in more realistic settings. Our next set of experiments focus on this exact aspect. B. MMIF on LFW (deep features). Unseen subject protocol. LFW [7] has received a lot of attention in the recent years, and algorithms have approached near human accuracy on the original testing protocol. In order to be able to effectively train under the scenario of general transformations and to challenge our algorithms, we define a new much harder protocol on LFW. Instead of evaluating on about 6000 pair wise matches, we pair wise match on all images of subjects not seen in training. We have no way of modelling these subjects whatsoever, making this a difficult task. We utilize 500 subjects and all their images for training and test on the remaining 5249 subjects and all of their images. To use maximum amount of data for training, we pick the top 500 subjects with the most number of images available (about 6,300 images). The test data thus contains about 7000 images. The number of test pairwise matches is about 49 million, four orders of magnitude larger than the 6000 matches that the original LFW testing protocol defined. The evaluation metric is defined to be the standard ROC curve with verification rate reported at 0.1% false accept rate. MMIF template set TG and X . We split the 500 subjects data into two parts of 250 subjects each. We use the 250 subjects with the most number of images as transformed template set TG and use the rest of the 250 subjects as X . Note that in this experiment, the transformations considered are very generic and highly non-linear making it a difficult experiment. We do not use any alignment for this experiment, and the faces were cropped according to [18]. 9 Protocol. For MMIF, we process the kernel features from the transformed template set T G exactly as in the previous experiment A. Similarly, we learn a total of 500 SVMs on subsets of X following the same protocol as the previous experiment. Results. Fig.3(b) shows the results of this experiment. We see that MMIF on VGG features significantly outperforms raw VGG on this protocol, boosting the VR at 0.1% FAR from 0.56 to 0.71. This suggests, that MMIF can be used in conjunction with pre-trained deep features. In this experiment, MMIF capitalizes on the non-linear transformations that exist in LFW, whereas in the previous experiment on the semi0synthetic dataset (Experiment A), the transformation was well-defined to be pose variation. This demonstrates that MMIF is able to generate invariance for highly non-linear transformations that are not well-defined rendering it useful in real-world scenarios where transformations are unknown but observable. 6 6.1 Additional Experiments Large-scale Semi Synthetic Mugshot Data Motivation: In the main paper, the transformations were observed only through unlabeled TG while X is only meant to provide labeled untransformed data. However, during our expeirments in the main paper, even though we do not explicitly pool over the transformations X , we utilize all transformations for training the SVMs. In order to be closer to our theoretical setting, we now run MMIF on raw pixels and VGG-Face features [13] while constraining the number of images the SVMs train on to 30 random images for each subject. 1 0.95 0.9 Verification Rate MMIF Template set TG and X : We utilize a large scale semi-synthetic face dataset to generate the template set TG for MMIF. The face textures are sampled from real faces and the poses are rendered using a 3D model fit to each face independently, making the dataset semisynthetic. This semi-synthetic dataset helps us evaluate our algorithm in a clean setting, where there exists only one challenging nuisance transformation (pose variation). Therefore G models pose variation in faces. We utilize the same pose variation dataset generation procedure as described in [11] in order for a fair comparison. The poses were rendered varying from −40◦ to 40◦ (yaw) and −20◦ to 20◦ (pitch) in steps of 5◦ using 3D-GEM [15]. The total number of images we generate is 153 x 1000 = 153,000 images. We align all faces by the two eye-center locations in a 168 × 128 crop. Unlike our experiment presented in the main paper on this dataset, the template set X is constrained to include only 30 randomly selected poses that TG contained . This is done to better simulate a real-world setting where through X we would only observe faces at a few random poses. 0.85 ∞ -DIKF (0.74) 1 -DIKF (0.61) NDP-∞ (0.41) NDP-1 (0.32) VGG Features (0.55) MMIF-raw (Ours) (0.78) MMIF-VGG (Ours) (0.61) MMIF-cons-raw (Ours) (0.43) MMIF-cons-VGG (Ours) (0.65) 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 False Accept Rate Figure 4: Pose-invariant face recognition results on the semi-synthetic large-scale mugshot database (testing on 114,750 images). Operating on deep features: MMIF-cons-VGG trained on VGG-Face features [13] produces a significant improvement in performance over pure VGG features even though it utilizes a constrained X set. Interestingly, MMIF-cons-VGG almost matches performance of MMIF-VGG while using less data. The numbers in the brackets represent VR at 0.1% FAR. MMIF-cons was trained on the entire TG but only 30 random transformations per subject in the X . Protocol: This experiment is a direct comparison with approaches similar in spirit to ours, namely l∞ -DIKF and l1 -DIKF [11] and NDP-l∞ and NDP-l1 [9, 1]. We call this setting for MMIF as MMIF-cons (constrained) for reference. We train on 250 subjects (38,250 images) and test each method on the remaining 750 subjects (114,750 images), matching all pose-varied images of a subject to each other. DIKF follows the same protocol as in [11]. For MMIF, we utilize the first 125 x 153 images (125 subjects with 153 poses each) as the template set TG . Thus, TG remains exactly the same as the protocol in the main paper. The template set X is generated by choosing 30 random poses (for every subject) of the next 125 subjects. A total of 500 SVMs are trained on X with a random subset of 5 subjects being labeled +1 and the rest labeled -1. 10 It’s important to note that since X does not contain transformations that are observed in its entirety, all explicit invariance properties are generated through integration over TG . For testing, we follow the same protocol as in the main paper. We compare all 153 images of the remaining unseen 750 subjects against each other (114,750 images). The algorithms are therefore tested on about 13 billion pair wise comparisons. Results are reported as ROC curves along with the VR at 0.1% FAR. For this experiment, we report results working on 1) raw pixels directly and 2) 4096 dimensional features from the pre-trained VGG-Face network [13]. As a baseline, we also report results on using the VGG-Face features directly. Results: Fig. 4 shows the ROC curves for this experiment. We find that even though we train SVMs for MMIF-cons-VGG on a constrained version of X , it outperforms raw VGG features. Although, we do observe that MMIF-cons-raw outperforms NDP methods thereby demonstrating superior discriminability, it fails to match the original MMIF-raw method performance. Interestingly however, MMIF-cons-VGG matches MMIF-VGG features in performance despite being trained on much lesser data (30 instead of 153 images per subject). Thus, we find that MMIF when trained on a good feature extractor can provide added benefits of discrimination despite having lesser labeled samples to train on. 6.2 IARPA IJB-A Janus In this experiment, we explore how the number of SVMs influences the recognition performance on a large scale real-world dataset, namely the IARPA Janus Benchmark A (IJB-A) dataset. Data: We work on the verification protocol (1:1 matching) of the original dataset IJB-A Janus. This subset consists of 5547 image templates that map to 492 distinct subjects with each template containing (possibly) multiple images. The images are cropped with respect to bounding boxes that are specified by the dataset for all labeled images. The cropped images are then re-sized to 244 x 244 pixels in accordance with the requirements of the VGG face model. Explicit pose invariance (MMIF) is then applied to these general face descriptors. Verification Rate MMIF Template set TG and X : In order to effectively train under the scenario of general transformations, we define a new protocol the 1 Janus dataset similar to the LFW protocol de0.95 fined in the main paper. This protocol is suited 0.9 for MMIF since we explicitly generate invari0.85 ance to transformations that exist in Janus data. 0.8 We utilize the first 100 subjects and all the tem0.75 plates that map to these subjects (23723 images) 0.7 for training MMIF and test on the remaining 392 VGG MMIF-100 SVMs (0.47) 0.65 VGG MMIF-250 SVMs (0.51) subjects (27363 images). To make use of the 0.6 VGG MMIF-500 SVMs (0.51) maximum amount of data for training, we pick 0.55 the top 100 subjects with the most number of 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 images, the rest are all utilized for testing. Our False Accept Rate training dataset is further split into templates TG and X similar to our LFW protocol in the main Figure 5: Results of MMIF trained on VGG-Face feapaper. We use the first 50 subjects (of the top tures on the IARPA IJB-A Janus dataset for 100, 250 100 subjects) as TG and the rest as X in order and 500 SVMs. The number in the bracket denotes VR to maximize the transformations that we gener- at 0.1% FAR. ate invariance towards. To showcase the ability of MMIF to be used in conjunction with deep learning techniques, similar to our LFW experiment in the main paper, we train and test on VGG-Face features [13] on the Janus data. Protocol: As in our LFW experiment, we split the training data into two templates - TG and X . Similarly to all MMIF protocols in this paper, we train a total of 100, 250 and 500 SVM’s on subsets of X following the same protocol. We perform pairwise comparisons for the entirety of the test data (∼ 750 million image comparisons) which far exceeds the number of comparisons defined in the original testing protocol (∼ 110, 000 template comparisons) thereby making this protocol much larger and harder. Recall that throughout this supplementary and the main paper we always test on 11 completely unseen subjects. The evaluation metric is defined to be the standard ROC curve using cosine distance. Results: Fig. 5 shows the ROC curves for this experiment with new much larger and harder protocol. We find that even with just 100 SVMs or 100 max-margin feature extractors, the performance is close to that of 500 feature extractors. This suggests, that though the SVMs provide enough discrimination, the invariant kernel provides bulk of the recognition performance by explicitly being invariant to the transformations in the TG . Hence, our proposed invariant kernel is effective at learning invariance towards transformations present in a unlabeled dataset. We provide these curves as baselines for future work focusing on the problem on learning unlabeled transformations from a given dataset. 7 7.1 Proofs of theoretical results Proof of Lemma 2.1 Proof. We have, g0 Z Z gω dg = G g 0 gω dg = Z G g 00 ω dg 00 = G Z gω dg G Since the normalized Haar measure is invariant, i.e. dg = dg 0 . Intuitively, g 0 simply rearranges the group integral owing to elementary group properties. 7.2 Proof of Lemma 2.2 Proof. We have, Z Z Z ΨT = ( g dg)T = g T dg = g −1 dg −1 = Ψ G G G Using the fact g ∈ G ⇒ g −1 ∈ G and dg = dg −1 . 7.3 Proof Lemma 2.3 Proof. We have, Z Z ΨΨ = gh dg dh (3) ZG ZG g 0 dg 0 dh Z Z = dh g 0 dg 0 = G (4) G G (5) G =Ψ (6) R Since the Haar measure is normalized ( G dg = 1), and invariant. Also for any ω, ω 0 ∈ Rd , we have R R hω, Ψω 0 i = G hω, gω 0 idg = G hg −1 ω, ω 0 idg −1 = hΨω, ω 0 i 7.4 Proof of Theorem 2.4 Proof. We have hφ(gx), φ(gy)i = hφ(x), φ(y)i = hgH φ(x), gH φ(y)i, since the kernel k is unitary. Here we define gH φ(x) as the action of gH on φ(x). Thus, the mapping gH preserves the dot-product in H while reciprocating the action of g. This is one of the requirements of a unitary operator, however gH needs to be linear. We note that linearity of gH can be derived from the linearity of the inner product and its preservation under gH in H. Specifically for an arbitrary vector p and a scalar α, we have ||αgH p − gH (αp)||2 = hαgH p − gH (αp), αgH p − gH (αp)i (7) (8) = ||αgH p||2 + ||gH (αp)||2 − 2hαgH p, gH (αp)i (9) 2 2 2 = |α|||p|| + ||αp|| − 2α hp, pi = 0 12 (10) Similarly for vectors p, q, we have ||gH (p + q) − (gH p + gH q)||2 = 0 We now prove that the set GH is a group. We start with proving the closure property. We have for any 0 fixed gH , gH ∈ GH 0 00 gH gH φ(x) = gH φ(g 0 x) = φ(gg 0 x) = φ(g 00 x) = gH φ(x) 00 0 00 Since g 00 ∈ G therefore gH ∈ GH by definition. Also, gH gH = gH and thus closure is established. Associativity, identity and inverse properties can be proved similarly. The set GH = {gH | gH : φ(x) → φ(gx) ∀g ∈ G} is therefore a unitary-group in H. 7.5 Proof of Theorem 2.5 Proof. Since ΨH ω ∗ is a perfect separator mini yi (ΨH φ(xi ))T (ΨH ω ∗ ) ≥ ρ0 ∀{xi , yi } ∈ X . for {ΨH φ(X )}, ∃ρ0 > 0, s.t. 0 Using Lemma 2.4 and Theorem 2.5, we have for any fixed gH ∈ GH , 0 (ΨH φ(xi ))T (ΨH ω ∗ ) = (gH φ(xi ))T (ΨH ω ∗ ) Hence, 0 min yi (gH φ(xi ))T (ΨH ω ∗ ) (11) 0 = min yi (ΨH φ(xi ))T (ΨH ω ∗ ) ≥ ρ0 ∀(gH ⇒ g) ∈ G (12) i i Thus, ΨH ω ∗ is perfect separator for {φ(XG )} with a margin of at-least ρ0 . It also implies that a max-margin separator of {ΨH φ(X )} is also a max-margin separator of {φ(XG )}. 7.6 Proof of Lemma 2.6 R R R Proof. We have hΨx0 , Ψω 0 i = h g gx0 , Ψω 0 idg = h g g 0 x0 , Ψω 0 idg = hg 0 x0 , Ψω 0 ) g dg = hg 0 x0 , Ψω 0 i In the second equality, we fix any group element g 0 ∈ G since the inner-product is invariant using the argument hω, Ψω 0 i = hg 0 ω, Ψω 0 i. This is true using Lemma 2.1 and the fact that G is unitary. Further, the final equality utilizes the fact that the Haar measure dg is normalized. 7.7 Proof of Theorem 3.1 0 K 0 Proof. Given TG and X , the MMIF feature is defined as MMIF(x P ) ∈ R for any test x with 0 each dimension k being computed as hl(x ), ωk i for ωk = . Further, j yj αj l(xj ) ∀xj ∈ X R l(x0 ) ∈ RM ∀x with each dimension i being l(x0 )i = hφ(x0 ), ΨH φ(ti )i. Here, ΨH = GH gH dgH where gH in the RKHS corresponds to the group action of g ∈ G acting in the space of X . We therefore have for the ith dimension of l(x0 ), l(x0 )i = hφ(x0 ), ΨH φ(ti )i Z 0 = hφ(x ), gH φ(ti )dgH i G Z H 0−1 = hφ(x0 ), gH gH φ(ti )dgH i GH Z 0−1 = hφ(x0 ), gH gH φ(ti )dgH i GH Z 0 = hgH φ(x0 ), gH φ(ti )dgH i (13) (14) (15) (16) (17) GH = hφ(g 0 x0 ), ΨH φ(ti )i 0 0 0 = l(g x )i ∀g ∈ G 13 (18) (19) Here, in line 15 we utilize the closure property of a group (since gH forms a group according to Theorem 2.4). Line 17 utilizes the fact that gH is unitary, and finally line 18 uses Theorem 2.4. 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arXiv:1703.03963v1 [] 11 Mar 2017 On Solving Travelling Salesman Problem with Vertex Requisitions∗ Anton V. EREMEEV Omsk Branch of Sobolev Institute of Mathematics SB RAS, Omsk State University n.a. F.M. Dostoevsky eremeev@ofim.oscsbras.ru Yulia V. KOVALENKO Sobolev Institute of Mathematics SB RAS, julia.kovalenko.ya@yandex.ru Received: / Accepted: Abstract: We consider the Travelling Salesman Problem with Vertex Requisitions, where for each position of the tour at most two possible vertices are given. It is known that the problem is strongly NP-hard. The proposed algorithm for this problem has less time complexity compared to the previously known one. In particular, almost all feasible instances of the problem are solvable in O(n) time using the new algorithm, where n is the number of vertices. The developed approach also helps in fast enumeration of a neighborhood in the local search and yields an integer programming model with O(n) binary variables for the problem. Keywords: Combinatorial optimization, System of vertex requisitions, Local search, Integer programming. MSC: 90C59, 90C10. 1 INTRODUCTION The Travelling Salesman Problem (TSP) is one of the well-known NP-hard combinatorial optimization problems [1]: given a complete arc-weighted digraph with n vertices, find a shortest travelling salesman tour (Hamiltonian circuit) in it. The TSP with Vertex Requisitions (TSPVR) was formulated by A.I. Serdyukov in [2]: find a shortest travelling salesman tour, passing at i-th position This research of the authors is supported by the Russian Science Foundation Grant (project no. 15-1110009). ∗ 1 a vertex from a given subset X i , i = 1, . . . , n. A special case where |X i | = n, i = 1, . . . , n, is equivalent to the TSP. This problem can be interpreted in terms of scheduling theory. Consider a single machine that may perform a set of operations X = {x1 , . . . , xn }. Each of the identical jobs requires processing all n operations in such a sequence that the i-th operation belongs to a given subset X i ⊆ X for all i = 1, . . . , n. A setup time is needed to switch the machine from one operation of the sequence to another. Moreover, after execution of the last operation of the sequence the machine requires a changeover to the first operation of the sequence to start processing of the next job. The problem is to find a feasible sequence of operations, minimizing the cycle time. TSP with Vertex Requisitions where |X i | ≤ k, i = 1, . . . , n, was called k-TSP with Vertex Requisitions (k-TSPVR) in [2]. The complexity of k-TSPVR was studied in [2] for different values of k on graphs with small vertex degrees. In [3] A.I. Serdyukov proved the NP-hardness of 2-TSPVR in the case of complete graph and showed that almost all feasible instances of the problem are solvable in O(n2) time. In this paper, we propose an algorithm for 2-TSPVR with time complexity O(n) for almost all feasible problem instances. The developed approach also has some applications to local search and integer programming formulation of 2-TSPVR. The paper has the following structure. In Section 2, a formal definition of 2-TSPVR is given. In Section 3, an algorithm for this problem is presented. In Section 4, a modification of the algorithm is proposed with an improved time complexity and it is shown that almost all feasible instances of the problem are solvable in time O(n). In Section 5, the developed approach is used to formulate and enumerate efficiently a neighborhood for local search. In Section 6, this approach allows to formulate an integer programming model for 2-TSPVR using O(n) binary variables. The last section contains the concluding remarks. 2 PROBLEM FORMULATION AND ITS HARDNESS 2-TSP with Vertex Requisitions is formulated as follows. Let G = (X, U) be a complete arc-weighted digraph, where X = {x1 , . . . , xn } is the set of vertices, U = {(x, y) : x, y ∈ X, x 6= y} is the set of arcs with non-negative arc weights ρ(x, y), (x, y) ∈ U . Besides that, a system of vertex subsets (requisitions) X i ⊆ X, i = 1, . . . , n, is given, such that 1 ≤ |X i | ≤ 2 for all i = 1, . . . , n. Let F denote the set of bijections from Xn := {1, . . . , n} to X, such that f (i) ∈ X i , i = 1, . . . , n, for all f ∈ F . The problem consists in finding such a mapping f ∗ ∈ F that ρ(f ∗ ) = n−1 P min ρ(f ), where ρ(f ) = ρ(f (i), f (i + 1)) + ρ(f (n), f (1)) for all f ∈ F . Later on the f ∈F i=1 symbol I is used for the instances of this problem. Any feasible solution uses only the arcs that start in a subset X i and end in X i+1 for some i ∈ {1, . . . , n} (we assume n + 1 := 1). Other arcs are irrelevant to the problem and we assume that they are not given in a problem input I. 2-TSPVR is strongly NP-hard [3]. The proof of this fact in [3] is based on a reduction of Clique problem to a family of instances of 2-TSPVR with integer input data, bounded by a polynomial in problem length. Therefore, in view of sufficient condition for non-existence of Fully Polynomial-Time Approximation Scheme (FPTAS) for strongly NP-hard problems [4], the result from [3] implies that 2-TSPVR does not admit an FPTAS, provided that P6=NP. The k-TSPVR with k ≥ 3 cannot be approximated with any constant or polynomial factor of the optimum in polynomial time, unless P=NP, as follows from [5]. 3 SOLUTION METHOD Following the approach of A.I. Serdyukov [3], let us consider a bipartite graph Ḡ = (Xn , X, Ū) where the two subsets of vertices of bipartition Xn , X have equal sizes and the set of edges is Ū = {{i, x} : i ∈ Xn , x ∈ X i }. Now there is a one-to-one correspondence between the set of perfect matchings W in the graph Ḡ and the set F of feasible solutions to a problem instance I: Given a perfect matching W ∈ W of the form {{1, x1 }, {2, x2 }, . . . , {n, xn }}, this mapping produces the tour (x1 , x2 , . . . , xn ). An edge {i, x} ∈ Ū is called special if {i, x} belongs to all perfect matchings in the graph Ḡ. Let us also call the vertices of the graph Ḡ special, if they are incident with special edges. Supposing that Ḡ is given by the lists of adjacent vertices, the special edges and edges that do not belong to any perfect matching in the graph Ḡ may be efficiently computed by the Algorithm 1 described below. After that all edges, except for the special edges and those adjacent to them, are slit into cycles. Note that the method of finding all special edges and cycles in the graph Ḡ was not discussed in [3]. Algorithm 1. Finding special edges in the graph Ḡ Step 1 (Initialization). Assign Ḡ′ := Ḡ. Step 2. Repeat Steps 2.1-2.2 while it is possible: Step 2.1 (Solvability test). If the graph Ḡ′ contains a vertex of degree 0 then problem I is infeasible, terminate. Step 2.2 (Finding a special edge). If the graph Ḡ′ contains a vertex z of degree 1, then store the corresponding edge {z, y} as a special edge and remove its endpoints y and z from Ḡ′ . Each edge of the graph Ḡ is visited and deleted at most once (which takes O(1) time). The number of edges |Ū | ≤ 2n. So the time complexity of Algorithm 1 is O(n). Algorithm 1 identifies the case when problem I is infeasible. Further we consider only feasible instances of 2-TSPVR and bipartite graphs corresponding to them. After the described preprocessing the resulting graph Ḡ′ is 2-regular (the degree of each vertex equals 2) and its components are even cycles. The cycles of the graph Ḡ′ can be computed in O(n) time using the Depth-First Search algorithm (see e.g. [6]). Note that there are no other edges in the perfect matchings of the graph Ḡ, except for the special edges and edges of the cycles in Ḡ′ . In what follows, q(Ḡ) denotes the number of cycles in the graph Ḡ (and in the corresponding graph Ḡ′ ). Each cycle j, j = 1, . . . , q(Ḡ), contains exactly two maximal (edge disjoint) perfect matchings, so it does not contain any special edges. Every perfect matching in Ḡ is uniquely defined by a combination of maximal matchings chosen in each of the cycles and the graph G = ( X8, X, U ) 1 x1 2 x2 3 x3 4 x4 5 x5 6 x6 7 x7 8 x8 cycle 1 special edges cycle 2 Figure 1: An instance I with n = 8 and system of vertex requisitions: X 1 = {x1 , x2 }, X 2 = {x1 , x2 }, X 3 = {x3 }, X 4 = {x3 , x4 }, X 5 = {x5 , x6 }, X 6 = {x6 , x7 }, X 7 = {x7 , x8 }, X 8 = {x6 , x8 }. Here the edges drawn in bold define one maximal matching of a cycle, and the rest of the edges in the cycle define another one. The special edges are depicted by dotted lines. The edges depicted by dashed lines do not belong to any perfect matching. The feasible solutions of the instance are f 1 = (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ), f 2 = (x1 , x2 , x3 , x4 , x5 , x7 , x8 , x6 ), f 3 = (x2 , x1 , x3 , x4 , x5 , x6 , x7 , x8 ), f 4 = (x2 , x1 , x3 , x4 , x5 , x7 , x8 , x6 ). set of all special edges (see Fig. 1). Therefore, 2-TSPVR is solvable by the following algorithm. Algorithm 2. Solving 2-TSPVR Step 1. Build the bipartite graph Ḡ, identify the set of special edges and cycles and find all maximal matchings in cycles. Step 2. Enumerate all perfect matchings W ∈ W of Ḡ by combining the maximal matchings of cycles and joining them with special edges. Step 3. Assign the corresponding solution f ∈ F to each W ∈ W and compute ρ(f ). Step 4. Output the result f ∗ ∈ F , such that ρ(f ∗ ) = min ρ(f ). f ∈F To evaluate the Algorithm 2, first note that maximal matchings in cycles are found easily in O(n) time. Now |F | = |W| = 2q(Ḡ) so the time complexity of Algorithm 2 of solving 2-TSPVR is O(n2q(Ḡ) ), where q(Ḡ) ≤ ⌊ n2 ⌋ and the last inequality is tight. 4 IMPROVED ALGORITHM In [3], it was shown that almost all feasible instances of 2-TSPVR have not more than n feasible solutions and may be solved in quadratic time. To describe this result precisely, let us give the following Definition 1 [3] A graph Ḡ = (Xn , X, Ū) is called “good” if it satisfies the inequality q(Ḡ) ≤ 1.1 ln(n). Note that any problem instance I, which corresponds to a “good” graph Ḡ, has at most 2 < n0.77 feasible solutions. Let χ̄n denote the set of “good” bipartite graphs Ḡ = (Xn , X, Ū ) , and let χn be the set of all bipartite graphs Ḡ = (Xn , X, Ū ). The results of A.I. Serdyukov from [3] imply 1.1 ln n Theorem 1 |χ̄n |/|χn | −→ 1 as n → ∞. The proof of Theorem 1 from [3] is provided in the appendix for the sake of completeness. According to the frequently used terminology (see e.g. [7]), this theorem means that almost all feasible instances I have at most n0.77 feasible solutions and thus they are solvable in O(n1.77 ) time by Algorithm 2. Using the approach from [8] we will now modify Algorithm 2 for solving 2-TSPVR in
q(Ḡ) O q(Ḡ)2 + n time. Let us carry out some preliminary computations before enumerating all possible combinations of maximal matchings in cycles in order to speed up the evaluation of objective function. We will call a contact between cycle j and cycle j ′ 6= j (or between cycle j and a special edge) the pair of vertices (i, i + 1) (we assume n + 1 := 1) in the left-hand part of the graph Ḡ, such that one of the vertices belongs to the cycle j and the other one belongs to the cycle j ′ (or the special edge). A contact inside a cycle will mean a pair of vertices in the left-hand part of a cycle, if their indices differ exactly by one, or these vertices are (n, 1). Consider a cycle j. If a contact (i, i + 1) is present inside this cycle, then each of the two maximal matchings w 0,j and w 1,j in this cycle determines the i-th arc of a tour in the graph G. Also, if the cycle j has a contact (i, i + 1) to a special edge, each of the two maximal matchings w 0,j and w 1,j also determines the i-th arc of a tour in the graph G. For each of the matchings w k,j , k = 0, 1, let the sum of the weights of arcs determined by the contacts inside the cycle j and the contacts to special edges be denoted by Pjk . If cycle j contacts to cycle j ′ , j ′ 6= j, then each combination of the maximal matchings of these cycles determines the i-th arc of a tour in the graph G for any contact (i, i + 1) between the cycles. If a maximal matching is chosen in each of the cycles, one can sum up the weights of the arcs in G determined by all contacts between cycles j and j ′ . This yields (0,0) (0,1) (1,0) (1,1) four values which we denote by Pjj ′ , Pjj ′ , Pjj ′ and Pjj ′ , where the superscripts identify the matchings chosen in each of the cycles j and j ′ respectively. (0,0) (0,1) (1,0) (1,1) Parameters Pj0 , Pj1 , Pjj ′ , Pjj ′ , Pjj ′ and Pjj ′ can be found as follows. Suppose that intermediate values of Pj0 , Pj1 for j = 1, . . . , q(Ḡ) are stored in one-dimensional arrays of (0,0) (0,1) (1,0) (1,1) size q(Ḡ), and intermediate values of Pjj ′ , Pjj ′ , Pjj ′ and Pjj ′ for j, j ′ = 1, . . . , q(Ḡ) are stored in two-dimensional arrays of size q(Ḡ) × q(Ḡ). Initially, all of these values are assumed to be zero and they are computed in an iterative way by the consecutive enumeration of pairs of vertices (i, i + 1), i = 1, . . . , n − 1, and (n, 1) in the left-hand part of the graph Ḡ. When we consider a pair of vertices (i, i + 1) or (n, 1), at most four parameters (partial sums) are updated depending on whether the vertices belong to different cycles or to the same cycle, or one of the vertices is special. So the overall time complexity of the pre-processing procedure is O(q 2 (Ḡ) + n). Now all possible combinations of the maximal matchings in cycles may be enumerated using a Grey code (see e.g. [9]) so that the next combination differs from the previous one by altering a maximal matching only in one of the cycles. Let the binary vector δ = (δ1 , . . . , δq(Ḡ) ) define assignments of the maximal matchings in cycles. Namely, δj = 0, if the matching w 0,j is chosen in the cycle j; otherwise (if the matching w 1,j is chosen in the cycle j), we have δj = 1. This way every vector δ is bijectively mapped into a feasible solution fδ to 2-TSPVR. In the process of enumeration, a step from the current vector δ̄ to the next vector δ changes the maximal matching in one of the cycles j. The new value of objective function ρ(fδ ) may be computed via the current value ρ(fδ̄ ) by the formula ρ(fδ ) = P P (δj ,δ ′ ) (δ̄j ,δ̄ ′ ) δ̄ δ ρ(fδ̄ ) − Pj j + Pj j − Pjj ′ j , where A(j) is the set of cycles contacting to Pjj ′ j + j ′ ∈A(j) j ′ ∈A(j) the cycle j. Obviously, |A(j)| ≤ q(Ḡ), so updating the objective function value for the next solution requires O(q(Ḡ)) time, and the
overall time complexity of the modified algorithm for q(Ḡ) solving 2-TSPVR is O q(Ḡ)2 +n . In view of Theorem 1 we conclude that using this modification of Algorithm 2 almost all feasible instances of 2-TSPVR are solvable in O(n0.77 ln n + n) = O(n) time. 5 LOCAL SEARCH A local search algorithm starts from an initial feasible solution. It moves iteratively from one solution to a better neighboring solution and terminates at a local optimum. The number of steps of the algorithm, the time complexity of one step, and the value of the local optimum depend essentially on the neighborhood. Note that neighborhoods, often used for the classical TSP (e.g. k-Opt, city-swap, Lin-Kernighan [10]), will contain many infeasible neighboring solutions if applied to 2-TSPVR because of the vertex requisition constraints. A local search method with a specific neighborhood for 2-TSPVR may be constructed using the relationship between the perfect matchings in the graph Ḡ and the feasible solutions. The main idea of the algorithm consists in building a neighborhood of a feasible solution to 2-TSPVR on the basis of a Flip neighborhood of the perfect matching, represented by the maximal matchings in cycles and the special edges. Let the binary vector δ = (δ1 , . . . , δq(Ḡ) ) denote the assignment of the maximal matchings to cycles as above. The set of 2q(Ḡ) vectors δ corresponds to the set of feasible solutions by a one-to-one mapping fδ . We assume that a solution fδ′ belongs to the Exchange neighborhood of solution fδ iff the vector δ ′ is within Hamming distance 1 from δ, i.e. δ ′ belongs to the Flip neighborhood of vector δ. Enumeration of the Exchange neighborhood takes O(q 2 (Ḡ)) time if the preprocessing described in Section 4 is carried out before the start of the local search (without the preprocessing it takes O(nq(Ḡ)) operations). Therefore, for almost all feasible instances I, the Exchange neighborhood may be enumerated in O(ln2 (n)) time. 6 MIXED INTEGER LINEAR PROGRAMMING MODEL The one-to-one mapping between the maximal matchings in cycles of the graph Ḡ and feasible solutions to 2-TSPVR may be also exploited in formulation of a mixed integer linear programming model. Recall that Pj0 (Pj1 ) is the sum of weights of all arcs of the graph G determined by the contacts inside the cycle j and the contacts of the cycle j with special edges, when the maximal (k,l) matching w 0,j (w 1,j ) is chosen in the cycle j, j = 1, . . . , q(Ḡ). Furthermore, Pjj ′ is the sum of weights of arcs in the graph G determined by the contacts between cycles j and j ′ , if the ′ maximal matchings w k,j and w l,j are chosen in the cycles j and j ′ respectively, k, l = 0, 1, j = 1, . . . , q(Ḡ) − 1, j ′ = j + 1, . . . , q(Ḡ). These values are computable in O(n + q 2 (Ḡ)) time as shown in Section 4. Let us introduce the following Boolean variables:  0, if matching w 0,j is chosen in the cycle j, dj = 1, if matching w 1,j is chosen in the cycle j, j = 1, . . . , q(Ḡ). The objective function combines the pre-computed arc weights for all cycles, depending on the choice of matchings in d = (d1 , . . . , dq(Ḡ) ): q(Ḡ)−1 q(Ḡ)  X  (0,0) (0,1) Pjj ′ (1 − dj )(1 − dj ′ ) + Pjj ′ (1 − dj )dj ′ X j=1 j ′ =j+1 q(Ḡ)−1 q(Ḡ) + X  X (1,0) Pjj ′ dj (1 − dj ′ ) + (1,1) Pjj ′ dj dj ′ j ′ =j+1 j=1  (1) q(Ḡ) + X j=1
Pj0 (1 − dj ) + Pj1 dj → min, dj ∈ {0, 1}, j = 1, . . . , q(Ḡ). (2) Let us define supplementary real variables in order to remove non-linearity of the objective (k) function: for k ∈ {0, 1} we assume that pj ≥ 0 is an upper bound on the sum of weights of arcs in the graph Ḡ determined by the contacts of the cycle j, if matching w k,j is chosen in this cycle, i.e. dj = k, j = 1, . . . , q(Ḡ) − 1. Then the mixed integer linear programming model has the following form: q(Ḡ)−1  X q(Ḡ) (0) pj + (1) pj j=1  + X j=1
Pj0 (1 − dj ) + Pj1 dj → min, q(Ḡ) q(Ḡ) (0) pj ≥ X j ′ =j+1 (0,0) Pjj ′ (1 − dj − dj ′ ) + X j ′ =j+1 (0,1) Pjj ′ (dj ′ − dj ) , (3) j = 1, . . . , q(Ḡ) − 1, q(Ḡ) q(Ḡ) (1) pj ≥ X (4) (1,0) Pjj ′ j ′ =j+1 (dj − d ) + j′ X (1,1) Pjj ′ (dj + dj ′ − 1) , j ′ =j+1 j = 1, . . . , q(Ḡ) − 1, (5) pj ≥ 0, k = 0, 1, j = 1, . . . , q(Ḡ) − 1, (6) dj ∈ {0, 1}, j = 1, . . . , q(Ḡ). (7) (k) Note that if matching w 0,j is chosen for the cycle j in an optimal solution of problem (3)(0) (1) (7), then inequality (4) holds for pj as equality and pj = 0. Analogously, if matching w 1,j (1) (0) is chosen for the cycle j, then inequality (5) holds for pj as equality and pj = 0. Therefore, problems (1)-(2) and (3)-(7) are equivalent because a feasible solution of one problem corresponds to a feasible solution of another problem, and an optimal solution corresponds to an optimal solution. The number of real variables in model (3)-(7) is (2q(Ḡ) − 2), the number of Boolean variables is q(Ḡ). The number of constraints is O(q(Ḡ)), where q(Ḡ) ≤ ⌊ n2 ⌋. The proposed model may be used for computing lower bound of objective function or in branch-and-bound algorithms, even if the graph Ḡ is not “good”. Note that there are a number of integer linear programming models in the literature on the classical TSP, involving O(n2 ) Boolean variables. Model (3)–(7) for 2-TSPVR has at most ⌊ n2 ⌋ Boolean variables and for almost all feasible instances the number of Boolean variables is O(ln(n)). 7 CONCLUSION We presented an algorithm for solving 2-TSP with Vertex Requisitions, that reduces the time complexity bound formulated in [3]. It is easy to see that the same approach is applicable to the problem 2-Hamiltonian Path of Minimum Weight with Vertex Requisitions, that asks for a Hamiltonian Path of Minimum Weight in the graph G, assuming the same system of vertex requisitions as in 2-TSP with Vertex Requisitions. Using the connection to perfect matchings in a supplementary bipartite graph and some preprocessing we constructed a MIP model with O(n) binary variables and a new efficiently searchable Exchange neighborhood for problem under consideration. Further research might address the existence of approximation algorithms with constant approximation ratio for 2-TSP with Vertex Requisitions. References [1] Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the theory of NPcompleteness. W.H. Freeman and Company, San Francisco, CA (1979) [2] Serdyukov, A.I.: Complexity of solving the travelling salesman problem with requisitions on graphs with small degree of vertices. Upravlaemye systemi. 26, 73–82 (1985) (In Russian) [3] Serdyukov, A.I.: On travelling salesman problem with prohibitions. Upravlaemye systemi. 17, 80–86 (1978) (In Russian) [4] Garey, M.R., Johnson, D.S.: Strong NP-completeness results: Motivation, examples, and implications. Journal of the ACM. 25, 499–508 (1978) [5] Serdyukov A.I.: On finding Hamilton cycle (circuit) problem with prohibitions. Upravlaemye systemi. 19, 57–64 (1979) (In Russian) [6] Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edition. MIT Press (2001) [7] Chvatal, V.: Probabilistic methods in graph theory. Annals of Operations Research. 1, 171–182 (1984) [8] Eremeev, A., Kovalenko, J.: Optimal recombination in genetic algorithms for combinatorial optimization problems: Part II. Yugoslav Journal of Operations Research. 24 (2), 165-186 (2014) [9] Reingold, E.M., Nievergelt, J., Deo, N.: Combinatorial algorithms: Theory and Practice. Englewood Cliffs, Prentice-Hall (1977) [10] Kochetov, Yu. A.: Computational bounds for local search in combinatorial optimization. Computational Mathematics and Mathematical Physics. 48 (5), 747-763 (2008) [11] Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. 1. John Wiley & Sons, New York, NY (1968) [12] Riordan, J.: An Introduction to Combinatorial Analysis. John Wiley & Sons, New York, NY (1958) APPENDIX Note that A.I. Serdyukov in [3] used the term block instead of term cycle, employed in Section 3 of the present paper. A block was defined in [3] as a maximal (by inclusion) 2-connected subgraph of graph Ḡ with at least two edges. However, in each block of the graph Ḡ, the degree of every vertex equals 2 (otherwise F = Ø because the vertices of degree 1 do not belong to blocks and the vertex degrees are at most 2 in the right-hand part of Ḡ). So, the notions block and cycle are equivalent in the case of considered bipartite graph Ḡ. We use the term block in the proof of Theorem 1 below, as in the original paper [3], in order to avoid a confusion of cycles in Ḡ with cycles in permutations of the set {1, . . . , n}. Theorem 1 [3] |χ̄n |/|χn | −→ 1 as n → ∞. Proof. Let Sn be the set of all permutations of the set {1, . . . , n}. Consider a random permutation s from Sn . By ξ(s) denote the number of cycles in permutation s. It is known n P 1 and the (see e.g. [11]) that the expectation E[ξ(s)] of random variable ξ(s) is equal to i i=1 variance V ar[ξ(s)] equals n P i=1 i−1 . i2 Let S̄n denote the set of permutations from Sn , where the number of cycles is at most 1.1 ln(l). Then, using Chebychev’s inequality [11], we get |S̄n |/|Sn | −→ 1 as n → ∞. (8) Now let Sn′ denote the set of permutations from Sn , which do not contain the cycles of (i) length 1, and let Sn be the set of permutations from Sn , which contain a cycle with element i, i = 1, . . . , n. Using the principle of inclusion and exclusion [12], we obtain |Sn \ Sn′ | = [ Sn(i) 1≤i≤n X Sn(i) 1≤i6=j6=k≤n \ Sn(j) \ = n X Sn(i) − i=1 X 1≤i6=j≤n Sn(i) \ Sn(j) + Sn(k) − . . . = n! − Cn2 (n − 2)! + Cn3 (n − 3)! − . . . ≤ n! n! 2 2 + = n! = |Sn |. 2 6 3 3 Therefore, 1 |Sn′ | ≥ |Sn |. 3 (9) Combining (8) and (9), we get |S̄n′ |/|Sn′ | 3|Sn′ \S̄n′ | 3|Sn \S¯n | |Sn′ \S̄n′ | ≥ 1 − ≥ 1 − −→ 1, =1− |Sn′ | |Sn | |Sn | n→+∞ (10) where S̄n′ = Sn′ ∩ S¯n . The values |χ̄n | and |χn \χ̄n | may be bounded, using the following approach. We assign any permutation s ∈ Sl′ , l ≤ n, a set of bipartite graphs χn (s) ⊂ χn as follows. First of all let us assign an arbitrary set of n − l edges to be special. Then the non-special vertices {i1 , i2 , . . . , il } ⊂ Xn of the left-hand part, where ij < ij+1 , j = 1, . . . , l − 1, are now partitioned into ξ(s) blocks, where ξ(s) is the number of cycles in permutation s. Every cycle (t1 , t2 , . . . , tr ) in permutation s corresponds to some sequence of vertices with indices {it1 , it2 , . . . , itr } belonging to the block associated with this cycle. Finally, it is ensured that for each pair of vertices {itj , itj+1 }, j = 1, . . . , r − 1, as well as for the pair {itr , it1 }, there exists a vertex in the right-hand part X which is adjacent to both vertices of the pair. Except for special edges and blocks additional edges are allowed in graphs from class χn (s). These edges are adjacent to the special vertices of the left-hand part such that the degree of any vertex of the left-hand part is not greater than two. Moreover, additional edges should not lead to creating new blocks. There are n! ways to associate vertices of the left-hand part to vertices of the righthand part, therefore the number of different graphs from class χn (s), s ∈ Sl′ , l ≤ n, is |χn (s)| = Cnl 2ξn! h(n, l), where function h(n, l) depends only on n and l, and ξ1 (s) is the 1 (s) number of cycles of length two in permutation s. Division by 2ξ1 (s) is here due to the fact that for each block that corresponds to a cycle of length two in s, there are two equivalent ways to number the vertices in its right-hand part. Let s = c1 c2 . . . cξ(s) be a permutation from set Sl′ , represented by cycles ci , i = 1, . . . , ξ(s), and let cj be an arbitrary cycle of permutation s of length at least three, j = 1, . . . , ξ(s). Permutation s may be transformed into permutation s1 , s1 = c1 c2 . . . cj−1 c−1 j cj+1 . . . cξ(s), (11) by reversing the cycle cj . Clearly, permutation s1 induces the same subset of bipartite graphs in class χn as the permutation s does. Thus any two permutations s1 and s2 from set Sl′ , l ≤ n, induce the same subset of graphs in χn , if one of these permutations may be obtained from the other one by several transformations of the form (11). Otherwise the two induced subsets of graphs do not intersect. Besides that χn (s1 ) ∩ χn (s2 ) = ∅ if s1 ∈ Sl′1 , s2 ∈ Sl′2 , l1 6= l2 . On one hand, if s ∈ S̄l′ , l ≤ n, then χn (s) ⊆ χ̄n . On the other hand, if s ∈ S̃l′ := Sl′ \S̄l′ , l < n, then either χn (s) ⊆ χ̄n or, alternatively, χn (s) ⊆ χn \χ̄n may hold. Therefore, |χ̄n | ≥ n X X l=2 s∈S̄l′ ≥ n X |S̄l′ | · l=2 |χn \χ̄n | ≤ n! Cnl ξ1 (s) ξ(s)−ξ1 (s) h(n, l) 2 2 n! Cnl 1.1ln(l) h(n, l) 2 n X l=⌊1.1ln(n)⌋ Now assuming ψ(n) = X s∈S̃l′ max l=⌊1.1ln(n)⌋,...,n n X ≥ n! Cnl ξ(s) 2 = n X X Cnl l=2 s∈S̄l′ |S̄l′ | · Cnl l=⌊1.1ln(n)⌋ ≤ n X n! h(n, l) ≥ 2ξ(s) n! 21.1ln(l) |S̃l′ | · Cnl l=⌊1.1ln(n)⌋ (12) h(n, l), n! 21.1ln(l) h(n, l). (13) |S̃l′ |/|S̄l′ | and taking into account (12), (13) and (10), we obtain |χn \χ̄n | ≤ ψ(n) → 0 as n → +∞. |χ̄n | Finally, the statement of the theorem follows from (14). Q.E.D. (14) 

PyCraters: A Python framework for crater function analysis Scott A. Norris arXiv:1410.8489v1 [physics.comp-ph] 28 Oct 2014 October 31, 2014 Abstract We introduce a Python framework designed to automate the most common tasks associated with the extraction and upscaling of the statistics of single-impact crater functions to inform coefficients of continuum equations describing surface morphology evolution. Designed with ease-of-use in mind, the framework allows users to extract meaningful statistical estimates with very short Python programs. Wrappers to interface with specific simulation packages, routines for statistical extraction of output, and fitting and differentiation libraries are all hidden behind simple, high-level user-facing functions. In addition, the framework is extensible, allowing advanced users to specify the collection of specialized statistics or the creation of customized plots. The framework is hosted on the BitBucket service under an open-source license, with the aim of helping non-specialists easily extract preliminary estimates of relevant crater function results associated with a particular experimental system. 1 Introduction Irradiation by energetic ions is a ubiquitous materials processing technique, used widely in laboratories and industry for doping, cleaning, and modification of materials surfaces. Under certain environmental conditions, ion irradiation is observed to induce the spontaneous formation of nanometer-scale structures such as ripples, dots, and holes [1]. In some contexts, such as the gradual ion-induced degradation of fusion reactor components, these structures are an artifact to be avoided [2], but more recent observations of wellordered, high-aspect ratio structures [3] has led to the consideration of ion irradiation as an inexpensive means of inducing such structures deliberately. With sufficient understanding, this process could serve as the basis of an inexpensive, high-throughput means of creating surfaces with desired mechanical, optical, or electronic properties, ready for immediate application across the existing installed base of ion beam facilities. A major barrier to predictive understanding of the ion-induced nanostructuring process has been a large number of competing physical mechanisms, and an accompanying difficulty in estimating their relative magnitudes in different parameter regimes. To the original modeling of energy deposition and its relationship to sputter yield by Sigmund [4, 5], and subsequent discovery of an sputter erosion-driven instability mechanism identified by Bradley and Harper [6], numerous additional mechanisms have since been added. It has since been discovered that atoms displaced during the collision cascade, but not sputtered from the surface, produce contributions to the evolution equations that directly compete with the erosive mechanism. First studied in 1D by Carter and Vishnyakov, and later in 2D dimensions by Davidovitch et al. [7, 8], this ”momentum transfer” or “mass redistribution” contribution effectively doubles the number of unknown parameters in the problem, as every erosive term has a redistributive counterpart, and the magnitudes of each must in principle be estimated. Intensifying this problem was the realization that strongly-ordered structures generating the most excitement seemed to be due to the presence of multiple components in the target [9, 10], whether through contamination, intentional co-deposition, or the use of a two-component target such as III-V compounds. To accurately model such equations requires the introduction of a second field to track concentrations [11] which, although it can overcome the barriers to ordered structures exhibited by one-component models [12, 13], again doubles the number of unknown parameters present in the problem. Recently, the "Crater Function" framework has emerged as a means of rigorously connecting surface morphology evolution over long spatial and temporal scales to the statistical properties of single ion impacts [14, 15, 16]. Given the “Crater Function” ∆h (x, y; S) describing the average surface modification due to an ion impact (with a parametric dependence on the surface properties via the argument S), the multi-scale 1 analysis within the framework [14] produces contributions to partial differential equations governing the surface morphology evolution. It can therefore be viewed as a way of estimating many of the unknown parameters present in these problems by means of atomistic simulation. Originally applied only to pure materials and using only data from flat surfaces [15], the framework has since been expanded to the case of binary materials [16], and to enable incorporation of simulation data from curved targets [17]. It has thus matured to the point where it may be of value to the general community as a parameter estimation tool. However, to use the framework to this end, one needs the capability to (a) perform numerical simulations of ion impacts over a variety of surface parameters, (b) extract the necessary statistics from the output of each simulations, (c) fit parameter-dependent statistics to various appropriate functional forms, and (d) combine and report the results. This manuscript describes a Python library to provide all of the above capabilities through a simple and user-friendly API. Access to various simulation tools is provided via wrappers that automatically create input files, run the solver, read the output, and save in a common format. A customizable set of statistical analyses are then run on the common-format output file, and saved for later use under a unique parameter-dependent filename. A flexible loading mechanism and general purpose fitting library make it easy to load statistics as a function of arbitrary parameters, and then fit the resulting data points to appropriate smooth nonlinear functions. Finally, functions utilizing these capabilities are provided to perform all current mathematical operations indicated by the literature to extract and plot PDE components. Using this library, example codes that re-obtain existing results within the literature are as short as 20 lines each, and can easily be modified by end users to begin studying systems of their choosing. 2 Theoretical Background Crater Functions. If the dominant effects of the impact-induced collision cascade can be assumed to take place near to the surface of the evolving film, then the normal surface velocity of an ion-bombarded surface can be represented by an integro-differential equation of the form [18], ˆ vn = I (φ (x0 )) ∆h (x − x0 ; S (x0 )) dx0 , (1) where I = I0 cos (φ) is the projected ion flux depending on the local angle of incidence φ (x), ∆h is the “crater function” describing the average surface response due to single ion impacts, and S describes an arbitrary parametric dependence of the crater function on the local surface shape. This form has advantages over more traditional treatments of irradiation-induced morphology evolution. Instead of separate, simplified models of the processes of ion-induced sputtering [5, 6] and impact-induced momentum transfer [7, 8] – both of which break down as the angle of incidence approaches grazing – the crater function ∆h naturally includes components due to both sputtered atoms and redistributed atoms (thus unifying the two approaches), and can in principle be obtained empirically (thus avoiding inaccuracy at high angles of incidence). A Generic Framework. Exploiting the typical experimental observation of a separation of spatial scales between the size of the impact (direct spatial dependence of ∆h) and the typical size of emergent structures (spatial dependence of φ and S), a multiple-scale analysis was conducted in which the integral in Eq. (1) is expanded into an infinite series of terms involving the moments of ∆h [14]: h i h i h i 1 vn = I M̃ (0) + ε∇S · I M̃ (1) + ε2 ∇S · ∇S · I M̃ (2) + . . . 2 (2) where I is the projected ion flux (a scalar), the ∇S are co-ordinate free surface divergences, and the M̃ (i) are “effective” moments of the crater function ∆h in increasing tensor order [14]. This result provides intuition as to which parts of the crater function ∆h are most important to understand morphology evolution, and represents a general solution for the multiple-scale expansion of Eq. (1) in the sense that it should apply for any parametric dependencies of the crater function on the surface shape S. (Note that in this co-ordinate free form, the effective moments are really combinations of the actual moments as described in Ref. [14]; however, in any linearization, they are equivalent). 2 Example Applications. Equation (2) is fully nonlinear and independent of the specific form of the crater function. Therefore, to study surface stability, one must first choose a form for the crater function ∆h, and then linearize the resulting specific instance of Equation (2) about a flat surface. This process was first demonstrated in Ref. [15], where for simplicity and consistency with available simulation data, the crater function ∆h = g (x − x0 ; φ) , (3) was chosen, which depends parametrically only on the local angle of incidence φ. Inserting this expression into the general result Eq. (2), linearizing, and then adopting a moving frame of reference to eliminate translational and advective terms, one obtains to leading order the PDE ∂2h ∂2h ∂h = SX (θ) 2 + SY (θ) 2 ∂t ∂x ∂y (4) where the angle-dependent coefficients SX (θ) and SY (θ) are related to the crater functions via the expressions i ∂ h I0 cos (θ) Mx(1) (θ) h∂θ i SY (θ) = I0 cos (θ) cot (θ) Mx(1) (θ) SX (θ) = (5) (1) where Mx is the component of the (vector) first moment in the projected direction of the ion beam. More recently, a similar approach has been applied to an extended crater function of the form [17] ∆h = g (x − x0 ; φ, K11 , K12 , K22 ) (6) depending additionally on the surface curvatures Kij near the point of impact. It was found that including this dependency within the crater function reveals additional terms in the coefficient values, which take the revised form i i ∂ h ∂ h SX (θ) = I0 cos (θ) Mx(1) (θ) + I0 cos (θ) M (0) ∂θ ∂K11 (7) h i i. ∂ h (1) SY (θ) = I0 cos (θ) cot (θ) Mx (θ) + I0 cos (θ) M (0) ∂K22 Implications. The practical consequence of such results is that one can directly connect atomistic simulations over small length- and time-scales to continuum equations governing morphology evolution over much longer scales. If the crater function ∆h and its moments M (i) can be identified as functions of the local surface configuration (angle, curvature, etc.) by simulation (e.g. [19]) or even experiment (e.g. [20]), then the expected continuum evolution of the system can be predicted via Eq. (4) with coefficients supplied by results such as Eqs. (5) or (7). Early applications of these ideas show significant promise for predicting the outcome of experiments [15] or determining the likely physical mechanisms driving experimental observations [16]. However, the steps required to do so represent a non-trivial task in simulation and data analysis: an effective procedure must address questions of (1) creation or selection of a simulation tool to perform many single-impact simulations (2) obtaining statistically converged moments at individual parameter combinations, (3) estimation of derivative values using data from adjacent parameter combinations (4) a smoothing mechanism to prevent uncertainties at step (2) from being amplified in step (3) An approach incorporating such steps was first demonstrated in Ref. [15], where molecular dynamics simulations using the PARCAS code [21] were performed for irradiation of Si by 250 eV Ar+ at 5-degree increments between 0◦ and 90◦ . Smoothing was accomplished by a weighted fitting of the simulation results to a trun(1) cated Fourier series, and fitted values of Mx (θ) were then inserted into Eqs. (5). Analyzing the resulting linear PDE of the form (4) (with additional terms describing ion-enhanced viscous flow), the most-unstable 3 wavelengths at each angle were compared to the wavelengths of experimentally-observe structures, with reasonable agreement. In the process, the relative sizes of the effects of erosion and redistribution were directly obtained and compared, and the effects of redistribution were unexpectedly found to be dominant for the chosen system. 3 Goals and Outline of the Framework The previous section outlines the general features of the Crater Function approach, including the potential promise for the general problem of coefficient estimation, and also the technical hurdles associated with its use. However, while the process of simulation, statistical analysis, fitting, and differentiation is timeconsuming, it is also in principle mechanical, suggesting the utility of an open-source library to centralize best practices and avoid repeated re-implementations. Here we describe the primary goals of such a library, and a summary of the structure of the resulting codebase. Motivation #1: Accessibility. A principal motivation for the present work is to automate the process described above in a generic and accessible way. As many of the procedures as possible should be performed automatically, with reasonable default strategies applied for users that do not wish to delve into the details of atomistic simulation, internal statistical data structures, or the optimal fitting functions for moment curves. Instead, a first-time user should be able to spend most of their effort on deciding what system and environmental parameters to study, after which the library should take care of subsequent mechanical operations. In particular, it should provide simple visualization routines for tasks expected to be common, such as plotting the calculated angle-dependent coefficient values. Motivation #2: Extensibility. A second motivation is to accommodate the desires of users who wish to move beyond the basic capabilities just described. For example, whereas first-time users may wish to call a high-level function to obtain a graph similar to one already published in the literature, a more advanced user might like to work directly with the raw statistical data, using the collected moments to extract particular quantities of interest. Finally, a very advanced user or researcher within the field may wish to customize the set of statistics to be gathered, or even modify the methods used to calculate and fit those customized statistical quantities. As much as possible, the goal should be to accommodate users at each of these levels of detail, and allowing each of them to use built-in utilities to simplify the calculation, loading, fitting, and plotting of whatever quantities are of interest to a given user. Motivation #3: Portability. A final motivation is to enable the collection of statistical data from as many sources as possible, with as little effort as possible. When first demonstrated in Ref. [15], simulations were performed by Molecular Dynamics (MD), using the PARCAS simulation code [21]. However, in principle any MD code could be used, such as the open-source LAAMPS code [22]. Furthermore, if the time required for MD simulation is a significant obstacle, then the simpler, faster Binary Collision Approximation (BCA) becomes an appealing approach. A variety of BCA codes exist, including many descended from the widelyused SRIM code [23, 24, 25] such as TRIDYN [26, 27], SDTRIMSP [28] and TRI3DST [29]. Importantly, these various codes have different and complementary capabilities, and so practitioners may need to employ different codes to answer different types of questions. To facilitate this, the library should in principle be compatible with as many as possible. Framework Summary. In response to these motivations, we have therefore implemented our library as a layered framework. All of the analysis routines are implemented using an internal, standardized set of data structures for holding the results of simulation output. These structures, themselves, are then hidden behind a set of commonly-used high-level routines for generating coefficient values, which may be easily used by practitioners without reproducing any of the underlying work. Furthermore, the framework is designed to be agnostic with respect to the simulation tool used to obtain the impact data. Any solver capable of producing such data can be “wrapped” within an input-output object whose job is simply to write the input files required by that solver, run the simulation, and read the resulting output files. Because input and output file formats vary widely between codes, the specifics of each wrapper are abstracted from the user. 4 High-level visualization routines for simple display of common results plot_pure_material_coefficient_summary() plot_binary_material_coefficient_summary() plot_phase_diagram() Mid-level analysis routines performing calculations indicated by theory calculate_PDE_coefficients_pure() calculate_PDE_coefficients_binary() calculate_transition_angles() Low-level utility routines for solver-independent data manipulation extract_statistics_from_output() load_data_slice_as_list() nonlinear_data_fit() # basic statistical analysis of simulation output # load arbitrary parametric data slice into 1D list # generic method to fit data to supplied function PARCAS LAAMPS MARLOWE TRI3DST SDTrimSP history of use with ion impacts open source crystalline targets curved target surfaces concentration depth profile Molecular Dynamics wrappers Binary Collision Approximation wrappers Figure 1: A schematic diagram of the organization of the PyCraters library. (a) At the lowest level are solverspecific wrappers that abstract the performance of individual simulations behind a standardized interface. Requests for simulations are performed, and results obtained, in a standard format. (b) Above this layer is a set of generic utilities for the collection of statistics from simulation data, the storage and loading of data in easy-to-use formats, and the fitting of data points to smooth curves. (c) Next is a layer for performing the real work associated with the crater function formalism – the loading of appropriate slices of data, fitting that data to appropriate curves, and differentiating the resulting fits to obtain coefficients of linearized equations. (d) Finally, a related set of visualization utilities plots commonly-sought coarse-grained quantities. On top of this framework, small end-user codes to perform basic surveys can easily be written by non-specialists, while specialists can directly invoke the lower-level routines to suit their particular needs. Instead, simulation parameters common to all solvers are available in a simplified, standardized, high-level user interface common to all wrappers, whereas features specific only to certain solvers can be specified on a solver-by-solver basis. The overall approach is illustrated in Figure 1, and consists of components at various levels of abstraction. From lowest-level to highest-level, these may be summarized as consisting of 1. Wrappers around individual Molecular Dynamics or Binary Collision Approximation solvers. 2. Generic, but extensible analysis routines for the extraction of moments and other statistics. 3. An customizable library for the fitting of these statistics to appropriate smooth functional forms. 4. Routines for performing calculations needed to convert fitted functions into PDE coefficients. 5. Plotting utilities to display reasonable summaries of various kinds of data. It should be stressed, however, that from the user’s perspective, these capabilities are visible in the opposite order. For example, a user desiring to plot the PDE coefficients for a given material would simply call an associated plotting function. This function checks to see if fits are available from a prior use of the script, and if not, requests such fits from a lower-level function. That function checks to see whether the statistical data are available, and if not, requests them from a still lower-level function. Finally, that function invokes a BCA or MD simulation tool wrapper, which performs all communication with the solver needed to obtain the statistics. 5 Algorithm 1 Common code needed to load the PyCraters libraries. 1 2 3 4 5 6 7 8 9 10 11 12 # import necessary libraries import sys import numpy as np import matplotlib . pyplot as plt import pycraters . wrappers . TRI3DST as wrap import pycraters . helpers as help import pycraters . IO as io # build solver wrapper and parameter object exec_location = sys . argv [1] wrapper = wrap . TRI3DST_Wrapper ( exec_location ) params = wrap . TRI 3D ST _Pa ra me ter s () We conclude this section by noting that our goal is not to completely wrap the functionality of sophisticated tools such as LAAMPS and TRIDYN – rather, it is to provide a consistent interface to these tools for users with the specific aim of using them to gather and apply single-impact crater function statistics. Python – a high-level scripting language with very mature numerical capabilities – is an ideal language in which to implement such an interface. We note that although certain kinds of parameter sweeping are built into some of the solvers we have discussed, such capabilities effectively represent miniature “scripting languages” unique to each solver. By using a mature language like Python to perform all scripting, the PyCraters library allows parameter sweeping and statistical analysis to be done in a uniform manner, regardless of the underlying tool used to perform the simulations, and therefore without having to learn the details of the scripting capabilities of each solver. 4 Usage Examples We will now proceed to briefly illustrate several examples of the framework in use. The focus of this section is not on any particular result obtained herein (which are subject to revision as theoretical approaches improve), nor on precisely documenting the codebase (which is subject to change in future software versions). Rather, it is to emphasize the general structure of the code as depicted in Figure 1, and illustrate the kinds of problems that the framework has been designed to investigate. In all cases, we will estimate PDE coefficients using the simpler Eqs. (5), procedures for which have been documented in the literature. A report on somewhat more complicated procedures for using the revised Eqs. (7) will be the topic of future work. 4.1 Preliminaries: Basic library loading and setup We begin with a very brief introduction to the code typically needed at the beginning of a PyCraters script, shown in Algorithm 1. It shows the loading of common mathematical libraries, as well as a wrapper and set of helper routines from the PyCraters library. The first code block simply loads all needed libraries, including the relevant parts of PyCraters itself. The second code block reads the executable location from the command line, and creates the two main objects needed to interact with the library – a solver wrapper and a parameter holder. The latter is tasked with describing the simulation environment, while the former abstracts each solver behind a uniform interface. Note that in these two code blocks, the user chosen the TRI3DST solver [29]. If a different solver were desired, only three lines of code would need to be changed. 4.2 Simplest example: Angle-dependence over one energy We now present a very simple illustration of the framework, by using it to obtain results of the kind reported in Ref [15] – PDE coefficients associated with the low-energy irradiation of a pure material. For narrative consistency with the next section, we choose 100 eV Ne+ → C. The code needed for this example appears 6 Algorithm 2 An example program listing using the PyCraters Python framework. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # set parameter values in high - level format params . target = [[" C " , 1.0]] params . beam = " Ne " params . energy = 100 params . angle = None params . impacts = 1000 # perform simulations over a series of angles angles = np . linspace (0 ,85 ,18) for aa in angles : params . angle = aa wrapper . go ( params ) # extract statistics , perform fits , and generate plots fitdata = help . e x t r a c t _ P D E _ c o e f f i c i e n t s ( params , angles ) results = help . p l o t _ c o e f f i c i e n t _ s u m m a r y ( fitdata ) plt . show () in Algorithm 2, and should be assumed to be preceded by the code in Algorithm 1. The first code block specifies the environmental parameters the user wishes to consider, including the ion and target species, the ion energy and incidence angle, and the number of impacts to perform (the incidence angle is left blank). The second code block sweeps over the incidence angle in 5-degree increments: for each angle, it updates the parameter holder, and calls the solver wrapper for the specified set of parameters. Finally, the third code block calls a routine that plots a simple summary of the results. The bulk of the work performed by the PyCraters framework is hidden behind just three function calls. The call to the wrapper.go() routine performs all interaction with the underlying BCA solver, including writing input files, calling the executable, reading output files, extracting moments, and storing the results on the hard disk under a unique file name constructed by the Params() object. Next, the call to the help.extract_PDE_coefficients() routine reads these files for all angles in the sweep, uses a self-contained library to fit each of the angle-dependent moments to appropriate functions of the incidence angle, performs differentiations needed to construct the coefficients, and stores both fits and coefficients under additional reconstructable filenames. Finally, the help.plot_coefficient_summary() routine is a relatively simple visualization routine containing plots likely to be of interest to a casual user. For the program listing in Algorithm 2, the output is exhibited in Fig. 2. 4.3 Angle-Energy Phase Diagrams In this section we present a slightly more complicated example, demonstrating the ease with which additional parameters may be swept to identify trends in statistical behavior. Specifically, we observe that because the stability of Eqn. 4 is determined by the signs of the angle-dependent coefficients SX (θ) and SY (θ), the sixth panel of Figure 2 describes the transition in expected behavior from flat surfaces (both coefficients positive) to ripples oriented in the x-direction (SX < 0). The points at which these curves cross the origin, and each other, thus serve to divide domains of different expected behavior. Using the PyCraters framework, it is only a matter of adding an extra for() loop to repeat this calculation for a variety of ion energies, and thereby obtain an angle-energy phase diagram. Furthermore, additional sweeps over ion and target species allow the associated phase diagrams for each ion/target combination to be compared, enabling the identification of trends in stability with respect to ion and target atom mass. The code used to perform these simulation is listed as Algorithm 3. We note the great similarity to the code listed in Algorithm 2 – with the exception of three extra nested for() loops, and the additional calls to functions like help.find_pattern_transitions() (which calculates curve intersections between {0, SX (θ) , SY (θ)}) and help.plot_energy_angle_phase_diagram() (which plots the results) – the majority of the code is similar. 7 0.05 0.00 (1) (θ) Meros . (0) (θ) Meros . 0.02 0.04 0.06 0.08 2.0 1.5 1.0 0.05 1.0 0.10 0.8 0.15 0.20 SX components SX,eros. SX,redist. SX,total SY(θ) 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 60 70 80 90 angle θ SY components 1.6 SY,eros. 1.4 SY,redist. 1.2 SY,total 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0 10 20 30 40 50 60 70 80 90 angle θ 0.6 0.4 0.0 0.2 0 10 20 30 40 50 60 70 80 90 angle θ 2.0 SX and SY 1.5 1.0 Figure 2: Output of the program listing in. 8 1st redist. moment fit data 0.2 fit data 0.35 0 10 20 30 40 50 60 70 80 90 angle θ 0.5 SX(θ) 1.2 0.30 0.10 0 10 20 30 40 50 60 70 80 90 angle θ 1.4 0.00 0.25 fit data 1st erosive moment (1) ( ) Mredist .θ Sputter Yield SX,Y(θ) 0.02 0.5 0.0 0.5 1.0 SX SY 1.5 0 10 20 30 40 50 60 70 80 90 angle θ Algorithm 3 A program for generating angle-energy phase diagrams over many ion/target combination. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 # identify targets = beams = energies = angles = finedeg = impacts = parameters to sweep over [ [[" C " , 1.0]] , [[" Si " , 1.0]] , [[" Ge " , 1.0]] , [[" Sn " , 1.0]]] [" Ne " , " Ar " , " Kr " , " Xe "] 10.**( np . linspace (2.0 , 4.0 , 21)) np . linspace (0 ,85 ,18) np . linspace (0 , 90 , 91) 1000 # perform the simulations and store the results for tt in targets : for bb in beams : tdatalist = [] for ee in energies : for aa in angles : params . target = tt params . beam = bb params . energy = ee params . angle = aa params . impacts = impacts wrapper . go ( params ) # extract coefficients and find transition angles fitdata = help . e x t r a c t _ P D E _ c o e f f i c i e n t s ( params , angles ) tdata = help . f i n d _ p a t t e r n _ t r a n s i t i o n s ( fitdata ) tdatalist . append ( tdata ) # after each energy sweep , plot and save the phase diagram help . p l o t _ e n e r g y _ a n g l e _ p h a s e _ d i a g r a m ( tdatalist ) plt . savefig ("% s -% s - phase - diagram . svg " % ( ion_species , target_species )) This reflects the aims of the framework of providing high-level functionality in easy-to-use functions, allowing the end user to focus on specifying the range of parameters to be explored. A sampling of the respective graphs are shown in Figure (3). In the first column, the ion mass is increased, and a corresponding increase in the size of the region for stable, flat surfaces is observed, as well as a decrease in the size of the region for perpendicular mode ripples. Because the stable regions are induced by redistribution, and the latter by erosion, this indicates an increasing relative strength of the redistributive effect as the ion mass increases. By contrast, in the second column, the target mass is increased, and the trend is reversed – the stable region shrinks, and the region of perpendicular mode ripples grows. This indicates that generically, as the target mass increases, the role of erosion grows relative to that of mass redistribution. Interestingly, for most ion/mass combinations, the effect of the ion energy seems to be small above around 500 eV. It should be stressed, however, that these results are not presented as a definitive prediction on behavior. They capture only the effect of the collision cascade – sputter erosion and mass redistribution – and do not capture phenomena such as stress implantation and relaxation [30, 31, 32, 33]. This effect also contributes to the coefficients SX (θ) and SY (θ), and its magnitude relative to collisional effects is not yet known. 4.4 Binary Materials The simulation and analysis of single-impact events on binary targets is easily performed within the PyCraters framework. In Ref. [16], the Crater Function approach was extended to binary compound targets, and estimates were made of several coefficients within a theory describing the irradiation of such targets [12, 34], 9 Ne+ C Xe+ C Ar+ C Xe+ Si Kr+ C Xe+ Ge Xe+ C Xe+ Sn Figure 3: A collection of angle-energy phase diagrams generated using the program listed in Algorithm 3.  Left column descending: increasing ion mass ( Ne+ , Ar+ , Kr+ , Xe+ → C). Right column descending: increasing target mass (Xe+ → {C, Si, Ge, Sn}). In each figure, the region between the left edge and the blue line indicates flat targets, the region between the blue and red lines indicates parallel mode ripples, and the region between the red line and the right edge indicates perpendicular mode ripples. 10 (a) (b) Figure 4: Estimates of the relative sputter yields for various amorphous compounds Ga1−x Sbx (a) if the target is homogeneous, (b) if the target exhibits a 1nm layer enriched in Sb of the form Ga1−y Sby , with y = 2x−x2 . The enrichment of the surface layer produces a dramatic shift in the equilibrium film concentration. for the case of GaSb irradiated by Ar at 250 eV. However, during this process, a notable discrepancy between simulations and experiments was observed – whereas experiments observe gradual buildup of excess Ga on the irradiated target, the simulations of impacts on GaSb indicated a slight preferential sputtering of Ga, which would lead to excess Sb. It was hypothesized that this discrepancy might be due to the effect of Gibbsian surface segregation, whereby the atom with the lower surface free energy (in this case Sb) migrates to the surface [35], where it is more easily sputtered due surface proximity. Because the in-situ composition profile is unknown, the targets used in the initial studies were homogeneous, and could not capture this effect. The PyCraters framework is an ideal tool to explore this hypothesis, and a code listing which performs the needed simulations is provided in Algorithm 4. In this example, the user must move beyond built-in functionality, and begin to provide some customization to the default settings. For instance, we specify a list of amorphous targets with varying stoichiometries of the form Ga1−x Sbx , and a simple user-supplied function that allows a surface layer 1nm thick to be modified by enriching it in Sb from x to a plausible level of y = 2x − x2 . After the target is then irradiated at 1 keV by Ar+, and the relative sputter yields of Ga and Sb (which appear in the zeroth moment of the crater function) are extracted directly from the storage files, and a few lines of code are written to plot these yields as functions of the base Sb composition. The results for both unmodified and modified targets are shown in Figure 4. Unsurprisingly, the modified target exhibits a greater sputter yield of Sb than the unmodified target – the top monolayer of this target contains more Antimony, and most sputtering occurs from the top monolayer. In fact, the new yields largely mirror the enriched layer composition. Interestingly, however, this relatively small modification is entirely sufficient to resolve the discrepancy between simulation and experiment just described. Though the actual composition profile remains as yet unknown, the results of this kind of experimentation with plausible model targets suggests that this mechanism is likely sufficient to explain the observed enrichment of Ga over time. 4.5 Future work: Comparison of Methodologies An important future goal of the framework is to facilitate the comparison of simulations performed using Molecular Dynamics to those using the Binary Collision Approximation. For instance, using MD, Ref. [15] found redistributed atoms to contribute far more to the shape and magnitude of coefficients in Eqs. (5) than did sputtered atoms. However, using the BCA, Ref. [36] reports a reduced redistributive signal for the environment studied in Ref. [15], and finds that sputtered atoms are dominant for most angles at higher energies. The use of MD vs BCA may be an important source of these conflicting results – in Ref. [37], it was found that the BCA reports significantly fewer displacements of small magnitude than molecular dynamics. Because of the very large number of such displacements, they may contribute significantly to the effect of mass redistribution, and hence the BCA approach may systematically under-report the strength of this effect. This question demands further study, and because the extraction of the statistics and the 11 Algorithm 4 Code to investigate the effect of segregation. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 # define a simple function of depth def c o n c e n t r a t i o n _ f u n c t i o n ( params , depth ): cbase = [ a [1] for a in params . target ] if depth > 10: return cbase else : bSb = cbase [1] lSb = 2.0* bSb - bSb **2 return [1 - lSb , lSb ] # basic parameter setup params . target = None params . beam = " Ar " params . angle = 0.0 params . energy = 1000 params . impacts = 1000 params . set_parameter (" cfunc " , c o n c e n t r a t i o n _ f u n c t i o n ) # set up targets at different concentrations targets = [ ] for phi in np . linspace (0.0 , 1.0 , 11): target = [[" Ga " , 1.0 - phi ] ,[" Sb " , phi ]] targets . append ( target ) # iterate over targets for tt in targets : params . target = tt wrapper . go ( params ) # make plots import libcraters . IO as io m0e_avg = io . array_range ( ’./ ’ , params , ’ target ’ , m0e_std = io . array_range ( ’./ ’ , params , ’ target ’ , Gaval = [ a [0] for a in m0e_avg ] Sbval = [ a [1] for a in m0e_avg ] Gaerr = [ b [0]/ np . sqrt ( params . impacts ) * 1.97 for Sberr = [ b [1]/ np . sqrt ( params . impacts ) * 1.97 for targets , ’ m0e_avg ’ ) targets , ’ m0e_std ’ ) b in m0e_std ] b in m0e_std ] plt . figure (1 , figsize =(4.5 , 3.0)) pctSb_list = [ a [1][1] for a in targets ] plt . errorbar ( pctSb_list , - np . array ( Gaval ) , yerr = Gaerr , label = ’ Ga ’ , fmt = ’ gs - ’ , linewidth plt . errorbar ( pctSb_list , - np . array ( Sbval ) , yerr = Sberr , label = ’ Sb ’ , fmt = ’ bs - ’ , linewidth plt . xlabel ( ’ at . pct . Sb ’) plt . ylabel ( ’ fractional sputter yield ’) plt . legend () plt . ylim (0 , max ( - np . array ( Tvals ))*1.7 ) plt . tight_layout () plt . show () plt . savefig ( ’ relative - sputter - yields . svg ’) 12 estimation of coefficients are generic processes independent of the solver within the PyCraters library, it will be straightforward to use those common resources to identify the implications of these observations for the specific statistics required by crater function analysis. In a related vein, several different procedures for extracting the crater function ∆h and its moments M (i) from a list of atomic positions have been suggested in the literature [38, 15, 39, 36], and the importance of the differences between the strategies has never been investigated. The PyCraters library provides a natural environment in which to conduct such studies. Because the framework provides the results of simulations in a common format, variations on the statistical extraction routine can be written in a way that is independent of both the underlying solver and the methods used to fit and differentiate the results. This should allow easy comparison of the different extraction methods for a variety of simulation environments, and aid the identification of best practices for this important procedure. 5 Summary In conclusion, we have introduced a Python framework designed to automate the most common tasks associated with the extraction and upscaling of the statistics of single-impact crater functions to inform coefficients of continuum equations describing surface morphology evolution. The framework has been designed to be compatible with a wide variety of existing atomistic solvers, including Molecular Dynamics and Binary Collision Approximation codes. However, in order to remain accessible to first-time users, the details of each of these solvers is abstracted behind a standardized interface, and much functionality can be accessed via high-level functions and visualization routines. Although the addition of much functionality is still in progress, the current codebase is able to reproduce many important results from the recent literature, and examples demonstrating these capabilities are included to facilitate modification and additional exploration by the community. The project is currently hosted on the BitBucket repository under a suitable open-source license, and is available for immediate download. 13 References [1] M. Navez, D. Chaperot, , and C. Sella. Microscopie electronique - etude de lattaque du verre par bombardement ionique. Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences, 254:240, 1962. [2] M. J. Baldwin and R. P. Doerner. 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Implementation of a Direct Coupling Coherent Quantum Observer including Observer Measurements arXiv:1603.01436v1 [quant-ph] 4 Mar 2016 Ian R. Petersen and Elanor H. Huntington Abstract— This paper considers the problem of constructing a direct coupling quantum observer for a quantum harmonic oscillator system. The proposed observer is shown to be able to estimate one but not both of the plant variables and produces a measureable output using homodyne detection. I. I NTRODUCTION A number of papers have recently considered the problem of constructing a coherent quantum observer for a quantum system; e.g., see [1]–[4]. In the coherent quantum observer problem, a quantum plant is coupled to a quantum observer which is also a quantum system. The quantum observer is constructed to be a physically realizable quantum system so that the system variables of the quantum observer converge in some suitable sense to the system variables of the quantum plant. The papers [4]–[7] considered the problem of constructing a direct coupling quantum observer for a given closed quantum system. In [4], the proposed observer is shown to be able to estimate some but not all of the plant variables in a time averaged sense. Also, the paper [8] shows that a possible experimental implementation of the augmented quantum plant and quantum observer system considered in [4] may be constructed using a non-degenerate parametric amplifier (NDPA) which is coupled to a beamsplitter by suitable choice of the NDPA and beamsplitter parameters. One important limitation of the direct coupled quantum observer results given in [4]–[8] is that both the quantum plant and the quantum observer are closed quantum systems. This means that it not possible to make an experimental measurement to verify the properties of the quantum observer. In this paper, we address this difficulty by extending the results of [4] to allow for the case in which the quantum observer is an open quantum linear system whose output can be monitored using homodyne detection. In this case, it is shown that similar results can be obtained as in [4] except that now the observer output is subject to a white noise perturbation. However, by suitably designing the observer, it is shown that the level of this noise perturbation can be made arbitrarily small (at the expense of slow observer This work was supported by the Australian Research Council (ARC) and the Chinese Academy of Sciences Presidents International Fellowship Initiative (No. 2015DT006). Ian R. Petersen is with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia. i.r.petersen@gmail.com Elanor H. Huntington is with the College of Engineering and Computer Science, The Australian National University, Canberra, ACT 0200, Australia. Email: Elanor.Huntington@anu.edu.au. convergence). Also, the results of [8] are extended to show that a possible experimental implementation of the augmented quantum plant and quantum observer system may be constructed using a non-degenerate parametric amplifier (NDPA) which is coupled to a beamsplitter by suitable choice of the NDPA and beamsplitter parameters. In this case, the NDPA contains an extra field channel as compared to the result in [8] and this extra channel is used for homodyne detection in the observer. II. D IRECT C OUPLING C OHERENT Q UANTUM O BSERVER WITH O BSERVER M EASUREMENT In this section, we extend the theory of [4] to the case of a direct coupled quantum observer which is also coupled to a field to enable measurements to be made on the observer. In our proposed direct coupled coherent quantum observer, the quantum plant is a single quantum harmonic oscillator which is a linear quantum system (e.g., see [9]–[13]) described by the non-commutative differential equation ẋp (t) zp (t) = = 0; xp (0) = x0p ; Cp xp (t) (1) where zp (t) denotes the system variable to be estimated by the observer and Cp ∈ R1×2 . This quantum plant corre qp sponds to a plant Hamiltonian Hp = 0. Here xp = pp where qp is the plant position operator and pp is the plant momentum operator. It follows from (1) that the plant system variables xp (t) will remain fixed if the plant is not coupled to the observer. We now describe the linear quantum system which will correspond to the quantum observer; see also [9]–[13]. This system is described by a quantum stochastic differential equation (QSDE) of the form dxo = Ao xo dt + Bo dw; xo (0) = x0o ; = Co xo dt + dw; = Kyo (2)   dQ where dw = is a 2 × 1 vector of quantum noises dP expressed in quadrature form corresponding to the input field for the observer and dyo is the corresponding output field; e.g., see [9], [11]. The observer output zo (t) will be a real scalar quantity obtained by applying homodyne detection 2×2 2×2 to the observer output field.  Ao ∈ R , Bo ∈ R , qo is a vector of self-adjoint Co ∈ R2×2 . Also, xo (t) = po system variables corresponding to the observer position and dyo zo (t) momentum operators; e.g., see [9]. We assume that the plant variables commute with the observer variables. The system dynamics (2) are determined by the observer system Hamiltonian and coupling operators which are operators on the underlying Hilbert space for the observer. For the quantum observer under consideration, this Hamiltonian is a self-adjoint operator given by the quadratic form: Ho = 1 T 2 xo (0) Ro xo (0), where Ro is a real symmetric matrix. Also, the coupling operator L is defined by a matrix Wo ∈ R2×2 so that   L + L∗ = Wo xo . (3) L−L∗ i Then, the corresponding matrices Ao , Bo and Co in (2) are given by 1 Ao = 2JRo + WoT JWo , Bo = JWoT J, Co = Wo (4) 2 where   0 1 J= ; −1 0 e.g., see [9], [11]. Furthermore, we will assume that the quantum observer is coupled to the quantum plant as shown in Figure 1. In addition, we define a coupling Hamiltonian Quantum Observer Quantum Plant Homdyne Detector yo zo zp Fig. 1. Plant Observer System. Then, we can write the QSDEs describing the closed loop system as follows: dxp dxo dyo = 2Jαβ T xo dt; 1 = 2ωo Jxo dt + 2JβαT xp dt + JWoT JWo xo dt 2 +JW T Jdw   κo √ 2ωo −2 xo dt + 2JβαT xp dt − κdw; = κ −2ωo − 2 = Wo xo dt + dw √ = κxo dt + dw; (9) e.g., see [9], [11]. Now it follow from (1) and (8) that zp = αT xp . Hence, it follows from the first equation in (9) that dzp = 2αT Jαβ T xo dt = 0. That is, the quantity zp remains constant even after the quantum plant is coupled to the quantum observer. In addition, we can re-write the remaining equations in (9) as  κ  √ −2 2ωo dxo = xo dt + 2Jβzp dt − κdw; −2ωo − κ2 √ (10) dyo = κxo dt + dw; To analyse the system (10), we first calculate the steady state value of the quantum expectation of the observer variables as follows:  κ −1 −2 2ωo < x̄o > = −2 Jβzp −2ωo − κ2   4 κ 4ωo Jβzp . = κ2 + 16ωo2 −4ωo κ Then, we define the quantity which defines the coupling between the quantum plant and the quantum observer: T Hc = xp (0) Rc xo (0). The augmented quantum linear system consisting of the quantum plant and the quantum observer is then a linear quantum system described by the total Hamiltonian Ha where xa =  = Hp + Hc + Ho 1 = xa (0)T Ra xa (0) 2 xp xo  , Ra =  0 RcT Rc Ro (5)  , (6) and the coupling operator L defined in (3). Extending the approach used in [4], we assume that we can write Rc = αβ T , (7) √ Ro = ωo I, Wo = κI where α ∈ R2 , β ∈ R2 , ωo > 0 and κ > 0. In addition, we assume α = CpT . (8) 4 x̃o = xo − < x̄o >= xo − 2 κ + 16ωo2  κ −4ωo 4ωo κ  Jβzp . We can now re-write the equations (10) in terms of x̃0 as follows   κ √ 2ωo −2 xo dt + 2Jβzp dt − κdw dx̃o = −2ωo − κ2  κ  2ωo −2 = x̃o dt −2ωo − κ2  κ −1  κ −2 2ωo 2ωo −2 Jβzp dt −2 −2ωo − κ2 −2ωo − κ2 √ +2Jβz dt − κdw  κ p  √ −2 2ωo = x̃o dt − κdw; κ −2ωo − 2 −1  κ √ √ 2ωo −2 Jβzp dt + dw dyo = κx̃o dt − 2 κ −2ωo − κ2  κ −1 √ −2 2ωo = −2 κ Jβzp dt + dwout (11) −2ωo − κ2 where dwout = √ κx̃o dt + dw. We now look at the transfer function of the system   κ √ 2ωo −2 x̃o dt − κdw; dx̃o = κ −2ωo − 2 √ dwout = κx̃o dt + dw (12) which is given by G(s) = −κ  s + κ2 2ωo −2ωo s + κ2 −1 . It is straightforward to verify that this transfer function is such that G(jω)G(jω)† = I for all ω. That is G(s) is all pass. Also, the matrix  − κ2 2ωo is Hurwitz and hence, the system (12) −2ωo − κ2 will converge to a steady state in which dwout represents a standard quantum white noise with zero mean and unit intensity. Hence, at steady state, the equation −1  κ √ 2ωo −2 Jβzp dt + dwout (13) dyo = −2 κ −2ωo − κ2 shows that the output field converges to a constant value plus zero mean white quantum noise with unit intensity. We now consider the construction of the vector K defining the observer output zo . This vector determines the quadrature of the output field which is measured by the homodyne detector. We first re-write equation (13) as dyo = ezp dt + dwout where √ e = −2 κ  − κ2 −2ωo 2ωo − κ2 −1 Jβ (14) However, if we choose K= eT kek2 (16) 1 . Hence, this value of then (15) is satisfied and kKk = kek K must be the optimal K. We now consider the special case of ωo = 0. In this case, we obtain   2 √ 4 0 e = 2 κ κ 2 Jβ = √ Jβ. 0 κ κ Hence, as κ → 0, kek → ∞ and therefore kKk → 0. This means that we can make the noise level on our measurement arbitrarily small by choosing κ > 0 sufficiently small. However, as κ gets smaller, the system (12) gets closer to instability and hence, takes longer to converge to steady state. III. A P OSSIBLE I MPLEMENTATION OF THE P LANT O BSERVER S YSTEM In this section, we describe one possible experimental implementation of the plant-observer system given in the previous section. The plant-observer system is a linear quantum system with Hamiltonian 1 Hc + Ho = xTp Rc xo + xTo Ro xo (17) 2 and coupling operator defined so that   L + L∗ ∗ = Wo xo . L−L i Furthermore, we assume that Rc = αβ T , Ro = ωo I, Wo = √ κI where α ∈ R2 , β ∈ R2 , ωo > 0 and κ > 0. In order to construct a linear quantum system with a Hamiltonian of this form, we consider an NDPA coupled to a beamsplitter as shown schematically in Figure 2; e.g., see [14]. is a vector in R2 . Then out B2 B2 dzo = Kezp dt + Kdwout . Hence, we choose K such that A Ke = 1 NDPA B1 (15) and therefore dzo = zp dt + dn a,b out A out B1 where dn = Kdwout will be a white noise process at steady state with intensity kKk2 . Thus, to maximize the signal to noise ratio for our measurement, we wish to choose K to minimize kKk2 subject to the constraint (15). Note that it follows from (15) and the Cauchy-Schwartz inequality that 1 ≤ kKkkek and hence kKk ≥ 1 . kek B1 A Beamsplitter Fig. 2. NDPA coupled to a beamsplitter. A linearized approximation for the NDPA is defined by a quadratic Hamiltonian of the form 1 ı H1 = (ǫa∗ b∗ − ǫ∗ ab) + ωo b∗ b 2 2 where a is the annihilation operator corresponding to the first mode of the NDPA and b is the annihilation operator corresponding to the second mode of the NDPA. These modes will be assumed to be of the same frequency but with a different polarization with a corresponding to the quantum plant and b corresponding to the quantum observer. Also, ǫ is a complex parameter defining the level of squeezing in the NDPA and ωo corresponds to the detuning frequency of the b mode in the NDPA. The a mode in the NDPA is assumed to be tuned. In addition, theNDPA corresponds to a vector of  √ κ1 a √ coupling operators L =  κ2 b . Here κ1 > 0, κ2 > 0, √ κ3 b κ3 > 0 are scalar parameters determined by the reflectance of the mirrors in the NDPA. From the above Hamiltonian and coupling operators, we can calculate the following quantum stochastic differential equations (QSDEs) describing the NDPA:   da db  = 0 ǫ 2 ǫ 2 0 γ1 2  a∗ b∗  dt   a dt − γ2 1 b 0 2 + 2 ıωo    dA  √ κ1 0 0  dB1  ; √ √ − 0 κ2 κ3 dB2     √   dA κ1 0 √ a  0 κ  dt +  dB1  ; √ 2 b κ3 dB2 0 (18)  dAout  dB1out  = dB2out  0 where γ1 = κ1 and γ2 = κ2 + κ3 . We now consider the equations defining the beamsplitter  A B1  =  cos θ −eıφ sin θ e−ıφ sin θ cos θ  Aout B1out  where θ and φ are angle parameters defining the beamsplitter; e.g., see [15]. This implies  Aout B1out  =  cos θ eıφ sin θ −e−ıφ sin θ cos θ  A B1  . Substituting this into the second equation in (18), we obtain  −ıφ   cos θ −e sin θ dA dB1 eıφ sin θ cos θ      √ a dA κ1 0 √ dt + = b dB1 0 κ2 and hence   −e−ıφ sin θ cos θ − 1  √  κ1 0 √ = κ2 0 cos θ − 1 eıφ sin θ  dA dB1  a dt. b We now assume that cos θ 6= 1. It follows that we can write   dA = dB1   1 cos θ − 1 e−ıφ sin θ 2(1 − cos θ) −eıφ sin θ cos θ − 1    √ a κ1 0 √ dt. × b κ2 0 Substituting this into the first equation in (18), we obtain   da = db  ∗   a 0 2ǫ dt ǫ 0 b∗ 2    γ1 a 0 2 dt − γ2 1 b 0 2 + 2 ıωo  √   1 cos θ − 1 e−ıφ sin θ κ1 0 √ − κ2 −eıφ sin θ cos θ − 1 2(1 − cos θ) 0   √  a κ1 0 √ × dt b 0 κ2   0 dB2 ; − √ κ3 √ dB2out = κ3 b + dB2 . These QSDEs can be written    da  db       da∗  = F  db∗     dB2out = H out∗  dB2 in the form  a   b  dB2  dt + G ; a∗  dB2∗ ∗ b  a   dB2 b   dt + dB2∗ a∗  ∗ b F2 F1#  √ κ1 κ2 e−ıφ sin θ 2(1−cos θ) − κ23 − 21 ıωo  where the matrix F is given by F = F1 =  F2 =   0 √ 0 ǫ 2 κ1 κ2 eıφ sin θ 2(1−cos θ)  ǫ 2 .  F1 F2# − 0 Also, the matrix G is given by  0  √κ3 G = −  0 0 and the matrix H is given by  √ 0 κ3 H= 0 0  0  0 ,  0 √ κ3 0 0 0 √ κ3  . and , It now follows from the proof of Theorem 1 in [16] that we can construct a Hamiltonian for this system of the form   a   b  1 ∗ ∗  a b a b M H=  a∗  2 b∗ where the matrix R is given by R = = Rc = where the matrix M is given by M1 M2 = ı 2 " = ı 2  0 0 ǫ √ κ κ2 e−ıφ sin θ − 1 1−cos θ √ κ1 κ2 eıφ sin θ 1−cos θ ǫ 0  −ıωo # and δ = . Also, we can construct the coupling operator for this system in the form   a  b    L = N1 N2   a∗  b∗   N1 N2 is given by where the matrix N = N2# N1# N = H. Hence, N1 =  0 √ κ3  Then we calculate H = = 1 qp 2 1 T xp 2 pp xTo 0 1 0 1  0 ı  . 0  −ı   qp   pp   qo po R   qo  po    xp R xo ℜ(ǫ) + ℜ(δ) ℑ(ǫ) − ℑ(δ)  Hence, 1 ωo xTo xo + xTp Rc xo . 2 Comparing this with equation (7), we require that   −ℑ(ǫ) − ℑ(δ) ℜ(ǫ) + ℜ(δ) = αβ T ℜ(ǫ) − ℜ(δ) ℑ(ǫ) − ℑ(δ) (19) and the condition (15) to be satisfied in order for the system shown in Figure 2 to provide an implementation of the augmented plant-observer system. We first observe that the matrix on the right hand side of equation (19) is a rank one matrix and hence, we require that   −ℑ(ǫ) − ℑ(δ) ℜ(ǫ) + ℜ(δ) det = |δ|2 − |ǫ|2 = 0. ℜ(ǫ) − ℜ(δ) ℑ(ǫ) − ℑ(δ) That is, we require that √ κ1 κ2 sin θ = |ǫ|. 1 − cos θ sin θ Note that the function 1−cos θ takes on all values in (−∞, ∞) for θ ∈ (0, 2π) and hence, this condition can always be satisfied for a suitable choice of θ. This can be seen in Figure sin θ 3 which shows a plot of the function f (θ) = 1−cos θ. , N2 = 0. We now wish to calculate the Hamiltonian H in terms of the quadrature variables defined such that     a qp  b     ∗  = Φ  pp   a   qo  b∗ po where the matrix Φ is given by  1 ı  0 0 Φ=  1 −ı 0 0 √ κ1 κ2 eıφ sin θ . 1−cos θ = , −ℑ(ǫ) − ℑ(δ) ℜ(ǫ) − ℜ(δ) H= 40 30 20 10 f(θ) and J =  M1 M2 M2# M1#
ı JF − F † J M= 2   I 0 . Then, we calculate M  0 −I where  † Φ  MΦ  0 Rc , RcT ωo I 0 −10 −20 −30 −40 0 1 2 Fig. 3. 3 θ (rad) 4 5 6 7 Plot of the function f (θ). Furthermore, we will assume without loss of generality that θ ∈ (0, π) and hence we obtain our first design equation |ǫ| sin θ . = √ 1 − cos θ κ1 κ2 (20) In practice, this ratio would be chosen in the range of |ǫ| √ κ1 κ2 ∈ (0, 0.6) in order to ensure that the linearized model which is being used is valid. We now construct the vectors α and β so that condition (19) is satisfied. Indeed, we let # "   1 −ℑ(ǫ) − ℑ(δ) , β= . α= ℜ(ǫ)−ℜ(δ) ℜ(ǫ) + ℜ(δ) −ℑ(ǫ)−ℑ(δ) For these values of α and β, it is straightforward to verify that (19) is satisfied provided that |ǫ| = |δ|. With this value of β, we now calculate the quantity e defined in (14) as follows:   4 κ3 4ωo e = Jβ κ23 + 16ωo2 −4ωo κ3    4 ℜ(ǫ) + ℜ(δ) κ3 4ωo . = − 2 ℑ(ǫ) + ℑ(δ) κ3 + 16ωo2 −4ωo κ3 Then, the vector K defining the quadrature measured by the homodyne detector is constructed according to the equation (16). In the special case that ωo = 0, this reduces to   4 ℜ(ǫ) + ℜ(δ) e=− . κ3 ℑ(ǫ) + ℑ(δ) In terms of complex numbers e = e(1) + ıe(2), we can write this as 4 e = − (ǫ + δ). κ3 Then, in terms of complex numbers K = K(1) + ıK(2), the formula (16) becomes K= κ3 (ǫ + δ) κ3 e =− =− |e|2 4(ǫ + δ)(ǭ + δ̄) 4(ǭ + δ̄) ¯ denotes complex conjugate. Also, as noted in where (·) Section II, the steady state measurement noise intensity is given by κ3 1 =− kek 4|ǫ + δ| which approaches zero as κ3 → 0. However, this is at the expense of increasingly slower convergence to steady state. IV. C ONCLUSIONS In this paper, we have shown that a direct coupling observer for a linear quantum system can be implemented in the case that the observer can be measured using a Homodyne detection measurement. This would allow the plant observer system to be constructed experimentally and the performance of the observer could be verified using the measured data. R EFERENCES [1] Z. Miao and M. R. James, “Quantum observer for linear quantum stochastic systems,” in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, December 2012. [2] I. Vladimirov and I. R. Petersen, “Coherent quantum filtering for physically realizable linear quantum plants,” in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, July 2013, arXiv:1301.3154. [3] Z. Miao, L. A. D. Espinosa, I. R. Petersen, V. Ugrinovskii, and M. R. James, “Coherent quantum observers for finite level quantum systems,” in Australian Control Conference, Perth, Australia, November 2013. [4] I. R. 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Noname manuscript No. (will be inserted by the editor) Multi-point Codes from the GGS Curves arXiv:1706.00313v3 [] 29 Jun 2017 Chuangqiang Hu · Shudi Yang Received: date / Accepted: date Abstract This paper is concerned with the construction of algebraic geometric codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated to some totally ramified places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation of the basis for Riemann-Roch spaces from GGS curves. Additionally, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Finally, we apply these results to find multi-point codes with excellent parameters. As one of the examples, a presented code with parameters [216, 190, > 18] over F64 yields a new record. Keywords Algebraic geometric codes · GGS curve · Weierstrass semigroup · Weierstrass pure gap Mathematics Subject Classification (2010) 14H55 · 11R58 1 Introduction In the early 1980s, Goppa [12] constructed algebraic geometric codes (AG codes for short) from algebraic curves. Since then, the study of AG codes becomes an important instrument in the theory of error-correcting codes. Roughly speaking, the parameters of an AG code are good when the underlying curve has many rational points with respect to its genus. For this reason C. Hu Yau Mathematical Science Center, Tsinghua University, Peking 100084, P.R. China E-mail: huchq@mail2.sysu.edu.cn S. Yang School of Mathematical Sciences, Qufu Normal University, Shandong 273165, P.R.China E-mail: yangshudi7902@126.com 2 Chuangqiang Hu, Shudi Yang maximal curves, that is curves attaining the Hasse-Weil upper bound, have been widely investigated in the literature: for example the Hermitian curve and its quotients, the Suzuki curve, the Klein quartic and the GK curve. In this work we will study multi-point AG codes on the GGS curves. In order to construct good AG codes we need to study Weierstrass semigroups and pure gaps. Their use dates back to the theory of one-point codes. For example, the authors in [26,14,28,29] examined one-point codes from Hermitian curves and develop efficient methods to decode them. Korchmáros and Nagy [19] computed the Weierstrass semigroup of a degree-three closed point of the Hermitian curve. Matthews [24] determined the Weierstrass semigroup of any r-tuple rational points on the quotient of the Hermitian curve. As is known, Weierstrass pure gap is also a useful tool in coding theory. Garcia, Kim and Lax improved the Goppa bound using arithmetical structure of the Weierstrass gaps at one place in [9,10]. The concept of pure gaps of a pair of points on a curve was initiated by Homma and Kim [15], and it had been pushed forward by Carvalho and Torres [4] to several points. Maharaj and Matthews [21] extended this construction by introducing the notion of the floor of a divisor and obtained improved bounds on the parameters of AG codes. We mention that Maharaj [20] showed that Riemann-Roch spaces of divisors from fiber products of Kummer covers of the projective line, can be decomposed as a direct sum of Riemann-Roch spaces of divisors of the projective line. Maharaj, Matthewsa and Pirsic [22] determined explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. Along this research line, Hu and Yang [16] gave other explicit bases for RiemannRoch spaces of divisors over Kummer extensions, which makes it convenient to determine the pure gaps. In this work, we focus our attention on the GGS curves, which are maximal curves constructed by Garcia, Güneri and Stichtenoth [8] over Fq2n defined by the equations ( xq + x = y q+1 , 2 yq − y = z m, where q is a prime power and m = (q n + 1)/(q + 1) with n > 1 to be an odd integer. Obviously the GGS curve is a generalization of the GK curve initiated by Giulietti and Korchmáros [11] where we take n = 3. Recall that Fanali and Giulietti [7] have investigated one-point AG codes over the GK curves and obtained linear codes with better parameters with respect to those known previously. Two-point and multi-point AG codes on the GK maximal curves have been studied in [6] and [3], respectively. Bartoli, Montanucci and Zini [2] examined one-point AG codes from the GGS curves. Inspired by the above work and [16,5], here we will examine multi-point AG codes arising from GGS curves. To be precise, an explicit basis for the corresponding RiemannRoch space is determined by constructing a related set of lattice points. The properties of AG codes from GGS curves are also considered. Then the basis is Multi-point Codes from the GGS Curves 3 utilized to characterize the Weierstrass semigroups and pure gaps with respect to several totally ramified places. In addition, we give an effective algorithm to compute the floor of divisors. Finally, our results will lead us to find new codes with better parameters in comparison with the existing codes in MinT’s Tables [25]. A new record-giving [216, 190, > 18]-code over F64 is presented as one of the examples. The remainder of the paper is organized as follows. Section 2 focuses on the construction of bases for the Riemann-Roch space from GGS curves. Section 3 studies the properties of the related AG codes. In Section 4 we determine the Weierstrass semigroups and the pure gaps. Section 5 devotes to the floor of divisors from GGS curves. Finally, in Section 6 we employ the results in the previous sections to construct multi-point codes with excellent parameters. 2 Bases for Riemann-Roch spaces from GGS curves Throughout this paper, we always let q be a prime power and n > 1 be an odd integer. The GGS curve GGS(q, n) over Fq2n is defined by the equations ( xq + x = y q+1 , (1) 2 yq − y = z m, where m = (q n +1)/(q +1). The genus of GGS(q, n) is (q −1)(q n+1 +q n −q 2 )/2 and there are q 2n+2 − q n+3 + q n+2 + 1 rational places, see [8] for more details. Especially when n = 3, the equation (1) gives the well-known maximal curve introduced by Giulietti and Korchmáros [11], the so-called GK curve, which is not a subcover of the corresponding Hermitian curve. Denote by Pα,β,γ the rational place P of this curve except for the one centered at infinity P∞ . Take Qβ := αq +α=β q+1 Pα,β,0 where β ∈ Fq2 . Then deg(Qβ ) = q. For later use, we write P0 := P0,0,0 and Q0 := P0 +P1 +· · ·+Pq−1 . The following proposition describes some principle divisors from GGS curves. Proposition 1 Let the curve GGS(q, n) be given in (1) and assume that αµ with 0 6 µ < q are the solutions of xq + x = 0. Then we obtain (1) div(x − αµ ) = m(q + 1)Pµ − m(q + 1)P∞ , (2) div(y − β) P= mQβ − mqP∞ for β ∈ Fq2 , (3) div(z) = β∈F 2 Qβ − q 3 P∞ . q For convenience, we use Qν (0 6 ν 6 q 2 − 1) to represent the divisors Qβ (β ∈ Fq2 ). In particular, Qν |ν=0 = Qβ |β=0 . For a function field F , the Riemann-Roch vector space with respect to a divisor G is defined by n o L(G) = f ∈ F div(f ) + G > 0 ∪ {0}. 4 Chuangqiang Hu, Shudi Yang Let ℓ(G) be the dimension of L(G). From the famous Riemann-Roch Theorem, we know that ℓ(G) − ℓ(W − G) = 1 − g + deg(G), where W is a canonical divisor and g is the genus of the associated curve. Pq−1 P 2 −1 In this section, we consider a divisor G := µ=0 rµ Pµ + qν=1 sν Qν + tP∞ from the GGS curve GGS(q, n). Actually, we can show that the Riemann-Roch space L(G) is generated by some elements, say Ei,j,k for some i, j, k, and the number of such elements equals ℓ(G). To see this, we proceed as follows. Let j = (j1 , j2 , · · · , jq−1 ) and k = (k1 , k2 , · · · , kq2 −1 ). For (i, j, k) ∈ q2 +q−1 Z , we define Ei,j,k := z i q−1 Y µ=1 jµ (x − αµ ) 2 qY −1 (y − βν )kν . (2) ν=1 Pq−1 Pq2 −1 Set |j| = µ=1 jµ and |k| = ν=1 kν . Here and thereafter, we denote |v| to be the sum of all the coordinates of a given vector v. By Proposition 1, one can compute the divisor of Ei,j,k : div(Ei,j,k ) =iP0 + q−1 X (i + m(q + 1)jµ )Pµ + µ=1 2 qX −1 (i + mkν )Qν ν=1
− q 3 i + m(q + 1)|j| + mq|k| P∞ . (3) For later use, we denote by ⌊x⌋ the largest integer not greater thanx and  α if by ⌈x⌉ the smallest integer not less than x. It is easy to show that j = β and only if 0 6 βj − α < β, where β ∈ N and α ∈ Z. Put r = (r0 , r1 , · · · , rq−1 ) and s = (s1 , s2 , · · · , sq2 −1 ). Let us define a set 2 of lattice points for (r, s, t) ∈ Zq +q , n Ωr,s,t := (i, j, k) i + r0 > 0, 0 6 i + m(q + 1)jµ + rµ < m(q + 1) for µ = 1, · · · , q − 1, or equivalently, 0 6 i + mkν + sν < m for ν = 1, · · · , q 2 − 1, o q 3 i + m(q + 1)|j| + mq|k| 6 t , n Ωr,s,t := (i, j, k) i + r0 > 0,   −i − rµ for µ = 1, · · · , q − 1, jµ = m(q + 1)   −i − sν kν = for ν = 1, · · · , q 2 − 1, m Multi-point Codes from the GGS Curves 5 o q 3 i + m(q + 1)|j| + mq|k| 6 t . (4) The following lemma is crucial for the proof of our key result. However, the proof of this lemma is technical, and will be completed later. Lemma 1 The number of lattice points in Ωr,s,t can be expressed as: #Ωr,s,t = 1 − g + t + |r| + q|s|, o n for t > 2g − 1 − q 3 w, where w = min 06µ6q−1 rµ , sν . 16ν6q2 −1 Pq−1 Pq2 −1 Let G := µ=0 rµ Pµ + ν=1 sν Qν + tP∞ . It is trivial that deg(G) = |r| + q|s| + t. Now we can easily prove the main result of this section. Pq−1 P 2 −1 sν Qν + tP∞ . The elements Ei,j,k Theorem 1 Let G := µ=0 rµ Pµ + qν=1 with (i, j, k) ∈ Ωr,s,t constitute a basis for the Riemann-Roch space L(G). In particular ℓ(G) = #Ωr,s,t . Proof Let (i, j, k) ∈ Ωr,s,t . It follows from the definition that Ei,j,k ∈ L(G), Pq−1 Pq2 −1 where G = µ=0 rµ Pµ + ν=1 sν Qν + tP∞ . From Equation (3), we have vP0 (Ei,j,k ) = i, which indicates that the valuation of Ei,j,k at the rational place P0 uniquely depends on i. Since lattice points in Ωr,s,t provide distinct values of i, the elements Ei,j,k are linearly independent of each other, with (i, j, k) ∈ Ωr,s,t . In order to indicate that they constitute a basis for L(G), the only thing is to prove that ℓ(G) = #Ωr,s,t . For the case of r0 sufficiently large, it follows from the Riemann-Roch Theorem and Lemma 1 that ℓ(G) = 1 − g + deg(G) = 1 − g + |r| + q|s| + t = #Ωr,s,t . This implies that L(G) is spanned by elements Ei,j,k with (i, j, k) in the set Ωr,s,t . For the general case, we choose r0′ > r0 large enough and set G′ := Pq−1 Pq2 −1 ′ r0 P0 + µ=1 rµ Pµ + ν=1 sν Qν + tP∞ , r ′ = (r0′ , r1 , · · · , rq−1 ). From above argument, we know that the elements Ei,j,k with (i, j, k) ∈ Ωr′ ,s,t span the whole space of L(G′ ). Remember that L(G) is a linear subspace of L(G′ ), which can be written as o n L(G) = f ∈ L(G′ ) vP0 (f ) > −r0 . Thus, we choose f ∈ L(G) and suppose that X ai Ei,j,k , f= (i,j,k)∈Ωr′ ,s,t 6 Chuangqiang Hu, Shudi Yang since f ∈ L(G′ ). The valuation of f at P0 is vP0 (f ) = minai 6=0 {i}. Then the inequality vP0 (f ) > −r0 gives that, if ai 6= 0, then i > −r0 . Equivalently, if i < −r0 , then ai = 0. From the definition of Ωr,s,t and Ωr′ ,s,t , we get that X f= ai Ei,j,k . i,j,k∈Ωr,s,t Then the theorem follows. ⊓ ⊔ We now turn to prove Lemma 1 which requires a series of results listed as follows. Definition 1 Let (a1 , · · · , ak ) be a sequence of positive integers such that the greatest common divisor is 1. Define di = gcd(a1 , · · · , ai ) and Ai = {a1 /di , · · · , ai /di } for i = 1, · · · , k. Let d0 = 0. Let Si be the semigroup generated by Ai . If ai /di ∈ Si−1 for i = 2, · · · , k, we call the sequence (a1 , · · · , ak ) telescopic. A semigroup is called telescopic if it is generated by a telescopic sequence. Lemma 2 (Lemma 6.4, [18]) If (a1 , · · · , ak ) is telescopic and M ∈ Sk , then there exist uniquely determined non-negative integers 0 6 xi < di−1 /di for i = 2, · · · , k, such that M= k X xi ai . i=1 We call this representation the normal representation of M . Lemma 3 (Lemma 6.5, [18]) For the semigroup generated by the telescopic sequence (a1 , · · · , ak ) we have lg (Sk ) = k X (di−1 /di − 1)ai , i=1 g(Sk ) = (lg (Sk ) + 1)/2, where lg (Sk ) and g(Sk ) denote the largest gap and the number of gaps of Sk , respectively. Lemma 4 Let m = (q n + 1)/(q + 1), g = (q − 1)(q n+1 + q n − q 2 )/2 for an odd integer n > 1. Let t ∈ Z. Consider the lattice point set Ψ (t) defined by n o (α, β, γ) 0 6 α < m, 0 6 β 6 q, γ > 0, q 3 α + mqβ + m(q + 1)γ 6 t , If t > 2g − 1, then Ψ (t) has cardinality #Ψ (t) = 1 − g + t. Multi-point Codes from the GGS Curves 7 Proof Let a1 = q 3 , a2 = mq, a3 = m(q + 1). It is easily verified that the sequence (a1 , a2 , a3 ) is telescopic. By Lemma 2 every element M in S3 has a unique representation M = a1 α + a2 β + a3 γ, where S3 is the semigroup generated by (a1 , a2 , a3 ). One obtains from Lemma 3 that lg (S3 ) = (q − 1)(q n + 1)(q + 1) − q 3 , 1 1 g(S3 ) = (lg (S3 ) + 1) = (q − 1)(q n+1 + q n − q 2 ) = g. 2 2 It follows that the set Ψ (t) has cardinality 1 − g + t provided that t > 2g − 1 = lg (S3 ), which finishes the proof. ⊓ ⊔ From Lemma 4, we get the number of lattice points in Ω0,0,t . Lemma 5 If t > 2g − 1, then the number of lattice points in Ω0,0,t is #Ω0,0,t = 1 − g + t. Proof Note that n Ω0,0,t := (i, j, k) i > 0,   −i for µ = 1, · · · , q − 1, jµ = m(q + 1)   −i for ν = 1, · · · , q 2 − 1, kν = m o q 3 i + m(q + 1)|j| + mq|k| 6 t . (5) Set i := α + m(β + (q + 1)γ) with 0 6 α < m, 0 6 β 6 q and γ > 0. Then Equation (5) gives that n o Ω0,0,t ∼ = (α, β, γ) 0 6 α < m, 0 6 β 6 q, γ > 0, q 3 α + mqβ + (q n + 1)γ 6 t . ∼ B means that two lattice point sets A, Here and thereafter, the notation A = B are bijective. Thus the assertion #Ω0,0,t = 1 − g + t is derived from Lemma 4. ⊓ ⊔ Lemma 6 The lattice point set Ωr,s,t as defined above is symmetric with respect to r0 , r1 , · · · , rq−1 and s1 , · · · , sq2 −1 , respectively. In other words, we
q2 −1
q−1 have #Ωr,s,t = #Ωr′ ,s′ ,t , where the sequences ri i=0 and si i=1 are equal
q2 −1
q−1 to ri′ i=0 and s′i i=1 up to permutation, respectively. Proof Recall that Ωr,s,t is defined by n Ωr,s,t := (i′ , j ′ , k′ ) i′ + r0 > 0,  ′  −i − rµ jµ′ = for µ = 1, · · · , q − 1, m(q + 1) 8 Chuangqiang Hu, Shudi Yang  −i′ − sν = m  for ν = 1, · · · , q 2 − 1, o q 3 i′ + m(q + 1)|j ′ | + mq|k′ | 6 t , kν′ ′ where j ′ = (j1′ , · · · , jq−1 ) and k′ = (k1′ , · · · , kq′ 2 −1 ). It is important to write ′ i = i + m(q + 1)l with 0 6 i < m(q + 1). Let jµ′ = jµ − l for µ > 1, kν′ = kν − (q + 1)l for ν > 1. Then n Ωr,s,t ∼ = (i, l, j, k) i + m(q + 1)l > −r0 , 0 6 i < m(q + 1),   −i − rµ jµ = for µ = 1, · · · , q − 1, m(q + 1)   −i − sν for ν = 1, · · · , q 2 − 1, kν = m o
q 3 i + m(q + 1) l + |j| + mq|k| 6 t , where j = (j1 , · · · , jq−1  ) and k = (k1 , · · · , kq2 −1 ). The first inequality in Ωr,s,t −i − r0 . So we write l = j0 + ι with ι > 0. Then gives that l > j0 := m(q + 1) n Ωr,s,t ∼ = (i, ι, j0 , j, k) 0 6 i < m(q + 1), ι > 0,   −i − rµ jµ = for µ = 0, 1, · · · , q − 1, m(q + 1)   −i − sν for ν = 1, · · · , q 2 − 1, kν = m o
q 3 i + m(q + 1) j0 + ι + |j| + mq|k| 6 t . The right hand side means that the number of the lattice points does not depend on the order of rµ , 0 6 µ 6 q − 1, and the order of sν , 1 6 ν 6 q 2 − 1, which concludes the desired assertion. ⊓ ⊔ Lemma 7 Let r = (r0 , r1 , · · · , rq−1 ) and s = (s1 , s2 , · · · , sq2 −1 ). The following equality holds: #Ωr,s,t = #Ω0,s,t + |r|, where rµ > 0 for µ > 0, sν > 0 for ν > 1, and t > 2g − 1. Proof Let us take the sets Ωr,s,t and Ωr′ ,s,t into consideration, where r ′ = (0, r1 , · · · , rq−1 ). It follows from the definition that the complement set ∆ := Ωr,s,t \Ωr′ ,s,t is given by n (i, j, k) − r0 6 i < 0,   −i − rµ jµ = for µ = 1, · · · , q − 1, m(q + 1) Multi-point Codes from the GGS Curves  −i − sν kν = m 9  for ν = 1, · · · , q 2 − 1, o q 3 i + m(q + 1)|j| + mq|k| 6 t . It is trivial that ∆ = ∅ if r0 = 0. To determine the cardinality of ∆ with r0 > 0, we denote i := α + m(β + (q + 1)γ) with α, β, γ satisfying 0 6 α < m, 0 6 β 6 q and γ 6 −1. Then jµ 6 −γ for µ > 1, kν 6 −β − (q + 1)γ for ν > 1. A straightforward computation shows q 3 i + m(q + 1)|j| + mq|k| 6 q 3 α + mqβ + m(q + 1)γ 6 q 3 (m − 1) + mq 2 − m(q + 1) = 2g − 1. So the last inequality in ∆ always holds for all t > 2g − 1, which means that the cardinality of ∆ is determined by the first inequality, that is #∆ = r0 . Then we must have #Ωr,s,t = #Ωr′ ,s,t + r0 , whenever r0 > 0. Repeating the above routine and using Lemma 6, we get #Ωr,s,t = #Ω0,s,t + |r|, where r = (r0 , r1 , · · · , rq−1 ). ⊓ ⊔ Lemma 8 Let s = (s1 , s2 , · · · , sq2 −1 ). The following identity holds: #Ω0,s,t = #Ω0,0,t + q|s|, where si > 0 for i = 1, 2, · · · , q 2 − 1 and t > 2g − 1. Proof For convenience, let us denote r := (s0 , s0 , · · · , s0 ) to be the q-tuple with all entries equal s0 , where s0 > 0, and write Ωr,s,t as Γs0 ,(s1 ,··· ,sq2 −1 ),t . To get the desired conclusion, we first claim that #Γs0 ,(s1 ,··· ,sq2 −1 ),t = #Γs′0 ,(s′1 ,··· ,s′ 2 q −1 ),t , (6)
q2 −1
q2 −1 where the sequence si i=0 is equal to s′i i=0 up to permutation. Note that Γs0 ,(s1 ,··· ,sq2 −1 ),t is equivalent to n (i′ , j ′ , k1′ , · · · , kq′ 2 −1 ) i′ + s0 > 0, 0 6 i′ + m(q + 1)j ′ + s0 < m(q + 1), 0 6 i′ + mkν′ + sν < m for ν = 1, · · · , q 2 − 1, o q 3 i′ + m(q + 1)(q − 1)j ′ + mq|k′ | 6 t , Pq2 −1 where |k′ | = ν=1 kν′ . By setting i′ := i + mκ where 0 6 i < m and kν′ := kν − κ for ν > 1, we obtain n Γs0 ,(s1 ,··· ,sq2 −1 ),t ∼ = (i, κ, j ′ , k1 , · · · , kq2 −1 ) 0 6 i < m, i + mκ + s0 > 0, 10 Chuangqiang Hu, Shudi Yang 0 6 i + mκ + m(q + 1)j ′ + s0 < m(q + 1), 0 6 i + mkν + sν < m for ν = 1, · · · , q 2 − 1, o q 3 i + m(q + 1)(q − 1)j ′ + mq(κ + |k|) 6 t .  −i − s0 . Put κ := k0 + ε, where ε := −(q + 1)j0 + η, 0 6 η < q + 1 and k0 := m One gets that 0 6 i + mk0 + s0 < m, which leads to 0 6 i + mκ + m(q + 1)j0 − mη+s0 < m. So the inequality 0 6 i+mκ+m(q+1)j0 +s0 < m+mη 6 m(q+1) holds because ε = −(q + 1)j0 + η. Thus we must have j0 = j ′ 6 0. Therefore n Γs0 ,(s1 ,··· ,sq2 −1 ),t ∼ = (i, j0 , η, k0 , k1 , · · · , kq2 −1 ) 0 6 i < m, j0 6 0,  0 6 η < q + 1,   −i − sν kν = for ν = 0, 1, · · · , q 2 − 1, m o q 3 i − m(q + 1)j0 + mq(k0 + η + |k|) 6 t . The right hand side means that the lattice points do not depend on the order of sν with 0 6 ν 6 q 2 − 1, by observing that kν is determined by sν . In other words, we have shown that the number of lattice points in Γs0 ,(s1 ,··· ,sq2 −1 ),t does not depend on the order of sν with 0 6 ν 6 q 2 − 1, concluding the claim we presented by (6). So it follows from (6) and Lemma 7 that #Γ0,(s1 ,s2 ,··· ,sq2 −1 ),t = #Γs1 ,(0,s2 ,··· ,sq2 −1 ),t = #Γ0,(0,s2 ,··· ,sq2 −1 ),t + qs1 . By repeatedly using Lemma 7, we get #Γ0,(s1 ,s2 ,··· ,sq2 −1 ),t = #Γ0,(0,0,··· ,0),t + q(s1 + s2 + · · · + sq2 −1 ), concluding the desired formula #Ω0,s,t = #Ω0,0,t + q|s|. ⊓ ⊔ With the above preparations, we are now in a position to give the proof of Lemma 1. o n Proof [Proof of Lemma 1] By taking w := min 06µ6q−1 rµ , sν , we obtain 16ν6q2 −1 from the definition that Ωr,s,t is equivalent to Ωr′ ,s′ ,t′ , where r ′ = (r0 − w, · · · , rq−1 − w), s′ = (s1 − w, · · · , sq2 −1 − w) and t′ = t + q 3 w. Hence #Ωr,s,t = #Ωr′ ,s′ ,t′ . On the other hand, by observing that rµ − w > 0, sν − w > 0 and t′ > 2g − 1, we establish from Lemmas 5, 7 and 8 that #Ωr′ ,s′ ,t′ = #Ω0,s′ ,t′ + |r ′ | = #Ω0,0,t′ + q|s′ | + |r ′ | = 1 − g + t′ + q|s′ | + |r ′ | = 1 − g + t + q|s| + |r|. Multi-point Codes from the GGS Curves 11 It then follows that #Ωr,s,t = 1 − g + t + q|s| + |r|, completing the proof of Lemma 1. ⊓ ⊔ We finish this section with a result that allows us to give a new form of the Pq−1 P 2 −1 basis for our Riemann-Roch space L(G) with G = µ=0 rµ Pµ + qν=1 sν Q ν + tP∞ . Denote λ := (λ1 , · · · , λq−1 ) and γ := (γ1 , · · · , γq2 −1 ). For (u, λ, γ) ∈ 2 Zq +q−1 , we define 2 qY −1 q−1 Y u λµ hγνν , Λu,λ,γ := τ fµ µ=1 µ=1 n−3 zq x − αµ y − βν , fµ := for µ > 1, and hν := for ν > 1. The x − α0 x − α0 y − β0 divisor of Λu,λ,γ is computed from Proposition 1 that where τ := div(Λu,λ,γ ) = q−1  X µ=1 q n−3 2 qX −1  u + m(q + 1)λµ − m|γ| Pµ + (q n−3 u + mγν )Qν ν=1   − (m(q + 1) − q n−3 )u + m(q + 1)|λ| + m|γ| P0 + uP∞ . There is a close relationship between the elements Λu,λ,γ and Ei,j,k explored as follows. Pq−1 Pq2 −1 Corollary 1 Let G := µ=0 rµ Pµ + ν=1 sν Qν + tP∞ . Then the elements Λu,λ,γ with (u, λ, γ) ∈ Θr,s,t form a basis for the Riemann-Roch space L(G), where the set Θr,s,t is given by n (u, λ, γ) u > −t, 0 6 q n−3 u + mγν + sν < m for ν = 1, · · · , q 2 − 1 0 6 q n−3 u + m(q + 1)λµ − m|γ| + rµ < m(q + 1) for µ = 1, · · · , q − 1, o (7) (m(q + 1) − q n−3 )u + m(q + 1)|λ| + m|γ| 6 r0 . In addition we have #Θr,s,t = #Ωr,s,t . Proof It suffices to prove that the set o n Λu,λ,γ (u, λ, γ) ∈ Θr,s,t equals the set o n Ei,j,k (i, j, k) ∈ Ωr,s,t . In fact, for fixed (u, λ, γ) ∈ Zq 2 +q−1 i = −(m(q + 1) − q , we obtain Λu,λ,γ equals Ei,j,k with n−3 )u − m(q + 1)|λ| − m|γ|, 12 Chuangqiang Hu, Shudi Yang jµ = u + |λ| + λµ for µ = 1, · · · , q − 1, kν = (q + 1)(u + |λ|) + |γ| + γν for ν = 1, · · · , q 2 − 1. On the contrary, if we set u = −q 3 i − m(q + 1)|j| − mq|k|, λµ = q 2 i + q n−1 |j| + m|k| + jµ for µ = 1, · · · , q − 1, γν = (q + 1)(i + q n−3 |j|) + q n−2 |k| + kν for ν = 1, · · · , q 2 − 1, then Ei,j,k is exactly the element Λu,λ,γ . Therefore, if we restrict (i, j, k) in Ωr,s,t , then we must have (u, λ, γ) is in Θr,s,t and vice versa. This completes the proof of the claim and hence of this corollary. ⊓ ⊔ In the following, we will demonstrate an interesting property of #Ωr,s,t for GK curves with a specific vector s. Corollary 2 Let n = 3 and the vectors r, s be given by r := (r0 , r1 , · · · , rq−1 ), s := (s′1 , s′2 , · · · , s′q2 −1 ) = (s1 , s1 , · · · , s1 , s2 , s2 , · · · , s2 , · · · , sq−1 , sq−1 , · · · , sq−1 ). {z } | {z } | {z } | q+1 q+1 q+1 Then the lattice point set Ωr,s,t is symmetric with respect to r0 , r1 , · · · , r
q−1 , t. q In other words, we have #Ωr,s,t = #Ωr′ ,s,t′ , where the sequence ri i=0 is
q equal to ri′ i=0 up to permutation by putting rq := t and rq′ := t′ . Proof Denote r := (r0 , ṙ) = (r0 , r1 , · · · , rq−1 ) and Ω(r0 ,ṙ),s,t := Ωr,s,t . By Lemma 6, it suffices to prove that #Ω(r0 ,ṙ),s,t = #Θ(r0 ,ṙ),s,t = #Ω(t,ṙ),s,r0 . The first identity follows directly from Corollary 1. Applying Corollary 1 again gives the set Θ(r0 ,ṙ),s,t as n (u, λ, γ) u + t > 0, 0 6 u + mγν + s′ν < m for ν = 1, · · · , q 2 − 1, 0 6 u + m(q + 1)λµ − m|γ| + rµ < m(q + 1) for µ = 1, · · · , q − 1, o q 3 u + m(q + 1)|λ| + m|γ| 6 r0 . From our assumption, it is obvious that |γ| is divisible by q + 1. So if we take |γ| i := u, jµ := λµ − for µ > 1 and kν := γν for ν > 1, then Θ(r0 ,ṙ),s,t is q+1 equivalent to n (i, j, k) i + t > 0, Multi-point Codes from the GGS Curves 13 0 6 i + mkν + s′ν < m for ν = 1, · · · , q 2 − 1, 0 6 i + m(q + 1)jµ + rµ < m(q + 1) for µ = 1, · · · , q − 1, o q 3 i + m(q + 1)|j| + m|k| 6 r0 . The last set is exactly Ω(t,ṙ),s,r0 by definition. Hence the second identity is just shown, completing the whole proof. ⊓ ⊔ 3 The AG codes from GGS curves This section settles the properties of AG codes from GGS curves. Generally speaking, there are two classical ways of constructing AG codes associated with divisors D and G, where G is a divisor of arbitrary function field F and D := Q1 + · · · + QN is another divisor of F such that Q1 , · · · , QN are pairwise distinct rational places, each not belonging to the support of G. One construction is based on the Riemann-Roch space L(G), n o CL (D, G) := (f (Q1 ), · · · , f (QN )) f ∈ L(G) ⊆ FN q . The other one depends on the space of differentials Ω(G − D), n o CΩ (D, G) := (resQ1 (η), · · · , resQN (η)) η ∈ Ω(G − D) . It is well-known the codes CL (D, G) and CΩ (D, G) are dual to each other. Further CΩ (D, G) has parameters [N, kΩ , dΩ ] with kΩ = N − k and dΩ > deg(G) − (2g − 2), where k = ℓ(G) − ℓ(G − D) is the dimension of CL (D, G). If moreover 2g − 2 < deg(G) < N then kΩ = N + g − 1 − deg(G). We refer the reader to [26] for more information. P In this section, we will study the linear code CL (D, G) with D := α,β,γ Pα,β,γ γ6=0 Pq2 −1 Pq−1 and G := µ=0 rµ Pµ + ν=1 sν Qν + tP∞ . The length of CL (D, G) is N := deg(D) = q n+2 (q n − q + 1) − q 3 . It is well known that the dimension of CL (D, G) is given by dim CL (D, G) = ℓ(G) − ℓ(G − D). (8) Set R := N + 2g − 2. Since deg(G) > R, we deduce from the Riemann-Roch Theorem and (8) that dim CL (D, G) = (1 − g + deg(G)) − (1 − g + deg(G − D)) = deg(G) − deg(G − D) = N, which implies that CL (D, G) is trivial. So we only consider the case 0 6 deg(G) 6 R. Now, we use the following lemmas to calculate the dual of CL (D, G). 14 Chuangqiang Hu, Shudi Yang Lemma 9 (Proposition 2.2.10, [27]) Let τ be an element of the function field of the curve X such that vPi (τ ) = 1 for all rational places Pi contained in the divisor D. Then the dual of CL (D, G) is CL (D, G)⊥ = CL (D, D − G + div(dτ ) − div(τ )). Lemma 10 (Proposition 2.2.14, [27]) Suppose that G1 and G2 are divisors with G1 = G2 + div(ρ) for some ρ ∈ F \{0} and supp G1 ∩ supp D = supp G2 ∩ supp D = ∅. Let N := deg(D) and ̺ := (ρ(P1 ), · · · , ρ(PN )) with Pi ∈ D. Then the codes CL (D, G1 ) and CL (D, G2 ) are equivalent and CL (D, G2 ) = ̺ · CL (D, G1 ). Theorem 2 Let A := (q n +1)(q−1)−1, B := mq 2 (q n −q 3 )+(q n +1)(q 2 −1)−1 Pi P n−3 n 2j 2 and ρ := 1 + i=1 z (q +1)(q−1) j=1 q . Then the dual code of CL (D, G) is given as follows. (1) The dual of CL (D, G) is represented as ⊥ CL (D, G) = ̺ · CL (D, q−1 X (A − rµ )Pµ + µ=0 2 qX −1 (A − sν )Qν + (B − t)P∞ ), ν=1 where ̺ := (ρ(Pα1 ,β1 ,γ1 ), · · · , ρ(PαN ,βN ,γN )) with Pαi ,βi ,γi ∈ D. (2) In particular, for n = 3, we have ρ = 1 and q−1 X ⊥ CL (D, G) = CL (D, (A − rµ )Pµ + µ=0 2 qX −1 (A − sν )Qν + (B − t)P∞ ). ν=1 Proof Define o n 2 H := z ∈ F∗q2n ∃y ∈ Fq2n with y q − y = z m . Consider the element τ := Y (z − γ). γ∈H Then τ is a prime element for all places Pα,β,γ in D and its divisor is X div(τ ) = div(z − γ) = D − deg(D)P∞ , γ∈H where D = P α,β,γ γ6=0 Pα,β,γ and N = deg(D) = q n+2 (q n − q + 1) − q 3 . Moreover by a same discussion as in the proof of Lemma 2 in [1], we have τ =1+ k−1 X i=0 w Pi j=0 q2j + Pk−1 j=0 q2j+1 + k−1 X i=0 w Pi j=0 q2j+1 , Multi-point Codes from the GGS Curves 15 where n = 2k + 1 (note that n > 1 is odd) and w = z (q straightforward computation shows dτ = w Pk−1 j=0 q2j+1 n +1)(q−1) . Then a k−1  X Pi 2j  w j=1 q dw 1+ i=1 =w qn −q q2 −1 dw = −z Let ρ := 1 + Pk−1 i=1 w Pi j=1 k−1  X Pi 2j  w j=1 q dw, 1+ i=1 (qn +1)(q−1)−1 q2j dz. and denote its divisor by div(ρ). Set A := m(q n − q) + (q n + 1)(q − 1) − 1, S := q 3 A − 2g + 2. Since div(dz) = (2g−2)P∞ (see Lemma 3.8 of [13]), it follows from Proposition 1 that div(dτ ) = A · div(z) + div(dz) + div(ρ)   X =A Qβ − q 3 A − 2g + 2 P∞ + div(ρ) β∈Fq2 =A q−1 X Pµ + A 2 qX −1 Qν − SP∞ + div(ρ). ν=1 µ=0 Let η := dτ /τ be a Weil differential. The divisor of η is div(η) = div(dτ ) − div(τ ) =A q−1 X Pµ + A µ=0 2 qX −1 Qν − D + ν=1   deg(D) − S P∞ + div(ρ). By writing B := deg(D) − S = mq 2 (q n − q 3 ) + (q n + 1)(q 2 − 1) − 1, we establish from Lemma 9 that the dual of CL (D, G) is CL (D, G)⊥ = CL (D, D − G + div(η)) = CL (D, q−1 X (A − rµ )Pµ + µ=0 2 qX −1 (A − sν )Qν + (B − t)P∞ + div(ρ)). ν=1 Denote ̺ := (ρ(Pα1 ,β1 ,γ1 ), · · · , ρ(PαN ,βN ,γN )) with Pαi ,βi ,γi ∈ D. Then we deduce the first statement from Lemma 10. The second statement then follows immediately. ⊓ ⊔ 16 Chuangqiang Hu, Shudi Yang Theorem 3 Suppose that 0 6 deg(G) 6 R. Then the dimension of CL (D, G) is given by ( #Ωr,s,t if 0 6 deg(G) < N, dim CL (D, G) = ⊥ N − #Ωr,s,t if N 6 deg(G) 6 R, ⊥ where Ωr,s,t := Ωr′ ,s′ ,B−t with r ′ = (A − r0 , · · · , A − rq−1 ) and s′ = (A − s1 , · · · , A − sq2 −1 ). Proof For 0 6 deg(G) < N , we have by Theorem 1 and Equation (8) that dim CL (D, G) = ℓ(G) = #Ωr,s,t . For N 6 deg(G) 6 R, Theorem 2 yields that ⊥ dim CL (D, G) = N − dim CL (D, G)⊥ = N − #Ωr,s,t . So the proof is completed. ⊓ ⊔ We mention that the minimum distance of CL (D, G) follows from Goppa bounds and Theorem 3. Techniques for improving the Goppa bounds will be dealt with in the next two sections. 4 Weierstrass semigroups and pure gaps In this section, we will characterize the Weierstrass semigroups and pure gaps over GGS curves, which will enables us to obtain improved bounds on the parameters of AG codes. We first briefly introduce some corresponding definitions and notations [23]. For an arbitrary function field F , let Q1 , · · · , Ql be distinct rational places of F , then the Weierstrass semigroup H(Q1 , · · · , Ql ) is defined by n (s1 , · · · , sl ) ∈ Nl0 ∃f ∈ F with (f )∞ = l X i=1 o si Q i , and the Weierstrass gap set G(Q1 , · · · , Ql ) is defined by Nl0 \H(Q1 , · · · , Ql ), where N0 := N ∪ {0} denotes the set of nonnegative integers. Homma and Kim [15] introduced the concept of pure gap set with respect to a pair of rational places. This was generalized by Carvalho and Torres [4] to several rational places, denoted by G0 (Q1 , · · · , Ql ), which is given by l o n X si Q i . (s1 , · · · , sl ) ∈ Nl ℓ(G) = ℓ(G − Qj ) for 1 6 j 6 l, where G = i=1 In addition, they showed that (s1 , · · · , sl ) is a pure gap at (Q1 , · · · , Ql ) if and only if ℓ(s1 Q1 + · · · + sl Ql ) = ℓ((s1 − 1)Q1 + · · · + (sl − 1)Ql ). A useful way to calculate the Weierstrass semigroups is given as follows, which can be regarded as an easy generalization of a result due to Kim [17] Multi-point Codes from the GGS Curves 17 Lemma 11 For rational places Q1 , · · · , Ql with 1 6 l 6 r, the set H(Q1 , · · · , Ql ) is given by l o n X l si Q i . (s1 , · · · , sl ) ∈ N0 ℓ(G) 6= ℓ(G − Qj ) for 1 6 j 6 l, where G = i=1 In the rest of this section, we will restrict our study to the divisor G := Pq−1 µ=0 rµ Pµ +tP∞ and denote r = (r0 , r1 , · · · , rq−1 ). Our main task is to determine the Weierstrass semigroups and the pure gaps at totally ramified places P0 , P1 , · · · , Pq−1 , P∞ . Before we proceed, some auxiliary results are presented in the following. Denote Ωr,0,t by Ω(r0 ,r1 ,··· ,rq−1 ),t for the clarity of description. Lemma 12 For the lattice point set Ω(r0 ,r1 ,··· ,rq−1 ),t , we have the following assertions. (1) #Ω(r0 ,r1 ,··· ,rq−1 ),t = #Ω(r0 −1,r1 ,··· ,rq−1 ),t + 1 if and only if  q−1  lr m X t + q 3 r0 r0 − rµ 0 + q(q − 1) 6 . m(q + 1) m m(q + 1) µ=1 (2) #Ω(r0 ,r1 ,··· ,rq−1 ),t = #Ω(r0 ,r1 ,··· ,rq−1 ),t−1 + 1 if and only if  q−1  n−3 X q t − rµ µ=1 m(q + 1)  n−3  r0 − q n−3 t q t + q(q − 1) 6 t+ . m m(q + 1) Proof Consider two lattice point sets Ω(r0 ,r1 ,··· ,rq−1 ),t and Ω(r0 −1,r1 ,··· ,rq−1 ),t , which are given in Equation (4). Clearly, the latter one is a subset of the former one, and the complement set Φ of Ω(r0 −1,r1 ,··· ,rq−1 ),t in Ω(r0 ,r1 ,··· ,rq−1 ),t is given by n Φ := (i, j, k) i + r0 = 0,   −i − rµ for µ = 1, · · · , q − 1, jµ = m(q + 1)   −i kν = for ν = 1, · · · , q 2 − 1, m o q 3 i + m(q + 1)|j| + mq|k| 6 t , where j = (j1 , j2 , · · · , jq−1 ) and k = (k1 , k2 , · · · , kq2 −1 ). It follows immediately that the set Φ is not empty if and only if −q 3 r0 + m(q + 1)  q−1  lr m X r0 − rµ 0 + mq(q 2 − 1) 6 t, m(q + 1) m µ=1 which concludes the first assertion. For the second assertion, we obtain from Corollary 1 that the difference between #Ω(r0 ,r1 ,··· ,rq−1 ),t and #Ω(r0 ,r1 ,··· ,rq−1 ),t−1 is exactly the same as the 18 Chuangqiang Hu, Shudi Yang one between #Θr,0,t and #Θr,0,t−1 . In an analogous way, we define Ψ as the complementary set of Θr,0,t−1 in Θr,0,t , namely n Ψ := (u, λ, γ) u = −t, 0 6 q n−3 u + mγν < m for ν = 1, · · · , q 2 − 1, 0 6 q n−3 u + m(q + 1)λµ − m|γ| + rµ < m(q + 1) for µ = 1, · · · , q − 1,   o − (m(q + 1) − q n−3 )u + m(q + 1)|λ| + m|γ| + r0 > 0 . The set Ψ is not empty if and only if m(q + 1)  q−1  n−3 X q t − rµ µ=1 m(q + 1)  n−3  q t + mq(q 2 − 1) 6 r0 + (q n + 1 − q n−3 )t, m completing the proof of the second assertion. ⊓ ⊔ We are now ready for the main results of this section dealing with Weierstrass semigroups and pure gap sets, which play an interesting role in finding codes with good parameters. For simplicity, we define    l  X rj − ri rj Wj (rl , t, l) := + (q − 1 − l) m(q + 1) m(q + 1) i=0 i6=j lr m t + q 3 rj j , − + q(q − 1) m m(q + 1) for j = 1, · · · , l, and  l  n−3 X q t − ri  n−3  q t + (q − 1 − l) W∞ (rl , t, l) := m(q + 1) m(q + 1) i=1  n−3  q t r0 − q n−3 t + q(q − 1) , −t− m m(q + 1) where rl = (r0 , r1 , · · · , rl ). Theorem 4 Let P0 , P1 , · · · , Pl be rational places as defined previously. For 0 6 l < q, the following assertions hold. (1) The Weierstrass semigroup H(P0 , P1 , · · · , Pl ) is given by o n Wj (rl , 0, l) 6 0 for 0 6 j 6 l . (r0 , r1 , · · · , rl ) ∈ Nl+1 0 (2) The Weierstrass semigroup H(P0 , P1 , · · · , Pl , P∞ ) is given by o n W (r , t, l) 6 0 for 0 6 j 6 l and j = ∞ . (r0 , r1 , · · · , rl , t) ∈ Nl+2 j l 0 Multi-point Codes from the GGS Curves 19 (3) The pure gap set G0 (P0 , P1 , · · · , Pl ) is given by n o (r0 , r1 , · · · , rl ) ∈ Nl+1 Wj (rl , 0, l) > 0 for 0 6 j 6 l . (4) The pure gap set G0 (P0 , P1 , · · · , Pl , P∞ ) is given by n o (r0 , r1 , · · · , rl , t) ∈ Nl+2 Wj (rl , t, l) > 0 for 0 6 j 6 l and j = ∞ . Proof The desired conclusions follow from Theorem 1, Lemmas 6, 11 and 12. ⊔ ⊓ The following corollary states the characterizations of Weierstrass semigroup and gaps at only one point. Corollary 3 With notation as before, we have the following statements.     n k q3 k o k (1) H(P0 ) = k ∈ N0 (q − 1) + q(q − 1) 6 . m(q + 1) m m(q + 1) (2) Let α, β, γ ∈ Z. Then α + m(β + (q + 1)γ) ∈ N is a gap at P0 if and only if exactly one of the following two conditions is satisfied: (i) α = 0, 0 < β 6 q − 1, 0 6 γ 6 q − 1 − β.   q3 α β and − (ii) 0 < α < m, 0 6 β 6 q, 0 6 γ 6 q 2 − 1 − β + q + 1 m(q + 1)   β q3 α β− 6 q 2 − 1. − q + 1 m(q + 1) Proof The first statement is an immediate consequence of Theorem 4 (1). We now focus on the second statement. It follows from Theorem 4 (3) that the Weierstrass gap set at P0 is     n k q3 k o k G(P0 ) = k ∈ N (q − 1) + q(q − 1) > . m(q + 1) m m(q + 1) Let k ∈ G(P0 ) and write k = α + m(β + (q + 1)γ), where 0 6 α < m, 0 6 β 6 q and γ > 0. We find that the case α + mβ = 0 does not occur, since otherwise we have k = m(q + 1)γ and     k q3 k k (q − 1) + q(q − 1) − m(q + 1) m m(q + 1) = (q − 1)γ + q(q − 1)(q + 1)γ − q 3 γ = −γ 6 0, which contradicts to the fact k ∈ G(P0 ). So α + mβ 6= 0. There are two possibilities. (i) If α = 0, then 0 < β 6 q. In this case, k = mβ + m(q + 1)γ is a gap at P0 if and only if     k q3 k k (q − 1) + q(q − 1) > , m(q + 1) m m(q + 1) 20 Chuangqiang Hu, Shudi Yang or equivalently, (q − 1)(γ + 1) + q(q − 1)(β + (q + 1)γ) > q 3 γ + q3 β , q+1 leading to the first condition 0 6 γ 6 q − 1 − β and 0 < β 6 q − 1. (ii) If 0 < α < m, then 0 6 β 6 q. In this case, we have similarly that k = α + mβ + m(q + 1)γ is a gap at P0 if and only if q3 α q3 β + , m(q + 1) q + 1   q3 α β . which gives the second condition 0 6 γ 6 q 2 − 1 − β + − q + 1 m(q + 1) Note that   β q3 α q2 − 1 − β + > q 2 − 1 − q − 1 > 0. − q + 1 m(q + 1) (q − 1)(γ + 1) + q(q − 1)(1 + β + (q + 1)γ) > q 3 γ + The proof is finished. ⊓ ⊔ 5 The floor of divisors In this section, we investigate the floor of divisors from GGS curves. The significance of this concept is that it provides a useful tool for evaluating parameters of AG codes. We begin with general function fields. Definition 2 ([22]) Given a divisor G of a function field F/Fq with ℓ(G) > 0, the floor of G is the unique divisor G′ of F of minimum degree such that L(G) = L(G′ ). The floor of G will be denoted by ⌊G⌋. The floor of a divisor can be used to characterize Weierstrass semigroups and pure gap sets. Let G = s1 Q1 + · · · + sl Ql . It is not hard to see that (s1 , · · · , sl ) ∈ H(Q1 , · · · , Ql ) if and only if ⌊G⌋ = G. Moreover, (s1 , · · · , sl ) is a pure gap at (Q1 , · · · , Ql ) if and only if ⌊G⌋ = ⌊(s1 − 1)Q1 + · · · + (sl − 1)Ql ⌋. Maharaj, Matthews and Pirsic in [22] defined the floor of a divisor and characterized it by the basis of the Riemann-Roch space. Theorem 5 ([22]) Let G be a divisor of a function field F/Fq and let b1 , · · · , bt ∈ L(G) be a spanning set for L(G). Then n o ⌊G⌋ = − gcd div(bi ) i = 1, · · · , t . The next theorem extends Theorem 3.4 of [4] by determining the lower bound of minimum distance in a more general situation. Multi-point Codes from the GGS Curves 21 Theorem 6 ([22]) Assume that F/Fq is a function field with genus g. Let D := Q1 + · · · + QN where Q1 , · · · , QN are distinct rational places of F , and let G := H + ⌊H⌋ be a divisor of F such that H is an effective divisor whose support does not contain any of the places Q1 , · · · , QN . Then the minimum distance of CΩ (D, G) satisfies dΩ > 2 deg(H) − (2g − 2). The following theorem provides a characterization of the floor over GGS curves, which can be viewed as a generalization of Theorem 3.9 in [22] related to Hermitian function fields. Pq−1 P 2 −1 Theorem 7 Let H := µ=0 rµ Pµ + qν=1 sν Qν + tP∞ be a divisor of GGS curve given by (1). Then the floor of H is given by ⌊H⌋ = q−1 X rµ′ Pµ + µ=0 2 qX −1 s′ν Qν + t′ P∞ , ν=1 where r0′ = max n o − i (i, j, k) ∈ Ωr,s,t , o − i − m(q + 1)jµ (i, j, k) ∈ Ωr,s,t for µ = 1, · · · , q − 1, o n s′ν = max − i − mkν (i, j, k) ∈ Ωr,s,t for ν = 1, · · · , q 2 − 1, o n t′ = max q 3 i + m(q + 1)|j| + mq|k| (i, j, k) ∈ Ωr,s,t . rµ′ = max n Pq−1 Pq2 −1 Proof Let H = µ=0 rµ Pµ + ν=1 sν Qν + tP∞ . It follows from Theorem 1 that the elements Ei,j,k of Equation (2) with (i, j, k) ∈ Ωr,s,t form a basis for the Riemann-Roch space L(H). Note that the divisor of Ei,j,k is iP0 + q−1 X (i + m(q + 1)jµ )Pµ + µ=1 2 qX −1 (i + mkν )Qν ν=1
− q 3 i + m(q + 1)|j| + mq|k| P∞ By Theorem 5, we get that o n ⌊H⌋ = − gcd div(Ei,j,k ) (i, j, k) ∈ Ωr,s,t . The desired conclusion then follows. ⊓ ⊔ 22 Chuangqiang Hu, Shudi Yang 6 Examples of codes on GGS curves In this section we treat several examples of codes to illustrate our results. The codes in the next example will give new records of better parameters than the corresponding ones in MinT’s tables [25]. Example 1 Now, we study codes arising from GK curves, that is, we let q = 2 and n = 3 in (1) given at the beginning of Section 2. This curve has 225 F64 -rational points and its genus is g = 10. Here we will study multi-point AG codes from this curve by employing our previous results. Let us take H = 3P0 + 4P1 + 11P∞ for example. Then it can be computed from Equation (4) that the elements (−i, −i − m(q + 1)j1 , −i − mk1 , −i − mk2 , −i − mk3 , q 3 i + m(q + 1)j1 + mq(k1 + k2 + k3 )), with (i, j1 , k1 , k2 , k3 ) ∈ Ω3,4,0,0,0,11 , are as follows ( 3, 3, 0, 0, 0, −6 ), ( 2, 2, −1, −1, −1, 2 ), ( 1, 1, −2, −2, −2, 10 ), ( 0, 0, 0, 0, 0, 0 ), (−1, −1, −1, −1, −1, 8 ), (−3, −3, 0, 0, 0, 6 ), (−6, 3, 0, 0, 0, 3 ), (−7, 2, −1, −1, −1, 11 ), (−9, 0, 0, 0, 0, 9 ). So we obtain from Theorem 7 that ⌊H⌋ = 3P0 + 3P1 + 11P∞ . Let D be a divisor consisting of N = 216 rational places away from the places P0 , P1 , Q1 , Q2 , Q3 and P∞ . According to Theorem 6, if we let G = H + ⌊H⌋ = 6P0 + 7P1 + 22P∞ , then the code CΩ (D, G) has minimum distance at least 2 deg(H) − (2g − 2) = 18. Since 2g − 2 < deg(G) < N , the dimension of CΩ (D, G) is kΩ = N +g−1−deg(G) = 190. In other words, the code CΩ (D, G) has parameters [216, 190, > 18]. One can verify that our resulting code improve the minimum distance with respect to MinT’s Tables. Moreover CΩ (D, G) is equivalent to CL (D, G′ ), where G′ = 2P0 + P1 + 8Q1 + 8Q2 + 8Q3 + 4P∞ , and its generating matrix can be determined by Theorem 2. Additionally, we remark that more AG codes with excellent parameters can be found by taking H = aP0 + bP1 + 7P∞ , where a, b ∈ {4, 5, 6} and 9 6 a+ b 6 12. The floor of such H is computed to be ⌊H⌋ = aP0 + bP1 + 6P∞ . Let D be as before. If we take G = H +⌊H⌋ = 2aP0 +2bP1 +13P∞ , then we can produce AG codes CΩ (D, G) with parameters [216, 212−2a−2b, > 2a+2b−4]. All of these codes improve the records of the corresponding ones found on MinT’s Tables. Example 2 Consider the curve GGS(q, n) of (1) with q = 2 and n = 5. This curve has 3969 F210 -rational points and its genus is g = 46. It follows from Multi-point Codes from the GGS Curves 23 Theorem 4 that n o (57, j, 3) 1 6 j 6 3 ⊆ G0 (P0 , P1 , P∞ ). Let D be a divisor consisting of N = 3960 rational places except P0 , P1 , Q1 , Q2 , Q3 and P∞ . Applying Theorem 3.4 of [4] (see also Theorem 1, [16]), if we take G = 113P0 + 3P1 + 5P∞ , then the three-point code CΩ (D, G) has length N = 3960, dimension N + g − 1 − deg(G) = 3884 and minimum distance at least 36. Thus we produce an AG code with parameters [3960, 3884, > 36]. Unfortunately, this F210 -code cannot be compared with the one on MinT’s Tables because the alphabet size given is at most 256. Acknowledgements This work is partially supported by China Postdoctoral Science Foundation Funded Project (Project No. 2017M611801). This work is also partially supported by Guangdong Natural Science Foundation (Grant No. 2014A030313161) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04). References 1. Abdón, M., Bezerra, J., Quoos, L.: Further examples of maximal curves. Journal of Pure and Applied Algebra 213, 1192–1196 (2009) 2. Bartoli, D., Montanucci, M., Zini, G.: AG codes and AG quantum codes from the GGS curve. arXiv:1703.03178 (2017) 3. 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Overcoming Exploration in Reinforcement Learning with Demonstrations arXiv:1709.10089v2 [cs.LG] 25 Feb 2018 Ashvin Nair12 , Bob McGrew1 , Marcin Andrychowicz1 , Wojciech Zaremba1 , Pieter Abbeel12 Abstract— Exploration in environments with sparse rewards has been a persistent problem in reinforcement learning (RL). Many tasks are natural to specify with a sparse reward, and manually shaping a reward function can result in suboptimal performance. However, finding a non-zero reward is exponentially more difficult with increasing task horizon or action dimensionality. This puts many real-world tasks out of practical reach of RL methods. In this work, we use demonstrations to overcome the exploration problem and successfully learn to perform long-horizon, multi-step robotics tasks with continuous control such as stacking blocks with a robot arm. Our method, which builds on top of Deep Deterministic Policy Gradients and Hindsight Experience Replay, provides an order of magnitude of speedup over RL on simulated robotics tasks. It is simple to implement and makes only the additional assumption that we can collect a small set of demonstrations. Furthermore, our method is able to solve tasks not solvable by either RL or behavior cloning alone, and often ends up outperforming the demonstrator policy. I. I NTRODUCTION RL has found significant success in decision making for solving games, so what makes it more challenging to apply in robotics? A key difference is the difficulty of exploration, which comes from the choice of reward function and complicated environment dynamics. In games, the reward function is usually given and can be directly optimized. In robotics, we often desire behavior to achieve some binary objective (e.g., move an object to a desired location or achieve a certain state of the system) which naturally induces a sparse reward. Sparse reward functions are easier to specify and recent work suggests that learning with a sparse reward results in learned policies that perform the desired objective instead of getting stuck in local optima [1], [2]. However, exploration in an environment with sparse reward is difficult since with random exploration, the agent rarely sees a reward signal. The difficulty posed by a sparse reward is exacerbated by the complicated environment dynamics in robotics. For example, system dynamics around contacts are difficult to model and induce a sensitivity in the system to small errors. Many robotics tasks also require executing multiple steps successfully over a long horizon, involve high dimensional control, and require generalization to varying task instances. These conditions further result in a situation where the agent so rarely sees a reward initially that it is not able to learn at all. All of the above means that random exploration is not a tenable solution. Instead, in this work we show that we can use demonstrations as a guide for our exploration. To test our 1 OpenAI, 2 University of California, Berkeley. method, we solve the problem of stacking several blocks at a given location from a random initial state. Stacking blocks has been studied before in the literature [3], [4] and exhibits many of the difficulties mentioned: long horizons, contacts, and requires generalizing to each instance of the task. We limit ourselves to 100 human demonstrations collected via teleoperation in virtual reality. Using these demonstrations, we are able to solve a complex robotics task in simulation that is beyond the capability of both reinforcement learning and imitation learning. The primary contribution of this paper is to show that demonstrations can be used with reinforcement learning to solve complex tasks where exploration is difficult. We introduce a simple auxiliary objective on demonstrations, a method of annealing away the effect of the demonstrations when the learned policy is better than the demonstrations, and a method of resetting from demonstration states that significantly improves and speeds up training policies. By effectively incorporating demonstrations into RL, we shortcircuit the random exploration phase of RL and reach nonzero rewards and a reasonable policy early on in training. Finally, we extensively evaluate our method against other commonly used methods, such as initialization with learning from demonstrations and fine-tuning with RL, and show that our method significantly outperforms them. II. R ELATED W ORK Learning methods for decision making problems such as robotics largely divide into two classes: imitation learning and reinforcement learning (RL). In imitation learning (also called learning from demonstrations) the agent receives behavior examples from an expert and attempts to solve a task by copying the expert’s behavior. In RL, an agent attempts to maximize expected reward through interaction with the environment. Our work combines aspects of both to solve complex tasks. Imitation Learning: Perhaps the most common form of imitation learning is behavior cloning (BC), which learns a policy through supervised learning on demonstration stateaction pairs. BC has seen success in autonomous driving [5], [6], quadcopter navigation [7], locomotion [8], [9]. BC struggles outside the manifold of demonstration data. Dataset Aggregation (DAGGER) augments the dataset by interleaving the learned and expert policy to address this problem of accumulating errors [10]. However, DAGGER is difficult to use in practice as it requires access to an expert during all of training, instead of just a set of demonstrations. Fig. 1: We present a method using reinforcement learning to solve the task of block stacking shown above. The robot starts with 6 blocks labelled A through F on a table in random positions and a target position for each block. The task is to move each block to its target position. The targets are marked in the above visualization with red spheres which do not interact with the environment. These targets are placed in order on top of block A so that the robot forms a tower of blocks. This is a complex, multi-step task where the agent needs to learn to successfully manage multiple contacts to succeed. Frames from rollouts of the learned policy are shown. A video of our experiments can be found at: http://ashvin.me/demoddpg-website Fundamentally, BC approaches are limited because they do not take into account the task or environment. Inverse reinforcement learning (IRL) [11] is another form of imitation learning where a reward function is inferred from the demonstrations. Among other tasks, IRL has been applied to navigation [12], autonomous helicopter flight [13], and manipulation [14]. Since our work assumes knowledge of a reward function, we omit comparisons to IRL approaches. Reinforcement Learning: Reinforcement learning methods have been harder to apply in robotics, but are heavily investigated because of the autonomy they could enable. Through RL, robots have learned to play table tennis [15], swing up a cartpole, and balance a unicycle [16]. A renewal of interest in RL cascaded from success in games [17], [18], especially because of the ability of RL with large function approximators (ie. deep RL) to learn control from raw pixels. Robotics has been more challenging in general but there has been significant progress. Deep RL has been applied to manipulation tasks [19], grasping [20], [21], opening a door [22], and locomotion [23], [24], [25]. However, results have been attained predominantly in simulation per high sample complexity, typically caused by exploration challenges. Robotic Block Stacking: Block stacking has been studied from the early days of AI and robotics as a task that encapsulates many difficulties of more complicated tasks we want to solve, including multi-step planning and complex contacts. SHRDLU [26] was one of the pioneering works, but studied block arrangements only in terms of logic and natural language understanding. More recent work on task and motion planning considers both logical and physical aspects of the task [27], [28], [29], but requires domainspecific engineering. In this work we study how an agent can learn this task without the need of domain-specific engineering. One RL method, PILCO [16] has been applied to a simple version of stacking blocks where the task is to place a block on a tower [3]. Methods such as PILCO based on learning forward models naturally have trouble modelling the sharply discontinuous dynamics of contacts; although they can learn to place a block, it is a much harder problem to grasp the block in the first place. One-shot Imitation [4] learns to stack blocks in a way that generalizes to new target configurations, but uses more than 100,000 demonstrations to train the system. A heavily shaped reward can be used to learn to stack a Lego block on another with RL [30]. In contrast, our method can succeed from fully sparse rewards and handle stacking several blocks. Combining RL and Imitation Learning: Previous work has combined reinforcement learning with demonstrations. Demonstrations have been used to accelerate learning on classical tasks such as cart-pole swing-up and balance [31]. This work initialized policies and (in model-based methods) initialized forward models with demonstrations. Initializing policies from demonstrations for RL has been used for learning to hit a baseball [32] and for underactuated swingup [33]. Beyond initialization, we show how to extract more knowledge from demonstrations by using them effectively throughout the entire training process. Our method is closest to two recent approaches — Deep Q-Learning From Demonstrations (DQfD) [34] and DDPG From Demonstrations (DDPGfD) [2] which combine demonstrations with reinforcement learning. DQfD improves learning speed on Atari, including a margin loss which encourages the expert actions to have higher Q-values than all other actions. This loss can make improving upon the demonstrator policy impossible which is not the case for our method. Prior work has previously explored improving beyond the demonstrator policy in simple environments by introducing slack variables [35], but our method uses a learned value to actively inform the improvement. DDPGfD solves simple robotics tasks akin to peg insertion using DDPG with demonstrations in the replay buffer. In contrast to this prior work, the tasks we consider exhibit additional difficulties that are of key interest in robotics: multi-step behaviours, and generalization to varying goal states. While previous work focuses on speeding up already solvable tasks, we show that we can extend the state of the art in RL with demonstrations by introducing new methods to incorporate demonstrations. III. BACKGROUND A. Reinforcement Learning We consider the standard Markov Decision Process framework for picking optimal actions to maximize rewards over discrete timesteps in an environment E. We assume that the environment is fully observable. At every timestep t, an agent is in a state xt , takes an action at , receives a reward rt , and E evolves to state xt+1 . In reinforcement learning, the agent must learn a policy at = π(xt ) toPmaximize expected T returns. We denote the return by Rt = i=t γ (i−t) ri where T is the horizon that the agent optimizes over and γ is a discount factor for future rewards. The agent’s objective is to maximize expected return from the start distribution J = Eri ,si ∼E,ai ∼π [R0 ]. A variety of reinforcement learning algorithms have been developed to solve this problem. Many involve constructing an estimate of the expected return from a given state after taking an action: Qπ (st , at ) = Eri ,si ∼E,ai ∼π [Rt |st , at ] = Ert ,st+1 ∼E [rt + γ Eat+1 ∼π [Qπ (st+1 , at+1 )]] (1) (2) We call Qπ the action-value function. Equation 2 is a recursive version of equation 1, and is known as the Bellman equation. The Bellman equation allows for methods to estimate Q that resemble dynamic programming. B. DDPG Our method combines demonstrations with one such method: Deep Deterministic Policy Gradients (DDPG) [23]. DDPG is an off-policy model-free reinforcement learning algorithm for continuous control which can utilize large function approximators such as neural networks. DDPG is an actor-critic method, which bridges the gap between policy gradient methods and value approximation methods for RL. At a high level, DDPG learns an action-value function (critic) by minimizing the Bellman error, while simultaneously learning a policy (actor) by directly maximizing the estimated action-value function with respect to the parameters of the policy. Concretely, DDPG maintains an actor function π(s) with parameters θπ , a critic function Q(s, a) with parameters θQ , and a replay buffer R as a set of tuples (st , at , rt , st+1 ) for each transition experienced. DDPG alternates between running the policy to collect experience and updating the parameters. Training rollouts are collected with extra noise for exploration: at = π(s) + N , where N is a noise process. During each training step, DDPG samples a minibatch consisting of N tuples from R to update the actor and critic networks. DDPG minimizes the following loss L w.r.t. θQ to update the critic: yi = ri + γQ(si+1 , π(si+1 )) (3) L= 1 X (yi − Q(si , ai |θQ ))2 N i (4) The actor parameters θπ are updated using the policy gradient: 1 X ∇a Q(s, a|θQ )|s=si ,a=π(s) ∇θπ π(s|θπ )|si ∇θ π J = N i (5) To stabilize learning, the Q value in equation 3 is usually computed using a separate network (called the target network) whose weights are an exponential average over time of the critic network. This results in smoother target values. Note that DDPG is a natural fit for using demonstrations. Since DDPG can be trained off-policy, we can use demonstration data as off-policy training data. We also take advantage of the action-value function Q(s, a) learned by DDPG to better use demonstrations. C. Multi-Goal RL Instead of the standard RL setting, we train agents with parametrized goals, which lead to more general policies [36] and have recently been shown to make learning with sparse rewards easier [1]. Goals describe the task we expect the agent to perform in the given episode, in our case they specify the desired positions of all objects. We sample the goal g at he beginning of every episode. The function approximators, here π and Q, take the current goal as an additional input. D. Hindsight Experience Replay (HER) To handle varying task instances and parametrized goals, we use Hindsight Experience Replay (HER) [1]. The key insight of HER is that even in failed rollouts where no reward was obtained, the agent can transform them into successful ones by assuming that a state it saw in the rollout was the actual goal. HER can be used with any off-policy RL algorithm assuming that for every state we can find a goal corresponding to this state (i.e. a goal which leads to a positive reward in this state). For every episode the agent experiences, we store it in the replay buffer twice: once with the original goal pursued in the episode and once with the goal corresponding to the final state achieved in the episode, as if the agent intended on reaching this state from the very beginning. IV. M ETHOD Our method combines DDPG and demonstrations in several ways to maximally use demonstrations to improve learning. We describe our method below and evaluate these ideas in our experiments. A. Demonstration Buffer First, we maintain a second replay buffer RD where we store our demonstration data in the same format as R. In each minibatch, we draw an extra ND examples from RD to use as off-policy replay data for the update step. These examples are included in both the actor and critic update. This idea has been introduced in [2]. B. Behavior Cloning Loss Second, we introduce a new loss computed only on the demonstration examples for training the actor. LBC = ND X 2 kπ(si |θπ ) − ai k (6) i=1 This loss is a standard loss in imitation learning, but we show that using it as an auxiliary loss for RL improves learning significantly. The gradient applied to the actor parameters θπ is: λ1 ∇θπ J − λ2 ∇θπ LBC (7) (Note that we maximize J and minimize LBC .) Using this loss directly prevents the learned policy from improving significantly beyond the demonstration policy, as the actor is always tied back to the demonstrations. Next, we show how to account for suboptimal demonstrations using the learned action-value function. C. Q-Filter We account for the possibility that demonstrations can be suboptimal by applying the behavior cloning loss only to states where the critic Q(s, a) determines that the demonstrator action is better than the actor action: LBC = ND X 2 kπ(si |θπ ) − ai k 1Q(si ,ai )>Q(si ,π(si )) (8) i=1 The gradient applied to the actor parameters is as in equation 7. We label this method using the behavior cloning loss and Q-filter “Ours” in the following experiments. D. Resets to demonstration states To overcome the problem of sparse rewards in very long horizon tasks, we reset some training episodes using states and goals from demonstration episodes. Restarts from within demonstrations expose the agent to higher reward states during training. This method makes the additional assumption that we can restart episodes from a given state, as is true in simulation. To reset to a demonstration state, we first sample a demonstration D = (x0 , u0 , x1 , u1 , ...xN , uN ) from the set of demonstrations. We then uniformly sample a state xi from D. As in HER, we use the final state achieved in the demonstration as the goal. We roll out the trajectory with the given initial state and goal for the usual number of timesteps. At evaluation time, we do not use this procedure. We label our method with the behavior cloning loss, Qfilter, and resets from demonstration states as “Ours, Resets” in the following experiments. V. E XPERIMENTAL S ETUP A. Environments We evaluate our method on several simulated MuJoCo [37] environments. In all experiments, we use a simulated 7-DOF Fetch Robotics arm with parallel grippers to manipulate one or more objects placed on a table in front of the robot. The agent receives the positions of the relevant objects on the table as its observations. The control for the agent is continuous and 4-dimensional: 3 dimensions that specify the desired end-effector position1 and 1 dimension that specifies the desired distance between the robot fingers. The agent is controlled at 50Hz frequency. We collect demonstrations in a virtual reality environment. The demonstrator sees a rendering of the same observations as the agent, and records actions through a HTC Vive interface at the same frequency as the agent. We have the option to accept or reject a demonstration; we only accept demonstrations we judge to be mostly correct. The demonstrations are not optimal. The most extreme example is the “sliding” task, where only 7 of the 100 demonstrations are successful, but the agent still sees rewards for these demonstrations with HER. B. Training Details To train our models, we use Adam [38] as the optimizer with learning rate 10−3 . We use N = 1024, ND = 128, λ1 = 10−3 , λ2 = 1.0/ND . The discount factor γ is 0.98. We use 100 demonstrations to initialize RD . The function approximators π and Q are deep neural networks with ReLU activations and L2 regularization with the coefficient 5×10−3 . The final activation function for π is tanh, and the output value is scaled to the range of each action dimension. To explore during training, we sample random actions uniformly within the action space with probability 0.1 at every step, and the noise process N is uniform over ±10% of the maximum value of each action dimension. Task-specific information, including network architectures, are provided in the next section. C. Overview of Experiments We perform three sets of experiments. In Sec. VI, we provide a comparison to previous work. In Sec. VII we solve block stacking, a difficult multi-step task with complex contacts that the baselines struggle to solve. In Sec. VIII we do ablations of our own method to show the effect of individual components. VI. C OMPARISON W ITH P RIOR W ORK A. Tasks We first show the results of our method on the simulated tasks presented in the Hindsight Experience Replay paper [1]. We apply our method to three tasks: 1) Pushing. A block placed randomly on the table must be moved to a target location on the table by the robot (fingers are blocked to avoid grasping). 2) Sliding. A puck placed randomly on the table must be moved to a given target location. The target is outside the robot’s reach so it must apply enough force that the puck reaches the target and stops due to friction. 3) Pick-and-place. A block placed randomly on the table must be moved to a target location in the air. Note 1 In the 10cm x 10cm x 10cm cube around the current gripper position Pushing 1.0 0.4 0.2 0.8 Success Rate 0.6 0.6 0.4 0.2 0.0 2M 4M 6M Timesteps 8M 10M 0.6 0.4 0.2 0.0 0M Pick and Place 1.0 Ours HER BC 0.8 Success Rate 0.8 Success Rate Sliding 1.0 0.0 0M 2M 4M 6M 8M 10M 0M Timesteps 2M 4M 6M 8M 10M Timesteps Fig. 2: Baseline comparisons on tasks from [1]. Frames from the learned policy are shown above each task. Our method significantly outperforms the baselines. On the right plot, the HER baseline always fails. that the original paper used a form of initializing from favorable states to solve this task. We omit this for our experiment but discuss and evaluate the initialization idea in an ablation. As in the prior work, we use a fully sparse reward for this task. The agent is penalized if the object is not at its goal position: ( 0, if ||xi − gi || < δ rt = (9) −1, otherwise where the threshold δ is 5cm. B. Results Fig. 2 compares our method to HER without demonstrations and behavior cloning. Our method is significantly faster at learning these tasks than HER, and achieves significantly better policies than behavior cloning does. Measuring the number of timesteps to get to convergence, we exhibit a 4x speedup over HER in pushing, a 2x speedup over HER in sliding, and our method solves the pick-and-place task while HER baseline cannot solve it at all. The pick-and-place task showcases the shortcoming of RL in sparse reward settings, even with HER. In pick-and-place, the key action is to grasp the block. If the robot could manage to grasp it a small fraction of the time, HER discovers how to achieve goals in the air and reinforces the grasping behavior. However, grasping the block with random actions is extremely unlikely. Our method pushes the policy towards demonstration actions, which are more likely to succeed. In the HER paper, HER solves the pick-and-place task by initializing half of the rollouts with the gripper grasping the block. With this addition, pick-and-place becomes the easiest of the three tasks tested. This initialization is similar in spirit to our initialization idea, but takes advantage of the fact that pick-and-place with any goal can be solved starting from a block grasped at a certain location. This is not always true (for example, if there are multiple objects to be moved) and finding such a keyframe for other tasks would be difficult, requiring some engineering and sacrificing autonomy. Instead, our method guides the exploration towards grasping the block through demonstrations. Providing demonstrations does not require expert knowledge of the learning system, which makes it a more compelling way to provide prior information. VII. M ULTI -S TEP E XPERIMENTS A. Block Stacking Task To show that our method can solve more complex tasks with longer horizon and sparser reward, we study the task of block stacking in a simulated environment as shown in Fig. 1 with the same physical properties as the previous experiments. Our experiments show that our approach can solve the task in full and learn a policy to stack 6 blocks with demonstrations and RL. To measure and communicate various properties of our method, we also show experiments on stacking fewer blocks, a subset of the full task. We initialize the task with blocks at 6 random locations x1 ...x6 . We also provide 6 goal locations g1 ...g6 . To form a tower of blocks, we let g1 = x1 and gi = gi−1 + (0, 0, 5cm) for i ∈ 2, 3, 4, 5. By stacking N blocks, we mean N blocks reach their target locations. Since the target locations are always on top of x1 , we start with the first block already in position. So stacking N blocks involves N −1 pick-and-place actions. To solve stacking N , we allow the agent 50 ∗ (N − 1) timesteps. This means that to stack 6 blocks, the robot executes 250 actions or 5 seconds. We recorded 100 demonstrations to stack 6 blocks, and use subsets of these demonstrations as demonstrations for stacking fewer blocks. The demonstrations are not perfect; they include occasionally dropping blocks, but our method can handle suboptimal demonstrations. We still rejected more than half the demonstrations and excluded them from the demonstration data because we knocked down the tower of blocks when releasing a block. B. Rewards Two different reward functions are used. To test the performance of our method under fully sparse reward, we Ours Stack 2, Sparse Stack 3, Sparse Stack 4, Sparse Stack 4, Step Stack 5, Step Stack 6, Step 99% 99% 1% 91% 49% 4% Ours, Resets 97% 89% 54% 73% 50% 32% BC HER 65% 1% 0% - 0% 0% 0% - BC+ HER 65% 1% 0% - Fig. 3: Comparison of our method against baselines. The value reported is the median of the best performance (success rate) of all randomly seeded runs of each method. reward the agent only if all blocks are at their goal positions: rt = min 1||xi −gi ||<δ i (10) The threshold δ is the size of a block, 5cm. Throughout the paper we call this the “sparse” reward. To enable solving the longer horizon tasks of stacking 4 or more blocks, we use the “step” reward : X rt = −1 + 1||xi −gi ||<δ (11) i Note the step reward is still very sparse; the robot only sees the reward change when it moves a block into its target location. We subtract 1 only to make the reward more interpretable, as in the initial state the first block is already at its target. Regardless of the reward type, an episode is considered successful for computing success rate if all blocks are at their goal position in their final state. C. Network architectures We use a 4 layer networks with 256 hidden units per layer for π and Q for the HER tasks and stacking 3 or fewer blocks. For stacking 4 blocks or more, we use an attention mechanism [39] for the actor and a larger network. The attention mechanism uses a 3 layer network with 128 hidden units per layer to query the states and goals with one shared head. Once a state and goal is extracted, we use a 5 layer network with 256 hidden units per layer after the attention mechanism. Attention speeds up training slightly but does not change training outcomes. D. Baselines We include the following methods to compare our method to baselines on stacking 2 to 6 blocks. 2 Ours: Refers to our method as described in section IV-C. Ours, Resets: Refers to our method as described in section IV-C with resets from demonstration states (Sec. IV-D). BC: This method uses behavior cloning to learn a policy. Given the set of demonstration transitions RD , we train the 2 Because of computational constraints, we were limited to 5 random seeds per method for stacking 3 blocks, 2 random seeds per method for stacking 4 and 5 blocks, and 1 random seed per method for stacking 6 blocks. Although we are careful to draw conclusions from few random seeds, the results are consistent with our collective experience training these models. We report the median of the random seeds everywhere applicable. Stack 3, Sparse Reward 1.0 Ours Ours, Resets No Q-Filter No BC No HER 0.8 Success Rate Task 0.6 0.4 0.2 0.0 0M 50M 100M 150M 200M 250M 300M 350M 400M Timesteps Fig. 4: Ablation results on stacking 3 blocks with a fully sparse reward. We run each method 5 times with random seeds. The bold line shows the median of the 5 runs while each training run is plotted in a lighter color. Note “No HER” is always at 0% success rate. Our method without resets learns faster than the ablations. Our method with resets initially learns faster but converges to a worse success rate. policy π by supervised learning. Behavior cloning requires much less computation than RL. For a fairer comparison, we performed a large hyperparameter sweep over various networks sizes, attention hyperparameters, and learning rates and report the success rate achieved by the best policy found. HER: This method is exactly the one described in Hindsight Experience Replay [1], using HER and DDPG. BC+HER: This method first initializes a policy (actor) with BC, then finetunes the policy with RL as described above. E. Results We are able to learn much longer horizon tasks than the other methods, as shown in Fig. 3. The stacking task is extremely difficult using HER without demonstrations because the chance of grasping an object using random actions is close to 0. Initializing a policy with demonstrations and then running RL also fails since the actor updates depend on a reasonable critic and although the actor is pretrained, the critic is not. The pretrained actor weights are therefore destroyed in the very first epoch, and the result is no better than BC alone. We attempted variants of this method where initially the critic was trained from replay data. However, this also fails without seeing on-policy data. The results with sparse rewards are very encouraging. We are able to stack 3 blocks with a fully sparse reward without resetting to the states from demonstrations, and 4 blocks with a fully sparse reward if we use resetting. With resets from demonstration states and the step reward, we are able to learn a policy to stack 6 blocks. VIII. A BLATION E XPERIMENTS In this section we perform a series of ablation experiments to measure the importance of various components of our method. We evaluate our method on stacking 3 to 6 blocks. We perform the following ablations on the best performing of our models on each task: No BC Loss: This method does not apply the behavior cloning gradient during training. It still has access to demonstrations through the demonstration replay buffer. Stack 4, Step Reward 0.5 0.6 0.4 0.35 Stack 6, Step Reward Ours Ours, Resets 0.30 0.4 0.3 0.2 0.25 0.20 0.15 0.10 0.2 0.1 0.0 0.05 0.0 100M 200M 300M 400M 500M 600M 700M 800M 0M 3.0 2.5 Reward 2.0 1.5 1.0 500M 1000M 1500M 4.0 4.0 3.5 3.5 3.0 3.0 2.5 2.5 2.0 1.5 1.0 0.5 100M 200M 300M 400M 500M Timesteps 600M 700M 800M 1000M 1500M 2000M 1000M 1500M 2000M 2.0 1.5 0.5 0.0 0M 500M 1.0 0.5 0.0 0.00 0M Reward 0M Reward 0.40 Ours Ours, Resets No Q-Filter 0.6 Success Rate 0.8 Success Rate Stack 5, Step Reward Ours Ours, Resets No Q-Filter No BC Success Rate 1.0 0.0 0M 500M 1000M 1500M 0M 500M Timesteps Timesteps Fig. 5: Ablation results on longer horizon tasks with a step reward. The upper row shows the success rate while the lower row shows the average reward at the final step of each episode obtained by different algorithms. For stacking 4 and 5 blocks, we use 2 random seeds per method. The median of the runs is shown in bold and each training run is plotted in a lighter color. Note that for stacking 4 blocks, the “No BC” method is always at 0% success rate. As the number of blocks increases, resets from demonstrations becomes more important to learn the task. No Q-Filter: This method uses standard behavioral cloning loss instead of the loss from equation Eq. 8, which means that the actor tries to mimic the demonstrator’s behaviour regardless of the critic. No HER: Hindsight Experience Replay is not used. A. Behavior Cloning Loss Without the behavior cloning loss, the method is significantly worse in every task we try. Fig. 4 shows the training curve for learning to stack 3 blocks with a fully sparse reward. Without the behavior cloning loss, the system is about 2x slower to learn. On longer horizon tasks, we do not achieve any success without this loss. To see why, consider the training curves for stacking 4 blocks shown in Fig. 5. The “No BC” policy learns to stack only one additional block. Without the behavior cloning loss, the agent only has access to the demonstrations through the demonstration replay buffer. This allows it to view highreward states and incentivizes the agent to stack more blocks, but there is a stronger disincentive: stacking the tower higher is risky and could result in lower reward if the agent knocks over a block that is already correctly placed. Because of this risk, which is fundamentally just another instance of the agent finding a local optimum in a shaped reward, the agent learns the safer behavior of pausing after achieving a certain reward. Explicitly weighting behavior cloning steps into gradient updates forces the policy to continue the task. B. Q-Filter The Q-Filter is effective in accelerating learning and achieving optimal performance. Fig. 4 shows that the method without filtering is slower to learn. One issue with the behavior cloning loss is that if the demonstrations are suboptimal, the learned policy will also be suboptimal. Filtering by Q-value gives a natural way to anneal the effect of the demonstrations as it automatically disables the BC loss when a better action is found. However, it gives mixed results on the longer horizon tasks. One explanation is that in the step reward case, learning relies less on the demonstrations because the reward signal is stronger. Therefore, the training is less affected by suboptimal demonstrations. C. Resets From Demonstrations We find that initializing rollouts from within demonstration states greatly helps to learn to stack 5 and 6 blocks but hurts training with fewer blocks, as shown in Fig. 5. Note that even where resets from demonstration states helps the final success rate, learning takes off faster when this technique is not used. However, since stacking the tower higher is risky, the agent learns the safer behavior of stopping after achieving a certain reward. Resetting from demonstration states alleviates this problem because the agent regularly experiences higher rewards. This method changes the sampled state distribution, biasing it towards later states. It also inflates the Q values unrealistically. Therefore, on tasks where the RL algorithm does not get stuck in solving a subset of the full problem, it could hurt performance. IX. D ISCUSSION AND F UTURE W ORK We present a system to utilize demonstrations along with reinforcement learning to solve complicated multi-step tasks. We believe this can accelerate learning of many tasks, especially those with sparse rewards or other difficulties in exploration. Our method is very general, and can be applied on any continuous control task where a success criterion can be specified and demonstrations obtained. An exciting future direction is to train policies directly on a physical robot. Fig. 2 shows that learning the pick-andplace task takes about 1 million timesteps, which is about 6 hours of real world interaction time. This can realistically be trained on a physical robot, short-cutting the simulationreality gap entirely. Many automation tasks found in factories and warehouses are similar to pick-and-place but without the variation in initial and goal states, so the samples required could be much lower. With our method, no expert needs to be in the loop to train these systems: demonstrations can be collected by users without knowledge about machine learning or robotics and rewards could be directly obtained from human feedback. A major limitation of this work is sample efficiency on solving harder tasks. While we could not solve these tasks with other learning methods, our method requires a large amount of experience which is impractical outside of simulation. To run these tasks on physical robots, the sample efficiency will have to improved considerably. We also require demonstrations which are not easy to collect for all tasks. If demonstrations are not available but the environment can be reset to arbitrary states, one way to learn goal-reaching but avoid using demonstrations is to reuse successful rollouts as in [40]. Finally, our method of resets from demonstration states requires the ability to reset to arbitrary states. Although we can solve many long-horizon tasks without this ability, it is very effective for the hardest tasks. Resetting from demonstration rollouts resembles curriculum learning: we solve a hard task by first solving easier tasks. If the environment does not afford setting arbitrary states, then other curriculum methods will have to be used. X. 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1 Minimizing the Maximum End-to-End Network Delay: Hardness, Algorithm, and Performance arXiv:1707.02650v3 [] 16 Feb 2018 Qingyu Liu, Lei Deng, Haibo Zeng, Minghua Chen Abstract—We consider the scenario where a source streams a flow at a fixed rate to a receiver across the network, possibly using multiple paths. Each link has a finite capacity constraint. Transmission over a link incurs an integer delay if the rate is within the link capacity, and an unbounded delay otherwise. The objective is to minimize the maximum end-to-end delay experienced by the flow. The problem, denoted as Min-MaxDelay, appears in various practical scenarios, e.g., delay-critical video conferencing using inter-datacenter networks. In this paper, we first show that Min-Max-Delay is NP-hard in the weak sense and develop an exact algorithm with pseudo-polynomial time complexity. We then propose a Fully Polynomial Time Approximation Scheme (FPTAS) that obtains a (1 + )-approximate solution in polynomial time. These results reveal fundamental difference between the Min-Max-Delay problem and a similar maximum latency problem studied in the literature, for which is APX-hard and no PTAS1 exists unless P = NP. Moreover, there exists no exact pseudo-polynomial-time algorithm or constantapproximate algorithm for the maximum latency problem. We demonstrate the effectiveness of our algorithms in the scenario of routing delay-critical video-conferencing traffic over multiple paths of inter-datacenter networks, using simulations based on Amazon EC2 inter-datacenter topology. Both of our algorithms achieve the optimal maximum delay performance in all simulation instances, consistently outperforming all stateof-the-art solutions which only obtain sub-optimal maximum delay performance in certain instances. Furthermore, simulation results show that our achieved optimal delay performance always meet the end-to-end delay requirement for video conferencing applications, while the sub-optimal delay performance obtained by the alternatives fail to satisfy the video-conferencing delay requirement for up to 15% of simulation instances between certain cross-continental source-receiver pair. Index Terms—Delay-aware network flow, maximum delay optimization, video traffic, inter-datacenter networks, exact pseudopolynomial time algorithm, approximate algorithm. I. I NTRODUCTION A. Motivation We consider the scenario where a source streams a flow at a fixed rate to a receiver across a multi-hop network, possibly Manuscript received ... Part of this work has been presented at the IEEE Information Theory Workshop (ITW), Kaohsiung, Taiwan, November 6 - 10, 2017 [1]. Q. Liu, and H. Zeng are with the Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, USA (e-mail: qyliu14@vt.edu, haibo.zeng@gmail.com). L. Deng is with the School of Electrical Engineering & Intelligentization, Dongguan University of Technology, China (e-mail: denglei@dgut.edu.cn). M. Chen is with the Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong, China (e-mail: minghua@ie.cuhk.edu.hk). 1 Unless P = NP, it holds that FPTAS ( PTAS in that the runtime of a PTAS is required to be polynomial in problem input but not 1/, while the runtime of an FPTAS is polynomial in both the problem input and 1/ [2]. using multiple paths, to minimize the maximum end-to-end delay. We denote the problem as Min-Max-Delay. Transmission over a link is assumed to experience an integer delay if the rate is within the finite link capacity, and unbounded delay otherwise. This models many practical applications, particularly the routing of delay-critical video conferencing traffic over inter-datacenter networks. We note that using multiple paths for delay minimization is necessary when the shortest path is insufficient to support the fix-rate flow due to the bandwidth limitation. According to recent reports from Microsoft [3] and Google [4], most real-world inter-datacenter networks are characterized by sharing link bandwidth for different applications with over-provisioned link capacities. (i) Real-world inter-datacenter networks nowadays are utilized to simultaneously support traffic from various services, some of which have stringent delay requirements (e.g., video conferencing) while others are bandwidth-hungry and less sensitive to delay (e.g., data backup and data maintenance). Link capacity is often reserved separately for different types of services depending on their characteristics, e.g., bandwidth-hungry (delayinsensitive) services are reserved with a larger capacity. (ii) Cloud providers typically over-provision their inter-datacenter link capacity by 2 − 3 times on a dedicated backbone to guarantee reliability, and the average link-capacity utilizations (the aggregate utilization of applications, not the bandwidthutilization of individual applications) for busy links are 30 − 60% [5]. As such, in most real-world inter-datacenter networks queuing delays are negligible and the constant propagation delays dominate the end-to-end delay, as evaluated by [5] in a realistic network of Amazon EC22 . These observations justify our link capacity and delay model, especially for the problem of routing delay-critical video-conferencing traffic over interdatacenter networks. Our optimization objective of minimizing the end-to-end maximum delay is motivated by increasing interests on supporting delay-critical traffic to serve various communication applications, e.g., the video conferencing services [5], [6]. It is reported that 51 million users per month attend WebEx meetings [7], 3 billion minutes of calls per day use Skype [8], and 75% of high-growth innovators use video collaboration [9]. Low cross-network delay is vital for the video conferencing applications. As recommended by the 2 Note again that there may not be over-provisioning for individual applications/services of their respective reserved link capacities. Since their traffic peaks may not appear at the same time, the overall effect is that within the allocated (reserved) bandwidth, traffics of individual applications experience constant propagation delays; otherwise, they experience unbounded delay. 2 International Telecommunication Union (ITU) [10], for video conferencing, it is desirable to keep the cross-network oneway delay as low as possible. A delay less than 150ms can provide a transparent interactivity while delays above 400ms are unacceptable. In the delay-sensitive communication field, it is well-known that flow problems with objectives/constraints related to both throughput and maximum delay are highly non-trivial. In fact, it still remains open to characterize the hardness of the problem Min-Max-Delay and design efficient algorithms with performance guarantee, in spite of the practical relevance of Min-Max-Delay for delay-sensitive communications. The problem Min-Max-Delay models the link delay as a constant if the link rate is below the capacity and unbounded otherwise. It is an important special case of the maximum latency problem in the literature [11]–[13] which assumes the link delay as a general function of the link traffic rate. It is known that the maximum latency problem is NP-hard, in particular APX-hard [12]. This implies that unless P = NP, no PTAS exists, and there is even no polynomial time algorithm that can obtain a constant approximation ratio. In contrast, we show that the problem Min-Max-Delay is NP-hard in the weak sense, and we design a pseudopolynomial time exact algorithm as well as an FPTAS. Hence, while Min-Max-Delay is a special case of the maximum latency problem, they are fundamentally different in terms of hardness and admitting good exact/approximation algorithms. We summarize the differences in Tab. I. B. Contributions In this paper we focus on the Min-Max-Delay problem and make following contributions: B We prove Min-Max-Delay is NP-hard (Thm. 1, Sec. III), but is in the weak sense (Thm. 3, Sec. IV). Specifically, we first prove it is NP-hard, and then develop a pseudopolynomial time algorithm (Algorithm 1, Sec. IV) to solve Min-Max-Delay optimally. The algorithm has a time complexity of O((N 2 dmax )3.5 log(N 2 dmax ) log R), where N , max{|V |, |E|}, R is the flow rate (or throughput equivalently) requirement, and dmax is the maximum link delay. The time complexity is pseudo-polynomial because it is polynomial in the numeric value of the input dmax , but is exponential in the bit length of dmax , i.e., log(dmax ) [14]. B We further propose an FPTAS (Algorithm 3, Sec. V) to solve Min-Max-Delay approximately. The algorithm achieves a (1 + )-approximation ratio for any  > 0 with a time complexity of O((N M )3.5 log R log(N M )(L + log N )), where M = |E|(1+1/) and L is the bit complexity for representing dmax . The time complexity is polynomial in the problem input and 1/. B We demonstrate the effectiveness of our algorithms in the scenario of delay-critical inter-datacenter video-conferencing traffic routing over multiple paths, using simulations based on Amazon EC2 inter-datacenter topology. Both of our algorithms achieve optimal maximum delay performance in all simulation instances, which always the meet practically acceptable endto-end delay requirement for video conferencing. In contrast, the state-of-the-art solutions only obtain sub-optimal maximum delay performance, which fail to satisfy the videoconferencing delay requirement for up to 15% of simulation instances between certain cross-continental source-receiver pair. C. Paper organization The rest of the paper is organized as follows. Sec. II gives our system model and defines the problem Min-Max-Delay. Sec. III proves Min-Max-Delay to be NP-hard. We propose a pseudo-polynomial time algorithm in Sec. IV and an FPTAS in Sec. V. The NP-hardness proof together with the proposed pseudo-polynomial time algorithm show that the problem MinMax-Delay is NP-hard in the weak sense. Sec. VI presents our experiments and Sec. VII reviews existing studies, followed by the conclusion in Sec. VIII. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider a multi-hop network modeled as a directed graph G , (V, E) with |V | nodes and |E| links. Each link e ∈ E has a non-negative capacity ce and a non-negative integer delay de . We define N , max{|V |, |E|} to describe the size of the network, and dmax , max de as the maximum e∈E link delay. A source s ∈ V needs to stream a flow at a positive rate R to a receiver t ∈ V \{s}, possibly using multiple paths. We denote P as the set of all paths from s to t. For any path p ∈ P , we denote its path delay as X dp , de , (1) e∈E:e∈p i.e., the summation of link delays along the path. A flow solution f is defined as the assigned flow rate over P , i.e., f , {xp : xp ≥ 0, p ∈ P }. For a flow solution f , we define X xe , xp (2) p∈P :e∈p as the link rate of link e ∈ E. We further denote the total flow rate of a flow solution f by |f |, namely X |f | = xp . (3) p∈P The maximum delay of a flow solution f is defined as D(f ) , max p∈P :xp >0 dp , (4) i.e., the maximum delay among paths with positive rates. Another well-known end-to-end delay metric for a flow in the literature is the total delay, defined as X X T (f ) , (xp · dp ) = (xe · de ), (5) p∈P e∈E i.e., the total delay experienced by all flow units. We consider the problem of finding a flow solution f to minimize the maximum delay D(f ) while satisfying both the flow rate requirement subject to link capacity constraints. 3 TABLE I: Summary of the difference in hardness and admitting exact/approximation algorithms of the Maximum Latency Problem studied in the literature and problem Min-Max-Delay studied in this paper. Problem Hardness Maximum latency problem APX-hard [12] Weakly NP-hard (Thm. 3, Sec. IV) Min-Max-Delay Exact algorithm with pseudo -polynomial time complexity 7 Polynomial-time algorithm with constant approximation ratio 7 3(Algorithm 1, Sec. IV) 3(Algorithm 3, Sec. V) We denote the problem as Min-Max-Delay which can be formulated as min D X s.t. xp = R, (6a) v1 w0 w1 (6b) p∈P X xe = p x ≤ ce , ∀e ∈ E, p∈P :e∈p p xp (d − D) ≤ 0, ∀p ∈ P, vars. p x ≥ 0, ∀p ∈ P, (6c) v2 vn w2 ... wn-1 wn Link with capacity 1 and positive delay Source Link with capacity 1 and delay 0 Receiver Fig. 1: Reduced network graph from partition problem. (6d) (6e) where (6a) together with (6d) define our objective to minimize the maximum path delay for s−t paths that carry positive flow rates (called flow-carrying paths). Constraint (6b) restricts that the source s sends R rate to the receiver t, and constraint (6c) requires that the flow rate on link e does not exceed its capacity ce . In this paper we use fMM to denote the Min-Max-Delay flow, namely the optimal flow to our problem Min-Max-Delay, and hence D(fMM ) is the optimal maximum delay supporting a flow rate of R. From formulation (6) we observe two difficulties to solve Min-Max-Delay: (i) the number of paths (number of variables) is in general exponential in the network size, and (ii) formulation (6) is non-convex due to constraint (6d). In fact, later in Sec. III we prove Min-Max-Delay is NP-hard. Therefore, we cannot obtain fMM within polynomial time unless P = NP. III. NP- HARDNESS In this section, we analyze the computational complexity of Min-Max-Delay. Specifically, we prove that Min-Max-Delay is NP-hard based on the reduction from the well-known NPcomplete partition problem [14]. Thus, even as a special case of the NP-hard maximum latency problem, Min-Max-Delay is still NP-hard and it is impossible to solve in polynomial time unless P = NP. The partition problem is known to be NP-complete [14] (in the weak sense). We now leverage it to prove that Min-MaxDelay is NP-hard. Theorem 1. Min-Max-Delay problem is NP-hard. Proof: For any partition problem (Definition 2), we can construct a graph G0 with (2n + 1) nodes and 3n links as in Fig. 1. All links in the graph have unit capacity. Each dashed link (wi−1 , wi ) has a delay of ai for any i = 1, · · · , n, and solid links (wi−1 , vi ) and (vi , wi ) have a delay of zero. Obviously it takes polynomial time to construct the graph G0 from any given partition problem. Now we consider the decision version of a Min-Max-Delay problem instance: in graph G0 with source s = w0 , receiver t = wn , and flow rate requirement R = 2, is there any feasible flow f such that the maximum delay D(f ) ≤ b? In the following we prove the partition problem answers “Yes” if and only if the decision version of above Min-MaxDelay problem instance answers “Yes”. If Part. If the decision problem of Min-Max-Delay answers “Yes”, then there exists a flow f such that D(f ) ≤ b. Since f is feasible, the total rate from w0 to wn in f is R = 2. Now due to the capacity constraint and flow conservation, any link must exactly be assigned a flow rate of 1 to satisfy the total requirement R = 2. The total delay in flow f is X xp dp = p∈P X xe de = e∈E X e∈E 1 · de = n X ai = 2b. (9) i=1 Definition 1 (Partition). Given a non-empty set A, its partition is a set of non-empty subsets such that each element in A is in exactly one of these subsets. Since D(f ) ≤ b, we have Definition 2 (Partition Problem [14]). Given a set of n positive integers A = {a1 , a2 , ..., an } with sum X ai = 2b. (7) Also, because the total flow rate is equal to 2, we have X X X 2b = xp dp = xp dp ≤ b × xp = 2b. dp ≤ D(f ) ≤ b, ∀p ∈ P with xp > 0. p∈P p∈P :xp >0 p∈P :xp >0 (11) ai ∈A The partition problem asks is there a partition {A1 , A2 } of A such that X X ai = aj = b? (8) ai ∈A1 (10) aj ∈A2 As both ends in (11) are the same, it must be dp = b, ∀p ∈ P with xp > 0. (12) Therefore, all flow-carrying paths have a path delay of b. We choose an arbitrary flow-carrying path p. Since all solid 4 links have a delay of 0, the path delay of p is the delay of all dashed links belonging to p. We consider the set A1 that contains ai if (wi−1 , wi ) ∈ p. Clearly, it holds that X ai = b. (13) ai ∈A1 We then define A2 = A\A1 . It shall be X X X ai = 2b − b = b. aj = ak − aj ∈A2 ak ∈A (14) ai ∈A1 A1 and A2 are thus a partition of set A and meet the requirement of the partition problem. Hence, the partition problem answers “Yes”. Only If Part. If the partition problem answers “Yes”, then there exists a partition {A1 , A2 } such that X X aj = b. (15) ai = ai ∈A1 A. DCMF: maximize the flow subject to delay constraint A closely-related problem to Min-Max-Delay is the DelayConstrained Maximum Flow problem [15], denoted as DCMax-Flow: given a graph G, a source s, a receiver t, and a deadline T which is a path delay upper bound for all flowcarrying paths from s to t, the problem requires to find a feasible s − t flow with a maximized flow rate. If we denote P T as the set of all s − t paths whose path delay does not exceed T , the DC-Max-Flow can be formulated as X xp , max (17a) p∈P T aj ∈A2 We then construct the flow f with only two flow-carrying paths p1 and p2 and set xp1 = xp2 = 1. Since p1 and p2 are disjoint, the capacity constraint is satisfied. Also, since xp1 + xp2 = R = 2, the rate requirement is satisfied. Thus f is a feasible flow with maximum delay D(f ) ≤ b. Therefore, the decision problem of Min-Max-Delay answers “Yes”. Since the partition problem is NP-complete [14] and the reduction can be done in polynomial time, the Min-Max-Delay problem is NP-hard. Similar to the maximum latency problem, Min-Max-Delay is NP-hard. However, since in the following section a pseudopolynomial time algorithm is proposed to solve Min-MaxDelay optimally, Min-Max-Delay is actually NP-hard in the weak sense which is fundamentally different from the maximum latency problem that has been proved to be APX-hard. IV. E XACT P SEUDO - POLYNOMIAL T IME A LGORITHM In this section we propose an exact pseudo-polynomial time algorithm to solve Min-Max-Delay optimally. This, combined with the NP-hardness result in Thm. 1, shows that Min-MaxDelay is NP-hard in the weak sense. This result, combined with the existence of FPTAS described later in our Sec. V, shows that the Min-Max-Delay problem is fundamentally different from the maximum latency problem studied in the literature, which is APX-hard and admits no PTAS exists unless P = NP. Moreover, in our experiments (Sec. VI), X xe = s.t. aj ∈A2 We now construct two paths p1 and p2 . • ∀i ∈ [1, n], if ai ∈ A1 , we put (wi−1 , wi ) into path p1 ; otherwise, we put (wi−1 , vi ) and (vi , wi ) into p1 . • Similarly, for any i, if ai ∈ A2 , we put (wi−1 , wi ) into p2 ; otherwise, we put (wi−1 , vi ) and (vi , wi ) into p2 . Due to the definition of a partition (see Definition 1), A1 and A2 are two disjoint sets, i.e., A1 ∩ A2 = ∅. Thus, we can easily see that p1 and p2 are two disjoint s − t paths, i.e., p1 and p2 do not share any common link. Furthermore, it clearly holds that X X dp1 = ai = b, dp2 = aj = b. (16) ai ∈A1 extensive simulations show that the proposed exact algorithm always generates the optimal solution in 0.5 second empirically, implying that the exact algorithm is practically efficient. xp ≤ ce , (17b) p∈P T :e∈p xp ≥ 0, ∀p ∈ P T . vars. (17c) Since the size of P T could increase exponentially in the network size, formulation (17) can have an exponential number of variables. However, Wang and Chen [15] show that DCMax-Flow can be solved efficiently by an edge-based flow formulation which is a linear program with at most |E|·T variables. They implicitly use the idea of hop-expanded graph [15] by converting a delay-constrained max-flow problem in the original graph into a delay-unconstrained max-flow problem in the hop-expanded graph. In [15], the links are assumed to have unit-delay, but it is easy to generalize the results to integer-delay links. Overall the algorithm to solve problem DC-Max-Flow optimally with input (G, s, t, T ), denoted by DCMF(G, s, t, T ) in this paper, is to solve the following linear program [15, Proposition 1], max T X X xe(d) (18a) e∈In(t) d=0 s.t. X e) x(d = e X xe(d) = e∈In(v) x(d) e , xe(d+de ) , e∈Out(v) ∀v ∈ V \{s, t}, d ∈ [0, T ] T X vars. d=0 x(d) e (18b) e∈In(t) d=0 e∈Out(s) X T X X x(d) e ≤ ce , ≥ 0, (18c) ∀e ∈ E (18d) ∀e ∈ E, d ∈ [0, T ] (18e) where In(v) , {e = (w, v) : e ∈ E, w ∈ V } is the set containing all incoming links of node v. Similarly, Out(v) , {e = (v, w) : e ∈ E, w ∈ V } is the set of outgoing links (d) of node v, and xe is the total flow rate that experiences a delay of d after passing link e from the source s. The objective (18a) is the total flow rate that arrives at the receiver t within the deadline T . Constraint (18b) requires the rate entering the network should equal to the rate leaving the network. Constraints (18c) are the flow conservation constraints in the 5 expanded graph. Note that by convention, for any link e ∈ E, (d) we set xe = 0 for d < 0 and d > T . Constraints (18d) are the link capacity constraints which describe that flow rate assigned on the same link but experiencing different delays from the source should jointly respect the link capacity constraint. B. MMD: minimize the delay subject to flow requirement Comparing the problem Min-Max-Delay to the problem DC-Max-Flow, from a source s to a receiver t in a graph G, if we denote d∗ (R) as the minimized maximum delay subject to a rate requirement R (the optimal value of Min-Max-Delay), and we denote r∗ (T ) as the maximized flow rate subject to a deadline constraint T (the optimal value of DC-Max-Flow), we have the following lemma which is helpful to design an exact algorithm to solve Min-Max-Delay. Lemma 1. d∗ (R) ≤ T if and only if r∗ (T ) ≥ R. Proof: If Part. If r∗ (T ) ≥ R, then there exists a flow solution over P T , i.e., f = {xp : p ∈ P T } such that X xp ≥ R. (19) p∈P T We can thus decrease the flow solution f to construct another flow solution f˜ such that X x̃p = R. (20) p∈P T Since f satisfies the capacity constraints, f˜ must also satisfy the capacity constraints. Thus, f˜ is a feasible solution to Min-Max-Delay with rate requirement R. In addition, since all flow-carrying paths in f˜ belong to the set P T , we have d∗ (R) ≤ D(f˜) ≤ T . Only If Part. If d∗ (R) ≤ T , then there exists a flow solution f where the path delay of any flow-carrying path does not exceed T . Thus all flow-carrying paths belong to P T and f is also a feasible solution to DC-Max-Flow with a delay bound T . Thus, it holds that X r∗ (T ) ≥ xp = R. (21) p∈P T Lem. 1 suggests a binary search scheme to solve Min-MaxDelay optimally, namely our Algorithm 1. Given a lower bound Tl (= 0 initially) and an upper bound Tu (= U initially and U can be the maximum delay of any feasible flow satisfying rate requirement R) of the optimal maximum delay, in each iteration we solve the problem DC-Max-Flow with input (G, s, t, T ) where T = d(Tl + Tu )/2e. We compare the optimal value of (18), i.e., r∗ (T ) with the rate requirement R. If r∗ (T ) ≥ R, we decrease the upper bound as Tu = T − 1. Otherwise, if r∗ (T ) < R, we increase the lower bound as Tl = T + 1. In the end we can achieve a feasible s − t flow with maximum flow-carrying path delay optimized and the given flow rate requirement satisfied. Theorem 2. Algorithm 1 solves Min-Max-Delay optimally and has a pseudo-polynomial time complexity of O((N U )3.5 log(N U ) log R) where N , max{|V |, |E|}. Algorithm 1 MMD(G, R, s, t, U ): solve Min-Max-Delay optimally in pseudo-polynomial time 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: Input: G = (V, E), R, s, t, U Output: fMM procedure Tl = 0, Tu = U , T ∗ = Tu while Tu ≥ Tl do T = d(Tl + Tu )/2e f = DCMF(G, s, t, T ) r∗ (T ) = |f | if r∗ (T ) ≥ R then fMM = f , T ∗ = T Tu = T − 1 else Tl = T + 1 end if end while return fMM end procedure Proof: The optimality of the binary search scheme directly follows from Lem. 1. Our binary search scheme terminates in O(log U ) iterations where U must be upper bounded by |E| · dmax even in the worst case. In each iteration given a T ≤ U , we need to solve the linear program (18). Obviously the number of variables in (18) is O(|E| · T ). The conservation constraint (18c) is formulated for v ∈ V \{s, t} and d ∈ [0, T ]. Thus the number of conservation constraint is O(|V | · T ). Clearly the number of capacity constraint (18d) is |E|. Overall, the number of constraints in formulation (18) is O(N T ). [16] has proposed an algorithm to solve a linear program within time O((g + h)1.5 hL) where g is the number of constraints, h is the number of variables, and L denotes the standard “bit complexity” of the linear program. Therefore, it takes O((N T )2.5 (log R + N T log(N T ))) time to solve formulation (18) based on the calculation of L [17] and the Hadamard’s inequality. Since T ≤ U , the overall time complexity is O((N U )2.5 (log R+N U log(N U )) log U ) which can be described by O((N U )3.5 log(N U ) log R). Without loss of generality, clearly it holds that U ≤ |E| · dmax which is an upper bound for the path delay of any simple source-receiver paths, and hence the time complexity in Thm. 2 can be described by O((N 2 dmax )3.5 log(N 2 dmax ) log R). The time complexity is pseudo-polynomial in the sense that it is polynomial in the numeric value of the input dmax , but it is exponential in the bit length of the dmax . In the next section we design an FPTAS for our problem Min-Max-Delay. For any  > 0, the FPTAS can find a (1+)-approximate solution and the time complexity is polynomial with the problem input and 1/. A direct result of Thm. 2 is as follows. Theorem 3. Min-Max-Delay is NP-hard in the weak sense. Proof: It follows from Thm. 1 and Thm. 2. Note that in the end of Algorithm 1, solving linear program 6 (18) gives an edge-based flow with the optimal maximum delay d∗ (R). We need to do a flow decomposition on the hop-expanded graph with a deadline constraint T = d∗ (R) in order to get a path-based flow. The number of nodes and links in the expanded graph are both O(N d∗ (R)). And since the flow decomposition in a graph Ĝ(V̂ , Ê) has a time complexity of O(|Ê|(|V̂ | + |Ê|)) [18], the time complexity to get the path-based flow by decomposition in the expanded graph will be O((N U )2 ), which does not affect the time complexity of Algorithm 1 presented in Thm. 2. V. F ULLY-P OLYNOMIAL T IME A PPROXIMATION S CHEME In this section, we propose an FPTAS to solve Min-MaxDelay (1 + )-approximately in polynomial time with all the problem input and 1/. The existence of FPTAS for the MinMax-Delay problem reveals its fundamental difference to the maximum latency problem, which is APX-hard and admits no PTAS unless P = NP. Recall that our exact algorithm solves Min-Max-Delay optimally by pursuing the minimum T ∗ = D(fMM ) ∈ {0, 1, 2, ..., U } iteratively such that the problem DC-MaxFlow with input (G, s, t, T ∗ ) can return a feasible s − t flow with a flow rate r∗ (T ∗ ) ≥ R. Due to the binary search scheme, the number of iterations to achieve the optimal solution is O(log U ), which is not a problem to achieve a polynomial time complexity considering that U ≤ |E| · dmax . The difficulty of developing polynomial time algorithms in fact is the pseudopolynomial size (O(N T ), T ∈ {0, 1, 2, ..., U }) of the linear program (18), namely in each iteration, it takes pseudopolynomial time to solve DC-Max-Flow with deadline T . In order to solve Min-Max-Delay approximately in polynomial time, a natural idea is to follow the similar binary search procedure of our proposed exact algorithm MMD(·) but in each iteration quantize T to make the size of the problem DC-Max-Flow independent of T and polynomial with all the problem input and 1/, which is challenging since we also need to maintain certain maximum delay performance guarantee simultaneously. Our proposed FPTAS in this section utilizes an adaptive quantization technique on link delay de and T to address the challenge, and in the end a (1 + )approximate solution is guaranteed in fully polynomial time. A. Quantized DC-Max-Flow algorithm Given a deadline T , we first introduce a procedure QDCMF(·) in Algorithm 2 which solves a DC-Max-Flow problem after the quantization of de and T . The procedure must return a feasible flow with a total flow rate no smaller than R in polynomial time, if T is an upper bound of the optimal solution to Min-Max-Delay, i.e. our Algorithm 2 must return a flow fˆ in polynomial time with |fˆ| ≥ R if T ≥ D[fMM ] (see our Lem. 3). The procedure will be used iteratively in a binary search scheme later in our FPTAS, similar to the structure of our exact algorithm MMD(·) (Algorithm 1). First, each link delay is quantized (line 5) to be the multiple of an appropriately defined ∆T (line 4). Then we define the quantized network Ĝ as the network G with each link delay de replaced by dˆe (line 7). Besides, T is also quantized to Algorithm 2 QDCMF(G, s, t, T, ): quantized DC-Max-Flow algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: input: G = (V, E), s, t, T ,  output: fˆ procedure ∆T = T /|E| dˆe = dde /∆T e, ∀e ∈ E T̂ = dT /∆T e + |E| Ĝ(V, E): G(V, E) with each de replaced by dˆe fˆ = DCMF(Ĝ, s, t, T̂ ) return fˆ end procedure be T̂ based on ∆T (line 6). Due to that T̂ = d|E|/e + |E| which is independent of link delay, clearly the DC-Max-Flow in the graph Ĝ with the input deadline T̂ is polynomial-timesolvable. Moreover, such T̂ can guarantee that the quantized deadline T̂ is still an upper bound for the maximum delay of fMM in the quantized network Ĝ, if the deadline T is an upper bound for the maximum delay of fMM in G, which will be proved later in our Lem. 3. Finally in line 8, the returned solution fˆ is the optimal solution to problem DC-Max-Flow in the quantized network Ĝ with the quantized deadline T̂ . Since Ĝ and G share the same network topology and link capacity, obviously a flow f is feasible in Ĝ if and only if it is feasible in G. Due to the difference of link delay in those two networks, the maximum delay of f in Ĝ may be different from the maximum delay of the same flow f in G. In order to avoid possible confusion in the following lemmas and theorems, we use DĜ (f ) to denote the maximum delay of f in the quantized network Ĝ. Similarly, DG (f ) is the maximum delay of f in the original network G. Without loss of generality, D(f ) is same to DG (f ) since G is the input to Min-Max-Delay. Now first we prove Lem. 2 which will be used later to prove the (1 + )-approximation ratio for our FPTAS. Then we give Lem. 3 to show the solution fˆ of algorithm QDCMF(G, s, t, T, ) (Algorithm 2) must satisfy |fˆ| ≥ R if D(fMM ) ≤ T . Lemma 2. It holds that DG (f ) ≤ ∆T · DĜ (f ) where Ĝ and ∆T is defined in Algorithm 2. Proof: Suppose Pf is the flow-carrying path set of f . Since the only difference between G and Ĝ is the link delay, clearly Pf does not change in G and in Ĝ. Now the lemma holds due to the following proof: DG (f ) = max X p:p∈Pf = ∆T · max p:p∈Pf e:e∈p X de ≤ max p:p∈Pf X [∆T dde /∆T e] e:e∈p dˆe = ∆T · DĜ (f ) e:e∈p Lemma 3. If T ≥ D(fMM ), QDCMF(G, s, t, T, ) must return a feasible flow solution fˆ and |fˆ| ≥ R. 7 Proof: Since T ≥ D(fMM ), by Lem. 1, the solution to problem DCMF(G, s, t, T ), denoted as f , must satisfy |f | ≥ R and DG (f ) ≤ T, (22) due to the assumption d∗ (R) = D(fMM ) ≤ T , namely there must exist a feasible flow f in G with a total flow rate lower bounded by R and a maximum delay upper bounded by T , assuming T is an upper bound for the optimal to Min-MaxDelay. As long as we can prove DĜ (f ) ≤ T̂ , namely the maximum delay of f in Ĝ is no larger than T̂ , we can prove Lem. 3 since f is a feasible solution to DCMF(Ĝ, s, t, T̂ ) with |f | ≥ R and furthermore fˆ is the optimal solution to DCMF(Ĝ, s, t, T̂ ) which maximizes the flow rate from s to t, leading to the conclusion of |fˆ| ≥ |f | ≥ R. The detailed proof is shown below assuming Pf is the flowcarrying path set of f : X DĜ (f ) = max dde /∆T e p:p∈Pf X ≤ max p:p∈Pf ≤ max (de /∆T + 1) p:p∈Pf X 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: input: G = (V, E), R, s, t, U ,  output: fFPTAS procedure Tu = U , Tl = 0, T ∗ = Tu while Tu ≥ Tl do T = d(Tu + Tl )/2e fˆ = QDCMF(G, s, t, T, ) if |fˆ| < R then Tl = T + 1 else T ∗ = T , Tu = T − 1 end if end while fFPTAS = QMMD(G, R, s, t, T ∗ , ) return fFPTAS end procedure Algorithm 4 QMMD(G, R, s, t, T, ): quantized Min-MaxDelay algorithm 1: (de /∆T ) + |E| e:e∈p X 1 · max de + |E| = ∆T p:p∈Pf e:e∈p = 1: 2: 3: 4: 5: 6: e:e∈p e:e∈p (a) Algorithm 3 FPTAS to solve Min-Max-Delay 1 · DG (f ) + |E| ∆T 2: 3: 4: 5: 6: 7: (b) ≤ dDG (f )/∆T e + |E| ≤ T̂ Inequality (a) holds since the number of links on any flowcarrying path is upper bound by |E| in f without loss of generality. Inequality (b) holds due to the inequality (22). Next we prove the polynomial time complexity for QDCMF(G, s, t, T, ). Lemma 4. Algorithm 2 has a fully polynomial time complexity of O((N M )3.5 log R log(N M )) where M = |E|(1+1/)+1. Proof: It is straightforward that the overall time complexity of QDCMF(G, s, t, T, ) is dominated by the time complexity of DCMF(Ĝ, s, t, T̂ ) (line 8). Now considering that T̂ = dT /∆T e + |E| = d|E|/e + |E| ≤ |E|(1 + 1/) + 1, following the similar procedure in Thm. 2 we can get the proposed time complexity. B. FPTAS to solve Min-Max-Delay We describe our FPTAS in Algorithm 3, which follows a similar structure as our MMD(G, R, s, t, U ) (Algorithm 1) that solves Min-Max-Delay optimally in pseudo-polynomial time, except for two differences: (i) in each iteration, FPTAS solves a quantized DC-MaxFlow problem optimally (line 7) while MMD solves the exact DC-Max-Flow problem optimally, and (ii) in the algorithm MMD, when the binary search terminates, associated T ∗ is the optimal maximum delay, and thus 8: 9: 10: input: G = (V, E), R, s, t, T ,  output: fˆ procedure ∆T = T /|E| dˆe = dde /∆T e, ∀e ∈ E T̂ = dT /∆T e + |E| Ĝ(V, E): G(V, E) with each de replaced by dˆe fˆ = MMD(Ĝ, R, s, t, T̂ ) return fˆ end procedure the solution to problem DC-Max-Flow with deadline T ∗ is the optimal solution to our Min-Max-Delay. However, when the binary search terminates in the FPTAS, associated T ∗ may not be the optimal maximum delay, and the (1 + )-approximate solution is the solution to problem QMMD(G, R, s, t, T ∗ , ) (line 14), which is the problem Min-Max-Delay (not the problem DC-Max-Flow) in the quantized network Ĝ with a maximum delay upper bound T̂ ∗ = dT ∗ /∆T ∗ e + |E|. The quantization approach in QMMD(·) (Algorithm 4) is same as in QDCMF(·) (Algorithm 2) with the size of formulation (18) to be d|E|/e + |E| which is independent of dmax and polynomial with problem input and 1/, clearly leading to a polynomial time complexity of our FPTAS. Next we will prove the (1 + )-approximate performance guarantee. Lemma 5. In the end of Algorithm 3, it holds that T ∗ ≤ D(fMM ). Proof: Suppose Topt is the maximum delay performance of the optimal solution to Min-Max-Delay, namely Topt = D(fMM ). According to Lem. 3, QDCMF(G, s, t, T, ) always returns a feasible flow fˆ and |fˆ| ≥ R for any T ∈ [Topt , U ], and may or may not return such a flow for T ∈ [0, Topt ). Considering the property of the binary search structure of our FPTAS, in the end, clearly that T ∗ ≤ Topt = D(fMM ). Note that in the end of our FPTAS, T ∗ may not be 8 the minimum integer among T ∈ [0, U ] such that QDCMF(G, s, t, T, ) returns a feasible flow with a total rate no smaller than R. In fact, our FPTAS only requires that T ∗ ≤ D(fMM ) to guarantee a (1 + ) approximation ratio, as proved in the following theorem. Theorem 4. Algorithm 3 is a (1 + )-approximate algorithm for the Min-Max-Delay problem, namely D(fFPTAS ) ≤ (1 + )D(fMM ) Proof: The theorem holds due to the following proof: (a) (b) DG (fFPTAS ) ≤ ∆T ∗ · DĜ (fFPTAS ) ≤ ∆T ∗ · DĜ (fMM ) X = ∆T ∗ · max dde /∆T ∗ e p:p∈PfMM ≤ ∆T ∗ · max p:p∈PfMM ≤ ∆T ∗ · max p:p∈PfMM Fig. 2: Topology of the 6 Amazon EC2 datacenters. TABLE II: Link delays (in ms) for Amazon EC2 [6], (OR: Oregon, VA: Virginia, IR: Ireland, TO: Tokyo, SI: Singapore, SP: Sao Paulo). e:e∈p X (de /∆T ∗ + 1) e:e∈p X (de /∆T ∗ ) + ∆T ∗ |E| e:e∈p OR VA IR TO SI SP OR N/A - VA 41 N/A - IR 86 54 N/A - TO 68 101 138 N/A - SI 117 127 117 45 N/A - SP 104 82 120 151 182 N/A = DG (fMM ) + ∆T ∗ |E| = DG (fMM ) + T ∗ (c) ≤ (1 + )DG (fMM ) Inequality (a) holds due to Lem. 2. Inequality (b) is true because fFPTAS is the optimal to problem Min-Max-Delay in the graph Ĝ (our Algorithm 4). Inequality (c) is true due to Lem. 5. Theorem 5. Algorithm 3 has a fully polynomial time complexity of O((N M )3.5 log R log(N M )(L + log N )) where M = |E|(1 + 1/) + 1 and L is the bit complexity for representing dmax . Proof: Assume we need L bits to represent dmax , namely dmax = O(2L ). According to Lem. 4, in each iteration of Algorithm 3, corresponding time complexity is O((N M )3.5 log R log(N M )). Due to the fact that the initial upper bound U for the optimal maximum delay is definitely upper bounded by |E| · dmax , namely U ≤ |E| · dmax , in the binary search scheme in our FPTAS we need at most O(log(|E| · dmax )) iterations, leading to the proposed time complexity in the end. Note that in the running time of QMMD(·) (line 14) of Algorithm 3 is not a dominating part according to Thm. 2 and does not affect the time complexity. Thm. 4 together with Thm. 5 shows that our proposed Algorithm 3 can guarantee an approximation ratio of (1 + ) within polynomial time for any  > 0, and hence Algorithm 3 is an FPTAS for the problem Min-Max-Delay. In contrast, as a generalization of our Min-Max-Delay, the maximum latency problem studied in the literature has been proved to be APXhard [12] and admits no PTAS unless P = NP. Thus, although Min-Max-Delay is an important special case of the maximum latency problem which has been proved to be quite difficult to approximate, there exist efficient approximate algorithms, e.g. our proposed Algorithm 3, to solve the Min-Max-Delay with an approximation ratio arbitrarily close to one in polynomial time. This result reveals a fundamental difference between the two problems. VI. P ERFORMANCE E VALUATION In this section we evaluate our proposed algorithms and see how the empirically achieved maximum delay performance meets the video-conferencing end-to-end delay requirement (400ms as suggested by [10]), comparing our algorithms to state-of-the-art approaches using a real-world continent-scale inter-datacenter network topology of 6 globally distributed Amazon EC2 datacenters as shown in Fig. 2, modeled as a complete undirected graph. Each undirected link is treated as two directed links that operate independently and have identical capacities, a common way to model an undirected graph by a directed one, e.g. in [19]. We set integer link delays (see Tab. II) and link capacities (see Tab. III) according to practical evaluations on Amazon EC2 from studies [5], [6]. We do extensive simulations to solve Min-Max-Delay problem instances belonging to 4 different experimental cases (see Tab. IV). For each case characterized by a specific sourcereceiver pair, we solve Min-Max-Delay instances with flow rate requirement enumerated from Rmin to Rmax with a unit step. In this paper, Rmin is set to be the minimum flow rate when multiple paths (≥ 2 paths) must be used for delay minimization while supporting the required flow rate, since otherwise the Min-Max-Delay problem can be reduced to the simple shortest path problem that is polynomial-time solvable by Dijkstra’s algorithm [20] and is not our focus. We set Rmax to be the maximum flow rate that can be sent from the source TABLE III: Link capacities (in Mbps) for Amazon EC2 [5], (OR: Oregon, VA: Virginia, IR: Ireland, TO: Tokyo, SI: Singapore, SP: Sao Paulo). OR VA IR TO SI SP OR N/A - VA 82 N/A - IR 86 72 N/A - TO 138 41 56 N/A - SI 74 52 44 166 N/A - SP 67 70 61 41 33 N/A 9 to the receiver. Our test environment is an Intel Core i5 (2.40 GHz) processor with 8 GB memory running Windows 64-bit operating system. All the experiments are implemented in C++ and linear programs are solved using CPLEX [21]. A. Baseline algorithms In order to evaluate the performance of our exact algorithm and FPTAS, we compare them to three existing approaches. 1) the System-Optimal flow algorithm (SO): the systemoptimal flow is a γ(L)-approximate solution to the maximum latency problem of minimizing the maximum delay with flowdependent link delay model, but theoretically it is not a good solution to Min-Max-Delay with link capacity constraint and integer link delay because its maximum delay performance can become infinitely large compared to the optimal. In the experiments we obtain the system-optimal flow by solving its well-known edge-based linear programming formulation: X min (de · xe ) (23a) e∈E s.t. X e∈Out(s) xe = R, (23b) xe , ∀v ∈ V \{s, t}, (23c) e∈In(t) X X xe = e∈Out(v) vars. X xe = e∈In(v) xe ≤ ce , ∀e ∈ E, (23d) xe ≥ 0, ∀e ∈ E. (23e) In order to get the path-based system-optimal flow and associated maximum delay result, we do flow decomposition [18] on the optimal solution to the linear program (23). 2) the heuristic Iterative Shortest Path algorithm (ISP) [22]: the ISP approach is proposed by [22] to solve the maximum latency problem, and it can be used to handle our Min-Max-Delay directly with additional link capacity constraint involved. However, ISP is a heuristic approach with no maximum delay performance guarantee both in the maximum latency problem and in our Min-Max-Delay. ISP finds the shortest path from source to receiver iteratively, and in each iteration it assigns as much flow rate as possible to corresponding returned path. In our experiments we use Dijkstra’s algorithm [20] to get the shortest path. 3) the heuristic Iterative Linear Relaxation algorithm (ILR) [5]: the study in [5] optimizes three different endto-end-delay-related objectives in a general multi-commodity multi-cast scenario with constant link delay and constant link capacity. And our Min-Max-Delay problem in the singleunicast scenario can be casted by one of the three problems in [5] with the following formulation: X min max{dp φ(xp )} + β φ(xp ) (24a) p∈P s.t. X p∈P p x = R, (24b) p∈P X vars. e∈E:e∈p p xp ≤ ce , ∀e ∈ E, x ≥ 0, ∀p ∈ P. (24c) (24d) TABLE IV: Respective failure (%) of baseline algorithms to satisfy the video-conferencing delay requirement for 4 different experimental cases. Source Receiver Flow rate, [Rmin , Rmax ] ISP Failure of ILR Algo. (%) SO Case 1 OR TO Case 2 VA IR Case 3 VA SI Case 4 IR SI [139, 442] [73, 317] [53, 317] [45, 319] 7 7 7 2 1 2 0 0 0 9 15 0 where φ(·) is an identity function defined as ( 1, if x > 0, φ(x) = 0, if x = 0. The work in [5] defines variables as the flow rate assigned on the source-receiver paths with upper bounded length to reduce the time complexity. But in our experiments in order to minimize the maximum delay, we consider all source-receiver paths, namely the path set P . In the objective (24a), β is used by [5] as a penalty when too many paths are utilized to carry flow rates because the authors of [5] try to get a sparse network flow solution. However, in our experiments β is set to be 0 to optimize the maximum delay performance. Formulation (24) is hard to solve because of the identity function φ(·). The study in [5] proposes the ILR heuristic approach to overcome the difficulty by solving linear programs which are relaxations to formulation (24) iteratively. Details of ILR are referred to the paper [5]. As discussed later in our related work section, ILR has neither maximum delay performance guarantee nor running time performance guarantee theoretically. B. Experiments of our pseudo-polynomial time algorithm We first evaluate the performance of our pseudo-polynomial time algorithm MMD (Algorithm 1) which can solve MinMax-Delay optimally. In the experiments, TU is set to be the maximum delay of the flow solution to ISP, and thus the running time of the heuristic approach ISP is part of the total running time of MMD. We do simulations for extensive Min-Max-Delay instances as described in our Tab. IV, using algorithms including SO, ISP, ILR and our MMD. Recall as discussed in our introduction, to guarantee a real-time interaction for video conferencing, 400ms is the maximum acceptable end-to-end delay budget according to the suggestion from ITU [10]. Note that an end-to-end delay is in fact the delay experienced by the video and audio through a video conferencing, i.e. from source camera to receiver screen, which is a sum of coding delay at source, decoding delay at receiver and the network transmission delay. In this paper, however, we only target minimizing the transmission delay. According to [23], [24], a normal delay for the source (receiver) to process the video data can be 70ms, and such a processing delay can be reduced to 35ms for devices with advanced low-latency technology, e.g. the Sensoray Model 2253 audio/video codec [24]. 10 TABLE V: Average empirical maximum delay improvement (%) comparing the optimal solution of Min-Max-Delay which can be obtained by our MMD (Algorithm 1) to each of the baselines in non-trivial problem instances with relatively large flow rate requirement. Source Receiver Flow rate Max-delay imprv. (%) ISP ILR SO Case 1 OR TO [422, 442] 13 11 13 Case 2 VA IR [305, 317] 11 6 11 Case 3 VA SI [296, 317] 8 8 0 Case 4 IR SI [296, 319] 7 8 0 We reasonably assume a 35ms processing delay both for source and receiver, and hence the transmission delay upper bound to guarantee an acceptable video-conferencing service will be 330ms. In any simulation instance, the optimal maximum delay performance which can be obtained by MMD is always below the 330ms budget, while the maximum delay achieved by any baseline algorithm violates the 330ms upper bound and hence fail to provide an acceptable video conferencing in multiple instances with detailed results of failure in percentage provided in Tab. IV. As shown in the table, baseline algorithms fail to satisfy the 330ms video-conferencing delay budget for up to 15% of simulation instances between certain source-receiver pair (the maximum failure in percentage of baselines are 7%, 2%, 0% and 15% respectively for the 4 experimental cases). Besides, the baseline ISP (resp. ILR and SO) fails for up to 9% (resp. 15% and 7%) of all simulation instances in order to route the video-conferencing traffic within acceptable delay budget from certain source to certain receiver. In our simulations, for trivial problem instances with small/media rate requirement, all algorithms have the same and optimal maximum delay performance since only two or three paths will be used. We provide the detailed maximum delay and running time results for non-trivial problem instances with relatively large flow rate requirement in Fig. 3, where clearly any of the three baselines fail to solve our problem optimally in certain instances. In Tab. V, we give the average maximum delay improvement, namely the average of (D(f ) − D(fMM ))/D(f ), comparing fMM which is the optimal to our problem Min-Max-Delay and is the solution of our MMD, to the flow f which is the solution of individual baselines, in those non-trivial problem instances. The running time results are also presented in Fig. 3. According to the figure, the running time of SO and ISP is so small that can be ignored, compared with the results of MMD and ILR. Although our MMD, which can always find the optimal solution, consumes more time than SO and ISP, it in fact runs pretty fast in our experiments (less than 0.5 second) for all simulation instances. C. Experiments of our FPTAS For our problem Min-Max-Delay, because the exact algorithm (Algorithm 1) has an pseudo-polynomial time complexity theoretically, in Sec. V-B, we develop an FPTAS (Algorithm 3) which can guarantee a (1 + )-approximate solution within fully polynomial time. In this section, we evaluate the performance of our FPTAS for traffic from Oregon to Tokyo (case 1 in Tab. V) and for traffic from North Virginia to Singapore (case 3 in Tab. V). In both cases, the approximation parameter  is increased from 0 to 3 with a step of 0.1, and we obtain the maximum delay and running time results of the FPTAS for problem instances with different flow rate requirements, as shown in Fig. 4. According to the figure, in both experimental cases either from Oregon to Tokyo or from North Virginia to Singapore, obviously larger flow rate requirement results in a solution flow with larger optimal maximum delay. Although theoretically the maximum delay performance of the FPTAS is only upper bounded by a ratio of (1 + ) compared to the optimal solution, empirically our FPTAS is able to achieve the optimal maximum delay in all experimental cases for any  ∈ (0, 3] (the maximum delay result remains to be a constant with  as shown in the figure). According to the running time results, clearly smaller rate requirement and larger  lead to better running time results, particularly true for  < 0.5 when theoretically our FPTAS always gives a feasible flow with the maximum delay no worse than 1.5 × OPT where OPT denotes the optimal maximum delay performance of our MinMax-Delay problem instance. VII. R ELATED WORK The maximum latency problem models link delay as a function with the link aggregated flow rate. Since our MinMax-Delay problem which is characterized by an integer link delay within finite link capacity can be viewed as a special case of the maximum latency problem, existing algorithms for the maximum latency problem can be applied to solve our problem by assuming the delay function to be an integer within capacity and go to infinity otherwise. Although most existing algorithms can solve the maximum latency problem efficiently, compared to the optimal delay performance, none of them can achieve theoretically upper bounded maximum delay for our Min-Max-Delay problem. Correa et al. in [11], [12] prove the maximum latency problem is NP-hard, in particular APX-hard, and propose that compared to the optimal maximum delay, a tight maximum delay gap of the well-known system-optimal flow is γ(L) and a tight maximum delay gap of the Nash equilibrium flow is α(L). Both γ(L) and α(L) are constants that rely on the link delay function. However, for the link delay model in our Min-Max-Delay problem, both γ(L) and α(L) are infinitely large, implying neither the system-optimal flow nor the Nash equilibrium flow can guarantee a maximum delay within a finite-ratio gap to the optimal solution to our problem Min-Max-Delay. Roughgarden [13] shows that the Nash equilibrium also admits a maximum delay no worse than a network-dependent ratio of (|V |−1) for the maximum latency problem. Obviously the gap (|V | − 1) is too large for practical applications. Devetak et al. [22] have introduced fast heuristic algorithms to optimize maximum delay, but the maximum delay performance of returned solutions has no performance guarantee compared to the optimal solution, theoretically. To our best knowledge, [5] is the only existing study targeting the maximum delay optimization scenario with constant 11 Fig. 3: Running time and achieved maximum delay results of problem instances with large flow rate requirement for 4 experimental cases, comparing our pseudo-polynomial time algorithm (MMD) to three baselines. Fig. 4: Running time and achieved maximum delay results of our FPTAS for problem instances with different flow rate requirements and approximation parameter . link delay and finite link capacity involved. The heuristic approach in [5] for minimizing the maximum delay has two limitations: (i) the time complexity could be high because the number of variables in the approach increases exponentially in the network size, and (ii) there is not yet theoretical performance guarantee of the achieved solution. VIII. C ONCLUSION We study a delay-critical information flow problem which optimizes the maximum end-to-end cross-network delay in the single-unicast scenario, denoted as the Min-Max-Delay problem. Transmission over a link incurs an integer delay if the rate is within the finite link capacity, and an unbounded delay otherwise. Many practical applications, e.g. the routing of video conferencing traffic over inter-datacenter networks, can be formulated as such problems. We prove Min-Max-Delay is NP-hard in the weak sense and propose two algorithms: (i) an exact algorithm that can find the optimal solution with a pseudo-polynomial time complexity of O((N 2 dmax )3.5 log(N 2 dmax ) log R), where N , max{|V |, |E|} describes the network size, R is the flow rate requirement, and dmax is the maximum link delay, and (ii) an FPTAS with a (1 + )-approximation ratio and a polynomial time complexity of O((N M )3.5 log R log(N M )(L + log N )) for any  > 0, where M = |E|(1 + 1/) + 1 and L is the bit complexity for representing dmax . Our results reveal 12 fundamental differences between the Min-Max-Delay problem and the maximum latency problem studied in the literature, which is APX-hard and admits no PTAS unless P = NP. Moreover, there has not yet been exact pseudo-polynomialtime algorithm or constant-approximate algorithm for the maximum latency problem. 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PROTEIN THREADING BASED ON NONLINEAR INTEGER PROGRAMMING Wajeb Gharib Gharibi Department of Computer Engineering & Networks Jazan University Jazan 82822-6694, Saudi Arabia Gharibi@jazanu.edu.sa Marwah Mohammed Bakri Department of Biology Jazan University Jazan 755, Saudi Arabia Marwah890@gmail.com Abstract PDB in the past three years have similar structural folds to the ones in PDB. Protein threading is a method of computational protein structure prediction used for protein sequences which have the same fold as proteins of known structures but do not have homologous proteins with known structure. The most popular algorithm is based on linear integer programming. In this paper, we consider methods based on nonlinear integer programming. Actually, the existing linear integer programming is directly linearized from the original quadratic integer programming. We then develop corresponding efficient algorithms. Keywords: Protein Threading, Protein Structure Alignment, Integer Programming, Relaxation 1 INTRODUCTION Protein structure prediction from amino acid sequence is a fundamental scientific problem and it is regarded as a grand challenge in computational biology and chemistry. Protein threading problem also referred as the holy grail of molecular biology on the second half of the genetic code is to determine the three-dimensional folded shape (protein structure prediction) of a protein (sequence of characters drawn from an alphabet of 20 letters). It is important because the biological function of proteins underlies all life, their function is determined by their three-dimensional shape, and their shape determined by one-dimensional sequence. The prediction is made by "threading" (i.e. placing, aligning) each amino acid contained in the target sequence to a position in the template structure, and evaluating how well the target fits the template. After the best-fit template is selected, the structural model of the sequence is built based on the alignment with the chosen template. The protein threading method is based on two basic observations. One is that the number of different folds in nature is fairly small (approximately 1000), and the other is that according to the statistics of the Protein Data Bank (PDB), 90% of the new structures submitted to A general paradigm of protein threading consists of the following four steps: the construction of a structure template database, the design of the scoring function, threading alignment and threading prediction. The third step is one of the major tasks of all threading-based structure prediction programs, which mainly dedicated to solving the optimal alignment problem derived from a scoring function considering pairwise contacts. As a formal presentation of the problem, let C called core be a set of m items , called segments of length . This set must be aligned to a sequence L of N characters from some finite alphabet. Let be the position in L where starts. An alignment is called feasible threading if: 1) for all i, 2) the length characters; i.e (called gap or loop) of uncovered is bounded, say . Each feasible threading ) is scored by a function where score the placement of the segment i to a given position and is used in some experiments for scoring the gap between two consecutive segments. If the problem now is to minimize f(t) over the set F of feasible threading, one can show the equivalents with the shortest path problem between two vertices of a very structured graph. The model of protein threading problem is to minimize the objective function Subject to Where m is the number of segments, (The number are the lengths of the segments increased by the minimal number of gaps between the segments k and k+1) is the number of possible placements of each segment relative to the end of the previous one, are binary variables with meaning the segment i starts from the obuolute position of the position sequence L. Many different algorithms have been proposed for finding the correct threading of a sequence onto a structure, though many make use of dynamic programming in some form. For full 3-D threading, the problem of identifying the best alignment is very difficult (it is an NP-hard problem). Researchers have made use of many combinatorial optimization methods to arrive at solutions. There are many algorithms, for example, the protein threading software RAPTOR, which is based on linear integer programming. In this paper, we focus on developing efficient algorithms. We notice that the mathematical models used in the literatures are normally a linear integer programming, which can actually be regarded as a linearization of a quadratic integer programming problem. This motivates us to study the original quadratic integer programming directly. Recently, quadratic integer programming becomes a hot research topic in optimization society. Many mathematical tools such as conic programming are developed, with which we can construct corresponding efficient algorithms. Now, consider the zero-one quadratic programming problem P : min C T x  x T Qx s.t. h x  x Gx  g x  X  {0,1}, T T (1, 1) smaller. More tight linearization strategies are proposed in this article for further improvement. This article is organized as follows. In section 2, we shortly describe the existing efficient linearization approach. In section 3, we introduce our approach and represent the linearized model. We conclude the paper in section 4. 2 THE EXISTING EFFICIENT LINEARIZATION APPROACH Define i  min / max  min / max{ Qi x : x  X }, i, (2.1) where Q i is the i-th row of Q, and X is any suitable relaxation of X such that the problem (2.1) can be solved relatively easily.  min / max be i components  min , / max the vector with i  1, 2, ..., n, and min/ max  diag ( i min/ max ) . Similarly, define i min / max  min / max{ Gi x : x  X }, i, (2.2) and i T min / max  (min / max , i  1,..., n) , i  min / max  diag (min / max , i  1,..., n). Sherali and Smith [14] reformulated Problem P as an equivalent bilinearly constrained bilinear problem by introducing   Qx and  Gx . Linearizing the terms x i  i and x i i by s i and z i respectively, they obtained (1, 2) BP : min cT x  eT S  s. t. Qx   (2,4) (1, 3) hT x  eT z  g (2,5) Gx   (2,6) where Q and G are general symmetric matrices of dimension n  n . This problem is a generalization of unconstrained zero-one quadratic problems, zero-one quadratic knapsack problems, quadratic assignment problems and so on. It is clearly NP-hard. Linearization strategies are to reformulate the zero-one quadratic programs as equivalent mixed-integer programming problems (1.1) and (1.3) with additional binary variables and/or continuous variables and continuous constraints, see [1, 2, 3, 6, 7, 8, 9, 10, 12, 13]. Recently, Sherali and Smith [14] developed small linearizations for (1.1) - (1.3), which is more general with structure. The linearization generated by our approach is i i  min xi  si'   max xi , i (2, 3) (2,7) imin xi  Si  imax xi (1  xi ), i, (2,8) imin xi  zi'  imax xi , i, (2,9) imin (1  xi )  (i  zi' )  imax (1  xi ), i, x X where e is a conformable vector of ones and the constrains (2.7) - (2.10) comes from multiplying (2, 10) (2,11)  min     max , min    max by xi and (1  xi ). (2.12) let x be part of an optimal solution to Problem BP. Then x solves Problem P. Besides, BP can be improved by the additional cuts BP (2.3) - (2.11) has the following equivalent compact formulation T BP: min cT x  eT s   min x (2.13) i i i ( min   min  wmax ) xi  si  zi  0, i, (2.31) which is derived from multiplying i max i   i  w i max by  max{(G i  Qi )x : x  X } . (2.14) x i where w T hT x  eT z  min xg (2.15) 3 Gx   (2.16) i i 0  si  ( max   min ) xi , i, (2.17) Motivated by [15], we first reveal the relation between general quadratic and piece-wise linear terms for zero-one variables. i i 0  yi  ( max   min )(1  xi ), i, (2.18) Lemma 3.1. Let s. t. Qx  y  s   min e 0  zi  ( i max   ) xi i min   (i  zi )   x X i min i max (2.19)  ( i max   ) xi , i, (2.20) i min (2.21) A REPRESENTATION APPROACH i i i xi Qi x  max{ min xi , Qi x   max xi   max }, (3.1) i i i xi Qi x  min{ max xi , Qi x   min xi   min } (3.2) via the linear transformation Proof. Suppose i si  si   min xi , i, clearly i yi   i  si   min (1  xi ), i, zi  zi   (2.22) x , i, i min i Since the optimization and constraint senses of BP tend to push the variables s to their lower bounds and z to their upper bounds, the final relaxed version of BP was written as T BP: min cT x  eT s   min x (2.23) x  X  {0,1}n . for all i  1,..., n , 0 x i  0, the left hand side of (3.1) is and the right hand i side of (3.1) reads max{ max , Qi x }  Qi x , which is equal to the left hand side. The proof of (3.1) is completed and (3.2) can be similarly verified. Corollary 3.1. Let x X  {0,1}n . for all i  1,..., n, i i i max{ min xi , Qi x   max xi   max }  si  0  y  [ max   min ](e  x) (2.25)  min{ s0 (2.26) (2.27) if and only if Gx  z  min (2.28) si  xi Qi x. 0  z  [ max   min ] x (2.29) x X , (2.30) T T min by deleting the upper bounding inequalities for s and   z in (2.17) and (2.20), and combining (2.16) with (2.20). It was shown in [14] that Problems BP and P are equivalent in the sense that for each feasible solution to one problem, there exists a feasible solution to the other problem having the same objective value. Furthermore, (3.3) x , Qi x   x   }, i max i h xe z  x  g T becomes x i  0, it must hold that x i  1, the right hand (2.24) s. t. Q x  y  s   min e side i max{0, Qi x   max }  0 . On the other hand, if i min i i min (3.4) Combining (3.1) with (3.2), we have i i i xi Qi x  max{ min xi , Qi x   max xi   max }  min{ x , Qi x   i max i (3.5) x   }, i min i i min The above results hold true for G i and  ¸ defined before. Linearization based on Corollary 3.1 is just BP (2.3) - (2.11), where the linear inequalities (2.7) - (2.8) is nothing but (3.3). We remark here the four inequalities implied by (3.3) were first introduced in [8]. Actually, not all inequalities (3,3) are necessary in the final linearized model. To see this, below we first introduce the principle of reformulating zero-one quadratic programs into piecewise linear programs. Generally, for continuous programs, we have n hT x   zi  g , (3.9) i 1 i zi  max xi , (3.10) i i zi  Gi x  min xi  min , (3.11) Proposition 3.1. Any convex program with linear or piece-wise linear objective function and constraints is equivalent to a linear program in the sense that there is a one-to-one projection between both feasible solutions. since (3.9)-(3.11) is a relaxation of (3.7) and (3.9)-(3.11) also implies (3.7). Proof. We notice that min f ( x ) is equivalent to Now we can obtain a linearization for (3.6)-(3.8), which is similarly to BP except that we do not require y  0 and z  0 . In other words, they are redundant in min t s.t. t  f ( x )  0 BP Without loss of generality we assume that the objective function is linear. The constraint set is convex and characterized by piece-wise linear inequalities. It follows that it is convex polyhedral, which must have linear expression. It is easy to see that the equivalence of Proposition 3.1 holds if we restrict the variables to be zeros or ones. Next we show the existence of such equivalent 'convex' piece-wise linear program for zero-one quadratic minimization problem. Proposition 3.2. For any zero-one quadratic minimization problem, there is an equivalent zero-one piece-wise linear program with convex objective function and constraints. Proof. Clearly, the maximum of several linear functions is convex and the minimum is concave. Then (3.1) and (3.2) in Lemma 3.1 provide the convex and concave formulations, respectively. Therefore, for any given zero-one quadratic minimization problem, we can obtain an equivalent convex piece-wise linear program by using (3.1) and/or (3.2). Note that we use (3.1) and (3.2) simultaneously only when handling equality constraints, see also Corollary 3.1. Now we can see that (1.1) - (1.3) has the following equivalent formulation n i i i min cT x   max{ min xi , Qi x   max xi   max } (3.6) i 1 n i i i s.t. hT x   min {max xi , Gi x  min xi  min }  g , (3.7) i 1 x  X  {0,1}n . Linearizing (3.6)-(3.8) becomes very easy. For example, (3.7) is equivalent to (3.8) . Finally, we point out that the non-necessity of inequalities such as y  0 and z  0 was also observed in [1, 2]. Actually, the linearization generated by our convex piece-wise approach coincides theirs. 4 CONCLUSIONS In this article, we defined the protein folding problem and discussed its solution through presenting small linearizations for the zero-one quadratic minimization problem. We present the equivalence of quadratic terms and piece-wise linear terms for zero-one variables. There are two piece-wise formulations, convex and concave cases. We show the smaller linearization is based on the convex piece-wise objective function and constraints. Linearization generated by our approach is smaller than that in [14]. Our approach can be easily extended to linearize polynomial zero-one minimization problems which have many applications, particularly in biological computing problems. 5 REFERENCES [1]. 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M. and Prokopyev O.A., “A new linearization technique for multiquadratic 0-1 programming problems,” Operations Research Letters, vol.32, pp.517-522, 2004. [7]. Fortet R., “L'algebre de boole et ses applications en recherche operationnelle,” Cahiers du Centred'Etudes de Recheche Operationnelle, vol.1, pp. 5-36, 1959. [8]. Glover F., “Imporved linear integer programming formulations of nonlinear integer problems,” Manage. Sci., vol. 22, no. 4, pp. 455-460, 1975. [9]. Glover F. and Woolsey E., ”Further reduction of zero-one polynomial programming problems to zero-one linear programming problems,” Oper. Res., vol. 21, no. 1, pp. 156-161, 1973. [10]. Glover F. and Woolsey E., “Converting the 0-1 polynomial programming problem to a 0-1 linear program,” Oper. Res., vol.22, no. 1, pp.180-182, 1974. [11]. McCormick P., ”Computability of global solution to factorable nonconvex problems: Part I - Convex underestimating problems,” Mathematical Programming, vol.10, pp.147-175, 1976. [12]. 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JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG 68 Timing and Code Size Optimization on Achieving Full Parallelism in Uniform Nested Loops Y. Elloumi, M.Akil and M.H. Bedoui Abstract- Multidimensional Retiming is one of the most important optimization techniques to improve timing parameters of nested loops. It consists in exploring the iterative and recursive structures of loops to redistribute computation nodes on cycle periods, and thus to achieve full parallelism. However, this technique introduces a large overhead in a loop generation due to the loop transformation. The provided solutions are generally characterized by an important cycle number and a gr eat code size. It represents the most limiting factors while implementing them in embedded systems. In this paper, we present a new Multidimensional Retiming technique, called “Optimal Multidimensional Retiming” (OMDR). It reveals the timing and dat a dependency characteristics of nodes, to minimize the overhead. The experimental results show that the average improvement on t he execution time of the nested loops by our technique is 19.31% compared to the experiments provided by an existent Multidimensional Retiming Technique. The average code size is reduced by 43.53% compared to previous experiments. Index Terms— Graph Theory, Multidimensional Applications, Optimization, Parallelism and concurrency. ——————————  —————————— 1 INTRODUCTION T he design of real time systems should respect many constraints such as the execution time and code size, which require using optimization techniques. The Retiming presents one of these techniques which can be used to add and remove registers in order to provide a more efficient circuit [1]. The increased complexity of such application leads to the frequent use of a nested iterative and recursive loops. Such applications can be modeled as “Multidimensional Data Flow Graph” (MDFG). The standard software pipelining techniques can only be used to optimize a one-dimensional loop. When they are applied to optimize nested loops, the performance improvement is very limited [8]. Other works are proposed to offer an optimization technique taking advantage of the multiple nested loops, which is called “Multi-Dimensional Retiming” (MDR). It aims achieving full parallelism of uniform nested loops. It consists in scheduling the MDFG with the minimum cycle period and modifying the execution order of nodes, such as each one is executed in a separate cycle. But, Achieving full parallelism requires adding a large code overhead [2],[3]. It dramatically increases the whole code size of the provided MDFG. Furthermore, this extra code requires a significant cycle number to be executed outside the loop body. Thus, the provided solution does not allow achieving an application with an adequate execution time and a code size. It represents a limiting factor to implement the provided MDFG in a real-time embedded system. We propose in this paper a new technique of MDR, called “Optimal Multidimensional Retiming”. It allows ———————————————— • Y. Elloumi is a P.H.D. student in Paris-Est University, 93162 Noisy le Grand Cedex, France. • M. Akil is Professor at computer science department, ESIEE, Paris, 93162 Noisy le Grand Cedex, France. • M.H. Bedoui is Professor in faculty of medicine of Monastir, University of Monastir, 5019, Monastir, Tunisia, redistributing optimally the nodes on cycle periods, while scheduling the MDFG with the minimal cycle period. This technique significantly allows optimizing the number of period cycles (notably the execution time) and the code size, by exploring the execution time and data dependency between nodes belonging to the MDFG. Thus, it provides enhanced solutions, compared to the existent techniques. The rest of the paper is organized as follow. In section 2, we give an overview of MDFG formalism. In section 3, we list the existent MDR techniques and their constraints and limits. In section 4, we present the theory of the “Optimal Multidimensional Retiming” technique by describing the principles and basics concepts, and proposing the correspondent algorithms. Experimental results are presented in section 5, followed by concluding remarks in section 6. 2 MULTIDIMENSIONAL DATA FLOW GRAPH The Multidimensional Data Flow Graph (MDFG) is an extension of the classic data flow graph that allows to represent a nested iterative and recursive structures. It is modeled by a node-weighted and edge-weighted directed graph such as 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡), where V is the set of computation nodes, 𝐸𝐶 𝑉 × 𝑉 is the set of edges, and 𝑑(𝑒𝑖 ) is a function from E to Z n , representing the multidimensional delay between two nodes, where n is the number of dimensions (loops), and 𝑡(𝑣𝑗 ) is a function from V to the positive integers, representing the computation time of the node 𝑣𝑗 . For a MDFG with n dimensions, each edge 𝑒 ∶ 𝑣𝑖 → 𝑣𝑗 is characterized by a delay where 𝑑(𝑒) = (𝑐1 , 𝑐2 , … , 𝑐𝑛 ). The value 𝑐𝑘 represents the difference between the execution iteration of 𝑣𝑗 and the execution iteration of 𝑣𝑖 of the loop 𝑘. We show in Fig.1.a a two-dimensional Data Flow Graph (2DFG) corresponding the Wave Digital Filter described in Algorithm 1, which is composed of two nested loops. The execution of each node in V exactly represents one iteration, which is the execution of one instance of the loop body. Each edge belonging to the 2DFG shown in Fig.1.a is labeled by a delay 𝑑(𝑒) = (𝑑. 𝑥, 𝑑. 𝑦). Both terms « 𝑑. 𝑥 » and « 𝑑. 𝑦 » JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG represent the difference between the iteration number executing 𝑣𝑗 and the iteration number executing 𝑣𝑖 , in the outermost loop as well as in the innermost loop [9]. For an edge 𝑒 : 𝑣𝑖 → 𝑣𝑗 , the delay 𝑑(𝑒) = (0, 𝑥) consists in the execution of 𝑣𝑖 and 𝑣𝑗 in the same iteration of the outermost loop. For the innermost loop, if the node vi is executed in the iteration 𝑘, the node 𝑣𝑗 is executed in the ALGORITHM 1 WAVE DIGITAL FILTER 0: For i from 0 to m do 1: For j from 0 to n do 2: D(i,j)= B(i -1 , j+1) × C(i -1 , j-1) 3: A(i,j)= A(i,j) × 5 4: B(i,j)= A(i,j) + 1 5: C(i,j)= A(i,j) + 2 6: End for 7: End for G illustrates the dependencies between copies of nodes representing the MDFG G, such as the CDG shown in Fig.2.b which corresponds to the MDFG G in Fig.1.a. A node in CDG is a computational cell that represents a complete iteration. The CDG of a nested loop is bounded by the loop indexes. A schedule vector s defines a sequence of execution in the cell dependency graph. The CDG shown in Fig.2.b, can be executed by a row-wise execution sequence, i.e., the schedule vector 𝑠 = (1,0). A legal MDFG 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡) is realizable if there exists a schedule vector s for the cell dependency graph in respect to G; i.e., 𝑠 × 𝑑(𝑒) ≥ 0, 𝑒 ∈ 𝐸, and no cycle exists in its corresponding CDG. Note that delay vectors (0,1) and (0, −1) are both legal in respect to the schedule vector 𝑠 = (1,0), but they create cycles in cell dependency graph. 3 MULTIDIMENSIONAL RETIMING iteration(𝑘 − 𝑥). An edge with zero delay 𝑑(𝑒) = (0,0) represents a data dependency in the same iteration, such as the edges 𝐷 → 𝐴, 𝐴 → 𝐵 and 𝐴 → 𝐶 as shown in Fig. 1.a. 𝑒 69 3.1 Principles The retiming technique consists in redistributing delays in We use the notation 𝑣𝑖 → 𝑣𝑗 to indicate that 𝑒 is an edge 𝑝 from 𝑣𝑖 node to 𝑣𝑗 node, and 𝑣𝑖 ⇒ 𝑣𝑗 to mean that 𝑝 is a path from 𝑣𝑖 node to 𝑣𝑗 node. The delay vector of a path 𝑝: 𝑣𝑖 𝑒𝑚 𝑒𝑚+1 𝑒𝑛 �� 𝑣𝑖+1 �⎯⎯� … → 𝑣𝑗 is 𝑑(𝑝) = ∑𝑘=𝑛 and the total 𝑘=𝑚 𝑑(𝑒𝑘 ) 𝑘=𝑗 computation time of a path 𝑝 is 𝑡(𝑝) = ∑𝑘=𝑖 𝑡(𝑣𝑘 ). The period during which all computation nodes in iteration are executed, according to existing data dependencies and without resource constraints, is called a cycle period. The cycle period C(G) of an MDFG is the maximum computation time among paths that have a zero delay. For example, assuming that each node is executed in one time unit 𝑡(𝐴) = 𝑡(𝐵) = 𝑡(𝐶) = 𝑡(𝐷) = 1, the MDFG of Fig.1.a has C(G) = 3. It can be measured through the paths 𝑝: 𝐷 → 𝐴 → 𝐵 or 𝑝: 𝐷 → 𝐴 → 𝐶, as shown in the iteration scheduling illustrated in Fig.1.b. Each set of nodes belonging to the same iteration are modeled by a different motif. The execution pattern of a nested loop can be illustrated by iteration space as shown in Fig.2.a. Each cell in the iteration space is a copy of the MDFG. The marked cell, labeled by (0,0), is the first iteration to be executed. This ……. B C A e2 (0,0) B C (1,-1) e4 e5 (1,1) (a) D B C A D B (0,0) e3 D T im e A (0,0) e1 C A D (b) Fig.1. (a) MDFG of Wave Digital Filter; (b) Iteration scheduling of the MDFG in Fig.1.a. graph is transformed on an acyclic graph, called cell dependency graph (CDG), allowing to show clearly the execution sequence of a nested loop. The CDG of an MDFG (a) (b) Fig.2. (a) Iteration space of the MDFG in Fig.1; (b) The cell dependency graph. the graph. This technique can be applied on a data flow graph to minimize the cycle period in a polynomial time. The delays are moved around in the graph in the following way: a delay unit is drawn from each of the incoming edges of 𝑣, and then added to each of the outgoing edges of 𝑣, or vice versa [1]. In the case of MDFG, it consists in redistributing the execution of nodes on the iterations. The retiming vector 𝑟(𝑢) of a node 𝑢 ∈ 𝐺 represents the offset between the original iteration containing 𝑢, and the one after retiming. Note that the retiming technique preserves data dependencies of the original MDFG. Therefore, we have 𝑑𝑟 (𝑒) = 𝑑(𝑒) + 𝑟(𝑢) − 𝑟(𝑣) for every edge and 𝑑𝑟 (𝑙) = 𝑑(𝑙) for every cycle 𝑙 ∈ 𝐺. After retiming, the execution of the node u in the iteration i is moved to the iteration 𝑖 − 𝑟(𝑢). We show in Fig.3.a the MDFG 𝐺𝑟 = (𝑉, 𝐸, 𝑑𝑟 , 𝑡) of the wave digital filter after applying the retiming function 𝑟(𝐷) = (0,1). When a delay is pushed through node 𝐷 to its outgoing edge as shown in Fig.3.a, the actual effect on the Algorithm 2 of the new MDFG is that the 𝑖 th copy of 𝐷 is shifted up and is executed with (𝑖 − (0,1)) th copy of nodes 𝐴, 𝐵, and 𝐶. The original zero-delay edge 𝐷 → 𝐴 in Fig.1.a now has a delay (0,1) after retiming as shown in Fig.3.a. Node 𝐷 in the new loop body has not any data dependency with other nodes executed in the same cycle. So, node 𝐷 can be executed in parallel to node 𝐴, as shown in the iteration scheduling of Fig.3.b. Thus, the cycle period is reduced from three to two time units. P P JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG 70 In fact, every retiming operation corresponds to a software pipelining operation. When 𝑟(𝑢) delay units are pushed through a node u, every copy of this node is moved by 𝑟(𝑢) iterations. Hence, a new iteration consists in redistributing the execution nodes into different iterations. ALGORITHM 2 WAVE DIGITAL FILTER AFTER RETIMING BY THE FUNCTION R(D)=(0,1) 0: For i from 0 to m do 1: D(i,0) = B(i-1 , 1) × C(i-1 , -1) 2: For j from 0 to n-1 do 3: D(i,j+1) = B(i-1 , j+2) × C(i-1 , j) 4: A(i,j) = D(i,j) × 5 5: B(i,j) = A(i,j) + 1 6: C(i,j) = A(i,j) + 2 7: End for 8: A(i,n) = D(i,n) × 5 9: B(i,n) = A(i,n) + 1 10: C(i,n) = A(i,n) + 2 11:End for (a) Fig.4. (a) The iteration space the retimed MDFG in Fig.3; (b) The cell dependency graph. Hence, the MDR technique aims to transform a realizable MDFG G on MDFG 𝐺𝑟 in a way that 𝐺𝑟 is still realizable. Using such concepts, the basic conditions for legal multidimensional retiming are defined in the following e2 (0,0) C (1,-1) e4 e5 (1,1) (0,1) e3 (a) D C (1,-3) e4 e5 (1,-1) D B C A D B C A D B C A D A D D (0,1) e3 (a) Prologue (b) Fig.5. (a) Fully parallelized graph of the MDFG in Fig.1; (b) Iteration scheduling of the MDFG in Fig.5.a lemma [2]. B C A D Lemma 1. Let 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡) be a realizable MDFG, r a multidimensional retiming, and s a schedule vector for the retimed graph 𝐺 𝑟 = (𝑉, 𝐸, 𝑑𝑟 , 𝑡), then T im e (0,0) e1 e2 (0,1) B ……. ……. r(D)=(0,1) A A (0,1) e1 r(D)=(0,2) r(A)=(0,1) T im e Some nodes are shifted out of the loop body to provide the necessary data for the iterative process, which is called prologue. Correspondingly, some nodes will be executed after the loop body to complete the process, which is called epilogue. Using MD retiming function r, we can trace the pipelined nodes and also measure the size of the prologue and epilogue. For node v with retiming 𝑟(𝑣) = (𝑖, 𝑗), there are i copies of node v appearing in the prologue of outer loop, and j copies of node v in the prologue of the innermost loop. The B (b) B C A D B C A D 1. 2. D Prologue (b) Fig.3. (a) MDFG of algorithm 2; (b) Iteration scheduling of the MDFG in Fig.3.a number of copies of a node in the epilogue can also be derived in a similar way. The iteration space of the retimed MDFG shown in Fig.4.a with retiming 𝑟(𝐷) = (0,1) clearly shows that one copy of node D is pushed out of the loop body on j-dimension, and becomes prologue for the innermost loop. The corresponding cell dependency graph is shown in Fig.4.b. It is known that an MDFG can always be fully parallelized by applying successively the MD retiming functions 𝑟(𝐷) = (0,2) and 𝑟(𝐴) = (0,1), which is illustrated in Fig.5.a. We note that the retimed MDFG has non-zerodelay on each edge. It implies that the nodes belonging to the same iteration in the original loop body are distributed into three cycle periods. The MDFG is then scheduled with the minimal cycle period equal to one time unit, as schematized the iteration scheduling of Fig.5.b. To achieve a realizable MDFG after retiming, the legality condition,𝑠 × 𝑑(𝑒) ≥ 0, has to be satisfying, and there should not exist any cycle in the cell dependency graph of the MDFG. 3. 4. for any path , we have 𝑑𝑟 (𝑝) = 𝑑(𝑝) + 𝑟(𝑢) − 𝑟(𝑣) for any cycle 𝑙 ∈ 𝐺 we have 𝑑𝑟 (𝑙) = 𝑑(𝑙) 𝑒 for any edge 𝑢 → 𝑣, 𝑑𝑟 (𝑒) × 𝑠 ≥ 0 there is no cycle in the DG equivalent to the MDFG G. The selection of a legal multidimensional Retiming function is based on the edge delay of the MDFG. The approach proposed in [2],[3] consists in defining a scheduling subspace S for a realizable MDFG 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡). It represents the space region where there exist schedule vectors that realize 𝐺; i.e., if schedule 𝑠 ∈ 𝑆 then 𝑠 × 𝑑(𝑒) ≥ 0 for any 𝑒 ∈ 𝐸. In fact, the multidimensional retiming technique means to decrease zero-delay edges. Thus, a strictly positive scheduling subspace 𝑠 + is the set al all vectors 𝑠 ∈ 𝑆 where 𝑠 × 𝑑(𝑒) > 0 for every 𝑑(𝑒) ≠ (0,0, … ,0). The method of predicting a legal multidimensional retiming is introduced in the next theorem. Theorem 1. let 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡) be a realizable MDFG, 𝑠 + a strictly positive scheduling sub-space of G, s a scheduling vector in 𝑠 + , and 𝑢 ∈ 𝑉 a node with all incoming edge having nonzero delay. A legal MD retiming r of u is any vector orthogonal to s. 3.2 Multidimensional Retiming techniques We describe in this section the existent multidimensional retiming techniques. They are characterized by achieving full parallelism by providing the MDFG with no zero-delay edge [2],[3],[7]. a) Incremental Multidimensional Retiming JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG This technique is based on selecting a set of nodes that can be retimed by the same multidimensional retiming function, as described in the following corollary. Corollary 1. Given a realizable MDFG 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡), 𝑠 + a strictly positive scheduling sub-space of G, s a scheduling vector ALGORITHM 5 CHAINED MULTIDIMENSIONAL RETIMING Input : a realizable MDFG G =(V,E,d,t), Output : a realizable MDFG Gr =(V,E,dr,t) without d(e)=(0,0, … ,0) 0: Begin 1: Find a legal MDR function r as describedin steps 2 and 3 in algorithm 3 2: Provide the multi -chain graph and the maximal length of chain K, as indicated in algorithm 4 3: For each node v with label i do 4: Apply the MDR function (k-i)× (r) 5: End for 6: End 71 nodes assigned by 𝐴𝑗 . Fig.7 illustrates the full parallelized MDFG of the IIR filter after applying the Incremental Multidimensional Retiming. The steps of algorithm 3 are repeated four times, where in each one a different MD retiming function is applied. The fully parallelized MDFG is showed in Fig.7 where all edges are non-zero delay. a) Chained Multidimensional Retiming [2],[3] This technique allows obtaining the full parallelism solution by defining just one MDR function. It is based on the following corollary. in 𝑠 + , and an MD retiming function r orthogonal to s, if a set 𝑋𝐶𝑉has all incoming edges nonzero, then 𝑟(𝑋) is a legal MD retiming. Thus, this technique consists in defining a schedule vector s as described in definition 1, and chooses an MDR function orthogonal to s. This chosen function is applied to each node respecting the previous corollary. Those steps are ALGORITHM 3 INCREMENTAL MULTIDIMENSIONAL RETIMING Input : a realizable MDFG G =(V,E,d,t) Output : a realizable MDFG Gr=(V,E,dr ,t) without d(e) = (0,0, … ,0) 0: Begin 1: While exist zero-delay edge in the graph Do 2: Find a scheduling vector s=(s.x,s.y), that s.x+s.y is minimum 3: Choose a MDR function 4: Apply the selected MDR function to any nodes that has all incoming edges with nonzero delays and at least one outgoing edge with zero delay 5: End while 6: End repeated incrementally, until all zero-delay edges are transformed, as described in algorithm 3. We apply the algorithm above to the 2DFG of Infinite Fig.6. MDFG of IIR Filter. Impulse Response filter (IIR) that is illustrated in Fig.6. It is composed by multiplier nodes assigned by 𝑀𝑖 and adder Fig.7. IIR Filter MDFG after Incremental MDR. Corollary 2. [2] Given 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡), 𝑆 + a strict positive scheduling subspace for G,s a scheduling vector 𝑆 + , a MD retiming function orthogonal to s, a set 𝑋𝐶𝑉 which all incoming edges nonzero, and an integer value 𝑘 > 1, then (𝑘 × 𝑟) (𝑋) is a legal MD retiming. Thus, it applies the MD retiming to successive nodes in a path where each node has a retiming function multiple of the selected retiming value smaller than its predecessor nodes. ALGORITHM 4 MULTI-CHAIN GRAPH CONSTRUCTION Input : a realizable MDFG G =(V,E,d,t) Output : Labeled Multi -chain Graph 0: Begin 1: Remove all non-zero delay edges from the MDFG 2: For each chainCH do 3: Compute the length L of CH 4: For each node starting from the last to the first do 5: Labeled the node by L 6: L=L-1 7: End For 8: End For 9: End This technique starts by transforming the MDFG on Multi-chain Graph as described in Algorithm 4. Each chain represents a node succession where all interconnected edges between them are zero-delay. Each node is labeled by a level whose the value is greater than its predecessor node and smaller than its successor ones. In the case of MDFG of IIR filter, the red integers above each node of Fig.8 represent the level values that are labeled after executing algorithm 4. Therefore, the multi-chain maximum length of 𝐺 is 4. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG So, The technique proceeds to retime all the labeled nodes by a MD retiming function (𝑘 − 𝑖) × (𝑟), as described in algorithm 5. We present in Fig.8 the full parallelized graph of the IIR filter after applying the chained multidimensional retiming. The algorithm starts by finding the MDR function which is equal to (1, −1). We note that all zero delay edges in the original MDFG are assigned by a delay vector equal to the MDR function. Until now, no research works have been interested in comparing results provided by the techniques described above. The random choice of the scheduling vector does not allow defining the technique providing the optimal solution. However, based on their approaches, Chained MDR is generally more performing than the Incremental MDR. The first one consists in defining just one scheduling vector, which is executed in 𝑂(|𝐸|); while the second requires defining scheduling vector for each iteration of algorithm 4, which is executed in 𝑂(|𝑉|). b) SPINE (Software PIpelining of NEsted loops) Multidimensional Retiming This technique tries to provide a more optimal MDFG in terms of execution time and code size than those described above. It proceeds to remove all delays such as (0, 𝑘) by merging them in a delay such as (𝑖, 𝑗). This modification is applicable only if the MDFG contains at least one edge with a 72 loop. Since this limit is directly associated to the size of the iteration space, it is called spatial constraint and it is formally defined as follows. ALGORITHM 6 SPINE-FULL ALGORITHM Input : a realizable MDFG G =(V,E,d,t) Output : a realizable MDFG Gr =(V,E,dr,t) fully parallelized with minimum code size 1: Begin 2: If s=(1,0) is legal then 3: Apply s=(1,0) 4: Else If s=(0,1) is legal and d(e)×(0,1)>=0 then 5: Apply s=(0,1) 6: Else If s=(1,1) is legal and d(e)×(1,1)>=0 then 7: Apply s=(1,1) 8: Else 9: Choose a legal scheduling vector s such as d(e)×s>=0, for any edge and | sx| +| sy| is minimal 10: End Definition 1. Let a MDFG G contains k-level nested loop N, controlled by the set of indices 𝐼 = {𝑖0 , 𝑖1 , … , 𝑖𝑘 }, whose values vary, in unitary increments, in the range 𝐿 = {𝑙0 , 𝑙1 , … , 𝑙𝑘 } to 𝑈 = {𝑢0 , 𝑢1 , … , 𝑢𝑘 } where L is the set of lower boundaries for the indices and U is the set of maximum values, such as 𝑙𝑗 ≤ 𝑖𝑗 ≤ 𝑢𝑗 , then the spatial constraint Sc of the loop is defined as: 𝑆𝑐 = [(𝑢0 − 𝑙0 + 1), (𝑢1 − 𝑙1 + 1), … , (𝑢𝑘 − 𝑙𝑘 + 1)] This definition allows establishing the relation between the maximum retiming operation and the spatial constraint according to the following lemma. Lemma 2. Given a k-level loop N with spatial constraint 𝑆𝑐 = [𝑠0 , 𝑠1 , … , 𝑠𝑘 ]. The multi-dimensional retiming technique will be able to achieve full parallelism of the loop body instructions if the maximum retiming vector r applied to any node u, 𝑟(𝑢) = (𝑟0 , 𝑟1 , … , 𝑟𝑘 ) satisfies the following condition: 𝑟𝑗 < 𝑠𝑗 𝑠𝑢𝑐ℎ 𝑎𝑠 0 ≤ 𝑗 ≤ 𝑘 Fig.8. IIR Filter MDFG after Chained MDR. delay equal to (𝑖, 𝑗) such as 𝑖 > 0. It consists in finding a scheduling vector 𝑠 and a retiming function 𝑟 orthogonal to 𝑠, as described in algorithm 6, to provide a minimal overhead. 3.3 Multidimensional Retiming Constraints [5] We describe in this paragraph the algorithmic constraints which must be taken into account to achieve full parallelism. These constraints come from the ratio between loop bounds and a number of MDR functions to apply. For example, consider the multi-dimensional data flow graph in Fig.1; it is easy to verify that if 𝑑(𝑒4) = (0, 𝑘) where 𝑘 < 3, retiming 𝐷 and 𝐴 by some vector (0, 𝑝) will not satisfy the goal of re-distributing the delays among all edges in the graph. The same will happen if 𝑑(𝑒5) = (𝑚, 0) where 𝑚 < 3 for any retiming vectors of the form (𝑞, 0). Thus, if the loop has only one occurrence, i.e., the loop boundaries are both 1, then no parallelism can be obtained. This last constraint is equally applicable to a software or hardware implementation of the retimed loop. This study begins by evaluating the constraints imposed by the limitation on the number of iterations comprising the 3.4 Limitations of existing techniques We have shown that nested loops can always be fully parallelized using MD retiming. The presented techniques of MD retiming are a polynomial time algorithm that fully parallelizes a given MDFG 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡) by selecting a legal schedule vector s with 𝑠 × 𝑑(𝑒) > 0, 𝑒 ∈ 𝐸, and a retiming vector 𝑟 where 𝑟 is orthogonal to 𝑠. Each MD retiming techniques presented above shows that the selected 𝑠 is a legal schedule vector for the retimed graph where 𝑑𝑟 (𝑒) ≠ 𝑑(𝑒) ± 𝑘. 𝑟. However, Multidimensional retiming techniques imply a large overhead of the generated code. It is caused by several aspects of loop transformation. First, the code size is increased because of the large code sections of the prologue and epilogue produced in all the loop dimensions. Second, the computation of the new loop bounds and loop indexes need to be recomputed [10]. Moreover, the execution of the prologue and epilogue section is not fully parallel, which requires a considerable period cycle number compared to that required by the loop body. Those disadvantages are aggravated in terms of the retiming vector value, and the number of the multidimensional retiming function. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG Each Multidimensional Retiming techniques have a specific approach to choose the retiming function in order to decrease the overhead size. Chained and incremental Multidimensional retiming techniques proceed to chose a scheduling vector 𝑠 = (𝑠. 𝑥, 𝑠. 𝑦), where 𝑠. 𝑥 + 𝑠. 𝑦 is minimum. The SPINE technique tries to modify the MDFG with the intention of applying an MDR function that skews the minimum column-wise or/and row-wise. Despite providing an optimal solution, it is reliable only in the particular case of MDFG. In opposite cases, it applies the same approach than other techniques. However, all existent techniques consist in retiming each node of the MDFG having an out-coming edge with zerodelay: if a path p as 𝑑(𝑝) = (0,0, … ,0) is composed by n nodes, any technique applies (𝑛 − 1) MDR function to achieve full parallelism. But, overhead consequences are sudden after applying each MDR function: the more the number of MDR function increases, the more the consequences are dramatic. As a result, the provided solution becomes very complicated and not sufficient to be implemented in embedded systems. Therefore, the existing MD retiming techniques, although achieving full parallelism, are not suitable for software nested loops. 4 THEORY OF OPTIMAL MULTIDIMENSIONAL RETIMING In this section, we present the theoretical foundation of our proposal MDR technique “Optimal Multidimensional Retiming”. It aims at minimizing MDR functions by exploring execution times and data dependency of nodes, while achieving full parallelism. 3.1 Principle Multidimensional retiming techniques allow scheduling the MDFG with a minimal cycle period. For any path p 𝑝: 𝑣𝑖 𝑒𝑚+1 body is executed in 5 cycles. We illustrate in Fig.9 the static schedule of such an iteration after achieving full parallelism by the chained multidimensional retiming which is retimed by the function 𝑟 = (−1,1). The nodes belonging to the same iteration are modeled by gray circles. Each gray node has not any data dependency with any other node executed in the same cycle period. However, these scheduling shows that provided data by some nodes are not consumed immediately. For example, nodes 𝐴5 and 𝐴7 are executed in just one time unit and their provided values are consumed two times units later. We conclude that cycle periods are not exploited optimally: (more than 66% of the cycle periods (𝑖 − 2, 𝑗 + 2), (𝑖 − 3, 𝑗 + 3) and (𝑖 − 4, 𝑗 + 4) are not used). However, they allow executing more than one node, due to the difference between execution time. Let us try to execute nodes 𝐴5 and 𝐴7 in the same cycle period as nodes 𝐴1, 𝐴2 and 𝐴6, as schematized in Fig.10. The correspond MDFG with new delay values is illustrated in Fig.11. This transformation results in a legal MDFG that respects all conditions of lemma 1. Furthermore, it still keeps a fully parallelized execution, while preserving data dependency for the whole application. Compared to Fig.8, this transformation can be considered as decreasing the number of MDR functions by depriving nodes 𝐴5 and 𝐴7 to be retimed. In fact, minimizing the number of MDR functions implies decreasing the overhead of the generated code. It consists in reducing the correspondent prologue, epilogue M5 M6 M7 M8 M1 M2 M3 A3 M5 M6 M7 M8 M1 M2 M3 Theorem 2. Let 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡) a MDFG, the minimal value of cycle 𝑐𝑚𝑖𝑛 for the G graph is: 𝑐𝑚𝑖𝑛 = 𝑚𝑎𝑥{𝑡(𝑣𝑖 ), 𝑣𝑖 ∈ 𝑉} We choose to model the MDFG of IIR filter with different execution time of nodes such as of 𝑡(𝑀𝑖 ) = 3 and 𝑡(𝐴𝑗 ) = 1. The minimal cycle 𝑐𝑚𝑖𝑛 of this graph is equal to the execution time of the multiplication node. Applying any MDR techniques to the IIR filter results in the fact that each iteration belonging to the original loop A4 A1 A2 A6 A5 A7 A8 Cycle (i-3 , j+3) M4 𝑒𝑛 �� 𝑣𝑖+1 �⎯⎯� … → 𝑣𝑗 of MDFG, they proceed to execute each node 𝑣𝑘 where 𝑖 ≤ 𝑘 ≤ 𝑗 in a period cycle separately. These approaches can be generalized into general-time cases [2] [4]. In fact, the computation nodes belonging to a data flow graph have not generally the same execution times. These depend on the kind of operation to be done; for example, a multiplication node needs usually more clock period than an adder node. In this case, the minimal cycle period should be fixed differently. A cycle period represents a time interval leading to execute computation nodes. The minimal value of a period cycle can be defined as the smaller time interval that allows executing any node belonging to the MDFG. Thus, the minimal cycle period should be equal to the maximal execution time of node, as described in theorem 2 [1]. Cycle (i-4 , j+4) M4 A3 A4 A1 A2 A6 A5 A7 A8 T im e 𝑒𝑚 73 M5 M6 M7 M8 M1 M2 M3 Cycle (i-2 , j+2) M4 A3 M5 M6 M7 M8 M1 M2 M3 M6 M7 M8 M1 M2 M3 A1 A2 A6 A5 A7 A8 Cycle (i-1 , j+1) M4 A3 M5 A4 A4 A1 A2 A6 A5 A7 A8 Cycle (i , j) M4 A3 A4 A1 A2 A6 A5 A7 A8 Fig.9. Iteration scheduling after chained MDR. and the loop bound and index instructions. Moreover, it results in decreasing the number of cycle periods required to execute any iteration from 5 to 4, while respecting a fully parallelized execution. This minimization of cycle periods implies a similar minimization on the execution time of the whole application. Thus, this minimization of MDR functions leads to improve the performance of the provided full parallel solution. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG Our technique proceeds to share the MDFG on zerodelay paths. It leads to maximize the node number in such a A8 M5 M6 M7 M8 M1 M2 M3 A3 A4 A2 A1 Cycle (i-2 , j+2) A7 A5 M4 A6 A8 M5 M6 M7 M8 M1 M2 M3 Cycle (i-1 , j+1) A7 A5 M4 T im e Such modification can be defined as applying the MDR to path p that can be composed of several nodes where 𝑡(𝑝) is smaller than 𝑐𝑚𝑖𝑛 . The more the paths contain nodes, the more the MDR function number decreases. Thus, we propose a new MDR approach which consists in applying MDR function to a path of nodes, which can be executed in the same period cycle, instead of applying the MDR to each node separately. Our approach is based on selecting the path with the maximal nodes to achieve full parallelism with a minimal number of MDR functions. We start by computing the 74 A3 A4 A2 A1 A6 A8 M5 M6 M7 M8 M1 M2 M3 A5 M4 A3 M5 M6 M7 M8 M1 M2 M3 A4 A5 M4 A4 A6 A2 A1 A6 M5 M6 M7 M8 M1 M2 M3 M4 A5 A8 A3 A4 A2 A1 Cycle (i-1 , j+1) A7 A6 M6 M7 M8 M1 M2 M3 M4 A5 A8 A3 A4 A1 Cycle (i , j) A7 A2 A6 A8 Fig.10. Iteration scheduling after collecting nodes in cycle (i-2, j+2). Fig.11. MDFG of Fig.10. minimal period cycle 𝑐𝑚𝑖𝑛 , extracting the paths with maximal nodes from the MDFG while keeping their execution in 𝑐𝑚𝑖𝑛 , and applying the MDR function to the extracted paths. 3.2 Basics concepts Our technique consists in retiming a path of nodes that can be executed in the same cycle period. It means that those nodes are executed in the same iteration of the original loop body; i.e., all edges belonging to the path have zero-delay. We call such a kind of path “zero-delay path”, which is indicated in theorem 3. 𝑒𝑚+1 M1 M2 M3 A4 A1 Cycle (i , j) A7 A5 M4 A2 A6 Definition 2. Let 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡), 𝑐𝑚𝑖𝑛 the minimal period cycle of G. The optimal multidimensional retiming consists in retiming a set of path p follows : 2. 3. 𝑒𝑛 Theorem 3. Let 𝑝: 𝑣𝑖 �� 𝑣𝑖+1 �⎯⎯� … → 𝑣𝑗 . p is zero-delay 𝑑(𝑝) = (0, … ,0),if and only if : ∀ 𝑒𝑙 : 𝑣𝑘 → 𝑣𝑘+1 , 𝑑(𝑒𝑙 ) = (0, … ,0); 𝑠𝑢𝑐ℎ 𝑎𝑠 𝑒𝑙 ∈ 𝐸, 𝑣𝑘 ∈ 𝑉, 𝑘 ∈ [𝑖, (𝑗 − 1)], 𝑙 ∈ [𝑚, 𝑛] 𝑚𝑎𝑥 (𝑐𝑎𝑟𝑑(𝑝)) 𝑑(𝑝) = (0, … ,0) 𝑡(𝑝) ≤ 𝑐𝑚𝑖𝑛 𝑒𝑚 𝑒𝑚+1 𝑒𝑛 Where 𝑝: 𝑣𝑖 �� 𝑣𝑖+1 �⎯⎯� … → 𝑣𝑗 as 𝑣𝑘 ∈ 𝑉 ∀ 𝑘 ∈ [𝑖, 𝑗] and card(p) is the number of nodes belonging to p such as 𝑐𝑎𝑟𝑑(𝑝) = 𝑗 − 𝑖 + 1. To preserve data dependency of MDFG, each p path should not contain any cycle in a way that no cycle between vp and vm where ∀ 𝑝, 𝑚 ∈ [𝑖, 𝑗] for each selected path 𝑒𝑚 𝑒𝑚 M8 path, while being executed in the minimal cycle period. We define our idea of optimal multidimensional retiming as described in definition 2. 1. M5 M7 Fig.15. Iteration scheduling of MDFG shown in Fig.14. Cycle (i-2 , j+2) A7 M6 A3 A8 T im e A3 Cycle (i-3 , j+3) A7 A2 A1 M5 𝑒𝑚+1 𝑒𝑛 Furthermore, multidimensional 𝑝: 𝑣𝑖 �� 𝑣𝑖+1 �⎯⎯� … → 𝑣𝑗 . retiming consists in executing several paths in the same cycle period. Thus, such a path should not have any cycle between nodes that they belong to. Our technique is based on retiming a zero-delay path. It means that all edges belonging to this path preserve the same delay (zero-delay). Only the delay of in-coming and outcoming edges of the whole path are changed. Thus, it means retiming the last node that it belongs to. Referring to [2] and [3], the multidimensional retiming function is defined from edges having a non-zero delay of the MDFG. However, there is not constraint that requires applying the MDR function to nodes with non-zero incoming edges. In the case of zero delay paths, it can be applied to any node belonging to them, as defined in theorem 4. Theorem 4. Let 𝐺 = (𝑉, 𝐸, 𝑑, 𝑡), and a zero delay path 𝑝: 𝑣𝑖 𝑒(0,…,0) 𝑒(0,..,0) �⎯⎯⎯� … �⎯⎯� 𝑣𝑗 . If r is a legal MDR function of 𝑣𝑖 ,then r is a legal MDR function of vk, where 𝑣𝑘 ∈ 𝑃 and 𝑘 ∈ [(𝑖 + 1) , 𝑗]. Proof. a strict positive scheduling sub-space S + contains all scheduling vectors s where 𝑑(𝑒) × 𝑠 > 0 for each 𝑑(𝑒) ≠ (0,0, … ,0), such as described in definition 2. This implies that 𝑣𝑖 and vk have the same sub-space 𝑆 + . But, a legal MDR 𝑟 of 𝑣𝑖 is any orthogonal vector to 𝑠, where 𝑠 ∈ 𝑆 +, as indicated in theorem 1. This means that a legal MDR 𝑟 of 𝑣𝑖 is a legal MDR of 𝑣𝑘 . So, we provide an MDR function as indicated in theorem 2. This function is applied to the last node of the zero delay path to retime. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG We conclude in section 3 that it is more sufficient to select an MDR function to apply it for all nodes to be retimed to achieve full parallelism, as described in corollary 1. Thus, we proceed by selecting every last node of a zero-delay path that will be retimed. Afterwards, we find a legal MDR function to be applied successively to the selected nodes while respecting data dependency of the MDFG. ALGORITHM 8 OPTIMAL MULTIDIMENSIONAL RETIMING Input : a realizable MDFG G =(V,E,d,t) Output : a realizable MDFG Gr =(V,E,dr,t) 0: Begin 1: Find a legal MDR function r 2: Provide the LMDFG and the maximal length M , as described in algorithm 7 3: For i from M down to 1 do 4: Select nodesv k that are labeled by i 5: For each v k do 6: Apply the MDR Retiming (i × r) 7: End for 8: End for 9: End 3.3 Path Extraction This step is based on exploring nodes belonging to the MDFG by testing their data dependency and execution time, to share them onto paths. We proceed by sweeping ALGORITHM 7 LABELED MULTIDIMENSIONAL DATA FLOW GRAPH CONSTRUCTION Input : a realizable MDFG G =(V,E,d,t) Output : Labeled Multidimensional Data Flow Graph (LMDFG), maximal length M 0: Begin 1: Compute cmin 2: Extract all nonzero delay edges from G 3: i =1 4: Add the elements (v j, t(v j)) to R, such as vj is a node without outcoming edge 5: While R is not empty do 6: For each v j of R do 7: Collect all predecessor ofv j 8: For each predecessorv p of v j do 9: If t(v p)+t(v j) <= cmin and respect data dependency then 10: Add (v p, t(v p)+t(v j)) to R 11: Else 12: Add (v p, t(v p)) to N E 13: End If 14: End for 15: End for 16: Label all nodes of N E by i 17: I= i+1 18: R=N E 19: End while 20: M = i 21: End 75 incrementally the MDFG from the opposite direction of edge. For each node, we verify that it respects the previous conditions, to execute it in the suitable cycle. Our process consists on extracting the last node of each zero delay path that will be retimed, and labeled by an increasing order starting from 1. The result is illustrated in a “Labeled Multidimensional Data Flow Graph” (LMDFG) taken from the MDFG, as described in algorithm 7. We proceed by exploring the MDFG in the opposite direction of data dependency. We start by extracting the nonzero delay edges from the MDFG and collecting all nodes 𝑣𝑘 without outcoming edge in 𝑅 list. For each node belonging to 𝑅, we define all the predecessor nodes 𝑣𝑝 to verify that can be executed in the same cycle time as nodes in 𝑅. For each 𝑣𝑝 , if 𝑡�𝑣𝑝 � + 𝑡(𝑣𝑘 ) < 𝑐𝑚𝑖𝑛 and if 𝑣𝑝 respects data dependency conditions described above, we consider 𝑣𝑝 as a node belonging to the path and we add it to the 𝑅 list. Else, 𝑣𝑘 is the last node of the previous path that should be retimed, and then we add then the node 𝑣𝑘 to 𝑁𝐸 list to label it. This test is repeated for all predecessor nodes of 𝑅 list element. The next step consists in labeling all the nodes of 𝑁𝐸 list by 1(first value of 𝑖), before replacing 𝑅 elements by 𝑁𝐸 elements. These steps are repeated until testing all nodes of MDFG. We show in Fig.12 the LMDFG of IIR filter with 𝑡(𝑀𝑖 ) = 3 and 𝑡(𝐴𝑗 ) = 1. The first iteration of algorithm 7 consists in labeling the nodes 𝑀1, 𝑀2, 𝑀3, 𝑀4, 𝐴3 and 𝐴4 by 1. The last iteration assigns nodes 𝑀5, 𝑀6, 𝑀7 and 𝑀8 by 2. Therefore, the multi-chain maximum length is 2. 3.4 Optimal Multidimensional Algorithm Our technique starts by finding a legal MDR r of MDFG, as indicated in theorem 2. After that, it provides the LMDFG and the maximal label by running algorithm 7. Then, it selects nodes with maximal label, and applies the MDR function (𝑖 × 𝑟). These steps are repeated by decreasing the label of the selected node until achieving retiming all the labeled nodes, as described in algorithm 8. Fig.12. Labeled Multidimensional Data Flow Graph (LMDFG) of IIR filter. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG As an example, we apply this algorithm to the MDFG of IIR filter. It starts by defining an MDR function 𝑟 = (1, −1), and providing the LMDFG shown in Fig.12. The first iteration of the algorithm retimes the nodes that are labeled by 2, by applying the retiming function 𝑟 = (2, −2) as illustrated in Fig.13. The second iteration retimes the other labeled nodes by 𝑟 = (1, −1), to provide the fully parallelized MDFG of IIR filter as shown in Fig.14. The nodes interconnected by a zero-delay edge are executed in the same cycle period. It means that nodes belonging to the same iteration in the original MDFG are executed in three cycle periods, as illustrated in the scheduling iteration of the fully parallelized MDFG in Fig.15. 76 TABLE 4 EVOLUTION OF THE CODE SIZE IN TERMS OF CYCLE PERIOD MDFG G1 G2 G3 G4 G5 G6 Optimal MDR 144 144 64 144 60 60 Code size Chained MDR 400 400 400 400 72 72 Improve 48% 48% 84% 48% 16.66% 16.66% 5 EXPERIMENTAL RESULTS In this section, we validate our MDR technique by comparing its provided MDFGs to those generated by the chained MDR. Four parameters are compared in our experimentation: the cycle period, the number of MDR functions, the execution time and the code size. We choose as an application the IIR filter graph and the wave digital filter after applying the Fettweis transformation [2], as illustrated in Fig.16. These two graphs are both composed of addition and multiplication nodes. Our experiments consists in modeling each MDFG frequently in different cases of the execution time 𝑡(𝑣𝑖 ) where 𝑣𝑖 𝜖 𝑉, whose values are indicated in Table.1. We present in the last column the minimum cycle period whose values are defined as theorem 2. Fig.14. Full parallelized IIR filter after applying optimal MDR. required to achieve full parallelism, for each MDFG, using both techniques. After providing the fully parallelized MDFGs, we generate their respective algorithms, to extract their time and code size parameters. We present in Table.3 the values of the period cycle number and the execution time of each MDFG For each MDFG in Table.1, we apply both chained MDR and optimal MDR techniques. Each one is characterized by a number of MDR functions applied to achieve full parallelism. The chained MDR generates the same fully parallelized MDFG with the same MDR function number, whatever the Fig.16. MDFG of wave digital filter illustrated in Table.1 in terms of MDR techniques. The column “improve” presents the improvement of the execution time of result generated by our approach compared to those generated by the chained MDR, which accounts for an average improvement of 19.31%. The code size of each MDFG provided in term of both of TABLE 1 CYCLE PERIOD IN TERMS OF NODE EXECUTION TIMES Application IIR Filter Fig.13. Retiming labeled nodes by 2. set of node execution time is. Contrariwise, optimal MDR provides a specific MDFG for each MDFG of Table.1, with different MDR function numbers and retimed nodes. To guaranty a reliable comparison, we use the same MDR function for both techniques, which we apply the functions 𝑟 1 = (1, −1) and 𝑟2 = (0,1) respectively to the IIR filter and wave digital filter. Table.2 illustrates the numbers of MDR functions WD filter MDFG G1 G2 G3 G4 G5 G6 t(A i ) 1 1 1 2 1 1 t(M j) 2 3 4 5 2 3 Cmin 2 3 4 5 2 3 the two MDR techniques are shown in Table.4. Each value presents the instructions number of the loop body and the overhead caused by an MDR transformation. The code size values mention that our technique proposes an average improvement equal to 43.53% of the code size. JOURNAL OF COMPUTING, VOLUME 3, ISSUE 7, JULY 2011, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG 6 CONCLUSION In this paper, we have proposed a new Multidimensional Retiming technique to achieve full parallelism of MDFGs. It allows providing an optimized MDFG compared to those provided by the existent techniques. It allows minimizing Multidimensional functions by exploring the execution times and data dependencies between the nodes. In the section above, we have applied our technique and the chained MDR on different cases of MDFG. The results have shown that our technique generates a more efficient solution MDFG, than those generated by the chained MDR, in terms of cycle number, execution time and code size. We have concluded that our technique provides a more efficient solution, which allows respecting timing and code constraints while implementing the nested loop in embedded TABLE 2 EVOLUTION OF MDR FUNCTION NUMBER IN TERMS OF CYCLE PERIOD MDFG G1 G2 G3 G4 G5 G6 MDR function number Chained MDR Optimal MDR 2 4 2 4 1 4 2 4 1 2 1 2 systems. As an optimization technique, we try in our future works to study using our MDR technique with other optimization approaches such as unrolling, loop fusion ... It consists in defining the applying order and the evolution of the performance parameter in terms of both approaches. However, MDR techniques are based on scheduling the MDFG with a minimum cycle period. This period cycle value does not mean providing an MDFG with minimum execution time. We will be interested to extending our MDR technique to provide the adequate period cycle and a TABLE 3 EVOLUTION OF CYCLE NUMBER AND EXECUTION TIME IN TERMS OF CYCLE PERIOD Execution time Cycle number MDFG Optimal Chained Optimal Chained MDR MDR MDR MDR G1 258 316 516 632 G2 258 316 774 948 G3 229 316 916 1264 G4 258 316 1290 1580 G5 500 600 1000 1200 G6 500 600 1500 1800 Improve 18.35% 18.35% 27.53% 18.35% 16.66% 16.66% scheduling approach which allows providing MDFG with minimum execution time. Also, in the case of real-time embedded system, the design consists in respecting the code size constraint, which should not be exceeded. This principle implies reducing execution time while achieving a limit value of the code size. Thus, based on the opposite evolution of the timing parameters and the code size in terms of MDR functions, we 77 will be interested on proposing an optimization approach using the MDR technique: it requires finding the set of MDR functions that provide a retimed MDFG with a best ratio between the mentioned parameters. REFERENCES [1] E. Leiserson & B. Saxe, “retiming synchronous circuitry”, algorithmica, Springer New York, volume 6, numbers 1-6, juin 1991 [2] N.L. Passos, E.H.M. Sha, “achieving full parallelism using multidimensional retiming”, IEEE Transactions on Parallel and Distributed Systems , Volume 7 , Issue 11 (November 1996) [3] N.L. Passos, E.H.M. Sha, ”Full Parallelism In Uniform Nested Loops Using Multi-Dimensional Retiming”, International Conference on Parallel Processing, 1994. [4] Q. Zhuge, C. Xue, M. Qiu, J. Hu and E. H.-M. Sha, "Timing Optimization via Nest-Loop Pipelining Considering Code Size", Journal of Microprocessors and Microsystems, Volume 32, Issue 7, October 2008. [5] by N. L. Passos , D. C. Defoe, R. J. Bailey, R. H. Halverson, and R. P. Simpson, “Theoretical Constraints on Multi-Dimensional Retiming Design Techniques", Proceedings of the AeroSense-Aerospace/Defense Sensing, Simulation and Controls, Orlando, FL, April, 2001. [6] Qingfeng Zhuge, Chun Jason Xue, Meikang Qiu, Jingtong Hu, Edwin H.-M. Sha, ”timing optimization via nested loop pipelinig considering code size”, journal of microprocessors and Microsystems 32, 351-363, 2008. [7] Qingfeng Zhuge, Zili Shao, Edwin H.-M. Sha, “timing optimization of nested loops considering code size for DSP applications”, Proceeding of the 2004 international conference on parallel processing, 2004. [8]Q. Zhuge, E.H.-M. Sha, C. Chantrapornchai, CRED: code size reduction technique and implementation for software-pipelined applications, in Proceedings of the IEEE Workshop of Embedded System Codesign (ESCODES), September, 2002, pp. 50–56. [9] T. C. Denk and K. K. Parhi, "Two-Dimensional Retiming," in the IEEE Transactions on VLSI, Vol. 7, No. 2, June, 1999, pp. 198-211. [10] N. L. Passos and E. H.-M. Sha, "Scheduling of Uniform MultiDimensional Systems under Resource Constraints," in the IEEE Transactions on VLSI Systems, December, 1998, Volume 6, Number 4, pages 719-730. Yaroub Elloumi is a PHD student since September 2009, registered in both Paris-Est university (France) and Sfax university (Tunisia). He is a member of Institut Gaspard-Monge, unité mixte de recherche CNRSUMLPE-ESIEE, UMR 8049. His research interests are High-level design of real-time system, optimization techniques, and H igh-level parameter estimation. Mohamed Akil received his PhD degree from the Montpellier university (France) in 1981 and hi s doctorat d'état from the Pierre et Marie curie University (Paris, France) in 1985. He currently teaches and does research with the position of Professor at computer science department, ESIEE, Paris. He is a member of Institut Gaspard-Monge, unité mixte de recherche CNRS-UMLPE-ESIEE, UMR 8049. His research interests are Architecture for image processing, Image compression, Reconfigurable architecture and FPGA, High-level Design Methodology for multi-FPGA, mixed architecture (DSP/FPGA), System on C hip (SoC) and parallel programming of 2D/3D topological operators. Dr. Akil has more than 80 research papers in the above areas. Mohamed Hedi Bedoui received his PhD degree from Lille University in 1993. He currently teaches with the position of Professor of biophysics in the Faculty of Medicine of Monastir (FMM), Tunisia. He is a member of Medical Technology and i mage processing team (TIM), UR 08-27. His research interests are real-time and embedded systems, image & signal processing and har dware/software design in medical field, electronic applications in biomedical instrumentation. He is the president of the Tunisian Association of Promotion of Applied Research. 

International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 DISTRIBUTED EVOLUTIONARY COMPUTATION: A NEW TECHNIQUE FOR SOLVING LARGE NUMBER OF EQUATIONS Moslema Jahan, M. M. A Hashem and Gazi Abdullah Shahriar Department of Computer Science and Engineering, Khulna University of Engineering and Technology, Khulna 9203, Bangladesh mjahan.cse@gmail.com, mma_hashem@hotmail.com and rana_kuet@yahoo.com ABSTRACT Evolutionary computation techniques have mostly been used to solve various optimization and learning problems successfully. Evolutionary algorithm is more effective to gain optimal solution(s) to solve complex problems than traditional methods. In case of problems with large set of parameters, evolutionary computation technique incurs a huge computational burden for a single processing unit. Taking this limitation into account, this paper presents a new distributed evolutionary computation technique, which decomposes decision vectors into smaller components and achieves optimal solution in a short time. In this technique, a Jacobi-based Time Variant Adaptive (JBTVA) Hybrid Evolutionary Algorithm is distributed incorporating cluster computation. Moreover, two new selection methods named Best All Selection (BAS) and Twin Selection (TS) are introduced for selecting best fit solution vector. Experimental results show that optimal solution is achieved for different kinds of problems having huge parameters and a considerable speedup is obtained in proposed distributed system. KEYWORDS Master-Slave Architecture, Linear Equations, Evolutionary Algorithms, Hybrid Algorithm, BAS selection method, TS selection method, Speedup. 1. INTRODUCTION In recent years, application of evolutionary algorithms is increasing to a greater extent. Evolutionary algorithms (EAs) are stochastic search methods that have been applied successfully in many search, optimization and machine learning problems. Successful use of evolutionary algorithm for solving linear equations is applied in [1], [2], [3]. However, it is often very difficult to estimate the optimal relaxation factor, which is the key parameter of the successive over relaxation (SOR) method. Optimal solution is achieved quickly as relaxation factors are adapted automatically in evolutionary algorithm. Equation solving abilities was extended in [2], [3], [4] by using time variant parameter. The invention of hybrid evolutionary algorithm [5], [6] brought a greater benefit to solve linear equations within very short time. Many problems with huge parameters such as Numerical Weather Forecasting, Chain Reaction, Astrophysics (Modelling of Black hole), Astronomical formation, Semiconductor Simulation, Sequencing of the human genome, Oceanography need high computational cost in case of single processor. One approach to overcome this kind of limitation is to formulate the problem into distributed computing structure. The main parallel achievements in the algorithmic families including the evolutionary computation, parallel models and parallel implementations are discussed in [7]. A distributed cooperative coevolutionary algorithm is developed in [8] which is beneficial for solving complex problems. As there are no free lunch theorems for optimization algorithms, a graceful convergence is the key challenge for designing an optimization algorithm. A number of “no free DOI : 10.5121/ijdps.2011.2604 31 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 lunch” (NFL) theorems [9] are presented for any algorithm, which state that any two algorithms are equivalent when their performance is averaged across all possible problems. On the other hand, there are coevolutionary free lunches theorems in [10]. The proposed technique follows coevolutionary theme. A distributed technique [11] is proposed for parallelizing fitness evaluation time. Fitness evaluation time is high but other operations of evolutionary algorithm take more time. This paper proposes a Distributed Evolutionary Computation (DEC) in which mutation and adaptation processes are also distributed. This is the Champion Selection technique, where best champion is selected within short period of time. Master (Server) Slaves (Clients) Figure 1. Distributed Master-Slave Architecture Basic master-slave architecture (Figure 1) is used in proposed distributed technique that follows a server-client paradigm where connections are closed after each request. Slaves are connected with master through local area network (LAN) to take the advantage of distributed processing power of slaves. The basic approach of this system is to split a large problem into many subproblems and to evolve subproblems separately. These subproblems are then combined and actual solution is achieved. This process continues until the less erroneous solution comes out compared to a threshold error level. The remainder of this paper is structured as follows: Section 2 represents the previous work related to the proposed work. Section 3 mentions Jacobi method of solving linear equations and distributed model. Timing calculation is discussed in section 4. Section 5 examines results of the experiment of various problems and provides a comprehensive comparison of single and distributed system on the basis of BAS and TS selection mechanism. Finally, Section 6 provides our concluding remarks. 2. RELATED WORK The intrinsically parallel and distributed nature of EAs did not escape the attention of early researchers. Grefenstette [12] was one of the first in examining a number of issues pertaining to the parallel implementations of GAs in 1981. Grosso [13] is another early attempt to introduce parallelism using spatial multipopulation model. Several attempts were made to have a better and fast system that is capable of doing Evolutionary computations in parallel fashion. DREAM (Distributed Resource Evolutionary Algorithm Machine) [14] is such a system that used island model architecture on peer-to-peer connection. Both island-model and master-slave architecture has been combined at ParadisEO (PARAllel and DIStributed Evolving Objects) [15]. But in either case, all genetic operations are not done with a distributed manner. Among different models and architectures, this paper follows master-slave architecture to develop parallel and distributed environment. 32 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 JDEAL (Java Distributed Evolutionary Algorithms Library) [16] is a master-slave architecture coded in Java platform. Paladin-Dec [17] was another Java implementation of genetic algorithms, genetic programming and evolution strategies, with dynamic load balancing and fault tolerance. Still the communications among the nodes in distributed architecture uphold an issue. ECJ [18] is a Java-based framework that doing its computation using Java TCP/IP sockets. MPI (Message Passing Interface) is used at [19] with C++ framework. The developed distributed EC system was integrated transparently with the C++ Open BEAGLE framework [20] in 2002. Parallel performance of MPI sorting algorithms is presented in [21]. By gathering all ideas this paper implement a hybrid algorithm combining the jacobi-based successive relaxation (SR) method with evolutionary computation techniques which follow Java-based framework with socket programming in distributed manner and related speedup is calculated. Same algorithm was implemented in single processing system [4] using C++ framework and the related speedup was calculated. Finally experimental result shows distributed system is more speedy than single system to solve the problems with huge parameters. Selection mechanism of an Evolutionary Computation technique has been a key part which brings up a significant computational cost. In 1982 Hector Garcia-Molina provides an election algorithm in [22] for two categories failure environments by which a coordinator will be selected when failure occurs in distributed computing system. A sum-bottleneck path algorithm is developed in [23] that allows the efficient solution of many variants of the problem of optimally assigning the modules of a parallel program over the processors of a multiple-computer system under some constraints on the structure of the partitions. An individual selection method is provided in [24] to select efficient individuals and whole system follows a distributed technique. In 2010, an improved system of an abstraction, all-pairs that fits the needs of several applications in biometrics, bioinformatics, and data mining is implemented in [25] shows the effect of campus grid system which is more secured than single system because of following distributed manner. Different papers followed various election mechanisms for choosing right population. Like these, our paper also provides two new selection mechanisms BAS which choose all best individual with less error rate and TS which choose one individual between adjacent two and develop twin copy of the individual with low error. 3. PROPOSED MODEL OF DISTRIBUTED EVOLUTIONARY ALGORITHM The new distributed technique is anticipated based on Jacobi Based Time- Variant Adaptive Hybrid Evolutionary algorithm [3] through cluster computing environment. The proposed algorithm initializes a relaxation factor in a given domain which is adapted with time and fitness of the solution vector. 3.1. Jacobi Method of Solving Linear Equations Consider the following linear equations: (1) Ax = b or Ax − b = 0 n n n Where A ∈ ℜ × ℜ and x, b ∈ ℜ Let, A is n × n matrix, where aii is the diagonal elements, aij is other elements of the A matrix and bi is elements of b matrix. For solving linear equations Jacobi method is used [26]. Let, Linear equations is Ax = b and A ≠ 0 (D + U + L )x = b , Where A = (D + U + L ) Dx = b − (U + L )x or, x = D −1b − D −1 (U + L) x Then or, or, x = H j x + V j Where H j = D −1 (− L − U ) and V j = D −1b The linear equation can be written as 33 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 n ∑a ij x j = b j , (i = 1,2,3,.....n ) (2) j =1 In Jacobi method by using SR technique [27] is given by, ω  x i(k +1) = x i(k ) + bi − a ii  n ∑a (k ) ij x j j =1  ,   (i = 1,2,3,.....n ) (3) In matrix form matrix-vector equation is x (k +1) = H ω x (k ) + Vω (4) Where H ω called Jacobi iteration matrix, and Vω are given successively by H ω = D −1 {(1 − ω )I − ω (L + U )} And −1 Vω = ωD b (5) (6) If ω is set at a value between 0 and 1, the result is weighted average of corresponding previous result and sum of other (present or previous) result. It is typically employed to make a nonconvergence system or to hasten convergence by dampening out oscillations. This approach is called successive under relaxation. For value of ω from 1 to 2, extra weight is placed. In this instance, there is an implicit assumption that the new value is moving in the correct direction towards the true solution but at a very slow rate. Thus, the added weight ω is intended to improve the estimate by pushing it closer to the truth. This type of modification, which is called over relaxation, is designed to accelerate the convergence of an already convergent system. This approach is called successive over relaxation (SOR). The combine approach, i.e. for value of ω from 0 to 2, is called successive relaxation or SR technique [26]. Iterative process is continued using equation (3) until the satisfactory result is achieved. Based on this method, different hybrid evolutionary algorithms are developed in [1] [2]. Parallel iterative solution method for large sparse linear equation systems is developed in [28]. 3.2. Proposed Distributed Technique The availability of powerful network computers represents a wealth of computing resources to solve problems with large computational effort. Proposed distributed technique uses masterslave architecture and classical numerical method with Evolutionary Algorithm for solving complex problems. In this system, large problems are decomposed into smaller subproblems and mapped into the computers available in a distributed system. Communication process between master and slaves follow message passing paradigm that is identical with [29]. This technique introduces the high performance cluster computation, linking the slaves through LAN to exploit the advantage of distributed processing of the subproblems. Here each slave in a cluster always solves same weighted subproblems although machine configuration is different. In this cluster computing environment, master processor defines number of subproblems according to slave number in a cluster. The workflow diagram of proposed algorithm is portrayed as in Figure 2. 34 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Start Set Parameters Initialization of Population Stop Y Error<Th reshold Begin Mutation Recombination Connection Setup with master processor Fitness Evaluation Determine number of Subpopulations Waiting for coming job N Adaptation Y Is BAS Method? Job assigned? Distribute all subpopulat ions? N (TS) N Twin selection Block of genetic operations Return subpopulation to master Send solution to master and update slave status Y Sleep until all solutions return from all slaves End Block of genetic operations Champion selection Master processor Slave processor Figure 2. The workflow diagram of DEC system The proposed workflow diagram can be mapped into master-slave paradigm. In Figure 3 all steps of proposed DEC are expressed using numbering system in each position. Step 1, 2, 7 are completed in master processor and step 3, 4, 5 occurred in slave processors but working principle of step 7 depends on selection method. For both methods, step 6 is dedicated for checking whether selected offspring from all slaves return back in master processor or not. 35 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Server (Master) Controller Evaluators (Slaves) Loopback 2 3 1 4 7 6 5 Figure 3. Working model of proposed technique 3.2.1. Initialization In step 1, master processor initializes problems with population and relaxation factor. Initial population X (0 ) = x1(0 ) , x 2(0 ) ,.....x N(0 ) is generated randomly using normal distribution at master. Here N is the population size. Let k ← 0 where k is the generation counter. Relaxation factors ω (0 ) = ω1(0 ) , ω 2(0 ) ,......, ω N(0 ) are also initialized on the basis of corresponding individuals. { { } } 3.2.2. Recombination In step 2, Recombination operation is performed on parent at master and population ( X (k + c ) = R X (k ) ) t ( ) is obtained, where R = rij n and N×N ∑r ij = 1 and rij ≥ 0 for 1 ≤ i ≤ N j =1 Matrix R is a stochastic matrix. Population generated after recombination operation is distributed among slaves. Then mutation, fitness evaluation and adaptation operations are performed on that distributed subpopulations. 3.2.3. Mutation After completing recombination operation step 3 provides Mutation operation on the subpopulation in slave processors and mutated subpopulation X (k + m ) is obtained. For each Mutation on subpopulation is N sub ∑x (k + m) i = H ω x i( k + c ) + Vω (7) i =1 3.2.4. Fitness Evaluation After completing mutation operation, fitness evaluation is performed in step 4. Error function is N sub E= ∑E (8) i i =1 N sub Where E = ∑ a (x ) − b ij j i , i = 1,2,....., N sub j =1 The fitness evaluation of an individual measures the accuracy of an individual for a particular problem and calculates the error rate which is used for selecting best individuals. 36 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 3.2.5. Adaptation In step 5, adaptation is performed on mutated offspring according to fitness evaluation. Consider two individuals X and Y , corresponding errors e x and e y and relaxation factors ω x and ω y . If e x > e y then ω x is adapted to ω y . ω xm = (0.5 + p x )(ω x + ω y ) Similarly if e y < e x then ω y is adapted to ω x and if e x = e y then no adaptation is performed. Px is denoted as adaptive (TVA) probability parameter of ω x . In step 6, controller checks whether offspring from all slaves have reached in master processor or not. In BAS method, step 6 starts its task after completion of adaptation operation but in TS method, after completing of partial selection of best offspring in each slave. 3.2.6. Selection The DEC system provides two selection methods named BAS and TS. In BAS method, best individuals are selected among parent and mutated offspring. At position 7, mutated offspring from all slaves are combined and selection mechanism is performed on parent and newly generated offspring by mutation. Mathematically, Selection(i ) = errmin {x1 , x 2 , x 3 ,.....x 2 N −i +1 } (9) N Selection = ∑ Selection(i ) (10) i =1 Individuals compete among themselves and select best individual based on error value. Before finding the optimized solution, overall system will be continued in same process. TS method provides a partial selection operation on mutated offspring in each slave where one best individual is selected between two consecutive individuals and developed twin copy of selected offspring. Mathematically, Selection(i ) = errmin {x i , x i +1 } (11) N sub Selection = ∑ Selection(i) (12) i =1 These selected offspring are combined in server and select best one. The selected copy will be the champion among all individuals or optimized solution for a particular problem if this fulfils the desired condition, otherwise fill up the whole archive and next generation will be continued. There is no direct communication among the subpopulations. All communications are performed between the subpopulation and the central server. Such communications take place in every generation before reaching the accurate result. 3.3. Scenario of “Champion Selection” The DEC system can be compared with a game where a champion will be selected on the basis of some criteria. At the starting moment, all players are presented at master which is a coordinator. The Coordinator divides all players into different teams and assigns these teams in different slaves. The number of players in a team is determined according to the slave number in 37 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 a game region that is a cluster. After reaching the team, some operations are performed on that team separately and simultaneously in each slave. Each player has enriched after operation and all players from each slave come back in master. After coming back, all players compete with each other and select best one. If the selected one fulfils the desired criteria then this is the champion of the game or game will be continued following same process. Subpopulation Individuals 1 Slave 1 2 3 Slave 2 4 5 Server (Master) Slave 3 6 Figure 4. The model of DEC A prototype model of DEC with six individuals over three slave computers is illustrated in Figure 4. The figure shows the division of main population into three subpopulations containing two individual in each according to the number of slave computers where master processor is the coordinator which will take the decision. Each slave accepts two individuals among six of them. Some Evolutionary mechanisms are operated separately in each slave. After completing the operations, subpopulations are returned back to master and individuals compete among themselves and a champion is coming out. If the champion is not best suited with the standard value, this process will be continued. Otherwise this champion is the winner of the game. 3.4. Explanation of Selection Methods This section represents two selection methods which will help to find out the optimized solution of a problem. Working mechanisms of these two methods are as follows: 3.4.1. BAS selection method In this method, best individual is selected between parent and mutated offspring in each iteration. After coming back all individuals from all slaves to master, they compete with each other and provide best half of the total individuals including parent and offspring and assign priority according to the error rate. On the basis of the priority, a champion is selected. If the error rate of the champion is equal or less to the standard value, this is the optimized solution of the problem. Otherwise the process will be going on. 38 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Parent a b c e3 e2 e5 Offspring d e8 e7 Recombination and Mutation e1 e3 e5 e1 e6 e4 Selection e4 Figure 5. Selection mechanism for BAS method In Figure 5: a, b, c and d patterns represent four parent individuals and the stripped patterns represent offspring. Error of each parent individual is e3, e2, e5, e7 and corresponding relaxation factors are ω3, ω2, ω5, ω7. Similarly error of each offspring is e8, e1, e6 and e4 and relaxation factors are ω8, ω1, ω6 and ω4. Here error number represents the value of the error. Now best half individuals are selected between parent and offspring according to error value. Here two individuals are selected from parent and two are selected from offspring. Corresponding relaxation factor is also rearranged. Next generation is started with these selected offspring and their respective relaxation factor. The time variant adaptive (TVA) parameters are defined as Px = E x × N (0,0.25) × Tω And Px is denoted as adaptive (TVA) probability parameter of ω x , and Py = E y × [N (0,0.25)]× Tω (13) (14) γ t  And Py is denoted as adaptive (TVA) probability parameter of ω y ; Where Tω = 1 −   T Here λ and γ are exogenous parameters, used for increased or decreased of rate of change of curvature with respect to number of iterations; t and T denote number of generation and maximum number of generation respectively. Also N (0,0.25) is the Gaussian distribution with mean 0 and standard deviation 0.25. Now E x and E y denote the approximate initial boundary of the variation of TVA parameters of ω x and ω y respectively. If ω ∗ is denoted as the optimal relaxation factor then E x = Px | max = ωy ~ ωx ( 2 ωx +ω y (15) ) ( ) So that ω xm = (0.5 + Px | max ) ω x + ω y ≈ ω y and ∗ E y = Py | max = ∗ ω ~ ωy ( 2 ω y −ωL ) or ω ~ ωy ωU − ω y (16) 39 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 ( ( ) ) ω y + Py |max ωU − ω y , ω y > ω x  so that, ω ∗ ≈ ω ym =   ω y + Py |max ω L − ω y , ω y < ω x  (17) 3.4.2. TS selection method Each slave contains a subpopulation and each subpopulation consists of at least two individuals. After completing adaptation operation, best individual is selected between two consecutive individuals. These individuals are returned back to the master and compete among themselves. After the competition, a best quality champion is selected according to the error value and checking with the standard value. If the error value is equal or less to the standard value, this is the optimized solution. Otherwise this best champion is cloned and fills the archive and continues the process. The corresponding relaxation factors are also rearranged which is required for adaptation operation of next generation. Parent a b c e3 e2 e5 Offspring d e7 e8 e1 Recombination and Mutation e1 e1 e1 e1 e6 e4 Selection e1 e1 e6 e6 Select best champion among all champions Figure 6. Selection mechanism for TS method In Figure 6: a, b, c and d patterns represent four parent individuals and the stripped designs represent offspring. Error of each offspring is e8, e1, e6, e4 and corresponding relaxation factors are ω8, ω1, ω6, ω4. There are two subpopulations in two slaves and each contains two individuals. The error rate of the individuals in 1st subpopulation is e1 and e8, and 2nd subpopulation is e6 and e4. Now, best individual is selected between two consecutive offsprings and made a twin of it. The individual with the error rate e1 is chosen from 1st individual and individual with the error rate e6 is chosen from 2nd subpopulation. These individuals are returned back to master and select a best quality champion according to the error rate. This champion is cloned to the equal number of the parent and next generation is started with these cloned individuals. Corresponding relaxation factor is also rearranged. The time variant adaptive (TVA) parameters are defined as Px = gauss x × Pmax × Tω (18) Px is denoted as adaptive (TVA) probability parameter of ω x , and Py = gauss y × Pmin × Tω (19) 40 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Py is denoted as adaptive (TVA) probability parameter of ω y Where 1   Tω = λ ln1 + , λ > 10  t+λ  Approximately, Pmax and Pmin are two exogenous parameters that are assumed as Pmax = 0.125 and Pmin = 0.0325 Here, λ and γ are exogenous parameters, used for increased or decreased of rate of change of curvature with respect to number of iterations; t and T denote number of generation and maximum number of generation respectively. Also, gauss x (0,0.25) is the Gaussian distribution with mean 0 and standard deviation 0.25. Now Pmax and Pmin denote the approximate initial boundary of the variation of TVA parameters of ω x and ω y respectively. And if ω ∗ is denoted as the optimal relaxation factor then ω y + Py | max ωU − ω y , ω y > ω x  ω ∗ ≈ ω ym =   ω y + Py | max ω L − ω y , ω y < ω x  ( ( ) ) 4. COMPUTATION TIME ANALYSIS DISTRIBUTED PROCESSORS (20) OF SINGLE PROCESSOR AND Timing is the main parameter which is compared for single and distributed processors. For a specific optimization problem, DEC system provides some notation for recombination time, mutation time, fitness evaluation time, adaptation time, and selection time can be denoted as Tr , Tm , T f , Ta and Ts . Total time in single processor is indicated by Tsin gle . Time in single processor is represented as follows, TSingle = Tr + Tm + T f + Ta + Ts (21) In case of distributed system, time is calculated in master and slaves separately. Then total time is calculated by combining master and slave time. 4.1. Time in master processor For a particular problem, Recombination operation is performed on initial population and time Tr is calculated. Then the server distributes individuals among all slaves connected to it. Number of subpopulations is p which is distributed in slaves. In this case, Marshalling and transmission time is considered. Marshalling time is the time to construct data stream from object and transmission time is the time to place data stream in channel. Marshalling time of i th individual is Tmi and Transmission time with i th individual is Ttransi . Marshalling and transmission time with p subpopulations = P[Tmi + Ttransi ] 4.2. Time in slave processors Mutation time, fitness evaluation time, adaptation time with p subpopulations is Tm , T f and Ta respectively. Unmarshalling time Tumi is considered here. Unmarshalling time is the time to create object from data stream. After completing adaptation operation, each slave sends mutated offspring to server when BAS selection method is considered but in TS method, slaves send individuals which are selected as best quality for a particular slave. In BAS method, selection operation is performed in master processor but partial selection operation is completed in slave processor in TS method. So this selection time for TS method is calculated in slave processor 41 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 and sends to master. The processing time of each slave is not equal because of communication delay. In experimental calculation, maximum time is considered for all cases. 4.3. Time in Master Processor for selection The basic difference between BAS and TS selection method is to perform selection operation in two different ways. So selection time for BAS and TS method is calculated in different manner. Selection time with 2 N subpopulation is Ts , N individuals for parent and N individuals for offspring. Maximum slave time is considered for calculating speedup. Consider m is the number of slaves. In BAS selection method, total time for distributed processors is as follows, [ )] ( TDistributed ( BAS ) = Tr + p Tmi + Ttransi + max 1....m Tm + T f + Ta + pTumi + Ts (22) Where Tdistributed represents total distributed time, Tr , Tmi , Ttransi , Tm , T f , Ta , Tumi , Ts represents recombination time, marshalling time, transmission time, mutation time, fitness evaluation time, adaptation time, unmarshalling time and selection time. Speedup for distributed technique using BAS method is: Speedup = TSingle T Distributed ( BAS ) (23) In TS selection method, selection operation is performed on mutated offspring in each slave where each best individual is selected between two consecutive individuals and corresponding time is obtained. Then slaves send selected offspring with selection time calculated in slave to master. Total selection time with p subpopulation = Ts Consider m is the number of slaves. In the case of TS method: [ TDistributed = Tr + p[Tmi + Ttransi ] + T f + Tm + Ta + Ts ] (24) Speedup for TS method: Speedup = TSingle T Distributed (25) Furthermore, percentage of improvement: %=Speedup/Number of computers (26) Computation time in distributed processors will be less than single processor. So, speedup will be gradually rising with increasing number of individuals in a population. 5. EXPERIMENTAL RESULTS The environment of experiment consists with 15 homogeneous computers and 100 Mbits/sec Ethernet switch. This system is implemented in Java on a personal computer with Intel Pentium IV and 512MB RAM. In our experiment, individual values are generated randomly. Here, random values are generated using normal distribution with the range from -15 to 15. In order to evaluate the effectiveness and efficiency of the proposed algorithm, various problems are tested. A problem which is tested for different approaches is shown in different graphs. The testing problem is: Ax = b Where, a ii = 20.0 , a ij ∈ (0,1) , bi = 10 * i and i = 1,2,3,......, n , j = 1,2,3,......, n 42 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 The parameter is n = 100 and the problem was solved with an error rate resides in the domain from 9 × 10 −9 to 1×10 −8 for both BAS and TS method. Different experiments were carried out using this problem. Table I gives the results produced by proposed distributed algorithm and single processor algorithm. Here, different problems are tested with comparing time between single and distributed processors. Table I provides the results using BAS method. Similarly, Table II summarizes experimental results of various problems with TS method. In these two tables, number of computers in a cluster is five as distributed processors. In all cases, optimum solution is achieved. It is possible to solve various benchmark problems using the proposed distributed system. It is very clear from Table Ι and Table II that performance of distributed processors is better than single processor. Although BAS and TS both are selection methods that are used in distributed system, BAS method shows better performance than TS method. Table 1. Experimental Results of Time for Different Problems in Single and Distributed Processors for BAS Method Problem Number Problem Individu al Number Parame ter Numbe r Time in Single Processor (seconds) Time in Distributed Processor (seconds) Error p1 a ii = 20.0; a ij ∈ (0,1); 40 170 4.02 × 10 −1 6.88 × 10 −2 9 × 10 −4 bi = 10 * i p2 a ii = 20n; a ij = j; bi 40 120 3.68 × 10 −1 6.75 × 10 −2 9 × 10 −9 p3 a ii = 2i 2 ; a ij = j; bi = 40 120 3.38 × 10 −1 6.16 ×10 −2 9 ×10 −9 20 100 1.18 × 10 −1 3.57 × 10 −2 9 × 10 −9 20 100 1.33 × 10 −1 3.45 × 10 −2 9 × 10 −9 40 100 1.85 × 10 0 2.13 × 10 −2 9 × 10 −4 p4 a ii ∈ (− 100,100 ); a ij ∈ (− 10,10 ); bi ∈ (− 100,100 ) p5 a ii ∈ (− 70,70 ); a ij ∈ (0,7 ); bi ∈ (0,70 ) p6 a ii = 70; a ij ∈ (− 10,10 ); bi ∈ (− 70,70 ) Table 2. Experimental Results of Time for Different Problems in Single and Distributed Processors for TS Method 43 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Proble m Numb er p1 Problem aii = 20.0; aij ∈ (0,1 Individual Number Parameter Number Time in Single Processor (seconds) 1.21 × 10 −1 Time in Distribute d Processor (seconds) 9.93 × 10 −2 20 100 Error 1× 10 −8 bi = 10 * i p2 a ii = 20n; a ij = j; b 20 100 1.28 × 10 −1 1.05 × 10 −1 1× 10 −8 p3 a ii = 2i 2 ; a ij = j; b 20 100 1.48 × 10 −1 1.22 × 10 −1 1× 10 −8 p4 aii ∈ ( −100,100 ) ; 20 100 1.46 × 10 −1 1.34 × 10 −1 1× 10 −5 20 50 1.06 × 10 −1 9.59 × 10 −2 1× 10 −4 20 10 1.12 × 10 −1 9.94 × 10 −2 1× 10 −4 aij ∈ ( −10,10 ) ; bi ∈ ( −100,100 ) p5 a ii ∈ (− 70,70 ); a ij ∈ (0,7 ); bi ∈ (0, p6 aii = 70; aij ∈ ( −10,10 ) ; bi ∈ ( −70, 70 ) 5.1. Speedup comparison between BAS and TS method To compare speedup, BAS and TS method follows the system standard with 40 individuals for 5 and 10 number of computers in a cluster as well as 30 individuals for 15 number of computers when parameters are 100 for each case. 8 BAS method TS method 7 Speed up 6 5 4 3 2 1 5 10 Number of computers in a cluster 15 Figure 7. Speed up measurement according to number of computers in a cluster In Figure 7, speedup is calculated using eqn (23) in BAS method and TS method uses eqn (25). Speedup is 3.36, 4.57, 6.72 in BAS method and 2.109, 1.999, 2.421 in TS method when number 44 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 of slave computers is 5, 10 and 15. Percentage of improvement is sequentially 67.2 %, 45.7 % and 44.8 % for BAS method as well as 21.09 %, 19.99 % and 24.21 % for TS method which is calculated based on eqn (26). It can be easily visualized that BAS method provides better performance than TS method. 0.02 Time(Seconds) 0.015 0.01 0.005 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Computer Number Figure 8. Time (in seconds) measurement according to computer number in a cluster Figure 8 shows the time required for performing genetic operation in each slave. Here, number of slave computers in a cluster is 15 and individual is 30 and parameters are 100. There is no load balancing system so different slave computers need different time. Intuitively, workload balancing for a distributed system can be difficult because the working environment in a network is often complex and uncertain. From Figure 8, we can see that computer number 1 needs the highest computation time among all slaves. For calculating total distributed time, this maximum value is considered in both methods. 5.2. Comparing time with dimension Figure 9 and Figure 10 show the time requirement on the basis of parameter number to solve the problems using BAS and TS method and compare the performance in single and distributed processors. The system runs with 5 and 20 slave computers in a cluster and parameters are 50, 60, 70, 80, 90 and 100. There is a fluctuation of time with increasing number of parameters because of random production. All parameters are same for two methods. Distributed system needs less time than single system in both cases but BAS method is better selection mechanism than TS selection method. 0.16 0.16 0.14 0.14 0.1 0.08 Time(Seconds) Time(Seconds) 0.12 Single Processor Distributed Processor 0.06 0.12 0.1 0.08 0.04 0.06 0.02 0.04 0 50 Single Processor 60 70 80 Number of parameters 90 100 Figure 9. Time measurement according to the number of parameters in BAS method 0.02 50 Distributed Processor 60 70 80 90 Number of parameters 100 Figure 10. Time measurement according to the number of parameters in TS method 45 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 5.3. Comparing time with individual Time is compared between single and distributed processors according to individuals using BAS and TS method; this is shown in Figure 11 and Figure 12. The number of slave computers in a cluster is 5 and number of parameters is 100 in all cases where individual number is varying with the value of 10, 20, 30, 40 and 50. Time is increasing with increasing number of individuals but more time is always needed in single system comparing with distributed system. 0.35 0.35 Single Processor Single Processor 0.3 Distributed Processor 0.25 Time(Seconds) Time(Seconds) 0.3 0.2 0.15 0.1 Distributed Processor 0.25 0.2 0.15 0.1 0.05 0.05 0 10 20 30 40 Number of individual 0 10 50 20 30 40 Number of individual 50 Figure 12. Measurement of Time with number of individuals in TS method Figure 11. Measurement of Time with number of individuals in BAS method. 5.4. Comparing error with generation Figure 13 and Figure 14 visualize the error rating with number of generations in BAS and TS method. In both cases, efficiency is compared between single and distributed. These two methods use 5 slave computers in a cluster but other parameters are different. Number of parameters is 100 and number of individuals is 40 for BAS method but TS method uses 90 parameters and 20 individuals for this experiment. From figures it is easily understandable that some cases of distributed system needs less generation than single system to go convergence. 0.01 0.015 Single Processor Single Processor Distributed Processor 0.008 Distributed Processor 0.01 Error Error 0.006 0.004 0.005 0.002 0 7 8 9 10 11 Generation 12 Figure 13. Error measurement according to generation in BAS method 13 0 10 11 12 13 14 Generation 15 16 Figure 14. Error measurement according to generation in TS method 46 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 In all cases, performance of distributed processor is better than single processor. In the mean time it is also noticeable that BAS selection method shows incredible performance than TS method although this method completes a partial selection in slave processor. A tricky point is that individuals are filtering primarily in each slave, so there may be discarded better quality individual from a slave. On the other hand, BAS method choose best individual from parent and all offspring coming back from all slaves. So achieving optimal result in BAS method is faster than TS method. Apparently, it is not realized because of distributed selection operation in TS method which is not in BAS method. But in some case, TS method provide better performance if the selection order is perfect for each slave. 6. CONCLUSIONS It is easy to solve linear equations with less number of parameters in single processor but high computational cost is required for large number of parameters. This cost can be drastically reduced using distributed system. This paper introduced a new distributed algorithm for solving large number of equations with huge parameters. It also introduced two new selection methods called BAS and TS for selecting offspring. In these methods, best individuals are selected and computation load is distributed in each slave and mutation, fitness evaluation, and adaptation operations are performed on distributed load. As distributed technique is used, computation time is reduced using these selection methods but BAS method provides better performance compared to TS method. For both selection methods, computational time is analyzed and hence speed up is calculated in this new distributed computing system. REFERENCES [1] Jun He, Jiyou Xu and Xin Yao, (September 2000) “Solving Equations by Hybrid Evolutionary Computation Techniques”, IEEE Transactions on Evolutionary Computations, Vol. 4, No. 3, pp. 229-235. [2] A.R.M. Jalal Uddin Jamali, Md. Bazlar Rahman and M.M.A. Hashem, (2005), “Solving Linear Equations by Classical Jacobi-SR Based Hybrid Evolutionary Algorithm with Time-Variant Adaptation Technique”, Ganit: Journal of Bangladesh Mathematical Society, Vol. 24, No. 24, pp. 67-79. [3] A.R.M. Jalal Uddin Jamali, M.M.A. Hashem and Md. Bazlar Rahman, (2004), “An approach to Solve Linear Equations Using a Time-Variant Adaptation Based Hbrid Evolutionary Algorithm”, Jahangirnagar University Journal of Science, Vol. 27, pp. 277-289. [4] Jamali A.R.M.J.U., (2004), “Study of Solving Linear Equations by Hybrid Evolutionary Computation Techniques”, M.Phil Thesis, Khulna University of Engineering & Technology, Khulna, Bangladesh. 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Paizeau, (2002), “Open BEAGLE: A New Versatile C++ framework for evolutionary computation”, in Late Breaking Papers of GECCO 2002, New York, NY, USA. [21] Alaa Ismail El-Nashar, (May 2011), “Parallel Performance of MPI Sorting Algorithms on DualCore Processor Windows-Based Systems”, International Journal of Distributed and Parallel Systems (IJDPS), Vol. 2, No. 3, pp. 1-14. [22] Hector Garcia-Molina, (1982), “Elections in a Distributed Computing System”, IEEE Transactions on Computers, Vol. 31, No. 1, pp. 48-59. [23] Shahid H. Bokhari, (1988), “Partitioning Problems in Distributed Parallel, Pipelined, and Distributed Computing”, IEEE Transactions on Computers, Vol. 37, No. 1, pp. 48-57. [24] Moslema Jahan, M. M. A Hashem and G. A. Shahriar, (2008), “A New Distributed Evolutionary Computation Technique for Solving Large Number of Equations”, IEEE 11th International Conference on Computer and Information Technology (ICCIT). [25] Christopher Moretti, Hoang Bui, Karen Hollingsworth, Brandon Rich, Patrick Flynn and Douglas Thain, (2010), “ All-Pairs: An Abstraction for Data-Intensive Computing on Campus Grids”, IEEE Transactions on Parallel and Distributed System, Vol. 21, No. 1, pp. 33-46. [26] Gerald, C. F. and P.O. Wheatley, (1994), “Applied Numerical Analysis (5th edition)”, Addison-Wesley, New York. [27] Engell-Mǘllges, G.E. and F. Uhlig, (1999), “Numerical Algorithms with C”, Springer-Verlag, Heidlberg. [28] Rashid Mehmood, Jon Crowcroft, (2005), “Parallel iterative solution method for large spare linear equation systems”, Technical Report, University of Cambridge Computer Library, Cambridge, UK, UCAM-CL-TR-650, ISSN 1476-2986. [29] Alaa Ismail El-Nashar, (March 2011), “To Parallelize or Not to Parallelize, Speed up Issue”, International Journal of Distributed and Parallel Systems (IJDPS), Vol. 2, No. 2, pp. 14-28. 48 International Journal of Distributed and Parallel Systems (IJDPS) Vol.2, No.6, November 2011 Authors Moslema Jahan achieved the B.Sc. Engg. Degree (with honours) in Computer Science and Engineering (CSE) from KhulnaUniversity of Engineering & Technology (KUET), Khulna, Bangladesh. Currently she is serving as a Lecturer in the Department of Computer Science & Engineering in Dhaka University of Engineering & Technology (DUET), Gazipur. Her research interest is on Parallel and Distributed computing, Evolutionary Computation, Networking and Image processing. Moslema Jahan is now Member of the Engineers Institution, Bangladesh (IEB). M. M. A. Hashem received the Bachelor’s degree in Electrical & Electronic Engineering from Khulna University of Engineering & Technology (KUET), Bangladesh in 1988. He acquired his Masters Degree in Computer Science from Asian Institute of Technology (AIT), Bangkok, Thailand in 1993 and PhD degree in Artificial Intelligence Systems from the Saga University, Japan in 1999. He is a Professor in the Department of Computer Science and Engineering, Khulna University of Technology (KUET), Bangladesh. His research interest includes Soft Computing, Intelligent Networking, Wireless Networking, Distributed Evolutionary Computing etc. He has published more than 50 referred articles in international Journals/Conferences. He is a life fellow of IEB and a member of IEEE. He is a coauthor of a book titled “Evolutionary Computations: New Algorithms and their Applications to Evolutionary Robots,” Series: Studies in Fuzziness and Soft Computing, Vol. 147, Springer-Verlag, Berlin/New York, ISBN: 3-540-20901-8, (2004). He has served as an Organizing Chair, IEEE 2008 11th International Conference on Computer and Information Technology (ICCIT 2008) and Workshops, held during 24-27 December, 2008 at KUET. Currently, he is working as a Technical Support Team Consultant for Bangladesh Research and Education Network (BdREN) in the Higher Education Quality Enhancement Project (HEQEP) of University Grants Commission (UGC) of Bangladesh. Gazi Abdullah Shahriar achieved the B.Sc. Engg. Degree in Computer Science and Engineering (CSE) from Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh. Now he is working at Secure Link Services BD Ltd as software engineer. His current research interest is in Evolutionary computation, Distributed computing and Bioinformatics. 49 

A Polynomial Time Algorithm for Finding Area-Universal Rectangular Layouts arXiv:1302.3672v7 [cs.CG] 15 Sep 2016 Jiun-Jie Wang State University of New York at Buffalo, Buffalo, NY 14260, USA. Email: jiunjiew@buffalo.edu Abstract A rectangular layout L is a rectangle partitioned into disjoint smaller rectangles so that no four smaller rectangles meet at the same point. Rectangular layouts were originally used as floorplans in VLSI design to represent VLSI chip layouts. More recently, they are used in graph drawing as rectangular cartograms. In these applications, an area a(r) is assigned to each rectangle r, and the actual area of r in L is required to be a(r). Moreover, some applications require that we use combinatorially equivalent rectangular layouts to represent multiple area assignment functions. L is called area-universal if any area assignment to its rectangles can be realized by a layout that is combinatorially equivalent to L. A basic question in this area is to determine if a given plane graph G has an area-universal rectangular layout or not. A fixed-parameter-tractable algorithm for solving this problem was obtained 2 in [4]. Their algorithm takes O(2O(K ) nO(1) ) time (where K is the maximum number of degree 4 vertices in any minimal separation component), which is exponential time in general case. It is an open problem to find a true polynomial time algorithm for solving this problem. In this paper, we describe such a polynomial time algorithm. Our algorithm is based on new studies of properties of area-universal layouts. The polynomial run time is achieved by exploring their connections to the regular edge labeling construction. 1 Introduction A rectangular layout L is a partition of a rectangle R into a set R(L) = {r1 , . . . , rn } of disjoint smaller rectangles by vertical and horizontal line segments so that no four smaller rectangles meet at the same point. An area assignment function of a rectangular layout L is a function a : R(L) → R+ . We say L is a rectangular cartogram for a if the area of each ri ∈ R(L) equals to a(ri ). We also say L realizes the area assignment function a. Rectangular cartograms were introduced in [14] to display certain numerical quantities associated with geographic regions. Each rectangle ri represents a geographic region. Two regions are geographically adjacent if and only if their corresponding rectangles share a common boundary in L. The areas of the rectangles represent the numeric values being displayed by the cartogram. In some applications, several sets of numerical data must be displayed as cartograms of the same set of geographic regions. For example, three figures in [14] are the cartograms of land area, population, and wealth within the United States. In such cases, we wish to use cartograms whose underlying rectangular layouts are combinatorially equivalent (to be defined later). Fig 1 (1) and (2) show two combinatorially equivalent layouts with different area assignments. The following notion was introduced in [4]. Definition 2. A rectangular layout L is area-universal if any area assignment function a of L can be realized by a rectangular layout that is combinatorially equivalent to L. A natural question is: which layouts are area-universal? A nice characterization of area-universal rectangular layouts was discovered in [4]: 1 s (1) (2) (3) Figure 1: Examples of rectangular layout: (1) and (2) are two combinatorially equivalent layouts with different area assignments. Both are area-universal layouts. (3) A layout that is not area-universal. Theorem 3. [4] A rectangular layout L is area-universal if and only if every maximal line segment in L is a side of at least one rectangle in L. (A maximal line segment is a line segment in L that cannot be extended without crossing other line segments in L.) In Fig 1, the layouts (1) and (2) are area-universal, but the layout (3) is not. (The maximal vertical line segment s is not a side of any rectangle.) For a plane graph G, we say a rectangular layout L represents G if the following hold: (1) The set of smaller rectangles of L one-to-one corresponds to the set of vertices of G; and (2) two vertices u and v are adjacent in G if and only if their corresponding rectangles in L share a common boundary. In other words, if L represents G, then G is the dual graph of small rectangles in L. Area-universal rectangular layout representations of graphs are useful in other fields [13]. In VLSI design, for example [17], the rectangles in L represent circuit components, and the common boundary between rectangles in L model the adjacency requirements between components. In early VLSI design stage, the chip areas of circuit components are not known yet. Thus, at this stage, only the relative positions of the components are considered. At later design stages, the areas of the components (namely, the rectangles in L) are specified. An area-universal layout L enables the realization of the area assignments specified at later design stages. Thus, the ability of finding an area-universal layout at the early design stage will greatly simplify the design process at later stages. The applications of rectangular layouts and cartograms in building design and in tree-map visualization can be found in [2, 1]. Heuristic algorithms for computing the coordinates of a rectangular layout that realizes a given area assignment function were presented in [16, 11]. A plane graph G may have many rectangular layouts. Some of them may be area-universal, while the others are not. Not every plane graph has an area-universal layout. In [15], Rinsma described an outerplanar graph G and an area assignment to its vertices such that no rectangular layout realizes the area assignment. Thus it is important to determine if G has an area-universal layout or not. Based on Theorem 3, Eppstein et al. [4] described an algorithm that finds an area-universal layout for G if 2 one exists. Their algorithm takes O(2O(K ) nO(1) ) time, where K is the maximum number of degree 4 vertices in any minimal separation component. For a fixed K, the algorithm runs in polynomial time. However, their algorithm takes exponential time in general case. In this paper, we describe the first polynomial-time algorithm for solving this problem. Our algorithm is based on studies of properties of area-universal layouts and their connection to the regular edge labeling construction. The paper is organized as follows. In §2, we introduce basic definitions and preliminary results. §3 outlines a Face-Addition algorithm with exponential time that determines if G has an areauniversal rectangular layout. §4 introduces the concepts of forbidden pairs, G-pairs and M-triples that are extensively used in our algorithm. In §5, we describe how to convert the Face-Addition algorithm with exponential time to an algorithm with polynomial time. 2 vN vN c b g vW h vE e a f d vS vN c b g vW c b h e a f b g vE vW h e a d vS (1) Lext vN c (2) Gext f d g vE vW h e a f vS vS (3) G1 (4) G2 vE d Figure 4: Examples of rectangular layout and REL. (1) Rectangular layout Lext ; (2) The graph corresponding to Lext with an REL R = {T1 , T2 }; (3) the graph G1 of R; (4) the graph G2 of R. 2 Preliminaries In this section, we give definitions and important preliminary results. Definitions not mentioned here are standard. A graph G = (V, E) is called planar if it can be drawn on the plane with no edge crossings. Such a drawing is called a plane embedding of G. A plane graph is a planar graph with a fixed plane embedding. A plane embedding of G divides the plane into a number of connected regions. Each region is called a face. The unbounded region is called the exterior face. The other regions are called interior faces. The vertices and edges on the exterior face are called exterior vertices and edges. Other vertices and edges are called interior vertices and edges. We use cw and ccw as the abbreviation of clockwise and counterclockwise, respectively. For a simple path P = {v1 , v2 , · · · , vp } of G, the length of P is the number of edges in P . P is called chord-free if for any two vertices vi , vj with |i − j| > 1, the edge (vi , vj ) ∈ / E. A triangle of a plane graph G is a cycle C with three edges. C divides the plane into its interior and exterior regions. A separating triangle is a triangle in G such that there are vertices in both the interior and the exterior of C. When discussing the rectangular layout L of a plane graph G, we can simplify the problem as follows. Let a, b, c, d be the four designated exterior vertices of G that correspond to the four rectangles in L located at the southwest, northwest, northeast and southeast corners, respectively. Let the extended graph Gext be the graph obtained from G as follows: 1. Add four vertices vW , vN , vE , vS and four edges (vW , vN ), (vN , vE ), (vE , vS ), (vS , vW ) into Gext . 2. Connect vW to every vertex of G on the exterior face between a and b in cw order. Connect vN to every vertex of G on the exterior face between b and c in cw order. Connect vE to every vertex of G on the exterior face between c and d in cw order. Connect vS to every vertex of G on the exterior face between d and a in cw order. See Figs 4 (1) and (2) for an example. It is well known [12] that G has a rectangular layout L if and only if Gext has a rectangular layout Lext , where the rectangles corresponding to vW , vN , vE , vS are located at the west, north, east and south boundary of Lext , respectively. Not every plane graph has rectangular layouts. The following theorem characterizes the plane graphs with rectangular layouts. Theorem 5. [12] A plane graph G has a rectangular layout L with four rectangles on its boundary if and only if: 1. Every interior face of G is a triangle and the exterior face of G is a quadrangle; and 2. G has no separating triangles. 3 A plane graph that satisfies the conditions in Theorem 5 is called a proper triangular plane graph. From now on we only consider such graphs. Our algorithm relies heavily on the concept of the regular edge labeling (REL) introduced in [9]. RELs have also been studied by Fusy [7, 8], who refers them as transversal structures. REL are closely related to several other edge coloring structures of planar graphs that can be used to describe straight line embeddings of orthogonal polyhedra [5, 6]. Definition 6. Let G be a proper triangular plane graph. A regular edge labeling REL R = {T1 , T2 } of G is a partition of the interior edges of G into two subsets T1 , T2 of directed edges such that: • For each interior vertex v, the edges incident to v appear in ccw order around v as follows: a set of edges in T1 leaving v; a set of edges in T2 leaving v; a set of edges in T1 entering v; a set of edges in T2 entering v. (Each of the four sets contains at least one edge.) • Let vN , vW , vS , vE be the four exterior vertices in ccw order. All interior edges incident to vN are in T1 and entering vN . All interior edges incident to vW are in T2 and entering vW . All interior edges incident to vS are in T1 and leaving vS . All interior edges incident to vE are in T2 and leaving vE . Fig 4 (2) shows an example of REL. (The green solid lines are edges in T1 . The red dashed lines are edges in T2 .) It is well known that every proper triangular plane graph G has a REL, which can be found in linear time [9, 10]. Moreover, from a REL of G, we can construct a rectangular layout L of G in linear time [9, 10]. Conversely, if we have a rectangular layout L for G, we can easily obtain a REL R of G as follows. For each interior edge e = (u, v) in G, we label and direct e according to the following rules. Let ru and rv be the rectangle in L corresponding to u and v respectively. • If ru is located below rv in L, the edge e is in T1 and directed from u to v. • If ru is located to the right of rv in L, the edge e is in T2 and directed from u to v. The REL R obtained as above is called the REL derived from L. (See Fig 4 (1) and (2)). Definition 7. Let L1 and L2 be two rectangular layouts of a proper triangular plane graph G. We say L1 and L2 are combinatorially equivalent if the RELs of G derived from L1 and from L2 are identical. Thus, the RELs of G one-to-one correspond to the combinatorially equivalent rectangular layouts of G. We can obtain two directed subgraphs G1 and G2 of G from an REL R = {T1 , T2 } as follows. • The vertex set of G1 is V . The edge set of G1 consists of the edges in T1 with direction in T1 , and the four exterior edges directed as: vS → vW , vS → vE , vW → vN , vE → vN . • The vertex set of G2 is V . The edge set of G2 consists of the edges in T2 with direction in T2 , and the four exterior edges directed as: vS → vW , vN → vW , vE → vS , vE → vN . Fig 4 (3) and (4) show the graph G1 and G2 for the REL shown in Fig 4 (2). For each face f1 in G1 , the boundary of f1 consists of two directed paths. They are called the two sides of f1 . Each side of f1 contains at least two edges. Similar properties hold for the faces in G2 [7, 8, 9, 10]. Definition 8. A REL R = {T1 , T2 } of G is called slant if for every face f in either G1 or G2 , at least one side of f contains exactly two directed edges. Theorem 3 characterizes the area-universal layouts in terms of maximal line segments in L. The following lemma characterizes area-universal layouts in term of the REL derived from L. 4 c e d f b e c f b d a a (1) (2) Figure 10: (1) a fan F(a, b, c) has the back boundary (a, b, c) and the front boundary (a, d, e, f, c); (2) a mirror fan M(a, f, e) has the back boundary (a, b, c, d, e) and the front boundary (a, f, e). Lemma 9. A rectangular layout L is area-universal if and only if the REL R derived from L is slant. Proof: Note that each face in G1 (G2 , respectively) corresponds to a maximal vertical (horizontal, respectively) line segment in L. (In the graph G1 in Fig 4 (3), the face f1 with the vertices f, e, g, c, h corresponds to the vertical line segment that is on the left side of the rectangle h in Fig 4 (1)). Assume L is area-universal. Consider a face f in G1 . Let lf be the maximal vertical line segment in L corresponding to f . Since L is area-universal, lf is a side of a rectangle r in L. Without loss of generality, assume r is to the left of lf . Then the left side of the face f consists of exactly two edges. Thus G1 satisfies the slant property. Similarly, we can show G2 also satisfies the slant property. Conversely, assume R is a slant REL. The above argument can be reversed to show that L is area-universal. The REL shown in Fig 4 (2) is not slant because the slant property fails for one G2 face. So the corresponding layout shown in Fig 4 (1) is not area-universal. By Lemma 9, the problem of finding an area-universal layout for G is the same as the problem of finding a slant REL for G. From now on, we consider the latter problem and G always denotes a proper triangular plane graph. 3 Face-Addition Algorithm with Exponential Time In this section, we outline a Face-Addition procedure that generates a slant REL R = {T1 , T2 } of G through a sequence of steps. The procedure starts from the directed path consisting of two edges vS → vE → vN . Each step maintains a partial slant REL of G. During a step, a face f of G1 is added to the current graph, resulting in a larger partial slant REL. When f is added, its right side is already in the current graph. The edges on the left side of f are placed in T1 and directed upward. The edges of G in the interior of f are placed in T2 and directed to the left. The process ends when the left boundary vS → vW → vN is reached. With this informal description in mind, we first introduce a few definitions. Then we will formally describe the Face-Addition algorithm (which takes exponential time). Consider a face f of G1 added during the above procedure. Because we want to generate a slant REL R, at least one side of f must be a path of length 2. This motivates the following definition. Figs 10 (1) and (2) show examples of a fan and a mirror fan, respectively. Definition 11. Let vl , vm , vh be three vertices of G such that vl and vh are two neighbors of vm and (vl , vh ) ∈ / E. Let Pcw be the path consisting of the neighbors {vl , v1 , . . . , vp , vh } of vm in cw order between vl and vh . Let Pccw be the path consisting of the neighbors {vl , u1 , . . . , uq , vh } of vm in ccw order between vl and vh . Note that since G has no separating triangles, both Pcw and Pccw are chord-free. 1. The directed and labeled subgraph of G induced by the vertices vl , vm , vh , v1 , . . . , vp is called the fan at {vl , vm , vh } and denoted by F(vl , vm , vh ), or simply g. 5 • The front boundary of g, denoted by α(g), consists of the edges in Pcw directed from vl to vh in cw order. The edges in α(g) are colored green. • The back boundary of g, denoted by β(g), consists of two directed edges vl → vm and vm → vh . The edges in β(g) are colored green. • The inner edges of g, denote by γ(g), are the edges between vm and the vertices v 6= vl , vh that are on the path Pcw . The inner edges are colored red and directed away from vm . 2. The directed and labeled subgraph of G induced by the vertices vl , vm , vh , u1 , . . . , uq is called the mirror fan at {vl , vm , vh } and denoted by M(vl , vm , vh ), or simply g. • The front boundary of g, denoted by α(g), consists of two directed edges vl → vm to vm → vh . The edges in α(g) are colored green. • The back boundary of g, denoted by β(g), consists of the edges in Pccw directed from vl to vh in ccw order. The edges in β(g) are colored green. • The inner edges of g, denote by γ(g), are the edges between vm and the vertices v 6= vl , vh that are on the path Pccw . The inner edges are colored red and directed into vm . Both F(vl , vm , vh ) and M(vl , vm , vh ) are called a gadget at vl , vm , vh . We use g(vl , vm , vh ) to denote either of them. The vertices other than vl and vh are called the internal vertices of the gadget. If a gadget has only one inner edge, it can be called either a fan or a mirror fan. For consistency, we call it a fan. We use g0 = F(vS , vE , vN ) to denote the initial fan, and gT = M(vS , vW , vN ) to denote the final mirror fan. The following observation is clear: Observation 12. For a slant REL, each face f of G1 is a gadget of G. The REL shown in Fig 4 (2) is generated by adding the gadgets: F(vS , vE , vN ), F(vS , d, h), F(f, h, c), M(e, b, vN ), F(vS , f, e), M(vS , vW , vN ). The following lemma is needed later. Lemma 13. The total number of gadgets in G is at most O(n2 ). Proof: Let deg(v) denote the degree of the vertex v in G. For each v, there are at most 2 · deg(v) · (deg(v) − 3) gadgets with v as its middle element. Thus the total number of gadgets of G is at most P 2 v∈V 2 · deg(v) · (deg(v) − 3) = O(n ). Definition 14. A cut C of G is a directed path from vS to vN that is the left boundary of the subgraph of G generated during the Face-Addition procedure. In particular, C0 = vS → vE → vN denotes the initial cut and CT = vS → vW → vN denotes the final cut. Let C be a cut of G. For any two vertices v1 , v2 of C, C(v1 , v2 ) denotes the subpath of C from v1 to v2 . The two paths C and C0 enclose a region on the plane. Let G|C denote the subgraph of G induced by the vertices in this region (including its boundary). Consider a cut C generated by Face-Addition procedure and a gadget g = g(vl , vm , vh ). In order for Face-Addition procedure to add g to C, the following conditions must be satisfied: A1: no internal vertices of α(g) are in C; and A2: the back boundary β(g) is contained in C; and A3: g is valid for C (the meaning of valid will be defined later). 6 If g satisfies the conditions A1, A2 and A3, Face-Addition procedure can add g to the current graph G|C by stitching β(g) with the corresponding vertices on C. (Intuitively we are adding a face of G1 .) Let G|C ⊗ g denote the new subgraph obtained by adding g to G|C . The new cut of G|C ⊗ g, denoted by C ⊗ g, is the concatenation of three subpaths C(vS , vl ), α(g), C(vh , vN ). The conditions A1 and A2 ensure that C ⊗ g is a cut. Any gadget g satisfying A1 and A2 can be added during a step while still maintaining the slant property for G1 . However, adding such a g may destroy the slant property for G2 faces. The condition A3 that g is valid for C is to ensure the slant property for G2 faces. (The REL shown in Fig 4 (2) is not slant. This is because the gadget F(f, h, c) is not valid, as we will explain later.) This condition will be discussed in §4. After each iteration of Face-Addition procedure, the edges of the current cut C are always in T1 and directed from vS to vN . All G1 faces f1 in G|C are complete (i.e. both sides of f1 are in G|C ). Some G2 faces in G|C are complete. Some other G2 faces f2 in G|C are open. (i.e. the two sides of f2 are not completely in G|C .) Definition 15. Any subgraph G|C generated during the execution of Face-Addition procedure is called a partial slant REL of G, which satisfies the following conditions: 1. Every complete G1 and G2 face in G|C satisfies the slant REL property. 2. For every open G2 face f in G|C , at least one side of f has exactly one edge. The intuitive meaning of a partial slant REL G|C is that it is potentially possible to grow a complete slant REL of G from G|C . The left boundary of a partial slant REL R is called the cut associated with R and denoted by C(R). Definition 16. 1. PSR(G) denotes the set of all partial slant RELs of G that can be generated by Face-Addition procedure. 2. G̃ = {g | g is a gadget in a R ∈ PSR(G)}. Observe that every slant REL R of G is in PSR(G). This is because R is generated by adding a sequence of gadgets g1 , . . . gT = M(vS , vW , vN ) to the initial gadget g0 = F(vS , vE , vN ). So if we choose this particular gi during the ith step, we will get R at the end. Thus G has a slant REL if and only if gT ∈ G̃. Note that Face-Addition procedure works only if we know the correct gadget addition sequence. Of course, we do not know such a sequence. The Face-Addition algorithm, described in Algorithm 1, generates all members in PSR(G). Algorithm 1: Face-Addition algorithm with Exponential Time 1.1 1.2 1.3 1.4 1.5 1.6 Initialize G̃ = {g0 }, and PSR(G) = {G|α(g0 ) }; repeat Find a gadget g of G and an R ∈ PSR(G) such that the conditions A1, A2 and A3 are satisfied for g and C = C(R); Add g into G̃, and add the partial slant REL R ⊗ g into PSR(G); until no such g and R can be found; G has a slant REL if and only if the final gadget gT ∈ G̃; Because |PSR(G)| can be exponentially large, Algorithm 1 takes exponential time. 4 Forbidden Pairs, G-Pairs, M-Triples, Chains and Backbones In this section, we describe the conditions for adding a gadget to a partial slant REL R ∈ PSR(G), while still keeping the slant REL property for G2 faces. (In other words, the condition A3.) 7 4.1 Forbidden Pairs Consider a R ∈ PSR(G) and its associated cut C = C(R). Let e be an edge of C. We use open-face(e) to denote the open G2 face in G|C with e as its open left boundary. The type of open-face(e) specifies the lengths of the lower side Pl and the upper side Pu of open-face(e): • Type (1,1): length(Pl ) = 1 and length(Pu ) = 1. • Type (1,2): length(Pl ) = 1 and length(Pu ) ≥ 2. • Type (2,1): length(Pl ) ≥ 2 and length(Pu ) = 1. • Type (2,2): length(Pl ) ≥ 2 and length(Pu ) ≥ 2. Note that the type of every open G2 face in a partial slant REL cannot be (2, 2). Based on the properties of REL, we have the following (see Fig 18): Observation 17. Let R ∈ PSR(G) and e be an edge on C(R). • If e is the last edge of α(g) of a fan or a mirror fan g, the type of open-face(e) is (2,1). • If e is a middle edge of α(g) of a fan g, the type of open-face(e) is (1,1). • If e is the first edge of α(g) of a fan or a mirror fan g, the type of open-face(e) is (1,2). (2, 1) (2, 1) (1, 1) (1, 1) (1, 2) (1, 2) a (1) (2) Figure 18: The types of open G2 faces: (1) Faces defined by edges on the front boundary of a fan; (2) Faces defined by edges on the front boundary of a mirror fan. Definition 19. A pair (g, g ′ ) of two gadgets of G is called a forbidden pair if either (1) the first edge of β(g) is the last edge of α(g ′ ); or (2) the last edge of β(g) is the first edge of α(g ′ ). Lemma 20. If a partial REL R contains a forbidden pair (g, g′ ), then R is not slant. Proof: Case 1: Suppose the first edge e1 of β(g) is the last edge of α(g ′ ) (see Fig 21 (1)). Let e2 be the first edge of α(g). The type of open-face(e1 ) is (2, 1) (regardless of whether g′ is a fan or a mirror fan). Note that open-face(e2 ) extends open-face(e1 ). The length of the upper side of open-face(e1 ) is increased by 1. Thus the type of open-face(e2 ) is (2, 2) and the slant property for G2 face fails. Case 2: Suppose the last edge e1 of β(g) is the first edge of α(g ′ ) (see Fig 21 (2)). Let e2 be the last edge of α(g). The type of open-face(e1 ) is (1, 2) (regardless of whether g′ is a fan or a mirror fan). Note that open-face(e2 ) extends open-face(e1 ). The length of the lower side of open-face(e1 ) is increased by 1. Thus the type of open-face(e2 ) is (2, 2) and the slant property for G2 face fails. In the REL R shown in Fig 4 (2), (F(f, h, c), F(vS , d, h)) is a forbidden pair. So R is not a slant REL. 8 (2, 1)  1 2
2 (2, 1) e1 (1, 2) (1, 1) g (1, 1)  (1, 2) a (2) (1) Figure 21: The proof of Lemma 20: (1) g is a fan; (2) g is a mirror fan. 4.2 The Condition A3 The following lemma specifies a necessary and sufficient condition for adding a fan into G̃, and a sufficient condition for adding a mirror fan into G̃. Lemma 22. Let R ∈ PSR(G) and C = C(R) be its associated cut. Let g L be a gadget and L = β(g L ). Suppose that the conditions A1 and A2 are satisfied for gL and C. 1. A fan gL can be added to R (i.e. g L satisfies the condition A3) if and only if there exists a gadget g R ∈ R such that β(g L ) ⊆ α(g R ). 2. A mirror fan gL can be added to R (i.e. g L satisfies the condition A3) if there exists a gadget g R ∈ R such that β(g L ) ⊆ α(g R ). e4 e3 e4 e'' e2 gL (2, 1) e' (1, 1) gR e1 e'' e'' e3 (2, 1) (1, 1) (1, 2) (1) a e2 e' e1 gO gU (2, 1) e2 (1, 1) R g gR gL e3 gL (2, 1) ek e'' e' (1, 2) e1 (1, 1) gB (1, 2) e' e1 (1, 1) a (3) (2) a (4) Figure 23: (1) and (2) open faces defined by edges on the front boundary of a fan g L ; (3) and (4) open faces defined by edges on the front boundary of a mirror fan g O . Proof: If part of (1): Suppose there exists a gadget gR ∈ R such that β(g L ) ⊆ α(gR ). (Figs 23 (1) and (2) show two examples. In Fig 23 (1), gR is a fan. In Fig 23 (2), gR is a mirror fan). Let e1 , . . . , ek be the edges in α(g L ). Let e′ , e′′ be the two edges in β(g L ). Let C ′ = C ⊗ g L be the new cut after adding gL . For each 2 ≤ i ≤ k − 1, the type of open-face(ei ) is (1, 1). • open-face(e1 ) extends open-face(e′ ), and add 1 to the length of the upper side of open-face(e′ ). 9 • open-face(ek ) extends open-face(e′′ ), and add 1 to the length of the lower side of open-face(e′′ ). Regardless of where e′ , e′′ are located on α(gR ), and regardless of whether g R is a fan (see Fig 23 (1)) or a mirror fan (see Fig 23 (2)), the type of open-face(e1 ) is (1, 2); and the type of open-face(ek ) is (2, 1). Thus R ⊗ g L ∈ PSR(G). Only if part of (1): Suppose that there exists no gadget gR ∈ R such that β(gL ) ⊆ α(gR ). Let e′ , e′′ be the two edges of β(g L ). e′ must be on the front boundary of some gadget g′ in R. e′′ must be on the front boundary of some gadget g′′ in R. Clearly g ′ 6= g′′ . (If g′ = g′′ , we would have β(g L ) ⊆ α(g′ )). Then either (g L , g′ ) or (g L , g ′′ ) must be a forbidden pair. By Lemma 20, gL cannot be added to R. (2) Let gL be a mirror fan. Suppose there exists a gadget gR ∈ R such that β(gO ) ⊆ α(gR ) (see Fig 23 (3)). Similar to the proof of the if part of (1), we can show R ⊗ g L ∈ PSR(G). By Lemma 22, the only way to add a fan g L to R is by the existence of a gadget g R ∈ R such that ⊆ α(gR ). For a mirror fan g, there is another condition for adding g to R which we discuss next. Let v1 = vS , v2 , . . . , vt−1 , vN be the vertices of C = C(R) from lower to higher order. Let e1 and et be the first and the last edge of C. Imagine we walk along C from vS to vN . On the right side of C, we pass through a sequence of gadgets in R whose front boundary (either a vertex or an edge) touches C. Let support(R) = (g1 , g2 , . . . , gk−1 , gk ), where e1 ∈ α(g1 ) and et ∈ α(gk ), denote this gadget sequence. Note that some gadgets in support(R) may appear multiple times in the sequence. (See Fig 26 (1) for an example.) Consider a mirror fan g O to be added to R. Note that L = β(g O ) is a subsequence of C. Let a and b be the lowest and the highest vertex of L. Let el be the first edge and eh be the last edge of L. When walking along L from a to b, we pass through a subsequence of the gadgets in support(R) on the right of L. Let support(L, R) = (g B = gp , gp+1 , . . . , gq−1 , gq = gU ) denote this gadget subsequence, where: β(gL ) • g B is the gadget such that el ∈ α(gB ). • g U is the gadget such that eh ∈ α(gU ). In Fig 26 (1), if we add a mirror fan g3 with L = β(g3 ) = (a, b, f, h, k, d), then support(L, R) = (g1 , g0 , g2 ). Lemma 24. Let R ∈ PSR(G) and C = C(R) be its associated cut. Let g O be a mirror fan and L = β(g O ). Suppose that the conditions A1 and A2 are satisfied for g O and C. Let support(L, R) = (gB , gp+1 , · · · , gq−1 , gU ). Then gO can be added to R (i.e. gO satisfies the condition A3) if and only if neither (gO , gB ) nor (g O , gU ) is a forbidden pair. Proof: First suppose that gO can be added to R to form a larger partial slant REL. Then, by Lemma 20, neither (gO , gB ) nor (gO , g U ) is a forbidden pair. Conversely, suppose that neither (g O , gB ) nor (gO , g U ) is a forbidden pair. Let e1 , . . . , ek be the edges of L. The type of open-face(e1 ) is either (1, 1) or (1, 2). The type of open-face(ek ) is either (1, 1) or (2, 1). (Fig 23 (4) shows an example.) Let e′ , e′′ be the two edges in α(gO ). After adding gO to R, the types of open-face(e′ ) and open-face(e′′ ) becomes (1, 2) and (2, 1), respectively. They still keep the slant property for G2 faces. Moreover, for each edge ei (2 ≤ i ≤ k − 1), open-face(ei ) becomes a valid complete G2 face after adding g O to R. Hence R ⊗ gO is a partial slant REL of G. 4.3 Connections, Chains and Backbones Given an R ∈ PSR(G) and a gadget g, it is straightforward to check if the conditions in Lemmas 22 and 24 are satisfied. However, as described before, maintaining the set PSR(G) requires exponential time. So we must find a way to check the conditions in Lemmas 22 and 24 without explicit representation of R. 10 i g5 2 g4 g7 j k VN i C(h, i) g6 g8 f VE C(c, f) e d g2 VE k c g3 f g1 e h g5 O1 g2 g0 g4 g 1 g3 b (1) h g0 d a c (2) VS b a Figure 26: (1) R ∈ PSR(G) is obtained by adding gadgets g1 , g2 , g3 , g4 , g5 , g6 , g7 , g8 , in this order, to g0 . C(R) = (vS , a, c, d, e, j, i, vN ); support(R) = (g0 , g1 , g3 , g2 , g8 , g7 , g5 , g4 , g0 ) and (g0 C g1 C g3 C g2 C g8 C g7 C g5 C g4 C g0 ). The pair (g0 , g1 ) belongs to the G-pair Λ1 = (g1 , g0 ), the triple (g1 , g3 , g2 ) belongs to the M-triple Λ2 = (g1 , g3 , g2 ), the triple (g2 , g8 , g7 ) belongs to the M-triple Λ3 = (g2 , g8 , g7 ), the pair (g7 , g5 ) belongs to the M-triple Λ4 = (g6 , g7 , g5 ), the pair (g5 , g4 ) belongs to the G-pair Λ5 = (g5 , g4 ) and the pair (g4 , g0 ) belongs to the G-pair Λ6 = (g4 , g0 ). (2) The G-pair Λ = (g1 , g0 ) is a fractional connection with two pockets: O1 is bounded by C(c, f ) and α(Λ)(c, f ) and O2 is bounded by C(h, i) and α(Λ)(h, i). Consider two R, R′ ∈ PSR(G) such that R = 6 R′ but support(R) = support(R′ ). Clearly this implies C(R) = C(R′ ). By Lemmas 22 and 24, a gadget g can be added to R if and only if g can be added to R′ . Thus, whether g can be added to an R ∈ PSR(G) is completely determined by the structure of gadgets in support(R). There may be exponentially many R′ ∈ PSR(G) with support(R′ ) = support(R). Instead of keeping information of all these R′ , we only need to keep the information of the structure of support(R). This is the main idea for converting Algorithm 1 to a polynomial time algorithm. In order to describe the structure of support(R), we need the following terms and notations. Definition 25. Let R ∈ PSR(G) and g be a gadget with L = β(g). • If support(L, R) contains only one gadget g R , and the conditions A1, A2 and A3 are satisfied, then (g, g R ) is called a G-pair. We use (gL , gR ) to denote a G-pair. • If support(L, R) contains at least two gadgets, and the conditions A1, A2 and A3 are satisfied, then (g B , g, gU ) is called a M-triple. We use (gB , gO , gU ) to denote a M-triple. • A G-pair (gL , gR ) or a M-triple (gB , gO , gU ) is called a connection and denoted by Λ. L , v L ), g R = g(v R , v R , v R ), the front bound• For a connection Λ = (g L , gR ), where gL = g(vlL , vm m h l h L R ary of Λ, denoted by α(g , g ) or α(Λ), is the concatenation of the paths α(gR )(vlR , vlL ), α(gL ), α(gR )(vhL , vhR ). B , v B ), g O = g(v O , v O , v O ), g U = • For a connection Λ = (g B , gO , g U ), where gB = g(vlB , vm m h l h B O U U U g(vl , vm , vh ), the front boundary of Λ, denoted by α(g , g , gU ) or α(Λ), is the concatenation of the paths α(g B )(vlB , vlO ), α(gO ), α(gU )(vhO , vhU ). It is tempting to think that if all gadgets in support(R) have been added into G̃, then R has been constructed. Unfortunately, this is not true. In order to form R, the gadgets in support(R) = 11 (g1 , g2 , . . . , gk−1 , gk ) must have been added to G̃ in the following way: When walking along C(R) from vS to vN , the gadgets in support(R) form a sequence (Λ1 , . . . , Λp ) of connections such that each consecutive pair (gi , gi+1 ) or triple (gi−1 , gi , gi+1 ) of gadgets belong to a Λj (1 ≤ j ≤ p); and each consecutive pair Λi , Λi+1 share a common gadget in support(R). (See Fig 26 (1) for an illustration). Note that when the pair (gi−1 , gi ) and the pair (gi , gi+1 ) belong to the same connection Λj , it means gi−1 and gi+1 are the same gadget and (gi , gi+1 ) = (gi , gi−1 ) is a G-pair. In this case, we keep only one Λj in the sequence Λ1 . . . , Λp . As seen in Fig 26 (1), in addition to these connections Λj (1 ≤ j ≤ p), some gadget pairs (or triples) that are not consecutive in support(R) may also form additional connections. (In Fig 26 (1), the gadgets g0 and g2 are not consecutive in support(R). But they form a G-pair (g2 , g0 )). Let Con(R) denote the set of connections formed by the gadgets in support(R). (By this definition, each Λ ∈ Con(R) has at least two gadgets in support(R)). It is the structure of Con(R) that determines if a new gadget g can be added to R or not. In general, the connections in Con(R) cannot be described as a simple linear structure. To describe it precisely, we need the following definitions. Consider a connection Λ ∈ Con(R). If α(Λ)∩C is a contiguous subpath of C, Λ is called a contiguous connection. If not, Λ is called a fractional connection. (In Fig 26 (1), the G-pair (g1 , g0 ) is a fractional connection. Because the cut C ∩ α(g1 , g0 ) are (a, b, c) and (f, h) and (i, k), they are not a contiguous subpath of C.) Consider a fractional connection Λ. Let u and v be the lowest and the highest vertices of C ∩ α(Λ) respectively. When walking along C from u to v, we encounter α(Λ) multiple times. The subpath C(u, v) can be divided into a number of subpaths that are alternatively on α(Λ), not on α(Λ), . . ., on α(Λ). There exist at least two vertices a, b in α(Λ) such that C(a, b) ∩ α(Λ)(a, b) = {a, b}. For each such pair of vertices a, b, the interior region bounded by the subpaths C(a, b) and α(Λ)(a, b) is called a pocket, denoted by O = (C(a, b), Λ), of Con(R). Fig 26 (2) shows a fractional connection Λ (the G-pair (g1 , g0 )) with two pockets O1 and O2 . A connection Λ ∈ Con(R) is called maximal if it is not contained in any pocket of Con(R). A maximal connection can be either contiguous or fractional. A non-maximal connection Λ′ ∈ Con(R) is either completely contained in some pocket O formed by a subpath of C and a maximal fractional connection Λ (namely all gadgets of Λ′ are contained in O); or partially contained in O (namely some gadget of Λ′ is contained in O and some gadget of Λ′ is shared with Λ). (In Fig 26 (2), the G-pair (g3 , g1 ) and the M-triple (g1 , g2 , g0 ) are partially contained in the pocket O1 . The M-triple (g4 , g5 , g2 ) and the G-pair (g4 , g3 ) are completely contained in O1 ). Note that a pocket may contain other smaller pockets. In general, the pockets of Con(R) are nested in a forest-like structure. The way to deal with fractional connections is very similar to contiguous connections. Hence in the following paragraphs, we will assume there are no fractional connections. Definition 27. Let two gadgets {g, g ′ } belong to a connection Λ. We say g precedes g ′ on C and write g C g′ if the following conditions hold: (1) ((α(g) ∩ C) ∪ (α(g′ ) ∩ C)) is contiguous on C; (2) When walking along C, we encounter the gadget g before g′ . Depending on the types of connections and their positions on a cut C, there are five cases for the relation C . (They are shown in Fig 28.) Definition 29. Given a partial slant REL R with its associated cut C, a sequence of gadgets (g1 , g2 , · · · , gk ) in support(R) is called a chain of C and denoted by chain(C) if the following conditions hold: 1. (g1 C g2 C · · · C gk ) and 2. for each 1 ≤ i ≤ k − 1, either (gi , gi+1 ) or (gi , gi+1 , gi+2 ) belongs to a connection Λ ∈ Con(R). In Fig 26 (1), (g0 C g1 C g3 C g2 C g8 C g7 C g5 C g4 C g0 ) is a chain of C where we have the G-pair (g1 , g0 ), the M-triple (g1 , g3 , g2 ), the M-triple (g2 , g8 , g7 ), the M-triple (g6 , g7 , g5 ), the G-pair (g5 , g4 ) and the G-pair (g4 , g0 ). Because the way a partial slant REL R is constructed, the following property is clear. 12 gU gR gU gU gL gO gO g gB gB (1) gB gR gL (2) (3) (4) (5) Figure 28: (1) Case 1: (g L , gR ) is a G-pair and gL C gR ; (2) Case 2: (gL , gR ) is a G-pair and gR C gL ; (3) Case 3: (gB , g O , gU ) is a M-triple and gB C gO C g U ; (4) Case 4: (gB , gO , gU ) is a M-triple and gO C g U ; (4) Case 5: (g B , gO , gU ) is a M-triple and g B C gO . Property 30. Given a partial slant REL R with its associated cut C, the support(R) = (g1 , g2 , · · · , gk ) is a chain of C. Given a partial slant REL R with its associated cut C, if we can add a gadget g to R, then it implies that back boundary L = β(g) of g is a part of C. Let support(L, R) be a subsequence of support(R) consisting of gadgets in support(R) that touch L. We can define an order L which is similar to C . Definition 31. Given a mirror fan gO with L = β(g O ), let two gadgets {g, g ′ } belong to a connection Λ. We say g precedes g′ on L and write g L g′ if the following conditions hold: (1) (α(g) ∩ L) ∪ (α(g ′ ) ∩ L) is contiguous on L; (2) When walking along L, we encounter the gadgets g before g′ . Definition 32. Let g O be a mirror fan with L = β(g O ). A backbone(L, R) consists of a sequence of gadgets (g1 = gB , g2 , · · · , gk = gU ) in support(L, R) such that 1. (g1 L g2 L · · · L gk ), 2. for each 1 ≤ i ≤ k − 1, either (gi , gi+1 ) or (gi , gi+1 , gi+2 ) belongs to a connection Λ ∈ Con(R) and 3. Neither (gO , gB ) nor (g O , g U ) is a forbidden pair. In Fig 26 (1), consider the mirror fan g3 with L = β(g3 ). We have: support(L) = (g1 , g0 , g2 ) where Λ1 is the G-pair (g1 , g0 ), Λ2 is the G-pair (g2 , g0 ) and g1 L g0 L g2 . Based on above discussion, we can restate Lemma 24 as follows: Lemma 33. Let R ∈ PSR(G) and C = C(R). Let g O be a mirror fan with L = β(gO ). Suppose that the conditions A1 and A2 are satisfied for gO and C. Let support(L, R) = (g B , gp+1 , · · · , gq−1 , gU ). Then (gB , gO , gU ) forms a M-triple (i.e. gO satisfies the condition A3) if and only if there exists a backbone(L, R) consisting of gadgets in support(L, R) and connections in Con(R). Note that each gadget and connection in backbone(L, R) belong to the same partial slant REL R. 5 Face-Addition Algorithm with Polynomial Time We will present our polynomial time Face-Addition algorithm in this section. In §5.1, we will describe the algorithm to find a superset of chains. In §5.2, we will give more details of key procedures in §5.1. In §5.3, we will present an example that Algorithm 2 may combine two subchains of two different partial 13 RELs into a chain which only satisfies the order C in Property 30 (there exist gadgets coming from different chains). In §5.4, we will describe a backtracking algorithm to check whether whether a chain in the superset of chains constructed by Algorithm 2 corresponds to a slant REL or not. Also, we will give runtime analysis of the backtracking algorithm. 5.1 Polynomial Time algorithm The polynomial time Face-Addition algorithm is described in Algorithm 2. Algorithm 2: Face-Addition Algorithm with Polynomial Time Input: A proper triangular plane graph G 2.1 Set Ṽ = {g0 = F(vS , vE , vN )} and V̂ = ∅; 2.2 repeat 2.3 Find a gadget g such that: either: there exist v-G-pairs (g, gR ) ∈ / V̂ with gR ∈ Ṽ (v-G-pairs are defined later); add g into Ṽ (if it’s not already in Ṽ); add all such v-G-pairs (g, gR ) into V̂; or: g is a mirror fan and there exist v-M-triples (gB , g, gU ) 6∈ V̂ with gB , gU ∈ Ṽ (v-M-triples are defined later); add g into Ṽ (if it’s not already in Ṽ); add all such v-M-triples (gB , g, g U ) into V̂; 2.4 2.5 2.6 2.7 2.8 2.9 until no such gadget g can be found; if gT = M(vS , vW , vN ) is not in Ṽ then G has no slant REL; else Backtrack each v-chain of gT (v-backbone of gT ) to check whether G corresponds a slant REL in Algorithm 4; end Algorithm 2 emulates the operations of Algorithm 1 without explicitly maintaining the set PSR(G). Instead, it keeps two sets: (1) a set Ṽ of gadgets of G which contains the gadgets in the set G̃ defined in §3, and (2) a set V̂ of connections of G which contains the connections in the set {Con(R)|R ∈ PSR(G)} defined in §4.3. In §4.3, many concepts (cut, chain, backbone . . . etc.) were defined referring to a R ∈ PSR(G). We now need counterparts of these concepts without referring to a specific R. For a concept x defined previously, we will use virtue x or simply v-x for the counterpart of x. (For example, v-cut for virtue cut, v-chain for virtue chain, v-backbone for virtue backbone). A v-G-pair (v-M-triple, respectively) is similar to a G-pair (M-triple, respectively) but without referring to a specific R ∈ PSR(G). Whenever Algorithm 1 adds a gadget g to G̃ through a G-pair (or a M-triple, respectively), Algorithm 2 adds g into Ṽ and add a corresponding v-G-pair (or v-M-triple, respectively) into V̂. Initially, V̂ is empty and Ṽ contains only the initial fan g0 = F(vS , vE , vN ). In each step, the algorithm finds either new v-G-pairs (g, g R ) with gR ∈ Ṽ; or new v-M-triples (gB , g, g U ) with gB , g U ∈ Ṽ. In either case, it adds g into Ṽ. But instead of using a R ∈ PSR(G), Algorithm 2 relies on the information stored in Ṽ and V̂ to find v-G-pairs and v-M-triples. Fix a step in Algorithm 2 and consider the sets Ṽ and V̂ after this step. Any simple path C in G from vS to vN is called a v-cut of G. A gadget pair (g, gR ) is called a v-G-pair if gR ∈ Ṽ and β(g) ⊆ α(gR ). 14 For a v-cut C, define: Ṽ(C) = {g ∈ Ṽ | α(g) intersects C} V̂(C) = {Λ ∈ V̂ | the frontiers of at least two gadgets of Λ intersect C} Let e1 and et be the first and the last edge of C. A subset of gadgets D ⊆ Ṽ(C) is called a v-support of C if the following conditions hold: • The gadgets in D can be arranged into a sequence (g1 , g2 , . . . , gk ) such that e1 ∈ α(g1 ), et ∈ α(gk ) and, when walking along C from vS toward vN , we encounter these gadgets in this order. • Any two (or three) consecutive gadgets (gi , gi+1 ) (or (gi−1 , gi , gi+1 )) belong to a connection in V̂. If a set S of connections formed by the gadgets in a v-support of C satisfies the structure property described in Definition 29, S is called a v-chain of C. Clearly, any chain is also a v-chain. Let g be a mirror fan with L = β(g). Let a and b be the lowest and the highest vertex of L, and el and eh the first and the last edge of L, respectively. Define: Ṽ(L) = {g ∈ Ṽ | α(g) intersects L} V̂(L) = {Λ ∈ V̂ | the frontiers of at least two gadgets of Λ intersect L} A subset of gadgets D ⊆ Ṽ(L) is called a v-support of L if the following conditions hold: • The gadgets in D can be arranged into a sequence (g B = gp , g2 , . . . , gq = g U ) such that el ∈ α(gB ), eh ∈ α(gU ) and, when walking along L from a toward b, we encounter these gadgets in this order. • Any two (or three) consecutive gadgets (gi , gi+1 ) (or (gi−1 , gi , gi+1 )) belong to a connection in V̂. If a set S of connections formed by the gadgets in a v-support of L only satisfies the order property L and the third property described in Definition 32, S is called a v-backbone of L. If there is a vbackbone(L), we call (g B , g, g O ) a v-M-triple. Both v-G-pairs and v-M-triples are called v-connections. First, we bound the number of loop iterations in Algorithm 2. By Lemma 13, the number of gadgets in G is at most N = O(n2 ). So the number of v-G-pairs is at most O(N 2 ) and the number of v-M-triples is at most O(N 3 ). Hence V̂ contains at most O(n6 ) elements. Since each iteration adds at least either a v-G-pair or a v-M-triple into V̂, the number of iterations is bounded by O(n6 ). We need to describe how to perform the operations in the loop body, which is clearly dominated by finding v-G-pairs and finding v-M-triples. Given two gadgets g, gR and the sets Ṽ and V̂, it is easy to check if (g, g R ) is a v-G-pair (i.e. gR ∈ Ṽ and β(g) ⊆ α(gR )) in polynomial time. However, finding v-M-triples (gB , g, g U ) is much more difficult. In §5.2, we show this can be done, in polynomial time, by finding a v-backbone(β(g)) consisting of connections in V̂. This will establish the polynomial run time of the repeat loop of Algorithm 2. Lemma 34. Let S be the set of all v-backbones of gT . (Because L = β(gT ) is a v-cut, each v-backbone of L is actually a v-chain of G.) For each R ∈ PSR(G) with its associated cut C = C(R), there exists a v-chain S ∈ S (which is a v-backbone of L) generated by Algorithm 1 such that S = chain(C). Proof: For each mirror fan g with L = β(g), if g is in some partial slant REL, then its backbone follows the order L . So we have G̃ ⊆ Ṽ and {Con(R)|R ∈ PSR(G)} ⊆ V̂. Since we use the two supersets Ṽ and V̂ of G̃ and Ĝ to find v-backbones of gT (chains of G), we immediately have: {chain(C)|R is a slant REL of G with its associated cut C = C(R)} ⊆ S. 15 In this subsection, we have described Algorithm 2 which constructs (1) the set S of v-chains such that chain(C) of each partial REL R ∈ PSR(G) with its associated cut C = C(R) is included in S, (2) the set V̂ of v-connections containing each connection Λ ∈ {Con(R)|R ∈ PSR(G)}, (3) the set Ṽ of gadgets containing each gadget g ∈ G̃. Lemma 34 states that any chain is a v-chain. But the reverse is not necessarily true. In §5.3, we provide an example that a v-chain is not equal to the chain of the associated cut of any partial slant REL. Thus we need to check whether a v-chain constructed by Algorithm 2 is really a chain of a slant REL R of G. In §5.4, we describe a backtracking algorithm to detect all such v-chains. 5.2 Algorithm for Finding v-Backbones and v-M-triples Consider a gadgets (g B , g O , gU ) with the back boundary L = β(gO ). Let a and b be the lowest and the highest vertex of L, e1 and e2 the first and the last edge of L. In this subsection, we show how to check whether (gB , g O , gU ) is a v-M-triple or not in polynomial time. By the definition of v-M-triples, this is equivalent to finding v-backbone(L)s by using the v-connections in V̂. Let Ṽ(L) be the set of gadgets in Ṽ that can be in any v-backbone(L). From the conditions described in Definition 32, Ṽ(L) contains the gadgets g ∈ Ṽ that satisfy the following conditions: • The front boundary of g intersects L and g belongs to some v-connection Λ ∈ V̂. • the front boundary of gB contains e1 ; and (g O , gB ) is not a forbidden pair. • the front boundary of gU contains e2 ; and; and (g O , gU ) is not a forbidden pair. To determine which gadgets in Ṽ(L) can form a v-backbone of L, we construct a directed acyclic graph HL = (VL , EL ) as follows: Definition 35. Given a triple (g B , gO , gU ) with L = β(g O ), the backbone graph HL of gO is defined as follows: • VL = {(l, g)|l = L ∩ α(g) and g ∈ Ṽ(L)} and • EL = {(l1 , g1 ) → (l2 , g2 )| {g1 , g2 } belongs to a v-connection Λ ∈ V̂(L) and l1 ∪ l2 is contiguous on L } A source (sink, respectively) vertex has no incoming (outgoing, respectively) edges in HL . The intuitive meaning of a directed path P ∈ HL=β(gO ) from the source to the sink is that P corresponds to a v-backbone of L = β(g O ) and for each vertex (l, g) ∈ P , g corresponds a gadget in a v-backbone and l is equal to the intersection L ∩ α(g) of L and the front boundary of g. Moreover, for different vM-triples of gO Λ1 = (gB1 , gO , g U1 ), Λ2 = (gB2 , gO , gU2 ) ∈ V̂(L), we have vertices vO = (L ∩ α(g O ), gO ) ′ = (L ∩ α(g O ), g O ) in H to represent g O such that Λ and Λ represent different subpaths and vO L 1 2 in HL : one is (L ∩ α(gB1 ), g B1 ) → vO = (L ∩ α(gO ), g O ) → (L ∩ α(gU1 ), gU1 ) and the other one is ′ = (L ∩ α(g O ), g O ) → (L ∩ α(g U2 ), g U2 ) where the vertex v represents the mirror (L ∩ α(gB2 ), gB2 ) → vO O ′ fan in Λ1 and vO represents the mirror fan in Λ2 . Lemma 36. HL is acyclic and can be constructed in O(|V̂|2 ) time. Proof: Consider a g ∈ Ṽ. Knowing L, we can easily determine if g is in Ṽ(L) in constant time. So we can identify the set VL in O(|V̂|) time. For two vertices (l, g) and (l′ , g′ ) in VL , the edge (l, g) → (l′ , g′ ) exists if and only if the following two conditions are satisfied: (1) g and g ′ belong to some v-connection in V̂(L); (2) l ∪ l′ is contiguous on L; and (2) when walking along L upwards, we encounter the gadgets g before g′ . These two conditions can be easily checked in constant time. So the set EL can be determined in O(|VL |2 ) = O(|V̂|2 ) time. Thus HL can be constructed in O(|V̂|2 ) time. The edge directions of HL are defined by the relation L . Since L is acyclic, HL is acyclic. 16 Lemma 37. Given a triple (gB , gO , g U ) with L = β(gO ), let HL be the graph defined in Definition 35. 1. Each directed path from the source to the sink in HL corresponds to the v-M-triple (gB , gO , gU ). 2. The v-M-triple (gB , gO , gU ) corresponds to a set of directed paths from g B to gU in HL . Proof: Statement 1. Consider any directed path (lB , gB ) → · · · → (lU , gU ) from the source (lB , gB ) to the sink (lU , g U ) in HL . Since each directed edge (l, g) → (l′ , g′ ) in HL follows the order L on L, each directed path from (lB , gB ) to (lU , gU ) is a v-backbone of gO and (g B , gO , g U ) is a v-M-triple. Statement 2: Consider a v-M-triple (gB , g O , gU ). This means that there exists a v-backbone gB L g2 L · · · L gk−1 L gU on L. Because each g L g′ on L is a directed edge (l, g) → (l′ , g′ ) in HL , we have that (lB , g B ) → (l2 , g2 ) → · · · → (lk−1 , gk−1 ) → (lU , gU ) is a directed path from (lB , gB ) to (lU , gU ) in HL . Note that there may exist multiple paths in HL from (lB , gB ) to (lU , g U ). All these paths correspond to the same v-M-triple (g B , g O , gU ). The intuitive meaning of this fact is as follows. When we add g O via the v-M-triple (gB , g O , gU ), even though the gadgets g B and g U are fixed, the v-connections and gadgets in the v-backbone(L)s between g B and gU may be different. But as long as they form a valid v-backbone(L), we can add g O . The following Algorithm 3 finds v-M-triples (gB , gO , gU ) by finding v-backbone(L)s. Algorithm 3: Find v-M-triples 3.1 3.2 3.3 Input: A triple (gB , gO , g U ) with L = β(gO ) and the set V̂ of v-connections From the connections in V̂, identify the set V̂(L); Construct the directed graph HL as in Definition 35; By using Lemma 37, return whether (gB , gO , gU ) is v-M-triple or not; Theorem 38. Given a gadgets triple (gB , gO , gU ), Algorithm 3 can successfully test whether (gB , g O , gU ) is a v-M-triple in polynomial time. Proof: The correctness of the algorithm follows from Lemma 37. By Lemma 36, the steps 1 and 2 can be done in polynomial time. Step 3: Since HL is acyclic, we can use breadth-first search to find whether (lU , gU ) is reachable from B (l , gB ). Then (gB , g, g U ) is a v-M-triple if and only if (lU , gU ) is reachable from (lB , gB ). This step is carried out by calling breadth-first search which takes polynomial time. So the total time for this step is polynomial. Note that the total number of source to sink paths in HL can be exponential. However, we only need to find one path from (lB , gB ) to (lU , gU ). 5.3 An Example that A v-Chain Does Not Have A Slant REL In this subsection, we present an example to show why a v-chain defined in the last subsection does not necessarily have a corresponding partial slant REL. Imagine that we have two v-chains C and C ′ . Suppose that C can be partitioned into C = (C1 , C2 , C3 ) and C ′ can be partitioned into C ′ = (C1′ , C2′ , C3′ ) such that C2 = C2′ , then we may have another two v-chains (C1 , C2 = C2′ , C3′ ) and (C1′ , C2 = C2′ , C3 ). However, both of the two v-chains can’t correspond any partial slant REL. Fig 39 (3) shows a v-chain (g1 , g2 , g3 , g4 , g5 , g6 ) which can’t have a corresponding partial slant REL where (g1 , g2 , g3 ) from R1 and (g3 , g4 , g5 , g6 ) from R2 shares a common gadget g3 . 17 g3 g2 g1 1 g4 g3 gA g0 (1) 2 g5 gF gE gD g g6 g4 g3 R g2 g5 g6 g1 C gB g0 (2) (3) Figure 39: (1) is a partial slant REL R1 which consists of gadgets {g0 , gA , gF , g1 , g2 , g3 }; (2) is a partial slant REL R2 which consists of gadgets {g0 , g3 , g4 , g5 , g6 , gB , gC , gD , gE }; (3) (g1 , g2 , g3 , g4 , g5 , g6 ) is a v-chain where (g1 , g2 , g3 ) is a subchain of R1 and (g3 , g4 , g5 , g6 ) is a subchain of R2 . However, the v-chain is not coming from the same partial slant REL. {g1 , g2 , g3 } can be added into R1 only when {g0 , gA } have been added into R1 . The order of added gadgets in R2 is: (g0 , gB , g6 , g5 , gC , gD , gE , g3 , g4 ). But, gA and each gadget of {gB , gC , gD , gE } can not coexist in the same REL because some faces of gA and each gadget of {gB , gC , gD , gE } overlap. 5.4 An Algorithm to Find Conflicting Gadgets via Backtracking In the last subsection, we know that each v-chain of gT only contains partial information of a complete slant RELR. In this subsection, we use a recursive constructive definition to define a hierarchal v-chain which represents sufficient information of a complete REL R and can be represented by a DAG as follows: Definition 40. Given the final mirror fan gT with C = β(gT ), a hierarchal v-chain J = (V (J ), E(J )) of C is a DAG recursively defined as follows: 1. The root J (r) ∈ V (J ) is a sequence of pairs ((C1 , g1 ), (C2 , g2 ), · · · , (Ck , gk )) where (a) (g1 , g2 , · · · , gk ) is a v-chain (g1 C g2 C · · · C gk ) of C and (b) each Ci = C ∩ α(gi ), 1 ≤ i ≤ k, is a portion of the front boundary α(gi ) of gi . 2. While(α(g0 ) * C) (a) select a gadget g such that i. α(g) ⊆ C and ii. there exist a sequence of pairs ((l1 , g) ∈ S1 , (l2 , g) ∈ S2 , · · · , (lh , g) ∈ Sh ) where each Si , 1 ≤ i ≤ h, is a vertex of J and l1 ∪ l2 ∪ · · · ∪ lh = α(g), (b) create a vertex S consisting of a sequence of pairs ((l1 , g1 ), (l2 , g2 ), · · · , (lh , gh )) where i. (g1 L g2 L · · · L gh ) is a v-backbone of L = β(g) and ii. each li = L ∩ α(gi ), 1 ≤ i ≤ h, is a portion of the front boundary α(g) of g, (c) add S into V (J ) and for each Si , 1 ≤ i ≤ h, add an arc Si → S into E(J ). And, (d) change C to C(vS , a) ∪ β(g) ∪ C(b, vN ) where a and b are the first and the last vertices of β(g), respectively. Intuitively a hierarchal v-chain J is a hierarchal decomposition of a complete slant REL R and the root J (r) of J represents a chain(C) of R’s associated cut C = C(R). In the following definition, a hierarchial structure H consists of a set of DAGs (backbone graphs) and H can implicitly store all possible hierarchal v-chains J . 18 Definition 41. H = (V (H), E(H)) is a DAG where 1. for each vertex v ∈ V (H), v represents a DAG H(v) = (V (H(v)), E(H(v))) over V (H(v)) where (a) every vertex w ∈ V (H(v)) is a pair (l, g) and l is a portion of the front boundary α(g) of g, (b) for each arc e = (ue → u′e ) in E(H(v)), e associates with a DAG H(v ′ ), v ′ ∈ V (H) (the associated DAG of e is denoted by H(e)), the source of H(e) is the starting vertex ue of e and the sink of H(e) is the ending vertex u′e of e. 2. an ordered pair (u, v) belongs to E(H) if there exists an arc e in E(H(u)) such that e’s associated DAG H(e) is equal to H(v). We call an ordered pair vertices (u, v) ∈ E(H) a super arc of H. Also, for each arc e ∈ E(H(u)), let He be the maximal subgraph of H which can be reached from H(e) via super arcs. From now on, (1) when we mention a DAG H(e) from an arc, it means that the arc e is in the DAG represented by a vertex in V (H), (2) when we mention a DAG H(v) represented by a vertex v, it means that the vertex v is a vertex in V (H), and (3) we use the term H(eC ) to represent the DAG in the root of the hierarchal structure H. Given a fan g and an arc e = (l1 , g1 ) → (l2 , g2 ) ∈ E(H(v)), (1) e is a complete arc on g if g = g1 = g2 , (2) e is a left partial arc on g if g 6= g1 and g = g2 , (3) e is a right partial arc on g if g = g1 and g 6= g2 , and (4) e is minimal if l1 ∪ l2 is a contiguous path. Algorithm 4 emulates the growing process of all hierarchal v-chains J as follows: 1. add the root v into H and let H(r) be the backbone graph HC=β(gT ) of gT . Now, V (H) = {r} and the root J (r) of each hierarchal v-chain J is a directed path P ∈ H(v), and vice versa. 2. iteratively selects a gadget g (to be defined in Definition 44) such that if g is a mirror fan and has a path P = (· · · , (lB , g B ), (l = α(g), g), (lU , gU ), · · · ) ∈ H(v), (1) add a vertex v ′ into V (H), (2) let H(v ′ ) be the backbone graph HL′ =β(g) of g, (3) change P to (· · · , (lB , gB ), (lU , gU ), · · · ) ∈ H(v), (4) add a super arc from v to v ′ in E(H) and (5) let H(e′ ) be H(v ′ ) where e′ = (lB , gB ) → (lU , gU ). The backbone graph of g is embedded into H(e′ ). See Fig 42 as an example. Otherwise, g is a fan. For each maximal path P = (v1 , v2 , · · · , vk ) ∈ H(v), v ∈ V (H) where the gadget gi of each vi = (li , gi ) is equal to g, (1) merge P , (2) add an arc e′ between v1 and vk and (3) set H(e′ ) = (β(g) ∩ α(g R ), g R ) (the backbone graph of g) where (g, g R ) is a v-G-pair in V̂. The backbone graph of g is embedded into H(e′ ). Figs 43 (1) and (2) show an example of P before merging P and Figs 43 (3) and (4) show an example of P after merging P . The next definition defines a removable gadget g which can be selected in Algorithm 4 and add the backbone graph HL=β(g) into H . Intuitively a removable gadget g means that all gadgets g ′ which are connections (g′ , g) in V̂ have been selected and removed from Algorithm 4. Definition 44. In Algorithm 4, we say a gadget g is removable from a DAG H(e) V (H) if there exists a vertex (l, g) ∈ H(e) and we cannot find a vertex (l′ , g′ ) from another DAG H(e′ ) in V (H) such that g and g′ belong to some connection Λ ∈ V̂. Note that the vertex (l′ , g′ ) can also be selected from H(e). Now we give the definition of a conflicting hierarchal v-chain J which cannot form a slant REL R. An example for a conflicting hierarchal v-chain J has been shown in Fig 39. Definition 45. A hierarchal v-chain J is conflicting on a gadget g if there exist pairs (l, g) ∈ S and (l′ , g′ ) ∈ S ′ where S and S ′ are two vertices in V (J ) such that 19 e2 e3 e1 e4 g g1 g3 g2 v1 e1 gB gU gB (2)  e1 1 2 g1  3  4   5 e4 v1 e1 g 6 e4 !  g2 g3 g v5 gU g (1)  e4 e3 e2 5 ,- 6 "# 1 g g $% 2 g1 (3) () &' 3 g2 4 *+ 5  g g g3 (4) Figure 42: (1) and (2) show a M-triple (gB , g, g U ) and suppose that HC has a directed path P = (· · · , e1 , e2 , e3 , e4 , · · · ); (3) and (4) show that after removing g, we add a new arc e′ into HeC and P becomes (· · · , e1 , e′ , e4 , · · · ) ∈ HC . And, (e′1 , e′2 , e′3 , e′4 , e′5 , e′6 ) is a directed path in H(e′ ) where (e′1 , e′2 , e′3 , e′4 , e′5 , e′6 ) is a v-backbone of L = β(g). 1. if g 6= g′ , g and g′ overlap at least one face. 2. Otherwise (g = g′ ), l and l′ overlap at least two vertices. Note that S might be equal to S ′ . Moreover, we say the vertex (l′ , g′ ) is conflicting to (l, g) on g if (l, g) and (l′ , g ′ ) satisfy one of the above two conditions. On the other hand, we say (l′ , g′ ) is compatible to (l, g) on g if (l′ , g′ ) is not conflicting to (l, g) on g. And, for a hierarchal v-chain J , we say J is compatible on g if J is not conflicting on g. Next we can start to define that H is compatible on a gadget g as follows: Definition 46. Given a hierarchal structure H = (V (H), E(H)) and a gadget g ∈ Ṽ, we say H is compatible on g if 1. there exists a directed path P ∈ H(eC ) such that for each vertex (l, g) ∈ P , each vertex (l′ , g ′ ) ∈ P other than (l, g) is compatible to (l, g) on g. And, 2. for each arc e ∈ P , the hierarchal substructure He of H is also compatible on g. We say (1) a directed path P ∈ H(eC ) is compatible on g if P satisfies the conditions 1 and 2. And, (2) a directed path P ∈ H(eC ) is conflicting on g if P violates the condition 1 or the condition 2. Moreover, an arc e ∈ P is compatible on g if He is compatible on g. On the other hand, e is conflicting on g if He is not compatible on g. From the above definition of a compatible hierarchal structure H, we immediately have a recursive procedure to check whether there exists a compatible path P on g in H(eC ) as follows: for each directed path P ∈ H(eC ), recursively check each arc e ∈ P whether the DAG H(e) ∈ He (H(e) is the root’s associated DAG in He ) has a compatible directed path on g or not. Then P is conflicting on g if and only if P becomes disconnected after removing all conflicting arcs e on g from H(eC ). It is stated in Property 54. 20 e2 e1 e1 e41 5 1 e11 e12 e21 e3 e22 e13 g e14 e4 e1 1 2 e1 e24 gR (1) e1 e3 e23 JK gI e2 2 2 e11 e51 e1 e2 1 1 1 e 1 e 2 /1 52 1 2 1 e3 4 4 1 1 0 1 2 3 1 e34 62 83 e3 e4 94 :5 e14 e4 e4 2 3 e13 e23 L3 44 34 1 2 1 (2) e4 e24 e 31 e1 e4 . e34 =1 e1 e11 1 e21 e3 1 ? 1 @ g (3) ;< >2 PQ 1 H4 MN 2 gC (4) e4 e24 e4 3 G5 gE Figure 43: (1) and (2) are an example of H, complete arcs and partial arcs; (3) and (4) are an example to explain how H changes its structure after removing a fan g; (1) and (2): (· · · , e1 , e2 , e3 , e4 , · · · ) is a directed path P ∈ H(eC ) where e1 is a left partial arc on (g, gR ), {e2 , e3 } are complete arcs on (g, gR ) and e4 is a right partial arc on (g, gR ). Also, (e11 , e12 , e13 , e14 , e15 ) is a directed path in H(e1 ), (e21 , e22 ) is a directed path in H(e2 ), (e31 , e32 ) is a directed path in H(e3 ) and (e41 , e42 , e43 ) is a directed path in H(e4 ); (3) and (4): after removing the fan g, change the vertices v2 = (l2 , g) and v4 = (l4 , g) to v2 = (β(g) ∩ α(gR ), gR ) and v4 = (β(g) ∩ α(gR ), gR ), respectively where β(g) ∩ α(gR ) is the intersection of the back boundary β(g) and the front boundary α(gR ), and (g, g R ) is a G-pair in Ṽ. Also, the arcs {e2 , e3 } are replaced by the arc e′ and H(e′ ) is the path (v2 , (β(g) ∩ α(gR ), g R ), v4 ). Briefly speaking, Algorithm 4 iteratively removes the root J (r) of a conflicting hierarchal-v-chain J from H(eC ). Also, we utilize Algorithms 5 and 6 to adjust the structure of H. Moreover, after recursively adjusting H (it means that via adjusting H(eC ) ∈ H, we also adjust the structure of He , e ∈ H(eC )), we have the following fact: for each H(e) ∈ H, if there does not exist a directed path between v1 and v2 in H(e) before removing g from H(e), then v1 remains disconnected to v2 in H(e) after removing g from H(e). At the end of Algorithm 4, we can conclude that each connected path P ∈ H(eC ) has a corresponding compatible hierarchal v-chain J . The intuitive meaning of a path P ∈ H(eC ) keeps its connectivity after removing g is that P can add g into its corresponding hierarchal v-chain J . How to efficiently check whether a directed path P ∈ H(eC ) is compatible on g or not? We can recursively check whether there exists a compatible directed path P ′ ∈ H(e) on g for each complete and partial arcs e ∈ H(eC ). In Observations 48 and 51, we describe the recursive formulas to check complete arcs and partial arcs whether they are compatible on g or not. After we check all complete arcs and partial arcs, we keep all compatible arcs on g in H(eC ) and check whether there exists a directed path from source to sink in H(eC ). (The root J (r) of a compatible hierarchal v-chain J .) The recursive procedures to check directed paths, complete arcs and partial arcs on g in H(eC ) are described in Lemmas 47, 50 and 53, respectively. The main task for EXPAND operation in Algorithm 5 is to add the backbone graph of a mirror fan into H. See Fig 42 as an example for EXPAND operation. In the following lemma, we describe the recursive structure of a compatible directed path P ∈ H(eC ). A simple way to explain Lemma 47 is that to recursively check a compatible directed path P in H(eC ) is equal to, for each arc e ∈ P , recursively check whether there exists a compatible directed path P ′ in H(e). In general, each directed path P ∈ H(eC ) can be decomposed into five parts: (1) the subpath from source which doesn’t have any partial arc and complete arc on g, (2) the subpath which only has a left partial arc on g, (3) the subpath which only has complete arcs on g, (4) the subpath which only has a right partial arc on g and (5) the subpath to sink which doesn’t have any partial arc and complete arc on g. Because each arc e in a compatible directed path P must be compatible on g , it implies that 21 Algorithm 4: Find Conflicting Gadgets via Backtracking Algorithm 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 Input: Sets Ṽ and V̂ Add the backbone graph HC=β(gT ) of gT into H. V (H) = {H(eC ) = HC }; while the initial fan g0 is not removable from H(eC ) do Find a removable gadget g from H(eC ); if g is a mirror fan then for each M-triple (g B , g, gU ) ∈ V̂ do EXPAND g in H by Algorithm 5; end else if g is a fan then for each G-pair (g, g R ) do Recursively check whether each complete arc and partial arc in H(eC ) are compatible on g or not by Algorithm 6; end Recursively delete all complete arcs on g from H(eC ); end Remove g from Ṽ and all connections (g, gR ) and (gB , g, gU ) from V̂; end if there exists a path P = (e1 , e2 , · · · , ek ) ∈ H(eC ) where each ei , 1 ≤ i ≤ k, is a complete arc on g0 then G have an area-universal rectangular layout; else G does not have any area-universal rectangular layout; end H(e) must have at least one directed path from source to sink which is compatible on g. We describe their recursive structures of complete arcs and partial arcs on g in Observations 48, 49, 51 and 52. From the above discussion, we immediately have Lemma 47. Lemma 47. Given a fan g, suppose there is a directed path P = (e1 , e2 , ez , ep , ec1 , ec2 , · · · , eckc , eq , e′1 , e′2 , · · · , e′z ′ ) ∈ H(eC ) where the arcs ep and eq are the left and right partial arcs on g, respectively, and each arc eci , 1 ≤ i ≤ kc , is a complete arc on g. Then, P is the root J (r) of a compatible hierarchal v-chain J on g if and and if • for each complete arc eci ∈ P, 1 ≤ i ≤ kc , there is a compatible directed path Pic ∈ H(eci ) on g (see Observations 48 and 49 for more details of a complete arc), • for the left partial arc ep ∈ P , there is a compatible directed path P p ∈ H(ep ) on g (see Observations 51 and 52 for more details of a left partial arc) and • for the right partial arc eq ∈ P , there is a compatible directed path P q ∈ H(eq ) on g (see Observations 51 and 52 for similar details of a right partial arc). Given a complete arc e = (l1 , g) → (l2 , g) ∈ H(eC ) on g, each arc e′ ∈ P is also a complete arc on g. And, we know that if we want to guarantee that a complete arc e is compatible, we must recursively check whether H(e) can have a directed path which only consists of compatible complete arcs on g. Obviously, to recursively check a compatible complete arc on g is a recursive procedure implemented by dynamic programming technique. Also, the base case for the recursive procedure is that a complete arc on g whose two end vertices (l1 , g) and (l2 , g) have that l1 ∪ l2 is contiguous on the front boundary of g. It means that (l1 , g) → (l2 , g) is compatible on g. See Fig 43 as an example of a complete arc. 22 Algorithm 5: EXPAND a mirror fan in H 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Input: The hierarchal structure H with the root H(eC ) and a M-triple (gB , g, gU ) ∈ V̂ Add the backbone graph HL=β(g) into H as a vertex v ∈ V (H); e1 e2 for each DAG H(e) ∈ V (H) such that there is a subpath (l1 , gB ) −→ (α(g), g) −→ (l2 , g U ) in H(e) where l1 and l2 are portions of α(gB ) and α(gU ), respectively do e e′ e 1 2 Replace (l1 , gB ) −→ (α(g), g) −→ (l2 , gU ) by (l1 , gB ) − → (l2 , gU ) in H(e); Set H(e′ ) = H(v) and add a super arc from H(e) to H(v) in H; Add an arc from the starting vertex of e to the source vertex of H(v) and an arc from the sink vertex of H(v) to the ending vertex of e in H(e); Remove the vertex (α(g), g) from H(e); end From the above discussion, we can describe recursive structures of a compatible complete arc on g in Observations 48 and 49: Observation 48. Given a fan g, a complete arc e is compatible on g if and only if there exists a directed path P = (ec1 , ec2 , · · · , eckc ) ∈ H(e) where • the source vertex of P is the starting vertex of e, • the sink vertex of P is the ending vertex of e and • each eci , 1 ≤ i ≤ kc , is a compatible complete arc on g. Observation 49. Given a fan g, a minimal complete arc e = (l1 , g) → (l2 , g) is compatible on g if and only if l1 ∪ l2 is contiguous on the front boundary α(g) of g (the last vertex of l1 overlaps the first vertex of l2 ). Based on Observations 48 and 49, we can check a complete arc e ∈ H(eC ) on g whether it is compatible on g or not via Lemma 50. Lemma 50. Given a fan g, we can recursively check whether each complete arc e ∈ H(eC ) on g satisfies structure described in Observations 48 and 49 as follows: 1. recursively check whether each arc e′ ∈ H(e) satisfies the structures in Observations 48 and 49, 2. keep all arcs passing the above tests in H(e), and 3. check whether H(e) has a directed path from source to sink. If yes, keep e in H(eC ). Otherwise, delete e from H(eC ). For a left partial arc e = (l1 , g1 ) → (l2 , g2 ) ∈ H(eC ) on g, because g2 is equal to g, each directed path P in H(e) can be partitioned into (1) the subpath that consists of neither complete arcs nor partial arcs on g, (2) the left partial arc on g and (3) the subpath that only consists of complete arcs on g. See Fig 43 as an example of a left partial arc. Similarly, to check a compatible left partial arc on g is a recursive procedure which can be implemented by dynamic programming technique. Also, the base case for the recursive procedure is a left partial arc (l1 , g1 ) → (l2 , g2 = g) on g which has (1) (g2 = g, g1 ) is a v-connection in V̂ and (2) l1 ∪ l2 is contiguous on the front boundary α(g2 , g1 ) of the connection (g2 , g1 ). It means that (l1 , g1 ) → (l2 , g2 = g) is compatible on g. See Fig 43 for examples of a left partial arc and a minimal left partial arc. From the above discussion, we can describe recursive structures of a compatible left partial arc on g in Observations 51 and 52: 23 Observation 51. Given a fan g and a left partial arc e on g, a left partial arc e is compatible on g if and only if there exists a directed path P = (e1 , e2 , · · · , ez , ep , ec1 , ec2 , · · · , eckc ) in H(e) where • the source vertex of P is the starting vertex of e, • the sink vertex of P is the ending vertex of e, • for each 1 ≤ i ≤ z, ei = (li , gi ) → (li+1 , gi+1 ) is an arc where g 6= gi and g 6= gi+1 , • the arc ep is a compatible left partial arc on g, and • each arc eci , 1 ≤ i ≤ kc , is a compatible complete arc on g. Observation 52. Given a fan g, a minimal left partial arc e = (l1 , g1 ) → (l2 , g2 = g) on g is a compatible left partial arc on g if and only if (1) (g2 = g, g1 ) is a connection in V̂ and (2) l1 ∪ l2 is contiguous on the front boundary α(g2 , g1 ) of the connection (g2 , g1 ) (the last vertex of l1 overlaps the first vertex of l2 ). Based on Observations 51 and 52, we can recursively check whether a partial arc e ∈ H(eC ) is compatible on g or not via Lemma 53. Lemma 53. Given a fan g, we can recursively check whether a partial arc e ∈ H(eC ) on g satisfies the structures in Observations 51 and 52 as follows: 1. recursively check whether each partial arc e′ ∈ H(e) on g satisfies the structures in Observations 51 and 52, 2. recursively check whether each complete arc e′ ∈ H(e) on g satisfies the structures in Observations 48 and 49, 3. keep all arcs passing the above tests in H(e), and 4. check whether H(e) has a directed path from source to sink. If yes, keep e in H(eC ). Otherwise, delete e from H(eC ). There are three main tasks of MERGE operation in Algorithm 6. The first one is to recursively check each complete arc on a removable gadget g ∈ Ṽ. The second one is to recursively check each partial arc on g. The final one is to remove g and maintain the connectivity for each compatible directed path in H(eC ). What we do in the final for loop is to reconnect a new arc between a left partial arc eL and a right partial arc eR if and only if there exists a compatible directed path from eL to eR . See Fig 43 as an examples for Algorithm 6. Note that the connectivity between vL and vR is based on the arcs which are compatible on g in Algorithm 6. There are two important properties for the correctness of Algorithm 4. The first one states that we can eliminate each conflicting directed path P (hierarchal v-chain J ) on g via removing g from H(eC ). The second one states that the number of directed paths (hierarchal v-chains) decreases during Algorithm 4 executes. Property 54. For each DAG H(e) ∈ V (H), a directed path P ∈ H(e) is conflicting on g if and only if P becomes disconnected after removing g from H(e). Property 55. For each DAG H(e) ∈ V (H), if any two vertices u, v ∈ H(e) are disconnected, then u and v remain disconnected after removing g from H(e). 24 Algorithm 6: MERGE H via Dynamic Programming 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Input: The hierarchal structure H with the root H(eC ) and a G-pair (g, gR ) ∈ V̂ for each complete arc ec ∈ H(eC ) on g do Recursively check ec whether ec is compatible on g or not (this recursive check follows Lemma 50); end for each partial arc ep ∈ H(eC ) on g do Recursively check ep whether ep is compatible on g or not (this recursive check follows Lemma 53); end for each pair of left partial arc eL = va → vL = (vL , g) and right partial arc eR = vR = (vR , g) → vb in H(eC ) such that vL and vR remain connected in H(eC ) do Change vL = (vL , g) and vR = (lR , g) to vL = (β(g) ∩ α(g R ), gR ) and vR = (β(g) ∩ α(g R ), gR ), respectively; Add an arc e′ = (vL → vR ) into the graph H(eC ) and let H(e′ ) be the directed path (vL → (β(g) ∩ α(g R ), gR ) → vR ); end From the above two properties, we can see that if we can recursively guaranteed that for each compatible arc e ∈ H(eC ) on g, H(e) has Properties 54 and 55, then each compatible directed path P ∈ H(eC ) on g is also a compatible hierarchal v-chain on g. And, in the final "for" loop of Algorithm 6, it connects a new arc between vL and vR if and only if there is a compatible directed path on g from vL and vR . Hence it guarantees that no pair of vertices (vL , vR ) turns into connected if vL and vR are disconnected before removing conflicting arcs on g. Theorem 56. Algorithm 4 can successfully check whether there exists a directed path P ∈ H(eC ) such that P has a corresponding slant REL. Proof: The correctness of Algorithm 4 is based on Properties 54 and 55. Clearly, from Property 54, each conflicting directed path P ∈ H(eC ) (the root J (r) of each hierarchal v-chain J ) on g becomes disconnected after removing a removable gadget g and from Property 55, it remains disconnected in the following steps. Hence in the final step of Algorithm 4, each connected directed path P ∈ H(eC ) has been proven that P corresponds to the root J (r) of a compatible hierarchal v-chain on g for every gadget g ∈ Ṽ. Also, if a directed path P ∈ H(eC ), corresponds to the root J (r) of a hierarchal v-chain J , keeps its connectivity after recursively removing a gadget g from H(eC ), then this hierarchal v-chain J can add the gadget g into its corresponding slant REL. Hence we can have that each connected directed path in H(eC ) has a corresponding slant REL R. Theorem 56 has proven that we can backtrack all directed paths P ∈ H(eC ) to know whether P represents the chain of a partial slant REL R. Theorem 57. The time complexity of Algorithm 4 is polynomial bound. Proof: Let K be the number of iterations in Algorithm 4 and Ni , 1 ≤ i ≤ K, be the size of H in the i-th iteration. The time analysis of backtracking is based on three parts: (1) K is polynomial bound, (2) each Ni , 1 ≤ i ≤ K, is polynomial bound and (3) time complexity T (Ni ) in each i-th iteration is polynomial bound. Obviously, the number K of total iterations is bounded by the number of connections V̂. By Lemma 13, the number of gadgets in G is at most N = O(n2 ) and the number of connections in G is at most O(N 3 ) = O(n6 ). Hence K polynomially grows with respect to the number of G’s vertices n. 25 For each i-th iteration, we either execute EXPAND or MERGE operations to adjust H’ structure. When Algorithm 4 executes EXPAND operation on a removable gadget g with the back boundary L = β(g), we add g’s backbone graph HL into H where HL ’s size (the number of vertices in HL ) is polynomial bound. Since the number K of total iterations is polynomial bound, the total vertices added into H in all EXPAND operations are bounded by the summation of all backbone graphs’s size. Hence, the summation of all backbone graphs’s size is polynomial bound. When Algorithm 4 executes MERGE operation on a removable gadget g with a G-pair (g, gR ), we replace each maximal compatible directed path P on g in H by an arc e′ = (gu , lu ) → (gv , lv ) between the two end vertices of P where each arc in P is a complete arc on g, and add H(e′ ) into H where |H(e′ )| consists of the newly-added vertex (gR , l). Note that the added vertex (gR , l) only connects to the two end vertices (gu , lu ) and (gv , lv ) of e, and cannot be connected to other vertices in H in following iterations. Also, (g R , l) is removed from H when removing a removable gadget gR from H. Hence the total number of newly-added arcs e′ are summation of all backbone graphs’s size and it is polynomial bound. Also, the size of all added H(e′ ) is polynomial bound. Because the total vertices added into H are polynomial bound during each iteration and the number K of iterations is polynomial bound, the maximum of H’s size is polynomial bound of each iteration in Algorithm 4. Hence each number Ni , i ≥ 1, of vertices of H in each i-th iteration is polynomial bound. Now we analyze the time complexity of each iteration. When Algorithm 4 executes EXPAND operation on a removable gadget g with the back boundary L = β(g), we take polynomial time to add the g’s backbone graph HL into H because HL ’s size is polynomial bound. When Algorithm 4 executes MERGE operation on a removable gadget g, the tasks of MERGE operation consist of (1) recursively checking each complete arc on g in H by Lemma 50, (2) recursively checking each partial arc on g in H by Lemma 53 and (3) check that for each DAG H(e), whether there exists a directed path from the source to the sink in H(e) after removing all conflicting complete and partial arcs from H(e). The total work of a MERGE operation can be simply described as follows: check each DAG H(e) ∈ V (H) whether H(e) has at least one connected directed path from the source to the sink in H(e). And, it can be done by executing a breadth-first search in H(e) since H(e) is a DAG. Hence the complexity of the total work of a MERGE operation is polynomial bound as poly(max1≤i≤K Ni ). Note that the order to check each DAG H(e) ∈ V (H) is a bottom-up traversal in H as follows: a DAG H(e) ∈ V (H) is ready to check if and only if every DAG H(e′ ), e′ ∈ E(H(e)), has been checked. Because each iteration takes polynomial time as T (Ni ) ≤ poly(max1≤i≤K Ni ), the total time complexity of all K iterations in Algorithm 4 is also polynomial bound. Theorem 57 has proven that the time complexity of Algorithm 4 is polynomial bound. 26 References [1] M. Bruls, K. Huizing, and J. J. van Wijk, Squarified treemaps, in Proceedings of the Eurographics and IEEE TCVG Symposium on Visualization, Springer, 2000, pp. 33-42. [2] C. F. Earl and L. J. March, Architectural applications of graph theory, Applications of Graph Theory, R. Wilson and L. Beineke, eds., Academic Press, London, 1979, pp. 327-355. 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Fusy, Transversal structures on triangulations: A combinatorial study and straight-line drawings, Discrete Math. 309, 2009, pp. 1870-1894. [9] X. He, On finding the rectangular duals of planar triangular graphs, SIAM J. Comput. 22, no. 6, 1993, 1218-1226. [10] G. Kant and X. He, Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems, Theoret. Comput. Sci. 172, (1997), pp. 175-193. [11] M. van Kreveld and B. Speckmann, On rectangular cartograms, Computational Geometry 37, 2007, 175-187. [12] K. Koźmiński and E. Kinnen Rectangular duals of planar graphs, Networks 5, 1985, pp. 145-157. [13] E. Mumford, Drawing graphs for cartographic applications, Ph.D. thesis, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2008. [14] E. Raisz, The rectangular statistical cartogram, Geographical Review 24 (2), 1934, 292-296. [15] I. Rinsma, Nonexistence of a certain rectangular floorplan with specified areas and adjacency, Environ. Plann. 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A NORMAL GENERATING SET FOR THE TORELLI GROUP OF A COMPACT NON-ORIENTABLE SURFACE Abstract. For a compact surface S, let I(S) denote the Torelli group of S. For a compact orientable surface Σ, I(Σ) is generated by BSCC maps and BP maps (see [10] and [11]). For a non-orientable closed surface N , I(N ) is generated by BSCC maps and BP maps (see [5]). In this paper, we give an explicit normal generating set for I(Ngb ), where Ngb is a genus-g compact non-orientable surface with b boundary components for g ≥ 4 and b ≥ 1. 1. Introduction For g ≥ 1 and b ≥ 0, let Ngb denote a genus-g compact connected non-orientable surface with b boundary components, and let Ng = Ng0 . In this paper, we regard Ngb as a surface obtained by attaching g Möbius bands to a sphere with g + b boundary components, as shown in Figure 1. We say each of these Möbius bands attached to this sphere a cross cap. The mapping class group M(Ngb ) of Ngb is the group consisting of isotopy classes of all diffeomorphisms over Ngb which fix each point of the boundary. The Torelli group I(Ngb ) of Ngb is the subgroup of M(Ngb ) consisting of elements acting trivially on the integral first homology group H1 (Ngb ; Z) of Ngb . The Torelli group of a compact orientable surface is generated by BSCC maps and BP maps (see [10] and [11]). In particular, Johnson [6] showed that the Torelli group of an orientable closed surface is finitely generated by BP maps. Hirose and the author [5] showed that I(Ng ) is generated by BSCC maps and BP maps for g ≥ 4. In this paper, we give an explicit normal generating set for I(Ngb ) consisting of BSCC maps and BP maps, for g ≥ 4 and b ≥ 1. attach { { arXiv:1601.06257v2 [math.GT] 14 Feb 2017 RYOMA KOBAYASHI g b Figure 1. A genus-g compact connected non-orientable surface with b boundary components. Let N be a compact connected non-orientable surface. For a simple closed curve c on N, we call c an A-circle (resp. an M-circle) if its regular neighborhood is an annulus (resp. a Möbius band), as shown in Figure 2. For an A-circle c, we can define the mapping class tc , called the Dehn twist about c, and the direction of the twist is indicated by a 1 2 R. KOBAYASHI (a) A-circles. (b) M-circles. Figure 2. c tc Figure 3. The Dehn twist tc about c. small arrow written beside c as shown in Figure 3. We can notice that an A-circle (resp. an M-circle) passes through cross caps even times (resp. odd times). Let α, β, β ′ , γ, δi , ρi , σij and σ̄ij be simple closed curves on Ngb as shown in Figure 4. The main result of this paper is as follows. Theorem 1.1. Let g ≥ 5 and b ≥ 0. In M(Ngb ), I(Ngb ) is normally generated by tα , b b tβ t−1 β ′ , tδi , tρi , tσij and tσ̄ij for 1 ≤ i, j ≤ b − 1 with i < j. In M(N4 ), I(N4 ) is normally generated by tα , tβ t−1 β ′ , tδi , tρi , tσij , tσ̄ij and tγ for 1 ≤ i, j ≤ b − 1 with i < j. β α σij ' δi β i j (a) Loops α, β, β ′ , δi and σ̄ij . σij ρi γ i j (b) Loops γ, ρi and σij . Figure 4. In this paper, for f, g ∈ M(Ngb ), the composition gf means that we first apply f and then g. 2. Basics on mapping class groups for non-orientable surfaces 2.1. On mapping class groups for non-orientable surfaces. Mapping class groups for orientable surfaces are generated by only Dehn twists. Lickorish showed that M(Ng ) is generated by Dehn twists and Y -homeomorphisms and that the subgroup of M(Ng ) generated by all Dehn twists is an index 2 subgroup of M(Ng ) (see [7, 8]). Hence M(Ng ) is not generated by only Dehn twists. On the other hand, A NORMAL GENERATING SET FOR I(Ngb ) 3 since a Y -homeomorphism acts trivially on H1 (Ng ; Z/2Z), M(Ng ) is not generated by only Y -homeomorphisms. Chillingworth [2] found a finite generating set for M(Ng ). M(N1 ) and M(N11 ) are trivial (see [3]). Finite presentations for M(N2 ), M(N21 ), M(N3 ) and M(N4 ) are obtained by [7], [12], [1] and [14], respectively. Paris-Szepietowski [9] obtained a finite presentation of M(Ngb ) for b = 0, 1 and g + b > 3. Let N be a non-orientable surface, and let a and m be an oriented A-circle and an M-circle on N respectively such that a and m mutually intersect transversely at only one point. We now define a Y -homeomorphism Ym,a . Let K be a regular neighborhood of a∪b in N, and let M be a regular neighborhood of m in the interior of K. We can see that K is homeomorphic to the Klein bottle with one boundary component. Ym,a is defined as the isotopy class of a diffeomorphism over N which is described by pushing M once along a and which fixes each point of the boundary and the exterior of K (see Figure 5). K a m Ym,a M Figure 5. The Y -homoemorphism Ym,a . 2.2. On Torelli groups for non-orientable surfaces. Let c be an A-circle on a non-orientable surface N such that N \ c is not connected. We call tc a bounding simple closed curve map, for short a BSCC map. For example, in Theorem 1.1, tα , tγ , tδi , tρi and tσij are BSCC maps. Let c1 and c2 be A-circles on N such that N \ci is connected, N \(c1 ∪c2 ) is not connected and one of its connected components is an orientable surface with two boundary components. We call tc1 t−1 c2 a bounding pair −1 map, for short a BP map. For example, in Theorem 1.1, tβ tβ ′ is a BP map. Hirose and the author [5] obtained the following theorem. Theorem 2.1 ([5]). For g ≥ 5, I(Ng ) is normally generated by tα and tβ t−1 β ′ in M(Ng ). −1 I(N4 ) is normally generated by tα , tβ tβ ′ and tγ in M(N4 ). Theorem 1.1 is a natural extension of Theorem 2.1. We can check that all BSCC maps and BP maps are in I(Ngb ). In addition, by Theorem 1.1, we have that I(Ngb ) is generated by BSCC maps and BP maps. We do not know whether or not I(Ngb ) can be finitely generated. 2.3. Capping, Pushing and Forgetful homomorphisms. Let N be a compact non-orientable surface. Take a point ∗ in the interior of N. Let M(N, ∗) denote the group consisting of isotopy classes of all diffeomorphisms over N which fix ∗ and each point of the boundary, and let I(N, ∗) denote the subgroup of M(N, ∗) consisting of elements acting trivially on H1 (N; Z) Take a point ∗ in the interior of Ngb−1 . We regard Ngb as a subsurface of Ngb−1 not containing ∗. The natural embedding Ngb ֒→ Ngb−1 induces the homomorphism Cgb : M(Ngb ) → M(Ngb−1 , ∗), called the capping homomorphism. We have the following lemma. Lemma 2.2 (cf. [4, 13]). ker Cgb is generated by tδb . 4 R. KOBAYASHI Since tδb is in I(Ngb ) we obtain the following. Corollary 2.3. ker Cgb |I(Ngb ) is generated by tδb . We remark that Cgb and Cgb |I(Ngb ) are not surjective. The pushing homomorphism Pgb−1 : π1 (Ngb−1 , ∗) → M(Ngb−1 , ∗) is defined as follows. For x ∈ π1 (Ngb−1 , ∗) take a representative oriented loop x̃ based at ∗. Pgb−1 (x) is described by pushing ∗ once along x̃ and fixes each point of exterior of neighborhood of x̃. Note that Pgb−1 is an anti-homomorphism, that is, for x, y ∈ π1 (Ngb−1 , ∗) we have Pgb−1 (xy) = Pgb−1 (y)Pgb−1 (x). The forgetful homomorphism Fgb−1 : M(Ngb−1 , ∗) → M(Ngb−1 ) is defined naturally. Note that Fgb−1 is surjective. We have the natural exact sequence Fgb−1 Pgb−1 π1 (Ngb−1 , ∗) −→ M(Ngb−1 , ∗) −→ M(Ngb−1 ) −→ 1. Since ImPgb−1 is in I(Ngb−1 , ∗) and Fgb−1 (I(Ngb−1 , ∗)) is equal to I(Ngb−1 ), we have the exact sequence π1 (Ngb−1 , ∗) −→ I(Ngb−1 , ∗) −→ I(Ngb−1 ) −→ 1. 3. A normal generating set fot Cgb (I(Ngb )) in Cgb (M(Ngb )) By the capping homomorphism Cgb : M(Ngb ) → M(Ngb−1 , ∗) we have that a normal generating set for I(Ngb ) in M(Ngb ) consists of tδb and lifts by Cgb of normal generators of Cgb (I(Ngb )) in Cgb (M(Ngb )). Thus in this section we consider a normal generating set for Cgb (I(Ngb )) in Cgb (M(Ngb )). Let αi and βj be oriented loops on Ngb−1 based at ∗ as shown in Figure 6, and let xi and yj be the elements of π1 (Ngb−1 , ∗) corresponding to αi and βj respectively. Note that π1 (Ngb−1 , ∗) is generated by xi and yj for 1 ≤ i ≤ g and 1 ≤ j ≤ b − 1. * αg β1 βb-1 { { α1 g b-1 Figure 6. The oriented loops αi and βj on Ngb−1 . Let p : π1 (Ngb−1 , ∗) → π1 (Ng , ∗) denote the natural surjection defined as p(xi ) = xi and p(yj ) = 1, for b ≥ 1. For x ∈ π1 (Ngb−1 , ∗) we can denote p(x) = xεi11 xεi22 · · · xεitt , where t is the word length of p(x) and εk = ±1. We define Oi (x) = ♯{i2k−1 | i2k−1 = i}, Ei (x) = ♯{i2k | i2k = i}, Γb−1 = {x ∈ π1 (Ngb−1 , ∗) | Oi (x) = Ei (x), 1 ≤ i ≤ g}. g −2 −1 3 −1 For example, for x = x1 y2 x2 x−1 ∈ π1 (Ngb−1 , ∗), since p(x) = 3 y5 y1 x1 x2 y4 x3 −1 −1 −1 x1 x2 x3 x1 x2 x3 , we have that Oi (x) and Ei (x) are equal to 1 (resp. 0) for i = 1, 2, 3 (resp. i ≥ 4), and hence x is in Γgb−1 . A NORMAL GENERATING SET FOR I(Ngb ) 5 In this section we prove the following three propositions. Proposition 3.1. Im Fgb−1 |Cgb (I(Ngb )) is equal to I(Ngb−1 ). Proposition 3.2. ker Fgb−1 |Cgb (I(Ngb )) is equal to Pgb−1 (Γb−1 g ). b−1 2 b−1 Proposition 3.3. Pgb−1 (Γb−1 g ) is the normal closure of Pg (xg ), Pg (yj ) and b b Pgb−1 (xg yj x−1 g ) for 1 ≤ j ≤ b − 1 in Cg (M(Ng )). By Proposition 3.1 and Proposition 3.2, we have the exact sequence Γb−1 −→ Cgb (I(Ngb )) −→ I(Ngb−1 ) −→ 1. g Hence Cgb (I(Ngb )) is the normal closure of Pgb−1 (Γgb−1 ) and lifts by Fgb−1 |Cgb (I(Ngb )) of normal generators of I(Ngb−1 ). In addition, by Proposition 3.3 we obtain the following. Corollary 3.4. In Cgb (M(Ngb )), Cgb (I(Ngb )) is normally generated by Pgb−1 (x2g ), Pgb−1 (yj ), b−1 Pgb−1 (xg yj x−1 |Cgb (I(Ngb )) of normal generators of I(Ngb−1 ), for 1 ≤ j ≤ g ) and lifts by Fg b − 1. 3.1. Proof of Proposition 3.1. To prove Proposition 3.1, it suffices to show that for any ϕ ∈ I(Ngb−1 ) there is ϕ e ∈ I(Ngb ) such that (Fgb−1 ◦ Cgb )(ϕ) e = ϕ. Let γi and δj be oriented loops on Ngb as shown in Figure 7, and let ci and di be the elements of H1 (Ngb ; Z) corresponding to γi and δj respectively. As a Z-module, H1 (Ngb , Z) has a presentation H1 (Ngb , Z) = hc1 , . . . , cg , d1, . . . , db | 2(c1 + · · · + cg ) + (d1 + · · · + db ) = 0i, and is isomorphic to Zg+b−1 . γg δ1 δb { { γ1 g b Figure 7. The oriented loops γi and δj on Ngb . For f ∈ M(Ngb ) we denote by f∗ the automorphism over H1 (Ngb ; Z) induced by f . Since f fixes each point of the boundary of Ngb , we have that f∗ (dj ) = dj for 1 ≤ j ≤ b. For any ϕ ∈ I(Ngb−1 ) there exists ψ ∈ M(Ngb ) such that (Fgb−1 ◦ Cgb )(ψ) = ϕ. For 1 ≤ i ≤ g there is an integer ni such that ψ∗ (ci ) = ci + ni db . Let γij and e γij be simple b closed curves on Ng as shown in Figure 8, and let τij = tγij tγeij , for 1 ≤ i < j ≤ g. τij is a mapping class which is described by pushing the b-th boundary component once along between γij and e γij . We can check   ci − db (t = i), cj + db (t = j), (τij )∗ (ct ) =  ct (t 6= i, j). 6 R. KOBAYASHI γij ~ γ ij i j Figure 8. The loops γij and e γij . n g−1 n2 n1 Let τ = τg−1g · · · τ2g τ1g and ϕ e = τ ψ. Since (Fgb−1 ◦ Cgb )(τ ) = 1 we have (Fgb−1 ◦ Cgb )(ϕ) e = ϕ. For 1 ≤ i ≤ g − 1 we calculate ϕ e∗ (ci ) = = = = = = τ∗ (ψ∗ (ci )) τ∗ (ci + ni db ) τ∗ (ci ) + ni τ∗ (db ) (τig )n∗ i (ci ) + ni db (ci − ni db ) + ni db ci . In addition, we calculate ϕ e∗ (cg ) = τ∗ (ψ∗ (cg )) = τ∗ (cg + ng db ) = τ∗ (cg ) + ng τ∗ (db ) = (cg + g−1 X nk db ) + ng db k=1 = cg + g X nk db . k=1 Let c = 2(c1 + · · · + cg ) + (d1 + · · · + db )(= 0). We see ϕ e∗ (c) = 2(ϕ e∗ (c1 ) + · · · + ϕ e∗ (cg )) + (ϕ e∗ (d1 ) + · · · + ϕ e∗ (db )) g X nk db ) + (d1 + · · · + db ) = 2(c1 + · · · + cg−1 + cg + k=1 g = c+2 X nk db k=1 = 2 g X nk db . k=1 Since ϕ e∗ (c) = 0 we have g X k=1 nk = 0, and hence ϕ e∗ (cg ) = cg . Hence we have that ϕ e∗ is the identity. Therefore we conclude that ϕ e is in I(Ngb ) with (Fgb−1 ◦ Cgb )(ϕ) e = ϕ. Thus we complete the proof of Proposition 3.1. 3.2. Proof of Proposition 3.2. A NORMAL GENERATING SET FOR I(Ngb ) 7 Note that Pgb−1 (x) can be lifted by Cgb if and only if the word length of p(x) ∈ π1 (Ng , ∗) is even. Let π1+ (Ngb−1 , ∗) denote the subgroup of π1 (Ngb−1 , ∗) consisting of x such that the word length of p(x) is even. For x ∈ π1+ (Ngb−1 , ∗), let x∗ denote the automorphism over H1 (Ngb , Z) induced by the natural lift by Cgb of Pgb−1 (x). For example, the natural lifts of Pgb−1 (x2g ) and Pgb−1 (yj ) are tρb and tσjb t−1 δj , respectively. −1 2 We can see that (xi )∗ = 1 and (yj )∗ = 1. Hence we have that (xi xj )∗ , (xi x−1 j )∗ , (xi xj )∗ , −1 −1 (x−1 i xj )∗ and (xj xi )∗ are mutually equal. Let xij = (xi xj )∗ for 1 ≤ i, j ≤ g. Note that  (γij )∗ (i < j), xij = −1 (γji )∗ (j < i). Lemma 3.5. For 1 ≤ i, j, k, l ≤ g, we have xij xkl = xkl xij . Proof. If (i, j, k, l) = (i, i, i, i), (i, i, k, i), (i, i, i, l), (i, i, k, k), (i, j, i, j), (i, j, j, i) and (i, i, k, l), it is clear that xij xkl = xkl xij . Hence we check the cases where (i, j, k, l) = (i, j, i, l), (i, j, k, j), (i, j, k, i) and (i, j, k, l). For any 1 ≤ i, j ≤ g since xij acts trivially on d1 , d2 , . . . , db , we check the actions on c1 , c2 , . . . , cg . For any mutually different indices 1 ≤ i, j, l ≤ g we see  x (c − db ) = ci − 2db (t = i),    ij i xij (cj ) = cj + db (t = j), xij xil (ct ) = xij (cl + db ) = cl + db (t = l),    x (c ) = ct (t 6= i, j, l), ij t  x (c − db ) = ci − 2db (t = i),    il i xil (cj + db ) = cj + db (t = j), xil xij (ct ) = xil (cl ) = cl + db (t = l),    x (c ) = c (t 6= i, j, l). il t t Hence we have xij xil = xil xij . For any mutually different indices 1 ≤ i, j, k ≤ g we see  x (c ) = ci − d b    ij i xij (cj + db ) = cj + 2db xij xkj (ct ) = xij (ck − db ) = ck − db    x (c ) = ct ij t  x (c − db ) = ci − db    kj i xkj (cj + db ) = cj + 2db xkj xij (ct ) = xkj (ck ) = ck − d b    x (c ) = c kj t Hence we have xij xkj = xkj xij . For any mutually different indices 1 ≤ i, j, k  x (c + db )    ij i xij (cj ) xij xki (ct ) = xij (ck − db )    x (c ) ij t  x (c − db )    ki i xki (cj + db ) xki xij (ct ) = xki (ck )    x (c ) ki t t (t = i), (t = j), (t = k), (t 6= i, j, k), (t = i), (t = j), (t = k), (t 6= i, j, k). ≤ g we see = = = = ci cj + d b ck − d b ct (t = i), (t = j), (t = k), (t 6= i, j, k), = = = = ci cj + d b ck − d b ct (t = i), (t = j), (t = k), (t 6= i, j, k). 8 R. KOBAYASHI Hence we have xij xki = xki xij . For any mutually different indices 1 ≤ i, j, k, l ≤ g we see  xij (ci ) = ci − db (t = i),     = cj + db (t = j),  xij (cj ) xij (ck − db ) = ck − db (t = k), xij xkl (ct ) =   xij (cl + db ) = cl + db (t = l),    x (c ) = ct (t 6= i, j, k, l), ij t  xkl (ci − db ) = ci − db (t = i),      xkl (cj + db ) = cj + db (t = j), xkl (ck ) = ck − db (t = k), xkl xij (ct ) =   xkl (cl ) = cl + db (t = l),    x (c ) = ct (t 6= i, j, k, l). kl t Hence we have xij xkl = xkl xij . Thus we obtain the claim.  For x ∈ π1+ (Ngb−1 , ∗) denote p(x) = xεi11 xεi22 · · · xεi2l2l . Since (yj )∗ = 1, we have x∗ = xi1 i2 xi3 i4 · · · xi2l−1 i2l . For 1 ≤ i ≤ g, let s = ♯{k | i2k−1 = i2k = i}, t = Oi (x) − s and u = Ei (x) − s. Since xii = 1 and xkl (ci ) = ci , by Lemma 3.5 we have x∗ (ci ) = = = = = xk(u)i · · · xk(1)i · xij(t) · · · xij(1) · xsii (ci ) xk(u)i · · · xk(1)i · xij(t) · · · xij(1) (ci ) xk(u)i · · · xk(1)i (ci − tdb ) (ci + udb ) − tdb ci + (u − t)db , using the suitable indices j(1), . . . , j(t) and k(1), . . . , k(u). Therefore we have that x ∈ Γb−1 if and only if x∗ = 1. g Note that ker Fgb−1 |Cgb (I(Ngb )) is equal to the intersection of ker Fgb−1 and Cgb (I(Ngb )). For b−1 b b any x ∈ Γb−1 g , since x∗ = 1, we have that Pg (x) is in Cg (I(Ng )). In addition, since Pgb−1 (x) is in ker Fgb−1 , we have that Pgb−1 (x) is in ker Fgb−1 |Cgb (I(Ngb )) . Hence we conclude b−1 |Cgb (I(Ngb )) . Pgb−1 (Γb−1 g ) ⊂ ker Fg b−1 For any ϕ ∈ ker Fg |Cgb (I(Ngb )) , since ϕ is in ker Fgb−1 there exists x ∈ π1 (Ngb−1 , ∗) such that ϕ = Pgb−1 (x). In addition, since ϕ is in Cgb (I(Ngb )) we have x∗ = 1, and hence x ∈ Γb−1 g . b−1 b−1 b−1 b−1 b−1 Hence ϕ is in Pg (Γg ). Therefore we conclude ker Fg |Cgb (I(Ngb )) ⊂ Pg (Γg ). Thus we complete the proof of Proposition 3.2. 3.3. Proof of Proposition 3.3. We have the exact sequence π1+ (Ngb−1 , ∗) −→ Cgb (M(Ngb )) −→ M(Ngb−1 ) −→ 1. b b Hence a normal generator of Pgb−1 (Γb−1 g ) in Cg (M(Ng )) is the image of a normal generator + of Γb−1 in π1 (Ngb−1 , ∗). Therefore we consider about the normal generators of Γb−1 in g g + b−1 π1 (Ng , ∗). Let πgb−1 denote the finitely presented group with the generators xi and yj , and with the relators x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ], where 1 ≤ i, i(1), i(2), i(3), i(4) ≤ g, 1 ≤ j ≤ b − 1 and [x, y] means xyx−1 y −1. There is the natural surjection ψ : π1 (Ngb−1 , ∗) → πgb−1 . We first show the following lemma. A NORMAL GENERATING SET FOR I(Ngb ) 9 Lemma 3.6. Γb−1 is equal to ker ψ. g Proof. It is clear that Γb−1 ⊃ ker ψ. We show Γb−1 ⊂ ker ψ. Namely, it suffices to show g g that x ≡ 1 modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ] for any x ∈ Γb−1 g . b−1 For any x ∈ Γg we can denote x ≡ xi1 xi2 · · · xi2l modulo x2i and yj . Since Oi1 (x) = Ei1 (x), there exists 1 ≤ t ≤ l such that i1 = i2t . Hence, modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ], we calculate x ≡ = ≡ = ≡ .. . = ≡ ≡ xi1 xi2 · · · xi2t · xi2t+1 · · · xi2l [xi1 xi2 , xi3 xi4 ]xi3 xi4 xi1 xi2 · xi5 xi6 · · · xi2t · xi2t+1 · · · xi2l xi3 xi4 xi1 xi2 · xi5 xi6 · · · xi2t · xi2t+1 · · · xi2l xi3 xi4 [xi1 xi2 , xi5 xi6 ]xi5 xi6 xi1 xi2 · xi7 xi8 · · · xi2t · xi2t+1 · · · xi2l xi3 xi4 xi5 xi6 xi1 xi2 · xi7 xi8 · · · xi2t · xi2t+1 · · · xi2l xi3 · · · xi2t−2 [xi1 xi2 , xi2t−1 xi2t ]xi2t−1 xi2t xi1 xi2 · xi2t+1 · · · xi2l xi3 · · · xi2t−1 · xi2t xi1 · xi2 · xi2t+1 · · · xi2l xi3 · · · xi2t−1 · xi2 · xi2t+1 · · · xi2l . Let x′ = xi3 · · · xi2t−1 · xi2 · xi2t+1 · · · xi2l . It immediately follows that the word length of ′ ′ ′ x′ is 2l − 2 and x′ is in Γb−1 g . Since Oi3 (x ) = Ei3 (x ), there exists 2 ≤ t ≤ l such that i3 = i2t′ . Then, similarly there exists x′′ such that the word length of x′′ is 2l − 4, x′′ is in Γgb−1 and x′ ≡ x′′ modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ]. Repeating the same operation, we have that x ≡ x′ ≡ x′′ ≡ · · · ≡ 1 modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ]. Therefore we have x ∈ ker ψ. Thus we obtain the claim.  We next show the following lemma. Lemma 3.7. πgb−1 has a presentation with πgb−1 = hx1 , . . . , xg , y1 , . . . , yb−1 | x21 , . . . , x2g , y1 , . . . , yb−1, (xi xj xk )2 , 1 ≤ i < j < k ≤ gi. Proof. It suffices to show that (xi xj xk )2 is a product of conjugations of some [xi(1) xi(2) , xi(3) xi(4) ] and that [xi(1) xi(2) , xi(3) xi(4) ] is a product of conjugations of some (xi xj xk )2 , modulo x21 , x22 , . . . , x2g . For any 1 ≤ i < j < k ≤ g, modulo x21 , x22 , . . . , x2g we see (xi xj xk )2 ≡ (xi xj xk )xj xj (xi xj xk ) = xi xj · xk xj · xj xi · xj xk ≡ xi xj · xk xj · (xi xj )−1 (xk xj )−1 = [xi xj , xk xj ]. If (i(1), i(2), i(3), i(4)) = (i, i, i, i), (i, i, i, j), (i, i, j, i), (i, j, i, i), (j, i, i, i) (i, i, j, j), (i, j, i, j), (i, j, j, i), (i, i, j, k) and (i, j, k, k), we have immediately that [xi(1) xi(2) , xi(3) xi(4) ] ≡ 1 modulo x21 , x22 , . . . , x2g . Hence we check the other cases. 10 R. KOBAYASHI For mutually different indices 1 ≤ i, j, k, l ≤ g, modulo x21 , x22 , . . . , x2g we see [xi xj , xi xk ] ≡ ≡ [xi xj , xk xj ] ≡ ≡ [xi xj , xk xi ] ≡ ≡ [xi xj , xj xk ] ≡ xi xj xi xk xj xi xk xi xi (xj xi xk )2 x−1 i , xi xj xk xj xj xi xj xk (xi xj xk )2 , xi xj xk xi xj xi xi xk (xi xj xk )2 , xi xj xj xk xj xi xk xj ≡ [xi xj , xk xl ] ≡ ≡ ≡ (xi xk xj )2 , xi xj xk xl xj xi xl xk (xi xj xk )2 xk xj xi · xi xj xl (xl xj xi )2 xl xk (xi xj xk )2 xk xl (xl xj xi )2 (xk xl )−1 For the relator (xi xj xk )2 , applying conjugations and taking their inverses, it suffices to consider the case i < j < k. Thus we obtain the claim.  By Lemma 3.6 and Lemma 3.7 we have the short exact sequence 1 −→ Γb−1 −→ π1 (Ngb−1 , ∗) −→ πgb−1 −→ 1. g Let (πgb−1 )+ denote the quotient group of π1+ (Ngb−1 , ∗) by Γb−1 g . Then we have the short exact sequence 1 −→ Γb−1 −→ π1+ (Ngb−1 , ∗) −→ (πgb−1 )+ −→ 1. g From presentations of π1+ (Ngb−1 , ∗) and (πgb−1 )+ , we obtain the normal generators of Γb−1 g in π1+ (Ngb−1 , ∗). −1 Lemma 3.8. π1+ (Ngb−1 , ∗) is the free group freely generated by xi x−1 g , xg xj , yk and xg yk xg for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1. Proof. We use the Reidemeister Schreier method. π1+ (Ngb−1 , ∗) is an index 2 subgroup of π1 (Ngb−1 , ∗). Let U = {1, xg }. Remark that U is a Schreier transversal for π1+ (Ngb−1 , ∗) in π1 (Ngb−1 , ∗). Let X = {x1 , . . . , xg , y1, . . . , yb−1 }. For u ∈ U and x ∈ X, ux = 1 if (u, x) = (1, yj ) or (xg , xi ) (resp. ux = xg if (u, x) = (1, xi ) or (xg , yj )). A generating set of −1 π1+ (Ngb−1 , ∗) is defined as B = {uxux−1 | u ∈ U, x ∈ X, ux ∈ / U}. We see 1xi 1xi = xi xg−1 , −1 xg xj xg xj −1 = xg xj , 1yk 1yk = yk , xg yk xg yk −1 = xg yk x−1 g , for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g −1 −1 is not in B. Thus we and 1 ≤ k ≤ b − 1. Since 1xg 1xg = xg x−1 g = 1 ∈ U, 1xg 1xg obtain the claim.  −1 Lemma 3.9. (πgb−1 )+ has a presentation with the generators xi x−1 g , xg xj , yk and xg yk xg for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1, and the following relators 2 (1) xi x−1 g · xg xi , xg for 1 ≤ i ≤ g − 1, (2) yj for 1 ≤ j ≤ b − 1, −1 −1 (3) xi x−1 g · xg xj · xk xg · xg xi · xj xg · xg xk for 1 ≤ i < j < k ≤ g, (4) xg xi · xi x−1 g for 1 ≤ i ≤ g − 1, (5) xg yj x−1 for 1 ≤ j ≤ b − 1, g −1 −1 (6) xg xi · xj xg · xg xk · xi x−1 g · xg xj · xk xg for 1 ≤ i < j < k ≤ g. A NORMAL GENERATING SET FOR I(Ngb ) 11 Proof. We apply the Reidemeister Schreier method for the presentation of πgb−1 in Lemma 3.7. By the argument similar to Lemma 3.8, (πgb−1 )+ is generated by xi x−1 g , −1 xg xj , yk and xg yk xg for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1. Let R be the set of the relators of πgb−1 in Lemma 3.7. A set of the relators of (πgb−1 )+ is defined as S = {uru−1 | u ∈ U, r ∈ R}, where U = {1, xg }. We see x2i = xi x−1 g · xg xi , 2 −1 −1 −1 2 −1 yj = yj , (xi xj xk ) = xi xg · xg xj · xk xg · xg xi · xj xg · xg xk , xg xi xg = xg xi · xi x−1 g , −1 2 −1 −1 −1 −1 xg yj x−1 = x y x , x (x x x ) x = x x · x x · x x · x x · x x · x x . Thus we g j g g i j k g i j g g k i g g j k g g g obtain the claim.  By Lemma 3.8 and Lemma 3.9, Γb−1 is normally generated by the relators of (πgb−1 )+ g b−1 2 of Lemma 3.9, in π1+ (Ngb−1 , ∗). Hence Pgb−1 (Γb−1 g ) is normally generated by Pg (xi ), b−1 −1 b−1 2 −1 Pgb−1 (yj ), Pgb−1 ((xi xj xk )2 ), Pgb−1 (xg x2i x−1 g ), Pg (xg yj xg ) and Pg (xg (xi xj xk ) xg ). 2 2 −1 bound a Möbius Representative loops of x2i , xg x2i x−1 g , (xi xj xk ) and xg (xi xj xk ) xg b−1 2 b−1 2 −1 b−1 band (see Figure 9). Therefore Pg (xi ), Pg (xg xi xg ), Pg ((xi xj xk )2 ) and b−1 2 −1 Pg (xg (xi xj xk ) xg ) are conjugate to Pgb−1 (x2g ). Thus we complete the proof of Proposition 3.3. * x12 xg2 (a) Representative loops of x21 , x22 , . . . , x2g bound a Möbius band. Similarly a representative loop of xg x2i x−1 g bounds a Möbius band for 1 ≤ i ≤ g. * (x1x2x3)2 (b) A representative loop of (x1 x2 x3 )2 bounds a Möbius band. Similarly representative loops of (xi xj xk )2 and xg (xi xj xk )2 xg−1 bound a Möbius band for 1 ≤ i < j < k ≤ g. Figure 9. 4. Proof of Theorem 1.1 We have the following short exact sequence: 1 → ker Cgb |I(Ngb ) → I(Ngb ) → Cgb (I(Ngb )) → 1. Hence I(Ngb ) is the normal closure of ker Cgb |I(Ngb ) and lifts by Cgb of normal generators of Cgb (I(Ngb )). By Corollary 2.3 and Corollary 3.4, we have that I(Ngb ) is normally generated −1 b b−1 b by tδb , the lift tρb by Cgb of Pgb−1 (x2g ), the lift tσjb t−1 δj by Cg of Pg (yj ), the lift tσ̄jb tδj by Cg b−1 of Pgb−1 (xg yj x−1 ◦ Cgb )|I(Ngb ) of normal generators of I(Ngb−1 ). We prove g ) and lifts by (Fg 12 R. KOBAYASHI Theorem 1.1 by induction on the number b of the boundary components of Ngb . We take the natural lift by (Fgb−1 ◦ Cgb )|I(Ngb ) of a normal generator of I(Ngb−1 ). At first, I(Ng ) is normally generated by tα , tβ t−1 β ′ and tγ (see [5]). Hence we have that −1 1 I(Ng ) is normally generated by tα , tβ tβ ′ , tγ and tδ1 , tρ1 . Similarly we have that I(Ng2 ) is normally generated by tα , tβ t−1 β ′ , tγ , tδ1 , tρ1 and tδ2 , tρ2 , tσ12 , tσ̄12 . For b ≥ 3, suppose that b−1 I(Ng ) is normally generated by tα , tβ t−1 β ′ , tγ , tδi , tρi , tσij and tσ̄ij for 1 ≤ i, j ≤ b − 1 b with i < j. Then we have that I(Ng ) is normally generated by tα , tβ t−1 β ′ , tγ , tδi , tρi , tσij , tσ̄ij and tδb , tρb , tσkb , tσ̄kb for 1 ≤ i, j, k ≤ b − 1 with i < j. Hence we obtain a normal generating set for I(Ngb ). In particular, for g ≥ 5 since I(Ng ) is normally generated by tα b and tβ t−1 β ′ (see [5]), we do not need tγ as a normal generator of I(Ng ) for g ≥ 5. Finally, proving the following lemma, we finish the proof of Theorem 1.1. Lemma 4.1. tδb , tρb , tσkb and tσ̄kb are not needed as normal generators of I(Ngb ). Proof. Let aij , bjk and ckl be simple closed curves on Ngb as shown in Figure 10, for 1 ≤ i, j ≤ g and 1 ≤ k, l ≤ b − 1. Let dm be a diffeomorphism defined by pushing the m-th boundary component once along an arrow as shown in Figure 11. Remark that the isotopy class of dm is not in M(Ngb ), since dm does not fix boundary. taij and tdm (aij ) are conjugate to tα . tbjk and tdm (bjk ) are conjugate to ether tρk or t−1 ρk . tckl and tdm (ckl ) are conjugate to tσkl , with m 6= k, l. tdk (ckl ) and tdl (ckl ) are conjugate to tσ̄−1 and tσ̄kl , kl respectively. Hence it suffices to show that tδb , tρb , tσkb and tσ̄kb are products of these Dehn twists. i j aij k l ckl bjk Figure 10. The loops aij , bjk and ckl . 1 g 1 b Figure 11. The diffeomorphism dm . For simplicity, we denote taij = ai,j , tbjk = bj,k , tckl = ck,l , tdm (aij ) = ai,j;m, tdm (bjk ) = bj,k;m and tdm (ckl ) = ck,l;m. We remark that ai,j , bj,k and ck,l are described as shown in Figure 12. A NORMAL GENERATING SET FOR I(Ngb ) 13 ai,j bj,k ck,l Figure 12. tδb , tρb and tσkb are explicitly described by products of Dehn twists as follows. tδb = (tδ1 · · · tδb−1 )−g−b+3 (a1,2 · · · a1,g · b1,1 · · · b1,b−1 ) · · · (ag−1,g · bg−1,1 · · · bg−1,b−1 )(bg,1 · · · bg,b−1 ) (c1,2 · · · c1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ), tρb = (tδ1 · · · tδb−1 )−g−b+4 (a1,2 · · · a1,g−1 · b1,1 · · · b1,b−1 ) · · · (ag−2,g−1 · bg−2,1 · · · bg−2,b−1 )(bg−1,1 · · · bg−1,b−1 ) (c1,2 · · · c1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ), tσkb = (tδ1 · · · tδk−1 · tδk+1 · · · tδb−1 )−g−b+4 (a1,2 · · · a1,g · b1,1 · · · b1,k−1 · b1,k+1 · · · b1,b−1 ) · · · (ag−1,g · bg−1,1 · · · bg−1,k−1 · bg−1,k+1 · · · bg−1,b−1 )(bg,1 · · · bg,k−1 · bg,k+1 · · · bg,b−1 ) (c1,2 · · · c1,k−1 · c1,k+1 · · · c1,b−1 ) · · · (ck−2,k−1 · ck−2,k+1 · · · ck−2,b−1 ) (ck−1,k+1 · · · ck−1,b−1)(ck+1,k+2 · · · ck+1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ). In addition, Since tσ̄kb = t−1 dk (σkb ) , tσ̄kb is a product of ai,j;k , bi,j;k , ci,j;k and tδi . Thus we obtain the claim.  Acknowledgement The author would like to express his thanks to Genki Omori for his valuable suggestions and useful comments. References [1] J.S. Birman, D.R.J. Chillingworth, On the homeotopy group of a non-orientable surface, Math. Proc. Camb. Phil. Soc. 71 (1972), 437–448. Erratum: Math. Proc. Camb. Phil. Soc. 136 (2004), 441–441. [2] D.R.J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Math. Proc. Camb. Phil. Soc. 65 (1969), 409–430. [3] D.B.A. Epstein, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966), 83–107. [4] B. Farb, D, Margalit, A primer on mapping class groups, Princeton Mathematical Series, 49. [5] S. Hirose, R. Kobayashi, A normal generating set for the Torelli group of a non-orientable closed surface, arXiv:1412.2222 [math.GT], 2015. 14 R. KOBAYASHI [6] D. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), 119–125. [7] W.B.R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307–317. [8] W.B.R. Lickorish, On the homeomorphisms of a non-orientable surface, Math. Proc. Camb. Phil. Soc. 61 (1965), 61–64. [9] L. Paris, B. Szepietowski, A presentation for the mapping class group of a nonorientable surface, arXiv:1308.5856v1 [math.GT], 2013. [10] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350. [11] A. Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007), 829–865. [12] M. Stukow, Dehn twists on nonorientable surfaces, Fund. Math. 189 (2006), no. 2, 117–147. [13] M. Stukow, Commensurability of geometric subgroups of mapping class groups, Geom. Dedicata 143 (2009), 117–142. [14] B. Szepietowski, A presentation for the mapping class group of the closed non-orientable surface of genus 4, J. Pure Appl. Algebra 213 (2009), no. 11, 2001–2016. Department of General Education, Ishikawa National College of Technology, Tsubata, Ishikawa, 929-0392, Japan E-mail address: kobayashi ryoma@ishikawa-nct.ac.jp 

Tensor network method for reversible classical computation Zhi-Cheng Yang,1 Stefanos Kourtis,1 Claudio Chamon,1 Eduardo R. Mucciolo,2 and Andrei E. Ruckenstein1 arXiv:1708.08932v2 [cond-mat.stat-mech] 5 Feb 2018 2 1 Physics Department, Boston University, Boston, Massachusetts 02215, USA Department of Physics, University of Central Florida, Orlando, Florida 32816, USA (Dated: February 6, 2018) We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs/outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times. I. INTRODUCTION Physics-inspired approaches have led to efficient algorithms for tackling typical instances of hard computational problems, shedding new light on our understanding of the complexity of such problems [1, 2]. The conceptual framework of these approaches is based on the realization that the solutions of certain computational problems are encoded in ground states of appropriate statistical mechanics models. However, the existence of either a thermodynamic phase transition into a glassy phase or a first-order quantum phase transition represent obstructions to reaching the ground state, often even for easy problems [3–7]. Recently, Ref. [8] introduced a new class of problems by mapping a generic reversible classical computation onto a two-dimensional vertex model with appropriate boundary conditions. The statistical mechanics model resulting from this mapping displays no bulk thermodynamic phase transitions and the bulk thermodynamics is independent of the classical computation represented by the model. Taken together, these features remove an obvious obstacle to reaching the ground state of a large class of computational problems and imply that the time-to-solution and the complexity of the problem are determined by the dynamics of the relaxation of the corresponding system to its ground state. However, when thermal annealing is employed, the resulting dynamics is found to be extremely slow, and even easy computational problems cannot be efficiently solved. Since any classical computation implemented as a reversible circuit can be formulated in this fashion, finding an algorithm that can solve the resulting vertex models efficiently would have far-reaching repercussions. In this paper, we introduce a tensor network approach that can treat vertex models encoding computational problems. Tensor networks are a powerful tool in the study of classical and quantum many-body systems in two and higher spatial dimensions, and are also used as compressed representations of large-scale structured data in “big-data” analytics [9–11]. Here we are interested in taking the trace of tensor networks [12–14], to count the number of solutions of a computational problem. As opposed to thermal annealing, which serially visits individual configurations, tensor network schemes sum over all configurations simultaneously. As a result, tensor-based approaches lead to a form of virtual parallelization [15], which, under certain circumstances, speeds up the computation of the trace. Most of the physics-driven applications have focused on tensor network renormalization group (TNRG) algorithms that coarse grain the network while optimally removing short-range entanglement [16– 29]. There are, however, two aspects of our physicsmotivated work that are qualitatively different from that of TNRG approaches. First, vertex models of computational gates are intrinsically not translationally invariant. Second, the trace over the tensor network, which counts the number of solutions of the computational circuit (the analogue of the zero temperature partition function of statistical mechanics models) must be computed exactly, within machine precision. (Approximations of the tensors lead to approximate counting, which in certain problems is no easier than exact counting [30].) Both features are naturally treated by the methods proposed in this paper. In our tensor network approach, the truth table of each vertex constraint corresponding to a computational gate is encoded in a tensor, such that the local compatibility between neighboring bits (or spins) is automatically guaranteed upon contracting the shared bond between two tensors. Summing over all possible unfixed boundary vertex states and contracting the entire tensor network give the partition function, which counts the total number of solutions compatible with the boundary conditions, a problem belonging to the class #P. Finding a solution can then be accomplished by fixing one boundary vertex at a time, with the total number of trials linear in the number of input bits [15]. Our tensor network method, which we refer to as the 2 iterative compression-decimation (ICD) algorithm, can be regarded as a set of local moves defining a novel dynamical path to the ground state of generalized vertex models on a square lattice. These moves can be shown to decrease or leave unchanged the bond dimensions of the tensors involved, thus achieving optimal compression (i.e., minimal bond dimension) of the tensor network on a lattice of fixed size. The algorithm’s first step is to propagate local vertex constraints across the system via repeated contraction-decomposition sweeps over all lattice bonds. These back and forth sweeps are the higher dimensional tensor-network analog of those employed in the one-dimensional finite-system density matrix renormalization group (DMRG) method [31]. For problems with non-trivial boundary conditions, such as those encountered in computation, these sweeps also propagate the boundary constraints into the bulk, thus progressively building the connection between opposite (i.e., input/output) boundaries. In the next step, the algorithm decreases the size of the lattice by coarse-graining the tensor network via suitable contractions. Repeated iterations of these two steps allow us to reach larger and larger system sizes while keeping the tensor dimensions under control, such that ultimately the full tensor trace can be taken. The computational cost of ICD hinges upon the maximum bond dimension of the tensors during the coarsegraining procedure. We identify the hardness of a given counting problem by studying the scaling of the maximum bond dimension as a function of the system size, the concentration of nontrivial constraints imposed by TOFFOLI gates, and the ratio of unfixed boundary vertices. We further present both the average and typical maximum bond dimension distributions over random instances of computations. While we cannot distinguish between polynomial and exponential scaling for the hardest regime of high TOFFOLI concentration, there exist certain regimes of the problem where the bond dimension grows relatively slowly with system size. Therefore, within this regime, we are able to count the exact number of solutions within a large search space that is intractable via direct enumerations. The rest of the paper is organized as follows. We first briefly introduce the tensor network representation of generic vertex models on a square lattice in Sec. II. Section III describes the ICD algorithm for coarse-graining and efficiently contracting the tensor network. In Section IV we apply the ICD method to reversible classical computational problems as encoded in the vertex model of computation introduced in Ref. [8], and discuss a relation between the number of solutions of the computational problem and the maximum bond dimension of the tensor network from an entanglement perspective. Section V presents the implementation of the ICD and the accompanying numerical scaling results for random computational networks defined by the concentration of Toffoli gates placed randomly at vertices of a tilted planar square lattice. Finally, we close with Sec. VI, where we outline future applications of our ICD algorithm to both computational and physics problems. II. TENSOR NETWORK FOR VERTEX MODELS We start by introducing the tensor network representation for a generic vertex model. In our formulation, discrete degrees of freedom reside on the edges of a regular lattice and they are coupled locally to their neighboring degrees of freedom. Couplings between degrees of freedom are denoted by vertices. The couplings at each vertex n = 1, . . . , Nsites , where Nsites denotes the total number of vertices, are encoded into a tensor T [n] whose rank will depend on the connectivity of the lattice. Fixing the state at all edges incident to a vertex collapses the corresponding tensor to a scalar. For concreteness, let us consider the square lattice as an example, as shown in Fig. 1; generalizations to other types of lattices are straightforward. Each tensor T [n] is therefore a rank-4 tensor T [n]ijkl , where i, j, k, l denote bond indices. i T [n] T [n]ijkl j l k Figure 1. (Color online) Vertex model on a square lattice. A local tensor T [n]ijkl is defined on each lattice site. The tensor representation is quite general. If, for example, one associates a Boltzmann weight with each combination of bond index values, one can encode statistical mechanics problems into the tensor network [17, 18, 22– 25]. Alternatively, by assigning boolean 1s to “compatible” combinations of bond index values and boolean 0s to “incompatible” ones, such that the tensor represent a vertex constraint or a truth table, one can either study statistical mechanical vertex models at zero temperature, or implement computational circuits with the tensor network (Sec. IV). Finally, one could even embed the weights of a discretized path integral for a 1+1D quantum problem in a two-dimensional network. For finite systems with boundaries, the boundary tensors will have a different rank from the bulk tensors. We define the tensor trace of the network as Y Z = tTr T [n]ijkl , (1) n where n runs over all lattice sites and tTr denotes full contractions of all bond indices. This trace may correspond 3 to the partition function for the 2D classical system, or the number of possible solutions of a computation, or the imaginary-time path integral for a 1D quantum system, etc. In general, a brute-force evaluation of the full tensor trace multiplies the dimensions of the tensors, thereby requiring a number of operations exponential in system sizes. It is therefore expedient for any strategy of evaluating the trace to keep the dimensions of tensors under control at intermediate steps, so that the tensor trace can be ultimately taken. Ideally, one would like a protocol that uses all available information — such as boundary conditions, compatibility constraints, energy costs or Boltzmann weights, depending on the particular problem at hand — to compress the tensor network as much as possible, while maintaining all the essential information therein. In Sec. III, we propose an efficient iterative scheme that achieves this goal. In particular, as we detail in Sec. IV, our algorithm provides a simple way to deal with finite systems without translational invariance, and subject to various types of boundary conditions. III. COMPRESSION-DECIMATION ALGORITHM In this section, we describe the compressiondecimation algorithm that facilitates the exact contraction of tensor networks. The algorithm consists of two steps. First, we perform sweeps on the lattice via a singular value decomposition (SVD) of pairs of tensors in order to eliminate short-range entanglement and propagate information from the boundary to the bulk, hence removing the redundancies in the bond dimensions. Due to its nature, we call this step compression. Next, we contract pairs of rows and columns of the lattice such that the system size is reduced. This step is referred to as decimation. The two steps are then repeated until the size and bond dimensions of the tensor network become small enough to allow an exact full contraction of the network. Locally, the sweeps remove redundancies due to either short-range entanglement or incompatibility in the local tensors, and compress the information into tensors with smaller bond dimensions. Globally, the sweeps propagate information about the boundary conditions to the bulk, thus imposing global constraints on the local bulk tensors. Moreover, since the sweeping is performed back and forth across the entire lattice, it does not differentiate between whether or not translational invariance is present. Therefore, our scheme may be thought of as a higher dimensional analog of the finite-system DMRG algorithm that applies to generic vertex models on finite lattices. A. Compression In this step, we visit sequentially each bond in the lattice and contract the corresponding indices of the two tensors sharing this bond. We then perform an SVD on the contracted bond and truncate the singular value spectrum keeping only those greater than a certain threshold δ. After that, the tensors are reconstructed with a smaller bond dimension. We define each forward plus backward traversal of all the bonds in the network as one sweep. The specific choice of the threshold δ depends on the desired precision, as well as the problem we are dealing with. For example, in formulating TNRG algorithms, δ can be chosen to be some small but finite number. On the other hand, for computational problems such as counting, δ is chosen to be zero within machine precision. Let us take two tensors with the shared bond labeled by i, T [1]a1 a2 a3 i and T [2]b1 ib2 b3 , as shown in Fig. 2a, where we denote the dimension of bond i as di . We would like to reduce di via a SVD. In principle, this can be achieved by directly contracting T [1] and T [2] along dimension i into a matrix MA,B = T [1]A,i T [2]i,B , where we have grouped the other three indices of each tensor into superindices A ≡ (a1 a2 a3 ) and B ≡ (b1 b2 b3 ), and then performing an SVD. However, to avoid decomposing the matrix MA,B with potentially large bond dimensions, we first do an SVD on each individual tensor (Fig. 2b): T [1]A,i = T [2]i,B = U [1]A,r Λ[1]r V [1]|r,i , U [2]i,r0 Λ[2]r0 V [2]|r0 ,B . (2a) (2b) Notice that the contraction of T [1] and T [2] can then be written as   T [1]T [2] = U [1]A,r Λ[1]r V [1]|r,i U [2]i,r0 Λ[2]r0 V [2]|r0 ,B . (2c) This implies that we can instead perform an SVD on fr,r0 = the part shown within brackets in Eq. (2c): M | Λ[1]r V [1]r,i U [2]i,r0 Λ[2]r0 , which has much smaller dimensions since dr ≤ min(dA , di ), dr0 ≤ min(dB , di ). Now we fr,r0 to obtain (Fig. 2c) perform an SVD on the matrix M fr,r0 = Ur,s Λs V | 0 . M s,r (2d) e(a a a ),s (Λs )1/2 , Te[1]a1 a2 a3 s ≡ U 1 2 3 | e T [2]sb1 b2 b3 ≡ (Λs )1/2 Ves,(b , 1 b2 b3 ) (2e) (At each SVD step described above, we discard singular values that are smaller than δ.) Therefore, after the above steps, the bond dimension ds ≤ min(dr , dr0 ) ≤ min(di , dA , dB ). Finally, we construct new tensors as (2f) where the dimension of the shared bond is reduced (Fig. 2d,e). Starting from one boundary, we visit sequentially each bond i ∈ 1, . . . , Nbonds , where Nbonds is the total number of bonds in the lattice, and perform the 4 steps outlined above, until we reach the opposite boundary. Then we repeat the procedure in the opposite direction, until we reach the original boundary. The sweeping can be repeated Nsweeps times, or until convergence of all bond dimensions. a1 (a) (b) b1 T [1] T [2] i a2 b3 a3 U [1] (a) b2 ⇤[1] V | [1] ⇤[2] V | [2] U [2] (b) U U [1] ⇤ V| V | [2] (c) U [1] U Figure 3. (Color online) (a) A column contraction involving pairs of tensors along the x direction. (b) A row contraction involving pairs of tensors along the y direction. The new tensors resulting from the contractions are denoted by pink dots, and the new bonds are denoted by orange lines. ⇤1/2 V | V | [2] ⇤1/2 (d) column contraction, we contract pairs of tensors along the x direction, and obtain a new tensor (see also Fig. 2a): a1 (e) a2 Te[1] b1 Te[2] s a3 b3 b2 Figure 2. (Color online) The contraction-decomposition step in the sweeping. (a) Two tensors T [1] and T [2] sharing a bond i. (b) Perform SVDs on individual tensors respectively. (c) Perform an SVD on the shaded part. (d) Split the resultant matrices into two pieces. (e) Construct new tensors Te[1] and Te[2]. B. Decimation The second step of the algorithm is to contract pairs of rows and columns of the tensor network, so as to yield a lattice with a smaller number of sites [22, 24]. As we show in Fig. 3, this step consists of a column contraction (Fig. 3a), followed by a row contraction (Fig. 3b). In the T(a1 b1 )a2 (a3 b2 )b3 = X T [1]a1 a2 a3 i T [2]b1 ib2 b3 . (3) i We then perform a row contraction similarly, during which pairs of tensors are contracted along the y direction. During this step, the dimensions of the bonds perpendicular to the current direction of contractions are multiplied and hence will inevitably grow. Therefore, after all columns/rows are contracted, we sweep back and forth again to reduce the bond dimensions. A simplified version of the compression-decimation scheme is presented as pseudocode in Algorithm 1. 5 Algorithm 1 Iterative Compression-Decimation Input: tensor network on a square lattice {T [n] | n ∈ 1, . . . , Nsites }; Nsweeps ≥ 1; δ ≥ 0 (SVD truncation parameter). Output: Z, as defined in Eq. (1) 1: repeat 2: for i = 1, . . . , Nsweeps do (compression) 3: for b = 1, . . . , Nbonds do (forward sweep) 4: Contract, SVD, and update tensors as in Eq. 2 5: end for 6: for b = Nbonds , . . . , 1 do (backward sweep) 7: Carry out backward sweep similarly 8: end for 9: end for 10: Perform column contractions by Eq.(3) (decimation) 11: Perform row contractions similarly (decimation) 12: until network is decimated to single site 13: Carry out tensor trace Eq. (1) IV. THE GENERAL TOFFOLI-BASED VERTEX MODEL In this section, we provide an example of a hard computational problem where our scheme can be applied to find solutions in cases that are otherwise intractable. The models we study here follow from the vertex model representation of reversible classical computations introduced in Ref. [8]. We remark that this general vertex model can address generic satisfiability problems, a statement that follows from a series of results already documented in the literature: 1. The circuit satisfiability (CSAT) problem is NPcomplete [33, 34]; 2. The CSAT problem can be formulated in terms of reversible circuits [35]; 3. Any reversible circuit can be constructed using only TOFFOLI gates [35]; A few remarks are in order. First, the lattice structure lends us more flexibility with the coarse-graining step since one does not have to contract every pair of rows and columns. For example, in cases of systems without translational invariance, representing either disordered statistical mechanics models or models encoding computational circuits, the bond dimensions are in general not distributed uniformly across the entire lattice. One could then perform the contractions selectively on rows and columns containing mostly tensors with small bond dimensions while leaving the rest for the next coarsegraining step. In practice, one could set an appropriate threshold in the algorithm depending on the specific problems. Second, the procedure described here is closely related to the TNRG algorithms where the key is to optimally remove short-range entanglement at each RG step. For example, Ref. [25] proposes a loop optimization approach for TNRG. An important step in that method is to filter out short-range entanglement within a plaquette via a QR decomposition, which we believe should be equivalent to our SVD-based sweeping. Moreover, as shall be shown in Sec. V C, the sweeps take into account the local environment around each tensor. The loop structure of short-range entanglement is eliminated (at least partially) when we visit each bond around the loop and sweep across the whole system. Whether or not more elaborate schemes [20–29, 32] for taking into account the tensor environment can improve the performance of the sweeps in the ICD scheme will not concern us in this work: we will see that even the simple sweep protocol described above is sufficient for the solution of complex generic computational problems. Third, our procedure is more apt for systems without translational invariance, e.g., spin glasses. Finally, the computational cost scales as O(χ5 ) for the SVD steps, and O(χ7 ) for the tensor contraction steps, where χ is the maximum bond dimension of the tensors. Hence the computational cost of our compression-decimation algorithm scales as O(χ7 ). 4. Any reversible circuit constructed out of TOFFOLI gates can be mapped onto our vertex model representation, with the addition of an appropriate number of identity and swap gates [8]. Hence, our vertex model can encode other satisfiability problems such as 3-SAT, which can be mapped into CSAT. (Indeed, it is possible to program 3-SAT with n variables and m clauses into a vertex model using a lattice of size n × 2m [36].) The vertex model is defined on a square lattice of finite size with periodic boundary conditions in the transverse direction, thus placing the model on a cylinder. Depending on the specific computation, different types of boundary conditions are imposed in the longitudinal direction. In addition, this model does not display translational invariance since different gates of the computational circuit are implemented by different vertices. This model can encode general computational problems, including any of the hard instances, and serves as an excellent candidate to benchmark the performance of our scheme. We start by giving a self-contained review of the general vertex model encoding reversible classical computations introduced in Ref. [8] and construct its tensor network representation. This is based on the fact that any Boolean function can be implemented using a reversible circuit constructed out of TOFFOLI gates, which are reversible three-bit logic gates taking the inputs (a, b, c) to (a, b, ab ⊕ c). To facilitate the coupling of far-away bits while maintaining the locality of TOFFOLI gates, we use two-bit SWAP gates to swap neighboring bits, (a, b) → (b, a), until pairs of distant bits are adjacent to one another. Bits that do not need to be moved are simply copied forward using two-bit Identity (ID) gates. To obtain a plane-covering tiling and thus a square-lattice representation of the circuit, we combine the SWAP and ID gates into the three-bit gates: ID-ID, ID-SWAP, SWAP-ID, SWAP-SWAP, and represent each of them as 6 well as the TOFFOLI gate as a vertex with three inputs and three outputs. The five types of vertices are shown in Fig. 4, with the input and output bits explicitly drawn on the links. a a a b c c b b a b c a a ab c TOFFOLI b c b SWAP-SWAP a ID-ID a a c b b b a c c b ID-SWAP c c SWAP-ID Figure 4. (Color online) Five types of vertices used for the vertex model representation of reversible classical computations. The input and output bits are denoted by blue squares on the links associated with a given vertex. Alternatively, one can think of bits as spin 1/2 particles located on the bonds between vertices, whereas each vertex imposes local constraints between “input” and “output” spins, such that only 23 = 8 out of the 26 = 64 total configurations are allowed. For all five types of vertices, one can write local one- and two-spin interaction terms, such that the allowed configurations are given by the ground-state manifold of the Hamiltonian comprised of all these terms [8]. The allowed configurations are then separated from the excited states by a gap set by the energy scale of the couplings. In the large-couplings limit, interactions can be equivalently thought of as constraints and one therefore needs only to consider the subspace where local vertex constraints are always satisfied. Using the five types of vertices introduced above, one can map an arbitrary classical computational circuit onto a vertex model on a tilted square lattice, as shown in Fig. 5. Bits at the left and right boundaries store the input and output respectively, and the horizontal direction corresponds to the computational “time” direction. The boundary condition along the transverse direction is chosen to be periodic. Spin degrees of freedom representing input and output bits associated with each vertex are placed on the links. This model can be shown to display no thermodynamic phase transition irrespective of the circuit realizations via a straightforward transfer matrix calculation [8]. When either only the input or only the output boundary bits are fully determined a priori, the physical system functions as a regular circuit: the solution can be ob- Figure 5. (Color online) Vertex model on a tilted square lattice encoding a generic classical computation. The left and right boundaries stores the input and output states, and periodic boundary condition is taken along the transverse direction. tained by passing the boundary state through the next column of gates, obtaining the output, then passing this output on to the next column of gates, repeating the procedure until the other boundary is reached. This mode of solution, which we shall call direct computation, is trivial and its computational cost scales linearly with the area of the system. On the other hand, by fixing only a subset of the left and right boundaries, a class of nontrivial problems can be encoded in the vertex model. For example, one can cast the integer factorization problem on a reversible multiplication circuit precisely in this way [8, 37]. In these cases, the boundary state cannot be straightforwardly propagated from the boundaries throughout the entire bulk, as the input or output of one or more gates is at most only partly fixed, and therefore direct computation unavoidably halts. Without any protocol of communication between the two partially fixed boundaries, one is left with trial-and-error enumeration of all boundary configurations, whose number grows exponentially with the number of unfixed bits at the boundaries. Even though it is sometimes possible to exploit special (nonuniversal) features of specific subsets of problems in order to devise efficient strategies of solution (e.g., factorization with sieve algorithms), general schemes that perform favorably in solving the typical instances in the encompassing class are important, both for highlighting the underlying universal patterns and as launchpads towards customized solvers for particular subsets of problems. The algorithm introduced in this work is of the latter general kind. A. Tensor network representation We shall now construct a tensor network representation of the vertex model, such that the full contraction of the tensors yields the total number of solutions satisfying the boundary conditions. In the statistical mechanics language, this is the partition function of the vertex model at zero temperature, which essentially counts the ground state degeneracy. 7 Bulk tensors. We define a rank-4 tensor associated with each vertex in the bulk, Tijkl , as shown in Fig. 6a. The tensor components are initialized to satisfy the truth table of the vertex constraint, meaning that Tijkl = 1 if (ij) → (kl) satisfies the vertex constraint, and Tijkl = 0 otherwise. Here the indices should be understood as integers labeling the spin (bit) states on each bond. Notice that the indices i, l correspond to double bonds on the lattice while j, k correspond to single bonds. Therefore, the original bond dimensions of the indices (i, j, k, l) are (4, 2, 2, 4). For concreteness, let us give an example of encoding the truth table of the TOFFOLI gate into the tensor Tijkl . First, recall that the gate function of TOFFOLI is (a, b, c) → (a, b, d = c ⊕ ab). Comparing Fig. 6a with Fig. 4, we identify on the input side, i ≡ (ab) = 21 b + 20 a, j = c; on the output side, k = a, l ≡ (bd) = 21 d+20 b. In Table I, we explicitly list the truth table of the TOFFOLI gate and its corresponding non-zero tensor components. All unspecified tensor components are set to zero. Tensors encoding the other four types of vertex constraints can be obtained in a similar fashion. meaning as the bulk tensors (Fig. 6b). Here we draw a distinction between boundary tensors whose vertex states are fixed and those that are not. For fixed boundary vertices, Tij = 1 only for one component corresponding to the fixed state, whereas for unfixed ones, Tij = 1 ∀ i, j. Under the above definitions of local tensors, local compatibility between spins shared by two vertices is automatically guaranteed when contracting the corresponding two tensors. Moreover, the unfixed boundary tensors already encode the information of all possible vertex states in a compact way, fulfilling a form of classical virtual parallelization [15]. Therefore, the full contraction of the tensor network — if it can be performed — will give the total number of solutions subject to a certain boundary condition. input output tensor component a b c a b d Tijkl ≡ T(ab)ca(bd) Figure 6. (Color online) Definition of (a) bulk and (b) boundary tensors. 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 T0000 T0102 T2001 T2103 T1010 T1112 T3013 T3111 =1 =1 =1 =1 =1 =1 =1 =1 Table I. Truth table and the corresponding tensor components for the TOFFOLI gate. On the input side, i ≡ (ab) = 21 b + 20 a, j = c; on the output side, k = a, l ≡ (bd) = 21 d + 20 b. All unspecified components are zero. Boundary tensors. The vertices at the boundary have only two bonds. Hence we define a rank-2 tensor Tij at the boundary, where the indices i, j have the same |ψi = = X {qin ,qout } X {qin ,qout } i Tijkl j (a) B. k i l j Tij i j (b) Entanglement and number of solutions Before moving on to the concrete application of the algorithm, let us try to gain some insights into the bond dimensions of local tensors needed to encode the information of the total number of solutions from an entanglement point of view [38]. Let us denote the collection of free vertex states at the input and output boundaries by {qin } and {qout }. We construct a weight W ({qin , qout }) which equals 1 if the state {qin , qout } is a solution, and 0 otherwise. The partition function is then given by X Z= W ({qin , qout }), (4) {qin ,qout } which equals the total number of solutions. Now we construct a quantum state as follows: W ({qin , qout }) |{qin , qout }i tTr (T [1]qin1 T [2]qin2 · · · T [i]T [i + 1] · · · T [N ]qoutL∂ ) |{qin }, {qout }i, where N is the total number of vertices, L∂ is the number of unfixed vertices on each boundary, and tTr denotes tracing over all internal indices of the tensors. Let us imagine taking a cut perpendicular to the peri- (5) odic direction and divide the system into two subsystems. The entanglement between subsystem left and right is determined by the singular value spectrum of the matrix W({qin }, {qout }) reshaped from the weight 8 W ({qin , qout }). W is a matrix whose entries are either 1 or 0, and there can be at most one entry in each row and column that equals 1 due to the reversible nature of the circuit. Thus the rank of the matrix W is Z, and the zeroth-order Rényi entropy S (0) = lnZ. The entanglement entropy of the quantum state (5) is hence upper bounded by S (1) ≤ S (0) = lnZ. Therefore, at least when there is only a small (nonextensive) number of solutions, the amount of entanglement is low and the information can be encoded in tensors with small bond dimensions. It may seem from the above argument that in the opposite limit of a large (extensive) number of solutions, the bond dimensions would necessarily be large. However, this is not true in general. Consider the open boundary condition under which every locally compatible configuration is a solution. In this extreme limit, the quantum state (5) is an equal amplitude superposition of all configurations, i.e., a product state. Such a state can be represented with tensors of bond dimension one in the ‘x-basis’. One thus expects that in cases of many solutions, the state should also be close to a product state with low entanglement, and hence can be represented with tensors of small bond dimensions. The above arguments indicate that, if there is a highly entangled regime where the bond dimensions required to represent the solution are large, then it must necessarily be for systems with an intermediate number of solutions. In Sec. V C, we show numerically that, even in the intermediate regime where the solutions of an arbitrary vertex model are more than just a few, it is possible to obtain an efficient and compressed tensor network representation of the allowedconfiguration manifold. Having argued that, for the case of problems with a small number of solutions, solutions of vertex models with partially fixed boundaries can be encoded into tensor networks with small bond dimensions, we set out to find this tensor-network representation. The pertinent motivating question is: given that there is a representation that can compress the full information of all solutions with relatively small bond dimensions, how can we find it efficiently? V. APPLICATION OF THE ICD TO THE RANDOM VERTEX MODEL In this section we apply ICD to N random instances of vertex lattices of fixed concentration of TOFFOLI gates, and fixed and equal concentrations of all four other types of gates shown in Fig. 4, with random input states for each instance. By evaluating the full tensor trace for each of these N instances and for various lattice sizes, we obtain information about the average scaling of performing the underlying classical computations by means of the ICD method. Moreover, we study the full distribution of the maximum bond dimension χ over random realizations and find that the typical behavior is generally different than the average, due to the presence of heavy tails in the bond-dimension distribution. Finally, we establish numerically that the scaling of the actual running time τ with the maximum bond dimension is always better than the worst-case estimate τ ∼ O(χ7 ). A. Local moves Since the vertex model is defined on a tilted square lattice, we first need to turn it into a lattice as shown in Fig. 1 in order to apply our algorithm in Sec. III. This can be done by performing local moves on the tilted lattice, which we explain below. (a) (b) (c) Figure 7. (Color online) Illustration of the local moves which turn the original lattice into a square lattice rotated by 45◦ . In (a), sites belonging to sublattice A and B are shown in blue and green dots, respectively. From (b) to (c), four sites belonging to a diamond are contracted into one. The tilted square lattice Fig. 7a is bipartite, with two sublattices A and B. Local tensor decompositions and contractions for tensors on each sublattice can rearrange the lattice into an “untilted” one, rotated by 45◦ with respect to the original lattice. We start by splitting each tensor on the original vertex lattice into two along either horizontal or vertical direction, depending on which sublattice the corresponding site belongs to. Let us take a bulk tensor Tijkl on the original lattice. If the site belongs to sublattice A, we decompose P the tensor horizontally into two rank-3 tensors, Tijkl = q Aijq Bklq ; if the site belongs to sublattice B, we instead decompose the tenP e e sor vertically, Tijkl = q A ikq Bjlq , as shown in Fig. 8. Such a decomposition can be achieved via an SVD on | the original tensors, T(ij),(kl) = U(ij),q Λq Vq,(kl) to yield | Aijq = U(ij),q (Λq )1/2 and Bklq = (Λq )1/2 Vq,(kl) . We visit each site and split the tensors in this way. This turns the 9 tensor network into the structure shown in Fig. 7b. We then further contract four tensors in a diamond into one and finally arrive at a new square lattice rotated by 45◦ with respect to the original one (Fig. 7c). With these local moves, which have to be carried out only once, we cast the problem into the form discussed in Sec. II. i k i j l j i k k (a) q l i k q (b) j l j l Figure 8. (Color online) Local moves that decompose each tensor on the original lattice into two along either horizontal or vertical direction, depending on whether the site belongs to sublattice A (a) or B (b). However, instead of doing an SVD on the original tensor, here we can use the fact that the tensors encode the truth tables of reversible gates and use an alternative method. Define a new set of tensors with an auxiliary inq dex q = 0, 1, . . . , 7 labeling the vertex state, Teijkl . Now the component of this rank-(4,1) tensor is one if and only if q is the same as the input state labeled by (i, j). Then, the desired decomposition can be achieved as follows: X q X q Aijq = Teijkl , Bklq = Teijkl , ij kl eikq = A X jl q Teijkl , ejlq = B X ik q Teijkl . (6) One can easily check that the contraction of the A and B tensors gives back the original tensor T , and hence this achieves the splitting shown in Fig. 8. The remaining steps of the algorithm are carried out exactly in the same way as before. By construction, the bonds between the resulting bulk tensors all have dimension 8. B. Control of bond dimensions We can now apply the compression-decimation algorithm to count the number of solutions for a given boundary condition. As we discuss in Sec. III A, a truncation threshold δ needs to be specified in the sweeping step of the algorithm. Since we are performing an exact counting, no approximation in the truncation of the bond dimensions is made during the coarse-graining procedure, i.e., we choose δ = 0 within machine precision. This is a key methodological difference of the ICD to TNRG methods, which approximate physical observables to within a certain accuracy by enforcing a finite δ. As mentioned above, from a statistical mechanics point-of-view, what we are computing is the zerotemperature partition function of the vertex model, which yields the ground state degeneracy. In the bulk, all locally compatible configurations are equally possible until they receive information from the boundary conditions. Therefore, the coarse-graining step effectively brings the boundaries close to one another, and the sweeping step propagates information from the boundary to the bulk and knocks out states encoded in local tensors that are incompatible with the global boundary conditions. The reason why the growth of bond dimensions remains controlled is that longer-range compatibility constraints over increasingly larger areas are enforced upon the coarse-grained tensors. These constraints are propagated to neighboring coarse-grained tensors upon sweeping, thus further reducing bond dimensions and compressing the tensor-network representation. For the trivial cases of either fixing all gates on one boundary or leaving them all free (open boundary condition), we have checked that the tensors converge to bond dimension one (scalars) after one sweep, without the need of coarse-graining. The tensor contraction is then simply reduced to multiplications of scalars, which can be trivially computed and indeed gives the correct counting. This demonstrates that the sweeping is responsible for propagating information from the boundary, and that the case of fully fixing one boundary is thus equivalent to direct computation, as described in Sec. IV. In cases of mixed boundary conditions, the sweeping on the original lattice scale will generally not be sufficient to propagate information across the whole system or establish full communications between the two boundaries. Thus one would expect that while the bond dimensions close to the boundary may be small, those deep in the bulk may be large. We therefore perform the contractions selectively on rows and columns containing mostly tensors with small bond dimensions while leaving the rest for the next coarse-graining step, as described in Sec. III. C. Numerical results The computational cost of the ICD algorithm is determined by the maximum bond dimension encountered during the coarse-graining and sweeping procedures. In this section, we study the scaling of the maximum bond dimension as function of the set of parameters defining an instance of the problem: the number of vertices in each column L, the total number of columns (circuit depth) W , the concentration of TOFFOLI gates c, and the number of unfixed boundary vertices L∂ . For a given set of parameters, we consider random tensor networks corresponding to typical instances of computational problems. 10 18 18 (a) 1.25 4 1.2 16 16 14 14 12 12 10 10 8 8 1.15 1.1 3 1.05 1 2 50 100 150 200 1 0.2 0 50 100 150 0.25 0.3 0.35 0.4 0.45 0.5 200 # sweeps Figure 9. (Color online) The average bond dimension of the entire lattice as a function of the number of sweeps in the compression-decimation steps. The bumps where the average bond dimension increases slightly correspond to the points where we coarse-grain the lattice via column and row contractions. Inset: zoom-in plot from the 20th sweeping step. By looking at the scaling of the bond dimensions, we gain some understanding of how the hardness of the problems depends on various parameters, which may serve as a guidance for designing and analyzing computational circuits for practical problems. Before looking into the scaling of the maximum bond dimensions, we first show the average bond dimension for the entire lattice as a function of the number of sweeps in the compression-decimation steps. As seen from Fig. 9, the average bond dimension indeed decreases as the sweeping is performed. The bumps in the plot correspond to the points where we coarse-grain the lattice via column and row contractions. At a given length scale, the average bond dimension converges after a few sweeps. As we increase the length scales, the average bond dimension may first increase, but will eventually drop again as we perform sweeps at the new length scale. This demonstrates that the sweeping is able to impose global constraints at the boundary into the bulk, hence keeping the bond dimensions of bulk tensors under control. We expect the maximum bond dimension to follow the scaling function χ = G(L∂ /L, c, L, W/L). Below we study the growth of maximum bond dimensions as a function of each system parameter numerically. First, we consider the scaling of χ as the ratio of unfixed boundary vertices L∂ /L is varied, with the other parameters fixed. As shown in Fig. 10a, the bond dimensions are small for both small and large L∂ /L. This is in agreement with our discussions in Sec. IV B, where we argued that in both regimes the states are close to product states and there should exist a representation in which the bond dimensions are small (the ‘z-basis’ and ‘x-basis’). For intermediate values of L∂ /L, the bond dimensions grow, indicating the existence of a hard regime where either there is no such a representation of small bond dimensions to fully encode the solutions, or it is very hard to 40 35 40 (b) 35 30 30 25 25 20 20 15 15 10 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 10. (Color online) Scaling of the average maximum bond dimension χ with (a) the ratio of unfixed boundary vertices L∂ /L, and (b) TOFFOLI concentration c, versus the scaling of the typical maximum bond dimension ehlnχi . The remaining parameters are fixed in each plot. The data are obtained by averaging over 2000 realizations of random tensor networks. find such a representation via tensor optimization algorithms. Fixing L∂ /L = 0.36, which corresponds to the hard regime in Fig. 10a, we plot the scaling of χ as a function of the TOFFOLI concentration c. The TOFFOLI gates impose nontrivial vertex constraints, which involve a nonlinear relationship between the input and output bits. In fact, in the absence of TOFFOLI gates, the vertex model can be expressed as 3L decoupled Ising chains whose dynamics are simple [8]. In the ICD algorithm, the maximum bond dimension indeed grows with increasing TOFFOLI concentrations, as depicted in Fig. 10b. Now let us look at the scaling of χ as a function of the input size L. Again, we fix L∂ /L = 0.36 to stay in the hard regime. Figure 11a shows that the maximum bond dimension increases with increasing input size, even when the aspect ratio W/L of the circuit is fixed. Because of the limited range of L we were able to analyze, we cannot draw any conclusion regarding the functional form of this scaling, which would determine the complexity of our algorithm. However, we can demonstrate that our algorithm is able to solve the problem in regimes that are still intractable using a naive enumeration of solutions. For this purpose, we move away from the hardest regime and choose L∂ /L = 0.5. As can be seen in 11 100 100 (a) 65 (b) 65 80 80 60 60 60 60 40 40 55 55 20 20 50 50 25 30 35 40 45 50 16 49 16 (c) 14 14 12 12 10 10 8 8 3 3.1 3.2 3.3 3.4 51 53 55 57 59 61 63 65 35 30 35 (d) 30 25 25 20 20 15 15 10 10 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1 Figure 11. (Color online) Scaling of the average maximum bond dimension hχi with L (a,b) and W/L (c,d), versus the scaling of the typical maximum bond dimension ehlnχi . The data are obtained by averaging over 500 to 2000 realizations of random tensor networks. In (a), the last point is averaged over 7 realizations and the error bar is not shown. The blue dotted line is a guide to the eye, and corresponds to a quadratic fitting. In all cases, the typical values stay below the average values, due to the presence of heavy tails in the distribution of χ. Fig. 11b, we are able to reach much larger values of L in this regime, and the bond dimensions, although still growing, increase at a much slower pace. In fact, we were able to reach L = 96 with an average maximum bond dimension hχi = 78.25 (data not shown). Since half of the input vertices are unknown, a direct trial-and-error enumeration would take 848 ≈ 1043 iterations to perform an exact counting, which is prohibitive even with parallelization. We have thus shown that there is a subset of nontrivial problems that can easily be solved by the ICD method, for which (a) direct enumeration is impossible to scale up and (b) efficient custom algorithms are not known. Finally, we show the scaling with the aspect ratio W/L. As we previously discussed, the key to the reduction of the bond dimensions of bulk tensors is the global constraint imposed at the boundary. The coarsegraining step brings the boundaries close together while the sweeping step helps propagate information. Therefore, one should expect the problem to become harder as the circuit depth W increases for a fixed L, since it takes more iterations of coarse-graining for the connection between the boundaries to be built, all the while the bond dimensions of the bulk tensors barely decrease. In Fig. 11c we show the scaling of χ as a function of W/L in the hard regime; χ indeed grows upon increasing W/L, as expected. For computational problems of practical interests, the vertex model representation often has the feature of low TOFFOLI concentration but large aspect ratio, e.g., the multiplication circuit [8, 37]. In Fig. 11d we show data for cases with this feature by lowering the TOFFOLI concentration to c = 5% and keeping L∂ /L = 0.5. We find that the bond dimensions grow in a similar fashion as in Fig. 11c, although a larger range of values of W/L now becomes amenable. The above results demonstrate the average scaling behavior of the ICD algorithm over random instances of computations. It is informative to compare this to the typical behavior, revealed by analyzing the full distribution of the maximum bond dimensions; see, e.g., Fig. 11(a). In Fig. 12, we present the probability distribution of the maximum bond dimension. A vertical cut at each L corresponds to the probability distribution over all instances for that L. For larger L we observe secondary peaks at larger χ, which gradually take up more weight, thus shifting both average and typical maximum bond dimension to higher values. Moreover, despite the fact that the highest weight is always encountered at small bond dimensions, a finite subset of hard instances generate much larger bond dimensions, leading to heavy tails in the distributions. Average values are sensitive to such tails, and hence do not faithfully represent the typical instances. In Fig. 10 and 11, we also plot the values of ehlnχi as an estimate of the typical behavior, in contrast to hχi. Indeed, the typical values stay below the average values in all cases studied. We point out that the pres- 12 (a) 10 9 8 7 6 5 2 3 4 5 6 (b) Figure 13. (Color online) Scatter plot of the (logarithm of) actual running time τ in units of seconds versus the (logarithm of) maximum bond dimension χ for 4600 instances. The calculations were performed using a Python implementation of the ICD algorithm, using NumPy / LAPACK for all linear algebra operations, on 2.0 GHz Intel Xeon Processors E7-4809 v3. Figure 12. (Color online) The full distribution of the maximum bond dimension over random instances. In (a), the color plot along the vertical direction shows the probability distribution at each L. The orange (diamond) points show the values of ehlnχi , giving an estimate of the typical instances in contrast to the average instances depicted in purple (triangle) points. In (b), we take the slice of L = 39 in (a) and plot the histogram of the distribution. ence of heavy tails in the distribution is ubiquitous in random satisfiability problems, and such instances could in principle be tackled with different strategies [39–42]. The efficiency of the ICD algorithm is controlled by the maximum bond dimension encountered in each instance, and in particular, the complexity of the algorithm is upper bounded by O(χ7 ) as discussed in Sec. III. Nevertheless, it is still useful to see whether the actual running time saturates this bound. In Fig. 13 we show the scatter plot for the actual running time τ versus χ for 4600 random instances. We see a clear clustering of the data points and a positive correlation between these two quantities. The fact that there is a spreading of τ for each χ can be understood by taking into account the nonuniform spatial distributions of the bond dimensions across the system. Unlike the TNRG algorithms, where the bond dimensions of all tensors and all tensor legs are frequently chosen to be uniform, bond dimensions of different tensors and of different legs of the same tensor are typically highly nonuniform in the ICD method. Therefore, running times for instances with the same maximum χ also depend on the number of bonds with dimension χ. The distribution of χ throughout the system is thus an important factor. Moreover, we find that the scaling of the running time with the maximum bond dimension τ ∼ χα has a power α < 7, which shows that the actual performance of the algorithm is generally better than the worst-case scenario estimate. VI. SUMMARY AND OUTLOOK We presented a method for contracting tensor networks that is well suited for the solution of statistical physics vertex models of universal classical computation. In these models, the tensor trace represents the number of solutions. Individual solutions can be efficiently extracted from the tensor network when the number of solutions is small. More generally, the method applies to any system, classical or quantum, whose quantity of interest is a tensor trace in an arbitrary lattice. Our scheme consists of iteratively compressing tensors through a contraction-decomposition operation that reduces their bond dimensions, followed by decimation, which increases bond dimensions but reduces the network size. By repeated applications of this two step process – compression followed decimation – one can gradually collapse rather large tensor networks. In the context of computation, the method allowed us to study relatively large classical reversible circuits represented by two dimensional vertex models. By contrast with thermal annealing, direct computation from a fully specified input boundary through the use of tensor networks occurs in a time linear in the depth of the circuit. For complex problems with partially fixed input/output boundaries tensor networks enable us to count solutions in problems where enumeration would otherwise take of order 850 operations. We close with an outlook of future directions motivated by this work. First, focusing on the method per se, the performance of our ICD algorithm could still be further improved. 13 There are enhancements that are simply operational in nature, such as parallelization of the sweeping step of the algorithm, which can be accomplished by dividing the tensors into separate non-overlapping sets. Second, at a more fundamental level, as we point out at the end of Sec. V C, a better understanding of the mechanism by which short-range entanglement is removed within the ICD method would require a systematic study of the evolution of the spatial distribution of bond dimensions. The goal would be to design more controlled bond dimension truncation schemes that involve the effect of the environment of local tensors, as proposed in Refs. [20, 22, 24, 25, 32]. More generally, we expect that our method can be applied to both classical and quantum many-body systems in two and higher dimensions. Third, in our study of computation-motivated problems, we focused on random tensor networks corresponding to random computational circuits. However, the ICD methodology should be used to address problems of practical interest, a research direction that is being currently explored. The results on the scaling of the bond dimensions presented above should inform the design and analysis of tractable computational circuits, such as circuits with W ∼ L and a moderate number of TOFFOLI gates. 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On completeness of logical relations for monadic types ⋆ Sławomir Lasota1 ⋆⋆ David Nowak2 Yu Zhang3 ⋆ ⋆ ⋆ arXiv:cs/0612106v1 [cs.LO] 21 Dec 2006 1 Institute of Informatics, Warsaw University, Warszawa, Poland 2 Research Center for Information Security, National Institute of Advanced Industrial Science and Technology, Tokyo, Japan 3 Project Everest, INRIA Sophia-Antipolis, France Abstract. Software security can be ensured by specifying and verifying security properties of software using formal methods with strong theoretical bases. In particular, programs can be modeled in the framework of lambda-calculi, and interesting properties can be expressed formally by contextual equivalence (a.k.a. observational equivalence). Furthermore, imperative features, which exist in most real-life software, can be nicely expressed in the so-called computational lambdacalculus. Contextual equivalence is difficult to prove directly, but we can often use logical relations as a tool to establish it in lambda-calculi. We have already defined logical relations for the computational lambda-calculus in previous work. We devote this paper to the study of their completeness w.r.t. contextual equivalence in the computational lambda-calculus. 1 Introduction Contextual equivalence. Two programs are contextually equivalent (a.k.a. observationally equivalent) if they have the same observable behavior, i.e. an outsider cannot distinguish them. Interesting properties of programs can be expressed using the notion of contextual equivalence. For example, to prove that a program does not leak a secret, such as the secret key used by an ATM to communicate with the bank, it is sufficient to prove that if we change the secret, the observable behavior will not change [18,3,19]: whatever experiment a customer makes with the ATM, he or she cannot guess information about the secret key by observing the reaction of the ATM. Another example is to specify functional properties by contextual equivalence. For example, if sorted is a function which checks that a list is sorted and sort is a function which sorts a list, then, for all list l, you want the expression sorted(sort(l)) to be contextually equivalent to the expression true. Finally, in the context of parameterized verification, contextual equivalence allows the verification for all instantiations of the parameter to be reduced to the ⋆ Partially supported by the RNTL project Prouvé, the ACI Sécurité Informatique Rossignol, the ACI jeunes chercheurs “Sécurité informatique, protocoles cryptographiques et détection d’intrusions”, and the ACI Cryptologie “PSI-Robuste”. ⋆⋆ Partially supported by the Polish K BN grant No. 4 T11C 042 25 and by the European Community Research Training Network Games. This work was performed in part during the author’s stay at LSV. ⋆⋆⋆ This work was mainly done when the author was a PhD student under an MENRT grant on ACI Cryptologie funding, École Doctorale Sciences Pratiques (Cachan). verification for a finite number of instantiations (See e.g. [6] where logical relations are one of the essential ingredients). Logical relations. While contextual equivalence is difficult to prove directly because of the universal quantification over contexts, logical relations [15,8] are powerful tools that allow us to deduce contextual equivalence in typed λ-calculi. With the aid of the so-called Basic Lemma, one can easily prove that logical relations are sound w.r.t. contextual equivalence. However, completeness of logical relations is much more difficult to achieve: usually we can only show the completeness of logical relations for types up to first order. On the other hand, the computational λ-calculus [10] has proved useful to define various notions of computations on top of the λ-calculus: partial computations, exceptions, state transformers, continuations and non-determinism in particular. Moggi’s insight is based on categorical semantics: while categorical models of the standard λcalculus are cartesian closed categories (CCCs), the computational λ-calculus requires CCCs with a strong monad. Logical relations for monadic types, which are particularly introduced in Moggi’s language, can be derived by the construction defined in [2] where soundness of logical relations is guaranteed. However, monadic types introduce new difficulties. In particular, contextual equivalence becomes subtler due to the different semantics of different monads: equivalent programs in one monad are not necessarily equivalent in another! This accordingly makes completeness of logical relations more difficult to achieve in the computational λ-calculus. In particular the usual proofs of completeness up to first order do not go through. Contributions. We propose in this paper a notion of contextual equivalence for the computational λ-calculus. Logical relations for this language are defined according to the general derivation in [2]. We then explore the completeness and we prove that for the partial computation monad, the exception monad and the state transformer monad, logical relations are still complete up to first-order types. In the case of the non-determinism monad, we need to restrict ourselves to a subset of first-order types. As a corollary, we prove that strong bisimulation is complete w.r.t. contextual equivalence in a λ-calculus with monadic non-determinism. Not like previous work on using logical relations to study contextual equivalence in models with computational effects [16,13,11], most of which focus on computations with local states, our work in this paper is based on a more general framework for describing computations, namely the computational λ-calculus. In particular, very different forms of computations like continuations and non-determinism are studied, not just those for local states. Plan. The rest of this paper is structured as follows: we devote Section 2 to preliminaries, by introducing basic knowledge of logical relations in a simple version of typed λ-calculus; then from Section 3 on, we move to the computational λ-calculus and we rest on a set-theoretical model. In particular, Section 3.4 sketches out the proof scheme of completeness of logical relations for monadic types and shows the difficulty of getting a general proof; we then switch to case studies and we explore, in Section 4, the completeness in the computational λ-calculus for a list of common monads: partial computations, exceptions, state transformers, continuations and the non-determinism; the last section consists of a discussion on related work and perspectives. 2 Logical relations for the simply typed λ-calculus 2.1 The simply typed λ-calculus λ→ Let λ→ be a simple version of typed λ-calculus: Types: Terms: τ, τ ′ , ... ::= b | τ → τ ′ t, t′ , ... ::= x | c | λx · t | tt′ where b ranges over a set of base types (booleans, integers, etc.), c over a set of constants and x over a set of variables. We write t[u/x] the result of substituting the term u for free occurrences of the variable x in the term t. Typing judgments are of the form Γ ⊢ t : τ where Γ is a typing context, i.e. a finite mapping from variables to types. We say that x : τ is in Γ whenever Γ (x) = τ . We write Γ, x : τ for the typing context which agrees with Γ except that it maps x to τ . Typing rules are as standard. We consider the set theoretical semantics of λ→ . The semantics of any type τ is given by a set Jτ K. Those sets are such that Jτ → τ ′ K is the set of all functions from Jτ K to Jτ ′ K, for all types τ and τ ′ . A Γ -environment ρ is a map such that, for every x : τ in Γ , ρ(x) is an element of Jτ K. We write ρ[x := a] for the environment which agrees with ρ except that it maps x to a. We write [x := a] for the environment just mapping x to a. Let t be a term such that Γ ⊢ t : τ is derivable. The denotation of t, w.r.t. a Γ -environment ρ, is given as usual by an element JtKρ of Jτ K. We write JtK instead of JtKρ when ρ is irrelevant, e.g., when t is a closed term. When given a value a ∈ Jτ K, we say that it is definable if and only if there exists a closed term t such that ⊢ t : τ is derivable and a = JtK. Let Obs be a subset of base types, called observation types, such as booleans, integers, etc. A context C is a term such that x : τ ⊢ C : o is derivable, where o is an observation type. We spell the standard notion of contextual equivalence in a denotational setting: two elements a1 and a2 of Jτ K, are contextually equivalent (written as a1 ≈τ a2 ), if and only if for any context C such that x : τ ⊢ C : o (o ∈ Obs) is derivable, JCK[x := a1 ] = JCK[x := a2 ]. We say that two closed terms t1 and t2 of the same type τ are contextually equivalent whenever Jt1 K ≈τ Jt2 K. Without making confusion, we shall use the same notation ≈τ to denote the contextual equivalence between terms. We also define a relation ∼τ : for every pair of values a1 , a2 ∈ Jτ K, a1 ∼τ a2 if and only if a1 , a2 are definable and a1 ≈τ a2 . 2.2 Logical relations Essentially, a (binary) logical relation [8] is a family (Rτ )τ type of relations, one for each type τ , on Jτ K such that related functions map related arguments to related results. More formally, it is a family (Rτ )τ type of relations such that for every f1 , f2 ∈ Jτ → τ ′ K, f1 Rτ →τ ′ f2 ⇐⇒ ∀a1 , a2 ∈ Jτ K . a1 Rτ a2 =⇒ f1 (a1 ) Rτ ′ f2 (a2 ) There is no constraint on relations at base types. In λ→ , once the relations at base types are fixed, the above condition forces (Rτ )τ type to be uniquely determined by induction on types. We might have other complex types, e.g., products in variations of λ→ , and in general, relations of these complex types should be also uniquely determined by relations of their type components. For instance, pairs are related when their elements are pairwise related. A unary logical relation is also called a logical predicate. A so-called Basic Lemma comes along with logical relations since Plotkin’s work [15]. It states that if Γ ⊢ t : τ is derivable, ρ1 , ρ2 are two related Γ -environments, and every constant is related to itself, then JtKρ1 Rτ JtKρ2 . Here two Γ -environments ρ1 , ρ2 are related by the logical relation, if and only if ρ1 (x) Rτ ρ2 (x) for every x : τ in Γ . Basic Lemma is crucial for proving various properties using logical relations [8]. In the case of establishing contextual equivalence, it implies that, for every context C such that x : τ ⊢ C : o is derivable (o ∈ Obs), JCK[x := a1 ] Ro JCK[x := a2 ] for every pair of related values a1 , a2 in Jτ K. If Ro is the equality, then JCK[x := a1 ] = JCK[x := a2 ], i.e., a1 ≈τ a2 . Briefly, for every logical relation (Rτ )τ type such that Ro is the equality for every observation type o, logically related values are necessarily contextually equivalent, i.e., Rτ ⊆ ≈τ for any type τ . Completeness states the inverse: a logical relation (Rτ )τ type is complete if every contextually equivalent values are related by this logical relation, i.e., ≈τ ⊆ Rτ for every type τ . Completeness for logical relations is hard to achieve, even in a simple version of λ-calculus like λ→ . Usually we are only able to prove completeness for types up to first order (the order of types is defined inductively: ord(b) = 0 for any base type b; ord(τ → τ ′ ) = max(ord(τ ) + 1, ord(τ ′ )) for function types). The following proposition states the completeness of logical relations in λ→ , for types up to first order: Proposition 1. There exists a logical relation (Rτ )τ type for λ→ , with partial equality on observation types, such that if ⊢ t1 : τ and ⊢ t2 : τ are derivable, for any type τ up to first order, t1 ≈τ t2 =⇒ Jt1 K Rτ Jt2 K. Proof. Let (Rτ )τ type be the logical relation induced by Rb = ∼b at every base type b and we show that it is complete for types up to first order. The proof is by induction over τ . Case τ = b is obvious. Let τ = b → τ ′ . Take two terms t1 , t2 of type b → τ ′ such that Jt1 K and Jt2 K are related by ≈b→τ ′ . Let f1 = Jt1 K and f2 = Jt2 K. Assume that a1 , a2 ∈ JbK are related by Rb , therefore a1 ∼b a2 since Rb = ∼b . Clearly, a1 and a2 are thus definable, say by terms u1 and u2 , respectively. Then, for any context C such that x : τ ′ ⊢ C : o (o ∈ Obs) is derivable, JCK[x := f1 (a1 )] = JC[xu1 /x]K[x := f1 ] = JC[xu1 /x]K[x := f2 ] = JCK[x := f2 (a1 )] = JC[t2 x/x]K[x := a1 ] = JC[t2 x/x]K[x := a2 ] = JCK[x := f2 (a2 )]. (since a1 = Ju1 K) (since f1 ≈b→τ ′ f2 ) (since f2 = Jt2 K) (since a1 ≈b a2 ) Hence f1 (a1 ) ≈τ ′ f2 (a2 ). Moreover, f1 (a1 ) and f2 (a2 ) are therefore definable by t1 u1 and t2 u2 respectively. By induction hypothesis, f1 (a1 ) Rτ ′ f2 (a2 ). Because a1 and a2 are arbitrary, we conclude that f1 Rb→τ ′ f2 . ⊓ ⊔ Note that an equivalent way to state completeness of logical relations is to say that there exists a logical relation (Rτ )τ type which is partial equality on observation types and such that, for all first-order types τ , ∼τ ⊆ Rτ . 3 Logical relations for the computational λ-calculus 3.1 The computational λ-calculus λComp From the section on, our discussion is based on another language — Moggi’s computational λ-calculus. Moggi defines this language so that one can express various forms of side effects (exceptions, non-determinism, etc.) in this general framework [10]. The computational λ-calculus, denoted by λComp , extends λ→ : Types: Terms: τ, τ ′ , ... ::= b | τ → τ ′ | Tτ t, t′ , ... ::= x | c | λx · t | tt′ | val(t) | let x ⇐ t in t′ An extra unary type constructor T is introduced in the computational λ-calculus: intuitively, a type Tτ is the type of computations of type τ . We call Tτ a monadic type in the sequel. The two extra constructs val(t) and let x ⇐ t in t′ represent respectively the trivial computation and the sequential computation, with the typing rules: Γ ⊢t:τ Γ ⊢ val(t) : Tτ Γ ⊢ t : Tτ Γ, x : τ ⊢ t′ : Tτ ′ Γ ⊢ let x ⇐ t in t′ : Tτ ′ Note that the let construct here should not be confused with that in PCF: in λComp , we bind the result of the term t to the variable x, but they are not of the same type — t must be a computation. Moggi also builds a categorical model for the computational λ-calculus, using the notion of monads [10]. Whereas categorical models of simply typed λ-calculi such as λ→ are usually cartesian closed categories (CCCs), a model for λComp requires additionally a strong monad (T, η, µ, t) be defined over the CCC. Consequently, a monadic type is interpreted using the monad T : JTτ K = T Jτ K, and each term in λComp has a unique interpretation as a morphism in a CCC with the strong monad [10]. Semantics of the two additional constructs can be given in full generality in a categorical setting [10]: the denotations of val construct and let construct are defined by the follwoing composites respectively: JΓ ⊢ val(t) : Tτ K : JΓ ⊢t:τ K ηJτ K JΓ K −−−−−→ Jτ K −−−→ T Jτ K, hidJΓ K ,JΓ ⊢t1 :Tτ Ki tJΓ K,Jτ K JΓ ⊢ let x ⇐ t1 in t2 : Tτ ′ K : JΓ K −−−−−−−−−−−→ JΓ K × T Jτ K −−−−−→ T JΓ K × Jτ K µJτ ′ K T JΓ,x:τ ⊢t2 :Tτ ′ K −−−−−−−−−−→ T T Jτ ′ K −−−→ T Jτ ′ K. In particular, the interpretation of terms in the computational λ-calculus must satisfy the following equations: Jlet x ⇐ val(t1 ) in t2 Kρ = Jt2 [t1 /x]Kρ, (1) Jlet x2 ⇐ (let x1 ⇐ t1 in t2 ) in t3 Kρ = Jlet x1 ⇐ t1 in let x2 ⇐ t2 in t3 Kρ,(2) Jlet x ⇐ t in val(x)Kρ = JtKρ. (3) We shall focus on Moggi’s monads defined over the category Set of sets and functions. Figure 1 lists the definitions of some concrete monads: partial computations, exceptions, state transformers, continuations and non-determinism. We shall write λPESCN Comp to refer to λComp where the monad is restricted to be one of these five monads. Partial computation: JTτ K = Jτ K ∪ {⊥} Jval(t)Kρ = JtKρ Jlet x ⇐ t1 in t2 Kρ = Exception: Continuation: Non-determinism: Jt2 Kρ[x := Jt1 Kρ], if Jt1 Kρ 6= ⊥ ⊥, if Jt1 Kρ = ⊥  Jt2 Kρ[x := Jt1 Kρ], if Jt1 Kρ 6∈ E Jt1 Kρ, if Jt1 Kρ ∈ E JTτ K = Jτ K ∪ E Jval(t)Kρ = JtKρ Jlet x ⇐ t1 in t2 Kρ = State transformer:  JTτ K = (Jτ K × St )St Jval(t)Kρ = λ s · (JtKρ, s) Jlet x ⇐ t1 in t2 Kρ = λ s · (Jt2 Kρ[x := a1 ])s1 , where a1 = π1 ((Jt1 Kρ)s), s1 = π2 ((Jt2 Kρ)s) Jτ K JTτ K = RR Jval(t)Kρ = λ kJτ K→R · k(JtKρ) Jlet x ⇐ t1 in t2 Kρ = λ kJτ2 K→R · (Jt1 Kρ)k′ where k′ is a function: λ v Jτ1 K · (Jt2 Kρ[x := v])k JTτ K = Pfin (Jτ K) Jval(t)Kρ = {JtKρ} S Jlet x ⇐ t1 in t2 Kρ = Jt2 Kρ[x := a] a∈Jt1 Kρ Fig. 1. Concrete monads defined in Set The computational λ-calculus is strongly normalizing [1]. The reduction rules in λComp are called βc-reduction rules in [1], which, apart from standard β-reduction in the λ-calculus, contains especially the following two rules for computations: let x ⇐ val(t1 ) in t2 →βc t2 [t1 /x], (4) let x2 ⇐ (let x1 ⇐ t1 in t2 ) in t →βc let x1 ⇐ t1 in (let x2 ⇐ t2 in t).(5) With respect to the βc rules, every term can be reduced to a term in the βc-normal form. Considering also the following η-equality rule for monadic types [1]: let x ⇐ t in t′ [val(x)/x′ ] =η t′ [t/x′ ], (6) we can write every term of a monadic type in the following βc-normal η-long form let x1 ⇐ d1 u11 · · · u1k1 in · · · let xn ⇐ dn un1 · · · unkn in val(u), where n = 0, 1, 2, . . ., every di (1 ≤ i ≤ n) is either a constant or a variable, u and uij (1 ≤ i ≤ n, 1 ≤ j ≤ kj ) are all βc-normal terms or βc-normal-η-long terms (of monadic types). In fact, the rules (4-6) just identify the equations (1-3) respectively. Lemma 1. For every term t of type Tτ in λComp , there exists a βc-normal-η-long term t′ such that Jt′ Kρ = JtKρ, for every valid interpretation J_Kρ (i.e., interpretations satisfying the equations (1-3)). Proof. Because the computational λ-calculus is strongly normalizing, we consider the βc-normal form of term t and prove it by the structural induction on t. – If t is either a variable, a constant or an application, according to the equation (3): JtKρ = Jlet x ⇐ t in val(x)Kρ. In particular, if t is an application t1 t1 , then t1 must be either a variable or a constant since t is βc-normal. Therefore, the term let x ⇐ t in val(x) is in the βc-normalη-long form. – If t is a trivial computation val(t′ ), by induction there is a βc-normal-η-long term t′′ such that Jt′ Kρ = Jt′′ Kρ, for every valid ρ, then Jval(t′ )Kρ = Jval(t′′ )Kρ as well. – If t is a sequential computation let x ⇐ t1 in t2 , since it is βc-normal, t1 should not be any val or let term — t1 must be of the form du1 · · · un (n = 0, 1, 2, . . .) with d either a variable or a constant. By induction, there is a βc-normal-η-long term t′2 such that Jt2 Kρ = Jt2 Kρ, for every valid ρ, then JtKρ = Jlet x ⇐ t′1 in t′2 Kρ and the latter is in the βc-normal-η-long form. ⊓ ⊔ 3.2 Contextual equivalence for λComp As argued in [3], the standard notion of contextual equivalence does not fit in the setting of the computational λ-calculus. In order to define contextual equivalence for λComp , we have to consider contexts C of type To (o is an observation type), not of type o. Indeed, contexts should be allowed to do some computations: if they were of type o, they could only return values. In particular, a context C such that x : Tτ ⊢ C : o is derivable, meant to observe computations of type τ , cannot observe anything, because the typing rule for the let construct only allows us to use computations to build other computations, never values. Taking this into account, we get the following definition: Definition 1 (Contextual equivalence for λComp ). In λComp , two values a1 , a2 ∈ Jτ K are contextually equivalent, written as a1 ≈τ a2 , if and only if, for all observable types o ∈ Obs and contexts C such that x : τ ⊢ C : To is derivable, JCK[x := a1 ] = JCK[x := a2 ]. Two closed terms t1 and t2 of type τ are contextually equivalent if and only if Jt1 K ≈τ Jt2 K. We use the same notation ≈τ to denote the contextual equivalence for terms. 3.3 Logical relations for λComp A uniform framework for defining logical relations relies on the categorical notion of subscones [9], and a natural extension of logical relations able to deal with monadic types was introduced in [2]. The construction consists in lifting the CCC structure and the strong monad from the categorical model to the subscone. We reformulate this construction in the category Set. The subscone is the category whose objects are binary relations (A, B, R ⊆ A × B) where A and B are sets; and a morphism between two objects (A, B, R ⊆ A × B) and (A′ , B ′ , R′ ⊆ A′ × B ′ ) is a pair of functions (f : A → A′ , g : B → B ′ ) preserving relations, i.e. a R b ⇒ f (a) R′ g(b). The lifting of the CCC structure gives rise to the standard logical relations given in Section 2.2 and the lifting of the strong monad will give rise to relations for monadic types. We write T̃ for the lifting of the strong monad T . Given a relation R ⊆ A × B and two computations a ∈ T A and b ∈ T B, (a, b) ∈ T̃ (R) if and only if there exists a computation c ∈ T (R) (i.e. c computes pairs in R) such that a = T π1 (c) and b = T π2 (c). The standard definition of logical relation for the simply typed λ-calculus is then extended with: (c1 , c2 ) ∈ RTτ ⇐⇒ (c1 , c2 ) ∈ T̃ (Rτ ). (7) This construction guarantees that Basic Lemma always holds provided that every constant is related to itself [2]. A list of instantiations of the above definition in concrete monads is also given in [2]. Figure 2 cites the relations for those monads defined in Figure 1. Partial computation: Exception: State transformer: Continuation: Non-determinism: c1 RTτ c2 ⇔ c1 Rτ c2 or c1 = c2 = ⊥ c1 RTτ c2 ⇔ c1 Rτ c2 or c1 = c2 ∈ E where E is the set of exceptions c1 RTτ c2 ⇔ ∀s ∈ St . π1 (c1 s) Rτ π1 (c2 s) & π2 (c1 s) = π2 (c2 s) where St is the set of states c1 RTτ c2 ⇔ c1 (k1 ) = c2 (k2 ) for every k1 , k2 such that ∀a1 , a2 . a1 Rτ a2 =⇒ k1 (a1 ) = k2 (a2 ) c1 RTτ c2 ⇔ (∀a1 ∈ c1 . ∃a2 ∈ c2 . a1 Rτ a2 ) & (∀a2 ∈ c2 . ∃a1 ∈ c1 . a1 Rτ a2 ) Fig. 2. Logical relations for concrete monads We restrict our attention to logical relations (Rτ )τ type such that, for any observation type o ∈ Obs, RTo is a partial equality. Such relations are called observational in the rest of the paper. Note that we require partial identity on To, not on o. But if we assume that denotation of val(_), i.e., the unit operation η, is injective, then that RTo is a partial equality implies that Ro is a partial equality as well. Indeed, let a1 Ro a2 , and by Basic Lemma, Jval(x)K[x := a1 ] RTo Jval(x)K[x := a2 ], that is to say ηJoK (a1 ) = ηJoK (a2 ). By injectivity of η, a1 = a2 . Theorem 1 (Soundness of logical relations in λComp ). If (Rτ )τ tional logical relation, then Rτ ⊆ ≈τ for every type τ . type is an observa- It is straightforward from the Basic Lemma. 3.4 Toward a proof on completeness of logical relations for λComp Completeness of logical relations for λComp is much subtler than in λ→ due to the introduction of monadic types. We were expecting to find a general proof following the general construction defined in [2]. However, this turns out extremely difficult although it might not be impossible with certain restrictions, on types for example. The difficulty arises mainly from the different semantics for different forms of computations, which actually do not ensure that equivalent programs in one monad are necessarily equivalent in another. For instance, consider the following two programs in λComp : let x ⇐ t1 in let y ⇐ t2 in val(x), let y ⇐ t2 in let x ⇐ t1 in val(x), where both t1 and t2 are closed term. We can conclude that they are equivalent in the non-determinism monad — they return the same set of possible results of t1 , no matter what results t2 produces, but this is not the case in, e.g., the exception monad when t1 and t2 throw different exceptions. Being with such an obstacle, we shall switch our effort to case studies in Section 4 and we explore the completeness of logical relations for a list of common monads, precisely, all the monads listed in Figure 1. But, let us sketch out here a general structure for proving completeness of logical relations in λComp . In particular, our study is still restricted to first-order types, which, in λComp , are defined by the following grammar: τ 1 ::= b | Tτ 1 | b → τ 1 , where b ranges over the set of base types. Similarly as in Proposition 1 in Section 2.2, we investigate completeness in a strong sense: we aim at finding an observational logical relation (Rτ )τ type such that if ⊢ t1 : τ and ⊢ t2 : τ are derivable and t1 ≈τ t2 , for any type τ up to first order, then Jt1 K Rτ Jt2 K. Or briefly, ∼τ ⊆ Rτ , where ∼τ is the relation defined in Section 2. As in the proof of Proposition 1, the logical relation (Rτ )τ type will be induced by Rb = ∼b , for any base type b. Then how to prove the completeness for an arbitrary monad T ? Note that we should also check that the logical relation (Rτ )τ type , induced by Rb = ∼b , is observational, i.e., a partial equality on To, for any observable type o. Consider any pair (a, b) ∈ RTo = T̃ (Ro ). By definition of the lifted monad T̃ , there exists a computation c ∈ T Ro such that a = T π1 (c) and b = T π2 (c). But Ro = ∼o ⊆ idJoK , hence the two projections π1 , π2 : Ro → JoK are the same function, π1 = π2 , and consequently a = T π1 (c) = T π2 (c) = b. This proves that RTo is a partial equality. As usual, the proof of completeness would go by induction over τ , to show ∼τ ⊆ Rτ for each first-order type τ . Cases τ = b and τ = b → τ ′ go identically as in λ→ . The only difficult case is τ = Tτ ′ , i.e., the induction step: ∼τ ⊆ Rτ =⇒ ∼Tτ ⊆ RTτ (8) We did not find any general way to show (8) for an arbitrary monad. Instead, in the next section we prove it by cases, for all the monads in Figure 1 except the non-determinism monad. The non-determinism monad is an exceptional case where we do not have completeness for all first-order types but a subset of them. This will be studied separately in Section 4.3. At the heart of the difficulty of showing (8), we find an issue of definability at monadic types in the set-theoretical model. We write def τ for the subset of definable elements in Jτ K, and we eventually show that the relation between def Tτ and def τ can be shortly spelled-out: def Tτ ⊆ T def τ (9) for all the monads we consider in this paper. This is a crucial argument for proving completeness of logical relations for monadic types, but to show (9), we need different proofs for different monads. This is detailed in Section 4.1. 4 Completeness of logical relations for monadic types 4.1 Definability in the set-theoretical model of λPESCN Comp As we have seen in λ→ , definability is involved largely in the proof of completeness of logical relations (for first-order types). This is also the case in λComp and it apparently needs more concern due to the introduction of monadic types. Despite we did not find a general proof for (9), it does hold for all the concrete monads in λPESCN Comp . To state it formally, let us first define a predicate Pτ on elements of Jτ K, by induction on types: – Pb = def b , for every base type b; – PTτ = T (def τ ∩ Pτ ); – Pτ →τ ′ = {f ∈ Pτ →τ ′ | ∀a ∈ def τ , f (a) ∈ Pτ ′ }. We say that a constant c (of type τ ) is logical if and only if τ is a base type or JcK ∈ Pτ . ESCN We then require that λP contains only logical constants. Note that this restriction Comp is valid because the predicates PT τ and Pτ →τ ′ depend only on definability at type τ . Some typical logical constants for monads in λPESCN are as follows: Comp – Partial computation: a constant Ωτ of type Tτ , for every τ . Ωτ denotes the nontermination, so JΩτ K = ⊥. – Exception: a constant raiseeτ of type Tτ for every type τ and every exception e ∈ E. raiseeτ does nothing but raises the exception e, so Jraiseeτ K = e. – State transformer: a constant updates of type Tunit for every state s ∈ St, where unit is the base type which contains only a dummy value ∗. updates simply changes the current state to s, so for any s′ ∈ St, Jupdates K(s′ ) = (∗, s). – Continuation: a constant callkτ of type τ → T bool for every τ and every continuation k ∈ RJτ K . callkτ calls directly the continuation k — it behaves somehow likey “goto” command, so for any a ∈ Jτ K and any continuation k ′ ∈ Rbool, q callkτ (a)(k ′ ) = k(a). – Non-determinism: a constant +τ of type τ → τ → Tτ for every non-monadic type τ . +τ takes two arguments and returns randomly one of them — it introduces the non-determinism, so for any a1 , a2 ∈ Jτ K, J+τ K(a1 , a2 ) = {a1 , a2 }. 1 We assume in the rest of this paper that the above constants are present in λPESCN Comp . Note that Pτ being a predicate on elements of Jτ K is equivalent to say that Pτ can be seen as subset of Jτ K, but in the case of monadic types, PTτ (i.e., T (def τ ∩ Pτ )) is not necessary a subset of JTτ K (i.e., T Jτ K). Fortunately, we prove that all the monads in preserves inclusions, which ensures that the predicate P is well-defined: λPESCN Comp Proposition 2. All the monads in λPESCN preserve inclusions: A ⊆ B ⇒ T A ⊆ T B. Comp Proof. We check it for every monad in λPESCN Comp : – Partial computation: according to the monad definition, if A ⊆ B, then for every c ∈ T A: c ∈ T A ⇐⇒ c ∈ A or c = ⊥ =⇒ c ∈ B or c = ⊥ ⇐⇒ c ∈ T B. – Exception: for every element c ∈ T A: c ∈ T A ⇐⇒ c ∈ A or c ∈ E =⇒ c ∈ B or c ∈ E ⇐⇒ c ∈ T B. – State transformer: for every a ∈ T A: c ∈ T A ⇐⇒ ∀s ∈ St . π1 (cs) ∈ A =⇒ ∀s ∈ St . π1 (cs) ∈ B ⇐⇒ c ∈ T B. A – Continuation: this is a special case because apparently T A = RR is not a subset of B T B = RR , since they contain functions that are defined on different domains, but we shall consider here the functions coinciding on the smaller set A as equivalent. We say that two functions f1 and f2 defined on a domain B coincide on A (A ⊆ B), written as f1 |A = f2 |A , if and only if for every x ∈ A, f1 (x) = f2 (x). Then for every c ∈ T A: ∀k1 , k2 ∈ RB . k1 = k2 =⇒ k1 |A = k2 |A =⇒ c(k1 ) = c(k2 ), so c is also function from RB to R, i.e., c ∈ T B. – Non-determinism: for every c ∈ T A: c ∈ T A ⇐⇒ ∀a ∈ c . a ∈ A =⇒ ∀a ∈ c . a ∈ B ⇐⇒ c ∈ T B. ⊓ ⊔ Introducing such a constraint on constants is mainly for proving (9). Let us figure out the proof. Take an arbitrary element c in def Tτ . By definition, there exists a closed term t of type Tτ such that JtK = c. While it is not evident that c ∈ T def τ , we are expecting to show that JtK ∈ T def τ , by considering the βc-normal-η-long form of t, since 1 It is easy to check that each of these constants is related to itself, except callkτ for continuations. However, we still assume the presence of callkτ for the sake of proving completeness, while we are not able to prove the soundness with it. Note that Theorem 1 and Theorem 2 still hold, but they are not speaking of the same language. λComp is strongly normalizing, Take the partial computation monad as an example, where T def τ = def τ ∪ {⊥}. Consider the βc-normal-η-long form of t: let x1 ⇐ d1 u11 · · · u1k1 in · · · let xn ⇐ dn un1 · · · unkn in val(u). We shall make the induction on n. It is clear that JtK ∈ T def τ when n = 0. For the induction step, we hope that the closed term d1 u11 · · · u1k1 (of type Tτ1 ) would produce either ⊥ (the non-termination), or a definable result (of type τ1 ) so that we can substitute x1 in the rest of the normal term with the result of d1 u11 · · · u1k1 and make use of induction hypothesis. The constraint on constants helps here: to ensure that after the substitution, the resulted term is still in the proper form so that the induction would go through. The following lemma shows that for every computation term t, JtK ∈ T def τ if t is in a particular form, which is a more general form of βc-normal-η-long form. Lemma 2. In λPESCN Comp , JtK ∈ T def τ , for every closed computation term t (of type Tτ ) of the following form: t ≡ let x1 ⇐ t1 w11 · · · w1k1 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w), where n = 0, 1, 2, . . . and ti (1 ≤ i ≤ n) is either a variable or a closed term such that P(Jti K) holds, and w, wij (1 ≤ i ≤ n, 1 ≤ j ≤ ki ) are valid λPESCN terms. Comp Proof. We prove it by induction on n, for every monad: – Partial computation (T def τ = def τ ∪ {⊥}): if n = 0, it is clear that JtK ∈ T def τ . When n > 0, because P(Jt1 K) holds (t1 must be closed), Jt1 w11 · · · w1k1 K ∈ T (def τ1 ∩ Pτ1 ). If Jt1 w11 · · · w1k1 K = ⊥, then JtK = ⊥ ∈ T def τ ; otherwise, assume that Jt′1 K = Jt1 w11 · · · w1k1 K where t′1 is a closed term of type τ1 (assuming that t1 w11 · · · w1k1 is of type Tτ1 ). According to the definition of P, P(Jt′1 K) holds. Let t′ be another closed term: ′ ′ ′ ′ in val(w[t′1 /x1 ]), in · · · let xn ⇐ t′n wn1 · · · wnk t′ ≡ let x2 ⇐ t′2 w21 · · · w2k n 2 ′ where t′i (2 ≤ i ≤ n) is either t′1 or ti , wij ≡ wij [t′1 /x1 ] (2 ≤ i ≤ n, 1 ≤ j ≤ ki ). ′ By induction, Jt K ∈ T def τ holds. Furthermore, Jt′ K = Jlet x2 ⇐ t2 w21 · · · w2k2 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K[x1 := Jt′1 K] = Jlet x1 ⇐ t1 w11 · · · w1k1 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K = JtK, hence JtK ∈ T def τ . – Exception (T def τ = def τ ∪ E): if n = 0, clearly JtK ∈ T def τ . When n > 0, because P(Jt1 K) holds, Jt1 w11 · · · w1k1 K ∈ T (def τ1 ∩ Pτ1 ). If Jt1 w11 · · · w1k1 K ∈ E, then JtK ∈ E ⊆ T def τ ; otherwise, exactly as in the case of partial computation, build a term t′ . Similarly, we prove that JtK = Jt′ K ∈ T def τ by induction. – State transformer (T def τ = (def τ × St)St ): when n = 0, for every s ∈ St, π 1 (JtKs) = JwK ∈ def τ hence JtK ∈ T def τ . When n > 0, for every s ∈ St, assume that Jts1 K = π 1 (Jt1 w11 · · · w1k1 Ks) where t′1 is a closed term of type τ1 (assuming that t1 w11 · · · w1k1 is of type Tτ1 ). According to the definition of P, P(Jts1 K) holds. Let ts be another closed term: s s s s in val(w[ts1 /x1 ]), in · · · let xn ⇐ tsn wn1 · · · wnk ts ≡ let x2 ⇐ ts2 w21 · · · w2k n 2 s where tsi (2 ≤ i ≤ n) is either ts1 or ti , wij ≡ wij [ts1 /x1 ] (2 ≤ i ≤ n, 1 ≤ j ≤ ki ). s By induction, Jt K ∈ T def τ holds. Furthermore, for every s ∈ St, JtKs = Jlet x1 ⇐ t1 w11 · · · w1k1 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)Ks = (Jlet x2 ⇐ t2 w21 · · · w2k2 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K[x1 := Jts1 K])s′ s ′ = Jt Ks , where s′ = π2 (Jt1 w11 · · · w1k1 Ks). Since Jts K ∈ T def τ for every s ∈ St, π1 (JtKs) = π1 (Jts Ks′ ) ∈ def τ , hence JtK ∈ T def τ . Jτ K def τ – Continuation (T def τ = RR ): we say that an element c ∈ JTτ K = RR is in Jτ K T def τ if and only if for every pair of continuations k1 , k2 ∈ R , k1 |def τ = k2 |def τ =⇒ c(k1 ) = c(k2 ). If n = 0, JtK = λ k.k(JwK) ∈ T def τ . When n > 0, according to the definition of the continuation monad: JtK = λ k · Jt1 w11 · · · wnkn K(k ′ ), where k ′ = λ a·(Jlet x2 ⇐ t2 w21 · · · w2k2 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K[x1 := a])k. For every continuations k1 , k2 ∈ RJτ K such that k1 |def τ = k2 |def τ let ki′ = λ a·(Jlet x2 ⇐ t2 w21 · · · w2k2 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K[x1 := a])ki , i = 1, 2. Because Jt1 w11 · · · w1k1 K ∈ T (Pτ1 ∩def τ1 ), if we can prove k1′ |Pτ1 ∩def τ1 = k2′ |Pτ1 ∩def τ1 , which implies JtK(k1 ) = JtK(k2 ), we can conclude JtK ∈ T def τ . For every a ∈ Pτ1 ∩def τ1 , let Jta1 K = a where ta1 is a closed term. Define another closed term ta : a a a a in val(w[ta1 /x1 ]), in · · · let xn ⇐ tan wn1 · · · wnk ta ≡ let x2 ⇐ ta2 w21 · · · w2k n 2 a where tai (2 ≤ i ≤ n) is either ta1 or ti , wij ≡ wij [ta1 /x1 ] (2 ≤ i ≤ n, 1 ≤ a ′ j ≤ ki ). By induction, Jt K ∈ T def τ , so k1 (a) = Jta Kk1 = Jta Kk2 = k2′ (a), i.e., k1′ |Pτ1 ∩def τ1 = k2′ |Pτ1 ∩def τ1 . – Non-determinism (T def τ = Pfin (def τ )): when n = 0, JtK = {JwK} ∈ T def τ . When n > 0, for every a ∈ Jt1 w11 · · · w1k1 K, assume that Jta1 K = a where t′1 is a closed term of type τ1 (assuming that t1 w11 · · · w1k1 is of type Tτ1 ). According to the definition of P, P(Jta1 K) holds. Let ta be another closed term: a a a a in val(w[ta1 /x1 ]), in · · · let xn ⇐ tan wn1 · · · wnk ta ≡ let x2 ⇐ ta2 w21 · · · w2k n 2 a where tai (2 ≤ i ≤ n) is either ta1 or ti , wij ≡ wij [ta1 /x1 ] (2 ≤ i ≤ n, 1 ≤ j ≤ ki ). a By induction, Jt K ∈ T def τ holds. Furthermore, JtK = Jlet S x1 ⇐ t1 w11 · · · w1k1 in · · · let xn ⇐ tn wn1 · · · wnkn in val(w)K Jlet x2 ⇐ t2 w21 · · · w2k2 in · · · = a∈Jt1 K xn ⇐ tn wn1 · · · wnkn in val(w)K[x1 := a] S let = Jta K. a∈Jt1 K Because Jta K ∈ T def τ holds for every a ∈ Jt1 w11 · · · w1k1 K, JtK ∈ T def τ . ⊓ ⊔ From the above lemma, we conclude immediately that for every closed βc-normalη-long computation term t in λPESCN with logical constants, JtK ⊆ T def τ . Comp Proposition 3. def Tτ ⊆ T def τ holds in the set-theoretical model of λPESCN with Comp logical constants. Proof. It follows from Lemma 2 by considering the βc-normal-η-long terms that define elements in JTτ K since λComp is strongly normalizing. ⊓ ⊔ 4.2 Completeness of logical relations in λPESC Comp for first-order types We prove (8) in this section for the partial computation monad, the exception monad, the state monad and the continuation monad. We write λPESC Comp for λComp where the monad is restricted to one of these four monads. Proofs depend typically on the particular semantics of every form of computation, but a common technique is used frequently: given two definable but non-related elements of JTτ K, one can find a context to distinguish the programs (of type Tτ ) that define the two given elements, and such a context is usually built based on another context that can distinguish programs of type τ . Lemma 3. Let (Rτ )τ type be a logical relation in λPESC Comp with only logical constants. ∼τ ⊆ Rτ =⇒ ∼Tτ ⊆ RTτ holds for every type τ . Proof. Take two arbitrary elements c1 , c2 ∈ JTτ K such that (c1 , c2 ) 6∈ RTτ , we prove that c1 6∼Tτ c2 for every monad in λPESC Comp : – Partial computation: the fact (c1 , c2 ) 6∈ RTτ amounts to the following two cases: • c1 , c2 ∈ Jτ K but (c1 , c2 ) 6∈ Rτ , then c1 6∼τ c2 . If one of these two values is not definable at type τ , by Proposition 3, it is not definable at type Tτ either. If both values are definable at type τ but they are not contextually equivalent, then there is a context x : τ ⊢ C : To such that JCK[x := c1 ] 6= JCK[x := c2 ]. Thus, the context y : Tτ ⊢ let x ⇐ y in C : To can distinguish c1 and c2 (as two values of type Tτ ). • c1 ∈ Jτ K and c2 = ⊥ (or symmetrically, c1 = ⊥ and c2 ∈ Jτ K), then the context let x ⇐ y in val(true) can be used to distinguish them. c1 6∼Tτ c2 in both cases. – Exception: the fact (c1 , c2 ) 6∈ RTτ amounts to three cases: • c1 , c2 ∈ Jτ K but (c1 , c2 ) 6∈ Rτ , then c1 6∼τ c2 . Suppose both values are definable at type τ , otherwise by Proposition 3, they must not be definable at type Tτ . Similar as in the case of partial computation we can build a context that distinguishes c1 and c2 as values of type Tτ , from the context that distinguishes c1 and c2 as values of type τ . • c1 ∈ Jτ K, c2 ∈ E. Consider the following context: y : Tτ ⊢ let x ⇐ y in val(true) : Tbool. When y is substituted by c1 and c2 , the context evaluates to different values, namely, a boolean and an exception. • c1 , c2 ∈ E but c1 6= c2 . Try the same context as in the second case, which will evaluate to two different exceptions that can be distinguished. c1 6∼Tτ c2 in all the three cases. – State transformer: because (c1 , c2 ) 6∈ RTτ , there exists some s0 ∈ St such that • either (π1 (c1 s0 ), π1 (c2 s0 )) 6∈ Rτ . Then by induction π1 (c1 s0 ) 6∼τ π1 (c2 s0 ). If π1 (ci s0 ) (i = 1, 2) is not definable, then by Proposition 3, ci is not definable either. If both π1 (c1 s0 ) and π1 (c2 s0 ) are definable, but π1 (c1 s0 ) 6≈τ π1 (c2 s0 ), then there is a context x : τ ⊢ C : To such that JCK[x := π1 (c1 s0 )] 6= JCK[x := π1 (c2 s0 )], i.e., for some state s′0 ∈ St , JCK[x := π1 (c1 s0 )](s′0 ) 6= JCK[x := π1 (c1 s0 )](s′0 ). Now we can use the following context: y : Tτ ⊢ let x ⇐ y in let z ⇐ updates′0 in C : T o, Let fi = s ∈ St, r z let x ⇐ y in let z ⇐ updates′0 in C [y := ci ], then for every r z fi (s) = let z ⇐ updates′0 in C [x := π1 (ci s)](π2 (ci s)) = JCK[x := π1 (ci s)](s′0 ), (i = 1, 2). f1 6= f2 , because when applied to the state s0 , they will return two different pairs, so the above context can distinguish the two values c1 and c2 ; • or π2 (c1 s0 ) 6= π2 (c2 s0 ). we use the context y : Tτ ⊢ let x ⇐ y in val(true) : Tbool, then Jlet x ⇐ y in val(true)K[y := ci ] = λs.(true, π2 (ci s)) (i = 1, 2). These two functions are not equal since they return different results when applied to the state s0 . In both cases, c1 6∼Tτ c2 . – Continuation: first say that two continuations k1 , k2 ∈ RJτ K are R-related, if and only if for every a1 , a2 ∈ Jτ K, a1 Rτ a2 =⇒ k1 (a1 ) = k2 (a2 ). The fact (c1 , c2 ) 6∈ RTτ means that there are two R-related continuations k1 , k2 such that c1 (k1 ) 6= c2 (k2 ). Because ∼τ ⊆ Rτ , for every definable value a ∈ def τ , clearly, a ∼τ a =⇒ a1 R a2 =⇒ k1 (a1 ) = k2 (a2 ), so k1 and k2 coincide over def τ . Suppose that both c1 and c2 are definable, then by Proposition 3, c1 (k1 ) = c1 (k2 ) and c2 (k1 ) = c2 (k2 ), hence c1 (k1 ) 6= c2 (k1 ). Consider the context y : Tτ ⊢ let x ⇐ y in callkτ 1 (x) : T bool. For every k ∈ RJboolK , q y k1 let x ⇐ q y inkcallyτ (x) [y := ci ](k) = ci (λ a · ( callτ 1 (x) [x := a])k) = ci (λ a · k1 (a)) = ci (k1 ). (i = 1, 2), Since c1 (k1 ) 6= c2 (k1 ), this context distinguishes the two computations, hence c1 6∼Tτ c2 . ⊓ ⊔ Theorem 2. In λPESC Comp , if all constants are logical and in particular, if the following constants are present – updates for the state transformer monad; – callkτ for the continuation monad, then logical relations are complete up to first-order types, in the strong sense that there exists an observational logical relation (Rτ )τ type such that for any closed terms t1 , t2 of any type τ 1 up to first order, if t1 ≈τ 1 t2 , then Jt1 K Rτ 1 Jt2 K. Proof. Take the logical relation (Rτ )τ type induced by Rb =∼b , for any base type b. We prove by induction on types that ∼τ ⊆ Rτ for any first-order type τ . In particular, the induction step ∼τ ⊆ Rτ =⇒∼Tτ ; ⊆ RTτ is shown by Lemma 3. ⊓ ⊔ 4.3 Completeness of logical relations for the non-determinism monad The non-determinism monad is an interesting case: the completeness of logical relations for this monad does not hold for all first-order types! To state it, consider the following two programs of a first-order type that break the completeness of logical relations: ⊢ val(λx.(true +bool false)) : T(bool → Tbool), ⊢ λx.val(true) +bool→Tbool λx.(true +bool false) : T(bool → Tbool). Recall the logical constant +τ of type τ → τ → Tτ : J+τ K(a1 , a2 ) = {a1 , a2 } for every a1 , a2 ∈ Jτ K. The two programs are contextually equivalent: what contexts can do is to apply the functions to some arguments and observe the results. But no matter how many time we apply these two functions, we always get the same set of possible values ({true, false}), so there is no way to distinguish them with a context. Recall the logical relation for non-determinism monad in Figure 2: c1 RTτ c2 ⇔ (∀a1 ∈ c1 . ∃a2 ∈ c2 . a1 Rτ a2 ) & (∀a2 ∈ c2 . ∃a1 ∈ c1 . a1 Rτ a2 ). Clearly the denotations of the above two programs are not related by that relation because the function Jλx.val(true)K from the second program is not related to the function in the first. However, if we assume that for every non-observable base type b, there is an equality test constant testb : b → b → bool (clearly, P(testb ) holds), logical relations for the non-determinism monad are then complete for a set of weak first-order types: τw1 ::= b | Tb | b → τw1 . Compared to all types up to first order, weak first-order types do not contain monadic types of functions, so it immediately excludes the two programs in the above counterexample. Theorem 3. Logical relations for the non-determinism monad are complete up to weak first-order types. in the strong sense that there exists an observational logical relation (Rτ )τ type such that for any closed terms t1 , t2 of a weak first-order type τw1 , if t1 ≈τw1 t2 , then Jt1 K Rτw1 Jt2 K. Proof. Take the logical relation R induced by Rb =∼b , for any base type b. We prove by induction on types that ∼τw1 ⊆ Rτ 1 for any weak first-order type τw1 . w Cases b and b → τw1 go identically as in standard typed lambda-calculi. For monadic types Tb, suppose that (c1 , c2 ) 6∈ RTb , which means either there is a value in c1 such that no value of c2 is related to it, or there is such a value in c2 . We assume that every value in c1 and c2 is definable (otherwise it is obvious that c1 6∼Tb c2 because at least one of them is not definable, according to Proposition 3). Suppose there is a value a ∈ c1 such that no value in c2 is related to it, and a can be defined by a closed term t of type b. Then the following context can distinguish c1 and c2 : x : Tτ ⊢ let y ⇐ x in testb (y, t) : Tbool since every value in c2 is not contextually equivalent to a, hence not equal to a.  Now let state and label be base types such that label is an observation type, whereas state is not. Using non-determinism monad, we can define labeled transition systems as elements of Jstate → label → TstateK, with states in JstateK and labels in JlabelK, as functions mapping states a and labels l to the set of states b such that l / b . The logical relation at type state → label → Tstate is given by [2]: a (f1 , f2 ) ∈ Rstate→label→Tstate ⇐⇒ ∀a1 , a2 , l1 , l2 · (a1 , a2 ) ∈ Rstate & (l1 , l2 ) ∈ Rlabel =⇒ (∀b1 ∈ f1 (a1 , l1 ) · ∃b2 ∈ f2 (a2 , l2 ) · (b1 , b2 ) ∈ Rstate ) & (∀b2 ∈ f2 (a2 , l2 ) · ∃b1 ∈ f1 (a1 , l1 ) · (b1 , b2 ) ∈ Rstate ) In case Rlabel is equality, f1 and f2 are logically related if and only if Rstate is a strong bisimulation between the labeled transition systems f1 and f2 . Sometimes we explicitly specify an initial state for certain labeled transition system. In this case, the encoding of the labeled transition system in the nondeterminism monad is a pair (q, f ) of Jstate × (state → label → Tstate)K, where q is the initial state and f is the transition relation as defined above. Then (q1 , f1 ) and (q2 , f2 ) are logically related if and only if they are strongly bisimular, i.e., Rstate is a strong bisimulation between the two labeled transition systems and q1 Rstate q2 . Corollary 1 (Soundness of strong bisimulation). Let f1 and f2 be transition systems. If there exists a strong bisimulation between f1 and f2 , then f1 and f2 are contextually equivalent. Proof. There exists a strong bisimulation between f1 and f2 , therefore f1 and f2 are logically related. By Theorem 1, f1 and f2 are thus contextually equivalent.  In order to prove completeness, we need to assume that label has no junk, in the sense that every value of JlabelK is definable. Corollary 2 (Completeness of strong bisimulation). Let f1 and f2 be transition systems which are definable. If f1 and f2 are contextually equivalent and label has no junk, then there exists a strong bisimulation between f1 and f2 . Proof. Let R be the logical relation given by Theorem 3. f1 and f2 are definable and contextually equivalent, so f1 Rstate→label→Tstate f2 . Moreover, because label has no junk, Rlabel is equality. Rstate is thus a strong bisimulation between f1 and f2 . ⊓ ⊔ 5 Conclusion The work presented in this paper is a natural continuation of the authors’ previous work [2,3]. In [2], we extend [9] and derive logical relations for monadic types which are sound in the sense that the Basic Lemma still holds. In [3], we study contextual equivalence in a specific version of the computational λ-calculus with cryptographic primitives and we show that lax logical relations (the categorical generalization of logical relations [14]) derived using the same construction is complete. Then in this paper, we explore the completeness of logical relations for the computational λ-calculus and we show that they are complete at first-order types, for a list of common monads: partial computations, exceptions, state transformers and continuations, while in the case of continuation, the completeness depends on a natural constant call, with which we cannot show the soundness. Pitts and Stark have defined operationally based logical relations to characterize the contextual equivalence in a language with local store [13]. This work can be traced back to their early work on the nu-calculus [12] which can be translated in a special version of the computational λ-calculus and be modeled using the dynamic name creation monad [17]. Logical relations for this monad are derived in [19] using the construction from [2]. It is also shown in [19] that such derived logical relations are equivalent to Pitts and Stark’s operational logical relations up to second-order types. An exceptional case of our completeness result is the non-determinism monad, where logical relations are not complete for all first-order types, but a subset of them. We effectively show this by providing a counter-example that breaks the completeness at first-order types. This is indeed an interesting case. A more comprehensive study on this monad can be found in [4], where Jeffrey defines a denotational model for the computational λ-calculus specialized in non-determinism and proves that this model is fully abstract for may-testing. The relation between our notion of contextual equivalence and the may-testing equivalence remains to be clarified. Recently, Lindley and Stark introduce the syntactic ⊤⊤-lifting for the computational λ-calculus and prove the strong normalization [7]. Katsumata then instantiates their liftings in Set [5]. The ⊤⊤-lifting of strong monads is an essentially different approach from that in [2]. It would be interesting to establish a formal relationship between these two approaches, and to look for a general proof of completeness using the ⊤⊤-lifting. References 1. P. N. Benton, G. M. Bierman, and V. C. V. de Paiva. Computational types from a logical perspective. J. Functional Programming, 8(2):177–193, 1998. 2. J. Goubault-Larrecq, S. Lasota, and D. Nowak. Logical relations for monadic types. In Proceedings of CSL’2002, volume 2471 of LNCS, pages 553–568. Springer, 2002. 3. J. Goubault-Larrecq, S. Lasota, D. Nowak, and Y. Zhang. Complete lax logical relations for cryptographic lambda-calculi. In Proceedings of CSL’2004, volume 3210 of LNCS, pages 400–414. Springer, 2004. 4. A. Jeffrey. A fully abstract semantics for a higher-order functional language with nondeterministic computation. Theoretical Computer Science, 228(1-2):105–150, 1999. 5. S. Katsumata. A semantic formulation of ⊤⊤-lifting and logical predicates for computational metalanguage. In Proceedings of CSL’2005, volume 3634 of LNCS, pages 87–102. Springer, 2005. 6. R. Lazić and D. Nowak. A unifying approach to data-independence. In Proceedings of CONCUR’2000, volume 1877 of LNCS, pages 581–595. Springer, 2000. 7. S. Lindley and I. Stark. Reducibility and ⊤⊤-lifting for computation types. In Proceedings of TLCA’2005, number 3461 in LNCS, pages 262–277. Springer, 2005. 8. J. C. Mitchell. Foundations of Programming Languages. MIT Press, 1996. 9. J. C. Mitchell and A. Scedrov. Notes on sconing and relators. In Proceedings of CSL’1992, volume 702 of LNCS, pages 352–378. Springer, 1993. 10. E. Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991. 11. P. W. O’Hearn and R. D. Tennent. Parametricity and local variables. J. ACM, 42(3):658–709, 1995. 12. A. Pitts and I. Stark. Observable properties of higher order functions that dynamically create local names, or: What’s new? In Proceedings of MFCS’1993, number 711 in LNCS, pages 122–141. Springer, 1993. 13. A. Pitts and I. Stark. Operational reasoning for functions with local state. In Higher Order Operational Techniques in Semantics, pages 227–273. Cambridge University Press, 1998. 14. G. Plotkin, J. Power, D. Sannella, and R. Tennent. Lax logical relations. In Proceedings of ICALP’2000, volume 1853 of LNCS, pages 85–102. Springer, 2000. 15. G. D. Plotkin. Lambda-definability in the full type hierarchy. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363–373. Academic Press, 1980. 16. K. Sieber. Full abstraction for the second order subset of an algol-like language. Theoretical Computer Science, 168(1):155–212, 1996. 17. I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77–107, 1996. 18. E. Sumii and B. C. Pierce. Logical relations for encryption. J. Computer Security, 11(4):521– 554, 2003. 19. Y. Zhang. Cryptographic logical relations. Ph. d. dissertation, ENS Cachan, France, 2005. 

arXiv:1303.6011v1 [math.FA] 25 Mar 2013 THE INVERSE FUNCTION THEOREM AND THE RESOLUTION OF THE JACOBIAN CONJECTURE IN FREE ANALYSIS J. E. PASCOE Abstract. We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices. 1. Introduction A free map is a function defined on some structured subset of tuples of matrices that respects joint invariance. For the purposes of introduction, the standard example is a free polynomial being evaluated on tuples of matrices. We give a formal defintion in Section 2. We consider the inverse function theorem for free maps. Classically, the inverse function theorem states that given a map f , if Df (x) is nonsingular for some x, then there is a neighborhood U of x such that f −1 is well-defined on f (U ). Our inverse function theorem is as follows. Theorem 1.1 (Inverse function theorem). Let f be a free map. The following are equivalent: (1) Df (X) is a nonsingular map for every X. (2) f is injective. (3) f −1 exists and is a free map. In the classical case, one may obtain a neighborhood of x where the derivative is nonsingular. The geometry of free analysis is not exactly topological due noted algebraic obstructions such as those observed in the work of D. S. KaliuzhnyiVerbovetskyi and V. Vinnikov [9], and Amitsur and Levitzki [2]. Thus, we assert nonsingularity of the derivative on the entire domain in our theorem, and, in return, we obtain a global result. We prove the inverse function theorem result in Section 3. We also consider a famous conjecture of Ott-Heinrich Keller, the so-called Jacobian conjecture [10]. Question 1.2. Let P : CN → CN be a polynomial map. If the Jacobian DP (x) is invertible for every x ∈ CN , is the map P itself invertible? James Ax [3] and Alexander Grothendieck [5] independently showed that if a polynomial map P : CN → CN is injective, then it must be surjective. Furthermore, 2010 Mathematics Subject Classification. Primary 46L52; Secondary 14A25, 47A56. 1 2 J. E. PASCOE a proof in [12] shows that the inverse must be given by a polynomial via techniques from Galois theory. We prove a free Ax-Grothendieck theorem as Theorem 4.4. The following is an immediate corollary of our inverse function theorem combined with the results in the preceeding paragraphs. Theorem 1.3 (Free Jacobian conjecture). Let M(C)N be the set of matrix N tuples. Suppose P : M(C)N → M(C)N is a free polynomial map. The following are equivalent: (1) DP (X) is a nonsingular map for each X. (2) P is injective. (3) P is bijective. (4) P −1 exists and P −1 |Mn (C)N agrees with free polynomial. We prove this result in Section 4. We remark the last condition could be conjectured to be that P −1 is a bona fide free polynomial. However, we caution that degree bounds in the Ax-Grothendieck theorem are very large [4], and the theory of polynomial identity rings supplies many low degree polynomial identities satisfied by all matrix tuples of a specific size [2]. This means that there are a plethora of maps that have polynomial formulas for each specific size of matrix, but are not actually given by some polynomial. Additionally, we immediately obtain a matrix version of the commutative Jacobian conjecture via another application of the inverse function theorem. In this case, we do obtain true polynomials for the inverse map, because commutative polynomials are determined by their values on the scalars. Theorem 1.4 (Commuting matrix Jacobian conjecture). Let P : CN → CN . The following are equivalent: (1) DP (X) is an nonsingular map for each commuting matrix N -tuple X. (2) P is injective. (3) P is bijective. (4) P −1 exists and P −1 is given by a polynomial. We prove this result in Section 4. We caution that the structure of free maps greatly simplifies their geometry. The Jacobian conjecture in the classical context is notoriously difficult, but in the matricial context is shown here to be tractable. Indeed, free maps have generally been observed to encode nonlocal data in many contexts which is in strong contrast to the classical case. For another example, compare between Putinar’s Positivstellensatz [11] to Helton’s noncommutative Positivstellensatz [6]. 1.1. Some examples of domains of invertibility. We briefly give some examples of applications of our main result, the inverse function theorem for free maps. 1.1.1. Domains of inveribility for squaring. Take the function f (X) = X 2 . Suppose we want to find a domain where f is invertible. We obtain a derivative for f given by the formula Df (X)[H] = XH + HX. Thus, by the inverse function theorem, we need the equation XH + HX = 0 THE INVERSE FUNCTION THEOREM 3 to have no nontrivial solutions for each X in our domain. This is a degenerate form of the famous Sylvester equation, so this will be nonsingular if X has no eigenvalues in common with −X. For a detailed account of the Sylvester equation, see Horn and Johnson [8]. Thus, if we take a subset H ⊂ C such that H ∩ −H = ∅, and lift to the set of matrices with spectrum in H, then f will be invertible there. In fact, these are all possible maximal domains for such an inverse. 1.1.2. The quadratic symmetrization map. Consider, f (X, Y ) = (X + Y, X 2 + Y 2 ). Taking the derivative, Df (X, Y )[H, K] = (H + K, HX + XH + KY + Y K). So we need to check the second coordinate of the derivative is nonzero when H = −K. So, we want H(X − Y ) + (X − Y )H = 0 to have no nontrivial solutions. By the same use of Sylvesters equation as in the first example, this exactly says X − Y needs to have spectrum disjoint from Y − X. 1.1.3. A more exotic quadratic map. Now consider the function f (X) = (X + X 2 + [X, Y ], Y + [X, Y ]). Taking the derivative, Df (X, Y )[H, K] = (H + HX + HX + [H, Y ] + [X, K], K + [H, Y ] + [X, K]). Suppose this had a nontrivial solution at some (X, Y ) for (H, K). Either kHk ≥ kKk or kKk ≥ kHk. In the case where kHk ≥ kKk, H 6= 0, and kH + HX + XH + [H, Y ] + [X, K]k ≥ kHk(1 − 4kXk − 2kY k). So, it must be that 1 − 4kXk − 2kY k ≤ 0. In the case where kKk ≥ kHk, K 6= 0, and kK + [H, Y ] + [X, K]k ≥ kKk(1 − 2kXk − 2kY k) So, it must be that 1 − 2kXk − 2kY k ≤ 0. Restricting f to the set of (X, Y ) such that 4kXk + 2kY k < 1 precludes Df from being singular. However, this fails to be a free domain since it is not closed with repect to direct sums. (See Section 2 for a formal definition of a free domain.) However, if we restrict f to the set of (X, Y ) such that kXk < 81 and kY k < 41 we do indeed obtain a free map, and thus by the inverse function theorem, the function f will be invertible there. 2. Free analysis S Let Mn be the n × n matrices. We denote MN = MN n. A free set D ⊂ MN is closed under direct sums and joint similarity. That is, (1) A, B ∈ D ⇒ A ⊕ B ∈ D, −1 (2) A ∈ D ∩ MN AS ∈ D. n , S ∈ GLn ⇒ S 4 J. E. PASCOE Where (A1 , A2 , . . . , AN ) ⊕ (B1 , B2 , . . . , BN ) = (A1 ⊕ B1 , A2 ⊕ B2 , . . . , AN ⊕ BN ), and S −1 (A1 , A2 , . . . , AN )S = (S −1 A1 S, S −1 A2 S, . . . , S −1 AN S). A prototypical example of such a set is the zero set of some free polynomial map. For example, the commuting tuples of matrices form a free set. We define a free domain D ⊂ MN to be either a free set or, if working over a local field, a set that is relatively open in its orbit under conjugation by invertible matrices and is closed under direct sums. Any function on a free domain extends to a function on a free set as in the envelope method described in [9]. A free map f : D → MN̂ obeys the following (1) f (A ⊕ B) = f (A) ⊕ f (B), (2) f (S −1 AS) = S −1 f (A)S, (3) D is a free domain. This definition of free sets and free maps is a direct generalization of the definition given in Helton-Klep-McCullough [7]; in the language of Agler-McCarthy [1] this generalizes functions on basic open sets in their free topology. Essentially, these maps are an emulation of the classical functional calculus for non-commuting tuples of operators. We note that we do not specify a ground ring for the matrices in the general inverse function theorem; it merely needs to have a multiplicative identity. 2.1. Derivatives in free analysis. Helton, Klep and McCullough differentiated free maps in [7]. They obtained the formula     X H f (X) Df (X)[H] (2.1) f = . 0 X 0 f (X) These types of formulas are pervasive throughout the free analysis literature. For other references see Voiculescu [14] and the recently completed tome by D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov [9]. Formula 2.1 can be seen to be true for formal differentiation satisfying Leibniz rule. Thus, we eschew any analytic means for obtaining the derivative, and instead use the above as our definition. That is, our results formally hold over any field, or indeed unital ring, and for sets that may have cusps or other exotic geometric features. We formalize the above in the following proposition. Proposition 2.2. Define  Hi , if i = j D(Xi )[Hj ] = , 0, if i 6= j and require D(f + g)[H] = D(f )[H] + D(g)[H], and D(f g)[H] = D(f )[H]g + f D(g)[H]. Equation 2.1 is satisfied. THE INVERSE FUNCTION THEOREM 5 Proof. We only need to prove this fact on monomials. This is obtained inductively via the following algebra:        m(X) Dm(X)[H] X H X i Hi X i Hi m = 0 m(X) 0 Xi 0 X 0 Xi =   Xi m(X) Xi Dm(X)[H] + Hi m(X) 0 Xi m(X) =   Xi m(X) D(Xi m(X))[H] . 0 Xi m(X)  3. The inverse function theorem We now prove the inverse function theorem. Proof. (¬1 ⇒ ¬2) Suppose Df is singular at some X and some direction H 6= 0. That is, Df (X)[H] = 0. So, applying 2.1 and the direct sum formula,     X H f (X) Df (X)[H] f = 0 X 0 f (X)   f (X) 0 0 f (X) =  X f 0 = 0 X  This equality witnesses the noninjectivity of f. (1 ⇒ 2) Suppose Df (X) is not singular at any X. Let X1 , X2 be two matrices of the same size such that f (X1 ) = f (X2 ). Let  1 0  S= 0 0 0 1 0 0 0 0 1 0  1 0 . 0 1 Let,  X1 0 X = 0 0 0 X2 0 0 0 0 X1 0  0 0 . 0 X2  X1  0 S −1 XS =  0 0 0 X2 0 0 0 0 X1 0  X1 − X2  0 ,  0 X2 Note, 6 J. E. PASCOE and, since  f (X1 ) 0  0 f (X 2) f (X) =   0 0 0 0  0 0   0  f (X2 ) 0 0 f (X1 ) 0 because f preserves direct sums, we obtain the fomula  f (X1 ) 0 0  0 f (X2 ) 0 −1  S f (X)S =  0 0 f (X1 ) 0 0 0  f (X1 ) − (X2 )  0 .  0 f (X2 ) So, via the similarity relation for free maps,  X1 0 f 0 0 0 X2 0 0 0 0 X1 0    f (X1 ) 0 0 f (X1 ) − f (X2 ) X1 − X2    0 f (X2 ) 0 0 0 . =    0 0 f (X1 ) 0 0 0 0 0 f (X2 ) X2 Thus, since we assumed f (X1 ) = f (X2 ),  X1 0 f 0 0 0 X2 0 0 0 0 X1 0    f (X1 ) 0 0 0 X1 − X2   0 0 f (X2 ) 0 0  = .   0 0 0 f (X1 ) 0  X2 0 0 0 f (X2 ) On the other hand, by 2.1,  X1 0 f 0 0 0 X2 0 0 0 0 X1 0        0 X1 − X2 X1 − X2 X1 0 X1 0 Df  f 0 X2  0 0  X2 0  0 = .    0 X1 0 0 f X2 0 X2 So, Df  X1 0 0 X2     0 X1 − X2 0 0 = . 0 0 0 0 Thus, X1 − X2 = 0, since we assumed Df (X)[H] is nonsingular for all X in the domain, or equivalently that Df (X)[H] = 0 implies H = 0. So, f is injective. (2 ⇔ 3) We leave this to the reader. It is similar to a proof in Helton-KlepMcCullough [7].  THE INVERSE FUNCTION THEOREM 7 4. Polynomial maps In this section we recall some classical results from algebraic geometry which we will use to prove the Jacobian conjecture for free polynomials and commuting matrix polynomials, and subsequently give these proofs. Ott-Heinrich Keller infamously suggested the following conjecture. Question 4.1 (Jacobian conjecture). Let P : CN → CN be a polynomial map. If DP (x) is nonsingular for every x, must P necesarily be invertible? Furthermore, can the inverse be taken to be a polynomial? James Ax and Alexander Grothendieck independently proved the following seemingly related result about polynomial maps. Theorem 4.2 (Ax-Grothendieck theorem [3], [5]). Let P : CN → CN be a polynomial map. If P is injective, then P is surjective. This reduces the Jacobian conjecture to showing the condition on the derivative implying global injectivity. Furthermore, this result has been refined to the following. Theorem 4.3 (Ax-Grothendieck theorem[12]). Let P : CN → CN be a polynomial map. If P is injective, then P is surjective and P −1 is given by a polynomial. This tacitly gives an equivalence between a polynomial map being invertible and having a polynomial inverse. This can be lifted to the free case. Theorem 4.4 (Free Ax-Grothendieck theorem). Let P : M(C)N → M(C)N be a free polynomial map. If P is injective, then P is surjective and P −1 |MN is given n by a free polynomial. Proof. For each size of matrix n, we view P as a tuple of dn2 commuting polynomials by replacing the free variables with indeterminant n by n matices. Since P |Mn (C)N is injective, P −1 |Mn (C)N is given by a tuple of dn2 commuting polynomials. Since the global P −1 is a free map by the inverse function theorem and continuous free maps are analytic, they have a power series of free polynomials[9], the restriction P −1 |Mn (C)N must agree with a free polynomial; the terms in the power series for P −1 must eventually vanish on all of Mn (C)N .  We now prove the the Jacobian conjecture for free polynomials, Theorem 1.3. Proof of Theorem 1.3. (1 ⇔ 2) follows from the inverse function theorem. (2 ⇔ 3 ⇔ 4) is the free Ax-Grothendieck theorem.  We now prove the the Jacobian conjecture for commuting matrix polynomials, Theorem 1.4. Proof of Theorem 1.3. (1 ⇔ 2) follows from the inverse function theorem. (2 ⇒ 4) The function P |M1 (C)N has a polynomial inverse P −1 by the AxGrothendieck theorem. Since P |M1 (C)N is equal to P as a polynomial, we indeed obtained global inverse. (The values on the scalars determine a commutative polynomial.) (4 ⇒ 3 ⇒ 2) is trivial.  8 J. E. PASCOE References [1] J. Agler and J.E. McCarthy. On the approximation of holomorphic functions in several noncommuting variables by free polynomials. in progress. [2] A. S. Amitsur and J. Levitzki. Minimal identities for algebras. Proc. Amer. Math. Soc., 1:449–463, 1950. [3] James Ax. The elementary theory of finite fields. Ann. of Math., 88(2):239–271, 1968. [4] Hyman Bass, Edwin H. Connell, and David Wright. The jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc., 7:287–330, 1982. [5] Alexander Grothendieck. Eléments de géométrie algébrique IV. Publications Mathematiques de l’IHES, 1966. [6] Bill Helton. Positive noncommutative polynomials are sums of squares. Ann. of Math., 159:675–694, 2002. [7] J. William Helton, Igor Klep, and Scott McCullough. Proper analytic free maps. Journal of Functional Analysis, 260(5):1476 – 1490, 2011. [8] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. [9] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov. Foundations of Noncommutative Function Theory. ArXiv e-prints, 2012. [10] Ott-Heinrich Keller. Ganze cremona-transformationen. Monatshefte für Mathematik und Physik, 47:299–306, 1939. [11] M. Putinar. Positive polynomials on compact sets. Indiana Math. J., 42:969–984, 1993. [12] Walter Rudin. Injective polynomial maps are automorphisms. American Mathematical Monthly, 102(6):540–543, 1995. [13] Dan-Virgil Voiculescu. Free analysis questions. I: Duality transform for the coalgebra of ∂X:B . International Math. Res. Notices, 16:793–822, 2004. [14] Dan-Virgil Voiculescu. Free analysis questions. II: The Grassmannian completion and the series expansions at the origin. J. Reine Angew. Math., 645:155–236, 2010. 

Hypercube LSH for approximate near neighbors Thijs Laarhoven arXiv:1702.05760v1 [] 19 Feb 2017 IBM Research Rüschlikon, Switzerland mail@thijs.com Abstract. A celebrated technique for finding near neighbors for the angular distance involves using a set of random hyperplanes to partition the space into hash regions [Charikar, STOC 2002]. Experiments later showed that using a set of orthogonal hyperplanes, thereby partitioning the space into the Voronoi regions induced by a hypercube, leads to even better results [Terasawa and Tanaka, WADS 2007]. However, no theoretical explanation for this improvement was ever given, and it remained unclear how the resulting hypercube hash method scales in high dimensions. In this work, we provide explicit asymptotics for the collision probabilities when using hypercubes to partition the space. For instance, two near-orthogonal vectors are expected to collide with probability ( π1 )d+o(d) in dimension d, compared to ( 12 )d when using random hyperplanes. Vectors at angle π3 collide √ with probability ( π3 )d+o(d) , compared to ( 32 )d for random hyperplanes, and near-parallel vectors collide with similar asymptotic probabilities in both cases. For c-approximate nearest neighbor searching, this translates to a decrease in the exponent ρ of localitysensitive hashing (LSH) methods of a factor up to log2 (π) ≈ 1.652 compared to hyperplane LSH. For c = 2, we obtain ρ ≈ 0.302 + o(1) for hypercube LSH, improving upon the ρ ≈ 0.377 for hyperplane LSH. We further describe how to use hypercube LSH in practice, and we consider an example application in the area of lattice algorithms. Keywords: (approximate) near neighbors, locality-sensitive hashing, large deviations, dimensionality reduction, lattice algorithms 1 Introduction Finding (approximate) near neighbors. A key computational problem in various research areas, including machine learning, pattern recognition, data compression, coding theory, and cryptanalysis [SDI05, Bis06, Dub10, DHS00, MO15, Laa15], is finding near neighbors: given a data set D ⊂ Rd of cardinality n, design a data structure and preprocess D in a way that, when given a query vector q ∈ Rd , one can efficiently find a near point to q in D. Due to the “curse of dimensionality” [IM98] this problem is known to be hard to solve exactly (in the worst case) in high dimensions d, so a common relaxation of this problem is the (c, r)-approximate near neighbor problem ((c, r)-ANN): given that the nearest neighbor lies at distance at most r from q, design an algorithm that finds an element p ∈ D at distance at most c · r from q. Locality-sensitive hashing (LSH) and filtering (LSF). A prominent class of algorithms for finding near neighbors in high dimensions is formed by locality-sensitive hashing (LSH) [IM98] and localitysensitive filtering (LSF) [BDGL16]. These solutions are based on partitioning the space into regions, in a way that nearby vectors have a higher probability of ending up in the same hash region than distant vectors. By carefully tuning (i) the number of hash regions per hash table, and (ii) the number of randomized hash tables, one can then guarantee that with high probability (a) nearby vectors will collide in at least one of the hash tables, and (b) distant vectors will not collide in any of the hash tables. For LSH, a simple lookup in all of q’s hash buckets then provides a fast way of finding near neighbors to q, while for LSF the lookups are slightly more involved. For various metrics, LSH and LSF currently provide the best performance in high dimensions [AR15, BDGL16, ALRW17, Chr17]. Near neighbors on the sphere. In this work we will focus on the near neighbor problem under the angular distance, where two vectors x, y are considered nearby iff their common angle θ is small [Cha02, STS+ 13, SSLM14, AIL+ 15]. This equivalently corresponds to near neighbor searching for the `2 -norm, where the entire data set is assumed to lie on a sphere. A special on the sphere, √ case of (c, r)-ANN √ often considered in the literature, is the random case r = 1c 2 and c · r = 2, in part due to a reduction from near neighbor under the Euclidean metric for general data sets to (c, r)-ANN on the sphere with these parameters [AR15]. 2 1.1 Thijs Laarhoven Related work Upper bounds. Perhaps the most well-known and widely used solution for ANN for the angular distance is Charikar’s hyperplane LSH [Cha02], where a set of random hyperplanes is used to partition the space into regions. Due to its low computational complexity and the simple form of the collision probabilities (with no hidden order terms in d), this method is easy to instantiate in practice and commonly achieves the best performance out of all LSH methods when d is not too large. For large d, both spherical cap LSH [AINR14, AR15] and cross-polytope LSH [TT07, ER08, AIL+ 15, KW17] are known to perform better than hyperplane LSH. Experiments from [TT07, TT09] showed that using orthogonal hyperplanes, partitioning the space into Voronoi regions induced by the vertices of a hypercube, also leads to superior results compared to hyperplane LSH; however, no theoretical guarantees for the resulting hypercube LSH method were given, and it remained unclear whether the improvement persists in high dimensions. Lower bounds. For the case of random data sets, lower bounds have also been found, matching the performance of spherical cap and cross-polytope LSH for large c [MNP07, OWZ11, AIL+ 15]. These lower bounds are commonly in a model where it is assumed that collision probabilities are “not too small”, and in particular not exponentially small in d. Therefore it is not clear whether one can further improve upon cross-polytope LSH when the number of hash regions is exponentially large, which would for instance be the case for hypercube LSH. Together with the experimental results from [TT07,TT09], this naturally begs the question: how efficient is hypercube LSH? Is it better than hyperplane LSH and/or cross-polytope LSH? And how does hypercube LSH compare to other methods in practice? 1.2 Contributions Hypercube LSH. By carefully analyzing the collision probabilities for hypercube LSH using results from large deviations theory, we show that hypercube LSH is indeed different from, and superior to hyperplane LSH for large d. The following main theorem states the asymptotic form of the collision probabilities when using hypercube LSH, which are also visualized in Figure 1 in comparison with hyperplane LSH. Theorem 1 (Collision probabilities for hypercube LSH). Let X, Y ∼ N (0, 1)d , let θ ∈ [0, π] denote the angle between X and Y , and let p(θ) denote the probability that X and Y are mapped to the same hypercube hash region. For θ ∈ (0, arccos π2 ) (respectively θ ∈ (arccos π2 , π3 )), let β0 ∈ (1, ∞) (resp. β1 ∈ (1, ∞)) be the unique solution to: p p     (β0 − cos θ) β02 − 1 (β1 + cos θ) β12 − 1 −1 1 arccos = , arccos = . (1) β0 β0 (β0 cos θ − 1) β1 β1 (β1 cos θ + 1) Then, as d tends to infinity, p(θ) satisfies:  d+o(d)  (β0 − cos θ)2   ,    πβ0 (β0 cos θ − 1) sin θ      d+o(d)    (β1 + cos θ)2   , πβ1 (β1 cos θ + 1) sin θ p(θ) =       1 + cos θ d+o(d)    ,   π sin θ      0, if θ ∈ [0, arccos π2 ]; if θ ∈ [arccos π2 , π3 ]; (2) if θ ∈ [ π3 , π2 ); if θ ∈ [ π2 , π]. Denoting the query complexity of LSH methods by nρ+o(1) , the parameter ρ for hypercube LSH is up to log2 (π) ≈ 1.65 times smaller than for √ hyperplane LSH. For large d, hypercube LSH is dominated by cross-polytope LSH (unless c · r > 2), but as the convergence to the limit is rather slow, in practice either method might be better, depending on the exact parameter setting. For the random Hypercube LSH for approximate near neighbors 3 1 Hyperplane LSH → p(θ)1/d Hypercube LSH ν 3 π 1 π 0 0 arccos( π2 ) π 3 π 2 π →θ √ Fig. 1. Asymptotics of collision probabilities for hypercube LSH, compared to hyperplane LSH. Here ν = π/(2 π 2 − 4), and the dashed vertical lines correspond to boundary points of the piecewise parts of Theorem 1. The blue line indicates hyperplane LSH with d random hyperplanes. 1 →ρ 0.5 0.2 0.1 0.05 1 Hyperplane LSH Hypercube LSH Cross-polytope LSH 2 2 2 2 4 →c Fig. 2. Asymptotics for√the LSH exponent ρ when using hyperplane LSH, hypercube LSH, and cross-polytope LSH, for (c, r)-ANN with c · r = 2. The curve for hyperplane LSH is exact for arbitrary d, while for the other two curves, order terms vanishing as d → ∞ have been omitted. setting, Figure 2 shows limiting values for ρ for hyperplane, hypercube and cross-polytope LSH. We again remark that these are asymptotics for d → ∞, and may not accurately reflect the performance of these methods for moderate d. We further briefly discuss how the hashing for hypercube LSH can be made efficient. Partial hypercube LSH. As the number of hash regions of a full-dimensional hypercube is often prohibitively large, we also consider partial hypercube LSH, where a d0 -dimensional hypercube is used to partition a data set in dimension d. Building upon a result of Jiang [Jia06], we characterize when hypercube and hyperplane LSH are asymptotically equivalent in terms of the relation between d0 and d, and we empirically illustrate the convergence towards either hyperplane or hypercube LSH for larger d0 . An important open problem remains to identify how large the ratio d0 /d must be for the asymptotics of partial hypercube LSH to be equivalent to those of full-dimensional hypercube LSH. Application to lattice sieving. Finally, we consider a specific use case of different LSH methods, in the context of lattice cryptanalysis. We show that the heuristic complexity of lattice sieving with 4 Thijs Laarhoven hypercube LSH is expected to be slightly better than when using hyperplane LSH, and we discuss how experiments have previously indicated that in this application, hypercube LSH is superior to other dimensions up to dimensions d ≈ 80. 2 Preliminaries Notation. We denote probabilities with P(·) and expectations with E(·). Capital letters commonly denote random variables, and boldface letters denote vectors. We informally write P(X = x) for continuous X to denote the density of X at x. For probability distributions D, we write X ∼ D to denote that X is distributed according to D. For sets S, with abuse of notation we further write X ∼ S to denote X is drawn uniformly at random from S. We write N (µ, σ 2 ) for the normal distribution with mean µ and variance σ 2 , and H(µ, σ 2 ) for the distribution of |X| when X ∼ N (µ, σ 2 ). For µ = 0 the latter corresponds to the half-normal distribution. We write X ∼ Dd to denote a d-dimensional qP 2 vector where each entry is independently distributed according to D. In what follows, kxk = i xi P denotes the Euclidean norm, and hx, yi = i xi yi denotes the standard inner product. We denote the angle between two vectors by φ(x, y) = arccoshx/kxk, y/kyki. Lemma 1 (Distribution of angles between random vectors [BDGL16, Lemma 2]). Let X, Y ∼ N (0, 1)d be two independent standard normal vectors. Then P(φ(X, Y ) = θ) = (sin θ)d+o(d) . Locality-sensitive hashing. Locality-sensitive hash functions [IM98] are functions h mapping a ddimensional vector x to a low-dimensional sketch h(x), such that vectors which are nearby in Rd are more likely to be mapped to the same sketch than distant vectors. For the angular distance1 φ(x, y), we quantify a set of hash functions H as follows (see [IM98]): Definition 1. A hash family H is called (θ1 , θ2 , p1 , p2 )-sensitive if for x, y ∈ Rd we have: – If φ(x, y) ≤ θ1 then Ph∼H (h(x) = h(y)) ≥ p1 ; – If φ(x, y) ≥ θ2 then Ph∼H (h(x) = h(y)) ≤ p2 . The existence of locality-sensitive hash families implies the existence of fast algorithms for (approximate) near neighbors, as the following lemma describes2 . For more details on the general principles of LSH, we refer the reader to e.g. [IM98, And09]. Lemma 2 (Locality-sensitive hashing [IM98]). Suppose there exists a (θ1 , θ2 , p1 , p2 )-sensitive log(p1 ) family H. Let ρ = log(p . Then w.h.p. we can either find an element p ∈ L at angle at most θ2 from 2) q, or conclude that no elements p ∈ L at angle at most θ1 from q exist, in time nρ+o(1) with space and preprocessing costs n1+ρ+o(1) . Hyperplane LSH. For the angular distance, Charikar [Cha02] introduced the hash family H = {ha : a ∼ D} where D is any spherically symmetric distribution on Rd , and ha satisfies: ( +1, if ha, xi ≥ 0; ha (x) = (3) −1, if ha, xi < 0. The vector a can be interpreted as the normal vector of a random hyperplane, and the hash value depends on which side of the hyperplane x lies on. For this hash function, the probability of a collision is directly proportional to the angle between x and y:
φ(x, y) Ph∼H h(x) = h(y) = 1 − . π For any two angles θ1 < θ2 , the above family H is (θ1 , θ2 , 1 − 1 2 θ1 π ,1 − (4) θ2 π )-sensitive. Formally speaking, the angular distance is only a similarity measure, and not a metric. Various conditions and order terms (which are commonly no(1) ) are omitted here for brevity. Hypercube LSH for approximate near neighbors 5 Large deviations theory. Let P {Z d }d∈N ⊂ Rk be a sequence of random vectors corresponding to an empirical mean, i.e. Z d = d1 di=1 U i with U i i.i.d. We define the logarithmic moment generating function Λ of Z d as: Λ(λ) = ln EU 1 [exphλ, U 1 i] . (5) Define DΛ = {λ ∈ Rk : Λ(λ) < ∞}. The Fenchel-Legendre transform of Λ is defined as: Λ∗ (z) = sup {hλ, zi − Λ(λ)} . (6) λ∈Rk The following result describes that under certain conditions on {Z 0d }, the asymptotics of the probability measure on a set F are related to the function Λ∗ . Lemma 3 (Gärtner-Ellis theorem [DZ10, Theorem 2.3.6 and Corollary 6.1.6]). Let 0 be contained in the interior of DΛ , and let Z d be an empirical mean. Then for arbitrary sets F , lim 1 d→∞ d ln P(z ∈ F ) = − inf Λ∗ (z). z∈F (7) The latter statement can be read as P(z ∈ F ) = exp(−d inf z∈F Λ∗ (z) + o(d)), and thus tells us exactly how P(z ∈ F ) scales as d tends to infinity, up to order terms. 3 Hypercube LSH In this section, we will analyze full-dimensional hypercube hashing, with hash family H = {hA : A ∈ SO(d)} where SO(d) ⊂ Rd×d denotes the rotation group, and hA satisfies: ( +1, if xi ≥ 0; hA (x) = (h1 (Ax), . . . , hd (Ax)), hi (x) = (8) −1, if xi < 0. In other words, a hypercube hash function first applies a uniformly random rotation, and then maps the resulting vector to the orthant it lies in. This equivalently corresponds to a concatenation of d hyperplane hash functions, where all hyperplanes are orthogonal. Collision probabilities for prescribed angles θ between x and y are denoted by: p(θ) = P(hA (x) = hA (y) | φ(x, y) = θ). (9) Above, the randomness is over hA ∼ H, with x and y arbitrary vectors at angle θ (e.g. x = e1 and y = e1 cos θ + e2 sin θ). Alternatively, the random rotation A inside hA may be omitted, and the probability can be computed over X, Y drawn uniformly at random from a spherically symmetric distribution, conditioned on their common angle being θ. 3.1 Outline of the proof of Theorem 1 Although Theorem 1 is a key result, due to space restrictions we have decided to defer the full proof (approximately 5.5 pages) to the appendix. The approach of the proof can be summarized by the following four steps: – Rewrite the collision probabilities in terms of (normalized) half-normal vectors X, Y ; – Introduce dummy variables x, y for the norms of these half-normal vectors, so that the probability can be rewritten in terms of unnormalized half-normal vectors; – Apply the Gärtner-Ellis theorem (Lemma 3) to the three-dimensional vector given by P 2 P 1 P Z = d ( i Xi Yi , i Xi , i Yi2 ) to compute the resulting probabilities for arbitrary x, y; – Maximize the resulting expressions over x, y > 0 to get the final result. The majority of the technical part of the proof lies in computing Λ∗ (z), which involves a somewhat tedious optimization of a multivariate function through a case-by-case analysis. 6 Thijs Laarhoven A note on Gaussian approximations. From the (above outline of the) proof, and the observation that the final optimization over x, y yields x = y = 1 as the optimum, one might wonder whether a simpler analysis might be possible by assuming (half-)normal vectors are already normalized. Such a computation however would only lead to an approximate solution, which is perhaps easiest to see by computing collision probabilities for θ = 0. In the exact computation, where vectors are normalized, hX, Y i = 1 implies X = Y . If however we do not take into account the norms of X and Y , and do not condition on the norms being equal to 1, then hX, Y i = 1 could also mean that X, Y are slightly longer than 1 and have a small, non-zero angle. In fact, such a computation would indeed yield p(θ)1/d 6→ 0 as θ → 0. 3.2 Consequences of Theorem 1 From Theorem 1, we can draw √ several conclusions. Substituting values for θ, we can find asymptotics π 1/d for p(θ), such as p( 3 ) = π3 + o(1) and p( π2 )1/d = π1 + o(1). We observe that the limiting function of Theorem 1 (without the order terms) is continuous everywhere except at θ = π2 . To understand the boundary θ = arccos π2 of the piece-wise limit function, note that two (normalized) half-normal vectors X, Y have expected inner product EhX, Y i = π2 . LSH exponents ρ for random settings. Using Theorem 1, we can explicitly compute LSH exponents √ ρ for given angles θ1 and θ2 for large d. As an example, consider the random setting3 with c = 2, corresponding to θ2 = π2 and θ1 = π3 . Substituting the collision probabilities from Theorem 1, we get ρ → 1 − 21 logπ (3) ≈ 0.520 as d → ∞. To compare, if we had used random hyperplanes, we would have gotten a limiting value ρ → log2 ( 32 ) ≈ 0.585. For the random case, Figure 2 compares limiting values ρ using random and orthogonal hyperplanes, and using the asymptotically superior cross-polytope LSH. Scaling at θ → 0 and asymptotics of ρ for large c. For θ close to 0, by Theorem 1 we are in the regime defined by β0 . For cos θ = 1−ε with ε > 0 small, observe that β0 ≈ 1 satisfies β0 > 1/ cos θ. Computing √ 2 2 3/2 a Taylor expansion around ε = 0, we eventually find β0 = 1 + ε + π ε + O(ε2 ). Substituting this value β0 into p(θ) with cos θ = 1 − ε, we find: √ p(θ) = 2√ 1− ε + O(ε) π !d+o(d) . (10) To compare this with hyperplane LSH, recall that the collision √ probability for d random hyperplanes θ d is equal to (1 − π ) . Since cos θ = 1 − ε translates to θ = 2ε(1 + O(ε)), the collision probabilities √ √ for hyperplane hashing in this regime are also (1 − π2 ε + O(ε))d . In other words, for angles θ → 0, the collision probabilities for hyperplane hashing and hypercube hashing are similar. This can also be observed in Figure 1. Based on this result, we further deduce that in random settings with large c, for hypercube LSH we have:  ln 1 − ρ→ √ 2 πc +O ln(1/π) 1 c2
 √ 2 = +O πc ln π  1 c2  0.393 ≈ +O c  1 c2  . (11) For hyperplane LSH, the numerator is the same, while the denominator is ln( 12 ) instead of ln( π1 ), leading to values ρ which are a factor log2 π + o(1) ≈ 1.652 + o(1) larger. Both methods are inferior to cross-polytope LSH for large d, as there ρ = O(1/c2 ) for large c [AIL+ 15]. 3 √ √ Here we assume that c · r → ( 2)− ,√i.e. c · r approaches 2 from below. Alternatively, one might interpret this as that if distant points lie at distance 2 ± o(1), then we might expect approximately half of them to lie at distance √ √ less than 2, with query complexity O(n/2)ρ+o(1) = nρ+o(1) . If however c · r ≥ 2 then clearly ρ = 0, regardless of d and c. Hypercube LSH for approximate near neighbors 7 1◆ ● ■ ▲ ▼ ● ■ ◆ ▲ ▼ → p(θ)1/d 3 4 1 2 ● ■ 1 4 ◆ ▲ ▼ 0 ● ▲ ◆ ■ ▼ ● ◆ ▼ ● ■ ▲ ▲ ■ ◆ ▼ ◆ ● ■ ▲ ● ■ ▲ ● ◆ ■ ◆ ▲ ◆ ● ■ ▲ ◆ ● ▲ ● ■ ▲ ● ■ ◆ ■ ▲● ◆ Full hypercube LSH ■ ◆ ▲◆ ● ■● 2-dimensional hypercube ■● ◆ ■● 4-dimensional hypercube ◆ ■● ◆ 8-dimensional hypercube ■● ◆ ■ 16-dimensional hypercube ◆ ● ■ 32-dimensional hypercube π 8 3π 8 π 4 π 2 →θ Fig. 3. Empirical collision probabilities for hypercube LSH for small d. The green curve denotes the exact collision probabilities for d = 2 from Proposition 1. 3.3 Convergence to the limit To get an idea how hypercube LSH compares to other methods when d is not too large, we start by giving explicit collision probabilities for the first non-trivial case, namely d = 2. Proposition 1 (Square LSH). For d = 2, p(θ) = 1 − 2θ π for θ ≤ π 2 and p(θ) = 0 otherwise. Proof. In two dimensions, two randomly rotated vectors X, Y at angle θ can be modeled as X = (cos ψ, sin ψ) and Y = (cos(ψ + θ), sin(ψ + θ)) for ψ ∼ [0, 2π). The conditions X, Y > 0 are then over the equivalent to ψ ∈ (0, π2 ) ∩ (−θ, π2 − θ), which for θ < π2 occurs with probability π/2−θ 2π randomness of ψ. As a collision can occur in any of the four quadrants, we finally multiply this probability by 4 to obtain the stated result. Figure 3 depicts p(θ)1/2 in green, along with hyperplane LSH (blue) and the asymptotics for hypercube LSH (red). For larger d, computing p(θ) exactly becomes more complicated, and so instead we performed experiments to empirically obtain estimates for p(θ) as d increases. These estimates are also shown in Figure 3, and are based on 105 trials for each θ and d. Observe that as θ → π2 and/or d grows larger, p(θ) decreases and the empirical estimates become less reliable. Points are omitted for cases where no successes occurred. Based on these estimates and our intuition, we conjecture that (1) for θ ≈ 0, the scaling of p(θ)1/d is similar for all d, and similar to the asymptotic behavior of Theorem 1; (2) the normalized collision probabilities for θ ≈ π2 approach their limiting value from below; and (3) p(θ) is likely to be continuous for arbitrary d, implying that for θ → π2 , the collision probabilities tend to 0 for each d. These together suggest that values for ρ are actually smaller when d is small than when d is large, and the asymptotic estimate from Figure 2 might be pessimistic in practice. For the random setting, this would suggest that ρ ≈ 0 regardless of c, as p(θ) → 0 as θ → π2 for arbitrary d. Comparison with hyperplane/cross-polytope LSH. Finally, [TT07, Figures 1 and 2] previously illustrated that among several LSH methods, the smallest values ρ (for their parameter sets) are obtained with hypercube LSH with d = 16, achieving smaller values ρ than e.g. cross-polytope LSH with d = 256. An explanation for this can be found in: – The (conjectured) convergence of ρ to its limit from below, for hypercube LSH; 8 Thijs Laarhoven – The slow convergence of ρ to its limit (from above) for cross-polytope LSH4 . This suggests that the actual values ρ for moderate dimensions d may well be smaller for hypercube LSH (and hyperplane LSH) than for cross-polytope LSH. Based on the limiting cases d = 2 and d → ∞, we further conjecture that compared to hyperplane LSH, hypercube LSH achieves smaller values ρ for arbitrary d. 3.4 Fast hashing in practice To further assess the practicality of hypercube LSH, recall that hashing is done as follows: – Apply a uniformly random rotation A to x; – Look at the signs of (Ax)i . Theoretically, a uniformly random rotation will be rather expensive to compute, with A being a real, dense matrix. As previously discussed in e.g. [Ach01], it may suffice to only consider a sparse subset of all rotation matrices with a large enough amount of randomness, and as described in [AIL+ 15, KW17] pseudo-random rotations may also be help speed up the computations in practice. As described in [KW17], this can even be made provable, to obtain a reduced O(d log d) computational complexity for applying a random rotation. Finally, to compare this with cross-polytope LSH, note that cross-polytope LSH in dimension d partitions the space in 2d regions, as opposed to 2d for hypercube hashing. To obtain a similar finegrained partition of the space with cross-polytopes, one would have to concatenate Θ(d/ log d) random cross-polytope hashes, which corresponds to computing Θ(d/ log d) (pseudo-)random rotations, compared to only one rotation for hypercube LSH. We therefore expect hashing to be up to a factor Θ(d/ log d) less costly. 4 Partial hypercube LSH Since a high-dimensional hypercube partitions the space in a large number of regions, for various applications one may only want to use hypercubes in a lower dimension d0 < d. In those cases, one would first apply a random rotation to the data set, and then compute the hash based on the signs of the first d0 coordinates of the rotated data set. This corresponds to the hash family H = {hA,d0 : A ∈ SO(d)}, with hA,d0 satisfying: hA,d0 (x) = (h1 (Ax), . . . , hd0 (Ax)), ( +1, if xi ≥ 0; hi (x) = −1, if xi < 0. (12) When “projecting” down onto the first d0 coordinates, observe that distances and angles are distorted: the angle between the vectors formed by the first d0 coordinates of x and y may not be the same as φ(x, y). The amount of distortion depends on the relation between d0 and d. Below, we will investigate how the collision probabilities pd0 ,d (θ) for partial hypercube LSH scale with d0 and d, where pd0 ,d (θ) = P(h(x) = h(y) | φ(x, y) = θ). 4.1 Convergence to hyperplane LSH First, observe that for d0 = 1, partial hypercube LSH is equal to hyperplane LSH, i.e. p1,d (θ) = 1 − πθ . For 1 < d0  d, we first observe that both (partial) hypercube LSH and hyperplane LSH can be modeled by a projection onto d0 dimensions: 0 – Hyperplane LSH: x 7→ Ax with A ∼ N (0, 1)d ×d ; 0 – Hypercube LSH: x 7→ (A∗ )x with A ∼ N (0, 1)d ×d . 4 [AIL+ 15, Theorem 1] shows that the leading term in the asymptotics for ρ scales as Θ(ln d), with a first order term scaling as O(ln ln d), i.e. a relative order term of the order O(ln ln d/ ln d). Hypercube LSH for approximate near neighbors 9 Here A∗ denotes the matrix obtained from A after applying Gram-Schmidt orthogonalization to the rows of A. In both cases, hashing is done after the projection by looking at the signs of the projected vector. Therefore, the only difference lies in the projection, and one could ask: for which d0 , as a function of d, are these projections equivalent? When is a set of random hyperplanes already (almost) orthogonal? This question was answered in [Jia06]: if d0 = o(d/ log d), then maxi,j |Ai,j −A∗i,j | → 0 in probability as d → ∞ (implying A∗ = (1 + o(1))A), while for d0 = Ω(d/ log d) this maximum does not converge to 0 in probability. In other words, for large d a set of d0 random hyperplanes in d dimensions is (approximately) orthogonal iff d0 = o(d/ log d). Proposition 2 (Convergence to hyperplane LSH). Let pd0 ,d (θ) denote the collision probabilities 0 for partial hypercube LSH, and let d0 = o(d/ log d). Then pd0 ,d (θ)1/d → 1 − πθ . As d0 = Ω(d/ log d) random vectors in d dimensions are asymptotically not orthogonal, in that case one might expect either convergence to full-dimensional hypercube LSH, or to something in between hyperplane and hypercube LSH. 4.2 Convergence to hypercube LSH To characterize when partial hypercube LSH is equivalent to full hypercube LSH, we first observe that if d0 is large compared to ln n, then convergence to the hypercube LSH asymptotics follows from the Johnson-Lindenstrauss lemma. Proposition 3 (Sparse data sets). Let d0 = ω(ln n). Then the same asymptotics for the collision probabilities as those of full-dimensional hypercube LSH apply. Proof. Let θ ∈ (0, π2 ). By the Johnson-Lindenstrauss lemma [JL84], we can construct a projection x 7→ Ax from d onto d0 dimensions, preserving all pairwise distances up to a factor 1 ± ε for ε = Θ((ln n)/d0 ) = o(1). For fixed θ ∈ (0, π2 ), this implies the angle φ between Ax and Ay will be in the interval θ ± o(1), and so the collision probability lies in the interval p(θ ± o(1)). For large d, this means that the asymptotics of p(θ) are the same. To analyze collision probabilities for partial hypercube LSH when neither of the previous two propositions applies, note that through a series of transformations similar to those for full-dimensional hypercube LSH, it is possible to eventually end up with the following probability to compute, where d1 = d0 and d2 = d − d0 : max P x,y,u,v,φ d1 1 X Xi Yi = xy cos φ, d1 i=1 d2 1 X Ui Vi = uvf (φ, θ), d2 i=1 d1 1 X Xi2 = x2 , d1 1 d2 i=1 d2 X i=1 Ui2 = u2 , d1 1 X Yi2 = y 2 , d1 i=1 ! d2 1 X Vi2 = v 2 . d2 (13) (14) i=1 Here f is some function of φ and θ. The approach is comparable to how we ended up with a similar probability to compute in the proof of Theorem 1, except that we split the summation indices I = [d] into two sets I1 = {1, . . . , d0 } of size d1 and I2 = {d0 +1, . . . , d} of size d2 . We then substitute Ui = Xd0 +i and Vi = Yd0 +i , and add dummy variables x, y, u, v for the norms of the four partial vectors, and a dummy angle φ for the angle between the d1 -dimensional vectors, given the angle θ between the d-dimensional vectors. Although the vector Z formed by the six random variables in (14) is not an empirical mean over a fixed number d of random vectors (the first three are over d1 terms, the last three over d2 terms), one may expect a similar large deviations result such as Lemma 3 to apply here. In that case, the function Λ∗ (z) = Λ∗ (z1 , . . . , z6 ) would be a function of six variables, which we would like to evaluate at (xy cos φ, x2 , y 2 , uvf (φ, θ), u2 , v 2 ). The function Λ∗ itself involves an optimization (finding a supremum) over another six variables λ = (λ1 , . . . , λ6 ), so to compute collision probabilities for 10 Thijs Laarhoven 1◆ ● ■ ▲ ▼ ■ ◆ ● ▲ ▼ → p(θ)1/d 3 4 1 2 ● ■ ◆ ▲ ▼ ● ■ ◆ ▲ ◆ ▼ ● ■ ▲ ◆ ▼ ● ■ ● ▲ ■ ▼ ◆ ▲ ◆ ● ■ ▼ ● ▲ ■ ▼ ◆ ● ▲ ■ ● ▼ ◆ ■ ▲ ● ■ ▲ ▼ ◆ ● ▼ ◆ ■ ▲ ● ■◆ ▼ ▲◆ ●● ▼◆ ▲ ■◆ ▼ ■ ■◆ ▲◆ ● ▲● ■◆ ■◆ ● ▼ ■ ▼▲ ▲● ▼▼ Partial hypercube LSH (d = 50) ▲ ▲ ▼ ▼ ● 2-dimensional hypercube ■ 4-dimensional hypercube ◆ 8-dimensional hypercube 1 4 ▲ 16-dimensional hypercube ▼ 32-dimensional hypercube 0 0 π 8 π 4 3π 8 π 2 →θ 0 Fig. 4. Experimental values of pd0 ,50 (θ)1/d , for different values d0 , compared with the asymptotics for hypercube LSH (red) and hyperplane LSH (blue). given d, d0 , θ exactly, using large deviations theory, one would have to compute an expression of the following form: ) ( min x,y,u,v,φ sup Fd,d0 ,θ (x, y, u, v, φ, λ1 , λ2 , λ3 , λ4 , λ5 , λ6 ) . (15) λ1 ,λ2 ,λ3 ,λ4 ,λ5 ,λ6 As this is a very complex task, and the optimization will depend heavily on the parameters d, d0 , θ defined by the problem setting, we leave this optimization as an open problem. We only mention that intuitively, from the limiting cases of small and large d0 we expect that depending on how d0 scales with d (or n), we obtain a curve somewhere in between the two curves depicted in Figure 1. 4.3 Empirical collision probabilities To get an idea of how pd0 ,d (θ) scales with d0 in practice, we empirically computed several values for fixed d = 50. For fixed θ we then applied a least-squares fit of the form ec1 d+c2 to the resulting data, and plotted ec1 in Figure 4. These data points are again based on at least 105 experiments for each d0 and θ. We expect that as d0 increases, the collision probabilities slowly move from hyperplane hashing towards hypercube hashing, this can also be seen in the graph – for d0 = 2, the least-squares fit is almost equal to the curve for hyperplane LSH, while as d0 increases the curve slowly moves down towards the asymptotics for full hypercube LSH. Again, we stress that as d0 becomes larger, the empirical estimates become less reliable, and so we did not consider even larger values for d0 . Compared to full hypercube LSH and Figure 3, we observe that we now approach the limit from above (although the fitted collision probabilities never seem to be smaller than those of hyperplane LSH), and therefore the values ρ for partial hypercube LSH are likely to lie in between those of hyperplane and (the asymptotics of) hypercube LSH. 5 Application: Lattice sieving for the shortest vector problem We finally consider an explicit application for hypercube LSH, namely lattice sieving algorithms for the P d shortest vector problem. Given a basis B = {b1 , . . . , bd } ⊂ R of a lattice L(B) = { i λi bi : λi ∈ Z}, Hypercube LSH for approximate near neighbors 11 the shortest vector problem (SVP) asks to find a shortest non-zero vector in this lattice. Various different methods for solving SVP in high dimensions are known, and currently the algorithm with the best heuristic time complexity in high dimensions is based on lattice sieving, combined with nearest neighbor searching [BDGL16]. In short, lattice sieving works by generating a long list L of pairwise reduced lattice vectors, where x, y are reduced iff kx − yk ≥ min{kxk, kyk}. The previous condition is equivalent to φ(x, y) ≤ π3 , and so the length of L can be bounded by the kissing constant in dimension d, which is conjectured to scale as (4/3)d/2+o(d) . Therefore, if we have a list of size n = (4/3)d/2+o(d) , any newly sampled lattice vector can be reduced against the list many times to obtain a very short lattice vector. The time complexity of this method is dominated by doing poly(d) · n reductions (searches for nearby vectors) with a list of size n. A linear search trivially leads to a heuristic complexity of n2+o(1) = (4/3)d+o(d) (with space n1+o(1) ), while nearest neighbor techniques can reduce the time complexity to n1+ρ+o(1) for ρ < 1 (increasing the space to n1+ρ+o(1) ). For more details, see e.g. [NV08, Laa15, BDGL16]. Based on the collision probabilities for hypercube LSH, and assuming the asymptotics for partial hypercube LSH (with d0 = O(d)) are similar to those of full-dimensional hypercube LSH, we obtain the following result. An outline of the proof is given in the appendix. Proposition 4 (Complexity of lattice sieving with hypercube LSH). Suppose the asymptotics for full hypercube LSH also hold for partial hypercube LSH with d0 ≈ 0.1335d. Then lattice sieving with hypercube LSH heuristically solves SVP in time and space 20.3222d+o(d) . As expected, the conjectured asymptotic performance of (sieving with) hypercube LSH lies in between those of hyperplane LSH and cross-polytope LSH. – – – – – – Linear search [NV08]: 20.4150d+o(d) . Hyperplane LSH [Laa15]: 20.3366d+o(d) . Hypercube LSH: 20.3222d+o(d) . Spherical cap LSH [LdW15]: 20.2972d+o(d) . Cross-polytope LSH [BL16]: 20.2972d+o(d) . Spherical LSF [BDGL16]: 20.2925d+o(d) . In practice however, the picture is almost entirely reversed [SG15]. The lattice sieving method used to solve SVP in the highest dimension to date (d = 116) used a very optimized linear search [Kle14]. The furthest that any nearest neighbor-based sieve has been able to go to date is d = 107, using hypercube LSH [MLB15, MB16]5 . Experiments further indicated that spherical LSF only becomes competitive with hypercube LSH as d & 80 [BDGL16, MLB17], while sieving with cross-polytope LSH turned out to be rather slow compared to other methods [BL16, Mar16]. Although it remains unclear which nearest neighbor method is the “most practical” in the application of lattice sieving, hypercube LSH is one of the main contenders. Acknowledgments. The author is indebted to Ofer Zeitouni for his suggestion to use results from large deviations theory, and for his many helpful comments regarding this application. The author further thanks Brendan McKay and Carlo Beenakker for their comments. The author is supported by the SNSF ERC Transfer Grant CRETP2-166734 FELICITY. 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Before starting the proof, we begin with a useful lemma regarding integrals of (exponentials of) quadratic forms. Hypercube LSH for approximate near neighbors 13 Lemma 4 (Integrating an exponential of a quadratic form in the positive quadrant). Let a, b, c ∈ R with a, c < 0 and D = b2 − 4ac < 0. Then:   b Z ∞Z ∞ π + 2 arctan √−D √ exp(ax2 + bxy + cy 2 ) dx dy = . (16) 2 −D 0 0 Proof. The proof below is based on substituting y = xs (and dy = x ds) before computing the integral over x. An integral over 1/(a + bs + cs2 ) then remains, which leads to the arctangent solution in case b2 < 4ac. Z ∞Z ∞ I= exp(ax2 + bxy + cy 2 ) dx dy (17) y=0 0  Z ∞ Z ∞
x exp (a + bs + cs2 )x2 dx ds (18) = s=0 0
#∞ Z ∞" exp (a + bs + cs2 )x2 = ds (19) 2(a + bs + cs2 ) 0 x=0  Z ∞ 1 = 0− ds (20) 2(a + bs + cs2 ) 0 Z 1 −1 ∞ ds. (21) = 2 0 a + bs + cs2 The last equality used the assumptions a, c < 0 and b2 < 4ac so that a + bs + cs2 < 0 for all s > 0. We then solve the last remaining integral (see e.g. [AS72, Equation (3.3.16)]) to obtain:   ∞ −1 2 b + 2cs √ √ I= arctan (22) 2 4ac − b2 4ac − b2 s=0    −1 b = √ −π − 2 arctan √ . (23) 2 4ac − b2 4ac − b2 Eliminating minus signs and substituting D = b2 − 4ac, we obtain the stated result. Next, we begin by restating the collision probability between two vectors in terms of half-normal vectors. Lemma 5 (Towards three-dimensional large deviations). Let H denote the hypercube hash family in d dimensions, and as before, let p be defined as: p(θ) = Ph∼H (h(x) = h(y) | φ(x, y) = θ). Let X̂, Ŷ ∼ H(0, 1)d and let the sequence {Z d }d∈N ⊂ R3 be defined as: ! d d d X X 1 X 2 2 Zd = X̂i Ŷi , X̂i , Ŷi . d i=1 i=1 (24) (25) i=1 Then:  p(θ) = 1 2 sin θ d+o(d) max P(Z d = (xy cos θ, x2 , y 2 )). x,y>0 (26) Proof. First, we write out the definition of the conditional probability in p, and use the fact that each of the 2d hash regions (orthants) has the same probability mass. Here X, Y ∼ N (0, 1)d denote random Gaussian vectors, and subscripts denoting what probabilities are computed over are omitted when implicit. p(θ) = Ph∼H (h(x) = h(y) | φ(x, y) = θ) = 2d · PX,Y ∼N (0,1)d (X > 0, Y > 0 | φ(X, Y ) = θ) = 2d · P(X > 0, Y > 0, φ(X, Y ) = θ) . P(φ(X, Y ) = θ) (27) (28) (29) 14 Thijs Laarhoven By Lemma 1, the denominator is equal to (sin θ)d+o(d) . The numerator of (29) can further be rewritten as a conditional probability on {X > 0, Y > 0}, multiplied with P(X > 0, Y > 0) = 2−2d . To incorporate the conditionals X, Y > 0, we replace X, Y ∼ N (0, 1)d by half-normal vectors X̂, Ŷ ∼ H(0, 1)d , resulting in: p(θ) = PX̂,Ŷ ∼H(0,1)d (φ(X̂, Ŷ ) = θ) (2 sin θ)d+o(d) = q(θ) (2 sin θ)d+o(d) . (30) To incorporate the normalization over the (half-normal) vectors X̂ and Ŷ , we introduce dummy √ √ variables x, y corresponding to the norms of X̂/ d and Ŷ / d, and observe that as the probabilities are exponential in d, the integrals will be dominated by the maximum value of the integrand in the given range: Z ∞Z ∞ P(hX̂, Ŷ i = x y d cos θ, kX̂k2 = x2 d, kŶ k2 = y 2 d) dx dy (31) q(θ) = 0 0   = 2o(d) max P hX̂, Ŷ i = x y d cos θ, kX̂k2 = x2 d, kŶ k2 = y 2 d . (32) x,y>0 Substituting Z d = d1 (hX̂, Ŷ i, kX̂k2 , kŶ k2 ), we obtain the claimed result. Note that Z1 , Z2 , Z3 are pairwise but not jointly independent. To compute the density of Z d at (xy cos θ, x2 , y 2 ) for d → ∞, we use the Gärtner-Ellis theorem stated in Lemma 3. Lemma 6 (Applying the Gärtner-Ellis theorem to Z d ). Let {Z d }d∈N ⊂ R3 as in Lemma 5, and let Λ and Λ∗ as in Section 2. Then 0 lies in the interior of DΛ , and therefore
P(Z d = (xy cos θ, x2 , y 2 )) = exp −Λ∗ (xy cos θ, x2 , y 2 )d + o(d) . (33) Essentially, all that remains now is computing Λ∗ at the appropriate point z. To continue, we first compute the logarithmic moment generating function Λ = Λd of Z d : Lemma 7 (Computing Λ). Let Z d as before, and let D = D(λ1 , λ2 , λ3 ) = λ21 − (1 − 2λ2 )(1 − 2λ3 ). Then for λ ∈ DΛ = {λ ∈ R3 : λ2 , λ3 < 21 , D < 0} we have:    λ1 Λ(λ) = ln π + 2 arctan √ − ln π − 12 ln(−D). (34) −D Proof. By the definition of the LMGF, we have: h  i Λ(λ) = ln EX̂1 ,Ŷ1 ∼H(0,1) exp λ1 X̂1 Ŷ1 + λ2 X̂12 + λ3 Ŷ12 . (35) We next compute the inner expectation over the random variables X̂1 , Ŷ1 , by writing out the double integral over the product of the argument with the densities of X̂1 and Ŷ1 . 
 EX1 ,Y1 exp λ1 X1 Y1 + λ2 X12 + λ3 Y12 (36)  2 Z ∞ r  2 Z ∞r
2 x 2 y = exp − dx exp − dy exp λ1 xy + λ2 x2 + λ3 y 2 (37) π 2 π 2 0 0 Z ∞Z ∞


2 = exp λ1 xy + λ2 − 12 x2 + λ3 − 21 y 2 dx dy . (38) π 0 0 Applying Lemma 4 with (a, b, c) = (λ2 − 21 , λ1 , λ3 − 12 ) yields the claimed expression for Λ, as well as the bounds stated in DΛ which are necessary for the expectation to be finite. We now continue with computing the Fenchel-Legendre transform of Λ, which involves a rather complicated maximization (supremum) over λ ∈ R3 . The following lemma makes a first step towards computing this supremum. Hypercube LSH for approximate near neighbors 15 Lemma 8 (Computing Λ∗ (z) – General form). Let z ∈ R3 such that z2 , z3 > 0. Then the Fenchel-Legendre transform Λ∗ of Λ at z satisfies  √ z2 z3 1 Λ∗ (z) = ln π + sup + + λ1 z1 − |λ1 |β z2 z3 + ln(β 2 − 1) + ln |λ1 | (39) 2 2 2 λ1 ,β β>1 )  λ1 p . − ln π + 2 arctan |λ1 | β 2 − 1   (40) Proof. First, we recall the definition of Λ∗ and substitute the previous expression for Λ: Λ∗ (z) = sup {hλ, zi − Λ(λ)} λ∈R3     √ λ1 = ln π + sup hλ, zi + ln −D − ln π + 2 arctan √ . −D λ∈R3 (41) (42) Here as before D = λ21 − (1 − 2λ2 )(1 − 2λ3 ) < 0. Let the argument of the supremum above be denoted by f (z, λ). We make a change of variables by setting t2 = 1 − 2λ2 > 0 and t3 = 1 − 2λ3 > 0, so that D becomes D = λ21 − t2 t3 < 0: z2 z3 t2 z2 t3 z3 + + λ 1 z1 − − 2 2 2 2  2 1 + 2 ln(t2 t3 − λ1 ) − ln π + 2 arctan √ f (z, λ1 , t2 , t3 ) = (43) λ1 t2 t3 −λ21  . (44) We continue by making a further change of variables u = t2 t3 > λ21 so that t2 = u/t3 . As a result the dependence of f on t3 is only through the fourth and p fifth terms above, from which one can p easily deduce that the supremum over t3 occurs at t3 = uz2 /z3 . This also implies that t2 = uz3 /z2 . Substituting these values for t2 , t3 , we obtain:    √ f (z, λ1 , u) = z22 + z23 + λ1 z1 − uz2 z3 + 21 ln(u − λ21 ) − ln π + 2 arctan √ λ1 2 . u−λ1 Finally, we use the substitution u = β 2 ·λ21 . From D < 0 it follows that u/λ21 = β > 1. This substitution and some rewriting of f leads to the claimed result. The previous simplifications were regardless of z1 , z2 , z3 , where the only assumption that was made during the optimization of t3 was that z2 , z3 > 0. In our application, we want to compute Λ∗ at z = (xy cos θ, x2 , y 2 ) for certain x, y > 0 and θ ∈ (0, π2 ). Substituting these values for z, the expression from Lemma 5 becomes:  x2 y 2 1 ∗ 2 2 Λ (xy cos θ, x , y ) = ln π + + + sup (λ1 cos θ − |λ1 |β)xy + ln(β 2 − 1) (45) 2 2 2 λ1 ,β β>1 )  λ1 p + ln |λ1 | − ln π + 2 arctan . |λ1 | β 2 − 1   (46) The remaining optimization over λ1 , β now takes slightly different forms depending on whether λ1 < 0 or λ1 > 0. We will tackle these two cases separately, based on the identity: n o Λ∗ (z) = max sup {hλ, zi − Λ(λ)} , sup {hλ, zi − Λ(λ)} = max{Λ∗+ (z), Λ∗− (z)}. λ∈R3 λ1 >0 λ∈R3 λ1 <0 Lemma 9 (Computing Λ∗ (z) for positive λ1 ). Let z = (xy cos θ, x2 , y 2 ) with x, y > 0 and θ ∈ (0, π2 ). For θ ∈ (0, arccos π2 ), let β0 = β0 (θ) ∈ (1, ∞) be the unique solution to (1). Then the FenchelLegendre transform Λ∗ at z, restricted to λ1 > 0, satisfies    πβ0 (β0 cos θ − 1)   ln , if θ ∈ (0, arccos π2 ); 2 x2 y 2 2(β − cos θ) ∗ 0 Λ+ (z) = + − 1 − ln(xy) + (47)  2 2  0, if θ ∈ [arccos 2 , π ). π 2 16 Thijs Laarhoven Proof. Substituting λ1 > 0 into (46), we obtain: Λ∗+ (xy cos θ, x2 , y 2 ) = ln π + n o x2 y 2 + + sup g+ (λ1 , β) , 2 2 λ1 >0 (48) β>1 g+ (λ1 , β) = (cos θ − β)λ1 xy + ln(β 2 − 1) 2    1 . + ln λ1 − ln π + 2 arctan p β2 − 1 (49) Differentiating w.r.t. λ1 gives (cos θ − β)xy + λ11 . Recall that β > 1 > cos θ. For λ1 → 0+ the derivative is therefore positive, for λ1 → ∞ it is negative, and there is a global maximum at the only root λ1 = 1/((β − cos θ)xy). In that case, the expression further simplifies and we can pull out more terms that do not depend on β, to obtain: n o x2 y 2 Λ∗+ (xy cos θ, x2 , y 2 ) = ln π + + − 1 − ln(xy) + sup g+ (β) , (50) 2 2 β>1   p 2 β −1    = ln h+ (β). g+ (β) = ln  (51) 1 (β − cos θ) π + 2 arcsin β p Here we used the identity arctan(1/ β 2 − 1) = arcsin(1/β). Now, for β → 1+ we have h+ (β) → 0+ , while for β → ∞, we have     1 1 2 1 h+ (β) = + cos θ − +O . (52) π πβ π β2 In other words, if cos θ ≤ π2 or θ ≥ arccos π2 , we have h+ (β) → ( π1 )− (the second order term is negative for cos θ = π2 ), while for θ < arccos π2 we approach the same limit from above as h+ (β) → ( π1 )+ . For θ < arccos π2 there is a non-trivial maximum at some value β = β0 ∈ (1, ∞), while for θ ≥ arccos π2 , we can see from the derivative h0+ (β) that h+ (β) is strictly increasing on (1, ∞), and the supremum is attained at β → ∞. We therefore obtain two different results, depending on whether θ < arccos π2 or θ ≥ arccos π2 . Case 1: arccos π2 ≤ θ < π2 . The supremum is attained in the limit of β → ∞, which leads to h+ (β) → π1 and the stated expression for Λ∗+ (xy cos θ, x2 , y 2 ). Case 2: 0 < θ < arccos π2 . In this case there is a non-trivial maximum at some value β = β0 , namely there where the derivative h0+ (β0 ) = 0. After computing the derivative, eliminating the (positive) denominator and rewriting, this condition is equivalent to (1).This allows us to rewrite g and Λ∗ in terms of β0 , by substituting the given expression for arcsin β10 , which ultimately leads to the stated formula for Λ∗+ . Lemma 10 (Computing Λ∗ (z) for negative λ1 ). Let z = (xy cos θ, x2 , y 2 ) with x, y > 0 and θ ∈ (0, π2 ). For θ ∈ (arccos π2 , π3 ), let β1 ∈ (1, ∞) be the unique solution to (1). Then the FenchelLegendre transform Λ∗ at z, restricted to λ1 < 0, satisfies   0, if θ ∈ (0, arccos π2 ];          πβ1 (β1 cos θ + 1)  2 2 y x ∗ , if θ ∈ (arccos π2 , π3 ); Λ− (z) = + − 1 − ln(xy) + ln (53) 2 2(cos θ + β ) 1  2 2        π   , if θ ∈ [ π3 , π2 ). ln 2(1 + cos θ) Proof. We again start by substituting λ1 < 0 into (46): n o x2 y 2 Λ∗− (xy cos θ, x2 , y 2 ) = ln π + + + sup g− (λ1 , β) , 2 2 λ1 <0 β>1 g− (λ1 , β) = (cos θ + β)λ1 xy + ln(β 2 2 − 1)   −1  + ln(−λ1 ) − ln π + 2 arctan p . β2 − 1 (54) Hypercube LSH for approximate near neighbors 17 Differentiating w.r.t. λ1 gives (cos θ + β)xy + λ11 . For λ1 → −∞ this is positive, for λ1 → 0− this is negative, and so the maximum is at λ1 = −1/((cos θ + β)xy). Substituting this value for λ1 , and pulling out terms which do not depend on β yields: Λ∗− (xy cos θ, x2 , y 2 ) = ln g− (β) = ln π  2 + n o x2 y 2 + − 1 − ln(xy) + sup g− (β) , 2 2 β>1 ! p β2 − 1 (cos θ + β) arccos β1 (55) = ln h− (β). p Above we used the identity π + 2 arctan(−1/ β 2 − 1) = 2 arccos β1 , where the factor 2 has been pulled outside the supremum. Now, differentiating h− w.r.t. β results in: p
2 − 1(β cos θ + 1) arccos 1 − β 2 − 1 (cos θ + β) β β β h0− (β) = . (56) β (β 2 − 1) (cos θ + β)2 arccos β1 Clearly the denominator is positive, while for β → 1+ the limit is negative iff cos θ < 21 . For β → ∞ we further have h0− (β) → 0− for cos θ ≤ π2 and h0− (β) → 0+ for cos θ > π2 . We therefore analyze three cases separately below. Case 1: π3 ≤ θ < π2 . In this parameter range, h0− (β) is negative for all β > 1, and the supremum 1 ∗ lies at β → 1+ with limiting value h− (β) → 1+cos θ . This yields the given expression for Λ− . Case 2: arccos π2 < θ < π3 . For θ in this range, h0− (β) is positive for β → 1+ and negative for β → ∞, and changes sign exactly once, where it attains its maximum. After some rewriting, we find that this is at the value β = β1 (θ) ∈ (1, ∞) satisfying the relation from (1). Substituting this expression for arccos β11 into h− , we obtain the result for Λ∗− . Case 3: 0 < θ ≤ arccos π2 . In this case h0− is positive for all β > 1, and the supremum lies at β → ∞. For β → ∞ we have h− (β) → π2 (regardless of θ) and we therefore get the final claimed result. Proof (Proof of Theorem 1). Combining the previous two results with Lemma 6 and Equation 47, we obtain explicit asymptotics for P(Z d ≈ (xy cos θ, x2 , y 2 )). What remains is a maximization over 2 2 x, y > 0 of p, which translates to a minimization of Λ∗ . As x2 + y2 − 1 − ln(xy) attains its minimum at x = y = 1 with value 0, we obtain Theorem 1. B Proof of Proposition 4 We will assume the reader is familiar with (the notation from) [Laa15]. Let t = 2ct d+o(d) denote the number of hash tables, and n = (4/3)d/2+o(d) . Going through the proofs of [Laa15, Appendix A] and replacing the explicit instantiation of the collision probabilities (1 − θ/π) by an arbitrary function p(θ), we get that the optimal number of hash functions concatenated into one function for each hash table, denoted k, satisfies k= ln t ct d √ . = 0 − ln p(θ1 ) d log2 (π/ 3) (57) The latter equality follows when substituting θ1 = π/3 and substituting the collision probabilities for partial hypercube LSH in some dimension d0 ≤ d. As we need k ≥ 1, the previous relation translates ct √ d. As we expect the collision probabilities to be closer to those of to a condition on d0 as d0 ≤ log (π/ 3) 2 full-dimensional hypercube LSH when d0 is closer to d, we replace the above inequality by an equality, and what remains is finding the minimum value ct satisfying the given constraints. By carefully checking the proofs of [Laa15, Appendix A.2-A.3], the exact condition on ct to obtain the minimum asymptotic time complexity is the following:   ct −cn = max log2 sin θ2 + π . (58) ρ( 3 , θ2 ) θ2 ∈(0,π) 18 Thijs Laarhoven Here cn = 21 log2 ( 43 ) ≈ 0.20752, and ρ(θ1 , θ2 ) = ln p(θ1 )/ ln p(θ2 ) corresponds to the exponent ρ for given angles θ1 , θ2 . Note that in the above equation, only ct is an unknown. Substituting the asymptotic collision probabilities from Theorem 1, we find a solution at ct ≈ 0.11464, with maximizing angle θ2 ≈ 0.45739π. This corresponds to a time and space complexity of (n · t)1+o(1) = 2(cn +ct )d+o(d) ≈ 20.32216d+o(d) as claimed. 

arXiv:1609.07450v3 [cs.DM] 21 Feb 2018 Finding long simple paths in a weighted digraph using pseudo-topological orderings Miguel Raggi ∗ mraggi@gmail.com Escuela Nacional de Estudios Superiores Universidad Nacional Autónoma de México Abstract Given a weighted digraph, finding the longest path without repeated vertices is well known to be NP-hard. Furthermore, even giving a reasonable (in a certain sense) approximation algorithm is known to be NP-hard. In this paper we describe an efficient heuristic algorithm for finding long simple paths, using an hybrid approach of heuristic depth-first search and pseudo-topological orders, which are a generalization of topological orders to non acyclic graphs, via a process we call “opening edges”. Keywords: long paths, graphs, graph algorithms, weighted directed graphs, long simple paths, heuristic algorithms. 1 Introduction We focus on the following problem: Given a weighted digraph D = (V, E) with weight w : E → R+ , find a simple path with high weight. The weight of a path is the sum of the individual weights of the edges belonging to the path. A path is said to be simple if it contains no repeated vertices. Possible applications of this problem include motion planning, timing analysis in a VLSI circuit and DNA sequencing. The problem of finding long paths in graphs is well known to be NP-hard, as it is trivially a generalization of HAMILTON PATH. Furthermore, it was proved by Björklund, Husfeldt and Khanna in [BHK04] that the longest path cannot be aproximated in polynomial time within n1−ε for any ε > 0 unless P = N P . ∗ Research supported in part by PAPIIT IA106316, UNAM. Orcid ID: 0000-0001-9100-1655. 1 While LONGEST SIMPLE PATH has been studied extensively in theory for simple graphs (for example in [Scu03], [ZL07], [PD12]), not many efficient heuristic algorithms exist even for simple undirected graphs, much less for weighted directed graphs. A nice survey from 1999 can be found at [Sch99]. A more recent comparison of 4 distinct genetic algorithms for approximating a long simple path can be found in [PAR10]. An implementation of the proposed algorithm won the Oracle MDC coding competition in 2015. In the problem proposed by Oracle in the challenge “Longest overlapping movie names”, one needed to find the largest concatenation of overlapping strings following certain rules, which could be easily transformed to a problem of finding the longest simple path in a directed weighted graph. The graph had around 13,300 vertices. Our contribution lies in a novel method of improving existing paths, and an efficient implementation of said method. The proposed algorithm consists of two parts: finding good candidate paths using heuristic DFS and then improving upon those candidates by attempting to either replace some vertices in the path by longer subpaths–or simply insert some subpaths when possible–by using pseudo-topological orders. The full C++ source code can be downloaded from http://github.com/mraggi/LongestSimplePath. In Section 2 we give some basic definitions. We describe the proposed algorithm in Section 3. Finally, we give some implementation details and show the result of some experimental data in Section 4. It should be noted that for this particular problem it is generally easy to quickly construct somewhat long paths, but only up to a point. After this point even minor improvements get progressively harder. 2 Preliminaries Definition 2.1. A directed acylic graph (or DAG) D is a directed graph with no (directed) cycles. In a directed acyclic graph, one can define a partial order ≺ of the vertices, in which we say v ≺ u iff there is a directed path from v to u. Definition 2.2. A topological ordering for a directed acyclic graph D is a total order of the vertices of D that is consistent with the partial order described above. In other words, it is an ordering of the vertices such that there are no edges of D which go from a “high” vertex to a “low” vertex. Definition 2.3. Given a digraph D, a strongly connected component C is a maximal set of vertices with the following property: for each pair of vertices x, y ∈ C, there exists a (directed) path from x to y and one from y to x. A weakly connected component is a connected component of the associated simple graph. 2 1 2 3 3 2 6 1 5 5 6 4 4 Figure 1: Two different topological orders of the same digraph Definition 2.4. Given a digraph D, the skeleton S of D is the graph constructed as follows: The vertices of S are the strongly connected components of D. Given x, y ∈ V (S), there is an edge x → y iff there exists a ∈ x and b ∈ y such that a → b is an edge of D. It can be observed that S is always a directed acyclic graph. Definition 2.5. Denote by v the connected component of D which contains v. Given a vertex v, we define the out-rank of v as the length of the longest path of S that starts at v. Similarly, we define the in-rank of v as the length of the longest path of S which ends at v. 2.1 Longest simple path on DAGs In case that the digraph is a acyclic, a well-known algorithm that uses dynamic programming can find the optimal path in O(n) time. We describe Dijkstra’s algorithm adapted to finding the longest simple path in a DAG. As this algorithm is an essential building block of the algorithm described in Section 3.2, we add a short description here for convenience. For a longer discussion see, for example, [SW11]. 1. Associate to each vertex v a real number x[v], which will end up representing the weight of the longest simple path that ends at v. 2. Find a topological ordering of the vertices of D. 3. In said topological order, iteratively set x[v] to the max of x[p] + w(p → v) where p → v is an edge, or 0 otherwise. 4. Once we have x[v] for every vertex v, reconstruct the path by backtracking, starting from the vertex v with the highest x[v]. In more detail, 3 Algorithm 1 Longest simple path in a DAG Input: A DAG D with weight function w : E(D) → R. Output: A longest simple path P in D. function LSP DAG(D) x is an array of size |V (D)|, initialized with zeroes. Find a topological order T for V (D) for v ∈ T do x[v] := max{xp + w(p → v) : p → v} v := argmax(x) P := path with only v T 0 := reverse(T ) while x[v] 6= 0 do u := an in-neighbor of v for which x[u] + w(u → v) = x[v] Add u to the front of P v := u return P This algorithm is simple to implement and efficient. Its running time is O(E + V ), where E is the number of edges of D. 3 The Algorithm In what follows we shall assume we have preprocessed the graph and found the weakly connected components, the strongly connected components, and have fast access to both the outgoing edges and incoming edges for each vertex. As we may perform the following algorithm on each weakly connected component, without loss of generality assume D is weakly connected. Our proposed algorithm has two main parts: In the first part we find long paths using heuristic depth first search, choosing in a smart way which vertices to explore first, and in the second part we discuss a strategy to improve the paths obtained in the first part. Since the idea based on DFS is somewhat standard or straightforward, the main contribution of this paper lies in the ideas presented in the second part. 3.1 Depth-first search We describe a variation on depth-first search (DFS). The standard way of implementing a depth-first search is to either use a stack (commonly refered to as the frontier or fringe) of vertices to store unexplored avenues, or to use recursive calls (effectively using the callstack in lieu of the stack). If the graph is not acyclic, DFS may get stuck on a cycle. The standard way of dealing with this problem, when one simply wishes to see every vertex (and not every simple path, 4 as in our case), is to store previously explored vertices in a data structure that allows us to quickly check if a vertex has been explored before or not. However, for the problem of finding the longest simple path, it’s not enough to simply ignore previously explored vertices, as we may need to explore the same vertex many times, as we may arrive at the same vertex from many different paths, and these need to be considered separately. Thus, for this problem, it is not possible to backtrack to reconstruct the path, as in many other problems. This could be solved simply by modifying DFS slightly: make the frontier data structure containing paths instead of only vertices. However, storing paths uses a large amount of memory and all the extra allocations might slow down the search considerably. We propose a faster approach that results from only modifying a single path inplace. This is very likely not an original idea, but an extensive search in the literature did not reveal any article that considers depth-first search in this manner. Probably because for most problems the recursive or stack implementations are quite efficient, as they only need to deal with stacks of vertices and not stacks of paths. In this approach, instead of maintaining a stack of unexplored avenues, assume for each vertex the outgoing edges are sorted in a predictable manner. Later we will sort the outgoing edges in a way that explores the vertices with high probability of producing long paths first, but for now just assume any order that we know. Since this approach modifies the path in place, always make a copy of the best path found so far before any operation that might destroy the path. Furthermore, assume we have a function NextUnexploredEdge(P, {u, v}) that takes as input a path P and an edge {u, v}, in which the last vertex of P is u, and returns the next edge {u, w} in the order mentioned above for which w ∈ / P . This can be found using binary search, or even adding extra information to the edge, so that each edge remembers its index in the adjacency list of the first vertex. If there is no such edge, the function should return null. If no parameter {u, v} is provided, it should return the first edge {u, w} for which w ∈ / P. We will construct a single path P and modify it repeatedly by adding vertices to the back of P . 5 Algorithm 2 Next Path in a DFS manner Input: A weighted digraph D and a path P , which will be modified in place Output: Either done or not done function NextPath(P ) last := last vertex of P t := NextUnexploredEdge(P ) while t = null and |P | > 1 do last := last vertex of P Remove last from P newLast := last vertex of P t := NextUnexploredEdge(P, {newLast, last}) if t = null then return done Add t to the back of P return not done By repeatedly applying this procedure we can explore every path that starts at a given vertex in an efficient manner, but there are still too many possible paths to explore, so we must call off the search after a specified amount of time, or perhaps after a specified amount of time has passed without any improvements on the best so far. Finally, we can do both forward and backward search with a minor modification to this procedure. So once a path cannot be extended forward any more, we can check if it can be extended backward. We found experimentally that erasing the the first few edges of the path before starting the backward search often produces better results. 3.1.1 Choosing the next vertex We give the details for efficiently searching forward, as searching backward is analogous. So we are left with the following two questions: At which vertex do we start our search at? And then, while performing DFS, which vertices do we explore first? That is, how do we find a “good” order of the outgoing edges, so that good paths have a higher chance of being found quickly? The first question is easily answered: start at vertices with high out-rank. To answer the second question, we use a variety of information we collect on each vertex before starting the search: 1. The out-rank and in-rank. 2. The (weighted) out-degree and (weighted) in-degree. 3. A score described below. Once we find the score of each vertex, order the out-neighbors by rank first and score second, with some exceptions we will mention below. The score should not depend on any path, only on local information about the vertex, and should be fast to calculate. 6 Formally, let k be a constant (small) positive integer. For each vertex v, let Ak (v) be the sum of weights of all paths of length k starting at v. For example, A1 (v) is the weighted out-degree. Given a choice of parameters a1 , a2 , ..., ak ∈ R+ , construct the (out) score for vertex v as scoreout (v) = k X ai Ai (v) i=1 Intuitively, the score of each vertex tries to heuristically capture the number (and quality) of paths starting at that vertex. High score means more paths start at a vertex. When performing forward search, perhaps counter-intuitively, giving high priority to vertices with low score (as long as it is not 0) consistently finds better paths than giving high priority to vertices with high score. The reason for this is that exploring vertices with low score first means saving the good vertices–those with high scores–for later use, once more restrictions are in place. Low scoring vertices are usually quickly discarded if there is no out, and so by leaving vertices with high degree for later, when the path is longer and so there are more restrictions about which vertices can be used, makes sense. An exception is if a vertex has degree 0. In this case, we give the vertex a low priority, as no further paths are possible. Another exception is to give higher priority to vertices with very low indegree (for example, indegree 1), since if they are not explored in a path when first finding their parents, they will never be used again later in the path. In addition, we also use the in-degree information in an analogous way. 3.2 Pseudo-topological order The idea behind the second part of the algorithm is to try to improve paths by either inserting some short paths in-between or replacing some vertices by some short paths in an efficient way that covers both. We begin by introducing some definitions. Definition 3.1. Given a digraph D, a weak pseudo-topological ordering ≺ of the vertices of D is a total order of the vertices in which whenever x ≺ y and there is an edge y → x, then x and y are in the same strongly connected component. In other words, a weak pseudo-topological order is a total order that is consistent with the partial order given by the skeleton. Definition 3.2. Given a digraph D, a strong pseudo-topological ordering ≺ of the vertices of D is a total order of the vertices in which whenever x ≺ y and there is an edge y → x, every vertex in the interval [x, y] is in the same strongly connected component. In other words, a strongly pseudo-topological order is a weakly connected component in which the strongly connected components are not intermixed. 7 3 7 1 4 2 6 8 5 Figure 2: A (strong) pseudo-topological ordering From here on, whenever we mention a pseudo-topological ordering, we mean a strong pseudo-topological ordering. An easy way to get a random pseudo-topological ordering is to get a random topological ordering of the skeleton of the graph, and then “explode” the strongly connected components, choosing a random permutation of vertices in each component. We can think of a pseudo-topological ordering as a topological ordering of the digraph in which we erase all edges that go from a “high” vertex to a “low” vertex, thus considering an acyclic subdigraph of the original. We call this graph the subjacent DAG of the pseudotopological order. Thus, we may apply Algorithm 1 to this acyclic digraph and find its longest simple path. As can be expected, the results obtained in this fashion are very poor compared to even non heuristic recursive slow depth-first search. However, if we combine the two approaches we get better paths. 3.2.1 Combining the two approaches Definition 3.3. Given a path P and a pseudo-topological ordering T , the imposition of P on T is a pseudo-topological ordering TP which every vertex not in P stays in the same position as in T . The vertices in P are permuted to match the order given by P . For example, say we start with path P = 3 → 1 → 5 → 8. Consider any pseudotopological ordering, say, T = (1, 8, 7, 4, 3, 6, 5, 2). Then imposing the order defined by P into T gives rise to T 0 = (3, 1, 7, 4, 5, 6, 8, 2). Lemma 3.4. TP as constructed above is also a (strong) pseudo-topological order. Proof. As T is a strong pseudo-topological order, consider S1 , S2 , ... , Sc the strongly connected components in the order they appear in T . Denote by s(v) index of the strongly connected component of v. It suffices to prove that the vertices only move inside their own strongly connected components when going from T to TP . 8 S1 S2 S3 S4 S5 Figure 3: No backward edges can jump between strongly connected components. Let (p1 , p2 , ..., pk ) be the vertices of path P . Note that there is no i < j for which s(pi ) > s(pj ) since this would mean there exists a path from a vertex in Ss(pi ) to a vertex in Ss(pj ) , but if s(pi ) > s(pj ), no such path is possible in a pseudo-topological order, since this violates the order in the skeleton. This means that when imposing the order of P into T to get TP , no vertex can jump out of their strongly connected component, and thus TP is also a strong pseudo-topological order. The previous lemma ensures that we may run algorithm 1 with order TP , and get a path that is at least as good (and hopefully better) as path P , since all edges of P remain in the subjacent DAG of TP . If after applying this technique we do find an improved path P 0 , we can repeat the process with P 0 , by again taking a random pseudo-topological ordering, imposing the order of P 0 on this new ordering, and so on, until there is no more improvement. The idea then is to construct long paths quickly with DFS and then use these paths as starting points for imposing on random pseudo-topological orders. This approach does indeed end up producing moderately better paths than only doing DFS, even when starting from scratch with the trivial path and a random pseudo-topological order, albeit taking longer. However, we can do better. 3.2.2 Opening the edges Again, we are in the setting where we have a path P which we wish to improve. Now, instead of just imposing path P on multiple random pseudo-topological orders to find one that gets an improvement, construct orders as follows: Pick an edge pi → pi+1 of path P and construct a random pseudo-topological order that is consistent with P and furthermore, for which pi and pi+1 are as far apart as possible. Figure 4: The process of opening an edge. This is achieved by putting all vertices not in P in all strongly connected components between pi and pj in between pi and pj . In the figure above, the “large” vertices are vertices in P and the “small” vertices are all other vertices not in P in the same connected component as 9 pi and pi+1 . If pi and pi+1 are not in the same connected component, then, place every vertex in either connected component, and also every vertex that belongs in a connected component between the component of pi and the component of pi+1 between the two vertices, in such a way that the order is still a strong pseudo-topological order. We may repeat this process for each edge in P . The process of opening an edge is relatively expensive, since we must run Algorithm 1 each time. We now make an attempt at explaining why opening edges works as well as it does. Consider: 1. If there exist a vertex v that can be inserted into P , opening the corresponding edge finds this improvement. 2. If there exists a vertex v that can replace a vertex p in P and make the path longer (by means of edge weights), this process will find it when opening the edge to either side of p. 3. Any (small) path of size k that can be inserted into P , perhaps even replacing a few vertices of P , has probability at least 1/k! of being found if the corresponding vertices in the small path happen to be in the correct order. In the next section we try to heuristically maximize the probability that inserting or replacing paths will be found. 3.2.3 Opening the edges eXtreme In the previous section, when opening up each edge, we put all the remaining vertices in the same connected component in a random order (consistent with pseudo-topological orders). We now consider the question of which order to give to those unused vertices. We discuss three different approaches: one heuristic that is quick and practical, one powerful and somewhat slow, and one purely theoretical using sequence covering arrays but which provides some theoretical guarantees. Let B be the inbetween vertices (i.e. vertices between pi and pi+1 when opening this edge). Since every other vertex will remain in their place, we face the problem of giving B an ordering with a good chance of delivering an improvement. We only consider orders of B that leaves the total order a strongly pseudo-topological order. That is, we only permute the vertices of B within their own strongly connected components of the full graph. Consider the induced subdigraph on B: Recall that running Algorithm 1 with a pseudo-topological order is equivalent to finding the longest path on the digraph that results by erasing all the edges that go backward. 10 Therefore, we must consider only orders of B that are themselves pseudo-topological orders of the induced subgraph on B. We describe three approaches to choosing an order of B. 3.2.4 The powerful approach The “powerful” approach is to recursively repeat the process on the induced subgraph on B. That is, repeat for B the whole process of finding the strongly connected components, performing DFS as described in a previous section, finding suitable pseudo-topological orders, opening their respective edges, and recursively doing the same for the induced subgraphs in order to find good pseudo-topological orders and so on. The problem with this approach is that with any standard cache-friendly graph data structure we would need to reconstruct the induced digraph (i.e. rename the vertices of B but keep track of their original names), and the whole process is slow. Of course, we would only need to do this process once per connected component and then we can use the results for each edge of the path. The advantage of this approach is that we are precisely recursively finding good pseudotopological orders of B, which means it’s likely many long paths can be inserted in our original P . 3.2.5 The heuristic approach Instead of attempting to repeat the whole algorithm on the induced subgraph, we try to mimic heuristically its results. Consider the following operation on the inbetween vertices: pick a vertex u at random, and exchange its position with some out-neighbor v of u which appeared before u in the pseudo-topological order. If no such neighbor exists, simply find pick another u. Repeat this operation (which is quite inexpensive) as many times as the time limit allows. The following theorem ensures that this process will likely end in a (weak) pseudo-topological order. Theorem 3.5. With probability approaching 1 as the number of times approaches infinity, the order constructed above is a weak pseudo-topological order of the induced subgraph B. Proof: For any digraph, given a total order of the vertices, call a pair of vertices (a, b) bad if a appears before b in the order, there is a path from b to a but not one from a to b. In other words, if the strongly connected component which contains b is (strictly) less than the strongly connected component which contains a in the partial order of the skeleton of D. Thus, we only need to prove that the number of bad pairs never increases after an operation, and that with some positive probability, it decreases. Suppose we do an operation, and u is the chosen vertex, which is exchanged with v (so u was after v in the order, but there is an edge from u to v). Then only pairs of vertices of the form (a, u) and (v, a) could have changed status (and only when a is between u and v in the order). 11 Let U, V, A be the strongly connected components containing u, v, a respectively. If A is before U , then indeed the pair (a, u) is now bad after the exchange, but since we are assuming there is an edge from u to v, if A is before U , then it is before V , and so the pair (v, a) used to be bad, but is now good. When (v, a) becomes bad, the process is analogous, and (a, u) becomes good. The above process then gives a random approximate algorithm to calculate weak pseudotopological orders without calculating the strongly connected components. 3.2.6 The theoretical approach Denote E(P ) the edge set of a path P . For a positive integer k, we wish to find P that is maximal in the sense that if Q is another path for which |E(Q) \ E(P )| ≤ k, then the total weight of Q is less than or equal to the total weight of P . Given the edge opening process, this problem can be reduced to the following: we wish for a minimal set of permutations of Sn for which every k-subset of n appears in every possible order. The idea is to try an edge opening for every edge in the path with the order of B given by each element of the set of permutations. This problem has been worked on by Colbourn et al. on [CCHZ13] and [MC14], where they named any such set of permutations a sequence covering array. They give a mostly impractical algorithm for finding covering arrays, that works in practice up to n ≈ 100. However, an easy probabilistic argument yields that taking Θ(log(n)) permutations randomly gives a non-zero chance of ending up with a covering array. This suggests that merely taking many random permutations would yield (probabilistically) the desired result. Unsurprisingly, this approach is not nearly as efficient as the other two. For k = 2, however, a covering array is easy to find: take any permutation and its reverse. So by opening every edge and taking any permutation of the inbetween vertices and its reverse, we ensure the found path is optimal in this very limited sense: There exists no other path with higher total weight all whose edges, except one, are a subset of the edges of P plus one more. 4 4.1 Some details about the implementation Preprocessing the graph The data structure we use for storing the graph is a simple pair of adjacency lists: one for the out-neighbors and one for the in-neighbors, so we can access either efficiently. The vertices are numbered from 0 to n − 1 and we store an array of size n, each element of which is an array storing both the neighbors of the corresponding vertex and the corresponding edge-weights. Next, we find connected components. While it is true that finding the weakly connected components, strongly connected components and skeleton might require some non-trivial 12 processing, this pales in comparison to the NP-hard problem of finding the longest path. An efficient implementation (for example, using C++ boost graph library) can find the weakly and strongly connected components on graphs with millions of edges in a fraction of a second. Then, we find the out-heuristic and the in-heuristic scores for each vertex, as described in Section 3.1, and sort the neighbors of each vertex according to the heuristic. In our experiments, the whole preprocessing step took about 0.2 seconds on the Oracle graph, which has ∼13,300 vertices. This time includes reading the file of strings, figuring out which concatenations are possible and constructing the graph. Experiments with randomly generated graphs of comparable size take less than 0.1 seconds if the graph is already in memory. If one has a training set of a class of graphs, one could use some rudimentary machine learning to try to find the optimal parameters so that on average good paths are found. In fact, for the contest, we did just that, which provided a slight boost. The code includes a trainer for this purpose, but experimental results on the benefits of this are sketchy at best and do not (usually) warrant the long time it takes to train. 4.2 Pseudo-Topological orders Once we have a pseudo-topological order T , we construct its inverse for fast access, so in addition of being able to answer the query “which vertex is in position i of T ?” we can also answer the query “at which position is vertex v in T ?” efficiently. Therefore, any operation we do on T must be mirrored to the inverse. In addition, since we are constantly changing T and having to rerun Algorithm 1, it is worth it to store xv for each v, and just reprocess from the first vertex whose order was modified and onwards. Fortunately, when performing the edge opening process, much of the order has been preserved, so we can use this information and recalculate from the first modification onwards, speeding up the calculation considerably. Finally, opening the edges is just a matter of rearranging the vertices to satisfy the condition, which is straightforward. We found experimentally that the process finds good paths faster if the edges of the path are opened in a random order and not sequentially. This makes intuitive sense. If a path cannot be improved by a certain edge opening, it’s unlikely (but possible) an edge that is near will yield an improvement. Our implementation of the “powerful” approach described in Section 3.2.4 was by constructing a completely new graph and running the algorithm on the subgraph recursively, and so it was prohibitely slow, although it did tend to find longer subpath insertions with fewer edge openings. Perhaps this can be improved. The implementation of the heuristic approach of Section 3.2.5 was considerably more efficient over the random approach described in Section 3.2.6. 13 5 Experimental data We compare this algorithm to one other found in the literature, by Portugal et. al. [PAR10], as the authors have kindly provided us with the code. There is a scarcity of heuristic algorithms for this problem, and the code for some, such as [Sch99] appears to have been lost, so a direct comparison turns out to be impractical. Unfortunately, an extensive literature search did not provide any other accessible source code for this problem, making the code in the link of the introduction the only open source and readily available implementation we are aware of that heuristically finds long simple paths. In [PAR10], the authors compare four approaches based on genetic algorithms. The biggest graph they used as an example consists of 134 vertices. The result was that their fastest algorithm was able to find the optimal solution more than half of the time in around 22 seconds. For comparison, our program only took 0.001 seconds for the same graph and found the longest path on 100% of the test runs. Please bear in mind that the comparison might be somewhat unfair, since their implementation was in Matlab instead of C++. Our algorithm took less than a millisecond for all the graphs in [PAR10] and found the longest simple path 100% of the test runs. 5.1 Tests in large random graphs where we know the longest path size Given n and m, consider the following graph generation process for a graph with n vertices and m edges. Consider any random permutation of the vertices v1 , v2 , ..., vn and add all edges vi → vi+1 . Then pick n − m + 1 other edges uniformly at random. All edge weights were set to 1, so we know for certain the longest simple path has size n − 1. For example, in our experiments, for n = 10, 000 and m = 100, 000, the whole process (including reading the file containing the graph) took on average 1.28 seconds to find the longest simple path. 6 Acknowledgements We would like to thank the organizers of the Oracle MDC coding challenge for providing a very interesting problem to work on (and for the prize of a drone and camera, of course). Furthermore, we would like to thank the other participants, specially Miguel Ángel Sánchez Pérez and David Felipe Castillo Velázquez for the fierce competition. Also, we are grateful to Marisol Flores and Edgardo Roldán for their helpful comments on the paper, as well as David Portugal for providing the source code from their work. This research was partially supported by PAPIIT IA106316. 14 References [BHK04] Andreas Björklund, Thore Husfeldt, and Sanjeev Khanna, Approximating longest directed paths and cycles, Automata, Languages and Programming, Springer, 2004, pp. 222–233. [CCHZ13] Yeow Meng Chee, Charles J Colbourn, Daniel Horsley, and Junling Zhou, Sequence covering arrays, SIAM Journal on Discrete Mathematics 27 (2013), no. 4, 1844–1861. [MC14] Patrick C Murray and Charles J Colbourn, Sequence covering arrays and linear extensions, Combinatorial Algorithms, Springer, 2014, pp. 274–285. [PAR10] David Portugal, Carlos Henggeler Antunes, and Rui Rocha, A study of genetic algorithms for approximating the longest path in generic graphs, Systems Man and Cybernetics (SMC), 2010 IEEE International Conference on, IEEE, 2010, pp. 2539–2544. [PD12] Quang Dung Pham and Yves Deville, Integration of ai and or techniques in contraint programming for combinatorial optimzation problems: 9th international conference, cpaior 2012, nantes, france, may 28 – june1, 2012. proceedings, ch. Solving the Longest Simple Path Problem with Constraint-Based Techniques, pp. 292–306, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. [Sch99] John Kenneth Scholvin, Approximating the longest path problem with heuristics: a survey, Ph.D. thesis, University of Illinois at Chicago, 1999. [Scu03] Maria Grazia Scutella, An approximation algorithm for computing longest paths, European Journal of Operational Research 148 (2003), no. 3, 584–590. [SW11] R. Sedgewick and K. Wayne, Algorithms, Pearson Education, 2011. [ZL07] Zhao Zhang and Hao Li, Algorithms for long paths in graphs, Theoretical Computer Science 377 (2007), no. 1, 25–34. 15 

Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture M. Dumnicki, A. Küronya∗, C. Maclean, T. Szemberg† arXiv:1304.0249v1 [math.AG] 31 Mar 2013 April 2, 2013 Abstract In this paper we relate the SHGH Conjecture to the rationality of onepoint Seshadri constants on blow ups of the projective plane, and explain how rationality of Seshadri constants can be tested with the help of functions on Newton–Okounkov bodies. Keywords Nagata Conjecture, SHGH Conjecture, Seshadri constants, Okounkov bodies Mathematics Subject Classification (2000) MSC 14C20 1 Introduction Nagata’s conjecture and its generalizations have been a central problem in the theory of surfaces for many years, and much work has been done towards verifying them [19], [8], [13], [23], [9]. In this paper we open a new line of attack in which we relate Nagata-type statements to the rationality of one-point Seshadri constants and invariants of functions on Newton–Okounkov bodies. We obtain as a consequence of our approach some evidence that certain Nagata-type questions might be false. Seshadri constants were first introduced by Demailly in the course of his work on Fujita’s conjecture [10] in the late 80’s and have been the object of considerable interest ever since. Recall that given a smooth projective variety X and a nef line bundle L on X, the Seshadri constant of L at a point x ∈ X is the real number ε(L; x) =def inf C L·C , multx C (1) where the infimum is taken over all irreducible curves passing through x. An intriguing and notoriously difficult problem about Seshadri constants on surfaces is the question whether these invariants are rational numbers, see [17, Remark 5.1.13] It follows quickly from √ their definition that if a Seshadri constant is irrational then it must be ε(L; x) = L2 , see e.g. [3, Theorem 2.1.5]. It is also known that Seshadri constants of a fixed line bundle L, take their maximal value on a subset in X which is a complement of at most countably many Zariski closed proper subsets of X. ∗ During this project Alex Küronya was partially supported by DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”, and the OTKA Grants 77476 and 81203 by the Hungarian Academy of Sciences. † Szemberg research was partially supported by NCN grant UMO-2011/01/B/ST1/04875 2 We denote this maximum by ε(L, 1). Similar notation is used for√multi-point 2 Seshadri constants, see [3, Definition 1.9]. In particular, if ε(L; x) √ = L at some point the same holds in a very general point on X and ε(L; 1) = L2 . From a slightly different point of view, Seshadri constants reveal information on the structure of the nef cone on the blow-up of X at x, hence their study is closely related to our attempts to understand Mori cones of surfaces. An even older problem concerning linear series on algebraic surfaces is the conjecture formulated by Beniamino Segre in 1961 and rediscovered, made more precise and reformulated by Harbourne 1986, Gimigliano 1987 and Hirschowitz 1988. (See [8] for a very nice account on this development and related subjects.) In particular it is known, [8, Remark 5.12] that the SHGH Conjecture implies the Nagata Conjecture. We now recall this conjecture, using by Gimigliano’s formuation, which will be the most convenient form for us [12, Conjecture 3.3]. SHGH Conjecture. Let X be the blow up of the projective plane P 2 in s general points with exceptional divisors E1 , . . . , Es . Let H denote the pullback to X of the hyperplane bundle OP2 (1) on P2 . Let the integers d, m1 > . . . > ms > −1 with d > m1 + m2 + m3 be given. Then the line bundle dH − s X mi Ei i=1 is non-special. The main result of this note is the following somewhat unexpected relation between the SHGH Conjecture and the rationality problem for Seshadri constants. Theorem 1.1. Let s > 9 be an integer for which the SHGH Conjecture holds true. Let X be the blow up of the projective plane P2 in s general points. Then a) either there exists on X an ample line bundle whose Seshadri constant at a very general point is irrational; b) or the SHGH Conjecture fails for s + 1 points. Note that it is known that the SHGH conjecture holds true for s 6 9, [8, Theorem 5.1]. It is also known that Seshadri constants of ample line bundles on del Pezzo surfaces (i.e. for s 6 8) are rational, see [22, Theorem 1.6]. In any case, the statement of the Theorem is interesting (and non-empty) for s = 9. (See the challenge at the end of the article.) Corollary 1.2. If all one-point Seshadri constants on the blow-up of P2 in nine general points are rational, then the SHGH conjecture fails for ten points. An interesting feature of our proof is that the role played by the general position of the points at which we blow up becomes clear. In a different direction, we study the connection between functions on Newton– Okounkov bodies defined by orders of vanishing, and Seshadri-type invariants. Our main result along these lines is the following. Theorem 1.3. Let X be a smooth projective surface, Y• an admissible flag, L a big line bundle on X, and let P ∈ X be an arbitrary point. If max x∈∆Y• (L) φordp (x) ∈ Q then ε(L, P ) ∈ Q . 3 Acknowledgements. We thank Cristiano Bocci, Sébastien Boucksom and Patrick Graf for helpful discussions. Part of this work was done while the second author was visiting the Uniwersytet Pedagogiczny in Cracow. We would like to thank the Uniwersytet Pedagogiczny for the excellent working conditions. 2 Rationality of one point Seshadri constants and the SHGH Conjecture In this section we prove Theorem 1.1: we start with notation and preliminary leamms. Let f : X → P2 be the blow up of P2 at s > 9 general points P1 , . . . , Ps with exceptional divisors E1 , . . . , Es . We denote as usual by H = f ∗ (OP2 (1)) the pull back of the hyperplane bundle and we let E = E1 + · · · + Es be the sum of exceptional divisors. We consider the blow up g : Y → X of X at P with exceptional divisor F . Whilst The following result is well known, we include it for the lack of a proper reference. P Lemma 2.1. If there exists a curve C ⊂ X in the linear system dH − si=1 mi Ei computing Seshadri constant of a Q-line bundle L = H − αE, then there exists a divisor Γ with multP1 Γ = . . . = multPs Γ = M computing Seshadri constant of L at P , i.e. L·C L·Γ = = ε(L; P ). multP Γ multP C Proof. Since the points P1 , . . . , Ps are general, there exist curves Cσ = dH − s X mσ(j) Ej j=1 for all permutations σ ∈ Σs . Since the point P is general, we may take all these curves to have the same multiplicity m at P . Summing over a cycle σ of length s in Σs , we obtain a divisor Γ= s X i=1 Cσi = sdH − s s X X i=1 j=1 mσi (j) Ej = sdH − M E, with M = m1 + . . . + ms . Note that the multiplicity of Γ at P equals sm. Taking the Seshadri quotient for Γ we have sd − αsM d − αM L·Γ = = = ε(L; P ) sm sm m hence Γ satisfies the assertions of the Lemma. The following auxiliary Lemma will be used in the proof of Theorem 1.1. We postpone its proof to the end of this section. Lemma 2.2. Let s > 9 be an integer. The function p √ √ f (δ) = (2 s + 1 − s) 1 − sδ2 + s(1 − s + 1)δ + s − 2 (2) is non-negative for δ satisfying √ 1 1 <δ< √ . s s+1 (3) 4 Proof of Theorem 1.1. Let δ be a rational number satisfying (3). Note that the SHGH Conjecture implies the Nagata Conjecture [8, Remark 5.12] so that 1 ε(OP2 (1); s) = √ s and hence the Q–divisor L = H − δE is ample. If ε(L; 1) is irrational, then we are done. √ So we proceed assuming that ε(L; 1) is rational and that it is not equal to L2 (this can be achieved changing δ a little bit if necessary). In particular, by Lemma 2.1 for a general point P ∈ X Seshadri constant ε(L; P ) there is a divisor Γ ⊂ P2 of degree γ with M = multP1 Γ = . . . = multPs γ and m = multP Γ whose proper e on X computes the Seshadri constant transform Γ ε(L; P ) = p e L·Γ γ − δsM = < 1 − sδ2 . m m This gives an upper bound on γ γ<m p 1 − sδ2 + δsM. (4) We need to prove that statement b) in Theorem 1.1 holds. Suppose not: the SHGH Conjecture then holds for s + 1 points in P2 . The Nagata Conjecture then also holds for s + 1 points and this gives a lower bound for γ, since for Γ we must have that 1 γ . >√ sM + m s+1 (5) γ > 2M + m. (6) γ < 2M + m. (7) We now claim that Suppose not. We then have that The real numbers √ √ 2 s+1−s s − δs s + 1 a := and b := 2 − δs 2 − δs are positive. Multiplying (4) by a and (7) by b and adding we obtain p √ sM + m 6 γ s + 1 < sM + (b + a 1 − sδ2 )m, where the first inequality follows from (5). Subtracting sM in the left and in the right term and dividing by m we obtain p 1 < b + a 1 − sδ2 . Plugging in the definition of a and b and rearranging terms we obtain that p √ √ (2 s + 1 − s) 1 − sδ2 + s − δs s + 1 < 2 − δs, which contradicts Lemma 2.2. Hence (6) holds. 5 It follows now from the SHGH conjecture for s + 1 points (in the form stated in the introduction) that the linear system γH − M E − mF on Y is non-special. Indeed the condition γ > 2M + m is (6) and the condition γ > √1s (because the Nagata Conjecture holds for s γ > 3M is satisfied since sM by hypothesis) and because we have assumed that s > 9. This system is also nonempty because the proper transform of Γ under g is its member. Thus by a standard dimension count 0 6 γ(γ + 3) − sM (M + 1) − m(m + 1). The upper bound on γ (4) together with the above inequality yields p p 0 6 (sδM + m 1 − sδ2 )(sδM + m 1 − sδ2 + 3) − m2 − m − sM − sM 2 . (8) Note that the quadratic term in (8) is a negative semi-definite form p (s2 δ2 − s)M 2 + 2sδ 1 − sδ2 M m − sδ2 m2 . Indeed, the restrictions on δ made in (3) imply that the term at M 2 is negative. The determinant of the associated symmetric matrix vanishes. These two conditions imply together that the form is negative semi-definite. In particular this term of (8) is non-positive. The linear part in turn is p (3sδ − s)M + (3 1 − sδ2 − 1)m, which is easily seen to be negative. This provides the desired contradiction and finishes the proof of the Theorem. Remark 2.3. As it is well known, Nagata’s conjecture can be interpreted in terms of the nef and Mori cones of the blow-up X of P2 at s general points. More precisely, consider the following question: for what t > 0 does the ray H − tE meet the boundary of the nef cone? The conjecture predicts that this ray should intersect the boundaries of the nef cone and the effective cone at the same time. Considering the Zariski chamber structure of X (see [5]), we see that this is equivalent to requiring that H − tE crosses exactly one Zariski chamber (the nef cone itself). Surprisingly, it is easy to prove that H − tE cannot cross more than two chambers. Proposition 2.4. Let f : X → P2 be the blow up of P 2 in s general points with exceptional divisors E1 , . . . , Es . Let H be the pull-back of the hyperplane bundle and E = E1 + . . . + Es . The ray R = H − tE meets at most two Zariski chambers on X. Proof. If ε = ε(OP2 (1); s) = √1s i.e. this multi-point Seshadri constant is maximal, then the ray crosses only the nef cone. P If ε is submaximal, then there is a curve C = dH − mi Ei computing this d P Seshadri constant, i.e. ε = mi . If this curve is homogeneous, i.e. m = m1 = · · · = ms , then we claim first that µ = µ(OP2 (1); E) = m/d. Indeed, this is an effective divisor on the ray R and it is not big (because big divisors on surfaces intersect all nef divisors positively, see [6, Corollary 3.3]), so it must be the point where the ray leaves the big cone. 6 Now, suppose that for some ε < δ < µ the ray R crosses another Zariski chamber wall. This means that there is a divisor eH −kE (obtained after possible symmetrization of a curve D with 0 intersection number with H − δE) with (H − δE) · (eH − kE) = 0. Hence e = δks < µks. On the other hand (eH − kE) · (dH − µE) = e − kµs < 0, implies that C is a component eH − kE which is not possible. Hence there are only two Zariski chambers meeting the ray R in this case. If the curve C is not homogeneous, then since the points are general there exist at least (and also at most) s different irreducible curves computing ε. All these curves are in the support of the negative part of the Zariski decomposition of H − λE for λ > ε. Hence their intersection matrix is negative definite and this is a matrix of maximal dimension (namely s) with this property. This implies that R cannot meet another Zariski chamber because the support of the negative part of Zariski decompositions grows only when encountering new chambers, see [5]. It is interesting to compare this result with the following easy example, which constructs rays meeting a maximal number of chambers. Example 2.5. Keeping the notation from Proposition 2.4 let L = ( s(s+1) + 1)H − 2 E1 − 2E2 − . . .− sEs is an ample divisor on X and the ray R = L + λE crosses s + 1 = ρ(X) Zariski chambers. Indeed, with λ growing, exceptional divisors E1 , E2 , . . . , Es join the support of the Zariski decomposition of L − λE one by one. We leave the details to the reader. We conclude this section with the proof of Lemma 2.2. √ s + 1) = 0 it is enough to show that f (δ) is inProof of Lemma √ 2.2. Since f (1/ √ creasing for 1/ s + 1 6 δ 6 1/ s. Consider the derivative   √ √ δ ′ (9) f (δ) = s 1 + √ (s − 2 s + 1) − s + 1 . 1 − sδ2 √ √ δ The function h(δ) = √1−sδ is increasing for 1/ s + 1 6 δ 6 1/ s since the nu2 merator is an increasing function of δ and the denominator is a decreasing function 1 ) = 1 so that h(δ) > 1 holds for all δ. Since the coefficient at of δ. We have h( √s+1 h(δ) in (9) is positive we have √ √ √
f ′ (δ) > s 1 + (s − 2 s + 1) − s + 1 = 1 + s − 3 s + 1 > 0, which completes the proof. 3 Rationality of Seshadri constants and functions on Okounkov bodies The theory of Newton–Okounkov bodies has emerged recently with work by Okounkov [21], Kaveh–Khovanskii [14], and Lazarsfeld–Mustaţă [18]. Shortly thereafter, Boucksom–Chen [7] and Witt-Nyström [20] have shown ways of constructing geometrically significant functions on Okounkov bodies, that were further studied 7 in [16]. In the context of this note the study of Okounkov functions was pursued by the last three authors in [16]. We refer to [16] for construction and properties of Okounkov functions. In this section we consider an arbitrary smooth projective surface X and an ample line bundle L on X. Let p ∈ X be an arbitrary point and let π : Y → X be the blow up of p with exceptional divisor E. Recall that the Seshadri constant of L at p can equivalently be defined as ε(L; p) = sup {t > 0 | π ∗ L − tE is nef } . There is a related invariant def µ(L; p) = sup {t > 0 | π ∗ L − tE is pseudo-effective} = sup {t > 0 | π ∗ L − tE is big} . The invariant ε(L; p) is the value of the parameter λ where the ray π ∗ L − λE meets the boundary of the nef cone of Y , and µ(L; p) is the value of λ where the ray meets the boundary of the pseudo-effective cone. The following relation between the two invariants is important in our considerations. Remark 3.1. If ǫ(L; p) is irrational, then ǫ(L; p) = µ(L; p) . In particular, if µ(L; p) is rational, then so is ǫ(L; p). Rationality of µ(L; p) implies rationality of the associated Seshadri constants on surfaces. This invariant appears in the study of the concave function ϕordp associated to the geometric valuation on X defined by the order of vanishing ordp at p. We fix some flag Y• : X ⊇ C ⊇ {x0 } and consider the Okounkov body ∆Y• (L) defined with respect to that flag. We define also a multiplicative filtration determined by the geometrical valuation ordP on the graded algebra V = ⊕k>0 Vk with Vk = H 0 (X, kL) by Ft (V ) = {s ∈ V : ordP (s) > t} , see [16, Example 3.7] for details. (All the above remains valid in the more general context of graded linear series.) There is an induced filtration F• (Vk ) on every summand of V and one defines the maximal jumping numbers of both filtrations as emax (V, F• ) = sup {t ∈ R : ∃kFktVk 6= 0} and emax (Vk , F• ) = sup {t ∈ R : Ft Vk 6= 0} respectively. Let ϕordP (x) = ϕF• (x) be the Okounkov function on ∆Y• (L) determined by filtration F• , see [16, Definition 4.8]. It turns out that µ(L; p) is the maximum of the Okounkov function ϕordP . Proposition 3.2. With notation as above we have that  max ordp (s) | s ∈ H 0 (X, OX (mL)) µ(L; p) = lim sup m m→∞ = max x∈∆Y• (L) φordp (x) . Proof. Observe that ordp (s) = ordE (π ∗ s) = max {m ∈ N | div(π ∗ s) − mE is effective} . Consequently, µ(L; p) = sup {t ∈ R>0 | π ∗ L − tE is pseudo-effective}  max ordp (s) | s ∈ H 0 (X, OX (mL) , = lim sup m m→∞ 8 which gives the first equality. For the second equality, we observe first that  max ordp (s) | s ∈ H 0 (X, OX (mL)) = emax (Vm , F• ) , and hence  max ordp (s) | s ∈ H 0 (X, OX (mL)) lim sup m m→∞ = emax (V, F• ) . Since emax (V, F• ) = max x∈∆Y• (L) ϕordp (x) by Theorem 3.4, we are done. 3.1 Independence of the maximum of an Okounkov function on the flag In the course of this section the projective variety X can have arbitrary dimension. Boucksom and Chen proved that though ϕF• and ∆(V• ) depend on the flag Y• , the integral of ϕF• over ∆(V• ) is independent of Y• , [7, Remark 1.12 (ii)]. We prove now that the maximum of the Okounkov function does not depend on the flag. This fact is valid in the general setting of arbitrary multiplicative filtration F defined on a graded linear series V• . Remark 3.3. Note that in general the functions ϕF• are only upper-semicontinuous and concave, but not continuous on the whole Newton–Okounkov body as explained in [16, Theorem 1.1]. They are however continuous provided the underlying body ∆(V• ) is a polytope (see again [16, Theorem 1.1]), which is the case for complete linear series on surfaces [15]. Theorem 3.4 (Maximum of Okounkov functions). With the above notation, we have that max ϕF• (x) = emax (L, F• ). x∈∆Y• (L) In particular the left hand side does not depend on the flag Y• . Proof. For any real t > 0, we consider the partial Okounkov body ∆t,Y• (L) associated the graded linear series Vt,k ⊂ H 0 (kL) given by def Vt,k = Fkt (H 0 (kL)) . Note that by definition emax (L, F• ) = sup{t ∈ R| ∪k Vt,k 6= 0.} In other words, emax (L, F• ) = sup{t ∈ R|∆t,Y• (L) 6= ∅}. Recall that by definition ϕF• (x) = sup{t ∈ R|x ∈ ∆t,Y• (L)}. and it is therefore immediate that ∀x ϕF• (x) 6 emax (L, F• ). 9 from which it follows that max x∈∆Y• (L) ϕF• (x) 6 emax (L, F• ). Since the bodies ∆t,Y• (L) form a decreasing family of closed subsets of Rd , we have that ∩t|∆t,Y• (L)6=∅ ∆t,Y• (L) 6= ∅. Consider a point y ∈ ∩t|∆t,Y• (L)6=∅ ∆t,Y• (L) We then have that y ∈ ∆t,Y• (L) ⇔ ∆t,Y• (L) 6= ∅ and hence sup{t ∈ R|y ∈ ∆t,Y• (L)} = sup{t ∈ R|∆t,Y• (L) 6= ∅} or in other words ϕF• (y) = emax (L, F• ) from which it follows that max x∈∆Y• (L) ϕF• (x) 6 emax (L, F• ). This completes the proof of the theorem. 4 The effect of blowing up on Okounkov bodies and functions We begin with an observation (valid in fact in arbitrary dimension, though we state and prove it here only for surfaces.) Proposition 4.1. Let S be an arbitrary surface with a fixed flag Y• and let f : X → S be the blow up of S at a point P not contained in the divisorial part of the flag, P ∈ / Y1 . Let E be the exceptional divisor. And finally let D be a big divisor on S. For any rational number λ such that 0 6 λ < µ(L; P ) we let Dλ be the Q–divisor f ∗ D − λE. There is then a natural inclusion ∆Y• (Dλ ) ⊂ ∆Y• (D). Moreover, a filtration F• on the graded algebra ⊕k>0 H 0 (S; kD) induces a filtration F•λ on the graded (sub)algebra ⊕H 0 (X, kDλ ), where the sum is taken over all k divisible enough. For associated Okounkov functions we have ϕF•λ (x) 6 ϕF• (x) (10) for all x ∈ ∆Y• (Dλ ). Remark 4.2. The best case scenario is that the functions φ are piecewise linear with rational coefficients over a rational polytope. Of these properties, some evidence for the first was given by Donaldson [11] in the toric situation. For the second condition, it was proven in [1] that every line bundle on a surface has an Okounkov body which is a rational polytope. 10 Proof. Note first that since the blow up center is disjoint from all elements in the flag, one can take Y• to be an admissible flag on X. (Strictly speaking one takes f ∗ Y• as the flag, but it should cause no confusion to identify flag elements upstairs and downstairs.) Then, if k is sufficiently divisible we have  H 0 (X, f ∗ kD − kλE) = s ∈ H 0 (S, kD) : ordP (s) > E ⊂ H 0 (S, kD). The inclusion of the Okounkov bodies follows immediately under this identification. We can view the algebra associated to f ∗ D − E as a graded linear series on S. The claim about the Okounkov functions follows from their definition, see [16, Definition 4.8]. Indeed, the supremum arising in the definition of ϕF•λ is taken over a smaller set of sections than it is for ϕF• . The following examples illustrate various situations arising in the setting of Proposition 4.1. Example 4.3. Let ℓ be a line in X0 = P2 and let P0 ∈ ℓ be a point. We fix the flag Y• : X0 ⊃ ℓ ⊃ {P0 } . Let D = OP2 (1). Then ∆Y• (D) is simply the standard simplex in R2 . 1 ∆Y• (D) 0 1 Let F• be the filtration on the complete linear series of D imposed by the geometric valuation ν = ordP0 and let ϕν be the associated Okounkov function. Then ϕν (a, b) = a + b. Indeed, given a point (a, b) with rational coordinates, we pass to the integral point (ka, kb). This valuation vector can be realized geometrically by a global section in H 0 (P2 , OP2 (k)) vanishing exactly with multiplicity ka along ℓ, exactly with multiplicity kb along a line passing through P0 different from ℓ and along a curve of degree k(1 − a − b) not passing through P0 . The next example shows that even when the Okounkov body changes in the course of blowing up, the Okounkov function may remain the same. Example 4.4. Keeping the notation from the previous Example and from Proposition 4.1 let f : X1 = BlP1 P2 → X0 = P2 be the blow up of the projective plane in a point P1 not contained in the flag line ℓ with the exceptional divisor E1 . We work now with a Q–divisor Dλ = f ∗ (OP2 (1)) − λE1 = H − λE1 , for some fixed λ ∈ [0, 1]. A direct computation using [18, Theorem 6.2] gives that the Okounkov body has the shape 11 1 ∆Y• (Dλ ) 0 1−λ Thus we see that the Okounkov body of Dλ is obtained from that of D by intersecting with a closed halfspace. For the valuation ν = ordP0 , we get as above ϕν (a, b) = a + b. Let now k be an integer such that the point (ka, kb) is integral and kλ is also an integer. Now we need to exhibit a section s in H 0 (P2 , OP2 (k)) satisfying the following conditions: a) s vanishes along ℓ exactly to order a; b) s vanishes in the point P1 to order at least kλ; c) s vanishes in the point P0 exactly to order b. We let the divisor of s to consist of a copies of ℓ (there is no other choice here), of b copies of the line through P0 and P1 , of kλ − b copies of any other line passing through P1 (if this number is negative then this condition is empty) and of a curve of degree k(1 − a − max {b, λ}) passing neither through P0 , nor through P1 . Remark 4.5. Note that in the setting of Proposition 4.1 Okounkov bodies of divisors Dλ always result from those of D by cutting with finitely many halfplanes. This is an immediate consequence of [15, Theorem 5]. We conclude by showing that the inequality in (10) can be sharp, i.e. the blow up process can influence the Okounkov function as well as the Okounkov body. Example 4.6. Keeping the notation from the previous examples, let f : X6 → P2 be the blow up of six general points P1 , . . . , P6 not contained in ℓ and chosen so that the points P0 , P1 , . . . , P6 are also general. Let E1 , . . . , E6 denote the exceptional divisors and set E = E1 +. . .+E6 . We consider the divisor D = H − 52 E. A direct computation using [18, Theorem 6.2] (this requires computing Zariski decompositions this time, see [2] for an effective approach) yields the triangle with vertices at the origin and in points (0, 1) and (1/25, 0) as ∆Y• (D). For the valuation ν = ordP0 we get now ϕν (a, b) 6 4/15 < a + b  for (a, b) ∈ Ω = (x, y) ∈ R2 : x ∈ [0, 11/360) and b ∈ (4/15 − a, 1 − 25a] . (11) 12 1 Ω 11 17 ( 360 , 72 ) 0 1 25 The reason for the above inequality is the following. Let (a, b) ∈ Ω be a valuation vector. Assume to the contrary that ϕ(a, b) > 4/15. It is well known that ε(OP2 (1), P0 , . . . , P6 ) = 38 , see for instance [23, Example 2.4]. On the other hand a section with the above valuation vector would have (after scaling to O(1)) multiplicities 2/5 at P1 , . . . , P6 and ϕν (a, b) > 4/15 at P0 . It would give Seshadri quotient 6· 2 5 3 1 < , 8 + ϕν (a, b) a contradiction. This proves (11). Since the SHGH Conjecture holds for 9 points, the first challenge arising in the view of our Theorem would be to compute the Okounkov body and the Okounkov function associated to ordP0 as above for the system 22H − 7(E1 + · · · + E9 ). References [1] Anderson, D., Küronya, A., Lozovanu, V.: Okounkov bodies of finitely generated divisors, International Mathematics Research Notices 2013; doi: 10.1093/imrn/rns286. [2] Bauer, Th.: A simple proof for the existence of Zariski decompositions on surfaces: J. Alg. Geom. 18 (2009), 789-793 [3] Bauer, Th., Di Rocco, S., Harbourne, B., Kapustka, M., Knutsen, A. L., Syzdek, W., Szemberg T.: A primer on Seshadri constants, Interactions of Classical and Numerical Algebraic Geometry, Proceedings of a conference in honor of A. J. Sommese, held at Notre Dame, May 22–24 2008. Contemporary Mathematics vol. 496, 2009, eds. D. J. Bates, G-M. Besana, S. Di Rocco, and C. W. Wampler, 362 pp. [4] Bauer, Th., Bocci, C., Cooper, S., Di Rocco, S., Dumnicki, M., Harbourne, B., Jabbusch, K., Knutsen, A.L., Küronya, A., Miranda, R., Roe, J., Schenck, H., Szemberg, T., Teitler, Z.: Recent developments and open problems in linear series, In ”Contributions to Algebraic Geometry”, 93–140, IMPANGA Lecture Notes (Piotr Pragacz , ed.), EMS Series of Congress Reports, edited by the European Mathematical Society Publishing House 2012. [5] Bauer, Th., Küronya, A., Szemberg, T.: Zariski chambers, volumes and stable loci, Journal für die reine und angewandte Mathematik, 576 (2004), 209–233. [6] Bauer, Th., Schmitz, D.: Volumes of Zariski chambers, arXiv:1205.3817 [7] Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series Compositio Math. 147 (2011), 1205–1229 13 [8] Ciliberto, C.: Geometric aspects of polynomial interpolation in more variables and of Waring’s problem. European Congress of Mathematics, Vol. I (Barcelona, 2000), 289-316, Progr. Math., 201, Birkhuser, Basel, 2001 [9] Cilibert, C., Harbourne, B., Miranda, R., Roé, J.: Variations on Nagata’s Conjecture, arXiv:1202.0475 [10] Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), Lect. Notes Math. 1507, Springer-Verlag, 1992, pp. 87–104 [11] Donaldson, S. K.: Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, 289–349. [12] Gimigliano, A.: Our thin knowledge of fat points. The Curves Seminar at Queen’s, Vol. VI (Kingston, ON, 1989), Exp. No. B, 50 pp., Queen’s Papers in Pure and Appl. Math., 83, Queen’s Univ., Kingston, ON, 1989 [13] Harbourne, B.: On Nagata’s conjecture. J. Algebra 236 (2001), 692-702 [14] Kaveh, K., Khovanskii, A.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Annals of Mathematics 176 (2012), 1–54 [15] Küronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors, Advances in Mathematics 229 (2012), no. 5, 2622–2639. [16] Küronya, A., Maclean, C., Szemberg, T.: Functions on Okounkov bodies coming from geometric valuations (with an appendix by Sébastien Boucksom), arXiv:1210.3523v2. [17] Lazarsfeld, R.: Positivity in Algebraic Geometry. I.-II. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48–49., Springer Verlag, Berlin, 2004. [18] Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series, Ann. Scient. Éc. Norm. Sup., 4 série, t. 42, (2009), 783–835. [19] Nagata, M.: On the 14-th problem of Hilbert. Amer. J. Math. 81 (1959), 766–772. [20] Nyström, D., W.: Transforming metrics on a line bundle to the Okounkov body. preprint, arXiv:0903.5167v1. [21] Okounkov, A.: Brunn-Minkowski inequalities for multiplicities, Invent. Math 125 (1996) pp 405–411. [22] Sano, T.: Seshadri arXiv:0908.4502v4. constants on rational surfaces with anticanonical pencils, [23] Strycharz-Szemberg, B., Szemberg, T.: Remarks on the Nagata Conjecture, Serdica Math. J. 30 (2004), 405-430 Marcin Dumnicki, Jagiellonian University, Institute of Mathematics, Lojasiewicza 6, PL-30-348 Kraków, Poland E-mail address: Marcin.Dumnicki@im.uj.edu.pl Alex Küronya, Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra, Pf. 91, H-1521 Budapest, Hungary. E-mail address: alex.kuronya@math.bme.hu Current address: Alex Küronya, Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstraße 1, D-79104 Freiburg, Germany. Catriona Maclean, Institut Fourier, CNRS UMR 5582 Université de Grenoble, 100 rue des Maths, F-38402 Saint-Martin d’Héres cedex, France E-mail address: catriona@fourier.ujf-grenoble.fr Tomasz Szemberg, Instytut Matematyki UP, Podchora̧żych 2, PL-30-084 Kraków, Poland. E-mail address: szemberg@up.krakow.pl 

A Supervisory Control Algorithm Based on Property-Directed Reachability⋆ arXiv:1711.06501v1 [] 17 Nov 2017 Koen Claessen1 , Jonatan Kilhamn1 , Laura Kovács13 , and Bengt Lennartson2 1 Department of Computer Science and Engineering, 2 Department of Electrical Engineering, Chalmers University of Technology 3 Faculty of Informatics, Vienna University of Technology {koen, jonkil, laura.kovacs, bengt.lennartson}@chalmers.se Abstract. We present an algorithm for synthesising a controller (supervisor) for a discrete event system (DES) based on the property-directed reachability (PDR) model checking algorithm. The discrete event systems framework is useful in both software, automation and manufacturing, as problems from those domains can be modelled as discrete supervisory control problems. As a formal framework, DES is also similar to domains for which the field of formal methods for computer science has developed techniques and tools. In this paper, we attempt to marry the two by adapting PDR to the problem of controller synthesis. The resulting algorithm takes as input a transition system with forbidden states and uncontrollable transitions, and synthesises a safe and minimally-restrictive controller, correct-by-design. We also present an implementation along with experimental results, showing that the algorithm has potential as a part of the solution to the greater effort of formal supervisory controller synthesis and verification. Keywords: Supervisory control ·Discrete-event systems ·Property-directed reachability ·Synthesis ·Verification ·Symbolic transition system 1 Introduction Supervisory control theory deals with the problems of finding and verifying controllers to given systems. One particular problem is that of controller synthesis: given a system and some desired properties—safety, liveness, controllability—automatically change the system so that it fulfills the properties. There are several approaches to this problem, including ones based on binary decision diagrams (BDD) [14, 6], predicates [11] and the formal safety checker IC3 [18]. In this work we revisit the application of IC3 to supervisory control theory. Namely, we present an algorithm for synthesising a controller (supervisor) for a discrete event system (DES), based on property-directed reachability [4] (PDR, a.k.a. the method underlying IC3 [2]). Given a system with a safety property and uncontrollable transitions, the synthesised controller is provably safe, controllable and minimally restrictive [16]. ⋆ The final publication is available at Springer via https://doi.org/10.1007/978-3-319-70389-3_8. 2 1.1 An illustrative example Let us explain our contributions by starting with an example. Figure 1 shows the transition system of a finite state machine extended with integer variables x and y. The formulas on the edges denote guards (transition cannot happen unless formula is true) and updates (after transition, x takes the value specified for x′ in the formula). This represents a simple but typical problem from the domain of control theory, and is taken from [17]. x = 0, y = 0 l0 a : y′ = 2 l2 b : y′ = 1 b:⊤ a:⊤ l1 l3 α:y =2∧x>2 l5 c : x′ = x + 1 α:y =2∧x≤2 l4 ω:⊤ Fig. 1. The transition system of the example. In a controller synthesis problem, a system such as this is the input. The end result is a restriction of the original system, i.e. one whose reachable state space is a subset of that of the original one. In this extended finite state machine (denoted as EFSM) representation, this is written as new and stronger guard formulas on some of the transitions. Our example has two more features: the location l5 , a dashed circle in the figure, is forbidden, while the event α is uncontrollable. The latter feature means that the synthesised controller must not restrict any transition marked with the event α. To solve this problem, we introduce an algorithm based on PDR [4] used in a software model checker (Section 3). Intuitively, what our algorithm does is to incrementally build an inductive invariant which in turn implies the safety of the system. This invariant is constructed by ruling out paths leading into the bad state, either by proving these bad states unreachable from the initial states, or by making them unreachable via strengthening the guards. In our example, the bad state l5 is found to have a preimage under the transition relation T in l3 ∧ y = 2 ∧ x > 2. The transition from l3 to l5 is uncontrollable, so in order to guarantee safety, we must treat this prior state as unsafe too. The transitions leading into l3 are augmented with new guards, so that the system may only visit l3 if the variables make a subsequent transition to l5 impossible. By applying our work, we refined Figure 1 with the necessary transition guards and a proof that the new system is safe. We show the refined system obtained by our approach in Figure 2. 3 x = 0, y = 0 a : y′ = 2 l0 l2 b : y′ = 1 b : y 6= 2 ∨ x ≯ 2 a : y 6= 2 ∨ x ≯ 2 l1 l3 α:y =2∧x>2 l5 c : x′ = x + 1 α:y =2∧x≤2 l4 ω:⊤ Fig. 2. The transition system from the example, with guards updated to reflect the controlled system. 1.2 Our Contributions 1. In this paper we present a novel algorithm based on PDR for controller synthesis (Section 3) and prove correctness and termination of our approach (Section 4). To the best of our knowledge, PDR has not yet been applied to supervisory control systems in this fashion. We prove that our algorithm terminates (given finite variable domains) and that the synthesised controller is safe, minimally-restrictive, and respects the controllability constraints of the system. Our algorithm encodes system variables in the SAT domain; we however believe that our work can be extended by using satisfiability modulo theory (SMT) reasoning instead of SAT. 2. We implemented our algorithm in the model checker Tip [5]. We evaluated our implementation on a number of control theory problems and give practical evidence of the benefits of our work (see Section 6). 2 Background We use standard terminology and notation from first-order logic (FOL) and restrict formulas mainly to quantifier-free formulas. We reserve P, R, T, I to denote formulas describing, respectively, safety properties, “frames” approximating reachable sets, transition relations and initial properties of control systems; all other formulas will be denoted with φ, ψ, possibly with indices. We write variables as x, y and sets of variables as X, Y . A literal is an atom or its negation, a clause a disjunction of literals, and a cube a conjunction of literals. We use R to denote a set of clauses, intended to be read as the conjunction of those clauses. When a formula ranges over variables in two or more variable sets, we take φ(X, Y ) to mean φ(X ∪ Y ). For every variable x in the control system, we assume the existence of a unique variable x′ representing the next-state value of x. Similarly, the set X ′ is the set {x′ |x ∈ 4 X}. As we may sometimes drop the variable set from a formula if it is clear from the context, i.e. write φ instead of φ(X), we take φ′ to mean φ(X ′ ) in a similar fashion. 2.1 Modelling Discrete Event Systems A given DES can be represented in several different ways. The simple, basic model is the finite state machine (FSM) [10]. A state machine is denoted by the tuple G = hQ, Σ, δ, Qi i, where Q is a finite set of states, Σ the finite set of events (alphabet), δ ⊆ Q × Σ × Q the transition relation, and Qi ⊆ Q the set of initial states. In this notation, a controller can be represented as a function C : Q → 2Σ denoting which events are enabled in a given state. For any σ ∈ Σ and q ∈ Q, the statement σ ∈ C(q) means that the controller allows transitions with the event σ to happen when in q; conversely, σ ∈ / C(q) means those transitions are prohibited. Extended Finite State Machine. The state machine representation is general and monolithic. In order to more intuitively describe real supervisory control problems, other formalisms are also used. Firstly, we have the extended finite state machine (EFSM), which is an FSM extended with FOL formulas over variables. In effect, we split the states into locations and variables, and represent the system by the tuple A = hX, L, Σ, ∆, li, Θi. Here, X is a set of variables, L a set of locations, Σ the alphabet, ∆ the set of transitions, li ∈ L the initial location and Θ(X) a formula describing the initial values of the variables. A transition in ∆ is now a tuple hl, a, mi where l, m are the entry and exit locations, respectively, while the action a = (σ, φ) consists of the event σ ∈ Σ and φ(X, X ′ ). The interpretation of this is that the system can make the transition from l to m if the formula φ(X, X ′ ) holds. Since the formula can include next-state variables—φ may contain arbitrary linear expressions over both X and X ′ —the transition can specify updated variable values for the new state. We have now defined almost all of the notation used in the example in Figure 1. In the figure, we write σ : φ to denote the action (σ, φ). Furthermore, the figure is simplified greatly by omitting next-state assignments on the form x′ = x, i.e. x keeping its current value. If a variable does not appear in primed form in a transition formula, that formula is implied to have such an assignment. Symbolic Representation. Moving from FSM to EFSM can be seen as “splitting” the state space into two spaces: the locations and the variables. A given feature of an FSM can be represented as either one (although we note that one purpose for using variables is to easier extend the model to cover an infinite state space). Using this insight we can move to the “other extreme” of the symbolic transition system (STS): a representation with only variables and no locations. The system is here represented by the tuple SA = hX̂, T(X̂, X̂ ′ ), I(X̂)i where X̂ is the set of variables extended by two new variables xL and xΣ with domains L and Σ, respectively. With some abuse of notation, we use event and variable names to denote formulas over those variables, such as ln for the literal xL = ln and ¬σ for the literal xΣ 6= σ. The initial formula I and transition formula T are constructed from the corresponding EFSM representation as I(X̂) = (xL = li ) ∧ Θ(X) and T(X̂, X̂ ′ ) = W ′ ′ hl,(σ,φ),mi∈∆ (l ∧ σ ∧ φ(X, X ) ∧ m ). 5 In this paper, we will switch freely between the EFSM and STS representations of the same system, depending on which is the best fit for the situation. Additionally, we will at times refer to X̂ as only X, as long as the meaning is clear from context. In either representation, we will use state to refer to a single assignment of location and variables, and path for a sequence of states s0 , s1 , ..., sk . 2.2 Supervisory Control The general problem of supervisory control theory is this: to take a transition system, such as the ones we have described so far, and modify it so that it fulfils some property which the unmodified system does not. There are several terms in this informal description that require further explanation. The properties that we are interested in are generally safety, non-blocking, and/or liveness, which can be seen as a stronger form of non-blocking. Controlling for a safety property means that in the controlled system, there should be no sequence of events which enables transitions leading from an initial state to a forbidden state. Non-blocking and liveness are defined relative to a set of marked state. The former means that at least one such state is reachable from every state which is reachable from the initial states. The latter, liveness, implies non-blocking, as it is the guarantee that the system not only can reach but will return to a marked state infinitely often. In this work we have reduced the scope of the problem by considering only safety. Furthermore, we talk about the property of controllability. This is the notion that some events in a DES are uncontrollable, which puts a restriction on any proposed controller: in order to be valid, the transitions involving uncontrollable events must not be restricted. Formally, in an (E)FSM it is enough to split the alphabet into the uncontrollable Σu ⊆ Σ and the controllable Σc = Σ \ Σu . In an STS, this is expressed by the transition relation taking the form T = Tu ∨ Tc , where Tc and Tu include literals xL = σ for, respectively, only controllable and only uncontrollable events σ. Finally there is the question of what form this “controlled system” takes, since a controller function C : Q → 2Σ can be impractical. A common method is that of designating a separate state machine as the supervisor, and taking the controlled system to be the synchronous composition of the original system and the supervisor [8]. In short, this means running them both in parallel, but only allowing a transition with a shared event σ to occur simultaneously in both sub-systems. However, the formidable theory of synchronised automata is not necessary for the present work. Instead, we take the view that the controlled system is the original system, either in the EFSM or STS formulation, with some additions. In the EFSM case, the controlled system has the exact same locations and transitions, but additional guards and updates may be added. In other words, the controlled system augments each controllable transition by replacing the original transition formula φ with the new formula φs = φ ∧ φnew . The uncontrollable transitions are left unchanged. In the STS case, the new transition function is TS = Tu ∨ TSc where TSc = Tc ∧ Tnew c . This way, all uncontrollable transitions are guaranteed to be unmodified in the controlled system. Finally, a controlled system, regardless of which properties the controller is set out to guarantee, is often desired to be minimally restrictive (eqiv. maximally permissive). 6 The restrictiveness of a controlled system is defined as follows: out of two controlled versions S1 and S2 of the same original system S, S1 is more restrictive than S2 if there is at least one state, reachable under the original transition function T, which is reachable under TS2 but unreachable under TS1 . A controlled system is minimally restrictive if no other (viable) controlled system exists which is less restrictive. The word “viable” in brackets shows that one can talk about the minimally restrictive safe controller, the minimally restrictive non-blocking controller and so on; for each combination of properties, the minimally restrictive controller for those properties is different. 3 PDRC: Property-Driven Reachability-Based Control Property-driven reachability (PDR) [4] is a name for the method underlying IC3 [2], used to verify safety properties in transition systems. In this paper we present PropertyDriven Reachability-based Control (PDRC), which extends PDR from verifying safety to synthesising a controller which makes the system safe. In order to explain PDRC, we first review the main ingredients of PDR. PDR works by successively blocking states that are shown to lead to unsafe states in certain number of steps. Blocking a state at step k here means performing SAT-queries to show that the state is unreachable from the relevant frame Rk . A frame Rk is a predicate over-approximating the set of states reachable from the initial states I in k steps. When a state is blocked—i.e. shown to be unreachable—the relevant frame is updated by excluding that state from the reachable-set approximation. If a state cannot be blocked at Rk , the algorithm finds its preimage s and proceeds to block s at Rk−1 . If a state that needs to be blocked intersects with the initial states, the safety property of the system has been proven false. Conversely, if two adjacent frames Ri , Ri+1 are identical after an iteration, we have reached a fixed-point and a proof of the property P in one of them entails a proof of P for the whole system. With PDRC, we focus on the step where PDR has found a bad cube s (representing unsafe states) in frame Rk , and proceeds to check whether it is reachable from the previous frame Rk−1 . If it is not, this particular cube was a false alarm: it was in the over-approximation of k-reachable states, but after performing this check we can sharpen that approximation to exclude s. If s was reachable, PDR proceeds to find its preimage t which is in Rk−1 . Note that t is also a bad cube, since there is a path from t to an unsafe state. However, in a supervisory control setting, there is no reason not to immediately control the system by restricting all controllable transitions from t to s. This observation is the basis of our PDRC algorithm. 3.1 Formal Description of PDRC As PDRC is very similar to PDR, this description and the pseudocode procedures draw heavily from [4]. Our PDRC algorithm is given in Algorithm 1. As input, we take a transition system that can be represented by a transition function T(X, X ′ ) = Tc ∨ Tu , i.e. one where each possible transition is either controllable or uncontrollable; and a safety property 7 Algorithm 1: Blocking and propagation for one iteration of N . 1 2 3 4 5 6 7 8 9 10 11 // finding and blocking bad states while SAT[RN ∧ ¬P] do extract a bad state m from the SAT model; generalise m to a cube s; recursively block s as per block(s, N ); // at this point R and/or T have been updated to rule out m end // propagation of proven clauses add new empty frame RN+1 ; for k ∈ [1, N ] and c ∈ Rk do if Rk  c′ then add c to Rk+1 ; end end P(X). The variables in X are boolean, in order to allow the use of a SAT solver – although see Section 3.2 describing an extension from SAT to SMT. Throughout the run of the algorithm, we keep a trace: a series of frames Ri , 0 ≤ i ≤ N . Each Ri (X) is a predicate that over-approximates the set of states reachable from I in i steps or less. R0 = I, where I is a formula encoding the initial states. V Each frame Ri , i > 0 can be represented by a set of clauses Ri = {cij }j , such that j cij (X) = Ri (X). An empty frame Rj = {} is considered to encode ⊤, i.e. the most over-approximating set possible. We maintain the following invariants: 1. Ri → Ri+1 2. Ri → P, except for i = N 3. Ri+1 is an over-approximation of the image of Ri under T Starting with N = 1 and R1 = {}, we proceed to do the first iteration of the blocking and propagation steps, as shown in Algorithm 1. The “blocking step” consists of the while-loop (lines 1–5) of Algorithm 1, and coming out of that loop we know that RN → P. The propagation step follows (lines 6–11), and here we consider for each clause in some frame of the trace whether it also holds in the next frame. Afterwards, we check for a fix-point in Ri ; i.e. two syntactically equal adjacent frames Ri = Ri+1 . Unless such a pair is found, we increment N by 1 and repeat the procedure. The most important step inside the while loop is the call to block (line 4). This routine is shown in Algorithm 2. Here, we take care of the bad states in a straightforward way. First, we consider its preimage under the controllable transition function Tc (line 2). The preimage cube t can be found by taking a model of the satisfiable query Rk−1 ∧ ¬s ∧ Tc ∧ s′ and dropping the primed variables. Each such cube encodes 8 Algorithm 2: The blocking routine, which updates the supervisor. 1 2 3 4 5 6 7 8 9 10 11 12 Data: A cube s and a frame index k // first consider the controllable transitions: while SAT[Rk−1 ∧ ¬s ∧ Tc ∧ s′ ] do extract and generalise a bad cube t in the preimage of Tc ; update Tc := Tc ∧ ¬t; end // then consider the uncontrollable transitions: while SAT[Rk−1 ∧ ¬s ∧ Tu ∧ s′ ] do if k=1 then throw error: system uncontrollable; end extract and generalise a bad cube t in the preimage of Tu ; call block(t, k − 1); end add ¬s to Ri , i ≤ k; states from which a bad state is reachable in one step. Thus, we update the supervisor to disallow transitions from those bad states (line 3). This accounts for the first while-loop in Algorithm 2. The second while-loop (lines 5–11)is very similar, but considers the uncontrollable transitions, encoded by Tu , instead. If a preimage cube is found here, we cannot rule it out by updating the supervisor. That preimage instead becomes a bad state on its own, to be controlled in the previous frame k − 1. Example 1. Example, revisited. Recall the example in Figure 1. Since it uses integer variables it seems to require an SMT-based version of PDRC. This particular example is so simple, however, that “bit-blasting” the problem into SAT by treating the proposition x < i as a separate boolean variable for each value of i in the domain of x will yield the same solution. PDRC requires 3 iterations to completely supervise the system. In the first, the clause ¬l5 is added to the first frame R1 , after proving that it is not in the initial states. In the second, ¬l5 is found again but this time the uncontrollable transition from l3 is followed backwards, and the clause ¬α ∨ ¬l3 ∨ y 6= 2 ∨ x ≯ 2 is also added to R1 , which allows us to add ¬l5 to R2 . Finally, in the third, the trace of preimages lead to the controllable transitions l1 → l3 and l2 → l3 , and we add new guards to both (technically, we add new constraints to the transition function). The updated system is the one shown in Figure 2. The third iteration also proves the system safe, as we have R1 = R2 . These frames then hold the invariant, (¬α∨¬l3 ∨y 6= 2 ∨ x ≯ 2) ∧ ¬l5 , which implies P and is inductive under the updated T. 3.2 Extension to SMT Our PDRC algorithm in Algorithm 1 uses SAT queries, and is straightforward to use with a regular SAT solver on systems with a propositional transition function. However, 9 like in [3, 9] it is possible to extend it to other theories, such as Linear Integer Arithmetic, using an SMT solver. The SAT query in Algorithm 1 provides no diffuculty, but some extra thought is required for the ones in the blocking procedure, which follow this pattern: while SAT[Ri ∧ ¬s ∧ T ∧ s′ ] do extract and generalise a bad cube t in the preimage of T; If one only replaces the SAT solver by an SMT solver capable of handling the theory in question, one can extract a satisfying assignment of theory literals. However, each of these might contain both primed and unprimed variables, such as the next-state assignment x′ = x + 1. These lines effectively ask the solver to generalise a state m—an assignment of theory literals satisfying some formula F—into a more general cube t, ideally choosing the t that covers the maximal amount of discrete states, while still guaranteeing t → F. In the SAT case, this is achieved by dropping literals of t that do not affect the validity of F(t). An alternate method based on ternary simulation, that is useful when the query is for a preimage of a transition function T, is given in [4]. For the SMT case, however, the extent of generalisation depends on the theory and the solver. In the worst case of a solver that cannot generalise at all, the algorithm is consigned to blocking a single state m in each iteration. This means that the state space simplification gained from using a symbolic transition function in the first place is lost, since the reachability analysis checks states one by one. In conclusion, PDRC could be implemented for systems with boolean variables using a SAT-solver with no further issues, while an SMT version would require carefully selecting the right solver for the domain. We leave this problem as an interesting task for future work. 4 Properties of PDRC In this section we prove the soundness and termination of our PDRC algorithm. 4.1 Termination Theorem 1. For systems with state variables whose domains are finite, the PDRC algorithm always terminates. The termination of regular PDR is proven in [4]. In the case of an unsafe system— which for us corresponds to an uncontrollable system—the counterexample proving this must be finite in length, and thus found in finite time. In the case of a safe system, the proof is based on the following observations: that each proof-obligation (call to block) must block at least one state in at least one frame; that there are a finite number of frames for each iteration (value of N ); that there are a finite number of states of the system; and that each Ri+1 must either block at least one more state than Ri , or they are equal. All these observations remain true for PDRC, substituting “uncontrollable” for “unsafe”. This means that the proof of termination from [4] can be used for PDRC with minimal modification. 10 4.2 Correctness We claim that the algorithm described above synthesises a minimally restrictive safe controller for the original system. Theorem 2. If there exists any safe controller for the system, the controller synthesised by the PDRC algorithm is safe. Proof. We prove Theorem 2 by contradiction. Assume there is an unsafe state s, i.e. we have ¬P(s), that is reachable from an I-state in k steps. We must then have k ≥ N , since invariant (2) states that Ri → P, i < N . Let M be the index of the discovered fix point RM = RM+1 . Invariant (1) (from Section 3.1) states that Ri → Ri+1 , and this applies for all values 0 ≤ i ≤ M . Repeated application of this means that any state in any Ri , i < M is also contained in RM . Invariant (3) states that Ri+1 is an over-approximation of the image of Ri . This means that any state reachable from RM should be in RM+1 . Since RM = RM+1 , such a state is also in RM itself. Repeated application of this allows us to extend the trace all the way to Rk = Rk−1 = · · · = RM . Now, for the bad state s, regardless of the number of steps k needed to reach it, we know that s is contained in Rk and therefore in RM . Yet when the algorithm terminated it had at one point found RM ∧ ¬P to be UNSAT. The state s, which is both in RM and ¬P, would constitute a satisfying assignment to this query. This contradiction proves that s cannot exist. ⊓ ⊔ Theorem 3. A controller synthesised by the PDRC algorithm is minimally restrictive. Proof. We prove Theorem 3 also by contradiction. Assume there is a safe path π = s0 , s1, . . . , sk through the original system (with transition function T), which is not possible using the controlled transition function TPDRC ; yet there exists another safe, controllable supervisor represented by TS where π is possible. By deriving a contradiction, we will prove that no such TS can exist. Consider the first step of π that is not allowed by TPDRC ; in other words, a pair (si , si+1 ) where we have ¬TPDRC (si , si+1 ) while we do have both TS (si , si+1 ) and T(si , si+1 ). The only way that TPDRC is more restrictive than T is due to strengthenings on the form TcPDRC = Tc ∧ ¬m, for some cube m. This means that si must be in some cube m that PDRC supervised in this fashion. This happened inside a call block(m, j). Since π is safe, this call cannot have been made because m itself encoded unsafe states. Instead, there must have been a previous call block(n, j + 1), where m is a minterm of the preimage of n under Tu . This cube n is either itself a bad cube, or it can be traced to a bad cube by following the trace of block calls. Since each step in this block chain only uses Tu , we can find a series of uncontrollable transitions, starting in some s̃i+1 ∈ n, leading to some cube p which is a generalisation of a satisfying assignment to the query RN ∧ ¬P. This proves that TS , whose TSc does not restrict transitions from si , allows for the system to enter a state s̃i+1 , from which there is an uncontrollable path to an unsafe state. This contradicts the assumption that TS was safe, proving that the combination of π and TS cannot exist. This proves that the controller encoded by TPDRC is minimally restrictive. ⊓ ⊔ 11 5 Implementation We have implemented a prototype of PDRC in the model checker Tip (Temporal Inductive Prover [5]). The input format supported by Tip is AIGER [1], where the transition system is represented as a circuit, which is not a very intuitive way to view an EFSM or STS. For this reason, our prototype also includes Haskell modules for creating a transition system in a control-theory-friendly representation, converting it to AIGER, and using the output from the Tip-PDRC to reflect the new, controlled system synthesised by PDRC. Finally, it also includes a parser from the .wmod format used by WATERS and Supremica [13], into our Haskell representation. Altogether, our implementation consists of about 150 lines of code added or changed in the Tip source, and about 1600 lines of Haskell code. Our tools, together with the benchmarks we used, is available through github.com/JonatanKilhamn/supermini and github.com/JonatanKilhamn/tipcheck. When converting transition systems into circuits, certain choices have to be made. Our encoding allows for synchronised automata with one-hot-encoded locations (e.g. location l3 out of 5 is represented by the bits [0, 0, 1, 0, 0]) and unary-encoded integer variables (e.g. a variable ranging from 0 to 5 currently having the value 3 is represented by [1, 1, 1, 0, 0]). Each of these encoding has a corresponding invariant: with one-hot, exactly one bit must be set to 1; with unary, each bit implies the previous one. However, these invariants need not be explicitly enforced by the transition relation (i.e. as guards on every transition), rather, it is enough that they are preserved by all variable updates. It should be noted that although the PDRC on a theoretical level works equally well on STS as EFSM, our implementation does assume the EFSM division between locations and variables for the input system. However, our implementation retains the generality of PDRC in how the state space is explored—the algorithm described in Section 3 is run on the circuit representation, where the only difference between the location variable xL and any other variable is the choice of encoding. 6 Experiments For an empirical evaluation, we ran PDRC on several standard benchmark problems: the extended dining philosophers (EDP) [15], the cat and mouse tower (CMT) [15] and the parallell manufacturing example (PME) [12]. The runtimes of these experiments are shown in Table 1 below. The benchmarks were performed on a computer with a 2.7 GHz Intel Core i5 processor and 8GB of available memory. 6.1 Problems For the dining philosophers, EDP(n, k) denotes the problem of synthesising a safe controller for n philosophers and k intermediary states that each philosopher must go through between taking their left fork and taking their right one. The transition system is written so that all philosophers respect when their neighbours are holding the forks, except for the even-numbered ones who will try to take the fork to their left even if it is held, which leads (uncontrollably) to a forbidden state. 12 For the cat and mouse problem, CMT(n, k) similarly denotes the problem with n floors of the tower, k cats and k mice. Again, the transition system already prohibits cats and mice from entering the same room (forbidden state) except by a few specified uncontrollable pathways. Finally, the parallel manufacturing example (PME) represents an automated factory, with an industrial robot and several shared resources. It differs from the other in that its scale comes mainly from the number of different synchronised automata. In return, it does not have a natural parameter that can be set to higher values to increase the complexity further. 6.2 Results We compare PDRC to Symbolic Supervisory Control using BDD (SC-BDD) [14, 6], which is implemented within Supremica. We wanted to include the Incremental, Inductive Supervisory Control (IISC) algorithm [18], which also uses PDR but in another way. However, the IISC implementation from [18] is no longer maintained. Despite this failed replication, we include figures for IISC taken directly from [18]—with all the caveats that apply when comparing runtimes obtained from different machines. Table 1 shows runtimes, where the problems are denoted as above and “×” indicates time-out (5 min). The parameters for EDP and CMT were chosen to show a wide range from small to large problems, while still mostly choosing values for which [18] reports runtimes for IISC. We see that while SC-BDD might have the advantage on certain small problems, PDRC quickly outpaces it as the problems grow larger. Table 1. Performance of PDRC (our contribution), SC-BDD and IISC on standard benchmark problems. Note that the IISC implementation was not reproducible by us; the numbers here are lifted from [18]. “×” indicates timeout (5 min), and “–” means this particular problem was not included in [18]. Model CMT(1,5) CMT(3,3) CMT(5,5) CMT(7,7) EDP(5,10) EDP(10,10) EDP(5,50) EDP(5,200) EDP(5,10e3) PME PDRC IISC[18] SC-BDD 0.09 0.13 0.007 1.3 0.43 1.12 8.3 0.73 × 30.02 0.98 × 0.03 0.98 0.031 0.15 – 0.10 0.03 0.12 0.26 0.06 0.12 × 0.19 0.12 × 0.72 2.3 8.1 7 Discussion In this section, we relate briefly how BDD-SC [14, 6] and IISC [18] work, in order to compare and contrast to PDRC. 13 7.1 BDD-SC BDD-SC works by modelling an FSM as a binary decision diagram (BDD). The algorithm generates a BDD, representing the safe states, by searching backwards from the forbidden states. However, the size of this BDD grows with the domain of the integer variables. The reason is that the size of the BDD is quite sensitive to the number of binary variables, but also the ordering of the variables in the BDD. Even when more recent techniques on partitioning of the problem are used [6], the size of the BDD blows up, and we see in Table 1 that BDD-SC very quickly goes from good performance to time-out. 7.2 IISC It is natural to compare PDRC to IISC [18], since the latter is also inspired by PDR (albeit under the name IC3). In theory, PDRC has some advantages. The first advantage is one of representation. IISC is built on the EFSM’s separation between locations and variables, as described in 2.1. PDRC, on the other hand, handles the more general STS representation. Specifically, IISC explicitly unrolls the entire substate-space spanned by the locations. This sub-space can itself suffer a space explosion when synchronising a large number of automata. To once again revisit our example (Figure 1): IISC would unroll the graph, starting in l0 , into an abstract reachability tree. Each node in such a tree can cover any combination of variable values, but only one location. Thus, IISC effectively does a forwards search for bad locations, and the full power of PDR (IC3) is only brought to bear on the assignment of variables along a particular error trace. Thus, a bad representation choice w.r.t. which parts of the system are encoded as locations versus as variables can hurt IISC, while PDRC is not so vulnerable. PDRC, in contrast, leverages PDR’s combination of forwards and backwards search: exploring the state space backwards from the bad states in order to construct an inductive invariant which holds in the initial states. One disadvantage of the backwards search is that PDRC might add redundant safeguards. For example, the safeguard on the transition from l1 to 13 in Figure 2 is technically redundant, as there is no way to reach l2 with the restricted variable values from the initial states. As shown in [18], IISC does not add this particular guard. However, since both methods are proven to yield minimally-restrictive supervisors, any extra guards added by PDRC are guaranteed not to affect the behaviour of the final system. The gain, on the other hand, is that one does not need to unroll the whole path from the initial state to the forbidden state in order to supervise it. Consider: each such error path must have a “point of no return”—the last controllable transition. When synthesising for safety, this transition must never be left enabled (our proof of Theorem 3 hinges upon this). In order to find this point, PDRC traverses only the path between the point of no return and the forbidden state, whereas IISC traverses the whole path. In a sense, PDRC does not care about how one might end up close to forbidden state, but only where to put up the fence. In practice, our results have IISC outperforming PDRC on both PDE and CMT. We believe the main reason is that unlike IISC which uses IC3 extended to SMT [3], 14 our implementation of PDRC works in SAT. This means that while both algorithms are theoretically equipped to abstract away large swathes of the state space, IISC does it much easier on integer variables than PDRC, which needs to e.g. represent each possible value of a variable as a separate gate. The one point where PDRC succeeds also in practice is on the PME problem. Here, most of the system’s complexity comes from the number of different locations across the synchronised automata, rather than from large variable domains. In order to further explore this difference in problem type, we would have liked to evaluate PDRC and IISC on more problems with more synchronised automata, such as EDP(10,10). Sadly, this was impossible since the IISC implementation is no longer maintained. 8 Conclusions and Future Work We have presented PDRC, an algorithm for controller synthesis of discrete event systems with uncontrollable transitions, based on property-driven reachability. The algorithm is proven to terminate on all solvable problem instances, and its synthesised controllers are proven to be safe and minimally restrictive. We have also implemented a prototype in the SAT-based model checker Tip. Our experiments show that even this SAT-based implementation outperforms a comparable BDD-based approach, but not the more recent IISC. However, since the implementation of IISC we compare against uses an SMT solver, not to mention that it is not maintained anymore, we must declare the algorithm-level comparison inconclusive. The clearest direction for future research would be to implement PDRC using an SMT solver, to see if this indeed does realise further potential of the algorithm like we believe. Both [3] and [9] provide good insights for this task. However, another interesting direction is to use both PDRC and IISC as a starting point to tackling the larger problem: safe and nonblocking controller synthesis. Expanding the problem domain like this cannot be done by a trivial change to PDRC, but hopefully the insights from this work can contribute to a new algorithm. Another technique to draw from is that of IICTL [7]. As discussed in Section 2.2, by restricting our problem to only safety, we remove ourselves from real-world applications. For this reason, we do not present PDRC as a contender for any sort of throne, but as a stepping stone towards the real goal: formal, symbolic synthesis and verification of discrete supervisory control. REFERENCES 15 References [1] Armin Biere. AIGER. 2014. 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Throughput-Optimal Broadcast in Wireless Networks with Dynamic Topology Abhishek Sinha Leandros Tassiulas Eytan Modiano Laboratory for Information and Decision Systems MIT Electrical Engg. and Yale Institute of Network Science Yale University Laboratory for Information and Decision Systems MIT arXiv:1604.00576v1 [] 3 Apr 2016 sinhaa@mit.edu leandros.tassiulas@yale.edu ABSTRACT We consider the problem of throughput-optimal broadcasting in time-varying wireless networks, whose underlying topology is restricted to Directed Acyclic Graphs (DAG). Previous broadcast algorithms route packets along spanning trees. In large networks with time-varying connectivities, these trees are difficult to compute and maintain. In this paper we propose a new online throughput-optimal broadcast algorithm which makes packet-by-packet scheduling and routing decisions, obviating the need for maintaining any global topological structures, such as spanning-trees. Our algorithm relies on system-state information for making transmission decisions and hence, may be thought of as a generalization of the well-known back-pressure algorithm which makes point-to-point unicast transmission decisions based on queue-length information, without requiring knowledge of end-to-end paths. Technically, the back-pressure algorithm is derived by stochastically stabilizing the networkqueues. However, because of packet-duplications associated with broadcast, the work-conservation principle is violated and queuing processes are difficult to define in the broadcast problem. To address this fundamental issue, we identify certain state-variables which behave like virtual queues in the broadcast setting. By stochastically stabilizing these virtual queues, we devise a throughput-optimal broadcast policy. We also derive new characterizations of the broadcastcapacity of time-varying wireless DAGs and derive an efficient algorithm to compute the capacity exactly under certain assumptions, and a poly-time approximation algorithm for computing the capacity under less restrictive assumptions. 1. INTRODUCTION The problem of efficiently disseminating packets, arriving at a source node, to a subset of nodes in a network, is known as the Multicast problem. In the special case when the packets are to be distributed among all nodes, the corresponding problem is referred to as the Broadcast problem. Multicast- ACM ISBN 978-1-4503-2138-9. DOI: 10.1145/1235 modiano@mit.edu ing and broadcasting is considered to be a fundamental network functionality, which enjoys numerous practical applications ranging from military communications [16], disaster management using mobile adhoc networks (MANET) [9], to streaming services for live web television [24] etc. There exists a substantial body of literature addressing different aspects of this problem in various networking settings. An extensive survey of various multicast routing protocols for MANET is provided in [12]. The authors of [8] consider the problem of minimum latency broadcast of a finite set of messages in MANET. This problem is shown to be NP-hard. To address this issue, several approximation algorithms are proposed in [11], all of which rely on construction of certain network-wide broadcast-trees. Cross-layer solutions for multi-hop multicasting in wireless network are given in [29] and [10]. These algorithms involve network coding, which introduces additional complexity and exacerbates end-to-end delay. The authors of [21] propose a multicast scheduling and routing protocol which balances load among a set of pre-computed spanning trees, which are challenging to compute and maintain in a scalable fashion. The authors of [26] propose a local control algorithm for broadcasting in a wireless network for the so called scheduling-free model, in which an oracle is assumed to make interference-free scheduling decisions. This assumption, as noted by the authors themselves, is not practically viable. In this paper we build upon the recent work of [23] and consider the problem of throughput-optimal broadcasting in a wireless network with time-varying connectivity. Throughout the paper, the overall network-topology will be restricted to a directed acyclic graph (DAG). We first characterize the broadcast-capacity of time-varying wireless networks and propose an exact and an approximation algorithm to compute it efficiently. Then we propose a dynamic link-activation and packet-scheduling algorithm that, unlike any previous algorithms, obviates the need to maintain any global topological structures, such as spanning trees, yet achieves the capacity. In addition to throughput-optimality, the proposed algorithm enjoys the attractive property of in-order packet-delivery, which makes it particularly useful in various online applications, e.g. VoIP and live multimedia communication [4]. Our algorithm is model-oblivious in the sense that its operation does not rely on detailed statistics of the random arrival or network-connectivity processes. We also show that the throughput-optimality of our algorithm is retained when the control decisions are made using locally available and possibly imperfect, state information. Notwithstanding the vast literature on the general topic of broadcasting, to the best of our knowledge, this is the first work addressing throughput-optimal broadcasting in timevarying wireless networks with store and forward routing. Our main technical contributions are the following: • We define the broadcast-capacity for wireless networks with time-varying connectivity and characterize it mathematically and algorithmically. We show that broadcastcapacity of time-varying wireless directed acyclic networks can be computed efficiently under some assumptions. We also derive a tight-bound for the capacity for a general setting and utilize it to derive an efficient approximation algorithm to compute it. • We propose a throughput-optimal dynamic routing and scheduling algorithm for broadcasting in a wireless DAGs with time-varying connectivity. This algorithm is of Max-Weight type and uses the idea of in-order delivery to simplify its operation. To the best of our knowledge, this is the first throughput-optimal dynamic algorithm proposed for the broadcast problem in wireless networks. • We extend our algorithm to the setting when the nodes have access to infrequent state updates. We show that the throughput-optimality of our algorithm is preserved even when the rate of inter-node communication is made arbitrarily small. • We illustrate our theoretical findings through illustrative numerical simulations. The rest of the paper is organized as follows. Section 2 introduces the wireless network model. Section 3 defines and characterizes the broadcast capacity of a wireless DAG. It also provides an exact and an approximation algorithm to compute the broadcast-capacity. Section 4 describes our capacity-achieving broadcast algorithm for DAG networks. Section 5 extends the algorithm to the setting of broadcasting with imperfect state information. Section 6 provides numerical simulation results to illustrate our theoretical findings. Finally, in section 7 we summarize our results and conclude the paper. 2. NETWORK MODEL link (i, j) is activated. Let r ∈ V be the source node. At slot t, A(t) packets arrive at the source. The arrivals, A(t), are i.i.d. over slots with mean E(A(t)) = λ. Our problem is to efficiently disseminate the packets to all nodes in the network. 2.1 Notations and Nomenclature: All vectors in this paper are assumed to be column vectors. For any set X
⊂ Rk , its convex-hull is denoted by conv(X ). Let U, V \ U be a disjoint partition of the set of vertices V of the graph G, such that the source r ∈ U and U ( V . Such a partition is called a proper-partition. To each proper partition corresponding to the set U , associate the propercut vector u ∈ Rm , defined as follows: ui,j = ci,j = 0 if i ∈ U, j ∈ V \ U otherwise (1) Denote the special, single-node proper-cuts by Uj ≡ V \ {j}, and the corresponding cut-vectors by uj , ∀j ∈ V \ {r}. The set of all proper-cut vectors in the graph G is denoted by U. The in-neighbours of a node j is defined as the set of all nodes i ∈ V such that there is a directed edge (i, j) ∈ E. It is denoted by the set ∂ in (j), i.e., ∂ in (j) = {i ∈ V : (i, j) ∈ E} (2) Similarly, we define the out-neighbours of a node j as follows  ∂ out (j) = i ∈ V : (j, i) ∈ E (3) For any two vectors x and y in Rm , define the componentwise product z ≡ x ⊙ y to be a vector in Rm such that zi = xi yi , 1 ≤ i ≤ m. For any set S ⊂ Rm and any vector v ∈ Rm , v⊙S, denotes the set of vectors obtained as the component-wise product of the vector v and the elements of the set S, i.e.,  v ⊙ S = y ∈ Rm : y = v ⊙ s, s ∈ S (4) Also, the usual dot product between two vectors x, y ∈ Rm is defined as, x·y = m X x i yi i=1 2.2 Model of Time-varying Wireless Connectivity First we describe the basic wireless network model without time-variation. Subsequently, we will incorporate timevariation in the basic model. A static wireless network is modeled by a directed graph G = (V, E, c, M), where V is the set of nodes, E is the set of directed point-to-point links1 , the vector c = (cij ) denotes capacities of the edges when the corresponding links are activated and M ⊂ {0, 1}|E| is the set of incidence-vectors corresponding to all feasible link-activations complying with the interference-constraints. The structure of the activation-set M depends on the interference model, e.g., under the primary or node-exclusive interference model [14], M corresponds to the set of all matchings on the graph G. There are a total of |V | = n nodes and |E| = m edges in the network. Time is slotted and at timeslot t, any subset of links complying with the underlying interference-constraint may be activated. At most cij packets can be transmitted in a slot from node i to node j, when Now we incorporate time-variation into our basic framework described above. In a wireless network, the channelSINRs vary with time because of random fading, shadowing and mobility [27]. To model this, we consider a simple ONOFF model where an individual link can be in one of the two states, namely ON and OFF. In an OFF state, the capacity of a link is zero 2 . Thus at a given time, the network can be in any one configuration, out of the set of all possible network configurations Ξ. Each element σ ∈ Ξ corresponds to a sub-graph G(V, Eσ ) ⊂ G(V, E), with Eσ ⊂ E, denoting the set of links that are ON. At a given time-slot t, one of the configuration σ(t) ∈ Ξ is realized. The configuration at time t is represented by the vector σ(t) ∈ {0, 1}|E| , where ( 1, if e ∈ Eσ(t) σ(e, t) = 0, otherwise. 1 We assume all transmit and receiving antennas to be directed and hence all transmissions to be point-to-point [3]. 2 Generalization of the ON-OFF model, to multi-level discretization of link-capacity is straight-forward. At a given time-slot t, the network controller may activate a set of non-interfering links that are ON. The network-configuration process {σ(t)}t≥1 evolves in discretetime according to a stationary ergodic process with the stationary distribution {p(σ)}σ∈Ξ [13], where X p(σ) = 1, p(σ) > 0, ∀σ ∈ Ξ (5) σ∈Ξ Since the underlying physical processes responsible for time-variation are often spatially-correlated [1], [19], the distribution of the link-states is assumed to follow an arbitrary joint-distribution. The detailed parameters of this process depend on the ambient physical environment, which is often difficult to measure. In particular, it is unrealistic to assume that the broadcast-algorithm has knowledge of the parameters of the process σ(t). Fortunately, our proposed dynamic throughput-optimal broadcast algorithm does not require the statistical characterization of the configurationprocess σ(t) or its stationary-distribution p(σ). This makes our algorithm robust and suitable for use in time-varying wireless networks. 3. DEFINITION AND CHARACTERIZATION OF BROADCAST CAPACITY Intuitively, a network supports a broadcast rate λ if there exists a scheduling policy under which all network nodes receive distinct packets at rate λ. The broadcast-capacity of a network is the maximally supportable broadcast rate by any policy. Formally, we consider a class Π of scheduling policies where each policy π ∈ Π consists of a sequence of actions {πt }t≥1 , executed at every slot t. Each action πt consists of two operations: • The scheduler observes the current network-configuration σ(t) and activates a subset of links by choosing a feasible activation vector s(t) ∈ Mσ(t) . Here Mσ denotes the set of all feasible link-activation vectors in the subgraph G(V, Eσ ), complying with the underlying interference constraints. As an example, under the primary interference constraint, Mσ is given by the set of all matchings [6] of the sub-graph G(V, Eσ ). Analytically, elements from the set Mσ will be denoted by their corresponding |E|-dimensional binary incidence-vectors, whose component corresponding to edge e is identically zero if e ∈ / Eσ . • Each node i forwards a subset of packets (possibly empty) to node j over an activated link (i, j) ∈ σ(t), subject to the link capacity constraint. The class Π includes policies that may use all past and future information, and may forward any subset of packets over a link, subject to the link-capacity constraint. To formally introduce the notion of broadcast capacity, we define the random variable Riπ (T ) to be the number of distinct packets received by node i ∈ V up to time T , under a policy π ∈ Π. The time average lim inf T →∞ Riπ (T )/T is the rate of packet-reception at node i. Definition 1. A policy π ∈ Π is called a “broadcast policy of rate λ” if all nodes receive distinct packets at rate λ, i.e., 1 min lim inf Riπ (T ) = λ, i∈V T →∞ T w.p. 1 (6) where λ is the packet arrival rate at the source node r. Definition 2. The broadcast capacity λ∗ of a network is defined to be the supremum of all arrival rates λ, for which there exists a broadcast policy π ∈ Π of rate λ. In the following subsection, we derive an upper-bound on broadcast-capacity, which immediately follows from the previous definition. 3.1 An Upper-bound on Broadcast Capacity Consider a policy π ∈ Π that achieves a broadcast rate of at least λ∗ − ǫ, for an ǫ > 0. Such a policy π exists due to the definition of the broadcast capacity λ∗ in Definition 2. Now consider any proper-cut U of the network G. By definition of a proper-cut, there exists a node i ∈ / U . Let sπ (t, σ(t)) = (sπe (t), e ∈ E) be the link-activation vector chosen by policy π in slot t, upon observing the currentconfiguration σ(t). The maximum number of packets that can be transmitted across the cut U in slot t is upperbounded by the total capacity P of all activated links across the cut-set U , which is given by e∈EU ce sπe (t, σ(t)). Hence, the number of distinct packets received by node i by time T is upper-bounded by the total available capacity across the cut U up to time T , subject to link-activation decisions of the policy π. In other words, we have Riπ (T ) ≤ T X X ce sπe (t, σ(t)) = u · T X sπ (t, σ(t)) (7) t=1 t=1 e∈EU i.e., Riπ (T ) ≤u· T   T 1 X π s (t, σ(t)) , T t=1 where the cut-vector u ∈ Rm , corresponds to the cut-set U , as in Eqn.(1). It follows that, λ∗ − ǫ (a) ≤ ≤ Rjπ (T ) Rπ (T ) ≤ lim inf i T →∞ j∈V T →∞ T T  X  T 1 lim inf u · sπ (t, σ(t)) , T →∞ T t=1 min lim inf (8) where (a) follows from the fact that π is a broadcast policy of rate at least λ∗ − ǫ. Since the above inequality holds for all proper-cuts u, we have   X T 1 sπ (t, σ(t)) (9) λ∗ − ǫ ≤ min lim inf u · u∈U T →∞ T t=1 The following technical lemma will prove to be useful for deriving an upper-bound on the broadcast-capacity. Lemma 1. For any policy π ∈ Π, and any propercut vector u, there exists a collection of vectors βσπ ∈
conv(Mσ ) σ∈Ξ , such that, the following holds w.p. 1  T 1 X π s (t, σ(t)) T →∞ T t=1 X  = min u · p(σ)βσπ min lim inf u · u∈U  u∈U σ∈Ξ The above lemma essentially replaces the minimum cutset bound of an arbitrary activations in (9), by the minimum cut-set bound of a stationary randomized activation, which is easier to handle. Combining Lemma 1 with Eqn. (9), we conclude that for the policy π ∈ Π, there exists a collection of vectors {βσπ ∈ conv(Mσ )}σ∈Ξ such that X  λ∗ − ǫ ≤ min u · (10) p(σ)βσπ u∈U σ∈Ξ  Maximizing the RHS of Eqn. (10) over all vectors βσ ∈ conv(Mσ ), σ ∈ Ξ and letting ǫ ց 0, we have the following universal upper-bound on the broadcast capacity λ∗ X  ∗ λ ≤ max min u · (11) p(σ)βσ βσ ∈conv(Mσ ) u∈U σ∈Ξ Specializing the above bound for single-node cuts of the form Uj = (V \ {j}) → {j}, ∀j ∈ V \ {r}, we have the following upper-bound  X (12) p(σ)βσ λ∗ ≤ max min uj · βσ ∈conv(Mσ ) j∈V \{r} σ∈Ξ It will be shown in Section 4 that in a DAG, our throughputoptimal policy π ∗ achieves a broadcast-rate equal to the RHS of the bound (12). Thus we have the following theorem Theorem 3.1. The broadcast-capacity λ∗DAG of a time-varying wireless DAG is given by: X  λ∗DAG = max min uj · p(σ)βσ (13) βσ ∈conv(Mσ ),σ∈Ξ j∈V \{r} σ∈Ξ The above theorem shows that for computing the broadcastcapacity of a wireless DAG, taking minimum over the singlenode cut-sets {uj , j ∈ V \ {r}} suffice (c.f. Eqn. (11)). 3.2 An Illustrative Example of Capacity Computation In this section, we work out a simple example to illustrate the previous results. Configuration σ4 Figure 1: A Wireless Network and its four possible configurations transmitted over a link if it is ON. Moreover, since the links are assumed to be point-to-point, even if both the links ra and rb are ON at a slot t (i.e., σ(t) = σ3 ), a packet can be transmitted over one of the links only. Hence, the sets of feasible activations are given as follows:     0 1 }, }, Mσ2 = { Mσ 1 = { 1 0     1 0 , }, Mσ4 = φ. Mσ 3 = { 0 1 Here the first coordinate corresponds to activating the edge ra and the second coordinate corresponds to activating the edge rb. To illustrate the effect of link-correlations on broadcastcapacity, we consider three different joint-distributions p(σ), all of them having the following marginal 1 2 1 p(rb = ON) = p(rb = OFF) = 2 p(ra = ON) = p(ra = OFF) = Case 1: Zero correlations. In this case, the links ra and rb are ON w.p. dently at every slot, i.e., p(σi ) = 1/4, Configuration σ1 indepen- i = 1, 2, 3, 4 (14) It can be easily seen that the broadcast capacity, as given in Eqn. (13), is achieved when in configurations σ1 and σ2 , the edges ra and rb are activated w.p. 1 respectively and in the configuration σ3 the edges ra and rb are activated with probability 12 and 12 . In other words, an optimal activation schedule of a corresponding stationary randomized policy is given as follows: βσ∗1 = 1 Wireless network 1 2
′ 0 , βσ∗2 = 0 1
′ 2
′ 1 1 , βσ∗3 = 2 The optimal broadcast capacity can be computed from Eqn. (13) to be λ∗ = 14 + 0 + 14 × 21 = 38 . Case 2: Positive correlations. In this case, assume that the edges ra and rb are positively correlated, i.e., we have Configuration σ2 Configuration σ3 Consider the simple wireless network shown in Figure (1), with node r being the source. The possible network configurations σi , i = 1, 2, 3, 4 are also shown. One packet can be p(σ1 ) = p(σ2 ) = 0; p(σ3 ) = p(σ4 ) = 1 2 Then it is clear that half of the slots are wasted when both the links are OFF (i.e., in the configuration σ4 ). When the network is in configuration σ3 , an optimal randomized activation is to choose one of the two links uniformly at random and send packets over it. Thus 1 1
′ βσ∗3 = 2 2 The optimal broadcast-capacity, computed from Eqn. (13) is λ∗ = 14 . Case 3: Negative correlations. In this case, we assume that the edges ra and rb are negatively correlated, i.e., we have 1 ; p(σ3 ) = p(σ4 ) = 0 2 It is easy to see that in this case, a capacity-achieving activation strategy is to send packets over the link whichever is ON. The broadcast-capacity in this case is λ∗ = 21 , the highest among the above three cases. In this example, with an arbitrary joint distribution of networkconfigurations {p(σi ), i = 1, 2, 3, 4}, it is a matter of simple ∗ calculation to obtain the optimal activations βσ in Eqn. i (13). However it is clear that for an arbitrary network with arbitrary activations M and configuration sets Ξ, evaluating (13) is non-trivial. In the following section we study this problem under some simplifying assumptions. p(σ1 ) = p(σ2 ) = 3.3 Efficient Computation of Broadcast Capacity In this section we study the problem of efficient computation of the Broadcast Capacity λ∗ of a wireless DAG, given by Eqn. (13). In particular, we show that when the number of possible network configurations |Ξ|(n) grows polynomially with n (the number of nodes in the network), there exists a strongly polynomial-time algorithm to compute λ∗ under the primary-interference constraint. Polynomially-bounded network-configurations arise, for example, when the set Ξ(n) consists of all subgraphs of the graph G with at most d number of edges, for some fixed integer d. In this case |Ξ(n)| can be bounded as follows ! d X m = O(n2d ), |Ξ|(n) ≤ k k=0 2 where m(= O(n )) is the number of edges in the graph G. Theorem 3.2 (Efficient Computation of λ∗ ). Suppose that there exists a polynomial q(n) such that, for a wireless DAG network G with n nodes, the number of possible network configurations |Ξ|(n) is bounded polynomially in n, i.e., |Ξ|(n) = O(q(n)). Then, there exists a strongly poly − time algorithm to compute the broadcast-capacity of the network under the primary interference constraints. Although only polynomially many network configurations are allowed, we emphasize that Theorem (3.2) is highly nontrivial. This is because, each network-configuration σ ∈ Ξ itself contains exponentially many possible activations (matchings). The key combinatorial result that leads to Theorem (3.2) is the existence of an efficient separator oracle for the matching-polytope for any arbitrary graph [22]. We first reduce the problem of broadcast-capacity computation of a DAG to an LP with exponentially many constraints. Then invoking the above separator oracle, we show that this LP can be solved in strongly polynomial-time. Proof. See Appendix 9.1. 3.4 Simple Bounds on λ∗ Using Theorem (3.2) we can, in principle, compute the broadcast-capacity λ∗ of a wireless DAG with polynomially many network configurations. However, the complexity of the exact computation of λ∗ grows substantially with the number of the possible configurations |Ξ|(n). Moreover, Theorem (3.2) does not apply when |Ξ|(n) can no longer be bounded by a polynomial in n. A simple example of exponentially large |Ξ|(n) is when a link e is ON w.p. pe independently at every slot, for all e ∈ E. To address this issue, we obtain bounds on λ∗ , whose computational complexity is independent of the size of |Ξ|. These bounds are conveniently expressed in terms of the broadcastcapacity of the static network G(V, E) without time-variation, i.e. when |Ξ| = 1 and Eσ = E, σ ∈ Ξ. Let us denote the broadcast-capacity of the static network by λ∗stat . Specializing Eqn. (13) to this case, we obtain λ∗stat = max min uj · β. β∈conv(M) j∈V \{r} (15) Using Theorem (3.2), λ∗stat can be computed in poly-time under the primary-interference constraint. Now consider an arbitrary joint distribution p(σ) such that each link is ON uniformly with probability p, i.e., X p(σ) = p, ∀e ∈ E. (16) σ∈Ξ:σ(e)=1 We have the following bounds: Lemma 2 (Bounds on Broadcast Capacity). pλ∗stat ≤ λ∗ ≤ λ∗stat . Proof. See Appendix 9.3. Generalization of the above Lemma to the setting, where the links are ON with non-uniform probabilities, may also be obtained in a similar fashion. Note that, in our example 3.2 the bounds in Lemma 2 are tight. In particular, here the value of the parameter p = 21 , the lower-bound is attained in case (2) and the upper-bound is attained in case (3). The above lemma immediately leads to the following corollary: Corollary 3.3. (Approximation-algorithm for computing λ∗ ). Assume that, under the stationary distribution p(σ), probability that any link is ON is p, uniformly for all links. Then, there exists a poly-time p-approximation algorithm to compute the broadcastcapacity λ∗ of a DAG, under the primary-interference constraints. Proof. See Appendix 9.4. In the following section, we are concerned with designing a dynamic and throughput-optimal broadcast policy for a time-varying wireless DAG network. 4. THROUGHPUT-OPTIMAL BROADCAST POLICY FOR WIRELESS DAGS The classical approach of solving the throughput-optimal broadcast problem in the case of a static, wired network is to compute a set of edge-disjoint spanning trees of maximum cardinality (by invoking Edmonds’ tree-packing theorem [20]) and then routing the incoming packets to all nodes via these pre-computed trees [21]. In the time-varying wireless setting that we consider here, because of frequent and random changes in topology, routing packets over a fixed set of spanning trees is no-longer optimal. In particular, part of the network might become disconnected from time-totime, and it is not clear how to select an optimal set of trees to disseminate packets. The problem becomes even more complicated when the underlying statistical model of the network-connectivity process (in particular, the stationary distribution {p(σ), σ ∈ Ξ}) is unknown, which is often the case in mobile adhoc networks. Furthermore, wireless interference constraints add another layer of complexity, rendering the optimal dynamic broadcasting problem in wireless networks extremely challenging. In this section we propose an online, dynamic, throughputoptimal broadcast policy for time-varying wireless DAG networks, that does not need to compute or maintain any global topological structures, such as spanning trees. Interestingly, we show that the broadcast-algorithm that was proposed in [23] for static wireless network, generalizes well to the time-varying case. As in [23], our algorithm also enjoys the attractive feature of in-order packet delivery. The key difference between the algorithm in [23] and our dynamic algorithm is in link-scheduling. In particular, in our algorithm, the activation sets are chosen based on current networkconfiguration σ(t). 4.1 Throughput-Optimal Broadcast Policy π ∗ All policies π ∈ Π, that we consider in this paper, comprise of the following two sub-modules which are executed at every time-slot t: • π(A) (Activation-module): activates a subset of links, subject to the interference constraint and the current network-configuration σ(t). • π(S) (Packet-Scheduling module): schedules a subset of packets over the activated links. Following the treatment in [23], we first restrict our attention to a sub-space Πin−order , in which the broadcast-algorithm is required to follow the so-called in-order delivery property, defined as follows in−order Definition 3 (Policy-space Π [23]). A policy π belongs to the space Πin−order if all incoming packets are serially indexed as {1, 2, 3, . . .} according to their order of arrival at the source r and a node can receive a packet p at time t, if and only if it has received the packets {1, 2, , . . . , p − 1} by the time t. As a consequence of the in-order delivery, the state of received packets in the network at time-slot t may be succinctly represented by the n-dimensional vector R(t), where Ri (t) denotes the index of the latest packet received by node i by time t. We emphasize that this succinct network-state representation by the vector R(t) is valid only under the action of policies in the space Πin−order . This compact representation of the packet-state results in substantial simplification of the overall state-space description. This is because, to completely specify the current packet-configurations in the network in the general policy-space Π, we need to specify the identity of each individual packets that are received by different nodes. To exploit the special structure that a directed acyclic graph offers, it would be useful to constrain the packet-scheduler π(S) further to the following policy-space Π∗ ⊂ Πin−order . Definition 4 (Policy-space Π∗ ⊂ Πin−order [23]). A broadcast policy π belongs to the space Π∗ if π ∈ Πin−order and π satisfies the additional constraint that a packet p can be received by a node j at time t if all in-neighbours of the node j have received the packet p by the time t. The above definition is further illustrated in Figure 2. The variables Xj (t) and i∗t (j) appearing in the Figure are defined subsequently in Eqn. (19). It is easy to see that for all policies π ∈ Π∗ , the packet Ra (t) = 18 Rj (t) = 10 Rb (t) = 15 Rc (t) = 14 Figure 2: Under a policy π ∈ Π∗ , the set of packets available for transmission to node j at slot t is {11, 12, 13, 14}, which are available at all in-neighbors of node j. The in-neighbor of j inducing the smallest packet deficit is i∗t (j) = c, and Xj (t) = 4. scheduler π(S) is completely specified. Hence, to specify a policy in the space Π∗ , we need to define the activationmodule π(A) only. Towards this end, let µij (t) denote the rate (in packets per slot) allocated to the edge (i, j) in the slot t by a policy π ∈ Π∗ , for all (i, j) ∈ E. Note that, the allocated rate µ(t) is constrained by the current network configuration σ(t) at slot t. In other words, we have µ(t) ∈ c ⊙ Mσ(t) , ∀t (17) This implies that, under any randomized activation Eµ(t) ∈ c ⊙ conv(Mσ(t) ), ∀t (18) In the following lemma, we show that for all policies π ∈ Π∗ , certain state-variables X(t), derived from the state-vector R(t), satisfy so-called Lindley recursion [15] of queuing theory. Hence these variables may be thought of as virtual queues. This technical result will play a central role in deriving a Max-Weight type throughput-optimal policy π ∗ , which is obtained by stochastically stabilizing these virtual-queues. For each j ∈ V \ {r}, define Xj (t) = min Ri (t) − Rj (t) i∈∂ in (j) i∗t (j) = arg min i∈∂ in (j)

Ri (t) − Rj (t) , (19) (20) where in Eqn. (20), ties are broken lexicographically. The variable Xj (t) denotes the minimum packet deficit of node j with respect to any of its in-neighbours. Hence, from the definition of the policy-space Π∗ , it is clear that Xj (t) is the maximum number of packets that a node j can receive from its in-neighbours at time t, under any policy in Π∗ . The following lemma proves a “queuing-dynamics” of the variables Xj (t), under any policy π ∈ Π∗ . Lemma 3 have ([23]). Under all policies in π ∈ Π∗ , we  Xj (t + 1) ≤ Xj (t) − + X µkj (t) k∈∂ in (j) X + µmi∗t (j) (t) account the time-variation of network configurations, which is the focus of this paper. To describe π ∗ (A), we first define the node-set Kj (t) = {m ∈ ∂ out (j) : j = i∗t (m)} where the variables are defined earlier in Eqn. (20). The activation-module π (A) is described in Algorithm 1. The resulting policy in the space Π∗ with the activationmodule π ∗ (A) is called π ∗ . Note that, in steps (1) and (2) above, the computation Algorithm 1 A Throughput-optimal Activation Module π ∗ (A) 1: To each link (i, j) ∈ E, assign a weight as follows: ( P Xj (t) − k∈Kj (t) Xk (t), if σ(i,j) (t) = 1 (23) Wij (t) = 0, o.w. 2: Select an activation s∗ (t) ∈ Mσ(t) as follows:
s∗ (t) ∈ arg max s · c ⊙ W (t) s∈Mσ(t) (21) m∈∂ in (i∗ t (j))
Lemma (3) shows that the variables Xj (t), j ∈ V \ {r} ∗ satisfy Lindley recursions in the policy-space Π . Interestingly, unlike the corresponding unicast problem [25], there is no “physical queue” in the system. Continuing correspondence with the unicast problem, the next lemma shows that any activation module π(A) that “stabilizes” the virtual queues X(t) for all arrival rates λ < λ∗ , constitutes a throughput optimal broadcast-policy for a wireless DAG network. Lemma 4. Suppose that, the underlying topology of the wireless network is a DAG. If under the action of a broadcast policy π ∈ Π∗ , for all arrival rates λ < λ∗ , the virtual queue process {X(t)}∞ 0 is rate-stable, i.e., X 1 lim sup Xj (T ) = 0, w.p. 1, T →∞ T j6=r then π is a throughput-optimal broadcast policy for the DAG network. Proof. See Appendix (9.5). Equipped with Lemma (4), we now set out to derive a dynamic activation-module π ∗ (A) to stabilize the virtual-queue ∗ process {X(t)}∞ 0 for all arrival rates λ < λ . Formally, the ∗ structure of the module π (A) is given by a mapping of the following form: π ∗ (A) : (X(t), σ(t)) → Mσ(t) Thus, the module π ∗ (A) is stationary and dynamic as it depends on the current value of the state-variables and the network-configuration only. This activation-module is different from the policy described in [23] as the latter is meant for static wireless networks and hence, does not take into (22) i∗t (m) ∗ (24) 3: Allocate rates on the links as follows: µ∗ (t) = c ⊙ s∗ (t) (25) of link-weights and link-activations depend explicitly on the current network-configuration σ(t). As anticipated, in the following lemma, we show that the activation-module π ∗ (A) stochastically stabilizes the virtual-queue process {X(t)}∞ 0 . Lemma 5. For all arrival rates λ < λ∗ , under the action of the policy π ∗ in a DAG, the virtual-queue process {X(t)}∞ 0 is rate-stable, i.e., 1 X Xj (T ) = 0, w.p. 1 lim sup T →∞ T j6=r The proof of this lemma is centered around a Lyapunovdrift argument [18]. Its complete proof is provided in Appendix (9.6). Combining the lemmas (4) and (5), we immediately obtain the main result of this section Theorem 4.1. The policy π ∗ is a throughput-optimal broadcast policy in a time-varying wireless DAG network. 5. THROUGHPUT-OPTIMAL BROADCASTING WITH INFREQUENT INTER-NODE COMMUNICATION In practical mobile wireless networks, it is unrealistic to assume knowledge of network-wide packet-state information by every node at every slot. This is especially true in the case of time-varying wireless networks, where network-connectivity changes frequently. In this section we extend the main results of section 4 by considering the setting where the nodes make control decisions with imperfect packet-state information that they currently possess. We will show that the dynamic broadcast-policy π ∗ retains its throughput-optimality even in this challenging scenario. State-Update Model. We assume that two nodes i and j can mutually update their knowledge of the set of packets received by the other node, only at those slots with positive probability, when the corresponding wireless-link (i, j) is in ON state. Otherwise, it continues working with the outdated packet stateinformation. Throughout this section, we assume that the nodes have perfect information about the current networkconfiguration σ(t). Suppose that, the latest time prior to time t when packetstate update was made across the link (i, j) is t − T(i,j) (t). Here T(i,j) (t) is a random variable, supported on the set of non-negative integers. Assume for simplicity, that the network configuration process {σ(t)}∞ 0 evolves according to a finite-state, positive recurrent Markov-Chain, with the stationary distribution {p(σ) > 0, σ ∈ Ξ}. With this assumption, T(i,j) (t) is related to the first-passage time in the finite-state positive recurrent chain {σ(t)}∞ 0 . Using standard theory P [7], it can be shown that the random variable T (t) ≡ (i,j)∈E T(i,j) (t) has bounded expectation for all time t . Analysis of π ∗ with Imperfect Packet-State Information. Consider running the policy π ∗ , where each node j now computes the weights Wij′ (t), given by Eqn.(23), of the incoming links (i, j) ∈ E, based on the latest packet-state information available to it. In particular, for each of its in-neighbour i ∈ ∂ in (j), the node j possess the following information of the number of packets received by node i: Ri′ (t) = Ri (t − T(ij) (t)) (26) ′ Now, if the packet-scheduler module π (S) of a broadcastpolicy π ′ takes scheduling decision based on the imperfect state-information R′ (t) (instead of the true state R(t)), it still retains the following useful property: Lemma 6. π ′ ∈ Π∗ . Figure 3: A 3 × 3 grid network. Wij′ (t), used by policy π ′ and the true link-weights Wij (t), as follows Lemma 7. There exists a finite constant C such that, the expected weight Wij′ (t) of the link (ij), locally computed by the node j using the random update process, differs from the true link-weight Wij (t) by at most C, i.e. |EWij′ (t) − Wij (t)| ≤ C (27) The expectation above is taken with respect to the random packet-state update process. Proof. See Appendix (9.8) From lemma (7) it follows that the policy π ′ , in which link-weights are computed using imperfect packet-state information is also a throughput-optimal broadcast policy for a wireless DAG. Its proof is very similar to the proof of Theorem (4.1). However, since the policy π ′ makes scheduling decision using W ′ (t), instead of W (t), we need to appropriately bound the differences in drift using the Lemma (7). The technical details are provided in Appendix (9.9). Theorem 5.1. The policy π ′ is a throughput-optimal broadcast algorithm in a time-varying wireless DAG. Proof. See Appendix (9.7). The above lemma states that the policy π ′ inherits the inorder delivery property and the in-neighbour packet delivery constraint of the policy-space Π∗ . From Eqn. (23) it follows that, computation of linkweights {Wij (t), i ∈ ∂ in (j)} by node j requires packet-state information of the nodes that are located within 2-hops from the node j. Thus, it is natural to expect that with an ergodic state-update process, the weights Wij′ (t), computed from the imperfect packet-state information, will not differ too much from the true weights Wij (t), on the average. Indeed, we can bound the difference between the link-weights 6. NUMERICAL SIMULATION We numerically simulate the performance of the proposed dynamic broadcast-policy on the 3 × 3 grid network, shown in Figure 3. All links are assumed to be of unit capacity. Wireless link activations are subject to primary interference constraints, i.e., at every slot, we may activate a subset of links which form a Matching [28] of the underlying topology. External packets arrive at the source node r according to a Poisson process of rate λ packets per slot. The following proposition shows that, the broadcast capacity λ∗stat of the static 3 × 3 wireless grid (i.e., when all links are ON with probability 1 at every slot) is 25 . 7. CONCLUSION Proposition 6.1. The broadcast-capacity λ∗stat of the static 3 × 3 wireless grid-network in Figure 3 is 25 . ′ Average Broadcast-Delay Dpπ (λ) See Appendix (9.10) for the proof. In our numerical simulation, the time-variation of the network is modeled as follows: link-states are assumed to evolving in an i.i.d. fashion; each link is ON with probability p at every slot, independent of everything else. Here 0 < p ≤ 1 is the connectivity-parameter of the network. Thus, for p = 1 we recover the static network model of [23]. We also assume that the nodes have imperfect packet-state information as in Section 5. Hence, two nodes i and j can directly exchange packet state-information, only when the link (i, j) (if any) is ON. ′ The average broadcast-delay Dpπ (λ) is plotted in Figure 4 as a function of the packet arrival rate λ. The broadcastdelay of a packet is defined as the number of slots the packet takes to reach all nodes in the network after its arrival. Because of the throughput-optimality of the policy π ′ (Theorem (5.1)), the broadcast-capacity λ∗ (p) of the network, for a given value of p, may be empirically evaluated from the ′ λ-intercept of vertical asymptote of the Dpπ (λ) − λ curve. As evident from the plot, for p = 1, the proposed dynamic algorithm achieves all broadcast rates below λ∗stat = 25 = 0.4. This shows the throughput-optimality of the algorithm π ′ . It is evident from the Figure 4 that the broadcast capacity λ∗ (p) is non-decreasing in the connectivity-parameter p, i.e., λ∗ (p1 ) ≥ λ∗ (p2 ) for p1 ≥ p2 . We observe that, with i.i.d. connectivity, the capacity bounds given in Lemma (2) are not tight, in general. Hence the lower-bound of pλ∗stat is a pessimistic estimate of the actual broadcast capacity ′ λ∗ (p) of the DAG. The plot also reveals that, Dpπ (λ) is nondecreasing in λ for a fixed p and non-increasing in p for a fixed λ, as expected. p = 0.4 p = 0.6 p=1 Packet Arrival Rate λ ′ Figure 4: Plot of average broadcast-delay Dpπ (λ), as a function of the packet arrival rates λ. The underlying wireless network is the 3 × 3 grid, shown in Figure 3, with primary interference constraints. In this paper we studied the problem of throughput-optimal broadcasting in wireless directed acyclic networks with pointto-point links and time-varying connectivity. We characterized the broadcast-capacity of such networks and derived efficient algorithms for computing it, both exactly and approximately. Next, we proposed a throughput-optimal broadcast policy for such networks. This algorithm does not require any spanning tree to be maintained and operates based on local information, which is updated sporadically. 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Prentice hall Upper Saddle River, 2001. J. Yuan, Z. Li, W. Yu, and B. Li. A cross-layer optimization framework for multihop multicast in wireless mesh networks. Selected Areas in Communications, IEEE Journal on, 24(11), 2006. APPENDIX 9.1 Proof of Theorem 3.2 Under the primary interference constraint, the set of feasible activations of the graphs are matchings [28]. To solve for the optimal broadcast capacity in a time-varying network, first we recast Eqn. (13) as an LP. Although this LP has exponentially many constraints, using a well-known separation oracle for matchings, we show how to solve this LP in strongly-polynomial time via the ellipsoid algorithm [2]. ′ For a subset of edges E ′ ⊂ E, let χE be the incidence ′ vector, where χE (e) = 1 if e ∈ E ′ and is zero otherwise. Let Pmatching (G(V, E)) = convexhull({χM |M is a matching in G(V, E)}) We have the following classical result by Edmonds [22]. Theorem 9.1. The set Pmatching (G(V, E)) is characterized by the set of all β ∈ R|E| such that : X βe ≥ 0 ∀e ∈ E βe ≤ 1 ∀v ∈ V βe ≤ (28) e∈∂ in (v)∪∂ out (v) X e∈E[U ] |U | − 1 ; 2 U ⊂ V, |U | odd Here E[U ] is the set of edge with both end points in U. Thus, following Eqn. (13), the broadcast capacity of a DAG can be obtained by the following LP : max λ (29) Subject to, λ ≤ X e∈∂ in (v) βσ ∈ ce X σ∈Ξ
p(σ)βσ,e , ∀v ∈ V \ {r} (30) Pmatching (G(V, Eσ )), ∀σ ∈ Ξ (31) The constraint corresponding to σ ∈ Ξ in (31) refers to the set of linear constraints given in Eqn.(28) corresponding to the graph G(V, Eσ ), for each σ ∈ Ξ. Invoking the equivalence of optimization and separation due to the ellipsoid algorithm [2], it follows that the LP (29) is solvable in poly-time, if there exists an efficient separatororacle for the set of constraints (30) and (31). With our assumption of polynomially many network configurations |Ξ|(n), there are only linearly many constraints (n − 1, to be precise) in (30) with polynomially many variables in each constraint. Thus the set of constraints (30) can be separated efficiently. Next we invoke a classic result from the combinatorial-optimization literature which shows the existence of efficient separators for the matching polytopes. Theorem 9.2. [22] There exists a strongly poly-time algorithm, that given G = (V, E) and β : E → R|E| determines if β satisfies (28) or outputs an inequality from (9.1) that is violated by β. Hence, there exists an efficient separator for each of the constraints in (9.1). Since there are only polynomially many network configurations, this directly leads to Theorem 3.2. 9.2 Proof of Lemma 1 Eqn. (37), (38) and Eqn. (38), we have Proof. Fix a time T . For each configuration σ ∈ Ξ, let σ be the index of the time-slots up to time T such {tσ,i }Ti=1 that σ(t) = σ. Clearly we have, X Tσ = T (32) u∈U T ր∞ = min u∈U σ∈Ξ = Hence, we can rewrite Tσ T X Tσ 1 X 1 X π sπ (tσ,i , σ) s (t, σ(t)) = T t=1 T T σ i=1 σ∈Ξ u· Since the process σ(t) is stationary ergodic, we have lim Tσ = p(σ), ∀σ ∈ Ξ, w.p. 1 T min lim inf u · u∈U T ր∞ = min u∈U X 1 T T X s (t, σ(t)) t=1 p(σ) lim inf u · T →∞ σ∈Ξ (36) Let an optimal solution to Eqn. (13) be obtained at
Ξ . Then from Eqn. (39), it follows that X p(σ)βσ∗ ∈ conv(M)  Tσ 1 X π s (tσ,i , σ) , Tσ i=1 w.p. 1 = min u∈U σ∈Ξ (37) ∈ π p(σ) lim u · ζσ,T σk k→∞ π p(σ) lim inf u · ζσ,T . σ Tσ →∞ π lim ζσ,T = βσπ , σk ∀σ ∈ Ξ Where βσπ ∈ Mσ , since Mσ is closed. Hence combining p(σ)βσ∗ σ∈Ξ max
min uj · β β∈conv(M) j∈V \{r} This proves the upper-bound. 9.3.2 Proof of the Lower-bound Since Mσ ⊂ M, the expression for the broadcast-capacity (13) may be re-written as follows: X X
p(σ)βσ (e)1(e ∈ σ) λ∗ = max min ce e∈∂ in (j) σ∈Ξ ∗ Let β ∈ M be the optimal activation, achieving the RHS of (15). Hence we can lower-bound λ∗ as follows X X
p(σ)1(e ∈ σ) λ∗ ≥ min ce β ∗ (e) j∈V \{r} e∈∂ in (j) p min j∈V \{r} X σ∈Ξ ce β ∗ (e) e∈∂ in (j) = p min uj · β ∗ (b) pλ∗stat = (38) X λ∗ ≤ λ∗stat βσ ∈M j∈V \{r} Let us denote k→∞ (39) βσ∗ , σ Using Eqn. (15), this shows that = X min uj · ≤ (a) σ∈Ξ w.p.1 ∀σ ∈ Ξ βσ ∈conv(Mσ ) j∈V \{r} Since s (tσ,i , σ) ∈ Mσ for all i ≥ 1, convexity of the set Mσ π ∈ Mσ for all Tσ ≥ 1. Since the set Mσ implies that ζσ,T σ is closed and bounded (and hence, compact) any sequence in Mσ has a converging sub-sequence. Consider any set of π } , σ ∈ Ξ such that, it converging sub-sequences {ζσ,T σk k≥1 achieves the following u∈U  βσ ∈ conv(Mσ ) =⇒ βσ ∈ conv(M) max π min σ∈Ξ Mσ ⊂ M,  Tσ 1 X π = s (tσ,i , σ) Tσ i=1 X p(σ)βσπ σ∈Ξ Since p(σ) > 0, ∀σ ∈ Ξ, the above implies that Tσ ր ∞ as T ր ∞∀σ, w.p.1. In the rest of the proof we will concentrate on a typical sample path {σ(t)}t≥1 having the above property. π } For each σ ∈ Ξ, define the sequence {ζσ,T σ Tσ ≥1 π ζσ,T σ u∈U X Hence we have, π  min u · Proof. 9.3.1 Proof of the Upper-bound Note that, for all σ ∈ Ξ, we have Eσ ⊂ E. Hence, it follows that (35) Hence from Eqn. (34) we have,  σ∈Ξ This in turn implies that Using countability of Ξ and invoking the union bound, we can strengthen the above conclusion as follows T →∞ p(σ)u · βσπ 9.3 Proof of Lemma 2  X   Tσ T Tσ 1 X π 1 X π s (t, σ(t)) = u· s (tσ,i , σ) (34) T t=1 T Tσ i=1 σ∈Ξ Tσ = p(σ), w.p. 1 ∀σ ∈ Ξ lim T →∞ T X  T 1 X π s (t, σ(t)) T t=1 (33) Hence,   min lim inf u · j∈V \{r} Equality (a) follows from the assumption (16) and equality (b) follows from the characterization (15). This proves the lower-bound. 9.4 Proof of Corollary 3.3 ∗ Consider the optimal randomized-activation vector β ∈ M, corresponding to the stationary graph G(V, E) (15). By Theorem (3.2), β ∗ can be computed in poly-time under the primary interference constraint. Note that, by Caratheodory’s theorem [17], the support of β ∗ may be bounded by |E|. Hence it follows that λ∗stat (15) may also be computed in poly-time. From the proof of Lemma (2), it follows that by randomly activating β ∗ (i.e., βσ (e) = β ∗ (e)1(e ∈ σ), ∀σ ∈ Ξ) we obtain a broadcast-rate equal to pλ∗stat where λ∗stat is shown to be an upper-bound to the broadcast capacity λ∗ in Lemma (2). Hence it follows that pλ∗stat constitutes a p-approximation to the broadcast capacity λ∗ , which can be computed in poly-time. Since P the variables Xi (t)’s are non-negative, we have j6=r Xj (t). Thus, for each node j Pk−1 i=1 Xui (t) ≤ T −1 T −1 1 X 1 X 1 1 X A(t) − A(t). Xj (T ) ≤ Rj (T ) ≤ T t=0 T T T t=0 j6=r Taking limit as T → ∞ and using the strong law of large numbers for the arrival process and Eqn. (40), we have Rj (T ) = λ, ∀j. T This concludes the proof. lim w.p. 1 T →∞ 9.6 Proof of Lemma (5) We begin with a preliminary lemma. Lemma 8. If we have 9.5 Proof of Lemma (4) Assume that under the policy π ∈ Π∗ , the virtual queues Xj (t) are rate stable i.e., limT →∞ Xj (T )/T = 0, a.s. for all j. Applying union-bound, it follows that, P j6=r Xj (T ) = 0, w.p. 1 (40) lim T →∞ T Now consider any node j 6= r in the network. We can construct a simple path p(r = uk → uk−1 . . . → u1 = j) from the source node r to the node j by running the following Path construction algorithm on the underlying graph G(V, E). Q(t + 1) ≤ (Q(t) − µ(t))+ + A(t) (42) + where all the variables are non-negative and (x) = max{x, 0}, then Q2 (t + 1) − Q2 (t) ≤ µ2 (t) + A2 (t) + 2Q(t)(A(t) − µ(t)). Proof. Squaring both sides of Eqn. (42) yields, Q2 (t + 1) ≤ (Q(t) − µ(t))+
2 + A2 (t) + 2A(t)(Q(t) − µ(t))+ ≤ (Q(t) − µ(t))2 + A2 (t) + 2A(t)Q(t), 2 where we use the fact that x2 ≥ (x+ ) , Q(t) ≥ 0, and µ(t) ≥ 0. Rearranging the above inequality finishes the proof. Algorithm 2 r → j Path Construction Algorithm Require: DAG G(V, E), node j ∈ V 1: i ← 1 2: ui ← j 3: while ui 6= r do 4: ui+1 ← i∗t (ui ); 5: i← i+1 6: end while Applying Lemma 8 to the dynamics (21) of Xj (t) yields, for each node j 6= r, Xj2 (t + 1) − Xj2 (t) ≤ B(t) + X X
2Xj (t) µkj (t) , µmi∗t (t) − m∈V At time t, the algorithm chooses the parent of a node ui in the path p as the one that has the least relative packet deficit as compared to ui (i.e. ui+1 = i∗t (ui )). Since the underlying graph G(V, E) is a connected DAG (i.e., there is a path from the source to every other node in the network), the above path construction algorithm always terminates with a path p(r → j). Note that the output path of the algorithm varies with time. The number of distinct packets received by node j up to time T can be written as a telescoping sum of relative packet deficits along the path p, i.e., =
− = − i=1 Xui (T ) + T −1 X X  X µmi∗t (t) (45) E[µij (t) | X(t), σ(t)] Xj (t) (46) Xj (t)E m∈V µkj (t) | X(t), σ(t) k∈V Xui (T ) + Rr (T ) k−1 X X j6=r = B|V | − 2 i=1 (a) ∆(X(t)|σ(t)) , E[L(X(t + 1) − L(X(t)) | X(t), σ(t)]
 X Xj2 (t + 1) − Xj2 (t) | X(t), σ(t) = E Rui (T ) − Rui+1 (T ) + Ruk (T ) i=1 =− where B(t) ≤ c2max + max{a2 (t), c2max } ≤ (a2 (t) + 2c2max ), a(t) is the number of exogenous packet arrivals in a slot, and cmax , maxe∈E ce is the maximum capacity of the links. We assume the arrival process a(t) has bounded second moments; thus, there
exists a finite constant B > 0 such that E[B(t)] ≤ E a2 (t) + 2c2max < B. the quadratic Lyapunov function L(X(t)) = PWe define 2 j6=r Xj (t). From (43), the one-slot Lyapunov drift ∆(X(t)), conditioned on the current network-configuration σ(t) yields ≤ B|V | + 2 k−1 X (44) k∈V j6=r Rj (T ) = Ru1 (T ) k−1 X (43) X  (i,j)∈E A(t), t=0 where the equality (a) follows the observation that Xui (T ) = Qui+1 ui (T ) = Rui+1 (T ) − Rui (T ). (41) − X Xk (t) k∈Kj (t) = B|V | − 2 X
(i,j)∈E Wij (t)E[µij (t) | X(t), σ(t)] (47) The broadcast-policy π ∗ is chosen to minimize the upperbound of conditional-drift, given on the right-hand side of (47) among all policies in Π∗ . Next, we construct a randomized scheduling policy π RAND ∈ ∗ ∗ Π . Let βσ ∈ conv(Mσ ) be the part of an optimal solution corresponding to σ(t) ≡ σ given by Eqn. 11. From Caratheodory’s theorem [17], there exist at most (|E| + 1) link-activation vectors sk ∈ σ and the associated nonPM |E|+1 negative scalars {ασk } with k=1 ασk = 1, such that |E|+1 βσ∗ = X ασk sσk . (48) k=1 Define the average (unconditional) activation vector X p(σ)βσ∗ β∗ = Note that this stationary randomized policy π RAND operates independently of the state of received packets in the network, i.e., X(t). However it depends on the current network-configuration σ(t). Since each network node j is relabelled as vl for some l, from (53) we have, for each node j 6= r, the total expected incoming transmission rate to the node j under the policy π RAND , averaged over all network states σ satisfies RAND X E[µπij X (t) | X(t)] = RAND E[µπij (t)] i:(i,j)∈E i:(i,j)∈E X = ql ce βe∗ e∈EUv l l . =λ+ǫ |V | (49) σ∈Ξ (54) Hence, from Eqn. (11) we have, λ∗ ≤ min U : a proper cut X ce βe∗ . (50) e∈EU Suppose that the exogenous packet arrival rate λ is strictly less than the broadcast capacity λ∗ . There exists an ǫ > 0 such that λ + ǫ ≤ λ∗ . From (50), we have X λ+ǫ≤ min ce βe∗ . (51) U : a proper cut e∈EU For any network node v 6= r, consider the proper cuts Uv = V \ {v}. Specializing the bound in (51) to these cuts, we have X λ+ǫ ≤ ce βe∗ , ∀v 6= r. (52) Equation (54) shows that the randomized policy π RAND provides each network node j 6= r with the total expected incoming rate strictly larger than the packet arrival rate λ via proper random link activations conditioned on the current network configuration. According to our notational convention, we have RAND X E[µπir (t) | X(t)] = E[ X RAND µπir (t)] = λ. i:(i,r)∈E i:(i,r)∈E (55) From (54) and (55), if node i appears before node j in the aforementioned topological ordering, i.e., i = vli < vlj = j for some li < lj , then e∈EUv Since the underlying network topology G = (V, E) is a DAG, there exists a topological ordering of the network nodes so that: (i) the nodes can be labelled serially as {v1 , . . . , v|V | }, where v1 = r is the source node with no in-neighbours and v|V | has no outgoing neighbours and (ii) all edges in E are directed from vi → vj , i < j [5]; From (52), we define ql ∈ [0, 1] for each node vl such that X l ql ce βe∗ = λ + ǫ , l = 2, . . . , |V |. (53) |V | e∈E Uv l Consider the randomized broadcast policy π RAND ∈ Π∗ working as follows: Stationary Randomized Policy π RAND : (i) If the observed network-configuration at slot t is σ(t) = σ, the policy π RAND selects 3 the feasible activation set sσk with probability ασk ; (ii) For each incoming selected link e = (·, vl ) to node vl such that se (t) = 1, the link e is activated independently with probability ql ; (iii) Activated links (note, not necessarily all the selected links) are used to forward packets, subject to the constraints that define the policy class Π∗ (i.e., in-order packet delivery and that a network node is only allowed to receive packets that have been received by all of its in-neighbors). X RAND E[µπki X RAND E[µπkj (t)] k:(k,j)∈E k:(k,i)∈E ≤− (t)] − ǫ . |V | (56) The above inequality will be used to show the throughput optimality of the policy π ∗ . The drift inequality (45) holds for any policy π ∈ Π∗ . The broadcast policy π ∗ observes the states (X(t), σ(t)) and and seek to greedily minimize the upper-bound of drift (47) at every slot. Comparing the actions taken by the policy π ∗ with those by the randomized policy π RAND in slot t in (45), we have ∗ ∆π (X(t)|σ(t)) X  π∗ E µij (t) | X(t), σ(t)]Wij (t) ≤ B|V | − 2 (57) (i,j)∈E ≤ B|V | − 2 X (i,j)∈E (∗) = B|V | − 2 X  RAND E µπij (t) | X(t), σ(t)]Wij (t) (i,j)∈E  RAND E µπij (t) | σ(t)]Wij (t) Taking Expectation of both sides w.r.t. (58) the stationary- i.e., process σ(t) and rearranging, we have ∗ ∆π (X(t)) (59) X ≤ B|V | − 2 E (i,j)∈E ≤ B|V | + 2 X  Xj (t) j6=r Xj (t) − cmax ET ≤ EXj′ (t) ≤ Xj (t) X m∈V 2ǫ X Xj (t). ≤ B|V | − |V | Where the expectation is with respect to the random update  fashion, since every in πRAND  X  πRAND  process at the node j. In a similar neighbour i of a node k ∈ ∂ out (j), is at most 2-hop away E µkj (t) E µmi∗t (t) − from the node j, we have k∈V (60) Ri (t) − T cmax ≤ Ri′ (t) ≤ Ri (t) j6=r Note that i∗t = arg mini∈In(j) Qij (t) for a given node j. Since node i∗t is an in-neighbour of node j, i∗t must lie before j in any topological ordering of the DAG. Hence, the last inequality of (60) follows directly from (56). Taking expectation in (60) with respect to X(t), we have     2ǫ E L(X(t + 1)) − E L(X(t)) ≤ B|V | − E||X(t)||1 , |V | where || · ||1 is the ℓ1 -norm of a vector. Summing the above inequality over t = 0, 1, 2, . . . T − 1 yields T −1 2ǫ X E L(X(T )) − E L(X(0)) ≤ B|V |T − E||X(t)||1 . |V | t=0   (64) RAND µπij (t)]Wij (t)   Dividing the above by 2T ǫ/|V | and using L(X(t)) ≥ 0, we have T −1 1 X B|V |2 |V | E[L(X(0))] E||X(t)||1 ≤ + T t=0 2ǫ 2T ǫ Rk (t) − T cmax ≤ Rk′ (t) ≤ Rk (t) It follows that for all i ∈ ∂ in (k) (Ri (t) − Rk (t)) − T cmax ≤ ≤ Ri′ (t) − Rk′ (t) (Ri (t) − Rk (t)) + T cmax Hence, Xk (t) − T cmax ≤ Xk′ (t) ≤ Xk (t) + T cmax Again taking expectation w.r.t. the random packet-state update process, Xk (t) − cmax ET ≤ EXk′ (t) ≤ Xk (t) + cmax ET (65) Combining Eqns (64) and (65) using Linearity of expectation and using Eqn. (23) we have Taking a lim sup of both sides yields T −1 B|V |2 1 XX E[Xj (t)] ≤ T t=0 j6=r 2ǫ Also, (61) −ncmax ET + Wij (t) ≤ EWij′ (t) ≤ Wij (t) + ncmax ET which implies that all virtual-queues Xj (t) are strongly stable [18]. Strong stability of Xj (t) implies that all virtual queues Xj (t) are rate stable [18, Theorem 2.8]. Thus the lemma (7) follows with C ≡ ncmax ET < ∞. 9.7 Proof of Lemma (6) To prove throughput-optimality of TheoremP (5.1), we work with the same Lyapunov function L(X(t)) = j6=r Xj2 (t) as in Theorem (4.1) and follow the same steps until Eqn. (47) to obtain the following upper-bound on conditional drift lim sup T →∞ Proof. Recall the definition of the policy-space Π∗ . For every node i, since Ri (t) is a non-decreasing function of t, if a packet p is allowed to be transmitted to a node j at time slot t, by the policy π ′ , it is certainly allowed to be transmitted by the policy π. This is because Ri′ (t) ≤ Ri (t), ∀j ∈ ∂ out (i) and hence outdated state-information may only prevent transmission of a packet p at a time t, which would otherwise be allowed by the policy π ∗ . As a result, the policy π ′ can never transmit a packet to node j which is not present at all in-neighbours of the node j. This shows that π ′ ∈ Π∗ . 9.8 Proof of Lemma (7) 9.9 Proof of Theorem (5.1) ′ ∆π (X(t)|X(t), X ′ (t), σ(t)) X ′ Wij (t)E(µπij (t)|X(t), X ′ (t), σ(t)) ≤ B|V | − 2 (i,j)∈E (66) Since the policy π ′ makes scheduling decision based on the locally computed weights Wij′ (t), by the definition of the policy π ′ , we have for any policy π ∈ Π: Consider the packet-state update process at node j. Since the capacity of the links are bounded by cmax , from Eqn. (26) and the fact that Ri (t) is non-decreasing, we have Ri (t) − T cmax ≤ Ri′ (t) in ≤ Ri (t), ∀i ∈ ∂ (j) (62) (63) ′ Wij′ (t)E(µπij (t)|X(t), X ′ (t), σ(t)) (i,j)∈E ≥ Hence, from Eq. (19), it follows that Xj (t) − T cmax ≤ Xj′ (t) ≤ Xj (t) X X Wij′ (t)E(µπij (t)|X(t), X ′ (t), σ(t)) (67) (i,j)∈E Taking expectation of both sides w.r.t. the random update process X ′ (t), conditioned on the true network state X(t) and the network configuration σ(t), we have X π′ Wij (t)E(µij (t)|X(t), σ(t)) Cn2 cmax /2 + Since the above holds for any stationary randomized policy π, we conclude λ∗stat ≤ (i,j)∈E (a) ≥ ′ 2 5 (69)  X EWij′ (t)E(µπij (t)|X(t), σ(t)) X EWij′ (t)E(µπij (t)|X(t), σ(t)) r a b r a b X Wij (t)E(µπij (t)|X(t), σ(t)) − Cn2 cmax /2 c d e c d e f g h f g h (i,j)∈E (b) ≥ (i,j)∈E (c) ≥ (i,j)∈E (68) Here the inequality (a) and (c) follows from Lemma (7) and the fact that |E| ≤ n2 /2 and µij (t) ≤ cmax . Inequality (b) follows from Eqn. (67). Thus from Eqn. (66) and (68), the expected conditional drift of the Lyapunov function under the policy π ′ , where the expectation is taken w.r.t. the random update and arrival process is upper-bounded as follows: X ′ Wij (t)E[µπij (t) | X(t), σ(t)] ∆π (X(t)|X(t), σ(t)) ≤ B ′ − 2 Matching M1 (i,j)∈E ′ Matching M2 r a b r a b c d e c d e f g h f g h 2 with the constant B ≡ B|V | + 2Cn cmax . Since the above inequality holds for any policy π ∈ Π, we can follow the exactly same steps in the proof of Theorem (4.1) by replacing an arbitrary π by π RAND and showing that it has negative drift. Matching M3 9.10 Proof of Proposition 6.1 r Like many proofs in this paper, this proof also has a converse and an achievability part. In the converse part, we obtain an upper bound of 25 for the broadcast capacity λ∗stat of the stationary grid network (i.e. when all links are ON w.p. 1). In the achievability part, we show that this upper bound is tight. Part (a): Proof of the Converse: λ∗stat ≤ b fra ≥ λ, fab ≥ λ Applying the primary interference constraint at node a, we then obtain fad ≤ 1 − 2λ Because of symmetry in the network topology, we also have fcd ≤ 1 − 2λ. However, to achieve a broadcast capacity of λ, the total allocated rate towards node d must be atleast λ. Hence we have, c d e f g c h f Matching M5 2 5 a b 2 5 1 5 2 5 1 5 2 5 d e 0 0 g 0 2 5 h ‘Time averaged’ Network Figure 5: Some feasible activations of the 3 × 3 grid network which are activated uniformly at random. The components corresponding to each edge in the resulting overall activation vector β is denoted by the numbers alongside the edges. Part (b): Proof of the Achievability: λ∗stat ≥ 25 : As usual, the achievability proof will be constructive. Consider the following five activations (matchings) M1 , M2 , . . . , M5 of the underlying graph as shown in Figure 5. Now consider a stationary policy π ∗ ∈ Π∗ that activates the matchings M1 , . . . , M5 at each slot uniformly at random with probability 51 for each matching. The resulting ‘time-averaged’ graph is also shown in Figure 5. Using Theorem 3.1, it is clear that λ∗stat ≥ 25 . Combining the above with the converse result in Eqn. (69), we conclude that, λ∗stat = 2 . 5 2 5 2 5 i.e. λ≤ r 2 5 We have shown earlier that for the purpose of achieving capacity, it is sufficient to restrict our attention to stationary randomized policies only. Suppose a stationary randomized policy π achieves a broadcast rate λ and it activates edge e ∈ E at every slot with probability fe . Then for the nodes a and b to receive distinct packets at rate λ, one requires 2(1 − 2λ) ≥ λ a Matching M4  2 5 

Spike Event Based Learning in Neural Networks arXiv:1502.05777v1 [] 20 Feb 2015 J. A. Henderson, T. A. Gibson, J. Wiles Abstract A scheme is derived for learning connectivity in spiking neural networks. The scheme learns instantaneous firing rates that are conditional on the activity in other parts of the network. The scheme is independent of the choice of neuron dynamics or activation function, and network architecture. It involves two simple, online, local learning rules that are applied only in response to occurrences of spike events. This scheme provides a direct method for transferring ideas between the fields of deep learning and computational neuroscience. This learning scheme is demonstrated using a layered feedforward spiking neural network trained self-supervised on a prediction and classification task for moving MNIST images collected using a Dynamic Vision Sensor. Keywords: Spiking Neural Networks, Learning, Vision, Prediction 1. Introduction Methods in deep learning for training neural networks (NNs) have been very successfully applied to a range of datasets, performing tasks at levels approaching human performance, such as image classification [1], object detection [1] and speech recognition [2, 3, 4]. Along with these experimental successes, the field of deep learning is rapidly developing theoretical frameworks in representation learning [5, 6, 7] including understanding the benefits of different types of non-linearities in neuron activation functions [8], disentanglement of inputs by projecting onto hidden layer manifolds, model averaging with techniques like maxout and dropout [9, 10] and assisting generalization through corruption of input with denoising autoencoders [11]. These types of experimental and theoretical work are necessary to effectively build and understand systems like brains that are capable of learning to solve real world problems. Many of the successes of deep learning are a Preprint submitted to arχiv February 23, 2015 result of a broad inspiration from biology; however, there is a large gap in understanding how the principles of deep learning are related to those of the brain. Some elements of deep learning may well inspire discoveries in brain function. Equally, deep learning systems are still inferior to the brain in aspects such as memory, thus efforts to develop models that bridge between deep learning and neuroscience are likely to be mutually beneficial. The neuron models commonly used in deep learning are abstracted away from neuron models that are used in computational neuroscience to model biological neurons. Spiking is a salient feature of biological neurons that is not typically present in deep learning networks. It is not yet understood why the brain uses spiking dynamics; for the purposes of machine learning it would be useful to know what if any advantages spiking dynamics confers spike based NN learning algorithms over other types of NN learning algorithms, rather than advantages that are otherwise useful in implementing algorithms in biology such as energy efficiency and robustness. Dynamical systems like spiking networks appear more naturally suited to processing continuous time temporal data than state machines, as deep networks are usually implemented, but this idea is yet to be demonstrated experimentally on machine learning tasks. In an effort to bridge this gap in understanding between spike, and nonspike based NN learning systems, and develop systems for processing event based, continuous time data, this paper develops a scheme for learning connectivity in a spiking neural network (SNN). The scheme is based upon learning conditional instantaneous firing rates, linking it to many of the statistical frameworks previously developed in deep learning that are based on conditional probabilities. However, our scheme is fundamentally different to most methods used in deep learning as the learning rules are based solely on the activity of the neurons in the network and are the same, independent of the choice of neuron dynamics or activation function unlike gradient descent methods [12], and they can be implemented online and do not require periods of statistical sampling from the model unlike energy based methods [13]. In addition, the learning scheme is local, meaning that modifying a connection only requires knowledge of the activity of the neurons it connects, not neurons from a distant part of the network, unlike gradient descent and energy based methods [12, 13]. From a perspective of biological plausibility, this means neurons do not have to make assumptions about, or communicate to each other their dynamics or activation function and associated parameters in order to correctly learn, and the system can be run online without 2 interrupting processing with periods of sampling for learning. This paper describes a general scheme for event based learning in SNNs. This scheme is demonstrated on a network similar to that commonly used in deep learning, specifically, a feedforward layered network architecture with rectified linear units, and piecewise constant temporal connectivity. Dropout is utilized to show that many ideas from deep learning can be directly imported into SNNs using this learning scheme. An event based dataset of moving MNIST digits collected using an DVS camera [14, 15, 16] is used to train the network for both prediction and classification tasks. 2. Learning Theory We begin by developing a method for learning the connectivity of a supervised output neuron. The discussion will be framed in reference to learning in a network operating continuously in time with temporally delayed connectivity and temporally encoded input signals since spiking neurons are usually modelled as dynamical systems; however, the results are also applicable to networks operating in discrete timesteps (as is usual in implementations of SNNs with current standard computer architecture), with or without temporally delayed connectivity and temporally encoded inputs, so they can also be applied to traditional artificial neural networks performing static image classification, for example. Figure 1 shows a general network containing input neurons whose activity is determined by an external source, hidden neurons, and supervised output neurons. At present we do not assume any particular connection architecture, nor do we specify the dynamics or activation functions of the individual neurons. We first consider learning the input, Qo (t), that a supervised output neuron o receives from the network at time t. We need to identify the mathematical quantity that o should learn - the quantity that Qo (t) should approximate. Note that Qo (t) may be calculated from two sets of quantities only. The first quantities are the weights W , that connect the activity of the network to o, potentially including self connections and connections that have a temporal delay. We call these connections weights as this is the terminology usually used in machine learning; however, these connections can also be imagined as propagator functions whose value varies with temporal delay. The second quantity is the history of activity of the neurons in the network, H, which since we are primarily interested in SNNs and event based learning, 3 Figure 1: Schematic of a general neural network consisting of input neurons controlled by an external source, hidden neurons and supervised output neurons. Connectivity between neurons is unrestricted, directed connections are allowed between every pair of neurons including self connections, as are temporally delayed connections. 4 we model as a set of one dimensional Dirac delta functions in time that may be normalized to non unit integral to allow a neuron’s spike strength to be a real value, instead of only binary as in many SNN models. However, since it allows the use of more intuitive terminology of spike rates, instead of spike strength rates, we assume that a spike with real valued strength is equivalent to a sum of simultaneous unit strength spikes and possibly one partial unit spike. Alternatively, this equivalence holds if spike strengths are restricted to a unit value and simultaneous spikes are not allowed. In any case the mathematical description and quantitative results are unchanged aside from a possible conversion function if simultaneous spikes are not considered to combine additively into a single real valued spike and vice-versa. Assuming W is fixed after learning, the only time varying quantity that can be used in calculating Qo (t) is H(t), so we re-parametrize Qo (t) to Qo (H). We then propose that a sensible output of the network to o is Q(o|H), the mean conditional instantaneous spike rate (during training) of supervised output neuron o, given activity H in the network. Integrating Q over a time period gives an expected number of spikes. Thus, in the case of a network operating in discretized time, as is common in the implementation of artificial SNNs in code, Q(o|H) can be interpreted as an expected number of spikes o given H, where the integral over a small discrete timestep is understood. That is, if we observe activity H in the network n times during training, and o spikes no times during timesteps coinciding with those n occurrences of H, then after training if we observe H again, the output that should have been learnt and produced by the network at that timestep is no /n. If only single unit strength spikes are allowed at each timestep, then this output can be interpreted as P (o|H), the probability of o spiking during the given timestep, given H. This interpretation is important in connecting the focus on probability distributions in machine learning with the focus on spike and rate coded networks in computational neuroscience [17]. Of course if H includes the full history of the network’s activity from inception, then only one sample trajectory of H will be observed and used for learning. However, we assume that W approaches zero as the connection time delay becomes large, meaning that only some recent history of activity in the network is used in calculating Q(o|H). Thus, a variety of different H will be observed during training. In any practical network W will be finitely parametrized, so for any particular Hk the parameters modified in learning Q(o|Hk ) will in effect learn for both a range of other H that are slight perturbations of Hk and also use the same parameters, as well as very 5 different H that only share a portion of the same parameters. If we assume the neurons in the network are spiking neurons, then there are only four events that occur within the network at which to apply learning rules that modify Qo (H). They are (i) when an input neuron spikes, (ii) when a hidden neuron spikes, (iii) when a supervised output neuron spikes due to supervision, and (iv) when a supervised output neuron spikes due to its own dynamics or activation function. Modification could also be made continuously at all times, at randomly generated time points, or according to a temporally periodic function; however, we proceed concentrating on the spiking events. Let D be a function that is applied to Qo (H) when H occurs (a combination of events (ii)), and let U be a function that is applied to Qo (H) for each supervised spike o (an event (i)) that co-occurs with H (see Fig. 2). After some period of training time t, we have a series of iterated applications of D and U applied to the initial value Q0o (H) Qto (H) = (D ◦ U ◦ ... ◦ U ) ◦ ... ◦ (D ◦ U ◦ ... ◦ U ) ◦ Q0o (H), (1) where the brackets group operations for each occurrence of H. To find a relation between U and D, suppose now that the initial value 0 Qo (H) = Q(o|H) as we desire. Clearly, we also require that Qto (H) ≈ Q(o|H), meaning that the application of the learning rules D and U do not cause the output to significantly deviate from the desired value. Note that we cannot require strict equality due to the stochastic nature of the event occurrences. If we choose t0 U = DJ(Qo (H)) , (2) where superscripts indicate composition, not exponentiation and J is an unknown function still to be determined, then with the initial value Q0o (H) = Q(o|H), Eq. (1) becomes     t00 t00 t0 t0 Qto (H) = D ◦ DJ(Qo (H)) ◦ ... ◦ DJ(Qo (H)) ◦...◦ D ◦ DJ(Qo (H)) ◦ ... ◦ DJ(Qo (H)) Q(o|H). (3) Let N be the number of occurrences of H. For Qto (H) ≈ Q(o|H), we require that all the applications of D and U approximately cancel, i.e. X 0 (4) N+ J(Qto (H)) ≈ 0. 6 Figure 2: Schematic showing the timing of events in the operation and learning of the network. Neuron spike events are indicated by crosses, learning rule applications are indicated by dots. Vertical dotted lines indicate timesteps of width τ used in the network’s implementation in code (the network can, given suitable hardware, be operated as a dynamical system in continuous time). Note that although variation in the location of spikes within a timestep occurs, this variation is not resolved by a time stepped implementation, additionally single neurons can spike multiple times within a single timestep. This diagram focuses upon learning for a particular hidden neuron activity pattern H1 indicated by red crosses. (i) For output neuron o, U1o is applied each time o spikes in conjunction with H1 , D1o is applied each time H1 is observed. (ii) For input neuron i, each time i spikes in conjunction with the beginning of H1 , U1i is applied at the conclusion of H1 as H1 must be observed in order to identify the connections to modify. D1i is applied each time H1i occurs. (iii) For prediction neuron p, each time p spikes time ∆t after H1 occurs, U1p is applied. D1p is applied each time H1 occurs. 7 The expected number of applications of U is N Q(o|H), and we require that Qto (H) ≈ Q(o|H) at all points in this sequence of applications of D and U , so we have J(Qo (H))N Q(o|H) ≈ −N. (5) However, since we do not know Q(o|H) a priori we use the network’s current estimate Qo (H) instead and set J(Qo (H)) = − 1 . Qo (H) (6) Using Eq. (2) we now have the following relation between the function D that is applied when H occurs, and the function U that is applied when o spikes due to supervision 1 (7) U = D− Qo . This requires that D has a unique inverse, and D−1 can be generalized in such a way as to be applied a fractional number of times. In the above we required that Qto (H) ≈ Q(o|H) at all points in a sequence of applications of D and the U . This implies that any single application of either D or U when Qo (H) ≈ Q(o|H), can only change Qo (H) by a small (but not necessarily fixed) amount Qo − U ≤ U (Qo ) ≤ Qo + U , (8) Qo − D ≤ D(Qo ) ≤ Qo + D , (9) which using (7) leads to the relation D = Qo U . (10) The required range of Qo is [0, ∞). To ensure that U remains small as Qo → 0, we require D be chosen so that in the limQ→0 , QD remains finite. Alternatively it would be possible to insert noise spikes, for example Poisson noise with rate m into the supervision to fix a minimum target value of Qo to m, hence bounding Qo > 0 and eliminating the divergence in Eq. (10). After learning this noise can be stopped and subtracted from the learnt value of Q(o|H). In most cases the maximum value of Q will be finite, and hence D and U can be chosen to give sufficiently small changes. To avoid Qo converging to an unwanted value, this learning scheme must have only a single globally stable fixed point Qo = Q(o|H). This means that 8 U (Q) and D(Q) cannot both have fixed points at any Q. We therefore adjust Eqs. (8) and (9) to Qo − U ≤ U (Qo ) < Qo or Qo < U (Qo ) ≤ Qo + U , (11) Qo < D(Qo ) ≤ Qo + D or Qo − D ≤ D(Qo ) < Qo . (12) and We choose between either the two left, or two right options in (11) and (12) by considering the stability of the fixed point Q(o|H) for each of these choices. Taking equalities in the above equations and using Eq. (10), the total change to Qo after N applications of D and an expected Q(o|H)N applications of U is ∆ ≈ ±N Qo U ∓ Q(o|H)N U . (13) If Qo > Q(o|H) we require ∆ < 0, and if Qo < Q(o|H) we require ∆ > 0. This implies the following choice for our learning rule restrictions Qo − D ≤ D(Qo ) < Qo , (14) Qo < U (Qo ) ≤ Qo + U , (15) that is, D slightly decreases Qo and U slightly increases Qo . 3. Application to Learning Layers of Autoencoders We now outline a demonstration of this learning scheme. A standard method for training an unsupervised deep feedforward network is to train each pair of layers successively as autoencoders [6] so that each layer encodes the activity of the layer below it, see Fig. 3. The learning rules described in Sec. 2 can be used to learn layers of autoencoders by replacing the supervised output neuron o , with an input neuron i that self-supervises, and by reversing the direction of connectivity so that i learns to output Q(i|H), where H is now the future activity of the hidden neurons in the layer above i, since causality is reversed from the previous case; the input layer causes activity in the hidden layer above, see Fig. 2. Using this method, the hidden layers learn so that by observing a period of hidden layer activity, the activity of the layer below at the beginning of 9 Figure 3: (a) Architecture of the feedforward layered network. (b) Illustration of piecewise constant connectivity between two neurons in Eq. (18). (c) Rectified linear unit activation function for hidden neurons used in this network. (d) Illustration of the spiking activity of a hidden neuron, vertical lines indicate the presence of a Dirac delta function, with height corresponding to different normalizations of each individual Dirac delta function. 10 that observation can be inferred. The activity of the hidden layer and the connectivity between layers acts like, and encodes a short term memory. In this demonstration we also include a layer of prediction neurons that predict the activity of the input layer at a specified time period in the future. These neurons are supervised by the activity of the input layer with the corresponding prediction time period delay, see Fig. 2. We also include a layer of digit classification neurons that are trained as for a supervised output neuron o, see Fig. 2. So far we have not needed to specify the dynamics or activation function of the hidden neurons in the network in order to develop this learning scheme. Spiking neuron models in computational neuroscience are often dynamical systems modeled using differential equations [18]. In contrast, neurons in machine learning are typically characterized by an activation function of the neuron’s input [6]. Any of these types of neuron models could be employed here; however, we choose rectified linear units (ReLUs) that are commonly used in deep learning networks [6]. The form we use here is A(I) = I, = 0, I > 0, I ≤ 0, (16) see Fig. 3c. 3.1. Weight Update Rules We have so far developed rules for learning a value Qo to approximate Q(o|H); however, we have not yet discussed rules for modifying W that are necessary for implementation in a network. Before these rules can be determined, the formula for calculating Qo from H and W needs to be chosen. The most common choice is to use the product of H and W summed across all neurons in the layer below and integrated across time in the case of time delay connections. We use this same choice here XZ t hj (t0 )wj (t − t0 )dt0 , (17) Qo (t) = j 0 where hj is a hidden neuron connected to o and wj is the corresponding connection between them. Other choices are possible and may have advantages over this choice, though this is left for future investigation. We also need to choose a parametrization for W . A wide variety of choices could 11 be made here such as sums of continuous functions, or convolution kernels acting across different sets of j, as is done in convolutional neural networks by modifying (17) to include a convolution across j as well as t. However as a first demonstration of this learning scheme we make a simpler choice of using a piecewise constant function (see Fig. 3b) that is easy to conceptualize and produces simple learning rules for the weight parameter updates wj (t) = K X ωk [S(t + (k − 1)τ ) − S(t + kτ )] , (18) k=1 where S is the Heaviside step function, τ defines the width of each of the K pieces of wj , and the ωk are modified by learning. We simplify this notation to use wjk = ωk [S(t + (k − 1)τ ) − S(t + kτ )] , (19) where wjk are effectively the time delayed weights in the network. For time delays greater than τ K, the connectivity weight is zero, meaning that only activity histories H of length τ K are used in calculating Qo . Assuming H is composed of spikes modeled as delta functions, Eq. (17) becomes a sum of weights multiplied by the numbers of spikes X Qo (t) = wjk hjt0 , (20) The following simple and fast weight update rules satisfy Eqs. (14) and (15), though other choices are possible. A weight update rule d that implements D when H occurs is d(w) = w − hQ, (21) and a corresponding weight update rule u that implements U when supervision spikes o occur is u(w) = w + ho, (22) where  is a hyperparameter of the learning rules and should be chosen to be appropriately small. These learning rules cause Qo to fluctuate within a small range of Q(o|H) and it may be useful to change  with time to allow a initial period of fast convergence from the initialization point, and then a reduced fluctuation error once Qo ≈ Q(o|H). Again, these rules are not specific to the ReLUs that we demonstrate with, these neurons could be replaced with sigmoid units, for example, without changing these weight update rules. We use the same weight update rules for learning to predict the activity of the input layer from the activity of the hidden layers, where during learning the prediction neurons are supervised by the future input, see Fig. 2. 12 3.2. DVS MNIST Event Based Dataset To demonstrate this learning scheme we use a dataset collected using a Dynamic Vision Sensor (DVS) [16]. The DVS is a type of video camera that collects event data, unlike conventional video cameras that collect frame data. In the camera an event is triggered by the light intensity impinging upon a pixel changing above a threshold amount. Upon such an event, the camera outputs the pixel coordinates, a timestamp (in µs) and the polarity of the change in intensity. The MNIST database [15] has been used extensively in the development of deep learning [6]. With the view of linking this work to previous work in deep learning, we demonstrate this learning scheme using a DVS version of the MNIST database [14, 15] in which the handwritten digits are displayed and moved on an LCD screen that is being recorded by a DVS camera. In this dataset the light intensity changes collected by the DVS camera are primarily edges of the moving MNIST digits; however, in general the camera also captures other scene changes such as changes in illumination. The resulting dataset is noisy. Viewing the recorded data reveals that the edges are often blurred, and the number of events captured is not uniform across a digit’s edges. The dataset also appears to contain some events that are not related to the movement of the MNIST digit on the LCD screen; however, these events are relatively few in number. The dataset contains recordings of 1000 handwritten digits for each integer from 0 to 9. We use the first 900 entries for training and the last 100 entries for testing. The DVS’s 128 × 128 array of pixels is cropped down to 23 × 23 pixels with each of these pixels mapped onto two input neurons, one for each polarity of light intensity change. The input training sequence was formed from a random selection of the individual MNIST digit sequences each separated by 15 timesteps or 75 ms of no input. Each individual MNIST event sequence has a duration of about 77 timesteps or about 2.3 s. Each pair of neurons have five ωk parameters encoding weights for connection delays kτ of width 30 ms corresponding to the network’s execution timesteps of 30 ms. In this demonstration we predict 15 timesteps or 450ms into the future. An additional ten output neurons are used to classify the current input as a digit from zero to nine. All connection weights ω between layers were initialized to small random values to the range [0, ] where  was initially set to 1 × 10−5 . The connection weights for the prediction and classification neurons were all initialized to zero and used an initial value of  of 2.5 × 10−6 , corresponding to the  value for the between layer connections di13 vided by the number of layers, since the prediction and classification neurons connect to all layers. The connections between hidden layers were trained one layer at a time for one pass through the training dataset, corresponding to 8.8 × 105 timesteps. After each pass through the hidden layers  was decreased by half for all connections and training was repeated, beginning at the first hidden layer. Note that the initial value, decay and decay period for  are not heavily optimized. The prediction and classification weights from all hidden layers were trained at every timestep. To demonstrate that many ideas used in deep learning are directly transferable to a spiking neural network that learns using this scheme, during training we use 50% dropout [10] for each hidden layer. The operation of the trained network is demonstrated in Fig. 4. The hidden layers are very active since in this demonstration the neurons have a threshold fixed at zero. Including a learnable threshold would produce a more sparse representation whilst also reducing the required cpu time as the network’s operation and learning are both dependent on the number of events that occur. The inference of the noisy input is significantly better than the prediction since the inference involves a memory of the input whereas the prediction does not. However, a smoothed version of the future input is usually identifiable in the prediction. The inference and predictions are often poor when the digit changes direction as the edges at these points are weak and the data are particularly noisy. The classification output correctly classifies the input digit 87.41% of the time. The classification error as a function of training time is shown in Fig. 5. Figure 6 shows receptive fields of neurons from the first hidden layer and predictive fields of neurons from all layers. Both positive (excitatory) and negative (inhibitory) weights are learnt. Initially all neurons are active and the small random weight vectors converge toward a time averaged input vector. Upon converging toward the time averaged input, the weight vectors are nearly identical; however, differences due to the small random initialization breaks their symmetry and the weights of different neurons begin to diverge toward other more specific features of the input. This process continues as these features themselves are further split into other even more specific features. After learning is stopped, some of the receptive fields are tuned toward responding to a small number of pixels, while others respond to a distributed pattern of pixels. Predictive fields tend to be composed of larger patches of the sensory field indicating that the encoding of the prediction is distributed across many neurons. Without dropout, denoising autoencoding 14 Figure 4: A demonstration of the feedforward network described in the text applied to the DVS MNIST dataset. (a) The present input to each neuron. Vertical red lines divide layers. Neurons 1 to 1058 are input neurons, thereafter each 1000 neurons form successive hidden layers. The number of presently active neurons in each layer are indicated at the top of this frame. (b) The activity of the DVS input delayed by 5 timesteps (corresponding to the maximum connection delay). (c) The inferred activity of the input Q(i|H) from the recent activity of first hidden layer. (d) The activity of the DVS input 15 timesteps into the future. (e) The network’s prediction Q(p|H) of the activity of the input 15 timesteps into the future. (f) Classification of the present input Q(c|H). (g)-(j) As for (b)-(e) with polarity removed by summing the activity of both polarities. (k) Sum of squared errors normalized by the sum of squares of the data at each timestep for the inferences and predictions in (c) and (e). An additional file is available to view this figure as a movie. 15 1 0.9 0.8 classification error 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 timesteps 1.5 2 7 x 10 Figure 5: Network classification error on the test set vs training time. Connections between layers are trained in a sequence from lowest to highest, vertical dashed lines indicate the end of each pass through the network, and the points at which  is halved. At the end of training the error on the test set is 12.59% 16 Figure 6: (a) Example receptive fields from eight neurons in the first hidden layer. (b)-(e) Example predictive fields for neurons in the input layer and hidden layers one to three respectively. In all frames each pixel is the sum of all temporal connection weights ω for that pixel. All fields have been normalized to have equal maximums. or another regularization method, the connectivity between hidden layers forms an identity mapping, with each neuron connecting only to a single neuron in the previous layer. 4. Summary This paper introduces an event based learning scheme for neural networks. The scheme does not depend on the specific form of the neuronal dynamics or activation function, and while this paper focuses on training spiking neural networks, this scheme may also be used to train traditional artificial neural networks, especially those that involve discontinuous activation functions that defeat gradient descent methods. The scheme may also 17 be applied to networks of neurons containing biologically inspired dynamics. Future work in this direction may inform theories of dynamics and learning in the brain. The broad applicability of this learning scheme provides an avenue to directly apply ideas from both deep learning and computational neuroscience and thus strengthen and inform the theoretical progress in both fields. References [1] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, L. Fei-Fei, Imagenet large scale visual recognition challenge (2014). arXiv:arXiv: 1409.0575. [2] G. Dahl, D. Yu, L. Deng, A. Acero, Context-dependent pre-trained deep neural networks for large-vocabulary speech recognition, Audio, Speech, and Language Processing, IEEE Transactions on 20 (1) (2012) 30–42. [3] L. Deng, G. Hinton, B. Kingsbury, New types of deep neural network learning for speech recognition and related applications: an overview, in: Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, 2013, pp. 8599–8603. [4] G. Hinton, L. Deng, D. Yu, G. Dahl, A. Mohamed, N. Jaitly, A. Senior, V. 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Salakhutdinov, Dropout: A simple way to prevent neural networks from overfitting, J. Mach. Learn. Res. 15 (1) (2014) 1929–1958. [11] P. Vincent, H. Larochelle, I. Lajoie, Y. Bengio, P.-A. Manzagol, Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion, J. Mach. Learn. Res. 11 (2010) 3371–3408. [12] D. Rumelhart, G. Hinton, R. Williams, Learning representations by back-propagating errors, Nature 323 (6088) (1986) 533–536. [13] G. E. Hinton, S. Osindero, Y.-W. Teh, A fast learning algorithm for deep belief nets, Neural Comput. 18 (7) (2006) 1527–1554. [14] T. Serrano-Gotarredona, B. Linares-Barranco, MNIST-DVS database, accessed: 27th Jan. 2015. URL http://www2.imse-cnm.csic.es/caviar/MNISTDVS.html [15] Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to document recognition, Proceedings of the IEEE 86 (11) (1998) 2278–2324. [16] T. Serrano-Gotarredona, B. 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arXiv:1711.02837v1 [stat.ML] 8 Nov 2017 Revealing structure components of the retina by deep learning networks Qi Yan, Zhaofei Yu, Feng Chen Center for Brain-Inspired Computing Research, Department of Automation, Tsinghua University {q-yan15,yuzf12}@mails.tsinghua.edu.cn chenfeng@mail.tsinghua.edu.cn Jian K. Liu Institute for Theoretical Computer Science, Graz University of Technology liu@igi.tugraz.at Abstract Deep convolutional neural networks (CNNs) have demonstrated impressive performance on visual object classification tasks. In addition, it is a useful model for predication of neuronal responses recorded in visual system. However, there is still no clear understanding of what CNNs learn in terms of visual neuronal circuits. Visualizing CNN’s features to obtain possible connections to neuronscience underpinnings is not easy due to highly complex circuits from the retina to higher visual cortex. Here we address this issue by focusing on single retinal ganglion cells with a simple model and electrophysiological recordings from salamanders. By training CNNs with white noise images to predicate neural responses, we found that convolutional filters learned in the end are resembling to biological components of the retinal circuit. Features represented by these filters tile the space of conventional receptive field of retinal ganglion cells. These results suggest that CNN could be used to reveal structure components of neuronal circuits. 1 Introduction Deep convolutional neural networks (CNNs) have been a powerful model for numerous tasks related to system identification [1]. By training a CNN with a large set of targeted images, it can achieve the human-level performance for visual object recognition. However it is still a challenge for understanding the relationship between computation and the underlying structure components learned within CNNs [2, 3]. Thus, visualizing and understanding CNNs are not trivial [4]. Inspired by neuroscience studies, a typical CNN model consists of a hierarchical structure of layers [5], where one of the most important properties for each convolutional (conv) layer is that one can use a conv filter as a feature detector to extract useful information from inputed images after the previous layer [6, 7]. Therefore, after learning, conv filters are meaningful. The features captured by these filters can be represented in the original natural images [4]. Often, one typical feature shares some similarities with part of natural images from the training set. These similarities are obtained by using a very large set of specific images. The benefit of this is that features are relative universal for one category of objects, which is good for recognition. However, it also causes the difficulty of visualization or interpretation due to the complex nature of natural images, i.e., the complex statistical structures of natural images [8]. As a result, the filters and features learned in CNNs are often not obvious to be interpreted [9]. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. On the other hand, researchers begin to adapt CNNs for studying the target questions from neuroscience. For example, CNNs have been used to model the ventral visual pathway that has been suggested as a route for visual object recognition starting from the retina to visual cortex and reaching inferior temporal (IT) cortex [10–12]. The prediction of neuronal responses recorded in monkey in this case has a surprisingly good performance. However, the final output of this CNN model is representing dense computations conducted in many previous conv layers, which may or may not be related to the neuronscience underpinnings of information processing in the brain. Understanding these network components of CNN are difficult given the IT cortex part is sitting at a high level of our visual system with abstract information, if any, encoded [13]. In principle, CNN models can also be applied to early sensory systems where the organization of underlying neuronal circuitry is relatively more clear and simple. Thus one expect knowledge of these neuronal circuitry could provide useful and important validation for such models. For instance, a recent study employs CNNs to predict neural responses of the retinal ganglion cells to white noise and natural images [14]. Here we move a step further in this direction by relating CNNs with single RGCs. Specifically, we used CNNs to learn to predict the responses of single RGCs to white noise images. In contrast to the study by [14] where one single CNN model was used to model a population of RGCs, in the current study, our main focus is based on single RGCs to revealing the network structure components learned by CNNs. Our aim is to study what kind of possible biological structure components in the retina can be learned by CNNs. This concerns the research focus of understanding, visualizing and interpreting the CNN components out of its black box. To the end, by using a minimal model of RGC, we found the conv filters learned in CNN are essentially the subunit components of RGC model. The features represented by these filters are fallen into the receptive field of modeled RGC. Furthermore, we applied CNNs to analyze biological RGC data recorded in salamander. Surprisingly, the conv filters are resembling to the receptive fields of bipolar cells that sit in the previous layer of RGC and pool their computations to a downstream single RGC. 2 Methods 2.1 RGC model and data A simulated RGC is modeled in Fig. 1 as previously [15]. The model cell has five subunits that each filter, similar to a conv filter in DNN, convolves the incoming stimulus image and then applies an nonlinearity of threshold-quadratic rectification. The subunit signals are then polled together by the RGC. The polled signal is applied with a threshold-linear output nonlinearity with a positive threshold at unity to make spiking sparse. The biological data of RGCs were recorded in salamander as described in [15]. Briefly RGC spiking activity were obtained by multielectrode array recordings as in [16]. The retinas was optically stimulated with spatiotemporal white noise images, temporally updated at a rate of 30 Hz and spatially arranged in a checkerboard layout with stimulus pixels of 30x30 mµ. 2.2 CNN model We adopt a CNN model containing two convolution layers and a dense layer as in [14]. Several sets of parameters in convolution layers, including the number of layers, the number and size of convolution filters were explored. The predication performance is robust against these changes of parameters. Therefore we used a filter size as of 15 × 15 to compare our results with those in [14]. The major difference between our model with that in [14] is that our CNN is for studying of single RGCs. For the RGC model, we used a data set consisting of 600k training samples of white noise images, and additional set of samples for testing. The training labels are a train of binary spikes with 0 and 1 generated by the model. For the biological RGCs recorded in salamander, we used the same data sets as in [15]. Briefly there are about 40k training samples and labels with the number of spikes as in [0 5] for each image. The test data have 300 samples, which are repeatedly presented to the retina for about 200 trials. The average firing rate of this test data is compared to the CNN output for performance calculation. 2 A B Subunit model CNN model convolution convolution dense labels Bernoulli process # Spikes Images Output nonlinearity Subunits C Subunit model RF Spikes Subunit nonlinearity D Conv filters CNN model Features Figure 1: The CNN filters are resembling to the subunits in RGC model. (A) An illustration of RGC model structure. Note there are five subunits playing the role of conv filters. (B) An illustration of CNN model that trains the same set of images to predicate the labels, here spikes, of all images. (C) Receptive fields (RFs) of modeled RGC and predication by CNN model. (D) Visualizing of CNN model components of both conv filters and average features represented by each filter. 3 Results Here we focus on single RGC that has the benefit to clarify the network structure components of CNNs. Recently, a variation of non-negative matrix factorization was used to analyze the RGC’ responses to white noise images and identify a number of subunits, resembling to the biopolar cells, within the receptive field of each RGC [15]. With this picture in mind, here we address the question that what types of network structure components can be revealed by CNNs when they are used to model the single RGC response. A previous study [14] focused on predicating neural response of RGC at the population level with one CNN model, and claimed that the features represented by conv filters are resembling to the receptive fields of bipolar cells (BCs). However, a careful examination reveals that this connection between CNN feature map and BCs is weak since the number of conv filters in the CNN is much less than that of BCs in the RGC population from the retina. By using a CNN model, one expect to reveal a more clear picture of this connection between CNNs and the retina. Here, we set up a single RGC model with conv subunits as in Fig. 1(A), which is resembling to a 2-layer CNN with one conv layer of subunits and one dense layer of single RGC. By training a CNN as in Fig. 1(B) with a set of white noise images to predicate the target labels as the simulated spikes generated by this RGC model, we found that the CNN model can predicate the RGC model response well with Pearson correlation coefficient (CC) up to 0.70 similar to the study by [14]. Interestingly, we also found the CNN model can predicate the receptive field (RF) well as in Fig. 1(C). Furtherer more, the conv filters learned by CNN are the exact subunits employed in the RGC model as shown in Fig. 1(C). A subset of the conv filters, that can be termed as effective filters, start from random shapes and converge to the exact subunits. Although we set up the filter size as 15x15 pixels, the resulting effective filters are sparse represented with a 6x6 pixel size. The rest of the filters are still random and close to zero. Therefore, these results show that CNN parameters are highly redundant. Such a redundancy of parameters, including conv filters, units/neurons and connections of conv and dense layers, is widely observed for deep learning models [17–19]. All together, These results suggest that the CNN model can identify the underlying hidden network structure components within the RGC model by only looking at the input stimulus images and the output response in terms of the number of spikes. 3 A BConv filters Data RF 100 m Features CNN 100 m 100 m C Data Firing rate (Hz) CNN 40 20 0 0 2 4 6 8 10 Time (sec.) Figure 2: The CNN reveals subunit structures in biological RGC data. (A) Receptive fields of the sample cell and CNN predication. (B) Visualizing of CNN model components of both conv filters and average features represented by each filter. (C) Neural response predicated well by CNN model visualized by RGC data spike rasters (upper) and CNN spike rasters (middle) and their average firing rates. To further characterizing these structure components in details, we use a CNN to learn the biological RGC data with the similar images of white noise and the spiking responses. Similar to the results of the RGC model above, the outputs of CNN model can recover the receptive field of data very well as in Fig. 2(A). We also found that the learned conv filters converge to a set of localized subunits whereas the rest of filters are noisy and close to zero as in Fig. 2(B). The size of these localized filters is comparable to that in bipolar cells around 100 mµ [15]. In addition, the features represented by these localized conv filers are also localized. Given the example RGC is a OFF type cell that response to the dark part of images strongly, most features have similar OFF peaks resulted from the OFF BC-like filters. These OFF features tile the space of receptive field of RGC. Interestingly, there are some features with ON peaks, which play a role as inhibition in the retinal circuit. A few features have some complex structures mixed with OFF and ON peaks, which are mostly resulted from the less localized filters. However, if the filters are pure noise, the resulting features are pure noise without any structure embedded. Besides filters and features, the CNN model generates a good predication of RGC response as in Fig. 2(C) with the CC up to 0.75. These observations are similar across different RGCs recorded. 4 Discussion Here by focusing on single RGCs, we shown that CNN can learn their parameters in an interpretable fashion. Both filters and features are close to the biological underpinnings within the retinal circuit. With the benefits of relative well-understood neuronal circuit of the retina ganglion cells, our preliminary results give a strong evidence that the building-blocks of CNNs are meaningful when they are applied to neuroscience for revealing network structure components. Our results extend the previous studies [11, 14] that focus on predication of neural responses. Furtherer more, our approach is suitable to address other difficult issues of deep learning, such as transfer learning, since the domain 4 of images seen by single RGCs is local and less complicated than those global structures of entire natural images. 5 Acknowledgements Q.Y., Z.Y. and F.C. were supported in part by the National Natural Science Foundation of China under Grant 61671266, 61327902, in part by the Research Project of Tsinghua University under Grant 20161080084, in part by National High-tech Research and Development Plan under Grant 2015AA042306. J.K.L. was partially supported by the Human Brain Project of the European Union #604102 and #720270. References [1] Yann Lecun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436–444, 2015. [2] Leslie N. Smith and Nicholay Topin. arXiv:1611.00847v3, 2016. Deep convolutional neural network design patterns. [3] Adam H. Marblestone, Greg Wayne, and Konrad P. Kording. Toward an integration of deep learning and neuroscience. Frontiers in Computational Neuroscience, 10, sep 2016. [4] Matthew D. Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European Conference on Computer Vision, pages 818–833, 2014. [5] Y Lecun, K Kavukcuoglu, and C Farabet. Convolutional networks and applications in vision. In IEEE International Symposium on Circuits and Systems, pages 253–256, 2010. [6] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. Computer Science, 2014. [7] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. In International Conference on Neural Information Processing Systems, pages 1097– 1105, 2012. [8] E. P. Simoncelli and B. A. Olshausen. Natural image statistics and neural representation. Annual Review of Neuroscience, 24(24):1193, 2001. [9] Matthew D. Zeiler, Graham W. Taylor, and Rob Fergus. Adaptive deconvolutional networks for mid and high level feature learning. In International Conference on Computer Vision, pages 2018–2025, 2011. [10] Daniel Yamins, Ha Hong, Charles Cadieu, and James J. Dicarlo. Hierarchical modular optimization of convolutional networks achieves representations similar to macaque it and human ventral stream. Advances in Neural Information Processing Systems, pages 3093–3101, 2013. [11] D. L. K. Yamins, H. Hong, C. F. Cadieu, E. A. Solomon, D. Seibert, and J. J. DiCarlo. Performanceoptimized hierarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619–8624, may 2014. [12] Charles F. Cadieu, Ha Hong, Daniel L. K. Yamins, Nicolas Pinto, Diego Ardila, Ethan A. Solomon, Najib J. Majaj, and James J. Dicarlo. Deep neural networks rival the representation of primate it cortex for core visual object recognition. Plos Computational Biology, 10(12):e1003963, 2014. [13] Daniel L K Yamins and James J DiCarlo. Using goal-driven deep learning models to understand sensory cortex. Nature Neuroscience, 19(3):356–365, feb 2016. [14] Lane McIntosh, Niru Maheswaranathan, Aran Nayebi, Surya Ganguli, and Stephen Baccus. Deep learning models of the retinal response to natural scenes. In Advances in Neural Information Processing Systems 29. 2016. [15] Jian K. Liu, Helene M. Schreyer, Arno Onken, Fernando Rozenblit, Mohammad H. Khani, Vidhyasankar Krishnamoorthy, Stefano Panzeri, and Tim Gollisch. Inference of neuronal functional circuitry with spiketriggered non-negative matrix factorization. Nature Communications, 8(1), jul 2017. [16] Jian K. Liu and Tim Gollisch. Spike-triggered covariance analysis reveals phenomenological diversity of contrast adaptation in the retina. PLOS Computational Biology, 11(7):e1004425, jul 2015. 5 [17] Misha Denil, Babak Shakibi, Laurent Dinh, MarcAurelio Ranzato, and Nando de Freitas. Predicting parameters in deep learning. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 26, pages 2148–2156. Curran Associates, Inc., 2013. [18] Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. Fiber, 56(4):3–7, 2015. [19] Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems 28, pages 1135–1143. Curran Associates, Inc., 2015. 6 

Addressing Expensive Multi-objective Games with Postponed Preference Articulation via Memetic Co-evolution Adam Żychowskia,∗, Abhishek Guptab , Jacek Mańdziuka , Yew Soon Ongb arXiv:1711.06763v1 [] 17 Nov 2017 a Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland b School of Computer Science and Engineering, Nanyang Technological University, Block N4, Nanyang Avenue, Singapore 639798 Abstract This paper presents algorithmic and empirical contributions demonstrating that the convergence characteristics of a co-evolutionary approach to tackle Multi-Objective Games (MOGs) with postponed preference articulation can often be hampered due to the possible emergence of the so-called Red Queen effect. Accordingly, it is hypothesized that the convergence characteristics can be significantly improved through the incorporation of memetics (local solution refinements as a form of lifelong learning), as a promising means of mitigating (or at least suppressing) the Red Queen phenomenon by providing a guiding hand to the purely genetic mechanisms of co-evolution. Our practical motivation is to address MOGs of a time-sensitive nature that are characterized by computationally expensive evaluations, wherein there is a natural need to reduce the total number of true function evaluations consumed in achieving good quality solutions. To this end, we propose novel enhancements to co-evolutionary approaches for tackling MOGs, such that memetic local refinements can be efficiently applied on evolved candidate strategies by searching on computationally cheap surrogate payoff landscapes (that preserve postponed preference conditions). The efficacy of the proposal is demonstrated on a suite of test MOGs that have been designed. Keywords: multi-objective games, Red Queen effect, surrogate-assisted memetic algorithm 1. Introduction Many practical problems can be modeled and resolved with game theory methods. In real world problems decisions are usually made with multiple objectives or lists of payoffs. The notion of vector payoffs for games was originally introduced by Blackwell [1] and later extended by Contini [2]. Such games are named Multi-Objective Games (MOGs). MOGs may have many practical applications in engineering, economics, cybersecurity [3] or Security ∗ Corresponding author Email address: a.zychowski@mini.pw.edu.pl (Adam Żychowski) Preprint submitted to Elsevier November 21, 2017 Games [4], where real-life situations can be easily modeled as a game, and their solutions help decision-makers make the right choice in multi-objective environments. Most existing studies about MOGs concentrate on differential games and defining the equilibrium for them. The most common approach is the Pareto-Nash equilibrium proposed in [5]. The Pareto-Nash equilibrium uses the concept of cooperative games, because subplayers under the same coalitions should, according to the Pareto notion, optimize their vector functions on a set of strategies. This notion also takes into account the concept of non-cooperative games, because coalitions are interested in preserving the Nash equilibrium between coalitions. First attempts to solving MOGs were using multi-parametric criteria linear programming, for example in [6] (for zero-sum games) and [2] (for non-zero-sum variants). Also artificial intelligence-based approaches, such as fuzzy functions, have been applied to MOGs. For example, in [7], the objectives are aggregated to one artificial objective for which the weights (players’ preferences towards the objectives) are modeled with fuzzy functions. Generally speaking, the most popular method of solving MOGs is to specify players objective preferences and define the utility function, for example a weighted sum, to transform the MOG into a surrogate Single Objective Game (SOG) [8, 6]. However, in real-life applications such an approach may not be sufficient, because preferences are often postponed until tradeoffs are revealed. Furthermore, in many cases decision-makers objectives are conflicting which makes specifying the utility function a priori more difficult. There is a lack of literature on the topic of MOGs, especially which involves players with postponed preference articulation. Thus, there is a significant gap in the availability of algorithms for tackling real-world MOGs. First of all, an efficient numerical scheme is needed that is able to present decision-makers with the optimal tradeoffs in competitive game settings comprising multiple conflicting objectives (somewhat similarly to the case of standard multi-objective optimization [9, 10, 11]). Only then can an informed postponed choice be made with regard to ascertaining the most preferred strategy to implement. The present paper takes a step towards filling this algorithmic void. First formalization of such MOGs can be found in [12]. In [13], the definition of rationalizable strategies in such games is provided together with a suggestion about how these strategies could be found (preliminary discussions are provided in Section 2 of this paper), analyzed, and used for choosing the preferred strategy. Co-evolutionary adaptation is a viable method for solving game theory problems and is successfully used in traditional SOGs [14, 15]. Preliminary works in [16] showed that in principle a co-evolutionary algorithm may be applied to MOGs as well, with the populationbased evolutionary algorithms being particularly well-suited for handling multiple objectives simultaneously (as a consequence of the implicit parallelism of population-based search). However, a canonical co-evolutionary approach to solving MOGs has certain drawbacks. One of them is the emergence of a phenomenon named the Red Queen effect which often hampers the convergence characteristics of the algorithm. In many time-sensitive applications involving computationally expensive evaluations, such a slowdown must be avoided. The Red Queen principle was first proposed by the evolutionary biologist L. van Valen in 1973 [17]. It states that populations must continuously adapt to survive against ever-evolving 2 competitors. It is based on a biologically grounded hypothesis that species continually need to change to keep up with the competitors (because when species stop changing, they will lose to the other species that do continue to change). The Red Queen effect could have positive as well as negative consequences. For example, co-evolution between predators and preys where the only way the predator can compensate for a better defense by the prey (e.g. rabbits running faster) is by developing a better offense (e.g. foxes running faster). This leads to improvement of the skills (running faster) of both species. In another example, consider trees in a forest which compete for access to sunlight. If one tree grows taller than its neighbours it can capture part of their sunlight. This causes the other trees to grow taller, in order not to be overshadowed. The effect is that all trees tend to become taller and taller, but still getting on average the same amount of sunlight. Optimizing access to sunlight for each individual tree does not lead to optimal performance for the forest as a whole [18]. Notably, such continuous adaptation as a result of the Red Queen effect occurs not only in species co-evolution, but also in disease mutations, business competitors or macroeconomics changes. In a co-evolutionary algorithm for solving MOGs, the emergence of the Red Queen effect may hamper the convergence characteristics of the algorithm, since many function evaluations are needed to overcome the continuous adaptations and drive the population to a more-or-less steady state of reasonably good solutions. It must be observed, however, that the presence of the Red Queen effect in a given co-evolutionary approach is - in general only hypothetical as tracking the Red Queen phenomenon is usually a complex and non obvious process [19]. Regardless of the detailed reasons, the decelerated convergence is often only an artefact of the co-evolutionary method, and may not be at all related to the underlying MOGs. For the cases where MOGs are time-sensitive and/or involve computationally expensive evaluations, the Red Queen effect is unaffordable from an algorithmic standpoint. Thus, in this paper, an approach to mitigating (or at least suppressing) the deleterious consequence of the Red Queen effect is proposed. It is achieved by applying a local solution refinement technique (alternatively known as lifelong learning of an individual in a population) - in the spirit of memetic algorithms. Canonical memetic algorithms [20, 21, 22] enhance population-based Evolutionary Algorithms (EA) by means of adding a distinctive local optimization phase. The underlying idea of memetics is to use local optimization techniques or domain knowledge to improve potential solutions (represented by individuals in a population) between consecutive EA generations. Drawing from their sociological interpretation, memes are seen as basic units of knowledge that are passed down from parents to offspring in the form of ideas (procedures) that serve as guidelines for superior performance. Thus, while genes leap from body to body as they propagate in the gene pool, memes are thought of as leaping from brain to brain as they propagate in the meme pool. A synergetic combination of simple local improvement schemes with evolutionary operators leads to complex and powerful solving methods which are applicable to a wide range of problems [23], and currently serve as one of the fastest growing subfields of Evolutionary Computation research. Local improvement of temporary solutions represented by genes is deemed to be of paramount importance in the context of the presumed existence of the Red Queen effect as it should strongly mit3 igate the rolling-horizon of the individual fitness evaluation. The main rationale for such a claim is that the lifelong learning module introduced by memetics can provide a guiding hand to the purely genetic mechanisms of co-evolutionary algorithms, thereby potentially suppressing the intensity of the Red Queen effect in our search for equilibrium solutions to the underlying MOG. Due to the complexity of rigorous theoretical analysis, we attempt to substantiate our claims experimentally in this paper. To summarize, the main contribution of this paper is a novel enhancement of co-evolutionary algorithms for MOGs. In particular, memetic local refinements are proposed on evolved candidate strategies, as a means of improving convergence behavior. Importantly, in order to make such local refinements computationally viable in competitive multi-objective game settings, we incorporate an approach that reduces sets of payoff vectors in objective space to a single representative point that preserves the postponed preference articulation condition (details are provided in Section 3). Thereafter, a surrogate model of the representative point can be built, which allows the local improvements to be carried out efficiently by searching on the surrogate landscape [24, 25]. As a result of the proposal, it is considered that MOGs of a computationally expensive nature can be effectively handled, at negligible computational overhead involved in surrogate modeling. The remainder of this paper is arranged as follows. Section 2 presents the general formulation and fundamentals of MOGs. An overview of the baseline co-evolutionary approach for solving MOGs based on [16] is presented in Section 3. Section 4 provides a more detailed description of the proposed surrogate-assisted memetic algorithm tailored for computationally expensive MOGs. In Section 5 experimental studies for both algorithms are carried out on a suite of test MOGs of varying degrees of complexity that have been designed based on an intuitively visualizable differential game. Results are discussed in the context of convergence and suppression of the Red Queen effect. The last section is devoted to conclusions and directions for future research. 2. Preliminaries on MOGs 2.1. Problem definition Single-act multi-objective games considered in this paper can be formally described as follows. Let P1 and P2 be the players competing against each other, and S1 = {s11 , s21 , . . . , sI1 }, S2 = {s12 , s22 , . . . , sJ2 } be complete sets of their possible strategies, respectively. Each cani(1) i(2) i(N ) didate strategy is a vector of decision parameters: si1 = [s1 , s1 , . . . , s1 1 ] ∈ RN1 , sj2 = j(1) j(2) j(N ) [s2 , s2 , . . . , s2 2 ] ∈ RN2 , where N1 and N2 are the numbers of players’ decision parameters. f¯i,j represents the payoff vector corresponding to a game evaluated using strategies si1 and sj2 . Without loss of generality, we assume that the goal of P1 is to minimize the payoff, while the goal of P2 is to maximize the payoff. Since both players do not know how the opponent evaluates their objectives in a postponed preference articulation setting, each player takes a worst-case approach. Thus, players must somehow take into account all possible moves available to the opponent. From P1 ’s perspective, the goal may be modeled as minimizing the objective function vector assuming the best possible opponent strategies: minsi1 ∈S1 maxsj ∈S2 f¯i,j . In contrast, player P2 2 4 aims at maximizing the objectives while considering the best strategies for the minimizer: maxsj ∈S2 minsi1 ∈S1 f¯i,j . Note that, the above max and min operators applied to vector-valued 2 payoffs are ill-defined. As one possible alternative, their meaning can be formalized by means of domination relations between payoff vectors, as described in subsection 2.3 below. 2.2. Solution approach Contrary to SOGs, in games with multiple objectives, a universally optimal strategy usually does not exist. For this reason, the notion of Pareto optimality is used based on domination relations between individual vectors as well as sets of vectors. Most importantly, such deductions can be found without the need to specify objective preferences, which aligns well with our basic premise of MOGs with postponed preference articulation. Definitions presented in the reminder of this section follow the discussions in [13]. Herein, we only provide a brief overview of the main ideas for the sake of brevity. 2.3. Domination relations To resolve domination relation between sets of vectors, first the domination relation between individual vectors must be defined. Definition 1. Domination relation between vectors Let f = [f1 , f2 , . . . , fK ] ∈ RK and h = [h1 , h2 , . . . , hK ] ∈ RK be two vectors in the objective max space. A vector f dominates vector h in a maximization problem (f  h), if fk ≥ hk for all k ∈ {1, . . . , K} and there exists k ∈ {1, . . . , K} for which fk > hk . With this, the domination relation between sets is defined as follows. Definition 2. Domination relation between sets Let F and H be sets of vectors from the objective space. Set F dominates set H in a max max maximization problem, F  H, if ∀h ∈ H, ∃f ∈ F , such that f  h. min Analogous definitions to the above are used for domination relations (  ) in minimization problems. In the context of MOGs, the notion of ’worst case domination’ emerges in addition to the simple domination relation between sets, because, while assessing the payoff of a particular strategy for P1 (or P2 ) in a competing game, the set of optimal strategies for the opponent must be taken into account. Accordingly, we define the worst-case domination relation. Definition 3. Worst-case domination w.c. min Set F worst-case dominates set H (F  H) in a maximization problem when H  F , max and set F worst-case dominates set H in a minimization problem when H  F . 5 2.4. Rationalizable strategies Given the worst-case domination relation, Pareto-optimality in a MOG is defined as follows. Definition 4. Pareto-optimality in MOGs A set Z ∗ is Pareto-optimal in a MOG if no other set exists that worst-case dominates ∗ Z . The set of all worst case non-dominated sets constitutes the Pareto set of sets P ∗ (also referred to as the Pareto Layer [26]): w.c. P ∗ = {F : ¬∃Hs.t.H  F } To elaborate from the point of view of the players in the MOG, when evaluating the i-th strategy of player P1 (si1 ), there are J possible strategies [s12 , s22 , . . . sJ2 ] of P2 to consider. If the objective preferences of P2 are not defined, then there is a set of non-dominated payoff vectors, which are in fact all possible best responses of P2 to the strategy si1 . This set is named the anti-optimal front (Fs−∗ i ) corresponding to the i-th strategy of P1 . 1 Definition 5. Irrational strategies A set of irrational strategies (S1irr ) of player P1 is defined as follows: w.c. 0  Fs−∗ ∀i ∈ {1, . . . , I}} S1irr = {si1 ∈ S1 | ∃si1 ∈ S1 Fs−∗ i i0 1 1 All strategies which are not irrational are called rationalizable. Definition 6. Rationalizable strategies A set of rationalizable strategies of player P1 is defined as S1R = S1 − S1irr . A detailed discussion with examples concerning domination relations and preferable outcomes can be found in the Appendix of [13]. The main conclusion stemming from that discussion is that, for a particular player in a MOG, if the anti-optimal front corresponding to strategy si worst-case dominates the anti-optimal front corresponding to sj , then si always produces a preferable outcome for that player. On the other hand, when two anti-optimal fronts are worst-case non-dominated, one of the strategies could be a better choice for a certain objective preference articulation, while the other strategy may be better under some other preferences. Therefore, in the latter case, their direct comparison is not possible. In this paper, the described worst-case domination approach is used as the basic tool to solve MOGs. All considered MOGs are assumed to have postponed preferences, and candidate strategies are accordingly analyzed from both players’ perspectives in the proposed co-evolutionary algorithms (described next). 6 Figure 1: Example of payoff vectors in two-dimensional objective space. The white circles representing the anti-optimal front corresponding to si1 worst-case dominate the white squares representing the anti-optimal front corresponding to sj1 in minimization sense. Black figures represent the opponent’s ideal points for the respective sets. 3. The Canonical Co-evolutionary Algorithm for MOGs Our implementation of the canonical co-evolutionary algorithm for MOGs (Canonical CoEvoMOG) is designed based on [16]. Each subpopulation in the algorithm caters to a unique player in the MOG. The key difference between our implementation and that of [16] is that, instead of considering the entire anti-optimal set of vectors while determining worstcase domination relations, we only consider the ideal point of the anti-optimal set as a single representative vector. Note that the term ”ideal” is used from the point of view of the opponent. Thus, without loss of generality, if the anti-optimal front corresponding to strategy si1 of the minimization player P1 is Fs−∗ i , then the ideal point is defined by 1 the maximum (extreme) individual objective values that occur in Fs−∗ i . An illustration is 1 depicted in Figure 1, where the set of white circles (representing the anti-optimal front corresponding to si1 ) worst-case dominate the set of white squares (representing the antioptimal front corresponding to sj1 ) in minimization sense. The maximizing opponent’s ideal points, given si1 or sj1 , are also shown in the figure in black. From Figure 1, we observe that while ascertaining the expected payoff of a particular strategy, the ideal point of the antioptimal front can be seen as a meaningful representative encompassing all possible moves of the opponent. It is worth mentioning that as the ideal point is composed of the extreme values of all objectives, no specific objective preference is assumed for the opponent. In that sense, the proposed simplification can be seen as preserving the postponed preference articulation condition of the MOG. We emphasize the rationale behind using only a single representative vector (instead of the entire anti-optimal set) following the observation presented in [27], which can also be stated through the theorem below. Theorem 1. If strategy si worst-case dominates sj , then the ideal point of the anti-optimal front of the former either dominates or is at least equal to the ideal point of the anti-optimal front of the latter strategy. 7 Proof. It follows from the definition of worst-case domination presented in the previous section. For brevity, we consider the strategies si and sj of the minimization player. Similar arguments can be applied to the maximization player as well. Thus, the antecedent statement of the theorem implies that the anti-optimal front of si is maximization dominated by the anti-optimal front of sj . Accordingly, there exist vector(s) in the anti-optimal set of sj that maximization dominate the extreme vectors of the anti-optimal front of si . As a result, if we assume that the opponent’s ideal point, given si , maximization dominates the opponent’s ideal point given sj , then we have a contradiction. In other words, the opponent’s ideal point, given si , must minimization dominate or be equal to the opponent’s ideal point given sj . From an algorithmic point of view, the first advantage of using the representative ideal point vector corresponding to a particular strategy is that it allows us to directly employ standard non-domination relations between vectors (as in Definition 1), bypassing the cumbersome process of comparing sets of vectors to determine worst-case domination relations. In other words, from minimization player P1 ’s standpoint, strategy si1 is preferred over sj1 simply if the ideal point of the anti-optimal front of si1 dominates that of sj1 in the minimization sense. Furthermore, the reduction of a set of vectors to a single point implies that simple diversity measures (such as the crowding distance [28]) may also be directly incorporated to facilitate a good distribution of alternative payoff vectors in the objective space of the MOG. Based on these basic ingredients, Figure 2 outlines the schematic workflow of the Canonical CoEvoMOG algorithm. In the Canonical CoEvoMOG algorithm, there are two subpopulations catering to the two players in the MOG. The method proceeds as in standard co-evolution for SOGs, where interactions are considered between all candidate strategies in the two subpopulations (forming a complete bipartite evaluation framework). The outcomes of the interactions are used to ”approximate” the ideal point of the anti-optimal front corresponding to every candidate strategy of both players. The approximated ideal point vectors are then used to calculate non-domination ranks and crowding distances of strategies within each subpopulation separately, similarly to the case of evolutionary multi-objective optimization [28]. The nondomination ranks and the crowding distances are considered lexicographically for selecting the most promising candidate strategies in each subpopulation that progress the search to the next generation through genetic operations of crossover and mutation. 4. The Memetic Co-evolutionary Algorithm for MOGs One of the drawbacks faced by the Canonical CoEvoMOG algorithm is the possible emergence of the Red Queen effect. This suggests that the convergence to the desired equilibrium strategies is impeded due to the continuously adapting subpopulations that endlessly try to keep pace with the changes in the opponent’s strategies. Notably, the slowdown is unlikely to be related to the underlying MOG, but is often an artefact of the co-evolutionary method itself. It is regarded that in MOG applications of a time-sensitive nature that may even involve computationally expensive evaluations, such a slowdown is not 8 Initialize randomly population S1 for player P1 and population S2 for player P2 . for all generations do Step 1: Create offspring populations S10 and S20 via crossover and mutation of parent individuals from S1 and S2 , respectively. Step 2: Set S100 as S1 ∪ S10 , and set S200 as S2 ∪ S20 . Step 3: Evaluate populations S100 and S200 by performing all interactions between candidate strategies in S100 and S200 (keep track of evaluations to prevent repetitions). Step 4: Obtain the ”approximate” ideal point corresponding to each candidate strategy in S100 and S200 based on the outcomes of all possible interactions. Step 5: Obtain non-domination rank and crowding distance of each strategy in S100 and S200 based on the approximated ideal point vectors. Step 6: Consider the non-domination ranks and crowding distances lexicographically to select the most promising candidate strategies from S100 and S200 to form S1 and S2 in the next generation. end for Figure 2: Pseudo-code of the Canonical Co-evolutionary MOG Algorithm. affordable. Therefore, in this section, we propose memetic local strategy refinements as a way of enhancing the purely genetic mechanisms of the Canonical CoEvoMOG algorithm, thereby potentially speeding up the convergence characteristics. Further, in order to maintain the computational feasibility of the method, a surrogate modeling of the representative payoff vector is proposed, which allows the local refinements to be carried out efficiently on the surrogate landscape. Our proposal is labeled as a Memetic CoEvoMOG algorithm, and involves a simple but important modification to the pseudo-code in Figure 2. Details of the modified procedure are presented in Figure 3. 4.1. Overview of Surrogate Modeling in MOGs A surrogate model is essentially a computationally cheap approximation of the underlying (expensive) function to be evaluated. By searching on the surrogate landscape instead of the original function, significant savings in computational effort can be achieved [29, 30]. However, before building the surrogate model, the function(s) to be approximated must first be ascertained. For a MOG, this task is in general unclear, as corresponding to a particular strategy, a set of optimal opponent strategies usually exist that constitute the anti-optimal front. It is at this juncture that the second, and perhaps most relevant, implication of using the representative ideal point vector (instead of the entire anti-optimal front) is revealed. Without the proposed modification, it is difficult to imagine an approach for incorporating memetics into the canonical co-evolutionary algorithm for MOGs. To elaborate, while creating a surrogate of an entire set of vectors is indeed prohibitive, surrogate models that map an individual strategy to its corresponding ideal point vector (of the anti-optimal set) can presumably be learned with relative ease. In the Memetic CoEvoMOG algorithm, the data generated for S1 and S2 at Step 6 of Figure 2 is used for iterative surrogate modeling. Candidate strategies in S1 and S2 9 Randomly generate initial population S1 for player P1 and S2 for P2 . Evaluate strategies in S1 and S2 considering all interactions possible. Train FNNs mapping candidate strategies to the corresponding ideal point approximations. for all generations do Step 1: Create offspring populations S10 and S20 via crossover and mutation of parent individuals from S1 and S2 , respectively. Step 2: Apply local refinements using the surrogate landscape on a subset of randomly chosen individuals from S10 and S20 (see Figure 4 for details). Step 3: Set S100 as S1 ∪ S10 , and set S200 as S2 ∪ S20 . Step 4: Evaluate populations S100 and S200 by performing all interactions between candidate strategies in S100 and S200 (keep track of evaluations to prevent repetitions). Step 5: Obtain the ”approximate” ideal point corresponding to each candidate strategy in S100 and S200 based on the outcomes of all possible interactions. Step 6: Obtain non-domination rank and crowding distance of each strategy in S100 and S200 based on the approximated ideal point vectors. Step 7: Consider the non-domination ranks and crowding distances lexicographically to select the most promising candidate strategies from S100 and S200 to form S1 and S2 in the next generation. Step 8: Retrain FNNs based on S1 , S2 and the corresponding ideal point approximations. end for Figure 3: Pseudo-code of the Memetic Co-evolutionary MOG Algorithm. serve as the inputs to the surrogate model, while the corresponding approximate ideal point objectives serve as outputs of interest. Note that separate surrogate models are learned for each player. Further, the models are learned repeatedly at every generation of the Memetic CoEvoMOG algorithm based on only the data generated during that generation (i.e., data is not accumulated across generations). The rationale behind this step is that the approximated ideal point vector tends to continuously adapt in a co-evolutionary algorithm in conjunction with the evolving strategies of the opponent, such that there may be little correlation in the data across generations. Finally, it must be mentioned that a simple feedforward neural network (FNN) is used for surrogate modeling in this paper, although any other preferred model type may also be incorporated with minimal change to the overall algorithmic framework. 4.2. Memetics via Local Refinement Memetics in stochastic optimization algorithms (such as EAs) are generally realized via a local solution refinement step as a form of lifelong learning of individuals. Since the original functions are assumed to be computationally expensive, herein, the local refinements are carried out in the surrogate landscape. Notably, since the ideal point is itself vector-valued, the local search is performed by first reducing the vector to a scalar value via a simple random weighting of objectives; we ensure that the randomly generated weights satisfy the partition of unity condition. It is important to mention here that the random weights are sampled from a uniform probability density function, such that no biased preference information 10 Let probability of local search be pls . for all individual in S1 do Step 1: Select the individual with probability pls . If not selected, then continue to next individual. Step 2: Generate a random weight vector satisfying partition of unity. Step 3: Combine the FNN surrogates using the random weights for scalarization. Minimize the resultant objective via the Nelder-Mead simplex algorithm where the individual’s strategy is taken as the starting point for local search. Step 4: Update the individual with the best solution found after the local search procedure. end for Figure 4: Pseudo-code of Memetics via Local Refinement. Steps are shown herein from the perspective of the minimization player P1 . The procedure is trivially generalized to the case of the maximization player P2 as well. is imposed (which preserves the postponed preference articulation condition of the MOG). The local search method used in this study is the popular derivative-free (bounded) NelderMead simplex algorithm, although any other algorithm may also be used. Thus, for the minimization player P1 , the simplex algorithm locally minimizes the randomly scalarized objectives, while for the maximization player P2 , the simplex algorithm locally maximizes the scalarized objectives. After offspring creation through genetic operations, a subset of them from both subpopulations of the Memetic CoEvoMOG algorithm are randomly selected with some user defined probability for local search. Once the local refinement is completed, i.e., the Nelder-Mead simplex algorithm converges to a point within the specified search space bounds, the improved solution (or strategy) is directly injected into the offspring population in the spirit of Lamarckian learning [31]. A brief overview of the steps involved in the memetic local refinement procedure is presented in Figure 4. 5. Numerical Experiments The proposed method was tested on a simple differential MOG named tug-of-war. The basic form of the game consists of a point with mass m placed at coordinates (0, 0). Two players P1 and P2 choose angles θ1 and θ2 , respectively, at which the respective forces with magnitudes F1 , F2 are applied (see Figure 5). The game outcome is the position (given by coordinates (x1 , x2 )) of the mass m after particular time tf . The objectives of player P1 are to minimize x1 and x2 and the objectives of player P2 are to maximize these two coordinates. The final coordinates can be computed with formulas: x1 = (F1 cos(θ1 ) + F2 cos(θ2 )) · 12 t2f , x2 = (F1 sin(θ1 ) + F2 sin(θ2 )) · 12 t2f . For simplification, the following assumptions are made: √ F1 = F2 = 1, tf = 2, and thus x1 = cos(θ1 ) + cos(θ2 ) and x2 = sin(θ1 ) + sin(θ2 ). In this game, the strategic decision is to choose the angles θ1 and θ2 , so the space of possible strategies is infinite, since θ1 , θ2 ∈ [0; 2π]. Accordingly, observe that the continuous search space of θ1 corresponds to the set S1 , and that of θ2 corresponds to the set S2 . 11 Figure 5: The tug-of-war game setting. Figure 6: Representative set of all points in the objective space that reflect the rationalizable strategies of the tug-of-war MOG. Rationalizable strategies for both players can be intuitively ascertained. Player P1 (minimizer) aims at having the mass in a position with negative coordinates (third quadrant), and therefore rationalizable strategies are in the range π ≤ θ1 ≤ 23 π. Similarly, player P2 (maximizer) wants the mass to be located in the first quadrant, thus the rationalizable strategies are in the range 0 ≤ θ2 ≤ 21 π. Due to postponed preference articulation, any move in the above ranges is a valid selection. Figure 6 shows all possible end game positions in the case of optimal performance, i.e. when both players select from the rationalizable range of strategies mentioned above. Due to the a priori known optimal performances of this MOG, the results of the algorithms applied to the tug-of-war game can be easily compared based on their approximation quality. Nevertheless, doing so is usually not possible in arbitrary MOGs where exact results are often too hard to compute in general practical settings. A more detailed description of 12 the tug-of-war game can be found in [16] where it was first used to test a version of the Canonical Co-evolutionary Algorithm similar to the one described in Section 3. 5.1. Experimental setup To make the tug-of-war MOG even more complex for the purpose of rigorous experimental testing, several synthetic functions have been artificially incorporated into the game formulation to create a number of alternate benchmarks. To elaborate, we define: x1 = F1 F2 F1 F2 cos(θ1 ) + cos(θ2 ), x2 = sin(θ1 ) + sin(θ2 ), 1 + φ(z1 ) 1 + φ(z2 ) 1 + φ(z1 ) 1 + φ(z2 ) where φ is the incorporated function, z1 and z2 are additional decision parameters introduced for P1 and P2 , respectively, while F1 , F2 ∈ [0, 1] are force magnitudes (which are now also treated as decision parameters). In this way, several tug-of-war variants can be created. Tested functions were chosen to check the efficacy of the proposed methods under varying conditions. Note that the selected functions are widely used in the literature to evaluate global optimization methods, including evolutionary techniques. In particular, the following 9 widely-known optimization functions were considered to serve as φ: Rosenbrock 2D, Rosenbrock 3D, Rastrigin 1D, Rastrigin 2D, Rastrigin 3D, Griewank 1D, Griewank 2D, Griewank 3D and Ackley 2D. Their plots and detailed description of properties can be found in [32]. It is worth mentioning that as all the selected functions have minimum value 0, the representation of rationalizable strategies of all MOG variants is the same as shown in Figure 6. 1. Rosenbrock nD φ1 (z) = n−1 X [100(z(i + 1) − z(i)2 )2 + (z(i) − 1)2 ] i=1 Global minimum: φ1 (1, . . . , 1) = 0. 2. Rastrigin nD n X φ2 (z) = 10n + [z(i)2 − 10cos(2πz(i))] i=1 Global minimum: φ2 (0, . . . , 0) = 0. 3. Griewank nD n φ3 (z) = X z(i)2 i=1 4000 − n Y z(i) cos( √ ) + 1 i i=1 Global minimum: φ3 (0, . . . , 0) = 0. 4. Ackley nD v u n n u1 X 1X 2 t φ4 (z) = −20 · exp[−0.2 z(i) ] − exp[ cos(2πz(i))] + 20 + e n n i=1 i=1 Global minimum: φ4 (0, . . . , 0) = 0. 13 In the experimental study, the canonical as well as the memetic co-evolutionay algorithms were run with the same hyperparameter settings in order to ensure that any performance differences are indeed a consequence of the proposed memetics module. The size of each subpopulation in the co-evolutionary algorithms was taken as 50, and the methods were run for 100 generations. For recombination operations, simulated binary crossover (SBX) [33] was used with distribution index of 20, and mutations were applied using the polynomial mutation operator [34] also with distribution index 20. In the Memetic CoEvoMOG algorithm, the probability of local search on the surrogate landscape (pls ) was set to 20% throughout. The test problems were assumed to be computationally expensive, such that the extra computational effort spent on building and searching the surrogate model was considered negligible in comparison to the cost of evaluations of the true underlying problem. For many real-world settings, such an assumption on the cost of surrogate assistance is reasonable, and is commonly used in most surrogate-assisted optimization studies. For this reason, the comparison plots presented in the next subsection are drawn on the basis of the results achieved over certain number of generations in the co-evolutionary algorithms, rather than explicitly taking computational time into account. 5.2. Experimental Results The Inverse Generational Distance (IGD) metric was used to measure the convergence characteristics (performance) achieved by the algorithms. IGD combines information about convergence and diversity of the obtained solutions. If P ∗ is a set of uniformly distributed points constituting the Pareto layer, and F is an approximation set of the Pareto layer obtained from the co-evolutionary algorithms, then P ∗ d(v, F ) IGD = v∈P ∗ |P | where d(v, F ) denotes minimum Euclidean distance between v and points in F , as measured in the objective space. Clearly, the lower the IGD values the better. Figures 7 and 8 present convergence comparison between Canonical CoEvoMOG and Memetic CoEvoMOG for 9 tested functions. Plots show the IGD convergence trends averaged over 20 independent runs. In all cases the Memetic CoEvoMOG algorithm’s convergence (dashed line) is noticeably faster than Canonical CoEvoMOG (doted line). Figures 9 and 10 present examples of the algorithms’ performance, in the objective space, as the search progresses through the generations. In each plot, each point represents the position of the mass as an outcome of the current strategies contained in the co-evolutionary subpopulations. Refer to Figure 6 for comparing these plots with the a priori known optimal solution (defined by the set of interactions between rationalizable strategies of both players). From generations 1 to 5, both algorithms produce mostly random solutions. The main difference can be noticed to emerge after 10-25 generations when plots of the memetic algorithm are closer to the optimal solution than those of the canonical approach, which agrees with the claim of faster Memetic CoEvoMOG convergence. Detailed numerical results after every 5 generations for each of the tested MOGs, in terms of average, minimum, maximum and standard deviation of IGD values are presented 14 Rosenbrock 2D Rosenbrock 3D Rastrigin 1D Rastrigin 2D Rastrigin 3D Griewank 1D Griewank 2D Griewank3D Figure 7: Convergence comparison between Canonical and Memetic Co-evolutionary Algorithms for tug-ofwar MOG variants with different synthetic functions φ. in the Appendix (Tables 1–9). It can be observed that not only is the convergence faster, but also the final results obtained after 100 generations are better in the case of Memetic 15 Ackley 2D Figure 8: Convergence comparison between Canonical and Memetic Co-evolutionary Algorithms for the tug-of-war MOG variant where φ = Ackley 2D. CoEvoMOG. Moreover, the Memetic CoEvoMOG algorithm appears to be more stable standard deviation values in most of the cases are lower. All results are proved to be statistically significant by the Wilcoxon Signed-Rank Test with p-value=0.05. The numbers of exact function evaluations were identical for both methods. Slower convergence of the Canonical CoEvoMOG algorithm is hypothesized to be caused by the existence of the Red Queen effect described in Section 1. In most cases, after a few generations of steep decrease of the IGD value, it is observed that the IGD value tends to rise for a brief period of time in the canonical case (as revealed in Figures 7 and 8). This surprising observation may be due to the continuous adaptations of subpopulations to the evolutions of each other, even though the overall performance may be far from the optimum (much like the trees in the forest as discussed in the introduction). In this respect, the local search steps included in the Memetic CoEvoMOG algorithm are seen to provide a guiding hand to the purely genetic mechanisms, thereby suppressing the Red Queen effect to a large extent and accelerating the convergence characteristics of the proposed algorithm. The above promising experimental results form a strong basis for attempts of solving more complex, real-life problems which can be represented as MOGs. In particular, multistep decision-making problems and problems characterized by payoffs changing over time seem to be perfect candidates for further evaluation of the Memetic CoEvoMOG algorithm. Such problems appear is various practical domains, including planning and decision-making under uncertainty or in adversarial environments, e.g. in the area of cyber security or homeland security (notably Security Games [35, 36]). 6. Conclusions This paper presents a new memetic co-evolutionary approach (Memetic CoEvoMOG) to finding strategies for multi-objective games under postponed objective preference articulation. The proposed method improves the canonical co-evolutionary model described in [16] by suppressing the Red Queen effect via the guiding light of lifelong learning. In particular, for ensuring the computational viability of lifelong learning in competitive multi-objective 16 Canonical gen. 1 Canonical gen. 10 Canonical gen. 15 Canonical gen. 20 Canonical gen. 25 Canonical gen. 50 Memetic gen. 1 Memetic gen. 10 Memetic gen. 15 Memetic gen. 20 Memetic gen. 25 Memetic gen. 50 Figure 9: The performance of Canonical CoEvoMOG and Memetic CoEvoMOG algorithms on the tug-of-war MOG variant with φ = Rastrigin 1D, after 1, 10, 15, 20, 25 and 50 generations. 17 Canonical gen. 1 Canonical gen. 10 Canonical gen. 15 Canonical gen. 20 Canonical gen. 25 Canonical gen. 50 Memetic gen. 1 Memetic gen. 10 Memetic gen. 15 Memetic gen. 20 Memetic gen. 25 Memetic gen. 50 Figure 10: The performance of Canonical CoEvoMOG and Memetic CoEvoMOG algorithms on the tug-ofwar MOG variant with φ = Ackley 2D, after 1, 10, 15, 20, 25 and 50 generations. 18 game settings, we incorporate an approach that reduces sets of payoff vectors in objective space to a single representative point without disrupting the postponed preference articulation condition. Thereafter, a surrogate model of the representative point is built, which allows the local improvements to be carried out efficiently by searching on the surrogate landscape. The reliability and effectiveness of our method is experimentally proved on a suite of testing MOG variants. It is demonstrated in the paper that incorporation of memetics improves the convergence characteristics and leads to better solutions in comparison with the canonical co-evolutionary algorithm. Consequently, in the proposed method the total number of function evaluations can be reduced with no harm to the overall quality of resultant strategies. Such time savings are of special importance in the case of time-sensitive and/or computationally expensive MOGs appearing in real-life applications. 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Karwowski, J. Mańdziuk, A new approach to security games, in: ICAISC’2015, Vol. 9120 of Lecture Notes in Artificial Intelligence, Springer-Verlag, 2015, pp. 402–411. [36] J. Karwowski, J. Mańdziuk, Mixed strategy extraction from UCT tree in security games, in: ECAI 2016, IOS Press, 2016, pp. 1746–1747. Appendix Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 65.28 108.77 89.17 72.94 65.48 57.10 50.49 48.01 38.29 37.67 33.65 31.71 31.44 32.01 30.76 30.32 31.04 30.44 29.24 29.45 CanonicalCoEvoMOG Min Max Std dev 39.39 133.60 30.45 35.07 249.93 81.80 31.81 262.70 68.31 27.08 244.94 63.69 27.66 208.32 53.05 22.71 203.50 52.95 22.95 198.35 52.39 21.29 176.34 45.65 21.80 128.73 32.03 20.70 102.80 24.04 20.57 76.62 16.01 21.75 74.82 15.75 22.37 65.54 13.45 21.74 68.39 14.49 22.99 64.73 13.26 21.63 64.61 13.16 21.88 58.89 11.87 21.64 57.39 11.35 20.99 60.32 11.61 20.82 58.14 11.36 Avg 54.37 49.92 50.89 43.86 36.77 30.68 28.29 25.72 24.13 23.95 22.75 22.57 22.09 21.58 21.26 20.92 21.09 21.07 20.89 20.96 MemeticCoEvoMOG Min Max Std dev 43.16 110.30 20.66 34.45 58.46 9.44 33.31 77.97 15.41 28.24 95.44 19.50 24.22 68.52 13.28 22.28 38.74 6.32 23.05 36.67 4.88 20.58 41.32 6.01 19.85 39.28 5.72 20.75 36.44 4.75 20.39 32.17 3.59 20.48 30.74 3.41 20.30 26.64 2.13 19.59 24.10 1.40 19.20 23.57 1.13 18.62 22.70 1.25 19.54 22.71 0.98 20.11 22.79 0.76 20.00 21.73 0.55 19.72 22.89 0.91 Table 1: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Rosenbrock 2D. 21 Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 221.43 139.15 163.41 142.60 156.23 156.85 148.23 133.30 140.77 130.92 131.97 123.37 118.80 121.05 108.20 112.03 110.11 109.76 116.73 112.51 CanonicalCoEvoMOG Min Max Std dev 62.10 422.71 121.20 94.28 235.24 40.96 85.68 348.19 77.72 40.23 333.12 85.41 34.67 325.21 83.47 37.83 392.68 99.68 31.43 388.38 98.04 33.33 334.02 91.26 32.21 356.72 97.36 30.93 356.50 95.12 27.86 359.18 98.65 26.57 351.25 95.70 26.21 345.35 93.30 26.22 401.82 110.77 27.95 292.76 81.13 26.12 302.42 82.19 25.34 302.54 81.86 24.38 325.80 85.75 23.13 319.10 84.37 22.08 318.41 84.95 Avg 159.70 127.09 96.67 107.87 88.77 91.32 81.51 74.29 67.55 53.79 47.50 45.11 41.47 39.08 38.25 38.89 36.36 32.38 32.46 31.13 MemeticCoEvoMOG Min Max Std dev 98.87 210.68 30.68 65.20 344.80 80.01 40.44 177.86 41.80 55.31 232.53 56.76 44.01 169.74 42.91 49.53 240.98 55.93 37.89 206.72 50.32 38.13 193.34 44.43 34.48 164.75 37.58 32.19 126.93 27.58 33.15 108.08 23.51 28.85 86.73 16.29 26.04 84.47 16.83 24.33 73.32 15.01 26.62 86.44 17.80 22.26 88.13 20.71 22.45 79.01 17.17 22.31 61.75 12.70 22.90 59.00 11.87 22.93 47.49 7.96 Table 2: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Rosenbrock 3D. Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 82.69 115.71 138.51 131.23 124.50 114.05 112.73 109.67 106.52 104.97 104.66 103.91 103.19 103.65 102.98 102.24 102.15 102.29 101.57 101.78 CanonicalCoEvoMOG Min Max Std dev 35.23 182.31 51.71 40.70 532.32 149.59 31.11 822.00 241.86 24.76 858.55 257.37 25.75 843.10 253.38 24.59 820.08 248.31 22.83 821.77 249.36 21.45 821.12 250.10 22.63 820.33 250.87 20.45 819.34 251.06 20.80 819.51 251.20 21.29 820.20 251.69 20.91 819.72 251.77 22.21 819.06 251.37 20.52 819.22 251.66 19.00 817.93 251.48 18.99 819.12 251.93 21.44 818.64 251.70 20.84 818.62 251.95 21.07 818.19 251.72 Avg 65.59 57.21 25.37 22.61 21.04 21.37 21.09 21.43 21.20 21.74 20.88 20.99 21.34 20.97 21.38 21.18 20.72 20.95 21.11 21.00 MemeticCoEvoMOG Min Max Std dev 34.88 128.11 36.63 28.04 100.62 26.31 19.83 37.58 5.31 20.40 30.55 3.02 19.59 22.69 1.11 20.13 22.45 0.76 20.27 22.50 0.80 20.36 22.84 0.75 19.02 22.45 1.02 19.67 23.46 0.96 19.91 22.35 0.64 19.35 22.76 1.05 19.87 22.10 0.64 19.91 21.90 0.75 20.20 22.76 0.85 19.12 22.76 1.08 19.22 21.88 0.95 19.94 22.48 0.79 19.84 23.26 1.00 18.74 22.56 1.26 Table 3: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Rastrigin 1D. 22 Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 267.25 112.35 129.75 130.90 108.05 81.50 64.85 48.32 44.62 36.43 32.11 29.49 28.88 26.92 26.61 26.55 25.01 24.29 23.86 23.72 CanonicalCoEvoMOG Min Max Std dev 105.65 401.83 97.01 30.70 320.32 96.15 37.76 424.67 126.87 35.54 492.63 142.76 24.44 374.31 117.24 21.98 306.10 87.68 21.36 195.60 58.29 20.80 121.05 33.20 21.74 97.24 28.69 21.50 76.02 17.43 21.30 60.89 13.70 20.61 50.33 10.26 21.08 46.79 9.63 21.45 39.10 6.44 20.68 40.84 7.53 21.64 42.22 6.31 21.43 33.32 3.94 21.40 28.88 2.93 19.51 30.49 3.28 20.77 29.40 3.01 Avg 167.62 72.66 47.83 58.24 42.87 36.85 44.76 51.40 33.58 41.13 52.31 42.92 50.44 52.59 49.01 36.21 48.01 49.40 46.31 47.22 MemeticCoEvoMOG Min Max Std dev 62.95 347.25 79.07 31.69 240.47 69.85 21.72 194.51 52.01 20.34 369.19 109.29 20.19 230.27 65.86 21.13 170.69 47.03 19.40 251.54 72.67 20.40 322.40 95.23 20.11 141.78 38.03 19.03 219.55 62.71 20.09 330.97 97.91 20.11 237.23 68.28 20.18 312.75 92.17 20.12 336.27 99.68 19.74 299.81 88.13 20.23 168.79 46.59 20.05 281.77 82.14 20.32 302.58 88.96 20.26 267.55 77.74 19.79 283.01 82.86 Table 4: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Rastrigin 2D. Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 539.35 297.53 246.47 253.62 253.72 239.35 226.37 200.78 208.77 189.19 171.68 161.08 170.20 161.04 170.60 183.36 156.82 170.47 158.73 162.51 CanonicalCoEvoMOG Min Max Std dev 264.70 793.37 170.54 109.05 497.63 122.76 47.36 574.07 177.01 45.83 514.96 172.03 43.73 559.89 182.96 32.67 555.56 185.00 24.05 520.98 187.45 25.86 570.16 185.05 24.15 487.22 185.19 23.75 483.84 180.69 23.93 431.40 170.07 23.21 460.39 161.00 22.22 499.57 182.10 23.58 464.79 171.77 23.12 455.10 183.43 22.01 549.62 209.91 22.85 400.55 173.16 21.48 450.96 192.60 21.84 434.21 176.66 20.60 492.29 188.31 Avg 473.73 195.57 164.72 128.32 115.86 103.92 140.85 100.02 101.88 121.58 106.75 119.49 100.48 131.46 129.83 125.35 130.58 118.76 99.18 121.98 MemeticCoEvoMOG Min Max Std dev 257.24 708.45 122.06 28.67 559.76 172.42 42.34 483.88 175.39 29.95 538.47 168.63 22.74 404.96 146.92 20.35 362.89 135.21 19.90 585.08 205.77 19.67 359.16 133.79 19.90 381.89 134.46 20.75 616.96 193.50 20.52 350.30 138.17 20.42 568.65 182.07 20.56 452.16 144.64 20.61 488.36 184.60 20.16 427.89 178.37 19.79 560.23 185.42 21.33 566.23 190.19 20.29 446.60 161.69 20.16 340.70 127.85 20.07 433.02 165.82 Table 5: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Rastrigin 3D. 23 Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 61.50 45.79 35.56 30.16 27.06 25.77 24.12 23.20 23.05 23.39 22.20 22.57 21.98 22.25 21.98 21.94 21.94 22.08 21.66 21.65 CanonicalCoEvoMOG Min Max Std dev 28.44 145.25 35.46 22.07 110.52 25.45 21.10 85.59 20.25 22.15 62.02 11.87 21.44 47.33 7.92 21.85 40.16 5.61 20.17 35.10 4.53 20.67 31.75 3.34 20.51 30.91 2.90 21.29 28.48 2.18 20.43 25.15 1.57 21.01 25.76 1.68 20.64 24.91 1.38 20.61 23.89 1.19 20.42 23.34 0.83 19.51 23.72 1.20 20.63 23.16 0.79 20.66 24.07 0.94 20.02 23.56 1.06 20.10 23.28 1.12 Avg 38.47 34.29 22.36 20.61 20.93 20.85 20.38 20.99 20.51 21.15 20.55 20.92 21.93 20.33 21.00 20.96 21.02 20.84 20.39 20.75 MemeticCoEvoMOG Min Max Std dev 30.15 83.04 15.51 20.95 46.41 7.90 20.52 28.91 2.58 20.21 23.69 1.02 19.66 22.52 1.03 19.59 22.33 0.83 20.12 22.01 0.67 19.47 22.19 0.92 19.69 23.65 1.36 19.62 21.89 0.89 19.51 22.29 0.93 19.79 22.49 0.88 20.73 22.90 0.75 19.75 23.45 1.16 19.38 22.49 1.00 19.72 21.81 0.71 20.18 22.81 0.85 19.90 21.68 0.59 19.71 22.63 0.98 19.86 21.80 0.69 Table 6: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Griewank 1D. Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 44.79 52.15 43.86 38.64 35.14 31.45 28.89 27.76 26.20 24.90 23.89 23.57 23.06 23.00 22.66 22.75 22.56 22.22 23.08 22.79 CanonicalCoEvoMOG Min Max Std dev 29.18 72.61 14.84 42.66 69.53 10.57 25.76 74.00 15.32 24.60 64.01 13.00 23.91 58.30 10.39 22.79 47.51 7.35 22.49 44.95 6.76 21.46 45.63 7.11 22.20 35.58 3.93 21.05 33.30 3.43 20.63 32.65 3.35 21.02 31.07 2.94 21.07 27.99 2.06 20.68 27.28 1.89 20.98 26.23 1.64 21.29 23.93 0.79 20.31 25.34 1.61 20.94 23.16 0.84 21.81 24.83 1.23 20.68 24.22 1.13 Avg 44.64 41.19 26.42 22.03 21.33 21.30 21.28 21.42 20.88 21.11 20.72 21.17 21.17 21.05 21.10 21.25 21.01 21.47 20.83 20.78 MemeticCoEvoMOG Min Max Std dev 28.02 72.71 13.84 30.52 50.83 6.56 21.14 37.49 4.74 20.17 24.37 1.41 20.24 24.37 1.14 19.44 22.20 0.84 19.82 21.95 0.69 19.47 22.31 0.94 19.87 22.15 0.73 19.47 24.04 1.27 19.76 22.16 0.67 20.11 22.59 0.76 19.45 22.16 0.79 20.10 21.96 0.60 20.27 22.10 0.66 20.72 22.08 0.45 18.76 22.23 1.17 20.02 22.76 0.92 20.19 22.05 0.69 19.84 22.29 0.71 Table 7: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Griewank 2D. 24 Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 50.06 64.26 44.51 32.81 28.11 25.05 23.85 23.47 23.81 22.87 22.58 22.79 23.85 22.80 22.61 22.06 22.65 22.42 22.36 22.00 CanonicalCoEvoMOG Min Max Std dev 31.19 65.34 11.21 34.18 117.82 25.48 26.98 70.69 12.33 26.98 66.00 11.97 21.81 52.40 8.96 21.83 34.17 3.71 20.58 31.20 2.89 20.72 33.44 3.76 20.25 29.19 2.57 20.80 25.21 1.48 21.18 25.99 1.37 21.33 26.38 1.63 21.12 27.83 2.28 20.89 27.65 1.96 20.35 25.07 1.47 19.95 24.24 1.17 20.03 24.90 1.64 21.07 23.86 0.90 20.57 25.39 1.47 20.92 24.34 1.06 Avg 45.12 38.02 25.65 22.74 21.88 21.42 21.62 21.79 21.31 21.97 21.07 21.62 21.28 20.53 21.77 21.18 21.34 21.32 21.39 21.04 MemeticCoEvoMOG Min Max Std dev 33.66 75.94 13.48 27.80 55.44 9.10 22.56 34.38 4.38 20.08 26.94 2.03 20.92 25.94 1.50 20.35 23.67 1.16 19.97 22.98 0.92 19.97 23.09 0.96 19.70 21.95 0.74 19.46 23.69 1.11 19.83 21.68 0.62 20.11 22.02 0.72 19.97 22.70 0.82 19.51 22.65 1.02 19.85 22.99 1.03 19.84 23.27 0.98 20.16 22.88 0.86 19.43 23.19 1.19 20.46 22.68 0.67 19.45 22.91 1.09 Table 8: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Griewank 3D. Generation 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Avg 226.76 199.62 182.56 159.47 137.04 114.79 98.50 94.67 68.56 55.32 52.46 51.33 43.51 39.83 37.38 32.96 33.67 29.98 29.84 27.13 CanonicalCoEvoMOG Min Max Std dev 99.59 383.74 95.09 42.24 409.33 129.43 48.71 398.80 106.92 39.99 427.36 116.25 32.40 336.20 93.39 30.98 283.01 83.79 27.48 203.53 63.03 25.11 261.62 72.15 23.42 145.00 41.81 22.41 122.20 32.63 21.91 93.02 26.93 20.29 118.80 30.57 21.67 76.99 20.07 22.65 82.61 19.24 21.74 96.71 23.09 21.08 71.85 15.29 21.33 67.30 15.18 21.45 55.10 10.64 21.26 64.97 13.10 22.15 44.12 6.91 Avg 126.66 48.78 31.29 23.55 21.37 21.12 21.16 20.87 20.83 20.62 21.14 21.10 21.07 20.86 20.92 20.63 21.08 21.17 20.82 21.00 MemeticCoEvoMOG Min Max Std dev 37.18 322.69 85.99 31.50 68.11 12.33 22.81 41.76 5.78 20.76 27.76 2.19 19.29 22.29 0.89 20.38 23.07 0.81 20.16 22.22 0.79 19.95 22.16 0.68 19.03 22.55 1.29 19.56 22.34 0.96 20.06 22.37 0.83 19.50 22.20 0.95 19.64 22.31 0.85 19.17 21.67 0.88 20.09 22.59 0.75 18.94 21.56 0.98 19.11 22.42 0.98 19.63 23.59 1.36 18.95 22.32 1.00 19.98 22.07 0.61 Table 9: Comparison between results (in terms of IGD) obtained by Cannonical and Memetic Coevolutionary Algorithms based on 20 independent runs of the tug-of-war MOG variant with φ = Ackley 2D. 25 

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 Algorithms for Scheduling Malleable Tasks arXiv:1501.04343v8 [cs.DC] 4 Feb 2018 Xiaohu Wu, and Patrick Loiseau Abstract—Due to the ubiquity of batch data processing in cloud computing, the related problems of scheduling malleable batch tasks have received significant attention recently. In this paper, we consider a fundamental model where a set of n tasks is to be processed on C identical machines and each task is specified by a value, a workload, a deadline and a parallelism bound. Within the parallelism bound, the number of machines assigned to a task can vary over time without affecting its workload. For this model, we first give two core results: the definition of an optimal state under which multiple machines could be utilized by a set of tasks with hard deadlines, and, an algorithm achieving such a state. The optimal utilization state plays a key role in the design and analysis of scheduling algorithms (i) when several typical objectives are considered, such as social welfare maximization, machine minimization, and minimizing the maximum weighted completion time, and, (ii) when the algorithmic design techniques such as greedy and dynamic programming are applied to the social welfare maximization problem. As a result, we give four new or improved algorithms for the above problems. F 1 I NTRODUCTION Cloud computing has become the norm for a wide range of applications and batch processing constitutes the most significant computing paradigm [1]. Applications such as web search index update, monte carlo simulations and bigdata analytics require executing a new type of parallel tasks on clusters, termed malleable tasks. Two basic features of malleable tasks are about workload and parallelism bound. There are multiple machines, and, throughout the execution, the number of machines assigned to a task can vary over time within the parallelism bound but its workload is not affected by the number of used machines [2], [3]. Beyond understanding how to schedule the fundamental batch task model, many efforts are also devoted to its online version [4], [5], [6] and its extension in which each task contains several subtasks with precedence constraints [7], [8]. In practice, for better efficiency, companies such as IBM have integrated these smarter scheduling algorithms for various time metrics [8] (than the popular dominant resource fairness strategy) into their batch processing platforms [9]. In scheduling theory, the above malleable task model can be viewed as an extension of the classic model of scheduling preemptive tasks on a single or multiple machines where the parallelism bound is one [10], [11]. When each task has to be completed by some deadline, the results from the special single machine case have already implied that the state of optimally utilizing machines plays a key role in the design and analysis of scheduling algorithms under several objectives [11]. In particular, the famous EDF (Earliest Deadline First) rule can achieve an optimal schedule for the single machine case. It is initially designed so as to find an exact algorithm for scheduling batch tasks to minimize the maximum task lateness (i.e., task’s completion time minus due date) [12]. So far, numerous applications of this rule • • Xiaohu Wu is with Fondazione Bruno Kessler, Trento, Italy. E-mail: xiaohuwu@fbk.eu Patrick Loiseau is with Univ. Grenoble Alpes, LIG, France and MPI-SWS, Germany. E-mail: patrick.loiseau@univ-grenoble-alpes.fr Manuscript received April 19, 2005; revised August 26, 2015. have been found, e.g., (i) to design exact algorithms for the extended model with release times [13] and for scheduling tasks with deadlines (and release times) to minimize the total weighted number of tardy tasks [14], and (ii) as a significant principle in the analysis of scheduling feasibility for real-time systems [15]. Similarly, we are convinced that, as far as malleable tasks are concerned, achieving such an optimal resource utilization state is also very important for designing and analyzing scheduling algorithms (i) under various objectives, or (ii) when different algorithmic design techniques such as greedy and dynamic programming are applied. The intuition for this is that, if the utilization state was not optimal in an algorithm, its performance could be improved by utilizing the machines optimally to allow more tasks to be completed. All these considerations motivate us to develop an theoretical framework proposed in this paper. Before this paper, a greedy algorithm was proposed in s−1 [3] that achieves a performance guarantee C−k C · s ; here, C is the number of machines, k is the maximum parallelism bound of all tasks, s is the minimum slackness of all tasks where each task’s slackness is defined to be the ratio of its deadline to its minimum execution time, which is the time when a task is always allocated the maximum number of machines during the execution. k is a system parameter and is assumed to be finite [16]. Intuitively, s characterizes the resource allocation urgency (e.g., s = 1 means that the maximum amount of machines have to be allocated to a task at every time slot to meet its deadline). 1.1 Our Results Core result (Section 3). The core result of this paper is to identify a sufficient and necessary condition under which a set of independent malleable tasks could be all completed by their deadlines on C machines, also referred to as boundary condition in this paper. In particular, by understanding the basic constraints of malleable tasks, we first identify and formally define a state in which C machines can be said to be optimally utilized by JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 a set of tasks with deadlines in terms of resource utilization. Then, we propose an optimal scheduling algorithm LDF(S ) (Latest Deadline First) that achieves such an optimal state. The LDF(S ) algorithm has a polynomial time complexity of O(n2 ) and is different from the EDF algorithm that gives an optimal schedule in the single-machine case. Here, the maximum deadline of tasks is assumed to be finitely bounded by a constant. Applications (Sections 4 and 5). The above core results have several applications to propose new or improved algorithmic design and analysis for scheduling malleable tasks under different objectives. The scheduling objectives considered in this paper include: (a) (b) (c) social welfare maximization: maximize the sum of values of tasks completed by their deadlines; machine minimization: minimize the number of machines needed to produce a feasible schedule for a set of tasks such that each task is completed by their deadline; maximum weighted completion time minimization: minimize the maximum weighted completion time of tasks. Here, the first and second objectives above have been considered in [2], [3], [8]. The second objective that concerns the optimal utilization of machines has been considered for other types of tasks [17] but we are the first to consider it for malleable tasks. After applying the core results above, we obtain the following algorithmic results: (i) (ii) (iii) (iv) an improved greedy algorithm GreedyRLM with a performance guarantee s−1 s for social welfare maximization with a time complexity of O(n2 ); the first exact dynamic programming algorithm for social welfare maximization with a pseudo-polynomial time complexity of O(max{ndL C L , n2 }), where L is the number of deadlines, D and d are the maximum workload and deadline of tasks; the first exact algorithm for machine minimization with a time complexity of O(n2 , Ln log n); a polynomial time (1+)-approximation algorithm for maximum weighted completion time minimization. In the greedy algorithm of [3], the tasks are considered in the non-decreasing order of their marginal values of tasks (i.e., the ratio of a task’s value to its size), and only if a task could be fully completed by its deadline according to the currently remaining resource, it will be accepted and allocated possibly different number of machines over time according to an allocation algorithm; otherwise, it will be rejected. In this paper, we also show that • • for social welfare maximization, s−1 s is the best possible performance guarantee that a class of greedy algorithms could achieve where they consider tasks in the non-increasing order of their marginal values. as a result, the proposed greedy algorithm of this paper is the best possible among this kind of greedy algorithms. The second algorithm for social welfare maximization can work efficiently when L is small since its time complexity is exponential in L. However, this may be reasonable 2 in a machine scheduling context. In scenarios like [7], tasks are often scheduled periodically, e.g., on an hourly or daily basis, and many tasks have a relatively soft deadline (e.g., finishing after four hours instead of three will not trigger a financial penalty). Then, the scheduler can negotiate with the tasks and select an appropriate set of deadlines {τ1 , τ2 , · · · , τL }, thereafter rounding the deadline of a task down to the closest τi (1 ≤ i ≤ L). By reducing L, this could permit to use the dynamic programming (DP) algorithm rather than GreedyRLM in the case where the slackness s is close to 1. With s close to 1, the approximation ratio of GreedyRLM approaches 0 and possibly little social welfare is obtained by adopting GreedyRLM while the DP algorithm can still obtain the almost optimal social welfare. Technical Difference. The second algorithm can be viewed as an extension of the pseudo-polynomial time exact algorithm in the single machine case [10] that is also designed via the generic dynamic programming procedure. However, before our work, how to enable this extension to malleable tasks was not clear as indicated in [2], [3]. This is mainly due to the lack of a notion of the optimal state of machines being utilized by malleable tasks with deadlines and the lack of an algorithm that achieves such a state. In contrast, the optimal state in the single machine case can be defined much more easily and achieved by the EDF algorithm. The core results of this paper are the enabler of a DP algorithm. The way of applying the core results to a greedy algorithm is less obvious since in the single machine case there is no corresponding algorithm to hint its role in the algorithmic design. For the above class of greedy algorithms, we manage to give a new algorithm analysis, figuring out what resource allocation features of tasks can benefit and determine the algorithm’s performance. This analysis is an extended analysis of the greedy algorithm for the standard knapsack problem [22] and it does not rely on the dualfitting technique, on which the algorithm in [3] is built. Here, the problem could be viewed as an extension of the knapsack problem where each item has two additional constraints in a two-dimensional space: a (time) window in which an item could be placed and a maximum width of the space that it could utilize at every moment. Two of the most important algorithms there are either based on the DP technique or of greedy type, that also considers items by their marginal values [22]; we give in this paper their counterparts in the scenario of malleable tasks. In the construction of the greedy and optimal scheduling algorithms, we are inspired by the algorithm in [3]. After our definition of the optimal state and a new analysis of the above class of greedy algorithms, we found that the algorithm in [3] could achieve an optimal resource utilization state from the maximum deadline of tasks d to some earlier time slot t. However, this is achieved by guaranteeing the existence of a time slot t0 earlier than t such that the number of available machines at t0 is ≥ k , which leads a suboptimal utilization of resources. In our algorithm, we only require t0 to be such that the number of available machines at t0 is ≥ 1, which leads to an optimal resource utilization. More details could be found in the remarks of Section 3.2. The above third and fourth algorithms are obtained by respectively applying the above core result to a binary JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3 search procedure, and the related results in [8]. 1.2 Related works Now, we introduce the related works. The linear programming approaches to designing and analyzing algorithms for the task model of this paper [2], [3] and its variants [4], [5], [7] have been well studied1 . All these works consider the same objective of maximizing the social welfare. In [2], Jain et al. proposed an algorithm with C an approximation ratio of (1 + C−k )(1 + ) via deterministic rounding of linear programming. Subsequently, Jain et al. [3] proposed a greedy algorithm GreedyRTL and used the dual-fitting technique to derive an approximation ratio C−k s−1 C · s . In [7], Bodik et al. considered an extension of our task model, i.e., DAG-structured malleable tasks, and, based on randomized rounding of linear programming, they proposed an algorithm with an expected approximation 1 ratio of α(λ) for every λ i> 0, where α(λ) = λ1 · e− λ · h (1−1/λ)C−k TABLE 1 Main Notation Notation C T Ti Di , di , vi ki yi (t) W (t) W (t) leni i si κ ·ln λ·(1− C ) 2ωκ . The online version of our task 1 − e− model is considered in [4], [5]; again based on the dual-fitting technique, two weighted greedy algorithms are proposed respectively for non-committed and committed scheduling 1 and achieve the competitive ratios of crA = 2 + O( ( √ 3 s−1)2 ) cr (s·ω(1−ω)) where s > 1 [3] and Aω(1−ω) where ω ∈ (0, 1) and 1 s > ω(1−ω) . In addition, Nagarajan et al. [8] considered DAGstructured malleable tasks and propose two algorithms with approximation ratios of 6 and 2 respectively for the objectives of minimizing the total weighted completion time and the maximum weighted lateness of tasks. Nagarajan et al. showed that optimally scheduling deadline-sensitive malleable tasks in terms of resource utilization is a key to the solutions to scheduling for their objectives. In particular, seeking a schedule for DAG tasks can be transformed into seeking a schedule for tasks with simpler chain-precedence constraints; then whenever there is a feasible schedule to complete a set of tasks by their deadlines, Nagarajan et al. proposed a nonoptimal algorithm where each task is completed by at most 2 times its deadline and give two procedures to obtain nearoptimal completion times of tasks in terms of the above two objectives. Technically, the works [2], [3], [4], [5], [7] formulate their problem as an Integer Program (IP) and relax the IP to a relaxed linear program (LP). The techniques in [2], [7] require to solve the LP to obtain a fractional optimal solution and then manage to round the fractional solution to an integer solution of the IP that corresponds to an approximate solution to their original problem. In [3], [4], [5], the dual fitting technique first finds the dual of the LP and then construct a feasible algorithmic solution X to the dual in some greedy way. This solution corresponds to a feasible solution Y to their original problems, and, due to the weak duality, the value of the dual under the solution X (expressed in the form of the value under Y multiplied by a parameter α ≥ 1) will be an upper bound of the optimal value of the IP, i.e., the optimal value that can be achieved in the original problem. Therefore, the approximation ratio 1. We refer readers to [11], [18] for more details on the general techniques to design scheduling algorithms. Explanation the total number of machines a set of tasks to be scheduled on C machines a task in T the workload, deadline, and value of a task Ti the parallelism bound of Ti , i.e., the maximum number of machines that can be allocated to and utilized by Ti simultaneously the number of machines allocated to Ti at a time slot t where yi (t) ∈ {0, 1, · · · , ki } and set all yi (t) to 0 initially the total number of machines thatP are allocated out to the tasks at t, i.e., W (t) = Ti ∈T yi (t) the total number of machines idle at t, i,e., W (t) = C − W (t) the minimum execution time of Ti where Ti is allocated ki machines in the entire execution i process, i.e., leni = d D e k s d, D vi0 {τ1 , · · · , τL } Di di the slackness of a task, i.e., len , measuring the i urgency of machine allocation to complete Ti by the deadline the minimum slackness of all tasks of T , i.e., minTi ∈T si the maximum deadline and workload of all tasks of T , i.e., d = maxTi ∈T di and D = maxTi ∈T Di vi the marginal value of Ti , i.e., vi0 = D i the set of the deadlines di of all tasks Ti of T , where 0 = τ0 < τ1 < · · · < τL = d all the tasks {Ti,1 , Ti,2 , · · · , Ti,ni } of T that have a deadline τi , 1 ≤ i ≤ L of the algorithm involved in the dual becomes clearly 1/α. Here, the approximation ratio is a lower bound of the ratio of the actual value obtained by the algorithm to the optimal value. A part of results of this paper appeared at the Allerton conference in the year 2015 [19], [20]. Following [19], [20], a recent work also gave a similar (sufficient and necessary) feasibility condition to determine whether a set of malleable tasks could be completed by their deadlines and showed that such a condition is central to the application of the LP technique to the three problems of this paper: greedy and exact algorithms for social welfare maximization and an exact algorithm for machine minimization. Guo & Shen first used the LP technique to give a new proof of this feasibility condition in the core result. Based on this condition, the authors gave a new formulation of the original problems as IP programs, different from the ones in [2], [3]. This new formulation enables from a different perspective proposing almost the same algorithmic results as this paper, e.g., for the machine minimization problem an exact algorithm with a time complexity O((n + d)3.5 Ls (log n + log k)), and for the social welfare maximization problem an exact algorithm with a complexity O(n·(C·d)d ), where Ls is the length of the LP’s input. In addition, we have shown that the best performance guarantee is s−1 s when a greedy algorithm considers tasks in the non-increasing order of their marginal values. Guo & Shen also considered another standard to determine the order of tasks, and proposed a greedy algorithm with a 2 performance guarantee C−k C and a complexity O(n + nd). JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2 M ODEL AND P ROBLEM D ESCRIPTION There are C identical machines and a set of n tasks T = {T1 , T2 , · · · , Tn }. The task Ti is specified by several characteristics: (1) value vi , (2) demand (or workload) Di , (3) deadline di , and (4) parallelism bound ki . Time is discrete and the time horizon is divided into d time slots: {1, 2, · · · , d}, where d = maxTi ∈T di and the length of each slot may be a fixed number of minutes. A task Ti can only utilize the machines located in time slot interval [1, di ]. The parallelism bound ki limits that, at any time slot t, Ti can be executed on at most ki machines simultaneously. Let k = maxTi ∈T ki be the maximum parallelism bound; here, ki is a system parameter and k is therefore assumed to be finite [16]. An allocation of machines to a task Ti is a function yi : [1, di ] → {0, 1, 2, · · · , ki }, where yi (t) is the number of machines allocated to task Ti at a time slot t ∈ [1, di ]. In this model, Di , di ∈ Z + for all Ti ∈ T . P For the system of C machines, denote by W (t) = Ti ∈T yi (t) the system’s workload at time slot t; and by W (t) = C − W (t) its complementary, i.e., the amount of available machines at time t. We say that time t is fully utilized if W (t) = 0, and is not fully utilized if W (t) > 0. In addition, we assume that the maximum deadline of tasks is bounded. Given the model above, the following three scheduling objectives are considered separately in this paper: • • • The first objective is social welfare maximization and it aims to choose an a subset S ⊆ T and produce a feasible schedule for S so as to maximize the P social welfare Ti ∈S vi (i.e., the sum of values of tasks completed by deadlines); here, the value vi of a task Ti is gained if P and only if it is fully allocated by the deadline, i.e., t≤di yi (t) ≥ Di , and partial execution of a task yields no value. The second objective is machine minimization, i.e., seeking the minimum number of machines needed to produce a feasible schedule of T on C machines such that the task’s parallelism bound and deadline constraints are not violated. The third objective is to minimize the maximum weighted lateness of tasks, i.e., minTi ∈T {vi ·(ti −di )}, where ti is the completion time of a task Ti . Furthermore, we denote by [l] and [l]+ the sets {0, 1, · · · , l} and {1, 2, · · · , l} for a positive integer l. Let leni = dDi /ki e denote the minimum execution time of Ti . Define by si = di leni the slackness of Ti , measuring the urgency of machine allocation (e.g., si = 1 may mean that Ti should be allocated the maximum amount of machines ki at every t ∈ [1, di ]) and let s = minTi ∈T si be the slackness of the least flexible vi task (s ≥ 1). Denote by vi0 = D the marginal value of task Ti , i i.e., the value obtained by the system per unit of demand. We assume that the demand of each task is an integer. Let D = maxTi ∈T {Di } be the demand of the largest task. Given a set of tasks T , the deadlines di of all tasks Ti ∈ T constitute a finite set {τ1 , τ2 , · · · , τL }, where L ≤ n, τ1 , · · · , τL ∈ Z + , and 0 = τ0 < · · · < τL = d. Let Di = {Ti,1 , Ti,2 , · · · , Ti,ni } PL denote the set of tasks with deadline τi , where i=1 ni = n (i ∈ [L]+ ). 4 The notation of this section is used in the entire paper and summarized in Table 1. Throughout this paper, we use i, j , m, l, or m0 as subscripts to index the element of different sets such as tasks and use t or t to index a time slot. 3 O PTIMAL S CHEDULE In this section, we identify a state under which C machines can be said to be optimally utilized by a set of tasks. We then propose a scheduling algorithm that achieves such an optimal state. Besides Table 1, the additional notation to be used in this section is summarized in Table 2. 3.1 Optimal Resource Utilization State In this paper, all tasks are denoted by a set T , and we denote by S ⊆ T an arbitrary subset of T ; all tasks of T with a deadline τl are denoted by Dl and we denote by Sl = S ∩Dl all tasks of S with a deadline τl (l ∈ [L]+ ). In this subsection, we define the maximum amount of workload of S that could be processed in a fixed time interval [τm +1, τL ] on C machines for all m ∈ [L − 1], where τL = d, i.e., the maximum deadline of tasks. Fig. 1. The green areas denote the maximum demand of Ti that need or could be processed in [τL−m + 1, τL ]. We first define the maximum amount of resource, denoted by λm (S), that could be utilized by S in [τL−m + 1, τL ] in an idealized case where there is an indefinite number of machines, i.e., C = ∞, for all m ∈ [L]+ . To define this, we clarify the maximum amount of resource that an individual task Ti can utilize in [τL−m + 1, τL ]. The basic constraints of malleable tasks with deadlines imply that: • • the deadline of Ti limits that Ti can only utilize the machines in [1, di ], and the parallelism bound limits that Ti can only utilize at most ki machines simultaneously at every time slot. The tasks with di ≤ τL−m cannot be executed in the interval [τL−m + 1, τL ]. Let us consider a task Ti with di ∈ [τL−m + 1, τL ]. The number of time slots available in [τL−m + 1, di ] is di − τL−m in the discrete case, and, also recall that leni the (minimum) execution time of Ti when it always utilizes the maximum number ki of machines throughout the execution. In the illustrative Fig. 1, the green area in the left (resp. right) subfigure denotes the maximum demand of a task, i.e., Di (resp. ki · (di − τL−m )), that could or need be processed in [τL−m + 1, τL ] in the case where the minimum execution time is such that leni ≤ di − τL−m (resp. leni > di − τL−m ). As a consequence of the observation above, λm (S) equals the sum of the maximum workload of every task JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 5 in S that could executed in [τL−m + 1, τL ] and is defined as follows. Definition 1. Initially, set λm (S) to zero for all m ∈ [L]. In the case where C = ∞ (i.e., the capacity constraint is ignored), for all m ∈ [L]+ , λm (S) is defined as follows: λm (S) ← λm (S) + βi , for every task Ti ∈ S , where βi is such that • • i.e., the size of all the colored areas in [τ0 + 1, τ2 ]. Generalizing the above process, we derived a recursive definition of λC m (S). Definition 2. In the case where C is finite (i.e., with the capacity constraint), for all m ∈ [L], the maximum amount of resource λC m (S) that could be utilized by S in [τL−m + 1, τL ] is defined by the following recursive procedure: if Ti ∈ S1 ∪ · · · ∪ SL−m where di ≤ τL−m , βi ← 0; if Ti ∈ SL−m+1 ∪ · · · ∪ SL where di ≥ τL−m + 1, as illustrated in Figure 1, – – in the case that leni ≤ di − τL−m , βi ← Dj ; otherwise, βi ← ki · (di − τL−m ). Here, βi represents the maximum workload of a task Ti that could be executed in [τL−m + 1, τL ]. Built on Definition 1, we move to the case where C is finite and define the maximum amount of resource λC m (S) that can be utilized by S on C machines in every [τL−m + 1, τL ], m ∈ [1, L]. • set λC 0 (S) to zero trivially; • C C set λ m (S) to the sum of λm−1 (S) and C min λm (S) − λm−1 (S), C · (τL−m+1 − τL−m ) . We finally state our definition that formalizes the concept of optimal utilization of C machines by a set S of malleable tasks with deadlines: Definition 3 (Optimal Resource Utilization State). We say that C machines are optimally utilized by a set of tasks S , if, for all m ∈ [L]+ , S utilizes λC m (S) resources in [τL−m + 1, d] on C machines. P C We define µC m (S) = Ti ∈S Di −λL−m (S) as the remaining (minimum) workload of S that needs to be processed after S has maximally utilized C machines in [τm + 1, τL ] for all m ∈ [L − 1]. Lemma 1 (Boundary Condition). If there exists a feasible schedule for S , the following inequality holds for all m ∈ [L − 1]: µC m (S) ≤ C · τm , which is referred to as boundary condition in this paper. Fig. 2. Derivation from the definition λm (S) to λC m (S). To help readers grasp the underlying intuition in the process of deriving λC m (S) from λm (S), we first illustrate this process in the case where L = 2 with the help of Fig. 2. Fig. 2 (left) illustrates the parameter λm (S) in Definition 1, where the green area denotes λ1 (S) and the green and blue areas together denote λ2 (S). As illustrated in Fig. 2 (right), due to the capacity constraint that C is finite, we have that (i) C · (τ2 − τ1 ) is the maximum possible workload that could be processed in [τ1 + 1, τ2 ] due to the capacity constraint, and λ1 (S) is the maximum available workload of S that needs to be processed in [τ1 + 1, τ2 ] due to the deadline and parallelism constraints. As a result, on C machines, the maximum workload λC 1 (S) of S that can be processed in [τ1 + 1, τ2 ] is the size of the green area in [τ1 + 1, τ2 ], i.e., λC 1 (S) = min{C · (τ2 − τ1 ), λ1 (S)} = C · (τ2 − τ1 ). (ii) After λC 1 (S) workload of S has been processed in [τ1 + 1, τ2 ], the remaining workload of S that needs to processed in [1, τ1 ] is λ2 (S) − λC 1 (S); the maximum workload that could be processed in [1, τ1 ] is C · τ1 due to the capacity constraint. As a result, λC 2 (S) is defined as follows: λC 2 (S) λC 1 (S) = + min{C · (τ1 − τ0 ), λ2 (S) − = min{C · (τ2 − τ0 ), λ2 (S)} = λ2 (S), λC 1 (S)} Proof. Recall the definition of λC L−m (S) in Definition 2. After S has maximally utilized the machines in [τm + 1, d] and been allocated the maximum amount of resource, i.e., λC L−m (S), if there exists a feasible schedule for S , the total amount of the remaining demands of S to be processed should be no more than the capacity C · τm in [1, τm ]. 3.2 Scheduling Algorithm In this section, we assume that S satisfies the boundary condition above, and, propose an algorithm LDF(S ) that achieves the optimal resource utilization state, producing a feasible schedule for S . 3.2.1 Overview of LDF(S ) Initially, for all Ti ∈ S and t ∈ [1, d], we set the allocation yi (t) to zero and LDF(S ) runs as follows: 1) 2) the tasks in S are considered in the non-increasing order of the deadlines, i.e., in the order of SL , SL−1 , · · · , S1 ; for a task Ti being considered, the algorithm Allocate-B(i), presented as Algorithm 2, is called to allocate Di resource to Ti under the constraints of deadline and parallelism bound. At a high level, we show in the following that, only if S satisfies the boundary condition and the resource utilization satisfies some properties upon every completion of AllocateB(·), all tasks in S will be fully allocated. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6 Fig. 3. The resource allocation state of Ti and the previous tasks S 0 respectively upon completion of Fully-Utilize(i), Fully-Allocate(i), and AllocateRLM(i, 1, t2 + 1) where L = m = 3: the blue area in the rectangle denotes the allocation to the previous tasks that satisfies Property 1 and Property 2 before executing Allocate-B(i) while the green area in the interval [1, τ3 ] denotes the allocation to Ti at every time slot. TABLE 2 Main Notation for the algorithms LDF(S ), Fully-Utilize(i), Fully-Allocate(i), and AllocateRLM(i, θ1 , x) Notation S Si λm (S) λC m (S) µC m (S) Ti S0 S0 t0 t1 t2 t0 t00 , t000 Explanation a set of tasks to be allocated by LDF(S ) and S⊆T the tasks of S with a deadline τi the maximum amount of resource that could be utilized by S in [τL−m + 1, τL ] in an idealized case where there is an indefinite number of machines, m ∈ [L]+ the maximum amount of resource that can be utilized by S on C machines in every [τL−m + 1, τL ], m ∈ [L]+ the remaining workload of S that needs to be processed after S has optimally utilized C machines in [τm + 1, τL ], i.e., P C µC m (S) = Ti ∈S Dj − λL−m (S), m ∈ [L − 1] a task that is being allocated by the algorithm LDF(S ); the actual allocation is done by Allocate-B(i) so far, all tasks that have been fully allocated by LDF(S ) and are considered before Ti S 00 = S 0 ∪ {Ti } a turning point defined in Property 2, with time slots respectively later than and no later than t0 having different resource utilization state similar to t0 , a turning point defined in Lemma 3 upon completion of Fully-Utilize(i) similar to t0 , a turning point defined in Lemma 6 upon completion of Fully-Allocate(i) the latest time slot in [1, τm ] with W (t0 ) > 0 a time slot that satisfies some property defined and only used in Section 3.2.3 Now, we begin to elaborate this high-level idea. In LDF(S ), when a task Ti is being considered, suppose that the allocated task Ti belongs to Sm and denote by S 0 ⊆ SL ∪ · · · ∪ Sm the tasks that have been fully allocated so far and are considered before Ti . Here, S satisfies the boundary condition and so do all its subsets including S 0 and S 0 ∪{Ti }. Before the execution of Allocate-B(i), we assume that the resource utilization satisfies the following two properties: Recall the optimal resource utilization state in Definitions 3, and the first property is that such an optimal resource utilization state of C machines is achieved by the current allocation to S 0 . 0 Property 1. For all l ∈ [L]+ , S 0 is allocated λC l (S ) resource in C 0 [τL−l + 1, d] where λl (S ) is defined in Definition 2. The second property is that a stepped-shape resource uti- lization state is achieved in [1, τm ] by the current allocation to S 0 . Property 2. If there exists a time slot t ∈ [1, τm ] such that W (t) > 0, let t0 be the latest slot in [1, τm ] such that W (t0 ) > 0; then we have W (1) ≥ W (2) ≥ · · · ≥ W (t0 ). If Property 1 and Property 2 hold, we will show in Section 3.2.2 and 3.2.3 that, there exists an algorithm AllocateB(i) such that, upon completion of Allocate-B(i), the following two properties are satisfied: Property 3. Ti is fully allocated. Property 4. The resource allocation to S 0 ∪ {Ti } still satisfies Property 1 and Property 2. Due to the existence of the above Allocate-B(i), only if S satisfies the boundary condition, S can be fully allocated by LDF(S ). The reason for this can be explained by induction. When the first task Ti in S is considered, S 0 is empty, and, before the execution of Allocate-B(i), Property 1 and Property 2 holds trivially. Further, upon completion of Allocate-B(i), Ti will be fully allocated by Allocate-B(i) due to Property 3, and Property 4 still holds. Then, assume that S 0 that denotes the current fully allocated tasks is nonempty and Property 1 and Property 2 hold; the task Ti being considered by LDF(S ) will still be fully allocated and Property 3 and Property 4, upon completion of AllocateB(i). Hence, all tasks in S will be finally fully allocated upon completion of LDF(S ). In the rest of this subsection, we will propose an algorithm Allocate-B(i) mentioned above such that, upon completion of Allocate-B(i), Property 3 and Property 4 holds, if, before the execution of Allocate-B(i), the resource allocation to S 0 satisfies Property 1 and Property 2 hold. Then, we immediately have the following proposition: Proposition 1. If S satisfies the boundary condition, LDF(S ) will produce a feasible schedule of S on C machines. Overview of Allocate-B(i). The construction of AllocateB(i) will proceed with two phases. In the first phase, we introduce what operations are feasible to make Ti fully allocated Di resource under Property 1 and Property 2. We will use two algorithms Fully-Utilize(i) and Fully-Allocate(i) to describe them, and the sketch of this phase is as follows: • From the deadline di towards earlier time slots, Fully-Utilize(i) makes Ti fully utilize the maximum JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 7 amount of machines available at every slot. Upon its completion, the resource allocation state is illustrated in Fig. 3 (left) and will be described in Lemma 3. If Ti is not fully allocated yet, as illustrated in Fig. 3 (middle), Fully-Allocate(i) transfers the allocation of the previous tasks S 0 at the time slots closest to di to the latest slots in [1, di ] that have idle machines, so that, ki machines are finally allocated to Ti at each of these slots closest to di ; as a result, Ti is fully allocated. Upon completion of Fully-Utilize(i), in the other case that the total allocation to S 00 is < C · τm , even for Ti , it may not be fully allocated. In this case, there exists a slot t ∈ [1, τm ] such that W (t) > 0, and let t1 denote the latest such time slot in [1, τm ]. In Fully-Utilize(i), upon completion of the allocation to Ti at t ∈ [1, t1 ], if Ti has not been fully allocated yet, it is allocated ki machines at t, i.e., yi (t) = ki , and these allocations in [1, t1 ] are non-decreasing, i.e., Upon completion of Fully-Allocate(i), the resource allocation state may not satisfy Property 1 and Property 2, as illustrated by Fig. 3 (middle). We therefore propose an algorithm AllocateRLM(i, η1 , x) in the second phase: Before executing Fully-Utilize(i), the numbers of idle machines have a stepped shape, i.e., W (1) ≥ · · ·P ≥ W (t0 ) by Property 2, where W (t) = C − W (t) = C − Tj ∈S 0 yj (t). Upon its completion, with yi (t) machines occupied by Ti , we conclude that • • the allocation of the previous tasks at every slot t closest to the deadline is again transferred to the latest slots that have idle machines, and, the allocation of Ti in the earliest slots is transferred to t; the final resource allocation state is illustrated in Fig. 3 (right). Following the above high-level ideas, the details of the first and second phases are respectively presented in Section 3.2.2 and Section 3.2.3. 3.2.2 Phase 1 Now, we introduce Fully-Utilize(i) and Fully-Allocate(i) formally. Before their execution, recall that we assume in the last subsection Ti ∈ Sm ; the allocation to the previously allocated tasks in S 0 satisfies Properties 1 and 2. The whole set of tasks S to be scheduled satisfies the boundary condition where S 0 S . Initially, set yi (t) to zero for all time slots, and, FullyUtilize(i) operates as follows: for every time slot t from the deadline di to 1, set P i yi (t) ← min{ki , Di − dt=t+1 yi (t), W (t)}. Pdi Here, ki is the parallelism bound, Di − t=t+1 yi (t) is the remaining workload to be processed upon completion of its allocations at slots t + 1, · · · , di , and W (t) is the number Pdi of machines idle at t; specially, y (t) is set to 0, t=di +1 i representing the allocation to Ti is zero before the allocation begins. Their minimum denotes the maximum amount of machines that Ti can or needs to utilize at t after the allocation to Ti at slots t + 1, · · · , di . Before executing Fully-Utilize(i), the resource allocation to the previous tasks S 0 satisfies Property 1. Its execution does not change the previous allocation to S 0 . Let • yi (1) ≤ yi (2) ≤ · · · ≤ yi (t1 ). Lemma 3. Upon completion of Fully-Utilize(i), in the case that the total allocation to S 00 is < C · τm , • for all t ∈ [1, t1 ], if the total allocation of Ti in [t, di ] is Pdi < the workload of Ti , i.e., Di − t=t yi (t) > 0, we have yi (t) = ki ; • the numbers of idle/unallocated machines in [1, t1 ] have a stepped shape, i.e., W (1) ≥ · · · ≥ W (t1 ) > 0. With the current resource allocation state shown in Lemma 3, we are enabled to propose the algorithm FullyAllocate(i) to make Ti fully allocated. Deducting the current resource allocated to Ti , let Ω denote the remaining workload of Ti to be allocated more resource, i.e., P Ω = Di − t≤di yi (t). For every slot t ∈ [1, t1 ], the number yi (t) of machines allocated to Ti at t is ki in the case that Ω > 0 by Lemma 3. The total workload Di is ≤ ki · di , and, with the parallelism bound, Fully-Allocate(i) considers each slot t from di towards t1 + 1 and operates as follows repeatedly at each t until Ω = 0: 1) 2) 3) S 00 = S 0 ∪ {Ti }. Since di = τm , the workload of Ti can only be processed in [1, τm ]; the maximum workload of S 00 that could be processed in [τm + 1, τL ] still equals its counterpart when S 0 is considered. We come to the following conclusion in order to not violate the boundary condition: Lemma 2. Upon completion of Fully-Utilize(i), all tasks of S would have been fully allocated in the caseP that theP total allocaτm tion to S 00 in [1, τm ] is C ·τm , i.e., C ·τm = Tj ∈S 00 t=1 yj (t). Proof. See the appendix for detailed proof. 4) ∆ ← min{ki − yi (t), Ω}. Notes. ki − yi (t) is the maximum number of additional machines that could be utilized at T with its previous allocation yi (t). Call Routine(∆, 1, 0, t), presented as Algorithm 1. Notes. Routine(·) aims to increase the number of available machines W (t) at t to ∆ by transferring the allocation of other tasks to an earlier time slot. Allocate W (t) more machines to Ti at t: yi (t) ← yi (t) + W (t), and, Ω ← Ω − W (t). Notes. Ω denotes the currently remaining workload to be processed; in this iteration, if Ω > 0 currently, ∆ = ki − yi (t) and the allocation yi (t) of Ti at t becomes ki . t ← t − 1. Now, we explain the existence of Ti0 in line 12 of Routine(·) and the reason why Ti will be finally fully allocated by Fully-Allocate(i). The only operation that changes the allocation to Ti occurs at the third step of FullyAllocate(i). Hence, we have JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 Algorithm 1: Routine(∆, η1 , η2 , t) 1 2 3 4 5 6 7 8 9 10 11 12 13 while W (t) < ∆ do t0 ← the current time slot earlier than and closest to t so that W (t0 ) > 0; if η1 = 1 then if there exists no such t0 then f lag ← 1, break; else 0 if t0 ≤ tth m , or there exists no such t then f lag ← 1, break; if η2 = 1 then Pt0 −1 if t=1 yi (t) ≤ W (t) then f lag ← 1, break; let i0 be a task such that yi0 (t) > yi0 (t0 ); yi0 (t) ← yi0 (t) − 1, yi0 (t0 ) ← yi0 (t0 ) + 1; Lemma 4. Fully-Allocate(i) never decreases the allocation yi (t) to Ti at any time slot t ∈ [1, di ] during its execution, compared with the yi (t) just before executing Fully-Allocate(i). We could also prove by contradiction that Lemma 5. When Routine(∆, 1, 0, t) is called, the task Ti0 in line 12 always exists if (i) the condition in line 4 is false, (ii) yi (t0 ) = ki , and (iii) yi (t) < ki and W (t) = 0. Proof. See the Appendix for the detailed proof. At each iteration of Fully-Allocate(i), if there exists a t0 such that W (t0 ) > 0 in the loop of Routine(·), with Lemmas 3 and 4, we have yi (t0 ) = ki . Since Ω > 0 and yi (t) < ki , when Routine(·) is called, we have W (t) = 0; otherwise, this contradicts Lemma 3. With Lemma 5, we will conclude that the task Ti0 in line 11 exists when it is called by Fully-Allocate(i). In addition, the operation at line 12 of Routine(·) does not change the total allocation to Ti0 , and violate the parallelism bound ki0 of Ti0 since the current yi0 (t0 ) is no more than the initial yi0 (t). Proposition 2. Upon completion of Fully-Allocate(i), the task Ti is fully allocated. Proof. Fully-Allocate(i) ends up with one of the following three events. The first is that the condition in line 4 of Routine(·) is true. Then, with Lemma 2, all tasks in S has been fully allocated. If the first event doesn’t happen, the second is Ω = 0 and Ti has been fully allocated. If the first and second events don’t happen, the third occurs after finishing the iteration of Fully-Allocate(i) at time slot t1 + 1; then, there is a slot t0 in [1, t1 + 1] that are not fully utilized. As a result, we have that Ti has been fully allocated; otherwise, Ω > 0, which implies yi (t1 + 1) = ki , and we have yi (t) = ki for all t ∈ [di ]+ due to Lemma 3, which contradicts Ω > 0. Finally, the theorem holds. Upon completion of Fully-Utilize(i), the resource allocation feature is described in Lemma 3 and illustrated in Fig. 3 (left). Built on this, Fully-Allocate(i) considers every slot from di to t1 +1; as illustrated in Fig. 3 (middle) and roughly 8 explained there, upon completion of Fully-Allocate(i), the resource allocation feature is described as follows. Lemma 6. Upon completion of Fully-Allocate(i), if there exists a t ∈ [1, τm ] such that W (t) > 0, let t2 be the latest such slot: • • for all t ∈ [1, t2 ], if the total allocation of Ti in [t, di ] is Pdi < Di (i.e., Di − t=t yi (t) > 0), we have yi (t) = ki ; the numbers of available machines in [1, t2 ] have a stepped shape, i.e, W (1) ≥ · · · ≥ W (t2 ) > 0. Here t2 ≥ t1 . Proof. See the Appendix for the formal proof. 3.2.3 Phase 2 Now, we introduce AllocateRLM(i, η1 , x). Recall that t0 always denotes the slot closest to but earlier than τm (i.e., the latest slot in [1, τm ]) such that W (t0 ) > 0 and, before executing AllocateRLM(·), t0 = t2 due to Lemma 6. The resource allocation feature before executing AllocateRLM(i, η1 , x) is described in Lemma 6 and illustrated in Fig. 3 (middle); the underlying intuition of AllocateRLM(i, η1 , x) is described in Section 3.2.1 and, upon its completion, the resource allocation feature is illustrated in Fig. 3 (right). Formally, AllocateRLM(i, η1 , x) considers each slot t from di to x and operates as follows repeatedly each t Pat t−1 until the total allocation of Ti in [1, t − 1], i.e., t=1 yi (t), equals zero, where η1 = 1 and x = t2 + 1 in this section: Pt−1 1) ∆ ← min{ki − yi (t), t=1 yi (t)}. Notes. ∆ denotes the maximum allocation of Ti before t that can be transferred to t with the parallelism constraint. 2) if ∆ = 0, go to the step 5; otherwise, execute the steps 3-5. 3) set f lag ← 0 and call Routine(∆, η1 , 1, t). Notes. Routine(·) aims to increase the number W (t) of available machines at t to ∆. With Lemma 6, the slots t0 earlier than but closest to t2 + 1 in Routine(·) will become fully utilized one by one and, together with the next step 4, upon completion of the iteration at t, for all t ∈ [t0 + 1, di ], W (t) = 0. 4) set θ ← W (t). Allocate θ more machines to Ti : yi (t) ← yi (t) + W (t), and reduce the allocations of Ti at the earliest slots by θ: in particular, let t00 be such a slot that Pt00 −1 Pt00 t=1 yi (t) ≥ θ , and execute t=1 yi (t) < θ and the following operations: Pt00 −1 a) set θ ← θ − t=1 yi (t), and, for every t ∈ [1, t00 − 1], yi (t) ← 0; b) yi (t00 ) ← yi (t00 ) − θ. 5) Notes. The number of idle machines at t becomes zero again, i.e., W (t) = 0. The allocation yi (t) of Ti at every t ∈ [1, t00 − 1] is zero. if Routine(∆, η1 , 1, t) does not change the value of f lag , i.e., f lag = 0, t ← t − 1; otherwise, exit AllocateRLM(i, η1 , x). Here, at each slot t, when Routine(·) is called, ∆ > 0, and yi (t) < ki . Further, we have W (t) = 0; otherwise, this JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 contradicts Lemma 6. Hence, with Lemma 5, we conclude that the task Ti0 in line 12 of Routine(·) exists. Based on our notes in the description of AllocateRLM(·), we conclude that Proposition 3. Upon completion of AllocateRLM(i, 1, x) where x = t2 + 1, the final allocation to S 00 can guarantee that Property 4 holds where S 00 = S 0 ∪ {Ti }. Proof. Fully-Utilize(i), Fully-Allocate(i) and AllocateRLM(i, η1 , x) never change the allocation at any slot in [τm + 1, d]. AllocateRLM(i, 1, x) ends up with one of the following four events. The first event occurs when the condition in line 4 of Routine(·) is true; then, the proposition holds trivially since all the slots t ∈ [1, di ] have been fully utilized, i.e., W (t) = 0. If the first event doesn’t occur, the second Pt−1event is that, for the first time, at some t ∈ [t2 + 1, di ], t=1 yi (t) = 0; then, we have that, Ti is fully allocated Di resource in [t, di ] ⊆ [t2 + 1, di ]. The third event occurs when the condition in line 10 of Routine(·) is true. In the following, we will analyze the resource utilization state when either of the second and third events occurs. Recall that t0 is defined in line 2 of Routine(·) where each slot in [t0 + 1, di ] will be fully utilized; when the second or third event occurs, all the slots in [t0 +1, di ] are fully utilized, i.e., W (t) = 0, for all t ∈ [t0 + 1, di ]. Upon completion of the iteration of AllocateRLM(·) at t when the third event occurs, or, at t+1 when the second event occurs, we have the following three points, in contrast to the allocation achieved just before executing Allocate-B(i), (i) (ii) (iii) (iv) t0 ∈ [1, t2 ] and the allocation to the previous tasks S 0 at every t ∈ [1, t0 − 1] is still the allocation achieved before executing Allocate-B(i); Pt0 −1 0 t=1 yi (t) = 0, i.e., the allocation to Ti in [1, t − 1] is zero and Ti is fully allocated Di resource in [t0 , di ]; the allocation to S 0 at t0 is not decreased; the allocation to Ti at t0 does not change. Noticing the above resource allocation state in [1, di ] where di = τm , since Property 2 holds before executing AllocateB(i), we conclude that Property 2 still holds upon its completion where t0 = t0 . Without loss of generality, assume that t0 ∈ [τm0 −1 + 1, τm0 ] for some m0 ∈ [m]+ . Then, all the slots in [τm0 + 1, di ] have been fully utilized and the allocation in [τm + 1, d] does not change at all; hence, we have that every interval [τl + 1, d], where m0 ≤ l ≤ L, is optimally utilized by S 0 ∪ {Ti } due to Property 1. Since the total allocation to S 0 in [1, τm0 −1 ] isn’t changed by Allocate-B(i) if m0 − 1 > 0, due to Property 1, the interval [τm0 −1 +1, d] is still optimally utilized by S 0 and the task Ti is fully allocated Di resource in this interval; hence, it is still optimally utilized by S 0 ∪ {Ti }. Further, every interval [τl + 1, d] is also optimally utilized where 1 ≤ l ≤ m0 − 1. Hence, the theorem holds. If the first three events don’t occur, the fourth event occurs upon completion of the iteration of AllocateRLM(·) at t = t2 + 1, i.e., the last iteration. In this case, we have that the conditions in lines 4 and 10 of Routine(·) are always false where at each iteration of AllocateRLM(·) there always exists such t0 (defined in line 2 of Routine(·) with W (t0 ) > 0); due to the current resource allocation state, we conclude that, at each of the slots in [t2 + 1, di ], Ti is allocated ki machines. 9 Upon completion of AllocateRLM(·), there exists a t0 defined in line 2 of Routine(·), and, let t000 denote the earliest slot at which yi (t000 ) 6= 0 where t000 ≤ t0 ; then, similar to our conclusion in the second and third events, we have that (i) (ii) (iii) the first point here is the same as the first and third points in the last paragraph; Ti is fully allocated Di resource in [t000 , di ]; if t000 > t0 , the allocation to Ti at each t ∈ [t000 + 1, t0 ] does not change and yi (t) = ki due to Lemma 6, and, the allocation to Ti at t000 is greater than zero. Similar to our analysis in the last paragraph for other events, we conclude that the proposition holds. Proposition 2 and Proposition 3 finish to show that Allocate-B(i) satisfies Property 3 and Property 4 and hence completes the proof of Proposition 1. We finally analyze the time complexity of Allocate-B(i). Lemma 7. The time complexity of Allocate-B(·) is O(n). Proof. See the Appendix for the proof. Algorithm 2: Allocate-B(i) 1 2 3 Fully-Utilize(i); Fully-Allocate(i); AllocateRLM(i, 1, t2 + 1); Since LDF(S ) considers a total of n tasks, its complexity is O(n2 ) with Lemma 7. Finally, we draw a main conclusion in this section from Lemma 1 and Proposition 1: Theorem 1. A set of tasks S can be feasibly scheduled and be completed by their deadlines on C machines if and only if the boundary condition holds, where the feasible schedule of S could be produced by LDF(S ) with a time complexity O(n2 ). In other words, if LDF(S ) cannot produce a feasible schedule for S on C machines, then S cannot be successfully scheduled by any algorithm; as a result, LDF(S ) is optimal. The relationships between the various algorithms of this paper are illustrated in Fig. 4 where GreedyRLM will be introduced in the next section. Remarks. We are inspired by the GreedyRTL algorithm [3] in the construction of LDF(·). In terms of the two algorithms themselves, LDF(·) considers tasks in the decreasing order of deadlines while the order is determined by the marginal values in GreedyRTL(·). In both algorithms, the allocation to a task Ti is considered from di to 1 (once in GreedyRTL, and possibly three times in LDF(·)); to make time slots t closest to the deadline of a task Ti being considered fully utilized, the key operations are finding a time slot t0 earlier than t such that there exists a task Ti0 with yi0 (t) > yi0 (t0 ) when W (t), and transferring a part of the allocation of Ti0 at t to t0 . In GreedyRTL(·), the existence of Ti0 requires that (i) the number W (t0 ) of available machines at t0 is ≥ k and (ii)2 W (t) < ki ; as a result, before doing any allocation to Ti at t, the existence could be proved by contradiction. In LDF(·), to achieve the optimality of resource utilization, 2. The particular condition there is W (t) Pdi y (t)}. t=t+1 i < min{ki , Di − JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10 Fig. 4. Relationship among Algorithms: for A → B , the blue and green arrows denote the relations that the algorithm A calls B , and, the algorithm B is executed upon completion of A. one requirement for such existence is relaxed to be that the number of available machines at t0 is ≥ 1. The existence is guaranteed by (i) first make every time slot from di to 1 fully utilized, as what Fully-Utilize(i) does, and (ii) a steppedshape resource utilization state in [1, di ] upon completion of the allocation to the last task, as described in Property 2. a total of C · d01 such tasks, where  ∈ (0, 1) is small enough; (iii) for all Tj ∈ D2 , vj0 = 1, kj = 1 and Dj = d02 − d01 + 1. Greedy will always fully allocate resource to the tasks in D1 , with all the tasks in D2 rejected to be allocated any resource. The performance guarantee of Greedy will be no C·d01 more than C·[(1+)(d0 −1)+1·(d 0 −d0 +1)] . Further, with  → 0, 1 2 1 this performance guarantee approaches 4 A PPLICATIONS : PART I In this section, we illustrate the application of the results in Section 3 to the greedy algorithm for social welfare maximization. In terms of the maximization problem, the general form of a greedy algorithm is as follows [22], [23]: it tries to build a solution by iteratively executing the following steps until no item remains to be considered in a set of items: (1) selection standard: in a greedy way, choose and consider an item that is locally optimal according to a simple criterion at the current stage; (2) feasibility condition: for the item being considered, accept it if it satisfies a certain condition such that this item constitutes a feasible solution together with the tasks that have been accepted so far under the constraints of this problem, and reject it otherwise. Here, an item that has been considered and rejected will never be considered again. The selection criterion is related to the objective function and constraints, and is usually the ratio of ’advantage’ to ’cost’, measuring the efficiency of an item. In the problem of this paper, the constraint comes from the capacity to hold the chosen tasks and the objective is to maximize the social welfare; therefore, the selection criterion here is the ratio of the value of a task to its demand that will refer to as the marginal value of this task. Given the general form of greedy algorithm, we define a class GREEDY of algorithms that operate as follows: 1) 2) Considers the tasks in the non-increasing order of the marginal value; assume without loss of generality that v10 ≥ v20 ≥ · · · ≥ vn0 ; Denoting by A the set of the tasks that have been accepted so far, a task Ti being considered is accepted and fully allocated iff there exists a feasible schedule for A ∪ {Ti }. In the following, we refer to the generic algorithm in GREEDY as Greedy. Proposition 4. The best performance guarantee that a greedy algorithm in GREEDY can achieve is s−1 s . Proof. Let us consider a special instance: (i) let Di = {Tj ∈ T |di = d0i }, where i ∈ [2]+ , d02 and d01 ∈ Z + , and d02 > d01 ; (ii) for all Tj ∈ D1 , vj0 = 1 + , Dj = 1, kj = 1, and, there is d02 d02 −d01 +1 s−1 s d01 −1 d02 . and = When s= Hence, the proposition holds. 4.1 d02 d01 d02 . In this instance, → +∞, d01 d02 = s−1 s . Notation Greedy will consider tasks sequentially. The first considered task will be accepted definitely and then it will use to the feasibility condition to determine whether or not to accept or reject the next task according to the current available resource and the characteristics of this task. To describe the process under which Greedy accepts or rejects tasks, we define the sets of consecutive accepted (i.e., fully allocated) and rejected tasks A1 , R1 , A2 , · · · . Specifically, let Am = {Tim , Tim , · · · , Tjm } be the m-th set of the adjacent tasks that are accepted by Greedy where i1 = 1 while Rm = {Tjm +1 , · · · , Tim+1 −1 } is the m-th set of the adjacent that are rejected tasks following the set Am , where m ∈ [K]+ for some integer K . Integer K represents the last step: in the K -th step, AK 6= ∅ and RK can be empty or non-empty. We also denote by cm the maximum deadline of all rejected tasks of ∪m l=1 Rl , i.e., cm = maxTi ∈∪m {di }, l=1 Rl and by c0m the maximum deadline of ∪m l=1 Al , i.e., c0m = maxTi ∈∪m {di }. l=1 Al While the tasks in Am ∪ Rm are being considered, we refer to Greedy as being in the m-th phase. Before the execution of Greedy, we refer to it as being in the 0-th phase. Upon completion of the m-th phase of Greedy, we define a threshold parameter tth m such that (i) (ii) if cm ≥ c0m , set tth m = cm , and 0 if cm < c0m , set tth m to any time slot in [cm , cm ]. m Here, di ≤ tth m for all Ti ∈ ∪j=1 Rj . For ease of the th subsequent exposition, we let t0 = 0 and tth K+1 = d. We also add a dummy time slot 0 but the task Ti ∈ T can not get any resource there, that is, yi (0) = 0 forever. We also let A0 = R0 = AK+1 = RK+1 = ∅. Besides the notation in Section 2, the additional key notation used for this section is also summarized in Table 3. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 TABLE 3 Main Notation for Section 4 Notation A1 , R1 , A2 , · · · , RK cm c0m tth m 4.2 Explanation the sets of consecutive accepted (i.e., fully allocated) and rejected SK tasks by Greedy where m=1 Am ∪ Rm = T the maximum deadline of all rejected tasks of ∪m l=1 Rl the maximum deadline of ∪m l=1 Al a threshold parameter such that (i) if cm ≥ c0m , set 0 th tth m = cm , and (ii) if cm < cm , set tm to any time slot in [cm , c0m ]; when introducing GreedyRLM, it will be set to a specific value A New Algorithmic Analysis We will show that as soon as the resource allocation done by Greedy satisfies some features, its performance guarantee can be deduced immediately, i.e., the main result of this subsection is Theorem 2. For all m ∈ [K]+ , upon completion of Greedy, we define the following two features that we want the allocation to ∪m j=1 Aj to satisfies: Sm Feature 1. The total allocation to j=1 Aj in [1, tth m ] is at least r · C · tth m , where r ∈ [0, 1]. Sm Feature 2. For each task Ti ∈ j=1 Aj , its maximum amount   of demand that can be processed in each tth l + 1, d is processed where m ≤ l ≤ K , i.e., di P t=tth l +1  yi (t) = min Di , ki (di − tth l ) . Theorem 2. If Greedy achieves a resource allocation structure that satisfies Feature 1 and Feature 2 for all m ∈ [K]+ , it gives an r-approximation to the optimal social welfare. 11 Lemma 9. The following schedule achieves an upper bound of the optimal social welfare of the MSW-II problem, ignoring the capacity constraint: 1) 2) Proof. See the appendix for the detailed proof. Lemma 10. For all m ∈ [K]+ , the total value generated by 0 executing the allocation to T10 , · · · , Tm is no larger than 1−r r times the total value generated by the allocation to ∪m l=1 Al in [1, tth m ]. Proof. See the appendix for the detailed proof. 0 In the case that m = K , the total value from T10 , · · · , TK is no larger than 1−r times the total value from the allocar th tion to ∪K l=1 Al in [1, tK ]. Hence, the total value generated 1 by the schedule in Lemma 9 is no larger than 1 + 1−r r = r times the total value generated by the allocation to all tasks of A1 , · · · , AK . By Lemmas 9 and 8, Theorem 2 holds. 4.3 Lemma 8. The optimal social welfare of the MSW-II problem is an upper bound of the optimal social welfare of the MSW-I problem. Proof. See the appendix for the detailed proof. Due to Feature 1, Feature 2, and the fact that the marginal 0 value of Tm is no larger than the ones of the tasks of ∪m l=1 Al , we derive the following two lemmas: Optimal Algorithm Design We now introduce the executing process of the optimal greedy algorithm GreedyRLM, presented as Algorithm 3: (1) (2) In the rest of Section 4.2, we prove Theorem 2; we will first provide an upper bound of the optimal social welfare. Proof Overview. We refer to the original problem of scheduling A1 , R1 , · · · , AK , RK on C machines to maximize the social welfare as the MSW-I problem. In the following, we define a relaxed version of the MSW-I problem. Assume that R0m consists of a single task 0 whose deadline is tth Tm m , whose size is infinite, and whose marginal value is the largest one of the tasks in Rm , denoted by v 0m ; here, different from the task in Rm , we assume 0 that there is no parallelism constraint on Tm whose bound 0 is C . In addition, partial execution of the task Tm and the tasks of A1 , · · · , AK can yield linearly proportional Pdi value, e.g., if a task Ti ∈ Al is allocated t=1 yi (t) < Di Pdi resource by its deadline, a value ( t=1 yi (t)/Di ) · vi will still be added to the social welfare. We refer to the problem of scheduling A1 , R01 , · · · , AK , R0K on C machines as the MSW-II problem. for all tasks of A1 , · · · , AK , their allocation is the same as the one achieved by Greedy with Features 1 and 2 satisfied; 0 for all m ∈ [K]+ , execute a part of task Tm such that  th the amount of processed workload in t + 1, tth m−1 m is
th (1 − r) · tth m − tm−1 · C . (3) considers the tasks in the non-increasing order of the marginal value. in P the m-th phase, for a task Ti being considered, if t≤di min{W (t), ki } ≥ Di , call Allocate-A(i), presented as Algorithm 4, where the details on FullyUtilize(i) and AllocateRLM(i, 0, tth m + 2) can be found in Section 3.2.2 and Section 3.2.3. if the allocation condition is not satisfied, set the threshold parameter tth m of the m-th phase that is defined by lines 8-15 of Algorithm 3. When the condition in line 5 of GreedyRLM is true, every accepted task can be fully allocated Di resource using FullyUtilize(i). The reason for the existence of Ti0 in Routine(·) is the same as the reason when introducing LDF(S ) since W (t0 ) > 0. Proposition 5. GreedyRLM gives an s−1 s -approximation to the optimal social welfare with a time complexity of O(n2 ). Now, we begin to prove Proposition 5. The time complexity of Allocate-A(i) depends on AllocateRLM(·). Using the time complexity analysis of AllocateRLM(·) in Lemma 7, we get that AllocateRLM(·) has a time complexity of O(n), and, the time complexity of GreedyRLM is O(n2 ). Due to Theorem 2, in the following, we only need to prove that Features 1 and 2 holds in GreedyRLM where r = s−1 s , which is given in Propositions 6 and 7. The utilization of GreedyRLM is derived mainly by analyzing the resource allocation state when a task Ti cannot be fully allocated (the condition in line 5 of GreedyRLM is not satisfied), and we have that JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 12 exceed Di . Finally, upon completion of the whole execution of Allocate-A(·), we have that Algorithm 3: GreedyRLM Input : n tasks with typei = {vi , di , Di , kj } Output: A feasible allocation of resources to tasks 1 2 3 4 5 6 7 8 9 • initialize: yi (t) ← 0 for all Ti ∈ T and 1 ≤ t ≤ d, m = 0, tth m = 0; sort tasks in the non-increasing order of the marginal values: v10 ≥ v20 ≥ · · · ≥ vn0 ; i ← 1; while i ≤ n do P if t≤di min{W (t), ki } ≥ Di then Allocate-A(i); // in the (m + 1)-th phase else if Ti−1 has ever been accepted then m ← m + 1; // in the m-th phase, the allocation to Am was completed; the first rejected task is Tjm = Ti 10 11 P while t≤di+1 min{W (t), ki+1 } < Di+1 do i ← i + 1; /* the last rejected task is Tim+1 −1 = Ti and Rm = {Tjm , · · · , Tim+1 −1 } 12 13 14 15 16 */ if cm ≥ c0m then tth m ← cm ; else set tth m to time slot just before the first time slot t with W (t) > 0 after cm or to c0m if there is no time slot t with W (t) > 0 in [cm , c0m ]; i ← i + 1; Algorithm 4: Allocate-A(i) 1 2 3 Fully-Utilize(i); if di ≥ tth m + 2 then AllocateRLM(i, 0, tth m + 2) where t2 = t1 that are defined in Section 3.2.2; Proposition 6. Upon completion of GreedyRLM, Feature 1 holds in which r = s−1 s . Proof. See the Appendix for the detailed proof. In GreedyRLM, when a task Ti is accepted (lines 5 and 6), Allocate-A(·) is called to make it fully allocated. In AllocateA(·), Fully-Utilize(·) and AllocateRLM(·) are sequentially called; both of them consider time slots t from the deadline towards earlier ones: (i) Fully-Utilize(·) makes Ti utilize the remaining (idle) machines at t, and it does not change the allocations of the previous tasks; (ii) at every t, if Ti does not utilize the maximum number of machines it can utilize (i.e., yi (t) < ki ), AlloacteRLM(·) (a) transfers the allocations of the previous allocated tasks to an earlier slot that is closest to t but not fully utilized (i.e., with idle machines), and (b) increases the allocation of Ti at t to the maximum (i.e., ki ) and, correspondingly reduce the equal allocations at the earliest slots, ensuring the total allocation to Ti does not the number of allocated machines at each slot does not decrease, in contrast to that amount just before executing AllocateA(·). For every accepted task Ti ∈ Am , upon completion of Allocate-A(·), time slot tth m + 1 is not fully utilized by th the definition of tth , i.e., W (t m m + 1) > 0. Further, we have that whenever Allocate-A(·) completes the allocation to a previous task Ti ∈ Am0 where m0 < m, tth m + 1 is also not fully utilized then. Based on this, we draw the following conclusion. Lemma 11. Due to the definition of tth m , we have for all m ≤ j ≤ K that Sm (1) [tth j + 1, d] is optimally utilized by l=1 Al upon completion of the allocation to it using Allocate-A(i); (2) for the total amount of the allocations to Ti in the interval [tth j + 1, d] just upon completion of Allocate-A(i), it does not change upon completion of GreedyRLM. Proof. We first prove the first point. Given a m0 ∈ [m]+ , for every Ti ∈ Am0 , upon completion of Allocate-A(i), W (tth j + 1) > 0 for all j ∈ [m, K]; based on this, we conclude that, Pdi in the case where di ≥ tth y (t) = Di if j + 1, either t=tth +1 i j th di − tth j > leni or yi (t) = ki for all t ∈ [tj + 1, di ] otherwise. The reason for this conclusion is similar to our analysis for the fourth event when proving Proposition 3; here, there always exists a slot tth j + 1 that is not fully utilized, i.e., th W (tj + 1) > 0, leading to that the t0 defined in line 2 of Routine(·) always exists where W (t0 ) > 0. Now, we prove the second point in Lemma 11. For every l ∈ [m0 , K], we observe the subsequent execution of Allocate-A(·) whose input is a task in Al and could conclude that, 1) 2) upon its completion, the allocations to Ti in [1, tth l ] are still the ones before executing Allocate-A(·); Allocate-A(·) can only change the allocations of Ti in the time range [tl0 +1, tl0 +1 ] where l0 ∈ [l, K] and the total amount of allocations in [tl0 + 1, tl0 +1 ] upon its completion is still the amount before its execution. As a result, we have that, upon completion of Allocate-A(i), every subsequent execution of Allocate-A(·) never change the total amount of allocations of Ti in [tl00 + 1, tl00 +1 ] for all l00 ∈ [m, K]. In the following, it suffices to prove the above two points. In the execution of Allocate-A(·), Fully-Utilize(·) is first called and it does not change the allocation to the previous tasks; then, AllocateRLM(·, 0, tth l ) is called in which only Routine(·) (i.e., its lines 12 and 13) in the step 3 can change the allocation to the previous tasks including Ti . In lines 12 and 13, a previous task Ti0 is found to change its allocations at t and t0 ; here, t0 is defined in lines 2 and 7 of 0 Routine(·) and tth l < t < t. As a result, Allocate-A(·) cannot change the allocations of the previous tasks in [1, tth l ]; for all th 0 t ∈ [tth l0 + 1, tl0 +1 ] where l ∈ [l, K], during the execution of the iteration of AllocateRLM(·) at t, we have t0 > tth l0 . Hence, the change to the allocations of the previous tasks can only th happen in the interval [tth l0 + 1, tl0 +1 ]. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 From the first and second points of Lemma 11, we could conclude that   Proposition 7. Given a m ∈ [1, K], tth l + 1, d is optimally Sm utilized by every task Ti ∈ j=1 Aj for all l ∈ [m, K]. 5 A PPLICATIONS : PART II In this section, we illustrate the applications of the results in Section 3 to (i) the dynamic programming technique for social welfare maximization, (ii) the machine minimization objective, and (iii) the objective of minimizing the maximum weighted completion time. 5.1 Dynamic Programming For any solution, there must exist a feasible schedule for the tasks selected to be fully allocated by this solution. So, the set of tasks in an optimal solution satisfies the boundary condition by Lemma 1. Then, to find the optimal solution, we only need address the following problem: if we are given C machines, how can we choose a subset S of tasks in T such that (i) this subset satisfies the boundary condition, and (ii) no other subset of selected tasks achieves a better social welfare? This problem can be solved via dynamic programming (DP). To propose a DP algorithm, we need to identify a dominant condition for the model of this paper [18]. Let F ⊆ T and recall that the notation λC m (F) in Section 3.1. Now, we define a L-dimensional vector C C C H(F) = (λC 1 (F) − λ0 (F), · · · , λL (F) − λL−1 (F)), + C where λC m (F) − λm−1 (F), m ∈ [L] , denotes the optimal resource that F can utilize on C machines in the segmented timescale [τL−m + 1, τL−m+1 ] after F has utilized λC m−1 (F) resource in [τL−m+1 + 1, τL ]. Let v(F) denote the total value of the tasks in F and then we introduce the notion of one pair (F, v(F)) dominating another (F 0 , v(F 0 )) if H(F) = H(F 0 ) and v(F) ≥ v(F 0 ), that is, the solution to our problem indicated by (F, v(F)) uses the same amount of resources as (F 0 , v(F 0 )), but obtains at least as much value. Algorithm 5: DP(T ) 1 2 3 4 5 6 7 8 9 10 11 12 F ← {T1 }; A(1) ← {(∅, 0), (F, v(F))}; for i ← 2 to n do A(j) ← A(i − 1); for each (F, v(F)) ∈ A(i − 1) do if {Ti } ∪ F satisfies the boundary condition then if there exist a pair (F 0 , v(F 0 )) ∈ A(i) so that (1) H(F 0 ) = H(F ∪ {Ti }), and (2) v(F 0 ) ≥ v(F ∪ {Ti }) then Add ({Ti } ∪ F, v({Ti } ∪ F)) to A(i); Remove the dominated pair (F 0 , v(F 0 )) from A(i); else Add ({Ti } ∪ F, v({Ti } ∪ F)) to A(i); return arg max(F ,v(F ))∈A(n) {v(F)}; 13 We now give the general DP procedure DP(T ), also presented as Algorithm 5 [18]. Here, we iteratively construct the lists A(i) for all i ∈ [n]+ . Each A(i) is a list of pairs (F, v(F)), in which F is a subset of {T1 , T2 , · · · , Ti } satisfying the boundary condition and v(F) is the total value of the tasks in F . Each list only maintains all the dominant pairs. Specifically, we start with A(1) = {(∅, 0), ({T1 }, v1 )}. For each i = 2, · · · , n, we first set A(i) ← A(i − 1), and for each (F, v(F)) ∈ A(i − 1), we add (F ∪ {Ti }, v(F ∪ {Ti })) to the list A(i) if F ∪ {Ti } satisfies the boundary condition. We finally remove from A(i) all the dominated pairs. DP(T ) will select a subset S of T from all pairs (F, v(F)) ∈ A(n) so that v(F) is maximum. Proposition 8. DP(T ) outputs a subset S of T = {T1 , · · · , Tn } such that v(S) is the maximum value subject to the condition that S satisfies the boundary condition; the time complexity of DP(T ) is O(ndL C L ). Proof. The proof is similar to the one in the knapsack problem [18]. By induction, we need to prove that A(i) contains all the non-dominated pairs corresponding to feasible sets F ∈ {T1 , · · · , Ti }. When i = 1, the proposition holds obviously. Now suppose it hold for A(i − 1). Let F 0 ⊆ {T1 , · · · , Ti } and H(F 0 ) satisfies the boundary condition. We claim that there is some pair (F, v(F)) ∈ A(i) such that H(F) = H(F 0 ) and v(F) ≥ v(F 0 ). First, suppose that Ti ∈ / F 0 . Then, the claim follows by the induction hypothesis and by the fact that we initially set A(i) to A(i − 1) and removed dominated pairs. Now suppose that Ti ∈ F 0 and let F10 = F 0 − {Ti }. By the induction hypothesis there is some (F1 , v(F1 )) ∈ A(i − 1) that dominates (F10 , v(F10 )). Then, the algorithm will add the pair (F1 ∪ {Ti }, v(F1 ∪ {Ti })) to A(i). Thus, there will be some pair (F, v(F)) ∈ A(i) that dominates (F 0 , v(F 0 )). Since the size of the space of H(F) is no more than (C · T )L , the time complexity of DP(T ) is ndL C L . Proposition 9. Given the subset S output by DP(T ), LDF(S ) gives an optimal solution to the welfare maximization problem with a time complexity O(max{ndL C L , n2 }). Proof. It follows from Propositions 8 and 1. Remark. As in the knapsack problem [18], to construct the algorithm DP(T ), the pairs of the possible state of resource utilization and the corresponding best social welfare have to be maintained and a L-dimensional vector has to be defined to indicate the resource utilization state. This seems to imply that we cannot make the time complexity of a DP algorithm polynomial in L. 5.2 Machine Minimization Given a set of tasks T , the minimal number of machines needed to produce a feasible schedule of T is exactly the minimum C ∗ such that the boundary condition is satisfied, by Theorem 1, where the feasible schedule could be produced with a time complexity O(n2 ). An upper bound of the minimum C ∗ is k · n and this minimum C ∗ can be obtained through a binary search procedure with a time complexity of log (k · n) = O(log n); the corresponding algorithm is presented as Algorithm 6. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 Lemma 12. In each iteration of the binary search procedure, the time complexity of determining the satisfiability of boundary condition (line 4 of Algorithm 6) is O(L · n) where L ≤ n. Proof. See the Appendix for the proof. With Lemma 12, the loop of Algorithm 6 has a complexity O(L · n · log n). Based on the above discussion, we conclude that Proposition 10. Algorithm 6 produces an exact algorithm for the machine minimization problem with a time complexity of O(n2 , L · n · log n). Algorithm 6: Machine Minimization) 1 2 3 4 5 L ← 1 , U ← k · n; // L and U are respectively the lower and upper bounds of the minimum number of needed machines mid ← L+U 2 ; while U − L ≤ 1 do if the boundary condition is satisfied with C = C ∗ then U ← mid; 7 else L ← mid; 8 mid ← 6 9 10 5.3 L+U 2 ; C∗ ← U ; // the optimal number of machines call the algorithm LDF(T ) to produce a schedule of T on C ∗ machines; Minimizing Maximum Weighted Completion Time Under the task model of this paper and for the objective of minimizing the maximum weighted completion time of tasks, a direction application of LDF(S ) improves the algorithm in [8] by a factor 2. In [8], with a polynomial time complexity, Nagarajan et al. find a completion time di for each task Ti that is 1 +  times the optimal in terms of the objective here; then they propose a scheduling algorithm where each task can be completed by the time at most 2 times di . As a result, an (2 + 2)-approximation algorithm is obtained. Instead, by using the optimal scheduling algorithm LDF(S ), we have that Proposition 11. There is a (1 + )-approximation algorithm for scheduling independent malleable tasks under the objective of minimizing the maximum weighted completion time of tasks. 6 C ONCLUSION In this paper, we study the problem of scheduling n deadline-sensitive malleable batch tasks on C identical machines. Our core result is a new theory to give the first optimal scheduling algorithm so that C machines can be optimally utilized by a set of batch tasks. We further derive four algorithmic results in obvious or non-obvious ways: (i) the best possible greedy algorithm for social welfare maximization with a polynomial time complexity of O(n2 ) that achieves an approximation ratio of s−1 s , (ii) the first dynamic 14 programming algorithm for social welfare maximization with a polynomial time complexity of O(max{ndL C L , n2 }), (iii) the first exact algorithm for machine minimization with a polynomial time complexity of O(n2 , L · n · log n), and (iv) an improved polynomial time approximation algorithm for the objective of minimizing the maximum weighted completion time of tasks, reducing the previous approximation ratio by a factor 2. Here, L and d are the number of deadlines and the maximum deadline of tasks. R EFERENCES [1] Han Hu, Yonggang Wen, Tat-Seng Chua, and Xuelong Li. ”Toward scalable systems for big data analytics: A technology tutorial.” IEEE Access (2014): 652-687. 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The Design of Approximation Algorithm. Cambridge University Press, 2011. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 [19] Xiaohu Wu, and Patrick Loiseau. ”Algorithms for scheduling deadline-sensitive malleable tasks.” In Proceedings of the 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 530-537. IEEE, 2015. [20] Xiaohu Wu, and Patrick Loiseau. ”Algorithms for Scheduling Malleable Cloud Tasks (Technical Report).” arXiv preprint arXiv:1501.04343v4 (2015). [21] Longkun Guo, and Hong Shen. ”Efficient Approximation Algorithms for the Bounded Flexible Scheduling Problem in Clouds.” IEEE Transactions on Parallel and Distributed Systems 28, no. 12 (2017): 3511-3520. [22] G. Brassard, and P. Bratley. Fundamentals of Algorithmics. Prentice-Hall, Inc., 1996. [23] G. Even, Recursive greedy methods, in Handbook of Approximation Algorithms and Metaheuristics, T. F. Gonzalez, ed., CRC, Boca Raton, FL, 2007, ch. 5. A PPENDIX Proof of Lemma 2. Before executing Fully-Utilize(i), the resource allocation to the previously allocated tasks S 0 satisfies Property 1. Its execution does not change the previous allocation to S 0 . Let S 00 = S 0 ∪ {Ti }. Since di = τm , the workload of Ti can only be processed in [1, τm ]; the maximum workload of S 00 that could be processed in [τm +1, τL ] still equals its counterpart when S 0 is considered, 00 C 0 i.e., λC L−m (S ) = λL−m (S ). Upon completion of FullyUtilize(i), if the total allocation to S 00 in [1, τm ] is C · τm , we could conclude that Ti is the last task of S being considered and all tasks in S have been fully allocated; otherwise, S 00 ( S , which contradicts the fact that S and its subset satisfy the boundary condition, which implies that after the maximum workload of S 00 has been processed in [τm +1, τL ], 00 the remaining workload µC m (S ) ≤ ·C · τm . Hence, we conclude that Proof of Lemma 3. During the execution of Fully-Utilize(i), upon completion of the allocation to Ti at t ∈ [1, t1 ], if Ti has not been fully allocated yet, it is allocated ki machines at this slot. The allocations to Ti at slots t1 , · · · , 1 are nonincreasing, i.e., yi (1) ≤ yi (2) ≤ · · · ≤ yi (t1 ). The reason for this is as follows: Fully-Utilize(i) allocates machines to Ti from di towards earlier slots and, after the allocation at every slot t ∈ [1, t1 ], yi (t) = min{ki , Di − Pdi y (t)} whose value is non-increasing with t. With t=t+1 i Property 2, before executing Fully-Utilize(i), the numbers of idle machines have a stepped shape, i.e., W (1) ≥ · · · ≥ W (t0 ). The execution of Fully-Utilize(i) does not change the previous allocation to S 0 and upon its completion the number of available machines W (t) at every slot t ∈ [1, τm ] will be no larger than its counterpart before executing Fully-Utilize(i); we thus have t0 ≥ t1 . Upon completion of Fully-Utilize(i), deducting the machines allocated to Ti , the numbers of idle machines still have a stepped shape in [1, t1 ]. Hence, the lemma holds. Proof of Lemma 5. Recall that W (t) is the sum of the allocations yj (t) of all tasks Tj ∈ S at t and W (t) + W (t) = C . Initially, we have the inequality that W (t) − yi (t) > W (t0 ) − yi (t0 ) due to the conditions (i)-(iii) of Lemma 5, and, there exists a Ti0 such that yi0 (t0 ) < yi0 (t); otherwise, that inequality would not hold. In the subsequent iteration of Routine(·), W (t) becomes > 0 since partial allocation of 15 Ti0 is transferred from t to t0 ; however, it still holds that W (t) < ∆ ≤ ki − yi (t). So, we have W (t)−yi (t) = C−W (t)−yi (t) > W (t0 )−ki = W (t0 )−yi (t0 ) and such Ti0 can still be found like the initial case. Proof of Lemma 6. If Ti has been allocated Di resource just upon completion of Fully-Utilize(·), Fully-Allocate(i) does nothing upon its completion and we have t2 = t1 and the lemma holds. Otherwise, within [1, τm ], by Lemma 3, only the time slots t in [1, t1 ] have available machines, i.e., W (t) > 0, and, at these time slots, yi (t) = ki ; for all t ∈ [t1 + 1, di ], W (t) = 0. So, only for each t in [t1 + 1, di ] and from di towards earlier time slots, Fully-Allocate(i) will reduce the allocations of the previous tasks of S 0 at t and transfer them to the latest time slot t0 in [1, t1 ] with W (t0 ) > 0 (see the step 2 of Fully-Allocate(i)); then, all the available machines at t will be re-allocated to Ti and W (t) is still zero again (see the step 3 of Fully-Allocate(i)), and, the number of available machines at t0 will be decreased to zero one by one from t1 toward earlier time slots. Due to Lemma 3, the lemma holds. Proof of Lemma 7. The time complexity of Allocate-B(i) depends on Fully-Allocate(i) or AllocateRLM(·). In the worst case, Fully-Allocate(i) and AllocateRLM(·) have the same time complexity from the execution of Routine(·) at every time slot t ∈ [1, di ]. In AllocateRLM(·) for every task Ti ∈ T , each loop iteration at t ∈ [1, di ] needs to seek the time slot t0 and the task Ti0 at most Di times. The time complexities of respectively seeking t0 and Ti0 are O(d) and O(n); the maximum of these two complexities is max{d, n}. Since di ≤ d and Di ≤ D, we have that both the time complexity of Allocate-B(i) is O(dD max{d, n}). Since we assume that d and k are finitely bounded where D ≤ d · k , we conclude that O(dD max{d, n}) = O(n). Proof of Lemma 8. Let us consider an optimal allocation to A1 , R1 , · · · , AK , RK for the MSW-I problem. If we replace an allocation to a task in Rm with the same allocation to a task in R0m and do not change the allocation to Am , this generates a feasible schedule for the MSW-II problem, which yields at least the same social welfare since the marginal value of the task in R0m is no smaller than the ones of the tasks in Rm ; hence, Lemma 8 holds. Proof of Lemma 9. We will show in an optimal schedule of the MSW-II problem that (i) only the tasks of R0m , th A1 , · · · , AK will be executed in [tth m−1 + 1, tm ], and (ii) the upper bound of the maximum workload of R0m that could th th th be processed in [tth m−1 + 1, tm ] is (1 − r) · (tm − tm−1 ) · C . As a result, the total value generated by executing all tasks of th A1 , · · · , AK and (1 − r) · (tth m − tm−1 ) · C workload of each 0 + Rm (m ∈ [K] ) is an upper bound of the optimal social welfare for the MSW-II problem. We prove the first point by contradiction. Given a m ∈ [K]+ , if m ≥ 2, all tasks of R01 , · · · , R0m−1 could th not be processed in [tth m−1 + 1, tm ] due to the deadline constraint. If m ≤ K − 1, the marginal value of the task in R0m is no smaller than the ones of R0m+1 , · · · , R0K ; instead th of processing R0m+1 , · · · , R0K in [tth m−1 + 1, tm ], processing 0 Rm could generate at least the same value or even a higher value. Hence, the first point holds. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 We prove the second point also by contradiction. If there th exists a m0 ∈ [1, K] such that more than (1−r)·(tth m0 −tm0 −1 )· th th 0 C workload of Rm0 is processed in [tm0 −1 + 1, tm0 ], let m denote the minimum such m0 . In the case where m = 1, due to Features 2 and 1, after the maximum workload of the th tasks of A1 has been processed in [tth 1 +1, tK ], the minimum remaining workload that could be processed in [1, tth 1 ] is at th 0 least r · tth 1 · C . If more than (1 − r) · t1 · C workload of R1 th is processed in [1, t1 ], this means that the total amount of th workload of A1 processed in [1, tth 1 ] is smaller than r · t1 · C ; in this case, we could always remove the allocation to R01 and add more allocation to A1 to increase the total value. As a result, the second point holds when m = 1. In the other case where m ≥ 2, since we are seeking for an upper bound, we could assume that for all l ∈ [m − 1]+ , (1 − r) · (tth l − 0 th th tth ) · C workload of R is processed in [t + 1, t ] . Again l−1 l l−1 l due to Features 2 and 1, similar to P the case where m = 1, the minimum available workload of Ti ∈∪m Al that could be l=1 th processed in [1, tth m ] is at least r ·tm ·C . In this case, we could stillPremove the allocation to R0m and add more allocation to Ti ∈∪m Al to increase the total value, with the second l=1 point holding when m ≥ 2. Proof of Lemma 10. It suffices to prove that, the total alloth cation to ∪m l=1 Al in [1, tm ] could be divided into m parts such that, for all l ∈ [1, m], (i) the l-th part has a size th r · (tth l − tl−1 ) · C , and (ii) the allocation of the l-th part is associated with marginal values no smaller than v 0l . Then, the total value generated by executing the l-th part is no times the total value generated by the smaller than 1−r r th allocation to R0l in [tth l−1 + 1, tl ]. As a result, the value th generated by the total allocation to ∪m l=1 Al in [1, tm ] is no 1−r smaller than r times the value generated by the allocation 0 . to T10 , · · · , Tm Due to Feature 1, the allocation to A1 achieves a utilization r in [1, tth 1 ] and we could use a part of this allocation as the first part whose size is r · tth 1 · C . Next, the allocation to A1 ∪ A2 achieves a utilization r in [1, tth 2 ]; we could deduct the allocation used for the first part and get a part of the remaining allocation to A1 ∪ A2 as the second part, whose th size is r · (tth 2 − t1 ) · C . Similarly, we could get the 3rd, · · · , m-th parts that satisfy the first point mentioned at the beginning of this proof. Since the marginal value of the task of R0l is no larger than the ones of the tasks in ∪ll0 =1 Al0 for all 1 ≤ l ≤ m, the second point mentioned above also holds. Proof of Proposition 6. We first show that the resource utilization of A1 ∪ · · · ∪ Am in [1, τm ] is r upon completion of the m-th phase of GreedyRLM; then, we consider a task Ti ∈ ∪m l=1 Rl such that di = cm . Since P Ti is not accepted when being considered, it means that t≤di min{ki , W (t)} < Di at that time and there are at most leni − 1 = d dsii e − 1 time slots t with W (t) ≥ ki in [1, cm ]. Then, we assume that the number of the time slots t with W (t) ≥ ki is µ. Since Ti isn’t fully allocated, we have the current resource utilization of A1 ∪ · · · ∪ Am0 in [1, cm ] is at least Cdi − µC − (Di − µki ) Cdi − Di − (leni − 1)(C − ki ) ≥ C · di C · di C(di − leni ) + (C − ki ) + (leni ki − Di ) s−1 ≥ ≥ ≥ r. C · di s 16 We assume that Ti ∈ Rm0 for some m0 ∈ [m]+ . Now, we show that, after Ti is considered and rejected, the subsequent resource allocation by Allocate-A(j ) to each task Tj of ∪L l=m0 +1 Al doesn’t change the utilization in [1, τm ]. FullyUtilize(j ) does not change the allocation to the previous accepted tasks; the operations of changing the allocation to other tasks in AllocateRLM(j , 0, tth m + 2) happen in its th call to Routine(∆, 0, 1, t) where we have cm0 ≤ tth m0 ≤ tl 0 for all m + 1 ≤ l ≤ L. Due to the function of lines 68 of Routine(∆, 0, 1, t), in the l-th phase of GreedyRLM, the call to any Allocate-A(j ) will never change the current allocation of A1 ∪ · · · ∪ Am0 in [1, cm ]. Hence, if tth m = cm , upon completion of GreedyRLM, the resource utilization of A1 ∪· · ·∪Am where m0 ≤ m; if tth m > cm , since each time slot th in [cm + 1, tth m ] is fully utilized by the definition of tm , the th resource utilization in [cm + 1, tm ] is 1 and the final resource utilization will also be at least r. Proof of Lemma 12. Recall the process of defining µC m (S) where S = T . In Definition 1 that defines λm (T ), n tasks are considered sequentially for each m ∈ [L]+ , leading to a complexity L · n. In Definition 2 that derives C C C λC m (T ) from λm (T ), λ1 (T ), λ2 (T ), · · · , λL (T ) are considered sequentially, leading to a complexity O(L). Finally, P C µC (T ) = D − λ (T ) . Hence, the time complexi m m Ti ∈T ity of determining the satisfiability of boundary condition depends on Definition 1 and is O(L · n). 

Distance-based Camera Network Topology Inference for Person Re-identification Yeong-Jun Cho and Kuk-Jin Yoon Computer Vision Laboratory, GIST, South Korea arXiv:1712.00158v1 [] 1 Dec 2017 {yjcho, kjyoon}@gist.ac.kr Abstract ID:337 1.23m/s In this paper, we propose a novel distance-based camera network topology inference method for efficient person re-identification. To this end, we first calibrate each camera and estimate relative scales between cameras. Using the calibration results of multiple cameras, we calculate the speed of each person and infer the distance between cameras to generate distance-based camera network topology. The proposed distance-based topology can be applied adaptively to each person according to its speed and handle diverse transition time of people between non-overlapping cameras. To validate the proposed method, we tested the proposed method using an open person re-identification dataset and compared to state-of-the-art methods. The experimental results show that the proposed method is effective for person re-identification in the large-scale camera network with various people transition time. ID:612 4.01m/s ID:578 2.04m/s 60 Distribution of person’s speed ID:337 ID:494 40 0.51m/s 20 0 0 ID:494 1 ID:612 ID:578 2 3 4 (m/s) Figure 1. Challenges in person re-identification based on the time-based camera network topology due to the diverse speeds of people. Each blob was marked at every 0.3 seconds interval and each color indicates a person identity. nectivities between cameras) into account; therefore, they have to examine every possible person candidate in the camera network to re-identify a person. Instead of examining every person, we can restrict and reduce the search space by inferring the spatio-temporal relation between cameras, referred to as a camera network topology. 1. Introduction Numerous surveillance cameras installed in public places (e.g., offices, stations, and streets) allow monitoring and tracking of people in large-scale environments. However, it is difficult for a person to individually observe each camera. To reduce the human efforts, person reidentification techniques which automatically identify people between multiple non-overlapping cameras can be used. Previous works have mainly focused on modeling people appearance information such as feature descriptor extraction [11, 21] and learning similarity metrics [10, 16] to perform person re-identification. Recently, many appearance modeling methods based on deep neural network [1, 42] have been proposed. However, in a large-scale camera network where the number of cameras and the number of people are large, the person re-identification problem becomes very challenging. In particular, it is difficult to effectively perform reidentification only by using the appearance-based methods in the large-scale camera network. This is because the conventional appearance-based methods do not take the structure of the camera network (e.g., spatial and temporal con- In recent years, several camera network topology inference methods [5, 26, 32] have been proposed. In general, camera network topology represents spatio-temporal relations and connections between cameras. The topology is represented as a graph Gt = (V, Et ), where vertices V denote cameras, and edges Et denote the transition distribution of people across cameras according to ‘time’. We name this topology Gt as a time-based camera network topology. Although there has been much progress in person re-identification using camera network topology, the time-based topology has difficulty in dealing with the diverse walking speed of the people. As shown in Fig. 1, the walking speed of people is very diverse. For example, if a person walks much faster or slower than the average speed, it is hard to predict its transition time between two non-overlapping cameras based on the time-based topology. To overcome the limitations of the previous works, we propose a novel distance-based camera network topology in1 ference method to perform efficient and accurate person reidentification. To estimate the speed of a moving person, we need to calibrate cameras in a network automatically. The progress in self-calibration methods [22, 25] enables us to estimate camera parameters without any off-line calibration steps [43]. However, the estimated camera extrinsic parameters (i.e., camera position t) are not accurate due to scale ambiguity in the self-calibration method, i.e. we can obtain the camera extrinsic parameters determined up to scale. In this work, we first estimate the relative scale between cameras based on human height information and correct the inaccurate camera calibration results. We then calculate speeds of all people in the camera network. Subsequently, we infer the distance between cameras by multiplying the speeds and transition times of people and build a distance-based camera network topology. The inferred distance-based topology can be applied adaptively to each person – when we divide the distance-based camera network topology according to the speed of a person, it gives a person-specific time-based camera network topology. The main idea of this work is simple but effective. To the best of our knowledge, this is the first attempt to infer the distance-based camera network for person re-identification. To validate the proposed method, we tested the proposed method using the SLP [6] dataset and compared with stateof-the-art methods. The results show that the proposed method is promising for person re-identification in the large-scale camera network with diverse speeds of people. The rest of the paper is organized as follows: In Sec. 2, we review previous works of person re-identification and camera network topology inference. In Sec. 3, we describe our proposed distance-based camera network topology inference and person re-identification methods. The dataset and evaluation methodologies are described in Sec 4. The experimental results are reported in Sec. 5 and we conclude the paper in Sec. 6. identification task. In addition, several works exploited additional information such as human pose prior [7, 33, 39] and group appearance model [20, 47] to improve the reidentification performance. In recent years, many re-identification methods based on learning deep convolutional neural network (CNN) [34, 40, 45] and Siamese convolutional network [1, 8, 36, 42] have been proposed for simultaneously learning both features and metrics. In addition, several works utilized recurrent neural network (RNN) [28, 48, 49] and long short term memory (LSTM) [41] network to perform multi-shot person re-identification. Although a lot of person re-identification methods have been proposed so far, the challenges of person re-identification in a large-scale camera network, e.g., spatio-temporal uncertainty between non-overlapping cameras and high computational complexity, still remain. 2.2. Camera Network Topology Inference Recently, many works have tried to infer a camera network topology and employ camera geometry to resolve the spatio-temporal ambiguities. Several works [4, 14, 30] assumed the camera network topology is given and showed the effectiveness of the spaito-temporal information between cameras. However, in practice, the camera network topology is not given. Thus, many works have tried to infer the camera network topology in an unsupervised manner. Makris et al. [26] proposed a topology inference method based on a simple event correlation model between cameras. This topology inference method was extended in many works [5, 29, 32]. Similarly, Loy et al. [23, 24] inferred a camera network topology by measuring mutual information between activity patterns of cameras. The previous topology inference methods [5, 23, 24, 26, 32] are practical since they do not require appearance matching steps such as re-identification or inter-camera tracking for topology inference. However, the inferred topologies are prone to be inaccurate since the topology is easily contaminated by false event correlations, which frequently occur when people pass through blind regions irregularly. On the other hand, several works [3, 6, 27] inferred the camera network topology using person re-identification results. These methods can be more robust to noise than the event-based approaches since the methods infer the topology by utilizing true correspondences between cameras. Especially, Cho et al. [6] iteratively solved re-identification and camera network topology inference. It achieved accurate results in both tasks thanks to its iterating strategy. The previous camera network topology inference methods, which have been proposed so far, mainly focused on to infer the transition time of people between cameras. However, the previous methods, so-called, ‘time-based topology’ inference methods cannot efficiently handle the diverse speeds of people as shown in Fig. 1. 2. Previous Works 2.1. Person Re-identification In general, most of the person re-identification methods rely on appearances of people to identify people across nonoverlapping views. To describe and classify the appearances of people, many works have tried to propose appearance modeling methods such as feature learning and metric learning methods. For the feature learning, [11, 21, 44] designed feature descriptors to well describe the appearance of people. For the metric learning, many methods [10, 16] have been proposed and used for person reidentification [9, 31, 38]. Several works [16, 31] extensively evaluated the feature and metric learning methods to show the effectiveness of those methods in the person re2 800 600 400 200 0 800 600 400 200 0 Distance-based camera network topology inference 𝑠 𝑘𝑙 0 -1000 Y parameters K [R t] 2000 0 1000 X distance -1000 Y 1000 0 People speeds 0.5 0 -0.5 1000 0 -1 1500 500 -1 -400 -200 0 3.2 m/s Distance = 0 200 400 600 800 1000 1000 3.0 m/s 500 0 -500 -1000 (Sec. 3.2) X speed × time Person re-identification via distance-based topology 𝑝𝑖 (Δ𝑡) People speeds in CAM l 𝐺𝑑 = (𝑉,𝐸𝑑) 𝑝(𝑑) 1.6 m/s 1500 1 0 -1000 1 Camera 1000 2000 1000 People speeds in CAM k 𝑝(𝑑) 1.4 m/s 3 4 ID: 1 2 2000 2000 Re-id results People heights in CAM l Distance -based topology -500 -400 -200 0 200 400 600 800 1000 𝑣𝑖 , 𝑣𝑗 (Sec. 3.3) 𝑣𝑖 distance Time 𝑣𝑗 𝑝 Δ𝑡 𝑗 Get adaptive search range Perform re-id People heights in CAM k 4 ID: 1 2 3 Z Z Re-identification results Relative scale estimation between cameras Time (Sec. 3.4) Figure 2. Overview of the proposed framework: distance-based camera network topology inference and person reidentification. 3. Proposed Methods topology as described in Sec 3.3. Finally, we perform person re-identification based on the inferred distance-based camera network topology in Sec. 3.4. The proposed framework is illustrated in Fig. 2. For reproducibility, the code of this work is available to the public at: https://. As mentioned above, many camera network topology inference methods have been proposed recently to perform efficient person re-identification in a large-scale camera network. They inferred a time-based camera network topology based on the people transition time between cameras. However, the speed of people can be very diverse, and it leads diverse people transition time between cameras as shown in Fig. 1. Therefore, the conventional time-based topology becomes ambiguous and inaccurate under the diverse speed of people. Actually, in surveillance videos, it is possible to extract additional cues such as camera parameters (position and viewpoint) and trajectories of people. Using the additional cues, we can also estimate walking speed of each person. Then, the challenge of re-identification due to diverse walking speed becomes more tractable. In this work, we fully exploit those additional cues and propose a new distancebased camera network topology inference, which does not depend on the speed of a person. 3.2. Relative Scale Estimation between Cameras In a general pinhole camera model, the relation between a 2D image (pixel coordinates [u, v]) and a 3D point (world coordinates [X, Y, Z]) can be represented by 3× 4 projection matrix P as > > [u, v, 1] = P [X, Y, Z, 1] , P = K [R t] , (1) where K, [R t] represent camera intrinsic and extrinsic parameters. Unfortunately, most surveillance cameras remain uncalibrated. In order to estimate the camera parameters, camera selfcalibration techniques [22, 25] can be employed. These methods do not require any pre-defined checkerboards and off-line calibration tasks [43]. Instead of using a checkerboard, they utilize a human height H as the checkerboard. A camera extrinsic parameter t (camera position) is determine by the value of H. However, H is unknown in general; thus Liu et al. [22] set H to any pre-defined specific value (e.g., the average height of humans: H=1.72m [35]). Therefore, although we can estimate the intrinsic and extrinsic parameters of each camera using the self-calibration technique, the camera extrinsic parameter t is not accurate since we use the inaccurate H value. In this work, we consider every camera in the camera network; thus each camera extrinsic parameter t should be adjusted to share the same world coordinate system. To this end, we estimate relative scale of human heights between cameras based on person re-identification results and adjust each camera’s extrinsic parameter t. We denote a person i in camera k as oki . Then, a matrix of people 3.1. Overall Proposed Framework We first obtain initial person re-identification results (i.e., person correspondence pairs) between cameras. To this end, we can apply any existing re-identification methods except for methods using prior knowledge of the camera network1 . The initial re-identification results are utilized in following proposed steps. In Sec. 3.2, we perform the relative scale estimation for each camera. Each camera in the camera network is calibrated using camera self-calibration techniques [22,25] and its camera parameters are adjusted based on the proposed method. After estimating relative scales between cameras, we calculate walking speeds of people in each camera. The speeds of people and re-identification results are used to infer the distance-based camera network 1 For example, person re-identification based on metric learning [10,16] requires true person correspondences between cameras to learn the distance metric. We aim to run our framework without any prior knowledge of the camera network. Thus we do not use the methods requiring prior knowledge of the camera network. 3 CAM A Unit (m) 4 ID:13 Z ID:7 ID:8 ID:1 ID:5 Head point Foot point 800 600 400 2 200 0 Z ID:13 4 2000 2 1000 X 2D image ID:7 ID:1 00 -2 -1000 Y ID:5 -2-1000 00 Y 21000 42000 3D world correspondences is defined as  Mkl = (mij ) |1 ≤ i ≤ N k , 1 ≤ j ≤ N l , ( 1 if oki corresponds to olj , mij = 0 otherwise S kl = arg min S l i=1 j=1 mij H(oki ) S− H(olj ) 1.9 m/s ID: 2 0.8 m/s 1.0 m/s ∆𝑡=46 1.2 m/s Figure 4. Example of estimating a distance between two cameras. The speed in the blind area is inferred by averaging two speeds from two cameras. The distance between cameras is estimated as 46m from both identities. calculated as follows, T q
2
2 1X k −Yk k − Xk + Yi,t Xi,t v̄ik = i,t−1 , (5) i,t−1 T t=1 (2) where the t is the time (second), and Xi ,Yi are the world coordinates (meter) of a person i’s foot position in camera k, respectively. In order to get the more reliable speed of a person, we average multiple speeds of the person along its trajectory in each camera. After calculating the speeds of people, we build a distance distribution based on person re-identification results between two cameras. For example, we have a correspondence between two cameras {oki , olj }. We then estimate the distance between two cameras by multiplying the speed and the time difference as follows, where N k and N l are the numbers of identities in camera k and l. To find the relative scale between two cameras, we find a scale ratio S that minimizes the following equation k 2.0 m/s ∆𝑡=23 X Figure 3. Examples of detected human heights in a 2D image and corresponding 3D human heights in a world coordinate system. N X N X CAM B ID: 1 2.1 m/s ID:8 CAM Human height Blind area ! , (3) where H(oki ) is an average height of a person oki along its moving path in the world coordinate system. Since the proposed Eq. (3) utilizes multiple correspondence pairs, we can prevent overfitting the value S. We assume that every person is on the planar ground plane. Inspired by [25], we can detect human foot and head points in a 2D image and compute the corresponding person’s height according to the projection matrix P as shown in Fig. 3. After estimating the scale ratio S between two cameras, the projection matrices of camera k and l are updated as   Pk = Kk Rk tk , (4)   Pl = Kl Rl S kl tl . 1 k (v̄ + v̄jl ) · ∆t, (6) 2 i where ∆t is a transition time of the person who appears in two cameras at different times. Note that it is impossible to directly observe the speed of the person in the blind area (i.e., area of between cameras). For that reason, to infer the speed in the blind area, we average the two speeds (v̄ik and v̄jl ) from two cameras. Figure 4 shows the example of the distance estimation between cameras. Although the speeds of two identities are different, the estimated distances are the same. Using multiple correspondences between cameras, we make a histogram of the distance and normalize the histogram by dividing with the number of the correspondences. We denote the distribution according to the distance between two cameras k and l as pkl (d). By performing distance distribution estimation between all camera pairs in the camera network, we obtain the distance-based camera network topology, and it is defined by a graph as follows, dkl = Through the proposed process, we can adjust all extrinsic camera parameters t in the camera network. Note that the proposed framework does not aim to find the absolute scale of the world coordinate system and real walking speeds of people, but to match the scales across cameras. 3.3. Distance-based Camera Network Topology Inference Gd = (V, Ed ) , To infer the distance-based camera network topology, we exploit the speeds of people in each camera. We assume that every person is on the planar ground plane (Z = 0, world XY plane). Based on this assumption and camera calibration result in Sec 3.2, the speed of a person oki is V ∈ {k|1 ≤ k ≤ Ncam } ,  Ed ∈ pkl (d)|1 ≤ k ≤ Ncam , 1 ≤ l ≤ Ncam , (7) where Ncam is the number of cameras in the camera network and V is a set of cameras and Ed is a set of distance 4 0.2 0 20 40 60 80 0 20 Transition time (sec) (a) Transition time distribution 40 60 (b) Distance distribution 20 40 60 80 (a) Person speed: 2.1 m/s 0.4 0.2 0 20 40 60 80 Transition time (sec) (b) Person speed: 0.8 m/s Figure 6. Inferring adaptive transition time distribution for each identity based on the speed of a person. the fixed time range [20, 60] (sec) for all test queries as shown in Fig. 5 (a). Although it has the much wider time range (40 seconds) than our method, it may fail to find a correct person who moves very slowly or fast. For example, its search range [20, 56.3) (sec) is redundant for the person who moves slowly (0.8 m/s), and it fails to search the person if the person reappears during (60, 73.8] (sec) in the other camera. distribution between cameras. As shown in Fig. 5, the variance of the distance distribution inferred by the proposed method (Fig. 5 (b)) is smaller than that of the conventional time-based transition distribution (Fig. 5 (a)). Note that it is difficult to reduce the search range when the variance of the distribution is large. 3.4. Person Re-identification via Distance-based Camera Network Topology Person re-identifiction. In this work, we utilize the LOMO feature extraction method [19], which shows promising reidentification performance, to describe the appearances of people. It divides a person image patch into six horizontal stripes and extracts a HSV color histogram from each stripe. It builds a descriptor based on Scale Invariant Local Ternary Pattern (SILTP). The descriptor from the 128×48 (pixel) image has 26,960 dimensions. In general, each person gives multiple appearances along with its trajectory. The multiple appearances provide rich information for re-identification. However, a lot of computations are needed to take into account all the multiple appearances. Inspired by [46], we employ an average feature pooling method. In average pooling, the feature vectors of multiple appearances are pooled into one by the averaged summation. Therefore, we can simply compare two identities and the similarity score between two identities is defined by In this section, we restrict the search range based on the inferred distance-based camera network topology and perform person re-identification. Search range restriction. Dividing the distance distribution pkl (d) by the speed of a person i in camera k gives a transition time distribution of the person i who moves from camera k to l as pkl (d) . v̄ik 0.2 Transition time (sec) Figure 5. Comparison of two distributions between cameras. The transition time distribution has a larger variance than that of the distance distribution. pkl i (∆t) = 0.4 0 80 Distance (m) 0.6 𝑝(∆𝑡) 0.4 0.2 CAM2(Zone4) - CAM1(Zone4) 0.6 𝑝(∆𝑡) 0.4 0 CAM2(Zone4) - CAM1(Zone4) CAM2(Zone4) - CAM1(Zone4) 0.6 𝑝(𝑑) 𝑝(∆𝑡) CAM2(Zone4) - CAM1(Zone4) 0.6 (8) Thus, we can adaptively give a camera network topology for each person depending on its speed as shown in Fig. 6. Based on the obtained transition time distribution pkl i (∆t), we restrict the search range for re-identification as follows: • Find a mean value of
the transition time distribution: m = mean pkl i (∆t) .
k l S oki , olj = e−kΦ(oi )−Φ(oj )k2 , • Set a search range Tr around the mean value to cover 95% of the distribution: [m − T2r , m + T2r ]. (9) where Φ (·) is a pooled feature vector of a person. The similarity score lies on [0, 1]. Note that it is possible to utilize any kind of feature extraction and pooling methods in our framework. In consequence, a person who moves fast (2.1 m/s) in the camera k will be searched within the time range [21.4, 28.1] (sec) in the camera l (Fig. 6 (a)). On the other hand, a person who moves slowly (0.8 m/s) in the camera k will be searched within the time range [56.3, 73.8] (sec) in the camera l (Fig. 6 (b)). The search range of each person is determined adaptively in this manner. Thus, the proposed adaptive search strategy according to the walking speed of a person becomes more effective under the large variation of people’s speed. On the other hand, a search strategy based on the conventional time-based camera network topology searches within 4. Dataset and Methodology 4.1. Dataset Over the past few years, numerous datasets of person re-identification have been published such as VIPeR [12], PRID 2011 [13], CUHK [17, 18], iLIDS-VID [37], MARS [45] and Airport [15]. However, most of the datasets do not provide camera synchronization information 5 2 3 C4 1 4 C6 4 6 5 2 1 3 1 C2 C1 3 5 4 2 1 3 the number of true matching results and Tgt is the total number of ground truth pairs in the camera network. In the SLP dataset, Tgt = 2,664 as summarized in Table. 1. In practice, rank-1 accuracy is the most important one among all ranks since other ranks (2, 3, ..., n) failed to find the correct correspondences at least one time. To evaluate the accuracy of the camera network topology inference, we draw a curve of the retrieval rate. The retrieval rate represents the retrieval accuracy of matching candidates derived from the camera network topology. For example, if the matching candidates include the true correspondence of a test query, it counts as a success, otherwise a fail. Naturally, when the topology gives a wide search range Tr , the retrieval rate becomes high since the matching candidates within the wide search range are likely to include a true correspondence. For unbiased evaluations, we draw a curve of the retrieval rate according to the average search range as shown in Fig. 9. If there are multiple links in the camera network, we measure the retrieval rates for all links and average them to get the final rate. Since we have no ground-truth of the camera network topology, measuring the retrieval rate is reasonable. 5 1 2 4 3 3 C5 C3 C7 3 4 2 1 C8 2 3 1 2 C9 2 1 4 1 2 Figure 7. Layout of the SLP dataset. Each blue ellipse in a camera is an entry–exit zone. Red lines are valid links between cameras (best viewed in color). Table 1. Numbers of transition identities (IDs) between two cameras. Link 1 2 3 4 CAM pairs # of IDs CAM1-CAM2 227 CAM2-CAM3 571 CAM3-CAM5 568 CAM3-CAM7 168 Link 5 6 7 8 CAM pairs # of IDs CAM4-CAM5 155 CAM5-CAM6 61 CAM7-CAM8 281 CAM8-CAM9 633 5. Experimental Results or time stamps of all frames; thus these datasets cannot be used for testing our framework. To validate our methods and compare to other stateof-the-art methods, we used the SLP re-identification dataset [6]. It is a large-scale person re-identification dataset containing nine synchronized outdoor cameras. The total number of identities in the dataset is 2,632. The layout of the camera network is shown in Fig. 7. It has eight valid links between cameras as summarized in Table. 1. It provides the ground truth detection and tracking information of every person including people positions (x,y locations) and sizes (height, width). We used the given detection and tracking results for our experiments, since we mainly focus on person re-identification and camera network topology inference problems. Most person re-identification researches follow this assumption and setting. We resized the every person image to 128×48 pixels and extracted feature descriptors from the resized images. In the experiments, we assume that the camera parameters are partially given – camera intrinsic parameter K and camera rotation R are given, but the camera translation is given by S · t, where S is unknown. Thus, the cameras in the camera network do not share the same world coordinate system, initially. To identify a pair of cameras is connected or not, we applied the connectivity check schemes in [6, 26] and we obtained initial person re-identification results based on [6] for our framework. Using the initial re-identification results, we perform two steps in our framework: 1) relative scale estimation between cameras, and 2) distance-based camera network topology inference. However, the initial results of the reidentification between cameras may include several false correspondences leading an inaccurate topology inference result. Therefore, we utilized only reliable people corre
spondences that have high similarity scores: S oki , olj > 0.7 and excluded the false correspondences in both steps 1) and 2). We empirically set the threshold as 0.7, but our framework does not highly depend on this threshold. As a result, we found seven links in the camera network. Unfortunately, we failed to find a link CAM5-CAM6 since the number of people is small due to the long distance from the camera (Fig. 7). Also, CAM6 is isolated from other cameras as shown in Table. 1. Figure 8 shows comparisons of two types of distribution between cameras: (left) transition time distributions in conventional time-based topology, (right) distance distributions in proposed distance-based topology. As we can see, the 4.2. Evaluation methodology To evaluate a person re-identification performance, many previous works plot a Cumulative Match Curve (CMC) [12] representing true match being found within the first n ranks. In general, to plot the CMC, the number of a gallery (i.e., matching candidates) should be fixed for all test queries. However, in our framework, the number of a gallery varies according to the camera network topology and test queries. Therefore, we cannot plot a complete CMC. In this work, we followed Cho’s [6] re-identification evaluation metric: P measuring the rank-1 accuracy by 100 · TTgt , where T P is 6 0.4 0.4 0.6 0.4 𝑝(𝑑) 0 20 40 0 60 (sec) 𝑝(𝑑) 0 20 40 0 60 (sec) 0.4 0.4 𝑝(𝑑) 0.6 0.8 Distance-based (gt) Distance-based (ours) Time-based Distance-based (error 25%) Distance-based (error 50%) 0.7 0.6 0 20 40 60 0 10 20 30 40 Average search range (sec) 50 60 (m) Figure 9. Topology accuracy: retrieval rate according to the average search range. 0 20 40 60 80 (m) 0.2 0 20 40 0 60 (sec) 0.6 0.4 0.4 𝑝(𝑑) 0.6 0.2 0 20 40 60 (m) 0 20 40 60 0.6 0.6 0.4 0.4 Distance-based (gt) Distance-based (ours) Time-based Distance-based (error 25%) Distance-based (error 50%) 50 30 0 60 (sec) 70 40 0.2 𝑝(𝑑) 𝑝(∆𝑡) (m) 0.9 0.2 0.2 𝑝(∆𝑡) 60 0.4 0.6 0 20 40 60 0 10 20 30 40 Average search range (sec) 50 60 (m) Figure 10. Re-identification accuracy: rank-1 accuracy according to the average search range. 0.2 0.2 0 40 0.6 0.2 0 20 0.2 0.4 0 0 Rank1 accuracy 𝑝(∆𝑡) CAM2-CAM3 0 60 (sec) 0.6 𝑝(∆𝑡) CAM3-CAM5 40 0.2 𝑝(∆𝑡) CAM4-CAM5 20 0.4 0 CAM3-CAM7 0 0.6 0 CAM8-CAM9 0.2 0.2 0 Retrieval rate 0.6 𝑝(𝑑) 𝑝(∆𝑡) CAM1-CAM2 1 0.6 0 20 40 60 (sec) (a) Transition time distribution 0 0 20 40 60 (m) To validate the effectiveness of the proposed method, we first evaluated the topology inference accuracy: retrieval rate according to the average search range. In this experiment, we compared a distance-based method with a timebased method. For a fair comparison, we applied the same baseline (e.g., feature extraction and pooling methods) to each method. As shown in Fig. 9, the proposed distancebased method (ours) shows superior performance than the conventional time-based method. To verify the proposed scale estimation method in Sec. 3.2, we also compared several distance-based methods with various experimental settings: • gt: a method using ground-truth camera parameters for all cameras, • error N%: a method using camera parameters with N percent of scale error. As we can see, distance-based methods with erroneous camera parameters (error 25%, 50%) show lower performance than the distance-based method with ground-truth camera parameters (gt). On the other hand, the proposed method (ours) shows a similar performance with the distance-based (gt). Interestingly, our method shows higher performance than distance-based (gt) around 5–10 second search times. In addition, a distance-based method (error 25%) shows better a retrieval rate than that of the time-based method. It implies that our method is robust to camera calibration error. We also evaluated the accuracy of re-identification based on each camera network topology. As shown in Fig. 10, the proposed distance-based method (ours) shows superior (b) Distance distribution Figure 8. Comparisons of two types of inferred distributions. proposed distance distributions show more clear peaks and small variances compared to those of transition time distributions. The result implies that the proposed distancebased topology is more effective than the time-based topology for person re-identification, since it is ambiguous to restrict search range with the unclear camera network topology. A CAM3-CAM5 pair has the negative values of both transition time and distance since they are overlapped. Ideally, a distance between two cameras should be one value if there is a single path between cameras. However, the proposed distance distributions did not converge to one value, but have some ranges e.g., CAM1-CAM2: [40, 55] and CAM4-CAM5: [35, 55]. This is because the speeds and paths of moving people in the blind region are totally unknown and can differ person to person. In addition, other values, which are used for inferring the topology, such as camera parameters and observed people positions in each view are not perfect due to noise. To overcome these limitations, we estimated the distance of the blind area by interpolating the information on both sides of cameras as in Eq. (6) and could obtain quite clear distance distributions. 7 Table 2. Performance comparison with state-of-the-art methods. Methods Makris’s [26] Nui’s [29] Chen’s [5] DNPR [27] Cai’s [3] Cho’s [6] ours–time ours–dist rank-1 accuracy 54.0 54.6 55.2 44.7 51.4 68.3 67.8 74.7 0.07 CAM3 CAM7 0.25 0.06 CAM7 CAM8 1 1 1 CAM8 CAM9 0.8 0.8 0.2 0.8 0.04 0.15 0.6 0.03 0.1 0.4 0.4 0.4 0.05 0.2 0.2 0.2 0.05 0.6 0.6 0.8 0.6 0.5 y 0.02 0.01 0 0 0 20 20 4040 6060 8080 100100 0 0 0 20 20 4040 6060 8080 100100 0.2 0 0 0 20 20 4040 6060 8080 100100 0 2.5 1 2 0.8 0.8 1.5 0.6 0.6 0.8 0.15 0.6 0.4 0.05 20 20 4040 6060 8080 100100 x 0.2 0 0 20 20 4040 6060 8080 100100 0 0.08 0.4 0.06 0.3 0.04 0.2 0.02 0.1 0 0 0 20 20 4040 6060 8080 (a) Makris’s [26] 100100 0 0 0 20 20 4040 6060 80 80 100100 1 0.4 0.4 0.5 0.2 0.2 0 0 0 20 20 4040 6060 80 80 0 100100 0 0 20 20 4040 6060 8080 (b) Nui’s [29] 100100 0 00 20 20 4040 x 6060 8080 100 100 0 00 0.8 0.8 0.8 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 6060 8080 (c) Chen’s [5] 100100 20 20 40 40 60 60 8080 100 100 20 20 40 40 60 60 8080 100 100 0.2 0 0 0.6 4040 100 100 0.4 0.8 20 20 8080 0.6 1 0 6060 0.8 1 0 40 40 0.5 1 0 20 20 1 0.6 0.5 y 0 0.2 0 0 y 0.1 0.4 0 0.4 0.2 0 0 20 20 4040 x 6060 8080 (d) Cho’s [6] 100 100 0 00 (e) ours–dist Figure 11. Comparison of inferred transition time distributions of previous methods and our distance distribution. First row – camera pair: CAM3-CAM7. Second row – camera pair: CAM7-CAM8. Third row – camera pair: CAM8-CAM9. re-identification performance than that of the time-based method. All tested re-identification performances increase up to a certain search range and then decrease. This is because a wide search range retrieves a lot of matching candidates that are likely to include a true correspondence, but they also include lots of irrelevant identities. Thus, it is important to set a proper search range. In our methods, we searches in the range of 20 seconds on average based on the propose search range restriction in Sec. 3.4. It is reasonable for both topology and re-identification performances as shown in Fig. 9 and 10. We compared the proposed method with several previous methods [3, 5, 6, 26, 27, 29] that infer a time-based camera network topology. Figure 11 shows comparison of inferred transition time distributions and an inferred our distance distribution. As we can see, the methods [5, 26, 29] show very unclear and noisy transition time distributions. This is because, to infer the topology, they used a simple correlation of people exiting–entering patterns instead of utilizing re-identification results. On the other hand, the method [6], which used re-identification results to infer the topology, shows reasonable results than other methods [5, 26, 29]. Compared to [6], our distance distributions between camera pairs show more clear peaks and have small variances. In Table. 2, we summarize the person re-identification results based on each inferred camera network topology. In this experiment, we have two results: ours–time and ours– dist. They share the same baseline except for the utilized camera network topology (time-, and distance-based) for re-identification. For a fair comparison, we set the same average search range (20 seconds) for both of our methods. Among the methods, which utilized the time-based camera network topology, the method [6] showed the highest reidentification performance. It employed the random forest algorithm [2] to perform accurate person re-identification. On the other hand, our methods (ours–time, ours–dist) used a simple feature pooling method for re-identification as described in Sec. 3.4. Although our method employed the simpler re-identification method, re-identification using the proposed distance-based topology shows superior performance than other state-of-the-art methods. 6. Conclusions In this paper, we proposed a novel distance-based camera network topology inference. We first estimate relative scale ratio between cameras based on the human heights information and infer the distance-based camera network topology. The proposed distance-based topology can be applied adaptively to each person according to its speed; therefore it can effectively handle the various people transition time between cameras. In order to validate the proposed method, we used a public synchronized large-scale re-identification dataset and compared our method with state-of-the-art methods. The results show that the proposed method is promising for person re-identification in largescale camera network with various people transition time between cameras. 8 References [21] C. Liu, S. Gong, C. C. Loy, and X. Lin. Person re-identification: What features are important? In ECCV, 2012. 1, 2 [22] J. Liu, R. T. Collins, and Y. Liu. Surveillance camera autocalibration based on pedestrian height distributions. In BMVC, 2011. 2, 3 [23] C. C. Loy, T. Xiang, and S. Gong. Time-delayed correlation analysis for multi-camera activity understanding. IJCV, 2010. 2 [24] C. C. Loy, T. Xiang, and S. Gong. 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Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Leveraging Time Series Data in Similarity Based Healthcare Predictive Models: The Case of Early ICU Mortality Prediction Full Paper Mohammad Amin Morid David Eccles School of Business University of Utah amin.morid@business.utah.edu Olivia R. Liu Sheng David Eccles School of Business University of Utah olivia.sheng@business.utah.edu Samir Abdelrahman Department of Biomedical Informatics University of Utah samir.abdelrahman@utah.edu Abstract Patient time series classification faces challenges in high degrees of dimensionality and missingness. In light of patient similarity theory, this study explores effective temporal feature engineering and reduction, missing value imputation, and change point detection methods that can afford similarity-based classification models with desirable accuracy enhancement. We select a piecewise aggregation approximation method to extract fine-grain temporal features and propose a minimalist method to impute missing values in temporal features. For dimensionality reduction, we adopt a gradient descent search method for feature weight assignment. We propose new patient status and directional change definitions based on medical knowledge or clinical guidelines about the value ranges for different patient status levels, and develop a method to detect change points indicating positive or negative patient status changes. We evaluate the effectiveness of the proposed methods in the context of early Intensive Care Unit mortality prediction. The evaluation results show that the k-Nearest Neighbor algorithm that incorporates methods we select and propose significantly outperform the relevant benchmarks for early ICU mortality prediction. This study makes contributions to time series classification and early ICU mortality prediction via identifying and enhancing temporal feature engineering and reduction methods for similarity-based time series classification. Keywords time-series classification, similarity-based classification, mortality prediction, directional change point. Introduction Patient time series data are collected over time at varying time intervals to update patient status and to support medical decisions, leading to a wide variety of patient time series data – e.g., vital signs, lab results, diagnoses, prescriptions and billings in Electronic Health Records (EHRs) and other healthcare information systems. Past studies have extracted and leveraged temporal patterns (e.g., temporal statistics, trends, transitions and similarity) from time series data for patient event, (e.g., readmission or mortality), risk (Johnson et al. 2012), cost prediction (Bertsimas et al. 2008), or performance prediction (Cho et al. 2008). Some of the past studies have reduced such problems to one of classifying one or multiple time series of the same entity into different outcome/decision classes, which is termed the time series classification (TSC) problem (Lee et al. 2012). Time series classification can be tackled by Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models engineering temporal features that provide meaningful representations of time series and predictive power at reduced dimensionality for use with general-purposed classification methods (Hippisley-Cox et al. 2009). The synergy between a certain temporal abstraction approach and classification algorithm varies amongst the vast choice space, and hence must be properly considered. To diagnose new patients, physicians are often influenced by previous similar cases with relevant clinical evidences, resulting in patient similarity theory in the clinical decision support literature (Rouzbahman and Chignell 2014). In light of this theory, this study focuses on enhancing similarity-based classification methods for patient time series classification. The challenges of time series data include high dimensionality and high degree of missingness. Temporal abstraction of time series, missing value imputation and feature reduction approaches provide options to address these challenges. In addition, changes or transitions in various contexts including time series classification have provided essential predictive power (Lin et al. 2014). To explore effective temporal abstraction, missing value imputation, feature weight assignment and change point detection methods that can afford the k-Nearest-Neighbor (kNN) algorithm with desirable synergy, this study asks these research questions: “What are the effective temporal abstraction, missing value imputation and feature reduction methods for similarity-based time series classification?" “What is an effective approach to define and detect patient status change points for similarity-based time series classification?” Proposed Framework for Similarity-based Patient Time Series Classification In this section, we introduce the methods this study has selected or proposed for patient time series classification and some background and justifications for these methods. Similarity based classification Similarity-based classifiers estimate the class label of a test or new sample based on the similarities between the test sample and a set of labeled training samples, and the pairwise similarities between the training samples (Chen et al. 2009). The most popular family of algorithms has grown out the k-NearestNeighbor (k-NN) algorithm where k is the number of training samples with maximal similarities to a test sample. Many advancements of the original k-NN including ENN (two-way similarity) (Tang and He 2015), CNN (condensed nearest neighbors) (Hart 1968) and kernel-based approaches (e.g., SVM-KNN) have been proposed (Chen et al. 2009) for similarity-based classification as well. To examine the usefulness of temporal features and reduction, and their synergy with similarity-based classification approaches, we made a conscientious decision to use the original k-NN in this study. The potentials of advancing the methods for patient time-series classification based on other similarity-based classification remain as future research directions. Past research on similarity measure has led to a wide variety of time series distance functions such as Dynamic Time Wrapping (DTW) (Berndt and Clifford 1994), Edit Distance with Real Penalty (ERP) (Chen and Ng 2004) and Longest Common Subsequence (LCSS) (Vlachos et al. 2002) to measure dissimilarities based on different considerations. Our empirical exploration of different distance functions shows that the simplicity of the Manhattan distance function offers desirable flexibility to leverage the joint benefits of distance function, PAA grain size and missing value imputation methods. We hence select the Manhattan distance function in k-NN for patient time series classification. Temporal abstraction Temporal abstraction decides on how to transform time series data into the input features of a classification model. The challenges of time series data include high dimensionality and high amount of missing data amongst others. Extant temporal abstraction methods vary in their considerations to address these challenges and the predictive power of the resulting temporal features (Fu 2011). One commonly used simple temporal abstraction approach, called piecewise aggregation approximation (PAA), segments a time series into a sequence of fixed-sized non-overlapping consecutive windows (or Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models intervals) (Lin et al. 2003). Each window is represented by the average of all data values time-stamped within the window. We regard the size of a PAA window as the grain size and divide the temporal features produced by PAA into fine-grain features versus coarse grain features. Table 1 compares the dimensionality, degrees of missingness and information loss of fine-grain versus coarse-grain features. Fine-Grain Coarse-Grain Window size Small Large Information loss Low High Dimensionality High Low Missingness High Low Table 1. Comparison of fine-grain versus coarse-grain PAA temporal abstraction The low information loss of fine-grain PAA could afford k-NN improved accuracy over coarse-grain PAA features. Exploring effective grain-size, dimensionality reduction and missing value imputation methods are necessary to realize additional predictive power of fine-grain PAA temporal features. PAA grain decision The optimal PAA grain decision is analytically intractable due to the apparent complexity of considering the interrelated factors including missing value imputation, feature reduction and distance function. Empirical comparison should be employed to decide on the grain size. To reduce the number of empirical experiments, we only compare the classification accuracy resulting from different grain sizes while holding other methods fixed at the selected or proposed settings for time series classification. Missing value imputation (MVI) Missing value handling methods such as propensity score imputation, predictive model–based imputation and hot-deck imputation can be found from past literature (Penny and Chesney 2006). Some of the time series distance functions such DTW also incorporates missing value imputation (Berndt and Clifford 1994). In particular, DTW considers prior and posterior values of a missing value at a time point when deciding on the similarity between two time series. Motivated by DTW, we propose an adjacency-based imputation method which replaces a missing value by its posterior value if its prior value is not available or by its prior value if its posterior value is not available. If both prior and posterior values are available, their average becomes the imputed value. The proposed imputation can be performed independent of the distance function of choice. Feature weighting (FW) Feature reduction for classification can utilize a variety of approaches such as information gain, Gini index and Chi-square metrics to calculate feature rankings or weights for feature selection or reduction (Singh et al. 2010). Because of the well-tested ability to improve accuracy, we adopt the Gradient Descent (GS) method (Modha and Spangler 2003; Wettschereck and Aha 1995) to assign weights to time series features. Change point detection (CPD) Many change point detection methods focus on detecting changes in the mean, variance or trend in a time series that follows a distribution – e.g., Gaussian, normal or regression (Hawkins and Zamba 2012). Such methods are not appropriate for detecting change points in patient time series due to the underlying data distribution assumptions. In addition, a change in the mean or variance of numeric patient time series, for example, of blood pressure may not be a change in patient status if both the values before and after a change point represent the same patient status – e.g. normal. Therefore, this paper uses a change point detection method based on clinical domain knowledge. Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Directional change point detection method Past research has emphasized the importance of change point in clinical guidelines and decision supports (Assareh et al. 2011; La Rosa et al. 2008; Sawaya et al. 2011). Few or many change points of a patient’s time series can easily differentiate a patient’s condition and outcomes. In this paper we propose to define and detect change points based on changes in the health status according to categories of values in the time series rather than measures like mean or variance. For instance, systolic blood pressure less than 120 is considered normal, while between 120 and 139 is considered Prehypertension (Le et al. 2013). A directional change point is defined as a change in a patients’ status category. We denote a directional change point of patient i for time series (TS) j as DCP(i, j). The input is the status(i, j, k) that indicates the health status level of patient i during time window k based on TS j. The status output is an integer, where the lowest value of status for a feature represents the worst category of health status, while the highest value of status for this feature represents the best category of health status. Status change of patient i at time window k in terms of TS j is computed as follow: StatusChange(i, j, k) = Positive, if status(i, j, k) > status (i, j, m), m<k = Negative, if status(i, j, k) < status (i, j, m), m<k = Stable, if status(i, j, k) = status (i, j, m), m<k where m is the time window of the most recent available status of the patient before window k. We propose three change point features for patient i based on TS j - the number of directional change points or 𝑁𝑢𝑚_𝑜𝑓_𝐷𝐶𝑃(𝑖, 𝑗), and the first and last status changes, or LastStatusChange(i,j), and FirstStatusChange(i,j). The following determines the value of directional change point of patient i status at time windows k in terms of TS j: DCP(i, j, k) = 1, If status(i, j, k) is opposite to status (i, j, r), r<k = 0, if status(i, j, k) = is not opposite to status (i, j, r), r<k where r is the most recent available status change of the patient before window k that is either positive or negative (i.e., Stable is not counted). DCP(i,j,1) for the first window is set to zero. Assume W is the number of time windows (e.g., 24), the number of directional change points is derived as: 𝑊 𝑁𝑢𝑚_𝑜𝑓_𝐷𝐶𝑃(𝑖, 𝑗) = ∑ 𝐷𝐶𝑃(𝑖, 𝑗, 𝑘) 𝑘=1 The most recent and most related study to this method is the change point detection method proposed by Lin et al. (2014)(Lin et al. 2014) for time-to-event prediction of chronic conditions using EHR data. To detect change in patients’ status, the numerical time series values are replaced by three nominal states (i.e., high, medium, low) based on numerical trend and two nominal trends (i.e., decrease, increase, stable) based on numerical value changes of each predictor. Their change point detection will be used as a benchmark for evaluating our proposed domain based directional change point detection method. kNN-TSC-FIWC We refer to the proposed patient time series classification method that combines the kNN algorithm with fine-grain temporal features (F), missing-value imputation (I), feature weight assignment (W) and change point detection (C) methods we select or propose to enhance similarity-based time series classification as kNN-TSC-FIWC. Figure 1 summarizes the flow of the training and testing phases of kNN-TSC-FIWC. Another benchmark of kNN-TSC-FIWC is Lee et al. (2012)(Lee et al. 2012) which proposes a similarity based time series classification algorithm - KNN-TSC to predict customer churn after 30 days. KNN-TSC divides each feature’s time series data into 15 equal size intervals and adopts the Discrete Fourier transform (DFT) technique for time series similarity calculation. It doesn’t assign feature weights. KNNTSC utilizes stratified average voting to estimate the churn decision of a test sample. In empirical evaluation, we will compare the accuracy of kNN-TSC-FIWC to that of KNN-TSC. Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Figure 1: An overview of kNN-TSC-FIWC Empirical Evaluation To evaluate the effectiveness of kNN-TSC-FIWC and its enabling methods, we compare them to benchmarks representing combinations of different temporal features, classification algorithms, similarity functions and dimension reduction methods in the context of early Intensive Care Unit (ICU) mortality prediction. Accurate ICU mortality prediction impacts medical therapy, triaging, end-of-life care, and many other aspects of ICU care (Gartman et al. 2009). To enhance the performance of ICU mortality prediction more sophisticated machine learning methods have been utilized recently. The PhysioNet/Computing in Cardiology 2012 Challenge aimed to provide a benchmark environment for early ICU mortality prediction (Silva et al. 2012). To the best of our knowledge, its winner (CCW hereafter) is the best early ICU mortality prediction benchmark using patients’ first 48 hours of ICU time series data. CCW utilizes a new Bayesian ensemble scheme comprising of 500 weak decision tree learners which randomly assigns an intercept and gradient to a randomly selected single feature (Johnson et al. 2012). Data and evaluation procedure To compare our results with CCW, we use the same experimental setup in the competition where patients were filtered to 22,561 patients who are younger than 16 years old and remained in the ICU for at least 48 hours. The data input consists of time series data of 36 variables (e.g., Glasgow Coma Score GCS) extracted from patients’ ICU stay, plus four static features (i.e., age, gender, height, and initial weight). The target variable is a binary feature showing whether or not the patient eventually dies in the hospital before discharge. While almost half of the ICU patients have died eventually, most of the deaths happened out of hospital. The problem we analyze in this study is the prediction of in hospital mortality, which has Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models an imbalanced distribution of 18% positive against 82% negative as shown in Table 2. It is interesting to observe that the difference in average ICU stays is very minor while the difference in outcomes is life versus death. Early mortality prediction may be able to help decision makers find ways to improve an ICU patient’s survival rate. For finding and tuning the parameters including finding the best k, 50% of the data was used, while the rest remained unseen for validation. In all experiments, 20-fold cross validation was used to evaluate the performance of each method based on the validation dataset. Classification performance was measured according to the average precision, recall, and F-measure across 20 folds. Number of Patients Average Hospital Stay Average ICU Stay Alive 11977 (53%) 9.68 5.21 Died 10584 (47%) 13.87 5.53 Died out of Hospital (no) 6516 (29%) 13.39 4.86 Died in Hospital (yes) 4068 (18%) 13.5 6.6 Died in ICU (yes) 3240 (14%) 10.53 7.49 Table 2: Data distribution over the target variable (mortality) Results Table 3 shows that using two hours’ time windows for the proposed method outperforms the same method with one, four and eight hours’ time windows. The performance of the smallest grain size suffers from high missingness and the resulting noises in two patients’ common PAA temporal values, while high loss of information details hurts the performance of large grain sizes. The best prediction performance is reached when the effect of missing values and information loss is balanced at window size of 2. Hence, the grain size chosen for the rest of the evaluations is 2 hours. Without change point detection (kNN-TSC-FIW) With change point detection (kNN-TSC-FIWC) Window size Accuracy F-Measure (yes) F-Measure (no) Accuracy F-Measure (yes) F-Measure (no) 1 0.72 0.56 0.81 0.80 0.69 0.91 2 0.78 0.66 0.89 0.82 0.77 0.93 4 0.75 0.63 0.76 0.80 0.69 0.91 8 0.74 0.61 0.83 0.75 0.63 0.86 12 0.66 0.48 0.75 0.64 0.45 0.74 24 0.55 0.30 0.7 0.55 0.30 0.70 48 0.44 0.19 0.62 0.43 0.16 0.60 Table 3. Window size (abstraction size effect) effect on performance Table 4 compares the performance of kNN-TSC-FIW where a few benchmarking distance functions replace the Manhattan distance function. Although the performance results are close, they do validate the performance benefit the simple Manhattan distance function offers. Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Without Change Point (kNN-TSC-FIW) With Change Point (kNN-TSC-FIWC) Accuracy F-Measure (yes) F-Measure (no) Accuracy F-Measure (yes) F-Measure (no) Manhattan 0.78 0.66 0.89 0.82 0.77 0.93 Euclidean 0.75 0.62 0.84 0.81 0.73 0.92 DTW 0.71 0.59 0.81 0.8 0.7 0.91 EDR 0.69 0.53 0.77 0.79 0.68 0.9 ERP 0.68 0.52 0.77 0.75 0.63 0.86 DFT 0.64 0.46 0.74 0.7 0.58 0.81 TSC 0.64 0.46 0.74 0.7 0.58 0.81 LCSS 0.57 0.33 0.71 0.66 0.48 0.75 Table 4. Comparison of different time-series distance function Table 5 compares the performance of kNN-FIW against some of the well-established data mining methods, including support vector machine (SVM), the original kNN without feature weight assignment, neural network (NN) and logistic regression (LR). The input features for these algorithms are derived based on the fine-grain PAA of 2-hr window size. The comparison validates the performance advantage of similarity-based classification over its non-similarity counter-parts for early ICU mortality prediction. Table 6 compares the performance of kNN combined with the selected or proposed methods starting with fine-grain time series abstraction (kNN-TSC-F), missing value imputation (kNN-TSC-FI), feature weighting (kNN-TSC-FIW) and change point detection (kNN-TSC-FIWC). The features based on the proposed fine-grain temporal abstraction, missing value imputation and feature weighting help the kNNTSC-FIW model outperform the CCW benchmark by increasing the F-measure of the “yes” class by 11%. The proposed change point features also further double this performance improvement. The significance and benefits of these performance improvements in early ICU mortality by the proposed classification features cannot be underestimated. Table 7 shows the significant effect of the proposed feature weighting technique on the proposed method (without considering change point features) against well-established feature weighting techniques, including Gini index, Chi-square and information gain, as well as the method proposed by Lee et al. (Lee et al. 2012). Without Change Point With Change Point Accuracy F-Measure (yes) F-Measure (no) Accuracy F-Measure (yes) F-Measure (no) kNN-TSC-FIW (left) kNN-TSC-FIWC (right) 0.78 0.66 0.89 0.82 0.77 0.93 CCW 0.7 0.55 0.81 0.75 0.62 0.84 CCW on fine-grain features 0.66 0.48 0.75 0.71 0.59 0.81 SVM 0.7 0.58 0.81 0.65 0.47 0.85 LR 0.61 0.42 0.74 0.72 0.56 0.81 kNN 0.68 0.5 0.77 0.68 0.52 0.77 NN 0.64 0.46 0.74 0.68 0.52 0.77 Table 5. Similarity based method against son-similarity based methods using fine-grain abstraction Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Method Grain MVI FW CPD Accuracy F-Measure (yes) F-Measure (no) CCW Coarse No No No 0.70 0.55 0.81 kNN-TSC-F Fine No No No 0.55 0.29 0.71 kNN-TSC-FI Fine Yes No No 0.64 0.40 0.83 kNN-TSC-FIW Fine Yes Yes No 0.78 0.66 0.89 kNN-TSCFIWC Fine Yes Yes Yes 0.82 0.77 0.93 Table 6. Full performance results by class Without Imputation With Imputation Accuracy F-Measure (yes) F-Measure (no) Accuracy F-Measure (yes) F-Measure (no) kNN-TSC-FW (left) kNN-TSC-FIW (right) Lee et al. 0.6 0.40 0.72 0.78 0.66 0.89 0.52 0.27 0.67 0.72 0.56 0.81 Manual weights 0.57 0.33 0.71 0.71 0.59 0.81 Chi Square 0.50 0.24 0.64 0.68 0.52 0.77 Information Gain 0.53 0.28 0.68 0.71 0.55 0.81 Gini Index 0.55 0.30 0.7 0.70 0.58 0.81 No Feature Weights 0.44 0.19 0.62 0.49 0.27 0.64 Table 7. Comparison of different feature weighting techniques on the proposed method This table shows the advantage of the selected Gradient Descent FW over methods that assign weights based on pre-calculated values as well as domain knowledge based weight assignment (i.e. Manual weights). The performance of a model without the proposed adjacency-based imputation on the left-hand side of Table 7 is significantly lower than the same model with imputation on the right-hand side, providing evidences for the effectiveness of the proposed minimalist imputation method. Table 8 compares the performance of kNN-TSC-FIWC using different change point detection (CPD) methods including parametric (M-G and V-G) and non-parametric (L-NP, S-NP, and LS-NP) CPD methods as well as the change point detection method proposed by Lin et al. (2014). Although nonparametric CPD approaches perform better than parametric CPD approaches, none of them nor the CPD method proposed by Lin et al. [16] could outperform kNN-TSC-FIWC. The comparison provides evidences that changes in patient time series cannot be detected without considering domain-based patient status categories. In addition, our post process analysis shows that patients with higher number of DCHP are more likely to die due to their unstable situation. Patients with negative last change points which indicate declining health status dominate the early death class. These patterns show the importance of the proposed change point detection features. Change Point Detection Method Accuracy F-Measure (yes) F-Measure (no) kNN-TSC-FIWC 0.82 0.77 0.93 Lin et al. L-NP S-NP LS-NP M-G V-G 0.75 0.79 0.75 0.8 0.75 0.75 0.63 0.68 0.63 0.7 0.62 0.62 0.86 0.9 0.76 0.91 0.84 0.84 Table 8. Comparison of kNN-TSC-FIWC using different Change Point Detection Methods Twenty-third Americas Conference on Information Systems, Boston, 2017 Leveraging Time Series Data in Similarity Based Healthcare Predictive Models Without Change Point With Change Point Window Size Num of Patients Died Accuracy F-Measure (yes) F-Measure (no) Accuracy F-Measure (yes) F-Measure (no) 24 33625 5102 (15%) 0.72 0.60 0.81 0.78 0.67 0.9 48 22561 4068 (18%) 0.78 0.66 0.89 0.82 0.77 0.93 72 15913 3389 (21%) 0.8 0.7 0.91 0.84 0.82 0.94 96 11976 2903 (24%) 0.75 0.63 0.86 0.82 0.75 0.92 12 9485 2647 (27%) 0.74 0.61 0.83 0.81 0.72 0.92 Table 9. Comparison of kNN-TSC-FIWC using different prediction windows Contributions and Limitations This study makes several contributions to the patient time series classification and the early ICU mortality prediction research fields: Based on the patient similarity theory, the study evaluates the effectiveness of the similarity-based patient time series classification approach. The study further evaluates and identifies effective fine-grain PAA temporal abstraction, similarity functions and proposes necessary enhancements via adjacency-based missing value imputation. The study also evaluates the effectiveness of the gradient decent feature weight assignment approach for reducing temporal dimensions and improving accuracy. To the best of our knowledge, the study is the first to propose directional patient status change point detection to extract effective features for patient time series classification. The study contributes to solutions to an important healthcare predictive problem – early ICU mortality prediction by significantly improving prediction accuracy with a new framework that embeds effective extant methods and new enhancements. Both intensive caregivers and patients’ families can benefit from this framework with the crucial decision on aggressive or supportive treatment. Also, unexpected deaths, which are still common despite evidence that patients often show signs of clinical deterioration hours in advance, can be detected. The main limitations of this study include the use of a single data set for evaluation, the difficulty of explaining a kNN model, and the need to examine additional methods appropriate for time series classification. Future research should pursue along the directions that could address these limitations. REFERENCES Assareh, H., Smith, I., and Mengersen, K. 2011. 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1 The Spatial Outage Capacity of Wireless Networks arXiv:1708.05870v2 [] 23 Jan 2018 Sanket S. Kalamkar, Member, IEEE, and Martin Haenggi, Fellow, IEEE Abstract We address a fundamental question in wireless networks that, surprisingly, has not been studied before: what is the maximum density of concurrently active links that satisfy a certain outage constraint? We call this quantity the spatial outage capacity (SOC), give a rigorous definition, and analyze it for Poisson bipolar networks with ALOHA. Specifically, we provide exact analytical and approximate expressions for the density of links satisfying an outage constraint and give simple upper and lower bounds on the SOC. In the high-reliability regime where the target outage probability is close to zero, we obtain an exact closed-form expression of the SOC, which reveals the interesting and perhaps counterintuitive result that all transmitters need to be always active to achieve the SOC, i.e., the transmit probability needs to be set to 1 to achieve the SOC. Index Terms Interference, outage probability, Poisson point process, spatial outage capacity, stochastic geometry, wireless networks. I. I NTRODUCTION A. Motivation In a wireless network, the outage probability of a link is a key performance metric that indicates the quality-of-service. To ensure a certain reliability, it is desirable to impose a limit on the outage probability, which depends on path loss, fading, and interferer locations. For example, in an interference-limited network, the outage probability of a link is the probability S. S. Kalamkar and M. Haenggi are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556 USA. (e-mail: {skalamka, mhaenggi}@nd.edu). This work is supported by the US National Science Foundation (grant CCF 1525904). Part of this work was presented at the 2017 IEEE International Conference on Communications (ICC’17) [1]. 2 that the signal-to-interference ratio (SIR) at the receiver of that link is below a certain threshold. The interference originates from concurrently active transmitters as governed by a medium access control (MAC) scheme. Clearly, if more transmitters are active, then the interference at a receiver is higher, which increases the outage probability. Hence, given the outage constraint, a natural and a fundamental question, which has surprisingly remained unanswered, is “What is the maximum density of concurrently active links that meet the outage constraint?” To rigorously formulate this question, we introduce a quantity termed the spatial outage capacity (SOC). The SOC has applications in a wide range of wireless networks, including cellular, ad hoc, device-to-device (D2D), machine-to-machine (M2M), and vehicular networks. In this paper we focus on the Poisson bipolar model, which is applicable to infrastructureless networks such as ad hoc, D2D, and M2M networks. B. Definition and Connection to SIR Meta Distribution Modeling the random node locations as a point process, formally, the SOC is defined as follows. Definition 1 (Spatial outage capacity). For a stationary and ergodic point process model where λ is the density of potential transmitters, p is the fraction of links that are active at a time, and η(θ, ǫ) is the fraction of links in each realization of the point process that have an SIR greater than θ with probability at least 1 − ǫ, the SOC is S(θ, ǫ) , sup λpη(θ, ǫ), (1) λ,p where θ ∈ R+ , ǫ ∈ (0, 1), and the supremum is taken over λ > 0 and p ∈ (0, 1]. The SOC formulation applies to all MAC schemes where the fraction of active links in each time slot is p and each link is active for a fraction p of the time. This includes MAC schemes where the events that nodes are transmitting are dependent on each other, such as carrier-sense multiple access (CSMA). In Def. 1 ǫ represents an outage constraint. Thus the SOC yields the maximum density of links that satisfy an outage constraint. Alternatively, the SOC is the maximum density of concurrently active links that have a success probability (reliability) greater than 1 − ǫ. Hence the SOC represents the maximum density of reliable links, where ǫ denotes a reliability threshold. We call the pair of λ and p that achieves the SOC as the SOC point. 3 We denote the density of concurrently active links that have an outage probability less than ǫ (alternatively, a reliability of 1 − ǫ or higher) as λǫ , λpη(θ, ǫ), (2) which results in S(θ, ǫ) = sup λǫ . Due to the ergodicity of the point process Φ, we can express λ,p λǫ as the limit 1 X 1 (P(SIRỹ > θ | Φ) > 1 − ǫ) , r→∞ πr 2 y∈Φ λǫ = lim kyk<r where ỹ is the receiver paired with transmitter y and 1(·) is the indicator function. From this formulation, it is apparent that the outage constraint results in a static dependent thinning of Φ to a point process of density λǫ . The probability η(θ, ǫ) in (1), termed meta distribution of the SIR in [2], is the complementary cumulative distribution function (ccdf) of the conditional link success probability which is given as Ps (θ) , P(SIR > θ | Φ), where the conditional probability is calculated by averaging over the fading and the medium access scheme (if random) of the interferers, and the SIR is calculated at the receiver of the link under consideration. Accordingly, the meta distribution is given as η(θ, ǫ) , P!t (Ps (θ) > 1 − ǫ), (3) where P!t (·) denotes the reduced Palm probability, given that an active transmitter is present at the prescribed location, and the SIR is calculated at its associated receiver. Under the expectation over the point process, it is the typical receiver. The meta distribution is the distribution of the conditional link success probability, which is obtained by taking an expectation over the point process. In other words, the meta distribution is the probability that the success probability of the transmission over the typical link is at least 1 − ǫ. As a result, as is standard in stochastic geometry, the calculation of the SOC is done at the typical user and involves averaging over the point process. Due to the ergodicity of the point process, η(θ, ǫ) corresponds to the fraction of reliable links in each realization of the point process. Hence we can calculate the SOC using the meta distribution framework.1 1 Note that the meta distribution provides the tool to analyze the SOC, but the meta distribution is not needed to define the SOC. 4 Fig. 1. The histogram of the empirical probability density function of the link success probability in a Poisson bipolar network with ALOHA channel access scheme for transmit probabilities p = 1/10 and p = 1. Both cases have the same mean success probability of ps (θ) = 0.5944, but we see a different distribution of link success probabilities for different values of the pair density λ and transmit probability p. For p = 1/10, the link success probabilities mostly lie between 0.4 and 0.8 (concentrated around their mean), while for p = 1, they are spread much more widely. The SIR threshold θ = −10 dB, distance between a transmitter and its receiver R = 1, path loss exponent α = 4, and λp = 1/3. The conditional link success probability Ps (θ) (and thus the meta distribution η(θ, ǫ)) allows us to directly calculate the standard (mean) success probability, which is a key quantity of interest in wireless networks. In particular, we can express the mean success probability as Z 1 !t ps (θ) , P(SIR > θ) = E (Ps (θ)) = η(θ, x)dx, 0 !t where the SIR is calculated at the typical receiver and E (·) denotes the expectation with respect to the reduced Palm distribution. The standard success probability can be easily calculated by taking the average of the link success probabilities. Hence, in a realization of the network, ps (θ) can be interpreted as a spatial average which provides limited information about the outage performance of an individual link. As Fig. 1 shows, for a Poisson bipolar network with ALOHA where each transmitter has an associated receiver at a distance R, depending on the network parameters, the distribution of Ps (θ) varies greatly for the same ps (θ). Hence the link success probability distribution is a much more comprehensive metric than the mean success probability that is usually considered. Since the SOC can be evaluated using the distribution of link success probabilities, it provides fine-grained information about the network. 5 C. Contributions This paper makes the following contributions: • We introduce a new notion of capacity—the spatial outage capacity. • For the Poisson bipolar network with Rayleigh fading and ALOHA, we give exact and approximate expressions of the density of reliable links. We also derive simple upper and lower bounds on the SOC. • We show the trade-off between the density of active links and the fraction of reliable links. • In the high-reliability regime where the target outage probability is close to 0, we give a closed-form expression of the SOC and prove that the SOC is achieved at p = 1. For Rayleigh distributed link distances, we show that the density of reliable links is asymptotically independent of the density of (potential) transmitters λ as ǫ → 0. D. Related Work For Poisson bipolar networks, the mean success probability ps (θ) is calculated in [3] and [4]. For ad hoc networks modeled by the Poisson point process (PPP), the link success probability Ps (θ) is studied in [5], where the focus is on the mean local delay, i.e., the −1st moment of Ps (θ) in our notation. The notion of the transmission capacity (TC) is introduced in [6], which is defined as the maximum density of successful transmissions provided the outage probability of the typical user stays below a predefined threshold ǫ. While the results obtained in [6] are certainly important, the TC does not represent the maximum density of successful transmissions for the target outage probability, as claimed in [6], since the metric implicitly assumes that each link in a realization of the network is typical. A version of the TC based on the link success probability distribution is introduced in [7], but it does not consider a MAC scheme, i.e., all nodes always transmit (p = 1). The choice of p is important as it greatly affects the link success probability distribution as shown in Fig. 1. In this paper, we consider the general case with the transmit probability p ∈ (0, 1]. The meta distribution η(θ, ǫ) for Poisson bipolar networks with ALOHA and cellular networks is calculated in [2], where a closed-form expression for the moments of Ps (θ) is obtained, and an exact integral expression and simple bounds on η(θ, ǫ) are provided. A key result in [2] is that, for constant transmitter density λp, as the Poisson bipolar network becomes very dense (λ → ∞) with a very small transmit probability (p → 0), the disparity among link success probabilities vanishes and all links have the same success probability, which is the mean success 6 probability ps (θ). For the Poisson cellular network, the meta distribution of the SIR is calculated for the downlink and uplink scenarios with fractional power control in [8], with base station cooperation in [9], and for D2D networks underlaying the cellular network (downlink) in [10]. Furthermore, the meta distribution of the SIR is calculated for millimeter-wave D2D networks in [11] and for D2D networks with interference cancellation in [12]. E. Comparison of the SOC with the TC The TC defined in [6] can be written as c(θ, ǫ) , (1 − ǫ) sup{λp : E!t (Ps (θ)) > 1 − ǫ}, while the SOC can be expressed as S(θ, ǫ) , sup{λpP(Ps (θ) > 1 − ǫ)}. λ,p The mean success probability ps (θ) , E!t (Ps (θ)) depends only on the product λp and is monotonic. Hence the TC can be written as c(θ, ǫ) , (1 − ǫ)p−1 s (1 − ǫ). The TC yields the maximum density of links such that the typical link satisfies the outage constraint. In other words, in the TC framework, the outage constraint is applied at the typical link, i.e., after averaging over the point process. This means that the outage constraint is not applied at the actual links, but at a fictive link whose SIR statistics correspond to the average over all links. The supremum is taken over only one parameter, namely λp. On the other hand, in the SOC framework, the outage constraint is applied at each individual link.2 It accurately yields the maximum density of links that satisfy an outage constraint. This means that λ and p need to be considered separately. We further illustrate the difference between the SOC and the TC through the following example. Example 1 (Difference between the SOC and the TC). For Poisson bipolar networks with ALOHA and SIR threshold θ = 1/10, link distance R = 1, path loss exponent α = 4, and target outage probability ǫ = 1/10, c(1/10, 1/10) = 0.0608 (see [13, (4.15)]), which is achieved at λp = 0.0675. At this value of the TC, ps (θ) = 0.9. But at p = 1, actually only 82% of the active links satisfy the 10% outage. Hence the density of links that achieve 10% outage is only 0.055. On the other hand, S(1/10, 1/10) = 0.09227 which is the actual maximum density 2 Hence the TC can be interpreted as a mean-field approximation of the SOC. 7 of concurrently active links that have an outage probability smaller than 10%. The SOC point corresponds to λ = 0.23 and p = 1, resulting in ps (θ) = 0.6984. Thus the maximum density of links given the 10% outage constraint is more than 50% larger than the TC. The version of the TC proposed in [7] applies an outage constraint at each link, similar to the SOC, but assumes that each link is always active (i.e., there is no MAC scheme) and calculates the maximum density of concurrently active links subject to the constraint that a certain fraction of active links satisfy the outage constraint. Such a constraint is not required by our definition of the SOC, and the SOC corresponds to the actual density of active links that satisfy the outage constraint. F. Organization of the Paper The rest of the paper is organized as follows. In Sec. II, we provide the network model, formulate the SOC, give upper and lower bounds on the SOC, and obtain an exact closed-form expression of the SOC in the high-reliability regime. In Sec. III, we consider the random link distance case where the link distances are Rayleigh distributed. We draw conclusions in Sec. IV. II. P OISSON B IPOLAR N ETWORKS WITH D ETERMINISTIC L INK D ISTANCE As seen from Def. 1, the notion of the SOC is applicable to a wide variety of wireless networks. To gain crisp insights into the design of wireless networks, in this paper, we study the SOC for Poisson bipolar networks where we consider deterministic as well as random link distances and obtain analytical results for both cases. Table I provides the key notation used in the paper. A. Network Model We consider the Poisson bipolar network model in which the locations of transmitters form a homogeneous Poisson point process (PPP) Φ ⊂ R2 with density λ [14, Def. 5.8]. Each transmitter has a dedicated receiver at a distance R in a uniformly random direction. In a time slot, each node in Φ independently transmits at unit power with probability p and stays silent with probability 1 − p. Thus the active transmitters form a homogeneous PPP with density λp. We consider a standard power law path loss model with path loss exponent α. We assume that a channel is subject to independent Rayleigh fading with channel power gains as i.i.d. exponential random variables with mean 1. 8 TABLE I S UMMARY OF N OTATION Notation Φ, Φt Definition/Meaning Point process of transmitters θ SIR threshold ǫ Target outage probability S(θ, ǫ) Spatial outage capacity (SOC) η(θ, ǫ) Fraction of reliable transmissions λ Density of potential transmitters µ Density of receivers for the random link distances case p Fraction of links that are active at a time λǫ Density of reliable transmissions Ps (θ) Conditional link success probability ps (θ) Mean success probability α Path loss exponent δ 2/α Mb (θ) R bth moment of the conditional link success probability Link distance in a bipolar network We focus on the interference-limited case, where the received SIR is a key quantity of interest. To the PPP, we add a (desired) transmitter at location (R, 0) and a receiver at the origin o. Under the expectation over the PPP, this link is the typical link. The success probability ps (θ) of the typical link is the ccdf of the SIR calculated at the origin. For Rayleigh fading, from [4], [14], it is known that
ps (θ) = exp −λpCθδ , (4) where C , πR2 Γ(1 + δ)Γ(1 − δ) with δ , 2/α. The model is scale-invariant in the following sense: The SIR of all links in any realization of the bipolar network with transmitter locations ϕ remains unchanged if the plane is scaled by an arbitrary factor a > 0. Such scaling results in transmitter locations aϕ and link distances aR. The density of the scaled network is λ/a2 . By setting a = 1/R to obtain unit distance links, the resulting density is λR2 . Hence without loss of generality, we can set R = 1. Applied to the meta distribution and the SOC, this means that the model with parameters (R, λ) behaves exactly the same as the model with parameters (1, λR2 ). 9 B. Exact Formulation Observe from Def. 1 that the SOC depends on η(θ, ǫ) = P(Ps (θ) > 1 − ǫ) whose direct calculation seems infeasible. But the moments of Ps (θ) are available in closed-form [2], from which we can derive exact and approximate expressions of λǫ and obtain simple upper and lower bounds on the SOC. Let Mb (θ) denote the bth moment of Ps (θ), i.e.,
Mb (θ) , E Ps (θ)b . (5) The mean success probability is ps (θ) ≡ M1 (θ). From [2, Thm. 1], we can express Mb (θ) as
Mb (θ) = exp −λCθδ Db (p, δ) , where  ∞   X δ−1 k b p , Db (p, δ) , k k−1 k=1 b ∈ C, (6) p, δ ∈ (0, 1]. (7) For b ∈ N, the sum is finite and Db (p, δ) becomes a polynomial which is termed diversity polynomial in [15]. The series in (7) converges for p < 1, and at p = 1 it is defined if b ∈ / Z− or b + δ ∈ / Z− and converges if ℜ(b + δ) > 0. Here ℜ(z) is the real part of the complex number z. For b = 1 (the first moment), D1 (p, δ) = p, and we get the expression of ps (θ) as in (4). We can also express Db (p, δ) using the Gaussian hypergeometric function 2 F1 as Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p). (8) Using the Gil-Pelaez theorem [16], the exact expression of λǫ = λpη(θ, ǫ) can be obtained in an integral form from that of η(θ, ǫ) given in [2, Cor. 3] as λp λp − λǫ = 2 π where j , √ Z∞ 0 sin(u ln(1 − ǫ) + λCθδ ℑ(Dju )) du, ueλCθδ ℜ(Dju ) (9) −1, Dju = Dju (p, δ), and ℑ(z) is the imaginary part of the complex number z. The SOC is then obtained by taking the supremum of λǫ over λ > 0 and p ∈ (0, 1]. C. Approximation with Beta Distribution We can accurately approximate λǫ in a semi-closed form using the beta distribution, as shown in [2]. The rationale behind such approximation is that the support of the link success 10 probability Ps (θ) is [0, 1], making the beta distribution a natural choice. With the beta distribution approximation, λǫ can be approximated as    µβ ,β , (10) λǫ ≈ λp 1 − Iǫ 1−µ R 1−ǫ where Iǫ (y, z) , 0 ty−1 (1 − t)z−1 dt/B(y, z) is the regularized incomplete beta function with B(·, ·) denoting beta function, µ = M1 , and β = (M1 − M2 )(1 − M1 )/(M2 − M12 ). The advantage of the beta approximation is the faster computation of λǫ compared to the exact expression without losing much accuracy [2, Tab. I, Fig. 4] (also see Fig. 7 of this paper). In general, it is difficult to obtain the SOC analytically due to the forms of λǫ given in (9) and (10). But we can obtain the SOC numerically with ease. We can also gain useful insights considering some specific scenarios, on which we focus in the following subsection. D. Constrained SOC 1) Constant λp in dense networks: For constant λp (or, equivalently, a fixed ps (θ)), we now study how the density of reliable links λǫ behaves in an ultra-dense network. Given θ, α, and ǫ, this case is equivalent to asking how λǫ varies as λ → ∞ while letting p → 0 for constant transmitter density λp (constant ps (θ)). We denote the constrained SOC by S̃(θ, ǫ). Lemma 1 (p → 0 for constant λp). Let ν = λp. Then, for constant ν while letting p → 0, the SOC constrained on the density of concurrent transmissions is   λp, if 1 − ǫ < ps (θ) S̃(θ, ǫ) =  0, if 1 − ǫ > ps (θ). (11) Proof: Applying Chebyshev’s inequality to (3), for 1 − ǫ < ps (θ) = M1 , we have η(θ, ǫ) > 1 − var(Ps (θ)) , ((1 − ǫ) − M1 )2 (12) where var(Ps (θ)) = M2 − M12 is the variance of Ps (θ). From [2, Cor. 1], for constant ν, we know that lim var(Ps (θ)) = 0. Hence the lower bound in (12) approaches 1, which leads to p→0 λp=ν η(θ, ǫ) → 1. This results in the SOC constrained on the density of concurrent transmissions equal to λp. On the other hand, for 1 − ǫ > M1 , η(θ, ǫ) ≤ var(Ps (θ)) . ((1 − ǫ) − M1 )2 (13) 11 As we let p → 0 for constant ν, the upper bound in (13) approaches 0, which leads to η(θ, ǫ) → 0. This results in the SOC constrained on the density of concurrent transmissions equal to 0. In fact, as var(Ps (θ)) → 0, the ccdf of Ps (θ) approaches a step function that drops from 1 to 0 at the mean of Ps (θ), i.e., at 1 − ǫ = ps (θ). This behavior is in agreement with (11). Remark: Lemma 1 shows that, if p → 0 while λp is fixed to the value ν at which ps (θ) equals the target reliability 1 − ǫ, the maximum value of the constrained SOC is the value of the TC times 1/(1 − ǫ), and that value of the TC is ν(1 − ǫ). This observation can be explained as follows: As p → 0 while keeping λp = ν, all links in a realization of the network have the same success probability, and that value of the success probability equals ps (θ) (i.e., the success probability of transmissions over the typical link) [2]. This implies that, from the outage perspective, each link in the network can now be treated as if it were the typical link, as in the TC framework. If ν is initially set to a value that results in ps (θ) > 1 − ǫ, we can always increase it till ps (θ) = 1 − ǫ while all active links satisfying the outage constraint (or, equivalently, the typical link satisfying the outage constraint with probability one). Accordingly, the value of the TC equals 1 − ǫ times the value of ν at which ps (θ) = 1 − ǫ. Fig. 2 shows that at small values of the target outage probability ǫ, the density of reliable transmissions monotonically increases with p. On the other hand, at larger values of ǫ, it first decreases with p. 2) λp → 0: For λp → 0, λǫ depends linearly on λp, which we prove in the next lemma. Lemma 2 (λǫ as λp → 0). As λp → 0, λǫ ∼ λp. p(δ−1) Proof: As λp → 0, M1 approaches 1 and thus var(Ps (θ)) = M12 (M1 − 1) approaches 0. Since ǫ ∈ (0, 1), we have 1−ǫ < M1 as λp → 0. Using Chebyshev’s inequality for 1−ǫ < M1 as in (12) and letting var(Ps (θ)) → 0, the lower bound in (12) approaches 1, leading to η(θ, ǫ) → 1. Lemma 2 can be understood as follows. As λp → 0, the density of active transmitters is very small. Thus each transmission succeeds with high probability and η(θ, ǫ) → 1. In this regime, the density of reliable links λǫ is directly given by λp. The case λp → 0 can be interpreted in two ways: 1) λ → 0 for constant p and 2) p → 0 for constant λ. Lemma 2 is valid for both cases, or any combination thereof. The case of constant 12 0.1 0.08 0.06 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 Fig. 2. The density of reliable links λǫ against the transmit probability p for λp = 1/10, θ = −10 dB, α = 4, and R = 1. The values at the curves are ǫ = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 (bottom to top). The mean success probability is ps (θ) = 0.855. p is relevant since it can be interpreted as a delay constraint: As p gets smaller, the probability that a node makes a transmission attempt in a slot is reduced, increasing the delay. Since the mean delay until successful reception is larger than the mean channel access delay 1/p, it gets large for small values of p. Thus, a delay constraint prohibits p from getting too small. Fig. 3 illustrates Lemma 2. Also, observe that, as p → 0 (p = 10−5 in Fig. 3), λǫ increases linearly with λp until λp reaches the value 0.0675 which corresponds to ps (θ) = 1 − ǫ = 0.9 and then drops to 0. This behavior is in accordance with Lemma 1. In general, as λp increases, λǫ increases first and then decreases after a tipping point. This is due to the two opposite effects of λp on λǫ : The density λp of active transmitters increases, but at the same time, more active transmissions cause higher interference, which in turn, reduces the fraction η(θ, ǫ) of links that have a reliability at least 1 − ǫ. The contour plot in Fig. 4(a) visualizes the trade-off between λp and η(θ, ǫ). The contour curves for small values of λp run nearly parallel to those for λǫ , indicating that η(θ, ǫ) is close to 1. Specifically, the contour curves for λp = 0.01 and λp = 0.02 match those for λǫ = 0.01 and λǫ = 0.02 almost exactly. This behavior is in accordance with Lemma 2. In contrast, for large values of λp, the decrease in η(θ, ǫ) dominates λǫ . Also, for larger values of λ (λ > 0.4 for Fig. 4(a)), λǫ first increases and then decreases with the increase in p. This behavior is due to the 13 0.1 0.08 0.06 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 Fig. 3. The density of reliable links λǫ given in (2) for different values of the transmit probability p for θ = −10 dB, α = 4, and ǫ = 1/10. Observe that the slope of λǫ is one for small λp. The dashed arrow points to the value of λp = 0.0675, which corresponds to 1 − ǫ = ps (θ) = 0.9. following trade-off in p. For a small p, there are few active transmitters in the network per unit area, but a higher fraction of links are reliable. On the other hand, a large p means more active transmitters per unit area, but also a higher interference which reduces the fraction of reliable links. For λ < 0.4, the increase in the density of active transmitters dominates the decrease in η(θ, ǫ), and λǫ increases monotonically with p. The three-dimensional plot corresponding to the contour plot in Fig. 4(a) is shown in Fig. 4(b). E. Bounds on the SOC In this subsection, we obtain simple upper and lower bounds on the SOC. Theorem 1 (Upper bound on the SOC). For any b > 0, the SOC is upper bounded as  1 1  , 0 < b ≤ 1, eπθ δ Γ(1−δ)Γ(1+δ) b(1−ǫ)b S(θ, ǫ) ≤ Γ(b) 1  , b > 1. eπθ δ Γ(1−δ) Γ(b+δ)(1−ǫ)b (14) Proof: Using Markov’s inequality, η(θ, ǫ) can be upper bounded as η(θ, ǫ) ≤ Mb (θ) , (1 − ǫ)b b > 0, (15) 14 1.8 1.6 1.8 0.04 X: 1 Y: 0.23 Z: 0.09227 1.4 1.6 1.2 0.1 SOC point 1 1.4 0.8 0.03 1.2 0.4 0.2 0.03 1 0.6 SOC point 0.8 0.02 0.05 0.04 0.6 0.1 0.4 0.01 0.2 0.02 0.05 0.06 0.07 0.02 0.2 0 1 0.08 0.09 0.01 0 0.05 0.01 0.4 0.6 0.8 0 1 0.5 2 1 0 (a) 3 (b) Fig. 4. (a) Contour plots of λǫ and the product λp for θ = −10 dB, α = 4, and ǫ = 1/10. The solid lines represent the contour curves for λǫ and the dashed lines represent the contour curves for λp. The numbers in “black” and “red” indicate the contour levels for λǫ and λp, respectively. The SOC is S(θ, ǫ) = 0.09227. The values of λ and p at the SOC point are 0.23 and 1, respectively, and the corresponding mean success probability is ps (θ) = 0.6984. The arrow corresponding to the “SOC point” points to the pair of λ and p for which the SOC is achieved. (b) Three-dimensional plot of λǫ corresponding to the contour plot. where Mb (θ) = e−λCθ δD b (p,δ) . Hence we can upper bound the SOC as S(θ, ǫ) ≤ Su , where δ e−λCθ Db (p,δ) Su = sup λp , (1 − ǫ)b λ,p (16) with C = πΓ(1 − δ)Γ(1 + δ) and Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p). Let us denote fu (λ, p) = λpe−λCθ δD b (p,δ) . We can then write ∂fu (λ, p) δ = pe−λCθ Db (p,δ) (1 − λCθδ Db (p, δ)). | {z } ∂λ >0 Setting ∂fu (λ,p) ∂λ = 0, we obtain the critical point as λ0 (p) = 1/(Cθδ Db (p, δ)). For any given p, the objective function is quasiconcave. Thus λ0 (p) is the global optimum for each p. As a result, the optimization problem in (16) reduces to 1 sup fu (λ0 (p), p) (1 − ǫ)b p 1 1 sup . = δ b eCθ b(1 − ǫ) p 2 F1 (1 − b, 1 − δ; 2; p) Su = (17) 15 For 0 < b < 1, 2 F1 (1 − b, 1 − δ; 2; p) monotonically increases with p. In this case, p → 0 solves (17). On the other hand, for b > 1, 2 F1 (1 − b, 1 − δ; 2; p) monotonically decreases with p. Thus p = 1 solves (17). Overall the value of p that solves (17) is   → 0, 0 < b < 1, p0  = 1, b > 1. (18) Note that the objective function in (17) is monotonic in p. Hence p0 in (18) is again the global optimum. Finally, for 0 < b < 1, the upper bound on the SOC is obtained by substituting p = 0 in the objective of (17). Since 2 F1 (1 − b, 1 − δ; 2; 0) = 1, Su = eπθδ Γ(1 1 1 , − δ)Γ(1 + δ) b(1 − ǫ)b Similarly since b 2 F1 (1 − b, 1 − δ; 2; 1) = Su = 1 eπθδ Γ(1 0 < b < 1. (19) b > 1. (20) Γ(b+δ) , Γ(b)Γ(1+δ) Γ(b) , − δ) Γ(b + δ)(1 − ǫ)b For b = 1, the hypergeometric function returns 1 irrespective of the other parameters, and thus (19) and (20) are identical. The tightest Markov upper bound can be obtained by minimizing Su in (19) and (20) over b. Now, the value of b that minimizes Su in (19) is bm = − 1 . ln(1 − ǫ) (21) Since Su takes two different values depending on whether 0 < b ≤ 1 or b > 1, to obtain the tightest Markov upper bound, we need to consider following two cases based on whether bm ∈ (0, 1] or bm > 1. 1) bm ∈ (0, 1]: If bm ∈ (0, 1], it is the optimum value of b that minimizes Su since Su in (19) is smaller than Su in (20) for 0 < b < 1, greater for b > 1, and equal to for b = 1. From (21), it is apparent that the case bm ∈ (0, 1] is equivalent to ǫ ∈ [0.6321, 1). Hence, if ǫ ∈ [0.6321, 1), the optimum b that gives the tightest Markov upper bound is given by (21). Substituting b = bm in (19), we get the exact closed-form expression of the tightest Markov upper bound as Sut = − ln(1 − ǫ) , − δ)Γ(1 + δ) πθδ Γ(1 if ǫ ∈ [0.6321, 1), (22) where ‘t’ in the superscript of Sut indicates the tightest bound. 2) bm > 1: If bm > 1, i.e., ǫ ∈ (0, 0.6321), the optimum b is the value of b that minimizes Su in (20). However, due to the form of Su in (20), the optimum b cannot be expressed in a 16 closed-form. Hence the tightest Markov upper bound also cannot be expressed in a closed-form, but it can be easily evaluated numerically. Furthermore, for b > 1, we can get a closed-form expression of the approximate tightest Markov bound by using the approximation Γ(b + δ) ≈ bδ Γ(b) (23) in (20). Then, for b > 1, we can express (20) as Su ≈ 1 eπθδ Γ(1 The value of b that minimizes (24) is given as b̄m = − − δ) bδ (1 1 . − ǫ)b (24) δ . ln(1 − ǫ) The corresponding closed-form expression of the tightest approximate Markov upper bound is obtained by substituting b̄m in (24) which is given as  δ − ln(1 − ǫ) e−(1−δ) t Su ≈ , θ πδ δ Γ(1 − δ) if ǫ ∈ (0, 0.6321). (25) Fig. 5 illustrates upper bounds on the SOC. Letting ǫ → 0, from (25), we observe that the lower tail of the SOC decreases exponentially, i.e., S(θ, ǫ) /  ǫ δ θ e−(1−δ) , πδ δ Γ(1 − δ) ǫ → 0, (26) where ‘/’ denotes an upper bound which gets tighter asymptotically (here as ǫ → 0). In the next subsection, we shall show that the bound in (26) is in fact asymptotically tight, i.e., (26) matches the exact expression of the SOC as ǫ → 0. We now obtain lower bounds on the SOC. Theorem 2 (Lower bound on the SOC). The SOC is lower bounded as    −(1−W ((1−ǫ)e))  1 − W ((1 − ǫ)e) e − (1 − ǫ) S(θ, ǫ) > , πθδ Γ(1 + δ)Γ(1 − δ) ǫ where W (·) denotes the Lambert W function. Proof: By the reverse Markov’s inequality, we have 1− E!t ((1 − Ps (θ))b ) < η(θ, ǫ), ǫb For b ∈ N we can lower bound the SOC as S(θ, ǫ) > Sl , b > 0. (27) 17 101 100 10-1 10 -2 0 0.2 0.4 0.6 0.8 1 Fig. 5. Analytical and numerical results for the SOC. The tightest Markov upper bound on the SOC obtained numerically uses (19) and (20), which are optimized over b. The tightest Markov upper bound obtained analytically uses (22) and (25). The SOC upper bound obtained analytically is quite close to that obtained numerically for the almost complete range of reliability threshold 1 − ǫ, except near 1 − ǫ = 0.3679 (which is due to the approximation in (23)). The classical Markov bounds are plotted using (19) and (20) for b = 1, b = 2, and b = 4. The lower bound for b = 1 is plotted using (27), while the lower bounds for b = 2 and b = 4 are plotted numerically using (28). θ = −10 dB and α = 4. where Sl = sup λp 1 − λ,p with Mk (θ) = e−λCθ δD k (p,δ) Pb b k=0 k
(−1)k Mk (θ) ǫb ! , (28) . For b = 1, (28) reduces to 1 − e−λpCθ Sl = sup λp 1 − ǫ λ,p δ ! . (29) Since λ and p appear together as their product λp, Sl can be obtained by taking the supremum over t = λp, i.e., ! δ 1 − e−tCθ . Sl = sup t 1 − ǫ t {z } | (30) f (t) Substituting the value of t that results in ∂f (t) ∂t = 0 in f (t), we get the desired expression in (27). 18 15 10 5 0 0 10 20 30 40 50 Fig. 6. The solid lines represent the exact Db (p, δ) as in (7), while the dashed lines represent the asymptotic form of Db (p, δ) as in (31). For the values of b ∈ R+ \ {1}, an analytical expression for Sl is difficult to obtain due to the form of (28), but we can easily obtain corresponding lower bounds numerically. Fig. 5 shows Markov lower bounds on the SOC for b = 1, b = 2, and b = 4. F. High-reliability Regime In this section, we investigate the behavior of λǫ and the SOC in the high-reliability regime, i.e., as ǫ → 0. To this end, we first provide an asymptote of Db (p, δ) as b → ∞, which will be used to obtain a closed-form expression of the SOC in the high-reliability regime. Then we state a simplified version of de Bruijn’s Tauberian theorem (see [17, Thm. 4.12.9]) which allows a convenient formulation of η(θ, ǫ) = P(Ps (θ) > 1 − ǫ) in terms of the Laplace transform as ǫ → 0. ‘.’ denotes an upper bound with asymptotic equality (here as b → ∞). Lemma 3 (Asymptote of Db (p, δ) as b → ∞). For b ∈ R, we have Db (p, δ) . pδ bδ , Γ(1 + δ) b → ∞, Proof: See Appendix A. Fig. 6 illustrates how quickly Db (p, δ) approaches the asymptote. (31) 19 Theorem 3 (de Bruijn’s Tauberian theorem [18, Thm. 1]). For a non-negative random variable Y , the Laplace transform E[exp(−sY )] ∼ exp(rsu ) for s → ∞ is equivalent to P(Y ≤ ǫ) ∼ exp(q/ǫv ) for ǫ → 0, when 1/u = 1/v + 1 (for u ∈ (0, 1) and v > 0), and the constants r and q are related as |ur|1/u = |vq|1/v . Theorem 4 (λǫ in the high-reliability regime). For ǫ → 0, the density of reliable links λǫ satisfies  θp λǫ ∼ λp exp − ǫ where κ = δ 1−δ = 2 α−2 κ (δλC ′ )κ/δ κ ! , ǫ → 0, (32) and C ′ = πΓ(1 − δ). Proof: Let Y = − ln(Ps (θ)). The Laplace transform of Y is E(exp(−sY )) = E(Ps (θ)s ) = Ms (θ). Using (6) and Lemma 3, we have   λC(θp)δ sδ , Ms (θ) ∼ exp − Γ(1 + δ) |s| → ∞. δ Comparing this expression with that in Thm. 3, we have r = − λC(θp) , u = δ, v = δ/(1−δ) = κ, Γ(1+δ) and thus q = 1 κ (δλC ′ )κ/δ (θp)κ , where C ′ = πΓ(1 − δ). Using Thm. 3, we can now write P(Y ≤ ǫ) = P(Ps (θ) ≥ exp(−ǫ)) (a) ∼ P(Ps (θ) ≥ 1 − ǫ), = exp − ǫ→0 ! (θp)κ (δλC ′ )κ/δ κǫκ , (33) where (a) follows from exp(−ǫ) ∼ 1 − ǫ as ǫ → 0. Since we have λǫ = λpP(Ps (θ) > 1 − ǫ), (34) the desired result in (32) follows from substituting (33) in (34). For the special case of p = 1 (all transmitters are active), P(Ps (θ) ≥ 1 − ǫ) in (33) simplifies to δλC ′ θδ P(Ps (θ) ≥ 1 − ǫ) ∼ exp − κǫκ
κ/δ ! , ǫ → 0, in agreement with [7, Thm. 2] where the result for this special case was derived in a less direct way than Thm. 4. Fig. 7 shows the behavior of (32) in the non-asymptotic regime and also the accuracy of the beta approximation given by (10). We now investigate the scaling of S(θ, ǫ) in the high-reliability regime. 20 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.5 0.6 0.7 0.8 0.9 1 Fig. 7. The solid line with marker ‘o’ represents the exact expression of λǫ as in (9), the dotted line represents the asymptotic expression of λǫ given by (32) as ǫ → 0, and the dashed line represents the approximation by the beta distribution given by (10). Observe that the beta approximation is quite accurate. θ = 0 dB, α = 4, λ = 1/2, and p = 1/3. Corollary 1 (SOC in high-reliability regime). For ǫ → 0,  ǫ δ e−(1−δ) , S(θ, ǫ) ∼ θ πδ δ Γ(1 − δ) (35) and the SOC is achieved at p = 1. Proof: For notational simplicity, let us define the rate-reliability ratio as ρ , ǫ/θ and ′ κ/δ denote ξρ , ρ−κ (δC κ) λǫ ∼ fρ (λ, p), and fρ (λ, p) , λp exp(−λκ/δ pκ ξρ ). From (32), we can then write ǫ → 0, and the SOC is S(θ, ǫ) ∼ sup fρ (λ, p), λ,p ǫ → 0. (36) We can then write  
∂fρ (λ, p) κξρ κ/δ κ κ/δ κ = p exp −λ p ξρ 1 − λ p . ∂λ δ {z } | >0 Setting ∂fρ (λ,p) ∂λ = 0, we obtain the critical point as δ/κ  δ . λ0 (p) = ξρ κpκ (37) 21 For any given p, the objective function is quasiconcave. Hence the optimization problem in (36) reduces to S(θ, ǫ) ∼ sup fρ (λ0 (p), p), p =  δ eκξρ δ/κ ǫ → 0, sup p1−δ , p ǫ → 0. Observe that fρ (λ0 (p), p) monotonically increases with p and thus attains the maximum at p = 1.  δ/κ and is given by (35) after simplification. Hence the SOC is achieved at p = 1 and λ = ξρδκ The equation (35) confirms the asymptotic bound on the SOC given in (26). Corollary 2 (The meta distribution at the SOC point). As ǫ → 0, the value of the meta distribution at the SOC point can be simply expressed as η(θ, ǫ) ∼ e−(1−δ) . (38) Proof: From Cor. 1, as ǫ → 0, the SOC can be expressed as S(θ, ǫ) ∼ λopt popt η(θ, ǫ), (39) where λopt = λ0 (given by (37)) and popt = 1 correspond to the SOC point as ǫ → 0. Then, comparing (39) with (35), we get the desired expression of η(θ, ǫ) as in (38). Corollary 3 (The mean success probability at the SOC point). As ǫ → 0, the mean success probability at the SOC point can be expressed as  ǫ δ ps,opt ∼ 1 − Γ(1 + δ). δ (40) Proof: Substituting λ = λ0 (given by (37)) and p = 1 in (4) and using e−x ∼ 1 − x as x → 0 yield the desired expression. We now provide few remarks pertaining to the high-reliability regime. Remarks: • Letting Cδ =
1 δ e−(1−δ) , δ Γ(1−δ) the density of transmitters λ∗ , S(θ, ǫ) that maximizes the density of active links that achieve a reliability at least 1 − ǫ behaves as  ǫ δ ∗ λ π ∼ Cδ , ǫ → 0. θ (41) 22 The coefficient Cδ depends only on δ. In the practically relevant regime 1/2 ≤ δ < 1, i.e., 2 < α ≤ 4, Cδ ≈ 1 − δ. In (41), the left side λ∗ π is the mean number of reliable receivers in a disk of unit radius in the network. Equation (41) reveals an interesting trade-off between the spectral efficiency (captured by θ) and the reliability (captured by ǫ), where only their ratio matters. For example, at low rates, ln(1 + θ) ∼ θ; thus, a 10× higher reliability can be achieved by lowering the rate by a factor of 10. • The (potential) transmitter density λopt that achieves the SOC is  ǫ δ 1 , ǫ → 0. (42) λopt ∼ δ θ πδ Γ(1 − δ) Here λopt π is the mean number of (potential) transmitters in a disk of unit radius in the network that achieves the SOC. • From (38), it is apparent that, at the SOC point, the fraction of links that satisfy the outage constraint depends only on the path loss exponent α, as δ , 2/α. • The mean success probability ps,opt at the SOC point (given by (40)) allows us to relate the SOC and the TC. Substituting q ∗ = 1 − ps,opt in [13, (4.29)], we can express the TC as  ǫ δ 1 , ǫ → 0, c(θ, ǫ) ∼ δ θ πδ Γ(1 − δ) which is the same as the expression of the optimum λ that achieves the SOC (given by (42)). Hence S(θ, ǫ) = c(θ, ǫ)e−(1−δ) if the TC framework used ps (θ) = ps,opt instead of ps (θ) = 1 − ǫ (given that p = 1 is optimum). • From Cor. 1, observe that the exponents of θ and ǫ are the same. The SOC scales in ǫ similar to the TC defined in [7], i.e., as Θ(ǫδ ), while the original TC defined in [6] scales linearly in ǫ. • For α = 4, the expression of SOC in (35) simplifies to  ǫ 1/2 S(θ, ǫ) ∼ 0.154 , ǫ → 0, θ and the meta distribution gives η ≈ 0.6. In other words, approximately 60% of active links satisfy the outage constraint if α = 4. Also, for α = 4, the mean success probability at the √ SOC point is simply given by ps,opt ∼ 1 − 1.2533 ǫ as ǫ → 0. Fig. 8 plots λǫ versus λ and p for ǫ = 0.007 and α = 4. In this case, the SOC is achieved at p = 1. III. P OISSON B IPOLAR N ETWORKS WITH R ANDOM L INK D ISTANCES We now consider the case of random link distance, where the link distances are i.i.d. random variables (which are constant over time). 23 0.05 SOC point X: 1 Y: 0.0701 Z: 0.04068 0.04 0.03 0.02 0.01 0 1 0.5 0 0.8 0.6 0.4 0.2 0 Fig. 8. Three-dimensional plot of λǫ for ǫ = 0.007, θ = −10 dB, and α = 4. Observe that p = 1 achieves the SOC. The mean success probability ps (θ) at the SOC point is 0.8964. A. Network Model Let Ri denote the random link distance between a transmitter i and its associated receiver in a √ Poisson bipolar network. We assume that Ri is Rayleigh distributed with mean 1/(2 µ) as it is the distribution of the nearest-neighbor distance in a PPP of density µ [19].3 This scenario can be interpreted as the one where an active transmitter tries to communicate to its nearest receiver in a network where the potential transmitters form a PPP Φt of density λ and the receivers form an another PPP (independent of Φt ) of density µ. Similar to the deterministic link distance case, we add a receiver at the origin o to the receiver PPP and an always active transmitter at location (Ro , 0), where Ro is the Rayleigh distributed link distance. Under the expectation over the point process, this link is the typical link. B. Exact Formulation of the SOC Lemma 4 (bth moment of the link success probability). For Rayleigh distributed link distances √ with mean 1/(2 µ), the bth moment of Ps (θ) is Mb (θ) = 3 µ . µ + λθδ Γ(1 + δ)Γ(1 − δ)Db (p, δ) (43) Generalizations to other link distance distributions are beyond the scope of this paper. This is because some new techniques may need to be developed, and it is unclear what other distribution to assume. 24 Proof: See Appendix B. For b = 1, using D1 (p, δ) = p, we get the expression of the mean success probability ps (θ). Moreover, for b ∈ N, (43) represents the joint success probability of b transmissions with random link distance, as obtained in [15, (23)]. As in the deterministic link distance case, using the Gil-Pelaez theorem, we can calculate the density of reliable links from (9), and the SOC is obtained by taking the supremum of λǫ over λ and p. Like the deterministic link distance case, the beta approximation is quite accurate. In the rest of the paper, we assume µ = 1 without loss of generality. C. Bounds on the SOC Theorem 5 (Upper bound on the SOC). For any b > 0, the SOC for Rayleigh distributed link distances is upper bounded as S(θ, ǫ) ≤    1 1 , θ δ Γ(1−δ)Γ(1+δ) b(1−ǫ)b Γ(b) 1 , θ δ Γ(1−δ) Γ(b+δ)(1−ǫ)b 0 < b ≤ 1, b > 1. (44) Proof: Again using Markov’s inequality, η(θ, ǫ) can be upper bounded as Mb (θ) , b > 0, η(θ, ǫ) ≤ (1 − ǫ)b 1 . Hence for any b > 0, we have where Mb (θ) = 1+λθδ Γ(1+δ)Γ(1−δ)D b (p,δ) S ≤ Su , where Su = 1 λp . sup b δ (1 − ǫ) λ,p 1 + λθ Γ(1 + δ)Γ(1 − δ)Db (p, δ) {z } | (45) Aλ,p Aλ,p is maximized at λ = ∞, and it follows that 1 1 Su = δ sup , θ Γ(1 + δ)Γ(1 − δ)b(1 − ǫ)b p 2 F1 (1 − b, 1 − δ; 2; p) where we have used Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p) as in (8). (46) Notice that the optimization problem in (46) is similar to that in (17). Thus, following the steps after (17) in the proof of Thm. 1, we get (44). Similar to the deterministic link distance case (as discussed in Sec. II-E after Thm. 1), from (44), for ǫ ∈ [0.6321, 1), we can obtain the exact closed-form expression of the tightest Markov bound as Sut = −e ln(1 − ǫ) . − δ)Γ(1 + δ) θδ Γ(1 (47) 25 102 10 1 100 10 -1 0 0.2 0.4 0.6 0.8 1 Fig. 9. Analytical and numerical results for the SOC. The tightest Markov upper bound on the SOC obtained numerically uses (44), which is optimized over b. The tightest Markov upper bound obtained analytically uses (47) when ǫ ∈ [0.6321, 1) and (48) when ǫ ∈ (0, 0.6321). Observe that the analytical approximation of the SOC upper bound provides a tight upper bound for the complete range of reliability threshold 1 − ǫ. The curve corresponding to the tightest Markov upper bound obtained analytically deviates from that obtained numerically at 1 − ǫ = 0.3679 due to the approximation as in (23). The classical Markov bounds are plotted using (44) for b = 1, b = 2, and b = 4. θ = −10 dB and α = 4. For ǫ ∈ (0, 0.6321), we can obtain the exact tightest Markov bound numerically. Alternatively, using the approximation in (23), we get a closed-form expression of the approximate tightest Markov bound as Sut ≈  − ln(1 − ǫ) θ δ eδ . δ δ Γ(1 − δ) (48) As Fig. 9 shows, the tightest Markov upper bound on the SOC obtained analytically using (48) deviates slightly from that obtained numerically at ǫ = 0.6321 due to the approximation in (23). As ǫ becomes smaller, i.e., 1 − ǫ becomes closer to 1, the approximation (48) becomes better. For ǫ < 0.2, the gap between the approximation of the upper bound and the beta approximation is less than 0.15 dB. Theorem 6 (Lower bound on the SOC). The SOC is lower bounded as √ (1 − 1 − ǫ)2 . S(θ, ǫ) > δ ǫθ Γ(1 + δ)Γ(1 − δ) Proof: The proof follows the proof of Thm. 2 with M1 (θ) = 1 . 1+λpθ δ Γ(1+δ)Γ(1−δ) (49) 26 Fig. 9 shows lower bounds on the SOC. Similar to the deterministic link distance case, the Markov lower bounds for b ∈ R+ \ {1} are analytically intractable. D. High-reliability Regime Theorem 7 (SOC in the high-reliability regime). For Rayleigh distributed link distances,  ǫ δ 1 , ǫ → 0, S(θ, ǫ) ∼ θ Γ(1 + δ)Γ(1 − δ) and the SOC is achieved at p = 1. Proof: As in the proof of Thm. 4, let Y = − ln(Ps (θ)) with its Laplace transform as LY (s) = Ms (θ). Asymptotically, (a) LY (s) ∼ where (a) follows from using Db (p, δ) ∼ A , sδ pδ bδ Γ(1+δ) |s| → ∞, (50) as |b| → ∞ in (43) and thus A = 1/(λθδ pδ Γ(1− δ)). We claim that the expression in (50) is equivalent to FY (ǫ) ∼ Aǫδ , Γ(1 + δ) ǫ → 0. (51) The proof that (50) and (51) are equivalent is given in Appendix C. As ǫ → 0, since FY (ǫ) = P(− ln(Ps (θ)) < ǫ) ∼ P(Ps (θ) > 1 − ǫ) = η(θ, ǫ), the density of reliable links in the high-reliability regime can be expressed as λǫ ∼ ǫδ p1−δ , θδ Γ(1 + δ)Γ(1 − δ) ǫ → 0. (52) Here, λǫ is independent of the density λ of (potential) transmitters. As a result, the SOC is S(θ, ǫ) ∼ ǫδ sup p1−δ , θδ Γ(1 + δ)Γ(1 − δ) p Setting p = 1 achieves the SOC, which is given as  ǫ δ 1 S(θ, ǫ) ∼ , θ Γ(1 + δ)Γ(1 − δ) ǫ → 0. ǫ → 0. 27 Similar to the deterministic link distance case, only the ratio of the spectral efficiency and the reliability matters. As we observe from (52), λǫ does not depend on λ. This is due to the fact that, in the high-reliability regime, the increase in λ decreases linearly the fraction of reliable links η(θ, ǫ). For example, a 2× increase in λ decreases η(θ, ǫ) by a factor of 2. Also, the SOC is a function of just two parameters, the reliability-to-target-SIR ratio and a constant that depends only on δ, i.e., on the path loss exponent α. IV. C ONCLUSIONS This paper introduces a new notion of capacity, termed spatial outage capacity (SOC), which is the maximum density of concurrently active links that meet a certain constraint on the success probability. Hence the SOC provides a mathematical foundation for questions of network densification under strict reliability constraints. Since the definition of the SOC is very general, i.e., it is not restricted to a specific point process model, link distance distribution, MAC scheme, transmitter-receiver association schemes, fading distribution, power control scheme, etc., it is applicable to a wide range of wireless networks. For Poisson bipolar networks with ALOHA and Rayleigh fading, we provide an exact analytical expression and a simple approximation for the density of reliable links λǫ . The SOC can be easily calculated numerically as the supremum of λǫ obtained by optimizing over the density of (potential) transmitters λ and the transmit probability p. In the high-reliability regime where the target outage probability ǫ of a link goes to 0, we give a closed-form expression of the SOC which reveals • the trade-off between the spectral efficiency and the reliability where only their ratio matters while calculating the SOC. • insights on the scaling behavior of the SOC where, for both deterministic and Rayleigh distributed link distance cases, we show that the SOC scales in ǫ as Θ(ǫδ ). Interestingly, p = 1 achieves the SOC in the high-reliability regime. This means that with ALOHA, all transmitters should be active in order to maximize the number of reliable transmissions in a unit area that succeed with a probability close to one. Hence, in the high-reliability regime, backing off is not SOC-achieving. This happens because the reduction in the density of active links with p cannot be overcome by the increase in the fraction of reliable links. For Rayleigh distributed link distances, in the high-reliability regime, we have shown that the density of reliable links does not depend on λ as the increase in λ is exactly offset by the 28 fraction of reliable links. To be precise, a t-fold increase in λ decreases the density of reliable links by a factor of t. As a future work, it is important to generalize the results obtained for Rayleigh fading to other fading distributions. However, since the current bounds on the SOC and the high-reliability regime results exploit a structure induced by Rayleigh fading assumption, one might need to develop new techniques depending on the fading distribution considered. A PPENDIX A P ROOF L EMMA 3 OF From (7), we have Db (p, δ) =  ∞   X δ−1 b k k=1 ∞  X =p k=1 | By Taylor’s theorem, Ak (p) = pk k−1   δ − 1 k−1 b . p k k−1 {z } (53) Ak (p) ∞ (j) X A (1) k j! j=0 (p − 1)j , (54) (j) where Ak (1) is the jth derivative of Ak (p) at p = 1. Let (k)j , k(k − 1)(k − 2) · · · (k − j + 1) (j) denote the falling factorial. Then Ak (1) can be written as  ∞   X δ−1 b (j) (k − 1)j Ak (1) = k−1 k k=1 = (a) . where (a) follows from Γ(b+δ−j) Γ(b−j) Γ(b + δ − j) (δ − 1)j Γ(b − j)Γ(1 + δ) bδ (δ − 1)j , Γ(1 + δ) (55) . bδ as b → ∞. From (54) and (55), ∞ Ak (p) . X (δ − 1)j bδ (p − 1)j . Γ(1 + δ) j=0 j! | {z } pδ−1 From (53) and (56), we get the desired result. (56) 29 A PPENDIX B P ROOF OF L EMMA 4 Let us denote the random link distance of the typical link by R. Then the probability density function of R is fR (a) = 2πµa exp(−πµa2 ). Let kzk denote the distance between a receiver and a potential interferer z ∈ Φt . Given Φt , the conditional link success probability Ps (θ) is   −α hR > θ | Φt = E (1(h > θRα I) | Φt ) , Ps (θ) = P I where I= X z∈Φt \{zo } hz kzk−α 1(z ∈ Φt ), where zo ∈ Φt denotes the desired transmitter. Conditioning on R and then averaging over fading and ALOHA results in Ps (θ) | R = Let f (r) = Y z∈Φt \{zo }   1+θ p  R kzk   α + 1 − p .
b + 1 − p . Then the bth moment of Ps (θ) is     Y R  Mb (θ) = E  f kzk z∈Φt \{zo }      Z ∞  R (a) = ER exp −2πλ t 1−f dt t 0   Z∞ Z∞    a 2 (b) dt e−µπa da = 2πµ a exp −2πλ t 1 − f t 0 0     Z∞ Z∞  1 2 (c) = 2πµ a exp −2πλa2 y 1 − f dy  e−µπa da y p 1+θr α 0 0 µ µ + 2λ 0 y (1 − f (1/y)) dy µ (d) = ,  b ! Z∞ pθr α 1− 1− µ + 2λ r −3 dr 1 + θr α {z } |0 = R∞ (57) Fb where (a) follows from the probability generating functional of the PPP [14, Chapter 4], (b) follows from the de-conditioning on R, (c) follows from the substitution y = t/a, and (d) 30 follows from the substitution y = 1/r and plugging f (r) back. With 1 as the upper limit of the integral and µ = λ, (57) reduces to the expression of the bth moment of the success probability in a Poisson cellular network as in [2, (27)]. With r α = x, the integral in (57) can be expressed as  b ! Z∞ pθx 1 1− 1− x−δ−1 dx Fb = α 1 + θx 0 (e) = ∞  X k=1 (f) = ∞  k Z xk−δ−1 (pθ) b (−1)k+1 dx α k (1 + θx)k 0 δ θ πδ Db (p, δ), 2 sin(πδ) where (e) follows from the binomial expansion of 1 − follows from and (−1)k+1 Z∞ xk−δ−1 δ−k  (58)
pθx b 1+θx and Fubini’s theorem, and (f) π Γ(k − δ) sin(πδ) Γ(k)Γ(1 − δ)  dx = θ (1 + θx)k 0

k−δ−1 δ−1 = . Finally, substituting (58) in (57) and using k−1 k−1 πδ sin(πδ) ≡ Γ(1 + δ)Γ(1 − δ), (43) follows. A PPENDIX C P ROOF THAT (50) AND (51) ARE EQUIVALENT The proof uses the Weierstrass approximation theorem that any continuous function f : [t1 , t2 ] → R can be approximated by a sequence of polynomials from above and below. In our case, t1 = 0 and t2 = 1. Thus, for any given t > 0, if f (y) is a continuous real-valued function on [0, 1], for n ≥ 1, there exists a sequence of polynomials Pn (y) and Qn (y) such that Pn (y) ≤ f (y) ≤ Qn (y) ∀y ∈ [0, 1], Z 1 (Qn (y) − f (y))dy ≤ t, (59) (60) 0 and Z 1 0 (f (y) − Pn (y))dy ≤ t. (61) Even if f (y) has a discontinuity of the first kind, we can still construct polynomials Pn (y) and Qn (y) that satisfy (59)-(61).4 4 See [20, Sec. 7.53] for the details of the construction of such polynomials. 31 To prove the desired result, we first show that Z ∞ Z ∞ A δ −sy −sy lim s e f (e )dFY (y) = y δ−1 f (e−y )e−y dy. s→∞ Γ(δ) 0 0 Pn k Let Qn (y) = k=0 ak y with ak ∈ R for k = 0, 1, . . . , n. We then have Z ∞ Z ∞ δ −sy −sy δ lim sup s e f (e ) dFY (y) ≤ lim s e−sy Qn (e−sy ) dFY (y) s→∞ s→∞ 0 = lim s→∞ (62) 0 n X ak s k=0 δ Z ∞ e−(k+1)sy dFY (y) 0 n X ak (k + 1)δ k=0 Z ∞ (b) A = y δ−1 e−y Qn (e−y ) dy Γ(δ) 0 Z ∞ (c) A = y δ−1 e−y f (e−y ) dy, Γ(δ) 0 (a) =A R∞ where (a) follows from lim sδ 0 e−sy dFY (y) = A, (b) follows from the definition of the s→∞R ∞ gamma function as Γ(δ) , 0 y δ−1 e−y dy, and (c) follows from the dominated convergence theorem as n → ∞. By a similar argument for Pn (y), we have Z ∞ Z ∞ A −sy −sy δ y δ−1 e−y f (e−y ) dy, e f (e ) dFY (y) ≥ lim inf s s→∞ Γ(δ) 0 0 and (62) follows. Now let   1, y f (y) =  0, 1 e ≤y≤1 0 ≤ y < 1e . (63) Letting s = 1/ǫ in (62) and using (63), we have Z ∞ −δ δ e−sy f (e−sy ) dFY (y) lim ǫ FY (ǫ) = lim s ǫ→0 s→∞ 0 Z 1 (d) A = y δ−1 dy Γ(δ) 0 A . = Γ(1 + δ) where (d) follows from (62) and (63). ACKNOWLEDGMENT The authors would like to thank Ketan Rajawat and Amrit Singh Bedi for their insights on the optimization problems in the paper. 32 R EFERENCES [1] S. S. Kalamkar and M. Haenggi, “Spatial outage capacity of Poisson bipolar networks,” in Proc. IEEE International Conference on Communications (ICC’17), (Paris, France), May 2017. [2] M. Haenggi, “The meta distribution of the SIR in Poisson bipolar and cellular networks,” IEEE Transactions on Wireless Communications, vol. 15, pp. 2577–2589, April 2016. [3] M. Zorzi and S. 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Optimal power dispatch in networks of high-dimensional models of synchronous machines arXiv:1603.06688v1 [math.OC] 22 Mar 2016 Tjerk Stegink and Claudio De Persis and Arjan van der Schaft Abstract— This paper investigates the problem of optimal frequency regulation of multi-machine power networks where each synchronous machine is described by a sixth order model. By analyzing the physical energy stored in the network and the generators, a port-Hamiltonian representation of the multimachine system is obtained. Moreover, it is shown that the openloop system is passive with respect to its steady states which implies that passive controllers can be used to control the multimachine network. As a special case, a distributed consensus based controller is designed that regulates the frequency and minimizes a global quadratic generation cost in the presence of a constant unknown demand. In addition, the proposed controller allows freedom in choosing any desired connected undirected weighted communication graph. I. INTRODUCTION The control of power networks has become increasingly challenging over the last decades. As renewable energy sources penetrate the grid, the conventional power plants have more difficulty in keeping the frequency around the nominal value, e.g. 50 Hz, leading to an increased chance of a network failure of even a blackout. The current developments require that more advanced models for the power network must be established as the grid is operating more often near its capacity constraints. Considering high-order models of, for example, synchronous machines, that better approximate the reality allows us to establish results on the control and stability of power networks that are more reliable and accurate. At the same time, incorporating economic considerations in the power grid has become more difficult. As the scale of the grid expands, computing the optimal power production allocation in a centralized manner as conventionally is done is computationally expensive, making distributed control far more desirable compared to centralized control. In addition, often exact knowledge of the power demand is required for computing the optimal power dispatch, which is unrealistic in practical applications. As a result, there is an increased desire for distributed real-time controllers which are able to compensate for the uncertainty of the demand. In this paper, we propose an energy-based approach for the modeling, analysis and control of the power grid, both for This work is supported by the NWO (Netherlands Organisation for Scientific Research) programme Uncertainty Reduction in Smart Energy Systems (URSES) under the auspices of the project ENBARK. T.W. Stegink and C. De Persis are with the Engineering and Technology institute Groningen (ENTEG), University of Groningen, the Netherlands. {t.w.stegink, c.de.persis}@rug.nl A.J. van der Schaft is with the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, 9747 AG Groningen, the Netherlands. a.j.van.der.schaft@rug.nl the physical network as well as for the distributed controller design. Since energy is the main quantity of interest, the portHamiltonian framework is a natural approach to deal with the problem. Moreover, the port-Hamiltonian framework lends itself to deal with complex large-scale nonlinear systems like power networks [5], [12], [13]. The emphasis in the present paper lies on the modeling and control of (networked) synchronous machines as they play an important role in the power network since they are the most flexible and have to compensate for the increased fluctuation of power supply and demand. However, the full-order model of the synchronous machine as derived in many power engineering books like [2], [6], [8] is difficult to analyze, see e.g. [5] for a port-Hamiltonian approach, especially when considering multi-machine networks [4], [9]. Moreover, it is not necessary to consider the full-order model when studying electromechanical dynamics [8]. On the other hand of the spectrum, many of the recent optimal controllers in power grids that deal with optimal power dispatch problems rely on the second-order (non)linear swing equations as the model for the power network [7], [11], [16], [17], or the third-order model as e.g. in [14]. However, the swing equations are inaccurate and only valid on a specific time scale up to the order of a few seconds so that asymptotic stability results are often invalid for the actual system [2], [6], [8]. Hence, it is appropriate to make simplifying assumptions for the full-order model and to focus on multi-machine models with intermediate complexity which provide a more accurate description of the network compared to the secondand third-order models [2], [6], [8]. However, for the resulting intermediate-order multi-machine models the stability analysis is often carried out for the linearized system, see [1], [6], [8]. Consequently, the stability results are only valid around a specific operating point. Our approach is different as the nonlinear nature of the power network is preserved. More specifically, in this paper we consider a nonlinear sixth-order reduced model of the synchronous machine that enables a quite accurate description of the power network while allowing us to perform a rigorous analysis. In particular, we show that the port-Hamiltonian framework is very convenient when representing the dynamics of the multi-machine network and for the stability analysis. Based on the physical energy stored in the generators and the transmission lines, a port-Hamiltonian representation of the multi-machine power network can be derived. More specifically, while the system dynamics is complex, the in- terconnection and damping structure of the port-Hamiltonian system is sparse and, importantly, state-independent. The latter property implies shifted passivity of the system [15] which respect to its steady states which allows the usage of passive controllers that steer the system to a desired steady state. As a specific case, we design a distributed realtime controller that regulates the frequency and minimizes the global generation cost without requiring any information about the unknown demand. In addition, the proposed controller design allows us to choose any desired undirected weighted communication graph as long as the underlying topology is connected. The main contribution of this paper is to combine distributed optimal frequency controllers with a high-order nonlinear model of the power network, which is much more accurate compared to the existing literature, to prove asymptotic stability to the set of optimal points by using Lyapunov function based techniques. The rest of the paper is organized as follows. In Section II the preliminaries are stated and a sixth order model of a single synchronous machine is given. Next, the multimachine model is derived in Section III. Then the energy functions of the system are derived in Section IV, which are used to represent the multi-machine system in portHamiltonian form, see Section V. In Section VI the design of the distributed controller is given and asymptotic stability to the set of optimal points is proven. Finally, the conclusions and possibilities for future research are discussed in Section VII. II. PRELIMINARIES Consider a power grid consisting of n buses. The network is represented by a connected and undirected graph G = (V, E), where the set of nodes, V = {1, ..., n}, is the set of buses and the set of edges, E = {1, ..., m} ⊂ V × V, is the set of transmission lines connecting the buses. The ends of edge l ∈ E are arbitrary labeled with a ‘+’ and a ‘-’, so that the incidence matrix D of the network is given by   +1 if i is the positive end of l Dil = −1 if i is the negative end of l (1)   0 otherwise. Each bus represents a synchronous machine and is assumed to have controllable mechanical power injection and a constant unknown power load. The dynamics of each synchronous machine i ∈ V is assumed to be given by [8] Mi ω̇i = Pmi − Pdi − Vdi Idi − Vqi Iqi δ̇i 0 0 Tdi Ėqi 0 0 Tqi Ėdi 00 00 Tdi Ėqi 00 00 Tqi Ėdi see also Table I. 0 0 = −Edi − (Xqi − Xqi )Iqi 0 00 0 00 = Eqi − Eqi + (Xdi − Xdi )Idi 0 00 0 00 = Edi − Edi − (Xqi − Xqi )Iqi , rotor angle w.r.t. synchronous reference frame frequency deviation mechanical power injection power demand moment of inertia synchronous reactances transient reactances subtransient reactances exciter emf/voltage internal bus transient emfs/voltages internal bus subtransient emfs/voltages external bus voltages generator currents open-loop transient time-scales open-loop subtransient time-scales TABLE I M ODEL PARAMETERS AND VARIABLES . Assumption 1: When using model (2), we make the following simplifying assumptions [8]: • • • • The frequency of each machine is operating around the synchronous frequency. The stator winding resistances are zero. The excitation voltage Ef i is constant for all i ∈ V. 00 The subtransient saliency is negligible, i.e. Xdi = 00 Xqi , ∀i ∈ V. The latter assumption is valid for synchronous machines with damper windings in both the d and q axes, which is the case for most synchronous machines [8]. It is standard in the power system literature to represent the equivalent synchronous machine circuits along the dqaxes as in Figure 1, [6], [8]. Here we use the conventional 00 00 00 00 00 00 +jEdi where E qi := phasor notation E i = E qi +E di = Eqi 00 00 00 Eqi , E di := jEdi , and the phasors I i , V i are defined likewise [8], [10]. Remark that internal voltages Eq0 , Ed0 , Eq00 , Ed00 as depicted in Figure 1 are not necessarily at steady state but are governed by (2), where it should be noted that, by definition, the reactances of a round rotor synchronous machine satisfy 0 00 0 00 Xdi > Xdi > Xdi > 0, Xqi > Xqi > Xqi > 0 for all i ∈ V [6], [8]. By Assumption 1 the stator winding resistances are negligible so that synchronous machine i can be represented by a subtransient emf behind a subtransient reactance, see Figure 2 [6], [8]. As illustrated in this figure, the internal and external voltages are related to each other by [8] 00 00 E i = V i + jXdi I i, i ∈ V. (3) III. MULTI-MACHINE MODEL = ωi 0 0 = Ef i − Eqi + (Xdi − Xdi )Idi δi ωi Pmi Pdi Mi Xqi , Xdi 0 , X0 Xqi di 00 , X 00 Xdi qi Ef i 0 , E0 Eqi di 00 , E 00 Eqi di Vqi , Edi Iqi , Idi 0 ,T0 Tqi di 00 , T 00 Tqi di (2) Consider n synchronous machines which are interconnected by RL-transmission lines and assume that the network is operating at steady state. As the currents and voltages of each synchronous machine is expressed w.r.t. its local dqreference frame, the network equations are written as [10] 00 I = diag(e−jδi )Y diag(ejδi )E . (4) j(Xd − Xd0 ) Td0 Ef j(Xd0 − Xd00 ) 0 Eq jXd00 00 Td00 Id Vq Eq To simplify the analysis further, we assume that the network resistances are negligible so that G = 0. By equating the real and imaginary part of (4) we obtain the following expressions for the dq-currents entering generator i ∈ V: X   00 00 00 Bik (Edk Idi = Bii Eqi − sin δik + Eqk cos δik ) , k∈Ni j(Xq − Xq0 ) Tq0 j(Xq0 − Xq00 ) 0 Ed jXq00 00 Tq00 Ed Iqi = Iq Vd Fig. 1: Generator equivalent circuits for both dq-axes [8]. For aesthetic reasons the subscript i is dropped. 00 jXdi 00 Ei Vi that D is the incidence matrix of the network defined by (1). 00 Ei jXT Vi Il X   00 00 Bik (Eqk sin δik − Edk cos δik ) , k∈Ni IV. ENERGY FUNCTIONS −1 T Here the admittance matrix1 Y := D(R + jX)P D satisfies YP ik = −Gik − jBik and Yii = Gii + jBii = k∈Ni Gik + j k∈Ni Bik where G denotes the conductance and B ∈ Rn×n denotes the susceptance of the network [10]. In ≤0 addition, Ni denotes the set of neighbors of node i. Remark 1: As the electrical circuit depicted in Figure 2 is 00 in steady state (3), the reactance Xdi can also be considered as part of the network (an additional inductive line) and is therefore implicitly included into the network admittance matrix Y, see also Figure 3. 00 jXdi − (5) where δik := δi − δk . By substituting (5) and (3) into (2) we obtain after some rewriting a sixth-order multi-machine model given by equation (6), illustrated at the top of the next page. Remark 2: Since the transmission lines are purely inductive by assumption, there are no energy losses in the transmission lines implying that the following energy conserP ∗ vation law holds: i∈V Pei = 0 where Pei = Re(E i I i ) = 00 00 Edi Idi + Eqi Iqi is the electrical power produced by synchronous machine i. Ii Fig. 2: Subtransient emf behind a subtransient reactance. 1 Recall 00 −Bii Edi 00 jXdk 00 V k Ek Fig. 3: Interconnection of two synchronous machines by a purely inductive transmission line with reactance XT . When analyzing the stability of the multi-machine system one often searches for a suitable Lyapunov function. A natural starting point is to consider the physical energy as a candidate Lyapunov function. Moreover, when we have an expression for the energy, a port-Hamiltonian representation of the associated multi-machine model (6) can be derived, see Section V. Remark 3: It is convenient in the definition of the Hamiltonian to multiply the energy stored in the synchronous machine and the transmission lines by the synchronous frequency ωs since a factor ωs−1 appears in each of the energy functions. As a result, the Hamiltonian has the dimension of power instead of energy. Nevertheless, we still refer to the Hamiltonian as the energy function in the sequel. In the remainder of this section we will first identify the electrical and mechanical energy stored in each synchronous machine. Next, we identify the energy stored in the transmission lines. A. Synchronous Machine 1) Electrical Energy: Note that, at steady state, the energy (see Remark 3) stored in the first two reactances2 of generator i as illustrated in Figure 1 is given by ! 0 0 00 2 − Ef i )2 (Eqi − Eqi ) 1 (Eqi Hedi = + 0 0 − X 00 2 Xdi − Xdi Xdi di ! (7) 0 2 0 00 2 1 (Edi ) (Edi − Edi ) Heqi = . 0 + X 0 − X 00 2 Xqi − Xqi qi qi Remark 4: The energy stored in the third (subtransient) reactance will be considered as part of the energy stored in the transmission lines, see also Remark 1 and Section IV-B. 2 In both the d- and the q-axes. Mi ∆ω̇i = Pmi − Pdi + X k∈Ni h i 00 00 00 00 00 00 00 00 Bik (Edi Edk + Eqi Eqk ) sin δik + (Edi Eqk − Eqi Edk ) cos δik δ̇i = ∆ωi 0 0 0 0 00 Tdi Ėqi = Ef i − Eqi + (Xdi − Xdi )(Bii Eqi − 0 0 Tqi Ėdi = 0 −Edi + (Xqi − 0 00 Xqi )(Bii Edi = 0 Edi − 00 Edi + 0 (Xqi − 00 00 Xqi )(Bii Edi B. Inductive Transmission Lines Consider an interconnection between two SG’s with a purely inductive transmission line (with reactance XT ) at steady state, see Figure 3. When expressed in the local dqreference frame of generator i, we observe from Figure 3 that at steady state one obtains3 00 jXl I l = E i − e−jδik E k , (8) where the total reactance between the internal buses of 00 00 generator i and k is given by Xl := Xdi + XT + Xdk . Note that at steady state the modified energy of the inductive transmission line l between nodes i and k is given by ∗ Hl = 21 Xl I l I l , which by (8) can be rewritten as 
1 00 00 00 00 Hl = − Bik 2 Edi Eqk − Edk Eqi sin δik 2
00 00 00 00 (9) −2 Edi Edk + Eqi Eqk cos δik  002 002 002 002 +Edi + Edk + Eqi + Eqk , where the line susceptance satisfies Bik = − X1l < 0 [10]. C. Total Energy The total physical energy of the multi-machine system is equal to the sum of the individual energy functions: X X Hp = (Hdei + Hqei + Hmi ) + Hl . (10) i∈V (6) k∈Ni 2) Mechanical Energy: The kinetic energy of synchronous machine i is given by 1 1 Hmi = Mi ωi2 = Mi−1 p2i , 2 2 where pi = Mi ωi is the angular momentum of synchronous machine i with respect to the synchronous rotating reference frame. 00 k∈N X  i  00 00 Bik (Edk − cos δik − Eqk sin δik ) ) 00 00 0 00 0 00 00 Tdi Ėqi = Eqi − Eqi + (Xdi − Xdi )(Bii Eqi − 00 00 Tqi Ėdi X   00 00 Bik (Edk sin δik + Eqk cos δik ) ) l∈E V. PORT-HAMILTONIAN REPRESENTATION Using the energy functions from the previous section, the multi-machine model (6) can be put into a port-Hamiltonian form. To this end, we derive expressions for the gradient of each energy function. 3 The mapping from dq-reference frame k to dq-reference frame i in the phasor domain is done by multiplication of e−jδik [10]. − X   00 00 Bik (Edk sin δik + Eqk cos δik ) ) k∈Ni X   00 00 Bik (Edk cos δik − Eqk sin δik ) ) k∈Ni A. Transmission Line Energy Recall that the energy stored in transmission line l between internal buses i and k is given by (9). It can P be verified that the gradient of the total energy HL := l∈E Hl stored in the transmission lines takes the form  ∂HL      00 E 00 Idi + Eqi Iqi Pei ∂δi  ∂H00L   di  =  Idi  , Idi  ∂Eqi  = ∂HL Iqi Iqi 00 ∂Edi where Idi , Iqi are given by P (5). Here it is used that the selfsusceptances satisfy Bii = k∈Ni Bik for all i ∈ V. 1) State transformation: In the sequel, it is more convenient to consider a different set of variable describing the voltage angle differences. Define for each edge l ∈ E ηl := δik where i, k are respectively the positive and negative ends of l. In vector form we obtain η = DT δ ∈ Rm , and observe that this implies D ∂HL ∂Hp =D = Pe . ∂η ∂η B. Electrical Energy SG Further, notice that the electrical energy stored in the equivalent circuits along the d- and q-axis of generator i is given by (7) and satisfies  " ∂Hedi #  0   0 0 0 Eqi − Ef i Xdi − Xdi Xdi − Xdi ∂Eqi = 0 00 00 0 ∂Hedi 0 Xdi − Xdi Eqi − Eqi 00 ∂Eqi " #   ∂Heqi   0 0 0 0 Xqi − Xqi Xqi − Xqi Edi ∂Edi . 0 00 00 0 ∂Heqi = 0 Xqi − Xqi Edi − Edi ∂E 00 di By the previous observations, and by aggregating the states, the dynamics of the multi-machine system can now be written in the form (11) where the Hamiltonian is given 0 0 0 00 by (10) and X̂di := Xdi − Xdi , X̂di := Xdi − Xdi , X̂d = 0 0 diagi∈V {X̂di } and X̂d , X̂q , X̂q are defined likewise. In addi0 tion, Td0 = diagi∈V {Tdi } and Td0 , Tq , Tq0 are defined similarly. Observe that the multi-machine system (11) is of the form ẋ = (J − R)∇H(x) + gu y = g T ∇H(x) (12) −D 0 0 0 0 0 0 −(Td0 )−1 X̂d 0 0 0 −(Tq0 )−1 X̂q 0 0 0 0 0 0 0  T g= I 0 0 0 0 0 .   0 ṗ  η̇  DT  0   Ėq   0   ẋp =   Ė 0  =  0  d  Ė 00   0  q Ėd00 y = g T ∇Hp , where J = −J T , R = RT are respectively the antisymmetric and symmetric part of the matrix depicted in (11). Notice that the dissipation matrix of the electrical part is positive definite (which implies R ≥ 0) if   0 X −X 0 Xdi −Xdi 2 diT 0 di 0 Tdi di  ∀i ∈ V, 0 0 00  > 0, Xdi −Xdi Xdi −Xdi 2 T0 T 00 di di which, by invoking the Schur complement, holds if and only if 0 00 0 0 00 4(Xdi − Xdi )Tdi − (Xdi − Xdi )Tdi > 0, ∀i ∈ V. (13) Note that a similar condition holds for the q-axis. Proposition 1: Suppose that for all i ∈ V the following holds: 0 00 0 4(Xdi − Xdi )Tdi − (Xdi 0 00 0 4(Xqi − Xqi )Tqi − (Xqi 0 00 − Xdi )Tdi 0 00 − Xqi )Tqi >0 > 0. (14) Then (11) is a port-Hamiltonian representation of the multimachine network (6). It should be stressed that (14) is not a restrictive assumption 00 0 00 , Tqi   Tdi as it holds for a typical generator since Tdi 0 Tqi , see also Table 4.2 of [6] and Table 4.3 of [8]. Because the interconnection and damping structure J − R of (11) is state-independent, the shifted Hamiltonian H̄(x) = H(x) − (x − x̄)T ∇H(x̄) − H(x̄) (15) acts as a local storage function for proving passivity in a neighborhood of a steady state x̄ of (12), provided that the Hessian of H evaluated at x̄ (denoted as ∇2 H(x̄)) is positive definite4 . Proposition 2: Let ū be a constant input and suppose there exists a corresponding steady state x̄ to (12) such that ∇2 H(x̄) > 0. Then the system (12) is passive in a neighborhood of x̄ with respect to the shifted external portvariables ũ := u − ū, ỹ := y − ȳ where ȳ := g T ∇H(x̄). Proof: Define the shifted Hamiltonian by (15), then we obtain ẋ = (J − R)∇H(x) + gu = (J − R)(∇H̄(x) + ∇H(x̄)) + gu = (J − R)∇H̄(x) + g(u − ū) = (J − R)∇H̄(x) + gũ ỹ = y − ȳ = g T (∇H(x) − ∇H(x̄)) = g T ∇H̄(x). 4 Observe that ∇2 H(x) = ∇2 H̄(x) for all x. (16) 0 0 −(Td0 )−1 X̂d 0 00 −1 0 −(Td ) X̂d 0  0  0   0  ∇Hp + g(Pm − Pd ), −(Tq0 )−1 X̂q    0 00 −1 0 −(Tq ) X̂q (11) As ∇2 H(x̄) > 0 we have that H̄(x̄) = 0 and H̄(x) > 0 for all x 6= x̄ in a sufficiently small neighborhood around x̄. Hence, by (16) the passivity property automatically follows where H̄ acts as a local storage function. VI. MINIMIZING GENERATION COSTS The objective is to minimize the total quadratic generation cost while achieving zero frequency deviation. By analyzing the steady states of (6), it follows that a necessary condition for zero frequency deviation is 1T Pm = 1T Pd , i.e., the total supply must match the total demand. Therefore, consider the following convex minimization problem: 1 T P QPm 2 m s.t. 1T Pm = 1T Pd , min Pm (17) where Q = QT > 0 and Pd is a constant unknown power load. Remark 5: Note that that minimization problem (17) is easily extended to quadratic cost functions of the form 1 T T n 2 Pm QPm + b Pm for some b ∈ R . Due to space limitations, this extension is omitted. As the minimization problem (17) is convex, it follows that Pm is an optimal solution if and only if the KarushKuhn-Tucker conditions are satisfied [3]. Hence, the optimal points of (17) are characterized by ∗ Pm = Q−1 1λ∗ , λ∗ = 1 T Pd T 1 Q−1 1 (18) Next, based on the design of [14], consider a distributed controller of the form T θ̇ = −Lc θ − Q−1 ω Pm = Q−1 θ − Kω (19) where T = diagi∈V {Ti } > 0, K = diagi∈V {ki } > 0 are controller parameters. In addition, Lc is the Laplacian matrix of some connected undirected weighted communication graph. The controller (19) consists of three parts. Firstly, the term −Kω corresponds to a primary controller and adds damping into the system. The term −Q−1 ω corresponds to secondary control for guaranteeing zero frequency deviation on longer time-scales. Finally, the term −Lc θ corresponds to tertiary control for achieving optimal production allocation over the network. Note that (19) admits the port-Hamiltonian representation ϑ̇ = −Lc ∇Hc − Q−1 ω Pm = Q−1 ∇Hc − Kω, Hc = 1 T −1 ϑ T ϑ, 2 (20) where ϑ := T θ. By interconnecting the controller (20) with (11), the closed-loop system amounts to       ẋp g J − R − RK GT = ∇H − P 0 d −G −Lc ϑ̇ (21)  −1  0 0 0 0 0 G= Q where J − R is given as in (11), H := Hp + Hc , and RK = blockdiag(0, K, 0, 0, 0, 0). Define the set of steady states of (21) by Ω and observe that any x := (xp , ϑ) ∈ Ω satisfies the optimality conditions (18) and ω = 0. Assumption 2: Ω 6= ∅ and there exists x̄ ∈ Ω such that ∇2 H(x̄) > 0. Remark 6: While the Hessian condition of Assumption 2 is required for proving local asymptotic stability of (21), guaranteeing that this condition holds can be bothersome. However, while we omit the details, it can be shown that ∇2 H(x̄) > 0 if the generator reactances are small compared to the transmission line reactances. • the subtransient voltage differences are small. • the rotor angle differences are small. Theorem 1: Suppose Pd is constant and there exists x̄ ∈ Ω such that Assumption 2 is satisfied. Then the trajectories of the closed-loop system (21) initialized in a sufficiently small neighborhood around x̄ converge to the set of optimal points Ω. Proof: Observe by (16) that the shifted Hamiltonian defined by (15) satisfies • H̄˙ = −(∇H̄)T blockdiag(R + RK , Lc )∇H̄ ≤ 0 where equality holds if and only if ω = 0, T −1 ϑ = θ = 1θ∗ for some θ∗ ∈ R, and ∇E H̄(x) = ∇E H(x) = 0. Here ∇E H(x) is the gradient of H with respect to the internal voltages Eq0 , Ed0 , Eq00 , Ed00 . By Assumption 2 there exists a compact neighborhood Υ around x̄ which is forward invariant. By invoking LaSalle’s invariance principle, trajectories initialized in Υ converge to the largest invariant set where H̄˙ = 0. On this set ω, η, θ, Eq0 , Ed0 , Eq00 , Ed00 are constant and, T more specifically, ω = 0, θ = 1λ∗ = 1 1T1QP−1d 1 corresponds to an optimal point of (17) as Pm = Q−1 1λ∗ where λ∗ is defined in (18). We conclude that the trajectories of the closed-loop system (21) initialized in a sufficiently small neighborhood around x̄ converge to the set of optimal points Ω. Remark 7: While by Theorem 1 the trajectories of the closed-loop system (21) converge to the set of optimal points, it may not necessarily converge to a unique steady state as the closed-loop system (21) may have multiple (isolated) steady states. VII. CONCLUSIONS We have shown that a much more advanced multi-machine model than conventionally used can be analyzed using the port-Hamiltonian framework. Based on the energy functions of the system, a port-Hamiltonian representation of the model is obtained. Moreover, the system is proven to be incrementally passive which allows the use of a passive controller that regulates the frequency in an optimal manner, even in the presence of an unknown constant demand. The results established in this paper can be extended in many possible ways. Current research has shown that the third, fourth and fifth order model as given in [8] admit a similar port-Hamiltonian structure as (11). It is expected that the same controller as designed in this paper can also be used in these lower order models. While the focus in this paper is about (optimal) frequency regulation, further effort is required to investigate the possibilities of (optimal) voltage control using passive controllers. Another extension is to include transmission line resistances of the network. 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Nominal Unification Revisited Christian Urban TU Munich, Germany urbanc@in.tum.de Nominal unification calculates substitutions that make terms involving binders equal modulo alphaequivalence. Although nominal unification can be seen as equivalent to Miller’s higher-order pattern unification, it has properties, such as the use of first-order terms with names (as opposed to alphaequivalence classes) and that no new names need to be generated during unification, which set it clearly apart from higher-order pattern unification. The purpose of this paper is to simplify a clunky proof from the original paper on nominal unification and to give an overview over some results about nominal unification. 1 Introduction The well-known first-order unification algorithm by Robinson [18] calculates substitutions for variables that make terms syntactically equal. For example the terms f hX, Xi =? f hZ, g hYii can be made syntactically equal with the substitution [X :=g hYi, Z := g hYi]. In first-order unification we can regard variables as “holes” for which the unification algorithm calculates terms with which the holes need be “filled” by substitution. The filling operation is a simple replacement of terms for variables. However, when binders come into play, this simple picture becomes more complicated: We are no longer interested in syntactic equality since terms like a.ha, ci ≈? b.hX, ci (1) should unify, despite the fact that the binders a and b disagree. (Following [19] we write a.t for the term where the name a is bound in t, and ht1 , t2 i for a pair of terms.) If we replace X with term b in (1) we obtain the instance a.ha, ci ≈ b.hb, ci (2) which are indeed two alpha-equivalent terms. Therefore in a setting with binders, unification has to be modulo alpha-equivalence. What is interesting about nominal unification is the fact that it maintains the view from first-order unification of a variable being a “hole” into which a term can be filled. As can be seen, by going from (1) to (2) we are replacing X with the term b without bothering that this b will become bound by the binder. This means the operation of substitution in nominal unification is possibly capturing. A result is that many complications stemming from the fact that binders need to be renamed when a capture-avoiding substitution is pushed under a binder do not apply to nominal unification. Its definition of substitution states that in case of binders σ (a.t) = a.σ (t) Maribel Fernandez (Ed.): 24th International Workshop on Unification (UNIF2010). EPTCS 42, 2010, pp. 1–11, doi:10.4204/EPTCS.42.1 2 Nominal Unification Revisited holds without any side-condition about a and σ . In order to obtain a unification algorithm that, roughly speaking, preserves alpha-equivalence, nominal unification uses the notion of freshness of a name for a term. This will be written as the judgement a # t. For example in (1) it is ensured that the bound name a on the left-hand side is fresh for the term on the right-hand side, that means it cannot occur free on the right-hand side. In general two abstraction terms will not unify, if the binder form one side is free on the other. This condition is sufficient to ensure that unification preserves alpha-equivalence and allows us to regard variables as holes with a simple substitution operation to fill them. Whenever two abstractions with different binders need to be unified, nominal unification uses the operation of swapping two names to rename the bound names. For example when solving the problem shown in (1), which has two binders whose names disagree, then it will attempt to unify the bodies ha, ci and hX, ci, but first applies the swapping (a b) to hX, ci. While it is easy to see how this swapping should affect the name c (namely not at all), the interesting question is how this swapping should affect the variable X? Since variables are holes for which nothing is known until they are substituted for, the answer taken in nominal unification is to suspend such swapping in front of variables. Several such swapping can potentially accumulate in front of variables. In the example above, this means applying the swapping (a b) to hX, ci gives the term h(a b)·X, ci, where (a b) is suspended in front of X. The substitution [X := b] is then determined by unifying the first components of the two pairs, namely a ≈? (a b)·X. We can extract the substitution by applying the swapping to the term a, giving [X := b]. This method of suspending swappings in front of variables is related to unification in explicit substitution calculi which use de Bruijn indices and which record explicitly when indices must be raised [7]. Nominal unification gives a similar answer to the problem of deciding when a name is fresh for a term containing variables, say a # hX, ci. In this case it will record explicitly that a must be fresh for X. (Since we assume a 6= c, it will be that a is fresh for c.) This amounts to the constraint that nothing can be substituted for X that contains a free occurrence of a. Consequently the judgements for freshness #, and also equality ≈, depend on an explicit freshness context recording what variables need to be fresh for. We will give the inductive definitions for # and ≈ in Section 2. This method of recording extra freshness constraints also allows us to regard the following two terms containing a hole (the variable X) a.X ≈ b.X as alpha-equal—namely under the condition that both a and b must be fresh for the variable X. This is defined in terms of judgements of the form {a # X, b # X} ` a.X ≈ b.X The reader can easily determine that any substitution for X that satisfies these freshness conditions will produce two alpha-equivalent terms. Unification problems solved by nominal unification occur frequently in practice. For example typing rules are typically specified as: (x, τ) ∈ Γ var Γ`x:τ Γ ` t1 : τ 1 → τ 2 Γ ` t2 : τ 1 app Γ ` t1 t2 : τ 2 (x, τ 1 )::Γ ` t : τ 2 x ∈ / dom Γ lam Γ ` λ x.t : τ 1 → τ 2 Assuming we have the typing judgement ∅ ` λ y.s : σ , we are interested how the lam-rule, the only one that unifies, needs to be instantiated in order to derive the premises under which λ y.s is typable. This leads to the nominal unification problem ∅ ` λ y.s : σ ≈? Γ ` λ x.t : τ 1 → τ 2 C. Urban 3 which can be solved by the substitution [Γ := ∅, t := (y x) · s, σ := τ 1 → τ 2 ] with the requirement that x needs to be fresh for s (in order to stay close to informal practice, we deviate here from the convention of using upper-case letters for variables and lower-case letters for names). Most closely related to nominal unification is higher-order pattern unification by Miller [14]. Indeed Cheney has shown that higher-order pattern unification problems can be solved by an encoding to nominal unification problems [4]. Levy and Villaret have presented an encoding for the other direction [12]. However, there are crucial differences between both methods of unifying terms with binders. One difference is that nominal unification regards variables as holes for which terms can be substituted in a possibly capturing manner. In contrast, higher-order pattern unification is based on the notion of captureavoiding substitutions. Hence, variables are not just holes, but always need to come with the parameters, or names, the variable may depend on. For example in order to imitate the behaviour of (1), we have to write X a b, explicitly indicating that the variable X may depend on a and b. If we replace X with an appropriate lambda-abstraction, then the dependency can by “realised” via a beta-reduction. This results in unification problems involving lambda-terms to be unified modulo alpha, beta and eta equivalence. In order to make this kind of unification problems to be decidable, Miller introduced restrictions on the form of the lambda-terms to be unified. With this restriction he obtains unification problems that are not only decidable, but also possess (if solvable) most general solutions. Another difference between nominal unification and higher-order pattern unification is that the former uses first-order terms, while the latter uses alpha-equivalence classes. This makes the implementation of higher-order pattern unification in a programming language like ML substantially harder than an implementation of nominal unification. One possibility is to implement elements of alpha-equivalence classes as trees and then be very careful in the treatment of names, generating new ones on the fly. Another possibility is to implement them with de-Bruijn indices. Both possibilities, unfortunately, give rise to rather complicated unification algorithms. This complexity is one reason that higher-order unification has up to now not been formalised in a theorem prover, whereas nominal unification has been formalised twice [19, 10]. One concrete example for the higher-order pattern unification algorithm being more complicated than the nominal unification algorithm is the following: higher-order pattern unification has been part of the infrastructure of the Isabelle theorem prover for many years [17]. The problem, unfortunately, with this implementation is that it unifies a slightly enriched term-language (which allows general beta-redexes) and it is not completely understood how eta and beta equality interact in this algorithm. A formalisation of Isabelle’s version of higher-order pattern unification and its claims is therefore very much desired, since any bug can potentially compromise the correctness of Isabelle. In a formalisation it is important to have the simplest possible argument for establishing a property, since this nearly always yields a simple formalisation. In [19] we gave a rather clunky proof for the property that the equivalence relation ≈ is transitive. This proof has been slightly simplified in [8]. The main purpose of this paper is to further simplify this proof. The idea behind the simplification is taken from the work of Kumar and Norrish who formalised nominal unification in the HOL4 theorem prover [10], but did not report about their simplification in print. After describing the simpler proof in detail, we sketch the nominal unification algorithm and outline some results obtained about nominal unification. 2 Equality and Freshness Two central notions in nominal unification are names, which are called atoms, and permutations of atoms. We assume in this paper that there is a countably infinite set of atoms and represent permutations as finite lists of pairs of atoms. The elements of these lists are called swappings. We therefore write permutations 4 Nominal Unification Revisited as (a1 b1 ) (a2 b2 ) . . . (an bn ); the empty list [] stands for the identity permutation. A permutation π acting on an atom a is defined as  if π · a = a1  a2 def def a1 if π · a = a2 π ·a = a (a1 a2 )::π · a =  π · a otherwise where (a1 a2 )::π is the composition of a permutation followed by the swapping (a1 a2 ). The composition of π followed by another permutation π 0 is given by list-concatenation, written as π 0 @ π, and the inverse of a permutation is given by list reversal, written as π −1 . The advantage of our representation of permutations-as-lists-of-swappings is that we can easily calculate the composition and the inverse of permutations, which are basic operations in the nominal unification algorithm. However, the list representation does not give unique representatives for permutations (for example (a a) 6= []). This is is different from the usual representation of permutations given for example in [9]. There permutations are (unique) bijective functions from atoms to atoms. For permutations-aslists we can define the disagreement set between two permutations as the set of atoms given by def ds π π 0 = {a | π · a 6= π 0 · a} and then regard two permutations as equal provided their disagreement set is empty. However, we do not explicitly equate permutations. The purpose of unification is to make terms equal by substituting terms for variables. The paper [19] defines nominal terms with the following grammar: trm ::= | | | | | hi ht1 , t2 i ft a a.t π·X Units Pairs Function Symbols Atoms Abstractions Suspensions In order to slightly simplify the formal reasoning in the Isabelle/HOL theorem prover, the function symbols only take a single argument (instead of the usual list of arguments). Functions symbols with multiple arguments need to be encoded with pairs. An important point to note is that atoms, written a, b, c, . . . , are distinct from variables, written X, Y, Z, . . . , and only variables can potentially be substituted during nominal unification (a definition of substitution will be given shortly). As mentioned in the Introduction, variables in general need to be considered together with permutations—therefore suspensions are pairs consisting of a permutation and a variable. The reason for this definition is that variables stand for unknown terms, and a permutation applied to a term must be “suspended” in front of all unknowns in order to keep it for the case when any of the unknowns is substituted with a term. Another important point to note is that, although there are abstraction terms, nominal terms are first-order terms: there is no implicit quotienting modulo renaming of bound names. For example the abstractions a.t and b.s are not equal unless a = b and t = s. This has the advantage that nominal terms can be implemented as a simple datatype in programming languages such as ML and also in the theorem prover Isabelle/HOL. In [19] a notion of equality and freshness for nominal terms is defined by two inductive predicates whose rules are shown in Figure 1. This inductive definition uses freshness environments, written ∇, which are sets of atom-and-variable pairs. We often write such environments as {a1 # X 1 , . . . , an # X n }. Rule (≈-abstraction2 ) includes the operation of applying a permutation to a nominal term, which can be recursively defined as C. Urban 5 ∇ ` a # hi (#-unit) ∇ ` a # t1 ∇ ` a # t2 ∇ ` a # ht1 , t2 i ∇ ` a # a.t (#-abstraction1 ) a 6= b ∇`a#b ∇ ` hi ≈ hi (≈-unit) ∇ ` t1 ≈ t2 ∇ ` a.t1 ≈ a.t2 ∇ ` a # t a 6= b ∇ ` a # b.t ∇ ` a # π·X ∇ ` s1 ≈ s2 ∇ ` ht1 , s1 i ≈ ht2 , s2 i ∇`a≈a (#-abstraction2 ) ∇ ` t1 ≈ t2 ∇ ` f t1 ≈ f t2 (≈-function symbol) ∇ ` t1 ≈ (a b) · t2 ∇ ` a.t1 ≈ b.t2 (≈-atom) (#-function symbol) (#-suspension) (≈-pair) a 6= b ∇ ` a # t2 (≈-abstraction1 ) ∇`a#ft (π −1 · a, X) ∈ ∇ (#-atom) ∇ ` t1 ≈ t2 ∇`a#t (#-pair) ∀ c∈ds π π 0. (c, X) ∈ ∇ ∇ ` π·X ≈ π 0·X (≈-abstraction2 ) (≈-suspension) Figure 1: Inductive definitions for freshness and equality of nominal terms. π · (hi) π · (ht1 , t2 i) π · (F t) π· (π 0·X) π · (a. t) def = def = def hi hπ · t1 , π · t2 i = F (π · t) def (π @ π 0)·X = def = (π · a). (π · t) where the clause for atoms is given in (2). Because we suspend permutations in front of variables (see penultimate clause), it will in general be the case that π · t 6= π 0 · t (3) even if the disagreement set of π and π 0 is empty. Note that permutations acting on abstractions will permute both, the “binder” a and the “body” t. In order to show the correctness of the nominal unification algorithm in [19], one first needs to establish that ≈ is an equivalence relation in the sense of (i) (ii) (iii) ∇`t≈t ∇ ` t1 ≈ t2 implies ∇ ` t2 ≈ t1 ∇ ` t1 ≈ t2 and ∇ ` t2 ≈ t3 imply ∇ ` t1 ≈ t3 (reflexivity) (symmetry) (transitivity) The first property can be proved by a routine induction over the structure of t. Given the “unsymmetric” formulation of the (≈-abstraction2 ) rule, the fact that ≈ is symmetric is at first glance surprising. Furthermore, a direct proof by induction over the rules seems tricky, since in the (≈-abstraction2 ) case one needs to infer ∇ ` t2 ≈ (b a) · t1 from ∇ ` (a b) · t2 ≈ t1 . This needs several supporting lemmas about freshness and equality, but ultimately requires that the transitivity property is proved first. Unfortunately, a direct proof by rule-induction for transitivity seems even more difficult and we did not manage to find one in [19]. Instead we resorted to a clunky induction over the size of terms (since size is preserved 6 Nominal Unification Revisited under permutations). To make matters worse, this induction over the size of terms needed to be loaded with two more properties in order to get the induction through. The authors of [8] managed to split up this bulky induction, but still relied on an induction over the size of terms in their transitivity proof. The authors of [10] managed to do considerably better. They use a clever trick in their formalisation of nominal unification in HOL4 (their proof of equivalence is not shown in the paper). This trick yields a simpler and more direct proof for transitivity, than the ones given in [19, 8]. We shall below adapt the proof by Kumar and Norrish to our setting of (first-order) nominal terms1 . First we can establish the following property. Lemma 1. If ∇ ` a # t then also ∇ ` (π · a) # (π · t), and vice versa. The proof is by a routine induction on the structure of t and we omit the details. Following [19] we can next attempt to prove that freshness is preserved under equality (Lemma 3 below). However here the trick from [10] already helps to simplify the reasoning. In [10] the notion of weak equivalence, written as ∼, is defined as follows t ∼ t0 hi ∼ hi a∼a f t ∼ f t0 t1 ∼ s1 t2 ∼ s2 t ∼ t0 ds π π 0 = ∅ ht1 , t2 i ∼ hs1 , s2 i a.t ∼ a.t 0 π·X ∼ π 0·X This equivalence is said to be weak because two terms can only differ in the permutations that are suspended in front of variables. Moreover, these permutations can only be equal (in the sense that is their disagreement set must be empty). One advantage of this definition is that we can show π · t ∼ π 0 · t provided ds π π 0 = ∅ (4) by an easy induction on t. As noted in (3), this property does not hold when formulated with =. It is also straightforward to show that Lemma 2. (i) If ∇ ` a # t1 and t1 ∼ t2 then ∇ ` a # t2 . (ii) If ∇ ` t1 ≈ t2 and t2 ∼ t3 then ∇ ` t1 ≈ t3 . by induction over the relations ∼ and ≈, respectively. The reason that these inductions go through with ease is that the relation ∼ excludes the tricky cases where abstractions differ in their “bound” atoms. Using these two properties together with (4), it is straightforward to establish: Lemma 3. If ∇ ` t1 ≈ t2 and ∇ ` a # t1 then ∇ ` a # t2 . Proof. By induction on the first judgement. The only interesting case is the rule (≈-abstraction2 ) where we need to establish ∇ ` a # d.t2 from the assumption (∗) ∇ ` a # c.t1 with the side-conditions c 6= d and a 6= d. Using these side-condition, we can reduce our goal to establishing ∇ ` a # t2 . We can also discharge the case where a = c, since we know that ∇ ` c # t2 holds by the side-condition of (≈abstraction2 ). In case a 6= c, we can infer ∇ ` a # t1 from (∗), and use the induction hypothesis to conclude with ∇ ` a # (c d) · t2 . Using Lemma 1 we can infer that ∇ ` (c d) · a # (c d)(c d) · t2 holds, whose left-hand side simplifies to just a (we have that a 6= d and a 6= c). For the right-hand side we can prove (c d)(c d) · t2 ∼ t2 , since ds ((c d)(c d)) [] = ∅. From this we can conclude this case using Lemma 2(i). 1 Their formalisation in HOL4 introduces an indirection by using a quotient construction over nominal terms. This quotient construction does not translate into a simple datatype definition for nominal terms. C. Urban 7 The point in this proof is that without the weak equivalence and without Lemma 2, we would need to perform many more “reshuffles” of swappings than the single reference to ∼ in the proof above [19]. The next property on the way to establish transitivity proves the equivariance for ≈. Lemma 4. If ∇ ` t1 ≈ t2 then ∇ ` π · t1 ≈ π · t2 . Also with this lemma the induction on ≈ does not go through without the help of weak equivalence, because in the (≈-abstraction2 )-case we need to show that ∇ ` π · t1 ≈ π · (a b) · t2 implies ∇ ` π · t1 ≈ (π·a π·b) · π · t2 . While it is easy to show that the right-hand sides are equal, one cannot make use of this fact without a notion of transitivity. Proof. By induction on ≈. The non-trivial case is the rule (≈-abstraction2 ) where we know ∇ ` π · t1 ≈ π · (a b) · t2 by induction hypothesis. We can show that π @ (a b) · t2 ∼ (π·a π·b) @ π · t2 holds (the corresponding disagreement set is empty). Using Lemma 2(ii), we can join both judgements and conclude with ∇ ` π · t1 ≈ (π·a π·b) · π · t2 . The next lemma relates the freshness and equivalence relations. Lemma 5. If ∀ a∈ds π π 0. ∇ ` a # t then ∇ ` π · t ≈ π 0 · t, and vice versa. Proof. By induction on t generalising over the permutation π 0. The generalisation is needed in order to get the abstraction case through. The crucial lemma in [10], which will allow us to prove the transitivity property by a straightforward rule induction, is the next one. Its proof still needs to analyse several cases, but the reasoning is much simpler than in the proof by induction over the size of terms in [19]. Lemma 6. If ∇ ` t1 ≈ t2 and ∇ ` t2 ≈ π · t2 then ∇ ` t1 ≈ π · t2 . Proof. By induction on the first ≈-judgement with a generalisation over π. The interesting case is (≈abstraction2 ). We know ∇ ` b.t2 ≈ (π · b).(π · t2 ) and have to prove ∇ ` a.t1 ≈ (π · b).(π · t2 ) with a 6= b. We have to analyse several cases about a equal equal with π · b, and b being equal with π · b. Let us give the details for the case a 6= π · b and b 6= π · b. From the assumption we can infer (∗) ∇ ` b # π · t2 and (∗∗) ∇ ` t2 ≈ (b π·b) · π · t2 . The side-condition on the first judgement gives us ∇ ` a # t2 . We have to show ∇ ` a # π · t2 and ∇ ` t1 ≈ (a π·b) · π · t2 . To infer the first fact, we use ∇ ` a # t2 together with (∗∗) and Lemmas 3 and 1. For the second, the induction hypothesis states that for any π we have ∇ ` t1 ≈ π · (a b) · t2 provided ∇ ` (a b) · t2 ≈ π · (a b) · t2 holds. We use the induction hypothesis with def the permutation π = (a π·b) @ π @ (a b). This means after simplification the precondition of the IH we need to establish is (∗∗∗) ∇ ` (a b) · t2 ≈ (a π·b) · π · t2 . By Lemma 5 we can transform (∗∗) to ∀ c ∈ ds [] ((b, π·b) @ π). ∇ ` c # t2 . Similarly with (∗∗∗). Furthermore we can show that ds (a b) ((a π·b) @ π) ⊆ ds [] ((b π·b) @ π) ∪ {a, π·b} holds. This means it remains to show that ∇ ` a # t2 (which we already inferred above) and ∇ ` π·b # t2 hold. For the latter, we consider the cases b = π · π · b and b 6= π · π · b. In the first case we infer ∇ ` π·b # t2 from (∗) using Lemma 1. In the second case we have that π · b ∈ ds [] ((b π·b) @ π). So finally we can use the induction hypothesis, which simplified gives us ∇ ` t1 ≈ (a π·b) · π · t2 as needed. With this lemma under our belt, we are finally in the position to prove the transitivity property. Lemma 7. If ∇ ` t1 ≈ t2 and ∇ ` t2 ≈ t3 then ∇ ` t1 ≈ t3 . 8 Nominal Unification Revisited Proof. By induction on the first judgement generalising over t3 . We then analyse the possible instances for the second judgement. The non-trivial case is where both judgements are instances of the rule (≈abstraction2 ). We have ∇ ` t1 ≈ (a b) · t2 and (∗) ∇ ` t2 ≈ (b c) · t3 with a, b and c being distinct. We also have (∗∗) ∇ ` a # t2 and (∗∗∗) ∇ ` b # t3 . We have to show ∇ ` a # t3 and ∇ ` t1 ≈ (a c) · t3 . The first fact is a simple consequence of (∗) and the Lemmas 1 and 3. For the other case we can use the induction hypothesis to infer our proof obligation, provided we can establish that ∇ ` (a b) · t2 ≈ (a c) · t3 holds. From (∗) we have ∇ ` (a b) · t2 ≈ (a b)(b c) · t3 using Lemma 4. We also establish that ∇ ` (a b)(b c) · t3 ≈ (b c)(a b)(b c) · t3 holds. By Lemma 5 we have to show that all atoms in the disagreement set are fresh w.r.t. t3 . The disagreement set is equal to {a, b}. For b the property follows from (∗∗∗). For a we use (∗) and (∗∗). So we can use Lemma 6 to infer (∗∗∗∗) ∇ ` (a b) · t2 ≈ (b c)(a b)(b c) · t3 . It remains to show that ∇ ` (a b) · t2 ≈ (a c) · t3 holds. We can do so by using (∗∗∗∗) and Lemma 2, and showing that (b c)(a b)(b c) · t3 ∼ (a c) · t3 holds. This in turn follows from the fact that the disagreement set ds ((b c)(a b)(b c)) (a c) is empty. This concludes the case. Once transitivity is proved, reasoning about ≈ is rather straightforward. For example symmetry is a simple consequence. Lemma 8. If ∇ ` t1 ≈ t2 then ∇ ` t2 ≈ t1 . Proof. By induction on ≈. In the (≈-abstraction2 ) we have ∇ ` (a b) · t2 ≈ t1 and need to show ∇ ` t2 ≈ (b a) · t1 . We can do so by inferring ∇ ` (b a)(a b) · t2 ≈ (b a) · t1 using Lemma 4. We can also show ∇ ` (b a)(a b) · t2 ≈ t2 using Lemma 5. We can join both facts by transitivity to yield the proof obligation. To sum up, the neat trick with using ∼ from [10] has allowed us to give a direct, structural, proof for equivalence of ≈. The formalisation of this direct proof in Isabelle/HOL is approximately half the size of the formalised proof given in [19]. 3 An Algorithm for Nominal Unification In this section we sketch the algorithm for nominal unification presented in [19]. We refer the reader to that paper for full details. The purpose of nominal unification algorithm is to calculate substitutions that make terms ≈-equal. The substitution operation for nominal terms is defined as follows: def σ (a) = a( π · σ (X) if X ∈ dom σ def σ (π·X) = π·X otherwise def σ (a.t) = a.σ (t) def σ (ht1 , t2 i) = hσ (t1 ), σ (t2 )i def σ (f t) = f σ (t) There are two kinds of problems the nominal unification algorithms solves: t1 ≈? t2 a #? t C. Urban 9 The first are called equational problems, the second freshness problems. Their respective interpretation is “can the terms t1 and t2 be made equal according to ≈?” and “can the atom a be made fresh for t according to #?”. A solution for each kind of problems is a pair (∇, σ ) consisting of a freshness environment and a substitution such that ∇ ` σ (t1 ) ≈ σ (t2 ) ∇ ` a # σ (t) hold. Note the difference with first-order unification and higher-order pattern unification where a solution consists of a substitution only. An example where nominal unification calculates a non-trivial freshness environment is the equational problem a.X ≈? b.X which is solved by the solution ({a # X, b # X}, []). Solutions in nominal unification can be ordered so that the unification algorithm produces always most general solutions. This ordering is defined very similar to the standard ordering in first-order unification. The nominal unification algorithm in [19] is defined in the usual style of rewriting rules that transform sets of unification problems to simpler ones calculating a substitution and freshness environment on the way. The transformation rule for pairs is {ht1 , t2 i ≈? hs1 , s2 i, . . . } =⇒ {t1 ≈? s1 , t2 ≈? s2 , . . . } There are two rules for abstractions depending on whether or not the binders agree. {a.t ≈? a.s, . . . } =⇒ {t ≈? s, . . . } {a.t ≈? b.s, . . . } =⇒ {t ≈? (a b) · s, a #? s, . . . } One rule that is also interesting is for unifying two suspensions with the same variable {π·X ≈? π 0·X,. . . } =⇒ {a #? X | a ∈ ds π π 0} ∪ {. . . } What is interesting about nominal unification is that it never needs to create fresh names. As can be seen from the abstraction rules, no new name needs to be introduced in order to unify abstractions. It is the case that all atoms in a solution, occur already in the original problem. This has the attractive consequence that nominal unification can dispense with any new-name-generation facility. This makes it easy to implement and reason about the nominal unification algorithm. Clearly, however, the running time of the algorithm using the rules sketched above is exponential in the worst-case, just like the simpleminded first-order unification algorithm without sharing. 4 Applications and Complexity of Nominal Unification Having designed a new algorithm for unification, it is an obvious step to include it into a logic programming language. This has been studied in the work about αProlog [5] and αKanren [1]. The latter is a system implemented on top of Scheme and is more sophisticated than the former. The point of these variants of Prolog is that they allow one to implement inference rule systems in a very concise and declarative manner. For example the typing rules for simply-typed lambda-terms shown in the Introduction can be implemented in αProlog as follows: 10 Nominal Unification Revisited type (Gamma, var(X), T) :- member (X,T) Gamma. type (Gamma, app(M,N), T2) :type (Gamma, M, arrow(T1, T2)), type (Gamma, N, T1). type (Gamma, lam(x.M) , arrow(T1, T2)) / x # Gamma :type ((x,T1)::Gamma, M, T2). member X X::Tail. member X Y::Tail :- member X Tail. The shaded boxes show two novel features of αProlog. Abstractions can be written as x.(−); but note that the binder x can also occur as a “non-binder” in the body of clauses—just as in the clauses on “paper.” The side-condition x # Gamma ensures that x is not free in any term substituted for Gamma. The novel features of αProlog and αKanren can be appreciated when considering that similarly simple implementations in “vanilla” Prolog (which, surprisingly, one can find in textbooks [15]) are incorrect, as they give types to untypable lambda-terms. An simple implementation of a first-order theorem prover in αKanren has been given in [16]. When implementing a logic programming language based on nominal unification it becomes important to answer the question about its complexity. Surprisingly, this turned out to be a difficult question. Surprising because nominal unification, like first-order unification, uses simple rewrite rules defined over first-order terms and uses a substitution operation that is a simple replacement of terms for variables. One would hope the techniques from efficient first-order unification algorithms carry over to nominal unification. This is unfortunately only partially the case. Quadratic algorithms for nominal unification were obtained by Calves and Fernandez [3, 2] and independently by Levy and Villaret [13]. These are the best bounds we have for nominal unification so far. 5 Conclusion Nominal unification was introduced in [19]. It unifies terms involving binders modulo a notion of alphaequivalence. In this way it is more powerful than first-order unification, but is conceptually much simpler than higher-order pattern unification. Unification algorithms are often critical infrastructure in theorem provers. Therefore it is important to formalise these algorithms in order to ensure correctness. Nominal unification has been formalised twice, once in [19] in Isabelle/HOL and another in [10] in HOL4. The latter formalises a more efficient version of nominal unification based on triangular substitutions. The main purpose of this paper is to simplify the transitivity proof for ≈. This in turn simplified the formalisation in Isabelle/HOL. There have been several fruitful avenues of research that use nominal unification as basic building block. For example the work on αLeanTap [16]. There have also been several works that go beyond the limitation of nominal unification where bound names are restricted to be constant symbols that are not substitutable [11, 6]. References [1] W. Byrd and D. Friedman. αKanren: A Fresh Name in Nominal Logic Programming Languages. In Proc. of the 8th Workshop on Scheme and Functional Programming, pages 79–90, 2007. C. Urban 11 [2] C. Calvès. Complexity and Implementation of Nominal Algorithms. PhD thesis, King’s College London, 2010. [3] C. Calvès and M. Fernández. A Polynomial Nominal Unification Algorithm. Theoretical Computer Science, 403(2-3):285–306, 2008. [4] J. Cheney. Relating Nominal and Higher-Order Pattern Unification. In Proc. of the 19th International Workshop on Unification (UNIF), pages 104–119, 2005. [5] J. Cheney and C. Urban. Alpha-Prolog: A Logic Programming Language with Names, Binding, and αEquivalence. In Proc. of the 20th International Conference on Logic Programming (ICLP), volume 3132 of LNCS, pages 269–283, 2004. [6] R. Clouston and A. Pitts. Nominal Equational Logic. In Computation, Meaning and Logic, Articles dedicated to Gordon Plotkin, volume 172 of ENTCS, pages 223–257. 2007. [7] G. Dowek, T. Hardin, C. Kirchner, and F. Pfenning. Higher-Order Unification via Explicit Substitutions: the Case of Higher-Order Patterns. In Proc. of the Joint International Conference and Symposium on Logic Programming (JICSLP), pages 259–273, 1996. [8] M. Fernández and J. Gabbay. Nominal Rewriting. Information and Computation, 205:917–965, 2007. [9] B. Huffman and C. Urban. A New Foundation for Nominal Isabelle. In Proc. of the 1st Interactive Theorem Prover Conference (ITP), volume 6172 of LNCS, pages 35–50, 2010. [10] R. Kumar and M. Norrish. (Nominal) Unification by Recursive Descent with Triangular Substitutions. In Proc. of the 1st Interactive Theorem Prover Conference (ITP), volume 6172 of LNCS, pages 51–66, 2010. [11] M. Lakin and A. Pitts. Resolving Inductive Definitions with Binders in Higher-Order Typed Functional Programming. In Proc. of the 18th European Symposium on Programming (ESOP), volume 5502 of LNCS, pages 47–61, 2009. [12] J. Levy and M. Villaret. Nominal Unification from a Higher-Order Perspective. In Proc. of the 19th International Conference on Rewriting Techniques and Applications (RTA), volume 5117 of LNCS, pages 246–260, 2008. [13] J. Levy and M. Villaret. An Efficient Nominal Unification Algorithm. In Proc. of the 21th International Conference on Rewriting Techniques and Applications (RTA), volume 6 of LIPIcs, pages 246–260, 2010. [14] D. Miller. A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification. Journal of Logic and Computation, 1(4):497–536, 1991. [15] J. C. Mitchell. Concepts in Programming Languages. CUP Press, 2003. [16] J. Near, W. Byrd, and D. Friedman. αLeanTAP: A Declarative Theorem Prover for First-Order Classical Logic. In Proc. of the 24th International Conference on Logic Programming (ICLP), volume 5366 of LNCS, pages 238–252, 2008. [17] T. Nipkow. Functional Unification of Higher-Order Patterns. In Proc. of the 8th IEEE Symposium of Logic in Computer Science (LICS), pages 64–74, 1993. [18] J. A. Robinson. A Machine Oriented Logic Based on the Resolution Priciple. JACM, 12(1):23–41, 1965. [19] C. Urban, A.M. Pitts, and M.J. Gabbay. Nominal Unification. Theoretical Computer Science, 323(1-3):473– 497, 2004. 

EXTENSIONS OF THE BENSON-SOLOMON FUSION SYSTEMS arXiv:1701.06949v1 [] 24 Jan 2017 ELLEN HENKE AND JUSTIN LYND To Dave Benson on the occasion of his second 60th birthday Abstract. The Benson-Solomon systems comprise the only known family of simple saturated fusion systems at the prime two that do not arise as the fusion system of any finite group. We determine the automorphism groups and the possible almost simple extensions of these systems and of their centric linking systems. 1. Introduction b is controlled b with normal subgroup G such that C b (G) 6 G, the structure of G Given a group G G by G and Aut(G). On the other hand, if Fb is a saturated fusion system with normal subsystem F such that CFb (F) 6 F, it is in general more difficult to describe the possibilities for Fb given F. For example, if F = FS (G) is the fusion system of the finite group G, then it is often the case that some extensions of F do not arise as extensions of G, but rather as extensions of a different finite group with the same fusion system. It is also possible that there are extensions of FS (G) that are exotic in the sense that they are induced by no finite group, but to our knowledge, no example of this is currently known. It has become clear that working with the fusion system alone is not on-its-face enough for describing extensions [Oli10, AOV12, Oli16, BMO16]; one should instead work with an associated linking system. In particular, when F = FS (G) and L = LcS (G) for some finite group G, there is a natural map κG : Out(G) → Out(L), and whether the extensions of F all come from finite groups has been shown to be related to asking whether κG′ is split surjective for some other finite group G′ with the same Sylow subgroup and with F = FS (G′ ) [AOV12, Oli16]. While this machinery is not directly applicable to the problem of determining extensions of exotic fusion systems, there are other tools for use, such as [Oli10]. There is one known family of simple exotic fusion systems at the prime 2, the Benson-Solomon systems. They were first predicted by Dave Benson [Ben98] to exist as finite versions of a 2local compact group associated to the 2-compact group DI(4) of Dwyer and Wilkerson [DW93]. They were later constructed by Levi and Oliver [LO02] and Aschbacher and Chermak [AC10]. The purpose of this paper is determine the automorphism groups of the Benson-Solomon fusion and centric linking systems, and use that information to determine the fusion systems having one of these as its generalized Fitting subsystem. This information is needed within certain portions of Aschbacher’s program to classify simple fusion systems of component type at the prime 2. In particular, it is presumably needed within an involution centralizer problem for Date: April 4, 2018. 2000 Mathematics Subject Classification. Primary 20D20, Secondary 20D05. Key words and phrases. fusion system, linking system, Benson-Solomon fusion system, group extension. Justin Lynd was partially supported by NSA Young Investigator Grant H98230-14-1-0312. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 707758. 1 these systems. Some of the work on automorphisms of these systems appears in the standard references [LO02, LO05, AC10], and part of our aim is to complete the picture. The plan for this paper is as follows. All maps are written on the left. In Section 2, we recall the various automorphism groups of fusion and linking systems and the maps between them, following [AOV12]. In Section 3, we look at automorphisms of the fusion and linking systems of Spin7 (q) and of the Benson-Solomon systems. We show in Theorem 3.10 that the outer automorphism group of the latter is a cyclic group of field automorphisms of 2-power order. Finally, we show in Theorem 4.3 that the systems having a Benson-Solomon generalized Fitting subsystem are uniquely determined by the outer automorphisms they induce on the fusion system, and that all such extensions are split. We would like to thank Jesper Grodal, Ran Levi, and Bob Oliver for helpful conversations. 2. Automorphisms of fusion and linking systems We refer to [AKO11] for the definition of a saturated fusion system, and also for the definition of a centric subgroup of a fusion system. Let F be a saturated fusion system over the finite p-group S, and write F c for the collection of F-centric subgroups. Whenever g is an element of a finite group, we write cg for the conjugation homomorphism x 7→ gx = gxg −1 and its restrictions. 2.1. Background on linking systems. Whenever ∆ is an overgroup-closed, F-invariant collection of subgroups of S, we have the transporter category T∆ (S) with those objects. This is the full subcategory of the transporter category TS (S) where the objects are subgroups of S, and morphisms are the transporter sets: NS (P, Q) = {s ∈ S | sP s−1 6 Q} with composition given by multiplication in S. A linking system associated to F is a nonempty category L with object set ∆, together with functors (2.1) δ π T∆ (S) −−−−→ L −−−−→ F. The functor δ is the identity on objects and injective on morphisms, while π is the inclusion on objects and surjective on morphisms. Write δP,Q for the corresponding injection NS (P, Q) → MorL (P, Q) on morphisms, write δP for δP,P , and use similar notation for π. The category and its structural functors are subject to several axioms which may be found in [AKO11, Definition II.4.1]. In particular, Axiom (B) states that for all objects P and Q in L and each g ∈ NS (P, Q), we have πP,Q(δP,Q (g)) = cg ∈ HomF (P, Q). A centric linking system is a linking system with ∆ = F c . Given a finite group G with Sylow p-subgroup S, the canonical centric linking system for G is the category LcS (G) with objects the p-centric subgroups P 6 S (namely those P whose centralizer satisfies CG (P ) = Z(P ) × Op′ (CG (P ))), and with morphisms the orbits of the transporter set NG (P, Q) = {g ∈ G | gP g−1 6 Q} under the right action of Op′ (CG (P )). 2.1.1. Distinguished subgroups and inclusion morphisms. The subgroups δP (P ) 6 AutL (P ) are called distinguished subgroups. When P 6 Q, the morphism ιP,Q := δP,Q (1) ∈ MorL (P, Q) is the inclusion of P into Q. 2.1.2. Axiom (C) for a linking system. We will make use of Axiom (C) for a linking system, which says that for each morphism ϕ ∈ MorL (P, Q) and element g ∈ NS (P ), the following identity holds between morphisms in MorL (P, Q): ϕ ◦ δP (g) = δQ (π(ϕ)(g)) ◦ ϕ. 2 2.1.3. Restrictions in linking systems. For each morphism ψ ∈ MorL (P, Q), and each P0 , Q0 ∈ Ob(L) such that P0 6 P , Q0 6 Q, and π(ψ)(P0 ) 6 Q0 , there is a unique morphism ψ|P0 ,Q0 ∈ MorL (P0 , Q0 ) (the restriction of ψ) such that ψ ◦ ιP0 ,P = ιQ0 ,Q ◦ ψ|P0 ,Q0 ; see [Oli10, Proposition 4(b)] or [AKO11, Proposition 4.3]. Note that in case ψ = δP,Q (s) for some s ∈ NS (P, Q), it can be seen from Axioms (B) and (C) that ψ|P0 ,Q0 = δP0 ,Q0 (s). 2.2. Background on automorphisms. 2.2.1. Automorphisms of fusion systems. An automorphism of F is, by definition, determined by its effect on S: define Aut(F) to be the subgroup of Aut(S) consisting of those automorphisms α αϕα−1 which preserve fusion in F in the sense that the homomorphism given by α(P ) −−−−→ α(Q) is in ϕ F for each morphism P − → Q in F. The automorphisms AutF (S) of S in F thus form a normal subgroup of Aut(F), and the quotient Aut(F)/ AutF (S) is denoted by Out(F). 2.2.2. Automorphisms of linking systems. A self-equivalence of L is said to be isotypical if it sends distinguished subgroups to distinguished subgroups, i.e. α(δP (P )) = δα(P ) (α(P )) for each object P . It sends inclusions to inclusions provided α(ιP,Q ) = ια(P ),α(Q) whenever P 6 Q. The monoid Aut(L) of isotypical self-equivalences that send inclusions to inclusions is in fact a group of automorphisms of the category L, and this has been shown to be the most appropriate group of automorphisms to consider. Note that Aut(L) has been denoted by AutItyp (L) in [AKO11,AOV12] and elsewhere. When α ∈ Aut(L) and P is an object with α(P ) = P , we denote by αP the automorphism of AutL (P ) induced by α. The group AutL (S) acts by conjugation on L in the following way: given γ ∈ AutL (S), consider the functor cγ ∈ Aut(L) which is cγ (P ) = π(γ)(P ) on objects, and which sends a morphism ϕ P − → Q in L to the morphism γϕγ −1 from cγ (P ) to cγ (Q) after replacing γ and γ −1 by the appropriate restrictions (introduced in §2.1.3). Note that when γ = δS (s) for some s ∈ S, then cγ (P ) is conjugation by s on objects, and cγ (ϕ) = δQ,sQ (s) ◦ ϕ ◦ δsP,P (s−1 ) for each morphism ϕ ∈ MorL (P, Q) by the remark on distinguished morphisms in §2.1.3. In particular, when L = LcS (G) for some finite group G, cγ is truly just conjugation by s on morphisms. The image of AutL (S) under the map γ 7→ cγ is seen to be a normal subgroup of Aut(L); the outer automorphism group of L is Out(L) := Aut(L)/{cγ | γ ∈ AutL (S)}; we refer to Lemma 1.14(a) and the surrounding discussion in [AOV12] for more details. This group is denoted by Outtyp (L) in [AKO11, AOV12] and elsewhere. 2.2.3. From linking system automorphisms to fusion system automorphisms. There is a group homomorphism (2.2) µ e : Aut(L) −→ Aut(F), given by restriction to S ∼ e induces a = δS (S) 6 AutL (S); see [Oli10, Proposition 6]. The map µ homomorphism on quotient groups µ : Out(L) −→ Out(F). We write µL (or µG when L = LcS (G)) whenever we wish to make clear which linking system we are working with; similar remarks hold for µ e. As shown in [AKO11, Proposition II.5.12], ker(µ) has an interesting cohomological interpretation as the first cohomology group of the center functor 3 ZF on the orbit category of F-centric subgroups, and ker(e µ) is correspondingly a certain group of normalized 1-cocycles for this functor. 2.2.4. From group automorphisms to fusion system and linking system automorphisms. We also need to compare automorphisms of groups with the automorphisms of their fusion and linking systems. If G is a finite group with Sylow p-subgroup S, then each outer automorphism of G is represented by an automorphism that fixes S. This is a consequence of the transitive action of G on its Sylow subgroups. More precisely, there is an exact sequence: incl g7→cg 1 → Z(G) −−→ NG (S) −−−→ Aut(G, S) → Out(G) → 1. where Aut(G, S) is the subgroup of Aut(G) consisting of those automorphisms that leave S invariant. For each pair of p-centric subgroups P, Q 6 S and each α ∈ Aut(G, S), α induces an isomorphism Op′ (CG (P )) → Op′ (CG (α(P ))) and a bijection NG (P, Q) → NG (α(P ), α(Q)). Thus, there is a group homomorphism κ eG : Aut(G, S) → Aut(LcS (G)) which sends α ∈ Aut(G, S) to the functor which is α on objects, and also α on morphisms in the way just mentioned. This map sends the image of NG (S) to {cγ | γ ∈ AutLcS (G) (S)}, and so induces a homomorphism κG : Out(G) → Out(LcS (G)) on outer automorphism groups. It is straightforward to check that the restriction to S of any member of Aut(G, S) is an automorphism of the fusion system FS (G). Indeed, for every α ∈ Aut(G, S), the automorphism α|S of FS (G) is just the image of α under µ eG ◦ κ eG . 2.2.5. Summary. What we will need in our proofs is summarized in the following commutative diagram, which is an augmented and updated version of the one found in [AKO11, p.186]. Z(F) incl 1 1 1  // Z(S)  b1 (O(F c ), ZF ) // Z  // lim1 (ZF ) e λ δS (2.3) Z(F)  // AutL (S)  // Aut(L) πS 1  // AutF (S)  1 ←− λ  // Out(L) // Aut(F)  1 // 1 µ µ e  // 1  // Out(F) // 1  1 All sequences in this diagram are exact. Most of this either is shown in the proof of [AKO11, Proposition II.5.12], or follows from the above definitions. The first and second rows are exact by this reference, except that the diagram was not augmented by the maps out of Z(F) (the center of F); exactness at Z(S) and AutL (S) is shown by following the proof there. Given [AKO11, Proposition II.5.12], exactness of the last column is equivalent to the uniqueness of centric linking 4 systems, a result of Chermak. In all the cases needed in this article, it follows from [LO02, Lemma 3.2]. The second-to-last column is then exact by a diagram chase akin to that in a 5-lemma for groups. 3. Automorphisms The isomorphism type of the fusion systems of the Benson-Solomon systems FSol (q), as q ranges over odd prime powers, is dependent only on the 2-share of q 2 − 1 by [COS08, Theorem B]. Since the centralizer of the center of the Sylow group is the fusion system of Spin7 (q), the same holds also for the fusion systems of these groups. For this reason, and because of Proposition 3.2 below, k it will be convenient to take q = 52 for the sequel. Thus, we let F be the algebraic closure of the field with five elements. 3.1. Automorphisms of the fusion system of Spin7 (q). Let H̄ = Spin7 (F). Fix a maximal torus T̄ of H̄. Thus, H̄ is generated by the T̄ -root groups X̄α = {xα (λ) : λ ∈ F} ∼ = (F, +), as α ranges over the root system of type B3 , and is subject to the Chevalley relations of [GLS98, Theorem 1.12.1]. For any power q1 of 5, we let ψq1 denote the standard Frobenius endomorphism of H̄, namely the endomorphism of H̄ which acts on the root groups via ψq1 (xα (λ)) = xα (λq1 ). Set H := CH̄ (ψq ). Thus, H = Spin7 (q) since H̄ is of universal type (see [GLS98, Theorem 2.2.6(f)]). Also, T := CT̄ (ψq ) is a maximal torus of H. For each power q1 of 5, the Frobenius endomorphism ψq1 of H̄ acts on H in the way just mentioned, and it also acts on T by raising each element to the power q1 . For ease of notation, we denote by ψq1 also the automorphism of H induced by ψq1 . The normalizer NH (T ) contains a Sylow 2-subgroup H [GL83, 10-1(2)], and NH (T )/T is isomorphic to C2 × S4 , the Weyl group of B3 . Applying a Frattini argument, we see that there is a Sylow 2-subgroup S of NH (T ) invariant under ψ5 , and we fix this choice of S for the remainder. The automorphism groups of the Chevalley groups were determined by Steinberg [Ste60], and in particular, (3.1) Out(H) = Outdiag(H) × Φ ∼ = C2 × C2k , where Φ is the group of field automorphisms, and where Outdiag(H) is the group of outer automorphisms of H induced by NT̄ (H) [GLS98, Theorem 2.5.1(b)]. We mention that S is normalized by every element of NT̄ (H). So we find canonical representatives of the elements of Φ and of Outdiag(H) in Aut(H, S). We need to be able to compare automorphisms of the group with automorphisms of the fusion and linking systems, and this has been carried out in full generality by Broto, Møller, and Oliver [BMO16] for groups of Lie type. Let FSpin (q) and LcSpin (q) be the associated fusion and centric linking systems over S of the group H, and recall the maps µH and κH from §§2.2.3, 2.2.4 Proposition 3.2. The maps µH and κH are isomorphisms, and hence k k Out(LSpin (52 )) ∼ = Out(FSpin (52 )) ∼ = C2 × C2k . Proof. That µH is an isomorphism follows from (2.3) and [LO02, Lemma 3.2]. Also, κH is an isomorphism by [BMO16, Propositions 5.14, 5.15].  5 3.2. Automorphisms of the Benson-Solomon systems. We keep the notation from the previous subsection. We denote by F := FSol (q) a Benson-Solomon fusion system over the 2group S ∈ Syl2 (H) fixed above, and by L := LcSol (q) an associated centric linking system with structural functors δ and π. Set T2 := T ∩ S, the 2-torsion in the maximal torus T of H. Then T2 is homocyclic of rank three and of exponent 2k+2 [AC10, §4]. Also, Z(S) 6 T2 is of order 2, and NF (Z(S)) = CF (Z(S)) = FSpin (q) is a fusion system over S. Since Z(S) is contained in every F-centric subgroup, by Definition 6.1 and Lemma 6.2 of [BLO03], we may take NL (Z(S)) = CL (Z(S)) for the centric linking system of Spin7 (q). By the items just referenced, CL (Z(S)) is a subcategory of L with the same objects, and with morphisms those morphisms ϕ in L such that π(ϕ)(z) = z. Further, CL (Z(S)) has the same inclusion functor δ, and the projection functor for CL (Z(S)) is the restriction of π. (This was also shown in [LO02, Lemma 3.3(a,b)].) Write Fz for FSpin (q) and Lz for CL (Z(S)) for short. Each member of Aut(F) fixes Z(S) and so Aut(F) ⊆ Aut(Fz ). So the inclusion map from Aut(F) to Aut(Fz ) can be thought of as a “restriction map” (3.3) ρ : Aut(F) −→ Aut(Fz ) given by remembering only that an automorphism preserves fusion in Fz . We want to make explicit in Lemma 3.5 that the map ρ of (3.3) comes from a restriction map on the level of centric linking systems. First we need to recall some information about the normalizer of T2 in L and Lz . Lemma 3.4. The following hold after identifying T2 with its image δT2 (T2 ) 6 AutL (T2 ). (a) AutLz (T2 ) is an extension of T2 by C2 × S4 , and AutL (T2 ) is an extension of T2 by C2 × GL3 (2) in which the GL3 (2) factor acts naturally on T2 /Φ(T2 ). In each case, a C2 factor acts as inversion on T2 . Also, T2 is equal to its centralizer in each of the above normalizers, Z(AutLz (T2 )) = Z(S), and Z(AutL (T2 )) = 1. (b) AutF (S) = Inn(S) = AutFz (S) and AutL (S) = δS (S) = AutLz (S). Proof. For part (a), see Lemma 4.3 and Proposition 5.4 of [AC10]. Since T2 is the unique abelian subgroup of its order in S by [AC10, Lemma 4.9(c)], it is characteristic. By the uniqueness of restrictions 2.1.3, we may therefore view AutL (S) as a subgroup of AutL (T2 ). Since AutL (T2 ) has self-normalizing Sylow 2-subgroups by (a), the same holds for AutL (S). Now (b) follows for L, and for F after applying π. This also implies the statement for Lz and Fz , as subcategories.  There is a 3-dimensional commutative diagram related to (2.3) that is the point of the next lemma. Lemma 3.5. There is a restriction map ρb: Aut(L) → Aut(Lz ), with kernel the automorphisms induced by conjugation by δS (Z(S)) 6 AutL (S), which makes the diagram Aut(L) ρb // Aut(Lz ) µ eLz µ eL   Aut(F) ρ 6 // Aut(Fz ) , commutative, which commutes with the conjugation maps out of AutL (S) id // AutL (S) z πS πS  AutF (S)  id // AutF (S), z and which therefore induces a commutative diagram Out(L) [b ρ] // Out(Lz ) µLz µL  Out(F)  [ρ] // Out(Fz ). Proof. Recall that we have arranged Lz ⊆ L. Thus, the horizontal maps in the second diagram are the identity maps by Lemma 3.4, and so the lemma amounts to checking that an element of Aut(L) sends morphisms in Lz to morphisms in Lz . For then, we can define its image under ρb to have the same effect on objects, and to be the restriction to Lz on morphisms. Now fix an arbitrary α ∈ Aut(L), objects P, Q ∈ F c = Fzc , and a morphism ϕ ∈ MorL (P, Q). Let Z(S) = hzi. By two applications of Axiom (C) for a linking system (§§2.1.2), (3.6) ιP,S ◦ δP (z) = δS (z) ◦ ιP,S and ια(P ),S ◦ δα(P ) (z) = δS (z) ◦ ια(P ),S , because π(ιP,S )(z) = π(ια(P ),S )(z) = z. Since αS is an automorphism of δS (S) ∼ = S, it sends δS (z) to itself. Thus, α sends the right side of the first equation of (3.6) to the right side of the second, since it sends inclusions to inclusions. Thus ια(P ),S ◦ α(δP (z)) = ια(P ),S ◦ δα(P ) (z). However, each morphism in L is a monomorphism [Oli10, Proposition 4], so we obtain (3.7) α(δP (z)) = δα(P ) (z), and the same holds for Q in place of P . Since ϕ ∈ Mor(Lz ), we have π(ϕ)(z) = z, so by two more applications of Axiom (C), (3.8) ϕ ◦ δP (z) = δQ (z) ◦ ϕ and α(ϕ) ◦ δα(P ) (z) = δα(Q) (π(α(ϕ))(z)) ◦ α(ϕ). After applying α to the left side of the first equation of (3.8), we obtain the left side of the second by (3.7). Thus, comparing right sides, we obtain δα(Q) (z) ◦ α(ϕ) = δα(Q) (π(α(ϕ)(z))) ◦ α(ϕ) Since each morphism in L is an epimorphism [Oli10, Proposition 4], it follows that δα(Q) (z) = δα(Q) (π(α(ϕ))(z)). Hence, π(α(ϕ))(z) = z because δα(Q) is injective (Axiom (A2)). That is, α(ϕ) ∈ Mor(Lz ) as required. The kernel of ρb is described via a diagram chase in (2.3). Suppose ρb(α) is the identity. Then, α is sent to the identity automorphism of S by µ eL , since ρ is injective. Thus, α comes from a normalized 1-cocycle by (2.3) and these are in turn induced by elements of Z(S) since lim1 (ZF ) ←− is trivial [LO02, Lemma 3.2].  7 Lemma 3.9. Let G be a finite group and let V be an abelian normal 2-subgroup of G such that CG (V ) 6 V . Let α be an automorphism of G such that [V, α] = 1 and α2 ∈ Inn(G). Then [G, α] 6 V , and if G acts fixed point freely on V /Φ(V ), then the order of α is at most the exponent of V . Proof. As [V, α] = 1, we have [V, G, α] 6 [V, α] = 1 and [α, V, G] = [1, G] = 1. Hence, by the Three subgroups lemma, it follows [G, α, V ] = 1. As CG (V ) 6 V , this means [G, α] 6 V. Assume from now on that G acts fixed point freely on V /Φ(V ). Write G∗ := G ⋊ hαi for the semidirect product of G by hαi. As [V, α] = 1 and [G, α] 6 V , the subgroup W := V hαi is an abelian normal subgroup of G∗ with [W, G∗ ] 6 V . As [V, α] = 1, it follows [V, α2 ] = 1. So α2 ∈ Inn(G) is realized by conjugation with an element −1 2 of CG (V ) = V . Pick u ∈ V with α2 = cu |G . This means that, for any g ∈ G, we have u α g = g in G∗ . So Z := hu−1 α2 i centralizes G in G∗ . Since W is abelian and contains Z, it follows that Z lies in the centre of G∗ = W G. Set G∗ = G∗ /Z. Because CG (V ) 6 V , the order of u equals the order of cu |G = α2 . Hence, Z ∩ G = 1 = Z ∩ hαi. So |ᾱ| = |α| and G ∼ = Ḡ. In particular, we have V̄ ∼ = V and Ḡ acts fixed point freely on 2 V̄ /Φ(V̄ ). Note also that ᾱ = ū. Hence |W̄ /V̄ | = 2 and Φ(W̄ ) 6 V̄ . Moreover, letting n ∈ N such that 2n is the exponent of V , we have |α| = |ᾱ| 6 2 · 2n = 2n+1 . Assume |ᾱ| = 2n+1 . Then ū = ᾱ2 has order 2n and is thus not a square in V̄ . Note that Φ(V̄ ) = {v 2 : v ∈ V̄ } and Φ(W̄ ) = {w2 : w ∈ W } = hᾱ2 iΦ(V̄ ) 6 V̄ . Hence, Φ(W̄ )/Φ(V̄ ) has order 2. As Ḡ normalizes Φ(W̄ )/Φ(V̄ ), it thus centralizes Φ(W̄ )/Φ(V̄ ) contradicting the assumption that Ḡ acts fixed point freely on V̄ /Φ(V̄ ). Thus |α| = |ᾱ| 6 2n which shows the assertion.  We are now in a position to determine the automorphisms of F = FSol(q) and L = LcSol (q). It is known that the field automorphisms induce automorphisms of these systems as we will make precise next. Recall that the field automorphism ψ5 of H of order 2k normalizes S and so ψ5 |S is an automorphism of Fz = FS (H). By [AC10, Lemma 5.7], the automorphism ψ5 |S is actually also an automorphism of F. We thus denote it by ψF and refer to it as the field automorphism of F induced by ψ5 . By Proposition 3.2, this automorphism has order 2k . By [LO02, Proposition 3.3(d)], there is a unique lift ψ of ψF under µ eL that is the identity on π −1 (FSol (5)) and restricts to κ eH (ψ5 ) on Lz . We refer to ψ as the field automorphism of L induced by ψ5 . Theorem 3.10. The map µL : Out(LcSol (q)) → Out(FSol (q)) is an isomorphism, and Out(Lc (q)) ∼ = Out(FSol (q)) ∼ =C k 2 Sol is induced by field automorphisms. Also, the automorphism group Aut(LcSol (q)) is a split extension of S by Out(LcSol (q)); in particular, it is a 2-group. More precisely, if ψ is the field automorphism of LcSol (q) induced by ψ5 , then ψ has order 2k and Aut(LcSol (q)) is the semidirect product of AutL (S) ∼ = S with the cyclic group generated by ψ. Proof. We continue to write L = LcSol (q), F = FSol (q), Lz = LcSpin (q), and Fz = FSpin (q), and we continue to assume that L has been chosen so as to contain Lz as a linking subsystem. Recall that T2 6 S is homocyclic of rank 3 and exponent 2k+2 . We first check whether the outer automorphism of Lz induced by a diagonal automorphism of H extends to L, and we claim that it doesn’t. A non-inner diagonal automorphism of H 8 is induced by conjugation by an element t of T̄ by [GLS98, Theorem 2.5.1(b)]. Its class as an outer automorphism has order 2, so if necessary we replace t by an odd power and assume that t2 ∈ T2 . Now T2 consists of the elements of T̄ of order at most 2k+2 , so t has order 2k+3 and induces an automorphism of H of order at least 2k+2 , depending on whether it powers into Z(H) = Z(S) or not (in fact it does, but this is not needed). For ease of notation, we identify T2 with δT2 (T2 ) 6 AutL (T2 ), and we identify s ∈ S with δS (s) ∈ AutL (S). Let τ = κ eH ([ct ]) ∈ Aut(Lz ), and assume that τ lifts to an element τb ∈ Aut(L) under the map ρb of Lemma 3.5. As ρb(b τ) = τ = κ eH (ct ), we have ρb(b τ 2) = τ 2 = κ eH (ct2 ), i.e. ρb(b τ 2 ) acts on every object and every morphism of Lz = LcS (H) as conjugation by t2 . Similarly, if we take the conjugation automorphism ct2 of L by t2 (or more precisely the conjugation automorphism cδS (t2 ) of L by δS (t2 )), then ρb(ct2 ) is just the conjugation automorphism of Lz by t2 . So according to the remark at the end of §§2.2.2, the automorphism ρb(ct2 ) acts also on Lz via conjugation by t2 , showing ρb(b τ 2 ) = ρb(ct2 ). By the description of the kernel in Lemma 3.5, we have thus τb2 = ct2 or 2 τb = ct2 z . Now set α := τbT2 ∈ Aut(AutL (T2 )). From what we have shown, it follows that α equals the conjugation automorphism ct2 or ct2 z of AutL (T2 ). Note that |ct2 z | = |t2 z| = |t2 | = |ct2 |, since Z(AutL (T2 )) = 1 by Lemma 3.4(a). Hence, (3.11) |α| = 2|α2 | = 2|t2 | = 2k+3 , On the other hand, α centralizes T2 , and we have seen that α2 is an inner automorphism of AutL (T2 ). Moreover, by Lemma 3.4(a), CAutL (T2 ) (T2 ) = T2 , and AutL (T2 ) acts fixed point freely on T2 /Φ(T2 ). The hypotheses of Lemma 3.9 thus hold for G = AutL (T2 ) and α ∈ Aut(G). So α has order at most 2k+2 by that lemma, contradicting (3.11). We conclude that a diagonal automorphism of Lz does not extend to an automorphism of L. The existence of the field automorphism ψF of F, and the fact that ψF has order 2k , now yields together with Proposition 3.2 that Out(F) ∼ = C2k is generated by the image of ψF in Out(F). Moreover, by [LO02, Lemma 3.2] and the exactness of the third column of (2.3), the maps µL and µLz are isomorphisms. Thus, Out(L) ∼ = Out(F) ∼ = C k. 2 Let ψ be the field automorphism of L induced by ψ5 as above. Then ψ is the identity on π −1 (FSol (5)) by definition. It remains to show that ψ has order 2k , since this will imply that Aut(L) is a split extension of AutL (S) ∼ = S by hψi ∼ = Out(L) ∼ = Out(F). k 2 The automorphism ψ maps to the trivial automorphism of F, and so is conjugation by an elek / Z(AutLcSol (5) (Ω2 (T2 ))) ment of Z(S) by (2.3). Now ψ 2 is trivial on AutLcSol (5) (Ω2 (T2 )), whereas z ∈ by Lemma 3.4(a) as Ω2 (T2 ) is the torus of LcSol (5). Thus, since a morphism ϕ is fixed by cz if and k only if π(ϕ)(z) = z (Axiom (C)), we conclude that ψ 2 is the identity automorphism of L, and this completes the proof.  4. Extensions In this section, we recall a result of Linckelmann on the Schur multipliers of the BensonSolomon systems, and we prove that each saturated fusion system F with F ∗ (F) ∼ = FSol (q) is a split extension of F ∗ (F) by a group of outer automorphisms. Recall that the hyperfocal subgroup of a saturated p-fusion system F over S is defined to be the subgroup of S given by hyp(F) = h[ϕ, s] := ϕ(s)s−1 | s ∈ P 6 S and ϕ ∈ O p (AutF (P ))i. 9 A subsystem F0 over S0 6 S is said to be of p-power index in F if hyp(F) 6 S0 and O p (AutF (P )) 6 AutF0 (P ) for each P 6 S0 . There is always a unique normal saturated subsystem on hyp(F) of p-power index in F, which is denoted by Op (F) [AKO11, §I.7]. We will need the next lemma in §§4.2. Lemma 4.1. Let F be a saturated fusion system over S, and let F0 be a weakly normal subsystem of F over S0 6 S. Assume that Op (AutF (S0 )) 6 AutF0 (S0 ). Then Op (AutF (P )) 6 AutF0 (P ) for every P 6 S0 . Thus, if in addition hyp(F) 6 S0 , then F0 has p-power index in F. Proof. Note that AutF0 (P ) is normal in AutF (P ) for every P 6 S0 , since F0 is weakly normal in F. We need to show that AutF (P )/ AutF0 (P ) is a p-group for every P 6 S0 . Suppose this is false and let P be a counterexample of maximal order. Our assumption gives P < S0 . Hence, P < Q := NS0 (P ), and the maximality of P implies that AutF (Q)/ AutF0 (Q) is a p-group. Notice that NAutF (Q) (P )/NAutF0 (Q) (P ) ∼ = NAutF (Q) (P ) AutF0 (Q)/ AutF0 (Q) 6 AutF (Q)/ AutF0 (Q) and thus NAutF (Q) (P )/NAutF0 (Q) (P ) is a p-group. If α ∈ HomF (P, S) with α(P ) ∈ F f then conjugation by α induces a group isomorphism from AutF (P ) to AutF (α(P )). As F0 is weakly normal, we have α(P ) 6 S0 and conjugation by α takes AutF0 (P ) to AutF0 (α(P )). So upon replacing P by α(P ), we may assume without loss of generality that P is fully F-normalized. Then P is also fully F0 -normalized by [Asc08, Lemma 3.4(5)]. By the Sylow axiom, AutS0 (P ) is a Sylow p-subgroup of AutF0 (P ). So the Frattini argument yields AutF (P ) = AutF0 (P )NAutF (P ) (AutS0 (P )) and thus AutF (P )/ AutF0 (P ) ∼ = NAutF (P ) (AutS0 (P ))/NAutF0 (P ) (AutS0 (P )). By the extension axiom for F and F0 , each element of NAutF (P ) (AutS0 (P )) extends to an automorphism of AutF (Q), and each element of NAutF0 (P ) (AutS0 (P )) extends to an automorphism of AutF0 (Q). Therefore, the map Φ : NAutF (Q) (P ) → NAutF (P ) (AutS0 (P )), ϕ 7→ ϕ|P is an epimorphism which maps NAutF0 (Q) (P ) onto NAutF0 (P ) (AutS0 (P )). Hence, AutF (P )/ AutF0 (P ) ∼ = NAutF (P ) (AutS0 (P ))/NAutF0 (P ) (AutS0 (P )) ∼ = NAut (Q) (P )/NAut (Q) (P ) ker(Φ). F F0 We have seen above that NAutF (Q) (P )/NAutF0 (Q) (P ) is a p-group, and therefore also NAutF (Q) (P )/NAutF0 (Q) (P ) ker(Φ) is a p-group. Hence, AutF (P )/ AutF0 (P ) is a p-group, and this contradicts our assumption that P is a counterexample.  10 4.1. Extensions to the bottom. A central extension of a fusion system F0 is a fusion system F such that F/Z ∼ = F0 for some subgroup Z 6 Z(F). The central extension is said to be perfect if F = O p (F). Linckelmann has shown that the Schur multiplier of a Benson-Solomon system is trivial. Theorem 4.2 (Linckelmann). Let F be a perfect central extension of a Benson-Solomon fusion system F0 . Then F = F0 . Proof. This follows from Corollary 4.4 of [Lin06a] together with the fact that Spin7 (q) has Schur muliplier of odd order when q is odd [GLS98, Tables 6.1.2, 6.1.3].  4.2. Extensions to the top. The next theorem describes the possible extensions (S, F) of a Benson-Solomon system (S0 , F0 ). The particular hypotheses are best stated in terms of the generalized Fitting subsystem of Aschbacher [Asc11], but they are equivalent to requiring that F0 E F and CS (F0 ) 6 S0 , where CS (F0 ) is the centralizer constructed in [Asc11, §6]. This latter formulation is sometimes expressed by saying that F0 is centric normal in F. k Theorem 4.3. Let F0 = FSol (52 ) be a Benson-Solomon system over S0 . (a) If F is a saturated fusion system over S such that F ∗ (F) = F0 , then F0 = O 2 (F), S splits over S0 , and the map S/S0 → Out(F0 ) induced by conjugation is injective. (b) Conversely, given a subgroup of A 6 Out(F0 ) ∼ = C2k , there is a saturated fusion system ∗ F over some 2-group S such that F (F) = F0 and the map S/S0 → Out(F0 ) induced by conjugation on S0 has image A. Moreover, the pair (S, F) with these properties is uniquely determined up to isomorphism. If L0 is a centric linking system associated to F0 , then AutL0 (S0 ) = S0 , and the p-group S can be chosen to be the preimage of A in Aut(L0 ) under the quotient map from Aut(L0 ) to Out(L0 ) ∼ = Out(F0 ). Proof. Let F be a saturated fusion system over S such that F ∗ (F) = F0 . Set F1 = F0 S, the internal extension of F0 by S, as in [Hen13] or [Asc11, §8]. According to [AOV12, Proposition 1.31], there is a normal pair of linking systems L0 E L1 , associated to the normal pair F0 E F1 . Furthermore, L0 E L1 can be chosen such that L0 is a centric linking system. There is a natural map from AutL1 (S0 ) to Aut(L0 ) which sends a morphism ϕ ∈ AutL1 (S0 ) to conjugation by ϕ. (So the restriction of this map to AutL0 (S0 ) is the conjugation map described in §§2.2.2.) The centralizer CS (F0 ) depends a priori on the fusion system F, but it is shown in [Lyn15, Lemma 1.13] that it does not actually matter whether we form CS (F0 ) inside of F or inside of F1 . Moreover, since F ∗ (F) = F0 , it follows from [Asc11, Theorem 6] that CS (F0 ) = Z(F0 ) = 1. Thus, conj by a result of Semeraro [Sem15, Theorem A], the conjugation map AutL1 (S0 ) −−→ Aut(L0 ) is injective. By Lemma 3.4, we have S0 = AutL0 (S0 ) via the inclusion functor δ1 for L1 . By Theorem 3.10, Aut(L0 ) is a 2-group which splits over S0 . Moreover, by the same theorem, we have that CAut(L0 ) (S0 ) 6 S0 and Out(L0 ) ∼ = Out(F0 ) is cyclic. Since (δ1 )S0 (S) ∼ = S is a Sylow 2-subgroup of AutL (S0 ) by [Oli10, Proposition 4(d)], we can conclude that S0 = AutL0 (S0 ) E AutL1 (S0 ) = S, via the inclusion functor δ1 for L1 . Moreover, it follows that S splits over S0 , and CS (S0 ) 6 S0 . The latter property means that the map S/S0 → Out(F0 ) is injective. In particular, S/S0 is cyclic as Out(F0 ) is cyclic. 11 Next, we show that O2 (F) = F0 . Fix a subgroup P 6 S, and let α ∈ AutF (P ) be an automorphism of odd order. Then α induces an odd-order automorphism of the cyclic 2-group P/(P ∩ S0 ) ∼ = P S0 /S0 6 S/S0 . This automorphism must be trivial, and so [P, α] 6 S0 . Hence, 2 [P, O (AutF (P ))] 6 S0 for all P 6 S. Since hyp(F0 ) = S0 , we have hyp(F) = S0 . Note that AutF (S0 ) is a 2-group as AutF (S0 ) 6 Aut(F0 ) and Aut(F0 ) is a 2-group by Theorem 3.10. Therefore O 2 (F) = F0 by Lemma 4.1. We conclude that F1 = F by the uniqueness statement in [Hen13, Theorem 1]. This completes the proof of (a). Moreover, we have seen that the following property holds for any normal pair L0 E L attached to F0 E F: (4.4) conj S0 = AutL0 (S0 ) E AutL1 (S0 ) = S and S −−−→ Aut(L0 ) is injective. Finally, we prove (b). Fix a centric linking system L0 associated to F0 with inclusion functor δ0 . Let S 6 Aut(L0 ) be the preimage of A under the quotient map to Out(F0 ). We will identify S0 with δ0 (S0 ) so that S0 = AutL0 (S0 ) by Lemma 3.4. Write ι : S0 → Aut(L0 ), s 7→ cs for map sending s ∈ S0 to the automorphism of L0 induced by conjugation with s in L0 . Then ι(S0 ) is normal in S. Let χ : S → Aut(S0 ) be the map defined by α 7→ ι−1 ◦ cα |ι(S0 ) ◦ ι; i.e. χ corresponds to conjugation in S if we identify S0 with ι(S0 ). We argue next that the following diagram commutes: (4.5) ι // Aut(L0 ) ✇;; incl ✇✇✇ α7→αS0 ι ✇ ✇✇   ✇✇✇ χ // Aut(S0 ) S S0 The upper triangle clearly commutes. Observe that α◦ι(s)◦α−1 = α◦cs ◦α−1 = cαS0 (s) = ι(αS0 (s)) for every s ∈ S0 and α ∈ S. Hence, for every α ∈ S and s ∈ S0 , we have (ι−1 ◦ cα |ι(S0 ) ◦ ι)(s) = ι−1 (α ◦ ι(s) ◦ α−1 ) = αS0 (s) and so the lower triangle commutes. We will now identify S0 with its image in S under ι, so that ι becomes the inclusion map and χ corresponds to the map S → Aut(S0 ) induced by conjugation in S. As the above diagram commutes, it follows then that the diagram in [Oli10, Theorem 9] commutes when we take Γ = S and τ : S → Aut(L0 ) to be the inclusion. Thus, by that theorem, there is a saturated fusion system F over S in which F0 is weakly normal, and there is a corresponding normal pair of linking systems L0 E L (in the sense of [AOV12, §1.5]) such that S = AutL (S0 ) has the given action on L0 (i.e. the automorphism of L0 induced by conjugation with s ∈ S in L equals the automorphism s of L0 ). By the same theorem, the pair (F, L) is unique up to isomorphism of fusion systems and linking systems with these properties. Since F0 is simple [Lin06b], F0 is in fact normal in F by a result of Craven [Cra11, Theorem A]. Thus, since CS (F0 ) 6 CS (S0 ) 6 S0 , it is a consequence of [Asc11, (9.1)(2), (9.6)] that F ∗ (F) = F0 . So it remains only to prove that (S, F) is uniquely determined up to an isomorphism of fusion systems. Let F ′ be a saturated fusion system over a p-group S ′ such that F ∗ (F ′ ) = F0 , and such that the map S ′ /S0 → Out(F0 ) induced by conjugation has image A. Then by (a), F0 = Op (F ′ ). So by [AOV12, Proposition 1.31], there is a normal pair of linking systems L′0 E L′ associated to the normal pair F0 E F ′ . Moreover, we can choose L′0 to be a centric linking system. Since a centric linking system attached to F0 is unique, there is an isomorphism θ : L′0 → L0 of linking systems. We may assume that the set of morphisms which lie in L′ but not in L′0 is disjoint from the set of morphisms in L0 . Then we can construct a new linking system from L′ by keeping every morphism of L′ which is not in L′0 and replacing every morphism ψ in L′0 by θ(ψ), and then carrying over the structure of L′ in the natural way. Thereby we may assume L′0 = L0 . So we are 12 given a normal pair L0 E L′ attached to F0 E F ′ . By (4.4) applied with L′ and F ′ in place of F and L, we have S0 = AutL0 (S0 ) E AutL′ (S0 ) = S ′ via the inclusion functor δ′ of L′ . Let τ : S ′ → Aut(L0 ) be the map taking s ∈ S ′ to the automorphism of L0 induced by conjugation with s in L′ . Again using (4.4), we see that τ is injective. Note also that τ restricts to the identity on S0 if we identify S0 with ι(S0 ) as above. Recall that the map S ′ /S0 → Out(F0 ) induced by conjugation has image A. So Theorem 3.10 implies τ (S ′ ) = S, i.e. we can regard τ as an isomorphism τ : S ′ → S. So replacing (S ′ , F ′ ) by (S, τF ′ ) and then choosing L0 E L′ as before, we may assume S = S ′ . So F ′ is a fusion system over S with F0 E F ′ , and L0 E L′ is a normal pair of linking systems associated to F0 E F ′ such that AutL′ (S0 ) = S via δ′ . Let s ∈ S. Recall that τ (s) is the automorphism of L0 induced by conjugation with s in L′ . Observe that the automorphism of S0 = AutL0 (S0 ) induced by τ (s) equals just the automorphism of S0 induced by conjugation with s in S. Similarly, the automorphism s of L0 equals the automorphism of L0 given by conjugation with s in L, and so induces on S0 = AutL0 (S0 ) just the automorphism given by conjugation with s in S. Theorem 3.10 gives CAut(L0 ) (S0 ) 6 S0 and this implies that any two automorphisms of L0 , which induce the same automorphism on S0 , are equal. Hence, τ (s) = s for any s ∈ S. In other words, S = AutL′ (S0 ) induces by conjugation in L′ the canonical action of S on L0 . The uniqueness of the pair (F, L) implies now F ′ ∼ = L. This shows that (S, F) is uniquely = F and L′ ∼ determined up to isomorphism.  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MR 0121427 E-mail address: ellen.henke@abdn.ac.uk E-mail address: justin.lynd@abdn.ac.uk Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB15 5LY, United Kingdom 14 

A generative model for sparse, evolving digraphs arXiv:1710.06298v1 [] 17 Oct 2017 Georgios Papoudakis and Philippe Preux and Martin Monperrus Abstract Generating graphs that are similar to real ones is an open problem, while the similarity notion is quite elusive and hard to formalize. In this paper, we focus on sparse digraphs and propose SDG, an algorithm that aims at generating graphs similar to real ones. Since real graphs are evolving and this evolution is important to study in order to understand the underlying dynamical system, we tackle the problem of generating series of graphs. We propose SEDGE, an algorithm meant to generate series of graphs similar to a real series. SEDGE is an extension of SDG. We consider graphs that are representations of software programs and show experimentally that our approach outperforms other existing approaches. Experiments show the performance of both algorithms. 1 Introduction We wish to generate artificial graphs that are similar to real ones: by “real”, we mean a graph that is observed in the real world; as we know, there is ample evidence that graphs coming from the real world are not Gilbert, or Erdös-Rényi graphs, but exhibit more structure. The motivations range from pure intellectual curiosity to, for instance, being able to test ideas on a set of graphs when only one is available (the WWW, a social network), or understanding which are the key properties of a graph. This paper is considering directed, un-looped, un-weighted, sparse graphs of moderate sizes (number of nodes ranging from 100 to a couple of thousands of nodes); by sparse, we mean that the number of edges is of the order of the number of vertices, and typically scales like aN, with a very small with regards to N (say Georgios Papoudakis e-mail: giwrpapoud@gmail.com Philippe Preux e-mail: firstname.lastname@inria.fr Université de Lille, CRIStAL & Inria, Villeneuve d’Ascq, France. Martin Monperrus KTH Royal Institute of Technology, Sweden e-mail: firstname.lastname@csc.kth.se 1 2 Georgios Papoudakis and Philippe Preux and Martin Monperrus a ≤ 10 to give an idea of its value). We assume weak connectivity of the graph. As a case study, we experiment with graphs extracted from software programs; beyond a better understanding of software programs, such graphs may be used e.g. to improve software development and track the sources of bugs [8, 7]. To generate a graph in this context, one may use an algorithm that builds a graph given the degree distribution of the real graph (see [5] and followers), or its adjacency matrix [3], or some other structure (see [11] and references therein). As we wish to understand and model the creation of the graph, and as real graphs are often dynamic, we are more interested in a second type of algorithms that build a graph incrementally. Another motivation is that we do not want to generate graphs that have the exact same number of vertices, or the exact same degree distribution, or anything identical to the real one. A reason for this is that if we consider the degree distribution, two graphs having the same degree distribution may be very different regarding their other properties; in the other way around, two graphs that have more or less slightly different degree distribution, may have very similar properties. The properties we are interested in are of various natures: connectivity, diameter, average path length, transitivity, modularity, assortativity, spectral properties, degree distribution. Furthermore, when considering degree distribution or spectral properties, it is not clear how to meaningfully measure the difference between two degree distributions: mean squared distance, Kolmogorov-Smirnov statistics, Kullback-Leibler divergence, Jensen-Shannon distance. Finally, an important property of the generator is its stability. We identify two types of stability: the first is that for a given set of parameters, the graphs that are generated should have approximately the same characteristics; the other is that the graphs generated by a set of parameters should not change too much when the value of the parameters change a bit (sort of continuity of the properties of the generated graphs in the space of parameters of the generator). Modeling and generating static graphs is important, but we are really interested in modeling the evolution of a graph. Though some works exist [4], the issues mentioned above take yet another aspect when considering the evolution of a graph. We see the evolution of a graph as time series of graphs, that is a set of couples {(t, g)}. We wish to generate the whole series of graphs with a single algorithm. Succeeding in this endeavor, we would access to general properties of the graph and the evolution process, as well as being able to predict the next graphs. The content of this paper is as follows: in section 2, we propose the Sparse Digraph Generator (SDG) which is an algorithm that generates graphs that fit our requirements; we then show that the degree distribution follows a power law distribution; we also show that the in-degree and the out-degree distributions are not identical, something often observed in real digraphs. Then, we put SDG to the test: we introduce the real graphs we work with and show how our generator performs. As we are interested in the modeling of the evolution of a dynamic graph, we introduce Sparse Evolving Digraph GEnerator (SEDGE) which is an incremental version of SDG in section 4 and put it to the test in section 5. Then, we conclude and draw some final remarks. A generative model for sparse, evolving digraphs 3 For the sake of reproducible research, all the experiments may be reproduced with the material freely available at https://github.com/papoudakis/sparse-digraph-generator. 2 The Sparse Digraph Generator: SDG We present a novel algorithm that aims at generating sparse digraphs. It is outlined in algorithm 1. SDG starts by creating a digraph made of N isolated nodes and then, at each iteration, it adds a link between two nodes. To add a link, SDG selects two nodes, one as output, and the other as an input node. The selection of either node is performed either at random or following a preferential attachment rule. Algorithm 1 Outline of SDG 1: Input: Number of nodes: N 2: Input: Number of edges: E (assumed to be ≪ N 2 ) 3: Input: Parameters e1 and e2 , both in the range [0, 1] 4: Output: Generated graph G 5: G ← DiGraph (with N nodes and no edge) 6: for t ∈ {1, ..., E} do 7: ⊲ Selection of the node that the edge will start from 8: With probability e1 : out ← select a node uniformly at random() 9: Otherwise: out ← select a node by preferential attachment 10: ⊲ Selection of the node that the edge will end to 11: With probability e2 : in ← select a node of in-degree 0() 12: Otherwise: in ← select a node by preferential attachment 13: G.add edge(out, in) return G We consider sparse digraphs in which the number of edges E is aN, where a ∈ (1, 10). Such digraphs are quite common in applications and they are quite specific with regards to their properties: for instance, there is usually a very small number of paths to navigate from one node to another. It is often the case that the in-degree and the out-degree distributions do not have the same shape. SDG achieves this: is e1 6= e2 , the parameters of the power law of in-degree and out-degree distributions are different. The selection of a node to connect to or from is either uniformly at random (among all nodes at line 8, among nodes of in-degree 0 at line 11), or with a probability proportional to the degree of the node, that is we use a linear preferential attachment rule. In the rest of this section, we derive the form of the in-degree and out-degree distributions resulting from SDG. We show that both distributions follow a power law, though of different parameters. 4 Georgios Papoudakis and Philippe Preux and Martin Monperrus 2.1 The in-degree distribution After the completion of the t th iteration of SDG, the graph is made of t edges. So, the probability for a node of degree k to be selected by linear preferential attachment is N k t . Additionally, we assume that e2 < E , so the expected number of nodes that have in-degree 0 is bigger than 0, E[N − e2 E] > 0 Let Dk (t) be the number of nodes with in-degree k at timestep t. For k > 1, Dk (t) decreases at timestep t only if a node with in-degree k is selected due to preferential attachment (line 12). So the probability that Dk decreases at iteration t is: (1 − e2) | {z } probability of selecting a node by preferential attachment k t |{z} Dk (t) (1) probability of choosing a degree k node Similarly, Dk (t) increases only if a node with in-degree k − 1 is selected due to preferential attachment. So the probability that Dk increases at iteration t is: (1 − e2 ) (k − 1) Dk−1 (t) t (2) Let dk (t) = E[Dk (t)]. It follows that the expected change in the number of nodes of degree k at iteration t is: dk (t + 1) − dk(t) = (1 − e2) (k − 1)dk−1 (t) − kdk (t) t (3) We set c2 = 1 − e2 and we assume that dk (t) = pk t so we get: Assuming that k ≫ 1 c2 pk = c2 ((k − 1)pk−1 − kpk ) (4)   (1 + c2)/c2 pk−1 pk = 1 − 1/c2 + k (5) and using the binomial approximation we come up with:     1+c2 (1 + c2 )/c2 k − 1 c2 pk−1 pk ≈ 1 − pk−1 ≈ k k (6) Finally, we calculate the values of p0 and p1 and we iterate the equation until k = 2. pk ≈  k−1 k  1+c2  c2 k−2 k−1  1+c2 c2   1+c2 1 c2 ... p1 2 (7) 1+c pk ≈ p1 k − c 2 2 (8) A generative model for sparse, evolving digraphs 5 2.2 The out-degree distribution In this section, Dk (t) is the number of nodes with out-degree k at iteration t. Starting with the same assumptions as before, we can write that the number of nodes with out-degree distribution k decreases if a node with out-degree k is selected due to preferential attachment with probability 1 − e1 or if such a node is selected from a uniform distribution with probability e1 . This second possibility is different from the analysis we did for the in-degree distribution. So, the probability that Dk decreases at iteration t is: Dk (t) Dk (t) e1 + (1 − e1)k (9) n t Similarly, Dk (t) increases with probability: e1 Dk−1 (t) Dk−1 (t) + (1 − e1)(k − 1) n t (10) After following the same steps as before we end up with: dk (t + 1) − dk(t) = (1 − e1) (k − 1)dk−1(t) − kdk (t) dk−1 (t) − dk (t) + e1 t N (11) Assuming that the solution is like dk (t) = pk t and by setting c1 = 1 − e1, we can prove that at the final timestep t = E: pk ≈ p1 (k + 1 (1 − c1) E − 1+c ) c1 c1 N (12) 2.3 Discussion & Related Work We have shown that the in-degree and the out-degree distributions of the graphs generated by SDG exhibit a power law. This may come as a surprise to the reader, well aware of earlier works, such as [1]. Indeed, our graph is not growing, keeping a set of N nodes, connecting them along the iterations of the algorithm. However, the departure from a power law is expected when the number of iterations is approximately N 2 , that is when the graph gets dense. However, as we emphasized it earlier, we only consider sparse graphs, and the number of iterations, hence the number of edges, remains O(N), hence much less than N 2 . It is worth noting that the power law coefficients of graphs generated by SDG are the same as those of graphs produced by Bollobas et al., though the algorithms are slightly different. Actually Bollobas et al. results come as special cases of our analysis. SDG departs from the usual Barabasi-Albert type of algorithms because it generates directed graphs. Strictly speaking, our algorithm generates a variant of a Price 6 Georgios Papoudakis and Philippe Preux and Martin Monperrus graph [9] and setting e1 to 0, e2 to 1, a kind of Price’s algorithm which adds one edge at a time is recovered. SDG comes very close to the one studied by Bollobas et al. [2] though only SDG is able to add two vertices at once, in a single iteration. 3 Experimental study of SDG In this experimental section, we mainly study two questions: • which algorithm performs the best to produce graphs that are similar to some real graphs? • the stability of SDG with regards to its parameters. We compare our algorithm with GDGNC [6] where it is shown to be the best graph generator available in the context of software graphs. We also compare our model with Bollobas et al.’s since they are quite similar: it is interesting to check how the small difference in these 2 algorithms convert into difference of performance. We have compared SDG with other algorithms (Kronecker graphs, ...) but since they perform poorly and due to space limitations, we do not report them. The experiments are performed with 10 major software programs taken from the maven dataset [10]. Table 1 summarizes the basic features of our dataset. Software (version) ant (1.5.1) findbugs (0.6.4) freemarker (1.5.3) hibernate (1.2) htmlunit (1.10) jasperreports (3.1.2) jparsec (0.2.2) ojb (0.5.200) pmd jdk14 (4.1.1) spring core (1.0.1) Nodes 266 56 76 365 219 1139 75 179 521 112 Edges Edges/Nodes Diameter 1427 5.36 6 183 3.27 5 358 4.71 7 1916 5.25 7 934 4.26 7 7460 6.54 7 203 2.71 5 766 4.28 6 3049 5.85 8 337 3.01 7 Table 1 Statistics of the dataset used in the experiments reported in section 3. In the literature, the measure of similarity between two graphs is not very well defined. In this paper, we measure the similarity between the generated graph (gg) and the original graph (go) using the following set of metrics: • The Kolmogorov-Smirnov statistic (KS) of the in-degree and out-degree distributions. Let CDFg denote the cumulative degree distribution function of a graph g, so that CDFg (k) = ∑i≤k Dk where Dk is the degree distribution of graph g. Then, KS = maxk |CDFgg (k) − CDFgo (k)|. We denote KSin (resp. KSout ) the KS statistics regarding in-degree (resp. out-degree) distribution. • The mean squared distance (MSD) of the sorted in-degree and out-degree distributions. For each generated graphs g we consider the in-degree and out-degree A generative model for sparse, evolving digraphs 7 of each node, sort these two lists to obtain din,g and dout,g . Then: MSDin = ∑i (din,gg (i)−din,go (i))2 N (d (i)−d (i))2 and MSDout = ∑i out,gg N out,go . The MSD can only be used for SDG and GDGNC because they generate the same number of nodes as the original graph. On the contrary, Bollobas et al.’ model does not necessarily produce graphs with the same number of nodes. We perform a grid search in order to determine the parameters of each model that best fit for each graph. SDG and GDGNC are optimized to minimize the maximum value between MSDin and the MSDout : minimize{max(MSDin , MSDout )}. As MSDin and MSDout are irrelevant for it, Bollobas et al. model is optimized to minimize the KS statistic. This may be seen as a caveat in our experiments, but we provide ample observations below to convince the reader that if we were tuning the parameters of the 3 models with the same metrics, the conclusions of the experiments would not change much. The experiments presented in table 2 below are performed with the optimal parameters for each software, averaged over 100 generated graphs. Table 2 provides the average value of KS and MSD for each model and each software. Software ant findbugs freemarker hibernate htmlunit jasperreports jparsec ojb pmd jdk14 spring core KSin KSout MSDin SDG GDGNC Bollobas SDG GDGNC Bollobas SDG GDGNC 0.26 0.24 0.39 0.16 0.17 0.34 17.4 30.45 0.29 0.41 0.37 0.33 0.35 0.37 2.32 3.74 0.23 0.23 0.4 0.48 0.49 0.38 3.11 6.14 0.38 0.41 0.33 0.22 0.32 0.32 14.38 21.87 0.37 0.37 0.42 0.31 0.36 0.44 12.67 20.68 0.24 0.24 0.28 0.35 0.43 0.29 32.37 97.72 0.22 0.22 0.41 0.36 0.47 0.42 0.69 2.72 0.26 0.25 0.44 0.21 0.27 0.4 3.6 6.36 0.27 0.28 0.28 0.5 0.56 0.41 14.92 114.9 0.36 0.4 0.4 0.23 0.34 0.3 2.54 4.67 MSDout SDG GDGNC 1.89 2.58 1.24 2.65 4.89 6.76 3.14 9.23 3.92 8.45 16.1 37.43 4.6 9.98 0.77 3.41 32.08 54.54 1.2 3.72 Table 2 Comparison of SDG with GDGNC and Bollobas et al. in terms of MSD and KS for 10 Java software graphs. Bold faces indicate best results. We can clearly see that SDG performs better than both GDGNC and Bollobas et al. model. Additionally, SDG is much more stable than the other models. That means that given the parameters of the generator the graphs that are produced are similar. In table 3, we give the average of the standard deviation for the experiments that appear in table 2. Model KSin KSout MSDin MSDout SDG 0.093 ± 0.012 0.084 ± 0.023 3.94 ± 2.55 1.33 ± 1.07 GDGNC 0.091 ± 0.01 0.081 ± 0.013 19.78 ± 26.6 4.01 ± 3.89 Bollobas 0.102 ± 0.029 0.099 ± 0.025 Table 3 Mean and standard deviation of standard deviation values of MSD and KS on 10 Java software graphs. 8 Georgios Papoudakis and Philippe Preux and Martin Monperrus From table 3 we can see the standard deviation values of SDG are on the same level or smaller than both GDGNC and Bollobas et al. But the most important property of SDG is that it can create graphs similar to the original one without the parameter optimization process, that both other models require in order to perform decently. For each software, we generate 100 graphs and we compute the average KSin , KSout , MSDin , and MSDout . All the experiments are performed with the same values e1 = 0.45 and e2 = NE − 0.05 for all software graphs; these values result from our experiments. Table 4 provides the results; in ()’s, we report the ratio between the SDG without and with tuning: e.g., 0.14(0.9) is the first row of column KSout means that KSout is 0.14 without tuning, and 0.14/0.9 with tuning. The value of KS without tuning may be smaller than with tuning because the parameter tuning is performed to minimize MSD. Software ant findbugs freemarker hibernate htmlunit jasperreports jparsec ojb pmd jdk14 spring core KSin 0.25 (1.0) 0.3 (1.0) 0.24 (1.0) 0.29 (0.8) 0.33 (0.9) 0.21 (0.9) 0.25 (1.1) 0.33 (1.3) 0.33 (1.2) 0.3 (0.8) KSout 0.14 (0.9) 0.34 (1.0) 0.46 (1.0) 0.3 (1.4) 0.29 (0.9) 0.43 (1.2) 0.42 (1.2) 0.27 (1.3) 0.54 (1.1) 0.25 (1.1) MSDin 20.54 (1.2) 2.66 (1.1) 3.43 (1.1) 27.16 (1.9) 12.84 (1.0) 119.42 (3.6) 1.52 (2.2) 13.47 (3.7) 61.67 (4.1) 2.63 (1.0) MSDout 0.89 (0.5) 1.34 (1.1) 5.31 (1.1) 13.27 (4.2) 5.24 (1.3) 49.16 (3) 8.41 (1.8) 2.36 (3.1) 45.43 (1.4) 2.43 (2.0) Table 4 MSD and KS without tuning parameters: numbers in ()’s gives the ratio between the measurement without tuning and the measurement with tuning. From table 4 we see that in most cases, SDG, without parameter tuning, performs better than both GDGNC and Bollobas et al. model after parameter tuning. Another very nice property is that the performance does not change very much as the value of a parameter is changing: there is some sort of continuity of the performance of SDG with regards to the value of parameters. This is a very nice property, as this implies that to tune the parameters of SDG, a coarse grid search is enough and computationally cheaper. Figure 1 provides a graphical illustration of these measurements: we plot the in-degree distribution, the out-degree distribution, and the spectra of the adjacency matrix for the real graph and for the graphs generated by each algorithm we compare to. To conclude this part, let us stress that SDG uses two pieces of information: the number of nodes N and the number of edges E. We have shown that SDG produces graphs which degree distributions follow power laws. When we want to generate graphs similar to a real one, both N and E are available, and we have shown that e1 and e2 , the parameters of SDG, are not that important to obtain satisfying graphs. A generative model for sparse, evolving digraphs 100 In-degree distribution of ant 100 9 Out-degree distribution of ant Sorted spectrum of ant 30 10 1 2 10 1 Eigenvalue 10 Cumulative frequency Cumulative frequency 20 10 10 0 2 10 10 3 100 10 101 102 In degree 103 104 20 3 100 101 102 Out degree 103 104 0 5 100 1
 Eigenvalue index 200 2 Fig. 1 In-degree distribution, out-degree distribution, and spectrum for the real graph and the generated graphs. Another point is that the occasional addition of 2 nodes instead of 1 seems beneficial since this is the only difference between SDG and Bollobas et al. approach. Finally, it is important to refer to the metrics we use to compare graphs and the metrics we use to optimize the parameters of the algorithm. As said earlier, it is not known how to assess the similarity of two graphs using a single metric; instead, we use a series of metrics (and more may be used) to formalize the idea of similarity between two graphs. The metrics we use are recognized as very important to characterize a graph: the degree distribution, and the spectrum. We have found that optimizing using the degree distributions leads to better results. We see that as a primary observation, other spectral information might be used, and other properties may be used too. Furthermore, a combination of metrics may be optimized or used to judge the similarity: this is left as future work. 4 SEDGE: modeling the evolution of a graph We consider a model of evolution of the real graph that is version-oriented. As the real graphs we consider are software, considering a sequence of versions of a software, the graphs along this sequence evolve by part: by that, we mean that the set of nodes and the set of edges evolve by chunks: from one graph to the next one (one version of a software to the next one), a set of nodes are added, some nodes are removed, and it is the same for the edges. So, we consider an algorithm that takes a graph as input, and then adds a set of nodes and a set of edges, possibly removing some existing nodes and edges. We propose the “Sparse Evolving Digraph GEnerator” SEDGE (see algorithm 2), a model to capture the evolution of software graphs based on the generative model that we proposed in section 2. SEDGE is an extension of SDG. It distinguishes existing nodes from new nodes. At each timestep, SEDGE chooses two nodes to connect, sampling them from either set of nodes, based on 2 parameters that act as probabilities α and β . 10 Georgios Papoudakis and Philippe Preux and Martin Monperrus Algorithm 2 SEDGE: a generative model for sparse digraph evolution. The SAM PLE A NODE samples nodes in exactly the same way algorithm 1 does. new nodes refers to the N new nodes that are added to the current graph. all nodes refers to all nodes of the new graph. 1: Input: Number of nodes to add N new 2: Input: Number of edges to add E new 3: Input: Parameters α , β , e1 , e2 , all in the range [0, 1] 4: Input: Current graph G cur 5: Output: Generated graph G new 6: function SAMPLE A NODE (so, si, e1 , e2 ) 7: With probability e1 : out ← select a node uniformly at random(so) 8: Otherwise: out ← select a node by preferential attachment (so) 9: With probability e2 : in ← select a node of in-degree 0(si) 10: Otherwise: in ← select a node by preferential attachment (si) 11: return (in, out) 12: End function 13: G new ← G cur.add nodes(N new) 14: for t ∈ {1, ..., E new} do 15: With probability α : (in, out) ← SAMPLE A NODE(all nodes, new nodes, e1 , e2 ) 16: With probability β : (in, out) ← SAMPLE A NODE(new nodes, all nodes, e1 , e2 ) 17: Otherwise: (in, out) ← SAMPLE A NODE(all nodes, all nodes, e1 , e2 ) 18: G new.add edge(out, in) return G new 5 Experimental study of SEDGE In this section, we evaluate the ability of SEDGE to capture the software evolution. For the experiments, we use 10 pairs of consecutive versions of software graphs1 from the maven dataset. With the term “first graph”, we refer to the first version of the software and with the term “second graph” to the second version. In each pair of these graphs, the second graph has at least 20% more nodes than the first graph. The degree distributions and the spectrum of the graphs of two successive versions are close. For this reason, in order to perform a better evaluation of SEDGE we compute KS and MSD only for the new nodes: doing so, we amplify the difference between the two versions. In table 5, we report on the values of KS and MSD averaged over 100 experiments, for each real graph, given the optimal parameters of the model. 1 (ant.1.4.1→ant.1.5), (commons collections.20030418.083655→commons collections.20031027. 000000), (hibernate.2.0.3→hibernate.2.1.1), (jasperreports.0.6.7→jasperreports.1.0.0), (jasperreports.1.0.3→jasperreports.1.1.0), (ojb.0.8.375→ojb.0.9), (ojb.0.9.5→ojb.0.9.6), (spring.1.0 →spring.1.1), (wicket.1.0.3→wicket.1.1), (wicket.1.1.1 → wicket.1.2) A generative model for sparse, evolving digraphs Nnew First Software 116 ant.1.4.1 commons.20030418 118 92 hibernate.2.0.3 jasperreports.0.6.7 170 jasperreports.1.0.3 117 100 ojb.0.8.375 120 ojb.0.9.5 199 spring.1.0 96 wicket.1.0.3 235 wicket.1.1.1 Enew 665 385 853 1100 1214 555 586 830 569 1800 KSin 0.29 (0.8) 0.41 (1.1) 0.39 (0.6) 0.23 (1.1) 0.19 (0.9) 0.31 (0.9) 0.47 (1.0) 0.36 (1.0) 0.36 (0.9) 0.25 (1.0) 11 KSout 0.4 (1.1) 0.39 (0.9) 0.29 (1.0) 0.2 (1.2) 0.25 (1.0) 0.39 (1.0) 0.36 (1.0) 0.4 (0.8) 0.32 (1.0) 0.2 (1.1) MSDin 5.57 (2.9) 0.99 (1.5) 3.52 (12.4) 15.39 (1.2) 22.1 (2.7) 5.72 (1.6) 1.51 (0.6) 2.66 (3.7) 1.21 (26.4) 4.92 (1.0) MSDout 2.37 (0.6) 1.05 (0.9) 3.22 (1.0) 6.08 (1.8) 9.2 (0.9) 1.27 (1.0) 2.4 (3.5) 1.17 (3.2) 1.93 (1.4) 2.75 (2.8) Table 5 MSD and KS for 10 evolutions of software graphs of SEDGE, averaged over 100 runs for each software. We also run the same experiments without tuning parameters: numbers in ()’s gives the ratio between the measurement without tuning and the measurement with tuning: a value below 1 means that it is better without tuning, above 1 that it is worse. SEDGE has the same fundamental property SDG has: it can capture the structure of the evolved network without tuning its parameters. As in table 4, the values in ()’s in table 5 gives the ratio between tuning and no tuning. We use α = 0.5, β = 0.4, e1 = 0.45 and e2 = NE − 0.05 in the non tuned parameters experiment. 6 Conclusion and future work In this paper, we consider the problem of generating graphs that are similar to real, sparse digraphs. We propose SDG which generates such graphs, exhibiting power law in their degree distributions. We show that SDG performs very well experimentally; furthermore, SDG is stable in terms of parameter tuning: we show that it behaves very well even if we do not perform parameter tuning. Then, we propose an extension named SEDGE which aims at generating series of sparse digraphs that is similar to a series of real graphs. The similarity between two graphs is not well defined; we have used different ways to measure it and we have discussed the influence on the final result of the generator. Other metrics can also be used and will be investigated in the future. We have used SDG and SEDGE with a type of graphs in mind; we have not defined these algorithms using any knowledge on the graphs being modeled: we have designed the algorithms, tested them on some real graphs, and observed the results. We think they may be used for many types of real graphs. More importantly, considering series of graphs is a very important aspect of our work. As real graphs are evolving, we think that we have to use dynamic models to deal with them to really capture something about the evolution of the real graph, and the understanding of the process underneath. 12 Georgios Papoudakis and Philippe Preux and Martin Monperrus Acknowledgements This work was partially supported by CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies 2015-2020, and the French Ministry of Higher Education and Research. We also wish to acknowledge the continual support of Inria, and the stimulating environment provided by the SequeL Inria project-team. References 1. Barabasi, A., Albert, R.: Emergence of scaling in random networks. Science 286 (1999) 2. Bollobas, B., Borgs, C., Chayes, J., Riordan, O.: Directed scale-free graphs. In: Proc. SODA, pp. 132–139 (2003) 3. Carstens, C.J., Berger, A., Strona, G.: Curveball: a new generation of sampling algorithms for graphs with fixed degree sequence (2016). Arxiv.org, 1609.05137 4. Holme, P.: Modern temporal network theory: a colloquium. The European Physical Journal B 88(9) (2015) 5. Kleitman, D., Wang, D.: Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Math. 6(1), 79–88 (1973) 6. Musco, V., Monperrus, M., Preux, P.: A generative model of software dependency graphs to better understand software evolution (2015). Arxiv, 1410.7921 7. Musco, V., Monperrus, M., Preux, P.: Mutation-based graph inference for fault localization. In: Proc. SCAM, pp. 97–106 (2016) 8. Musco, V., Monperrus, M., Preux, P.: A large-scale study of call graph-based impact prediction using mutation testing. Software Quality Journal 25(3), 921–950 (2017) 9. Newman, M.: The structure and function of complex networks. SIAM Review 45(2), 167–256 (2003) 10. Raemaekers, S., Deursen, A.v., Visser, J.: The maven repository dataset of metrics, changes, and dependencies. In: Proc MSR, pp. 221–224. IEEE Press (2013) 11. Staudt, C.L., Hamann, M., Safro, I., Gutfraind, A., Meyerhenke, H.: Generating Scaled Replicas of Real-World Complex Networks, pp. 17–28. Springer International Publishing (2017) 

1 Use of a speed equation for numerical simulation of hydraulic fractures Alexander M. Linkov Institute for Problems of Mechanical Engineering, 61, Bol’shoy pr. V. O., Saint Petersburg, 199178, Russia Presently: Rzeszow University of Technology, ul. Powstancow Warszawy 8, Rzeszow, 35-959, Poland, e-mail: linkoval@prz.edu.pl Abstract. This paper treats the propagation of a hydraulically driven crack. We explicitly write the local speed equation, which facilitates using the theory of propagating interfaces. It is shown that when neglecting the lag between the liquid front and the crack tip, the lubrication PDE yields that a solution satisfies the speed equation identically. This implies that for zero or small lag, the boundary value problem appears ill-posed when solved numerically. We suggest - regularization, which consists in employing the speed equation together with a prescribed BC on the front to obtain a new BC formulated at a small distance behind the front rather than on the front itself. It is shown that regularization provides accurate and stable results with reasonable time expense. It is also shown that the speed equation gives a key to proper choice of unknown functions when solving a hydraulic fracture problem numerically. Keywords: hydraulic fracturing, numerical simulation, speed equation, ill-posed problem, regularization 1. Introduction Hydraulic fracturing is a technique used extensively to increase the surface to or from which a fluid flows in a rock mass. It is applied for various engineering purposes such as stimulation of oil and gas reservoir recovery, increasing heat production of geothermal reservoirs, measurement of in-situ stresses, control of caving in the roof of coal and ore excavations, enhancing efficiency of CO2 sequestration and isolation of toxic substances in rocks. In natural conditions, a similar process occurs when a pressurized melted substance fractures impermeable rock leading to the formation of veins of mineral deposits. Beginning with researchers such as Khristianovich & Zheltov (1955), Carter (1957), Perkins & Kern (1961), Geertsma & de Klerk (1969), Howard & Fast (1970), Nordgren (1970), Spence &Sharp (1985), Nolte (1988) numerous studies have been published on the theory and numerical simulation of hydraulic fracturing (see, e. g., Desroches et al. 1994; Lenoach 1995; Garagash & Detournay 2000; Adachi & Detournay 2002; Detournay et.al. 2002; Savitski & Detournay 2002; Jamamoto et al. 2004; Pierce & Siebrits 2005; Garagash 2006; Adachi et al. 2007; Mitchel et al. 2007; Kovalyshen & Detournay 2009; Kovalyshen 2010; Hu & Garagash 2010; Garagash et al. 2011 and detailed reviews in many of them). The review by Adachi et al. (2007) is specially organized to give a comprehensive report on the computer simulation; it actually covers the present state of the art as well. Thus there is no need to dwell on the historical background, detailed analysis of the processes in a near-tip zone resulting in particular asymptotics and regimes of flow, general equations and general approaches used to date. Being interested in computer aided simulation of hydraulic fracturing, in this paper we would rather focus on a key area that needs to be addressed for further progress in numerical simulation (Adachi et al. 2007, p. 754); this is the need “to dramatically speed up” simulators of fracture propagation. Naturally, reasonable accuracy of results should be guaranteed. The goal cannot be reached without clear understanding of underlying computational difficulties which strongly influence the accuracy and stability of numerical results and robustness of procedures. The paper addresses this issue. It presents in detail the results of brief communications by the author (Linkov 2011a, b). Our prime objective is to delineate and to overcome the computational difficulty caused, in essence, by strong non-linearity of the lubrication equation and by the moving boundary. In contrast with the cited previous publications which employed the global form of the mass 2 balance to trace liquid front propagation in time, we explicitly write and use the local speed equation (SE). The speed function entering the SE may serve to employ methods developed in the theory of propagating surfaces (Sethian 1999). The SE gives also a key to proper choice of unknown functions, which are analytical up to the front. We show that at points where the lag between the liquid front and the crack tip is zero, the lubrication equation yields that its solution identically satisfies the SE. This implies that for zero or small lag, the problem will appear ill-posed when solved numerically for a fixed front at a time step, and consequently it requires appropriate regularization to have accurate and stable numerical results. We suggest a method of regularization that employs the very source of the difficulty to overcome it. The SE and a BC at the front are used together to derive a new BC formulated at a small distance behind the front rather than on the front itself. This leads to regularization which provides accurate and stable numerical results. The Nordgren model serves to clearly display the computational features discussed. It is also used to obtain benchmarks with five correct significant digits, at least. 2. Global mass balance, speed equation and ill-posed problem for hydraulically driven fracture Reynolds transport theorem (e. g. Crowe et al. 2009), applied to the mass of an arbitrary volume of a medium in a narrow channel between closely located boundaries, after averaging over the channel width, reads: dM w w dS wvn dL , (2.1) dt t t Sm L where dM / dt is the external mass coming into or out of the considered volume per unit time, S m is the middle surface of the volume, w is the width (opening) of the channel, L is its contour, ρ is the mass density averaged over the width, vn is the normal to L component of the particle velocity also averaged over the width; the normal n is assumed to be in the plane tangent to S at a considered point. Applying (2.1) to the total volume of the medium with the middle surface S t and contour Lt at a moment t we have: dM e w dS w vn dL . (2.2) dt t t St Lt Here, and henceforth, the star denotes that a value refers to the medium front. When writing (2.2) we assume that there is no significant sucking or vaporization through the front. Then the speed V* of the front propagation coincides with the normal component vn of the particle velocity. Thus we have the key equation: dxn * , (2.3) V* vn dt where xn* is the normal to the front component of a position vector of a point on the front; in global coordinates, x x * (t ) is a parametric equation of the front contour Lt . As a rule, the tangent component of the particle velocity is small as compared with the normal component at the front. In this case, the front moves with the speed exactly equal to the velocity of fluid particles comprising it, and we have V* vn* v * , where v * is the vector of the particle velocity averaged over the front width. For incompressible homogeneous liquid, ρ = const, and (2.2) may be written as the equation of the liquid volume balance: dVe w dS w vn dL , (2.4) dt St t Lt where Ve Me / . 3 Comment 1. For 1-D case, the volume conservation equation (2.4) yields x (t ) * w( x, t )dx Ve (t ) . 0 Then for a „rigid-wall‟ channel ( w w(x) , w / t 0 ), integration gives the front location as a function of time x* (t ) f (Ve (t )) . It is easily seen from the last expression that if the width w(x) decreases fast enough with growing x, then for a prescribed influx Ve (t ) , the front coordinate turns to infinity at a finite time t* ( x* (t* ) ). A solution does not exist for t t* . For instance, in the case of the constant influx rate q0 ( Ve (t ) q0t ) and exponentially decreasing width ( w a exp( x) , a, α > 0), the front turns to infinity as t t* a /( q0 ) . There is no solution for t t* . This clearly indicates that problems involving flow of incompressible liquid in a thin channel are quite tricky. A solution may not exist or it might be difficult to find the solution numerically, especially when the rigidity of channel walls is very high (according to Pierce & Siebrits (2005), high rigidity leads to a stiff system of ordinary differential equations, when solving a boundary value problem by finite differences). By definition of the flux through the channel width (e.g. Batchelor 1967), we have q wv . Then in (2.4) w vn qn is the normal to L component of the flux through the channel width at the liquid front, and equation (2.4) may be written as q n* (x * ) . (2.5) V* v n (x * ) w* (x * ) This is the speed equation (SE). Its right hand side (r. h. s) defines the so-called speed function. Emphasize that the SE is general. It is not influenced by viscous properties of a particular incompressible liquid and by the presence of leak-off or influxes through the walls of a thin channel. In some cases, for instance, when a liquid flows in a fracture without a lag between the liquid front and the crack tip, both the flux and opening turn to zero at the liquid front. Then (2.5) takes the limit form: q ( x) . (2.6) vn (x* ) lim n x x w(x) It is highly apparent that even when w* ( x * ) = 0 and qn* (x* ) = 0, the limit on the r. h. s. of (2.6) should be finite to exclude the front propagation with infinite velocity. This suggests using the particle velocity as an unknown in numerical calculations, because it is non-singular in entire flow region including the front. What is also beneficial, the velocity is non-zero except for flows with return points. In formulations of problems for hydraulic fracturing, the average particle velocity v(x) q(x) / w(x) does not enter equations. Rather, the total flux q through a cross-section is used. Specifically, the divergence theorem applied to (2.1) in the case of incompressible liquid yields the continuity equation in terms of the flux q and opening: w div q qe , (2.7) t where qe is the prescribed intensity of distributed external sources of liquid, q is the flux vector, defined in the tangent plane to St at a considered point. The Poiseuille law, used for a flow of incompressible liquid in a narrow channel, connects the flux q with the pressure gradient (2.8) q D(w, p)gradp . Herein, D is a prescribed function or operator, gradient is also confined to the coordinates in the tangent plane. Substitution of (2.8) into (2.7) gives the lubrication partial differential equation (PDE) at points of liquid: w (2.9) div D( w, p)gradp qe 0 . t Some initial (normally zero-opening) condition is assumed to account for the presence of the time derivative in (2.9). Being of the second order and elliptic in spatial derivatives, equation (2.9) requires 4 a boundary condition (BC) on the liquid contour Ll. Normally it is the condition of the prescribed flux q0 at a part Lq and of the prescribed pressure p0 at the remaining part Lp of the contour Ll: (2.10) qn (x* ) q0 (x* ) x* Lq ; p(x* ) p0 (x* ) x* L p . We see that neither partial differential equations (PDE) (2.7)-(2.9), nor BC (2.10) involve the particle velocity. The latter enters only the SE (2.5) or its limit form (2.6) for points on the front. At these points, it is defined by formula (2.3). Since according to (2.8) qn D(w, p) p / n for any direction n in a tangent plane, the flux entering the numerator on the r. h. s. of (2.5) is p qn D( w, p ) . (2.11) nx x * Then for the hydraulic fracturing, the SE (2.5) is specified as: 1 p vn D( w, p ) , w* (x * ) nx x (2.12) * where n is the outward normal to Lt in the tangent plane at a considered point of the liquid front. The r. h. s. of (2.12) specifies the speed function, which is the basic concept of the theory of propagating interfaces (Sethian 1999). It may serve to employ level set methods and fast marching methods. In hydraulic fracture problems, the opening is unknown. To have a complete system, the lubrication PDE (2.9) is complemented with an equation of solid mechanics connecting the opening w and pressure p: (2.13) A(w, p) 0 . As a rule, the operator A in (2.13) is prescribed by using the theory of linear elasticity. In addition, to let the fracture propagate, we need a fracture criterion (Otherwise, the liquid front reaches the crack contour and stops.). Commonly, at points of the crack contour, authors impose the condition of linear fracture mechanics: (2.14) K I K Ic , where K I is the stress intensity factor (SIF), K Ic its critical value. At the crack contour Lc , the opening is set zero: (2.15) w(xc ) 0 . In general, for physical reasons, excluding negative pressure, which tends to minus infinity at points of the front, where the latter coincides with the crack contour, the liquid surface St is solely a part of the crack surface Sc so that the liquid contour Lt is within the crack contour Lc . Thus, in general there is a lag between Lt and Lc . In this case, the second of the conditions (2.10) is prescribed at the boundary of the propagating liquid front: (2.16) p(x* ) p0 (x* ) . In many cases, the change of the pressure p0 (x* ) along the liquid front is small. Consequently, the tangential derivative p / of the pressure is small as compared with the normal derivative p / n , and (2.8) implies that the tangential component of the flux is small as compared with its normal component. Since v q / w , it means that the tangential component of the fluid velocity at the front may be neglected. Then, as mentioned above, the front velocity equals to the particle velocity itself. This case will be under discussion further on. We see that the existence of a lag, whatever small it is, has important physical implications for fracture propagation. Normally the lag is small and accounting for it strongly complicates a problem, while neglecting it may be justified (e. g., Garagash & Detournay 2000). For these reasons, many papers on hydraulic fracturing (e. g., Spence & Sharp 1985; Lenoach 1995; Garagash & Detournay 2000; Adachi & Detournay 2002; Detournay et. al. 2002; Savitski & Detournay 2002; Jamamoto et al. 2004; Pierce & Siebrits 2005; Adachi et al. 2007; Mitchel et al. 2007; Hu & Garagash 2010; Garagash et al. 2011) assume that the lag is zero. Then at all points of the propagating liquid front, coinciding in this case 5 with the crack contour, the flux is zero: qn* (x* ) 0 . The latter condition is met in view of (2.15), because the operator D(w, p) in (2.11) is such that D(0, p) 0 . Still, to satisfy both (2.15) and (2.14), the elasticity equations require specific asymptotic behavior of the opening with the coefficient of the asymptotic proportional to the SIF, when the latter is not zero. The SIF depends on the pressure (see, e. g. Spence & Sharp 1985). As a result, at points of the front we have the boundary condition (2.14) which, similar to (2.16), involves the pressure. The fact that the lag is small and it is commonly neglected has significant consequences for numerical calculations. To show it, consider the problem in the local Cartesian coordinates x1 ' O1 ' x2 ' with the origin O1 ' at a point x* at the liquid front and the axis x1 ' opposite to the direction of external normal to the front at this point. We employ aforementioned advantages of using the particle velocity v q / w . In the local system, for points close to the front, the normal component of the 1 p velocity is v . Then the lubrication PDE (2.9) at points near the front takes the form: D( w, p) w x1 ' v ln w ln w (2.17) (v v* ) qe 0 , x1 ' x1 ' t x ' const 1 where the partial time derivative is evaluated under constant x1 ' . Using ln w serves to account for an 0 ) at the front. In arbitrary power asymptotic behavior of the opening w( x1 ' , t ) C (t )(x1 ' ) ( particular, in the case of Newtonian liquid, for zero lag, the exponent α = 2/3 when in (2.14) K Ic 0 and α = ½ when K Ic 0 (Spence & Sharp 1985). In the case of non-Newtonian liquid, formulae for α are given by Adachi & Detournay (2002). For the Nordgren problem discussed below, α = 1/3. Note that for a flow with Carter‟s leak-off, so-called intermediate asymptotics may appear (e.g., Lenoach 1995; Kovalyshen & Detournay 2010). These asymptotics manifest themselves at some distance from a crack contour rather than at the contour itself. For this reason, we shall not use them below in equations involving a point on the front. When the opening has the power asymptotic near the front, it is reasonable, in addition to the particle velocity, to use also the variable y w1 / , which is linear in x1 ' near the front. The PDE (2.17) then becomes v v* y v 1 y (2.18) qe 0 . x1 ' y x1 ' y t x ' const 1 Note that the derivative v / x1 ' and, under the supposed asymptotic, the derivatives y / x1 ' , the multiplier (v v * ) / y and the term (1 / y) y / t in (2.18) are finite at the liquid front. Therefore, the D( w, p ) grad p and y w1 / present a proper choice of unknown functions for w problems of hydraulic fracturing. Write (2.18) as v y y (2.19) y (v v* ) yqe 0 . x1 ' x1 ' t x1' const variables v Equation (2.19) implies that v v* at any point, where the opening w, and consequently y w1 / , is zero. Such are points at the front for zero lag. This means that when neglecting the lag, the SE (2.12) is satisfied identically by a solution of the PDE (2.18). Obviously, the same holds for a solution of the starting PDE (2.9). We see that for zero lag, when solving the boundary value (BV) problem for (2.9), one implicitly has satisfied the SE (2.12) additional to the prescribed BC of zero opening (2.15). Note that for the mentioned power asymptotic of the opening, the SE may be re-written in terms of the normal 6 derivative of y w1/ as y x1 ' x Bv * , where B is a function of time only. Then at a point of the 1 0 front, we have satisfied two equations: y ( x* ) 0 and y n Bv* . Recall that the operator x x* div D( w, p)grad p is elliptic and requires only one BC, whereas actually there are two BC. Consequently, we have a Cauchy problem for the elliptic operator. As known (e. g. Lavrent‟ev & Savel‟ev 1999), such a problem is ill-posed in the Hadamard sense (1902). To have accurate and stable results when solving an ill-posed problem numerically, one needs its proper regularization (e. g. Tychonoff 1963; Lavrent‟ev & Savel‟ev 1999). In the case when the lag is not neglected, we come to similar conclusions if the lag is small enough. In this case, at a point of the liquid front, we have the BC of (2.16) type. As the lag is small, the opening w* (x* ) at the liquid front is small, as well. Then from (2.19) it follows that for a solution of the lubrication PDF, the SE is met approximately. Therefore, at points of the liquid front, in addition 1 p D( w, p ) vn to the BC (2.16) for the pressure, we have the approximate equation w* (x * ) nx x * for its normal derivative. Hence, the problem of solving the lubrication PDF will appear ill-posed in numerical calculations having the accuracy less than that of the approximate equation. It is reasonable to have a method of regularization, which removes computational difficulties caused by the discussed feature. Comment 2. If a problem is self-similar and the lag is neglected, then integration of the lubrication equation in the automodel coordinate (e. g. Spence & Sharp 1985; Adachi & Detournay 2002) from the liquid front removes the difficulty: it is sufficient to seek the solution by taking into account for asymptotic representation of the opening and pressure near the liquid front. In papers by Spence & Sharp (1985) and Adachi & Detournay (2002), such representations meet both conditions (2.15) and (2.6), which uniquely define the coefficients of the asymptotics at one point (the crack tip). In fact, the authors solve an initial value (Cauchy) well-posed problem. The boundary condition of the prescribed flux at another point (the inlet) is not used: the corresponding influx is found after obtaining the solution of the Cauchy problem. Similar approach is applicable to the Nordgren problem. It serves us to obtain the benchmarks in Sec. 4. Unfortunately, this method cannot be applied in a general case when a self-similar formulation is not available. 3. Nordgren problem. Evidence of ill-posed problem Consider the Nordgren (1972) model to see unambiguously that the BV problem is ill posed, to find a proper means for its regularization and to obtain accurate numerical results, which may serve as benchmarks. The analysis also confirms that the front velocity does satisfy the asymptotic equation (2.6) despite that both the opening w(x ) and the flux qn (x ) are zero at the front. 3.1. Problem formulation Recall the assumptions of the Nordgren (1972) problem. Similar to the Perkins-Kern (1961) model, it is assumed that a vertical fracture of a height h (Fig. 1) is in plane-strain conditions in vertical cross sections perpendicular to the fracture plane. The cross section is elliptical and the maximal opening w decreases along the fracture. Nordgren‟s improvement of the model includes finding the fracture length x (t ) as a part of the solution. Nordgren also accounts for the fluid loss due to leak-off. The corresponding term is actually a prescribed function of time; we neglect it to not overload the analysis. In this case, the continuity equation (2.7) reads q / x w / t 0 , where w is the average opening in a vertical cross section, q is the flux through a cross section divided by the prescribed height h. The 7 liquid is assumed Newtonian with the dynamic viscosity kl . Then in (2.8), D( w, p) kl w3 , where 1 /( 2 ) in the case of an elliptic cross section, considered by Nordgren (1972); for an arbitrary thin plane channel, the Poiseuille value kl 1 /(12 ) is often used (e. g., Garagash & Detournay 2000; Savitski & Detournay 2002; Garagash 2006; Mitchel et al. 2007; Hu & Garagash 2010). Thus the equation (2.8) becomes: p q kl w3 . (3.1) x The dependence (2.13) between the average opening and pressure is taken in the simplest form p kr w , (3.2) found from the solution of a plane strain elasticity problem for a crack of the height h; is the Poisson‟s ratio. Therefore, for a k r (2 / h) E /(1 2 ) , E is the rock elasticity modulus, non-negative opening, the pressure is non-negative behind the front. With lag neglected, the condition (2.15) in view of (3.2) implies that the pressure becomes zero at the liquid front. The opening (as well as the pressure) should then be positive behind the front and zero ahead of it. Under these assumptions, there is no need in the fracture criterion (2.14). In view of (3.1) and (3.2), the continuity equation (2.9) becomes the Nordgren PDE: 2 4 1 w w kl k r 0. (3.3) 2 4 t x It is solved under the initial condition of zero opening (3.4) w(x,0) 0 for any x along the prospect path of the fracture. The BC for the partial differential equation (PDF) (3.3) includes the condition of the prescribed influx q0 at the fracture inlet x = 0: (3.5) q(0, t ) q0 and the condition that there is no lag between the crack tip and the liquid front: w( x , t ) 0 . (3.6) The solution should be such that the opening is positive behind the front and zero ahead of it: w( x, t ) 0 0 x x , w( x, t ) 0 x x . (3.7) The Nordgren problem consists in finding the solution of PDE (3.3) under the zero-opening initial condition (3.4) and the BC (3.5), (3.6). The solution should comply with (3.7). 3.2. Speed equation, self-similar problem formulation, clear evidence that the problem is ill-posed Nordgren used the conditions (3.7) to find the front propagation, rather than the global mass balance commonly employed for this purpose (e. g., Howard & Fast 1970; Spence & Sharp 1985; Adachi & Detournay 2002; Savitski & Detournay 2002; Jamamoto et al. 2004; Garagash 2006; Adachi et al. 2007; Mitchel et al. 2007; Hu & Garagash 2010; Garagash et al. 2011). For our purposes, we employ the SE (2.12). In the case considered, it becomes: 1 w3 v kl k r , (3.8) 3 x x x where according to (2.3) v dx / dt . Introduce dimensionless variables: x x t xd , xd , td ,v d xn xn tn dx d dtd v , wd vn w , pd wn p , qd pn q , q0d qn q0 qn where xn (kl k r )1 / 5 qn3 / 5t n3 / 5 , v n xn / tn , wn qnt n / xn , pn 4k r wn , and t n , qn are arbitrary scales of the time and flux respectively. In terms of dimensionless values, the equations (3.1), (3.2) 8 pd , pd xd become, respectively, read qd wd 3 wd4 . Then the PDE (3.3) and the BC (3.5), (3.6) xd 4wd ; hence, qd 2 w4 x2 w4 x w t 0, (3.9) q0 , (3.10) x 0 w( x , t ) 0 . (3.11) From this point on, we omit the subscript d at variables and consider only dimensionless values. The homogeneous conditions (3.4) and (3.7) do not change their form. Nordgren (1972) solved the problem by finite differences not using the SE. In dimensionless variables, the SE (3.8) takes the form: 4 w3 v . (3.12) 3 x x x In compliance with the said in Sec. 2, the SE (3.12) gives a key to proper choice of the unknown function. Indeed, to have the front velocity finite, the partial derivative w3 / x should not be singular at the liquid front. This yields that w 3 is an analytical function of x and it may be represented by a x at any instant t: w3 ( x, t ) power series in x* j 0 a j (t )( x* x) j . The zero-opening condition (3.11) gives a0 (t ) 0 , while the SE (3.12) gives a1 (t ) 0.75v* (t ) . Then at the vicinity of the crack tip we have w( x, t ) w3 ( x, t ) a11/ 3 ( x* a1 (t )( x* x) O ( x* x)1/ 3 O ( x* x) 2 what means that the opening behaves as x) 2 / 3 . Therefore, near the crack tip, in contrast with w3 , the derivative w / x of w is singular as ( x* x) 2/3 . We see that using w 3 avoids unfavorable asymptotic behavior of w, while the SE (3.12) governs the linear asymptotic behavior of w 3 . Hence, it is reasonable to use w3 as the unknown function, rather than w as used by Nordgren (1972) or w 4 : in contrast with the derivatives of w3 , the 2 4 2 3 w w 4 w w3 w w derivatives and are singular at the front x x . In terms of w3 2 x 3 x x x x2 PDE (3.9) becomes: 2 2 w3 1 w3 1 w3 0. (3.13) x x 2 3w3 4 w3 t For further discussion we use the fact that the problem is self-similar which serves to reduce PDF (3.13) to an ordinary differential equation (ODE). We express the variables x and w via automodel variables and function y( ) as x 3 t 4 / 5 , w( x) t1 / 5 ( xt 4 / 5 ) . Then (3.13) becomes ODE with the unknown ( ): d2y where a( y, dy / d , ) (dy / d dy 3 20 0, d 0.6 ) /(3 y) . The BC (3.10) and (3.11) read: 2 a( y, dy / d , ) dy 0.75 0 y( ) q0 , 3 y (0) 0, (3.14) (3.15) (3.16) 9 and the SE (3.12) becomes: dy 0.6 . (3.17) Re-write (3.14) by using the expression for a( y, dy / d , ) as d2y 1 dy 0.6 3 dy 3 y 20 (3.18) 0, d 2 In limit , for a solution, satisfying the BC (3.16), ODE (3.18) turns into the SE (3.17). Hence, for ODE (3.14), at the point , we have imposed not only the BC (3.16) for unknown function y , but also the BC (3.17) for its derivative dy/ d . Note that equations (3.16), (3.17) imply that the factor a in (3.14) is finite at the liquid front: lim a( y, dy / d , ) 1 /(3 ) . y It is easy to check by direct substitution that if y1 ( 1 ) is the solution of the problem (3.14)-(3.17) for q0 q01 with y1 ( 2 k ) / k is the solution of the problem (3.14)-(3.17) 1 , then y 2 ( 2 ) k 5 / 6q01 with for q02 C ( q0 ) 0.6 / and C0 2 y ( 0) / 1/ 2 k ; herein, k is an arbitrary positive number. This implies that are constants not dependent on the prescribed influx q 0 . As (q0 ) / C , it is a matter of convenience to prescribe q 0 or . A particular value of q 0 or may be also taken as convenient. Indeed, with the solution y1 ( 1 ) for q0 q01 , we find the solution 0.6 for any q0 : y( ) y1 ( k ) / k , where k (q01 / q0 ) 6 / 5 , 1 / k ). 1/ k ( Let us fix . According to (3.16), (3.17), at the point , we have prescribed both the function y and its derivative dy / d . Thus, for the ODE of the second order (3.14) we have a Cauchy (initial value) problem. Naturally, its solution defines y(0) and dy / d 0 and consequently the flux q 0 at 0 . Hence, even a small error when prescribing q 0 in (3.15), excludes the existence of the solution of the BV problem (3.14)-(3.16). Therefore, by Hadamard (1902) definition (see also Tychonoff 1963; Lavrent‟ev & Savel‟ev 1999), the BV problem (3.14)-(3.16) is ill-posed. It cannot be solved without a proper regularization (Tychonoff 1963; Lavrent‟ev & Savel‟ev 1999). To make conclusions on the accuracy of numerical results obtained without and with regularization, it is reasonable to obtain benchmarks. 4. Benchmark solution The initial value (Cauchy) problem (3.14), (3.16), (3.17) is well-posed. Thus its solution provides the needed benchmarks. To solve the system we transformed the problem (3.14), (3.16), (3.17) to the equivalent problem in two unknowns Y1 ( ) y ( ) and Y2 ( ) dy / d . This yields the system of two ODE: dY1 Y2 d (4.1) dY2 3 a(Y1 , Y2 , )Y2 d 20 under the Cauchy conditions at the point , corresponding to (3.16) and (3.17), Y1 ( ) 0 , Y2 ( ) 0.6 . (4.2) The Cauchy problem (4.1), (4.2) is solved by using the fourth order Runge-Kutta scheme (see, e.g. Epperson 2002). Calculations were preformed with double precision. For certainty, we set = 1= 1 10 (calculation with other values of gave the same results for C and C0 to the seventh significant digit including). The integration step was changed from 10-2 to 10-5, and the number of steps was consequently changed from 102 to 105 to reach the inlet point 0 . The number of iterations for the nonlinear factor a( y, dy / d , ) was changed from 20 to 1000. We could see that the step 10-5 and the number of iterations 50 are sufficient to guarantee at least six correct digits. The values of the constants C and C0 are evaluated to the accuracy of seven digits: C 0.7570913 , C0 0.5820636. For values depending on , we shall use the subscript 1 when they correspond to = 1 = 1. Thus we have: q01 C*5 / 3 0.6288984 , 1 (0) 3 C0 =0.8349418. Values for an arbitrary flux q0 may be obtained as q0 0.6 /C 1.3208446q0 0.6 ( 0) , 3 C0 2 1.0051356 q00.4 . (4.3) 1.0073486 , For the value q0 2 / , used by Nordgren (1972), equations (4.3) give (0) 0.8390285 against the values given by this author to the accuracy of about one percent: 1.01 , (0) 0.83. The values of 1 3 y1 and d 13 / d 1 are presented in Table 1 with five correct digits. Values of ( ) and d 3 / d for an arbitrary prescribed flux q0 may be obtained from those in the Table 1 3 as y1 ( k)/k , d 3 /d 3 1 (d / d 1) / k k 1 with k (q01 / q0 )1.2 0.5731872 / q10.2 . Comment 3. Table 1 shows that the derivative d 3 / d , defining the particle velocity, is nearly constant along the entire liquid being close to its limit value at the front 3 3 approximately linear in . Next, from the BC (3.16) we obtain ( ) / (0) 1 approximate analytical solution of the Nordgren problem is given by the equation: w( x, t ) / w(0, t ) (1 x / x )1/ 3 0.2 3 . This implies that / is . Hence, the (4.4) 0.8 with w(0, t ) t are found (0) and x (t ) t ; for a given flux q0 , the values of (0) and from (4.3). As clear from Table 1, the error of (4.4) does not exceed one percent. The graph corresponding to the approximate solution (4.4) is indistinguishable from that given by Nordgren (1972). Naturally, the asymptotic of the solution (4.4) agrees with the predicted asymptotic behavior a11 / 3 ( x* w( x, t ) , and consequently, w 3 is proportional to x proportional to The (d 3 1 accurate / d 1) / k 1 0.6 1.3208446q0 , data k with k x. (0) 1.0051356 q00.4 , ( ) , ( ) and d 3 /d 0.2 , w( x, t ) d 3 /d are known, we can find the front location x (t ) , the front velocity v (t ) , the opening w( x, t ) and the particle velocity 0.8 t and 0.5731872 / q10.2 serve to estimate errors when solving the Nordgren problem as a BV problem. When v (t ) 3 is x)1 / 3 near the crack tip because the conditions (4.2) guarantee that t 0.2 ( xt 0.8 ) and ( x, t ) 4 0.2 d t 3 d ( x, t ) as x (t ) 3 xt 0.8 . 5. Straightforward solving self-similar BV problem. Method of regularization 5.1. Straightforward integration by finite differences t 0.8 , 11 Forget for a while about the SE and all the said on its influence on a BV problem. Let us see what happens when solving the BV problem (3.14)-(3.16) in a common way by finite differences. We performed hundreds of numerical experiments with various numbers of nodal points and iterations and different values of the prescribed influx q01 at the inlet. Finite difference approximations of second order for d 2 y / d 2 and dy/ d were combined with iterations for a( y, dy / d , ) . Up to 100 000 nodal points and up to 1500 iterations were used in attempts to reach the accuracy of three correct digits, at least. The attempts failed: by no means could we have more than two correct digits. Moreover, the results always strongly deteriorate near the liquid front. The numerical results clearly demonstrate that the BV problem (3.14)-(3.16) is ill-posed. It cannot be solved accurately without regularization. As illustration, the dashed line in Fig. 2 presents a typical graph of d 3 / d , obtained under the Nordgren boundary value q01 2 / . For comparison, the benchmark values of d 3 / d , calculated by using the Table 1, are shown by the solid line with markers. Obviously, the results strongly deteriorate near the liquid front (in the considered example, the benchmark value of equals 1.0073486). Comment 4. Using the variable 3 ( ) , which is linear near the liquid front, removes a suggestion that the deterioration is caused by singularity of d / d and d 2 / d 2 at the point . Comment 5. It is worth noting that the accuracy of two correct digits was obtained at points not too close to the front even when using a rough mesh with a hundred or even only ten nodes. This indicates that using a rough mesh may serve to regularize a problem when high accuracy is not needed. 5.2. - regularization. The numerical experiments evidently confirm that the considered ill-posed BV problem (3.14)(3.16) cannot be solved accurately without regularization. A regularization method is suggested by the conditions (3.16), (3.17). Indeed, we may use them together to get the approximate equation y 0.6 ( ) near the front. Hence, instead of prescribing the BC (3.16) at the liquid front , where it is implicitly complimented by the SE (3.17), we may impose the boundary condition, which combines (3.16) and (3.17) at a point (1 ) at a small relative distance 1 / from the front: 0 .6 2 . (5.1) The BV problem (3.14), (3.15), (5.1) is well-posed and may be solved by finite differences. Numerical implementation of this approach shows that with = 10-3, 10-4 the results for the step / = 10-3, 10-4, 10-5, 10-6 coincide with those of the benchmark solution. The time expense is fractions of a second. The results are stable if and are not simultaneously too small (both and -5 are greater than 10 ). As could be expected, the results deteriorate when both the regularization parameter and the step become too small. Specifically, when = = 10-6, the results are completely wrong. Actually, in this case, to the accuracy of computer arithmetic, the problem is solved without regularization. We could also see that with growing step , the accuracy decreases and for a coarse mesh it actually does not depend on the regularization parameter. In particular, for a quite coarse mesh with the step = 0.1, the accuracy is about one percent, and the results stay the same to this accuracy for any from 10-2 to 10-9. The essence of the suggested regularization consists in using the SE together with a prescribed BC to formulate a BC at a small distance behind the liquid front rather than on the front itself. We call such an approach -regularization. The next section contains its extension to the cases when a selfsimilar formulation is not available or is not used. y( ) 12 6. Straightforward solution of starting BV problem. Regularization 6.1. Straightforward integration by time steps with finite differences on a time step Forget again about the SE and its influence on numerical solution of a BV problem. Try to solve the starting Nordgren problem by common finite differences. Nordgren (1972) used straightforward numerical integration of the problem (3.9)-(3.11) under the zero-opening initial condition with the conditions that opening is positive behind the liquid front and zero ahead of it. This author applied Crank-Nicolson finite difference scheme to approximate PDF (3.9) and to meet the BC (3.10), (3.11). The resulting non-linear tridiagonal system was linearized by employing linear approximation of w 4 . Nordgren (1972) does not include details of calculations on the initialization, the time step, the number of nodes in spatial discretization, the number of iterations, stability of numerical results and expected accuracy. To obtain knowledge on these issues, we also solved the system (3.9)-(3.11) in a straightforward way by using the Crank-Nicolson scheme. The results are as follows. Actually performing 20 iterations to account for the non-linear term w 4 is sufficient to reproduce four digits of the fracture opening, except for close vicinity of the liquid front (Increasing the number to 100 iterations does not improve the solution for all tested time and spatial steps.). For various time -2 -3 -4 -2 -3 -4 steps ( t 10 , 10 , 10 ) and different spatial steps ( x 10 , 10 , 10 ) taken in various combinations, the results are stable along the main part of the interval [0, x (t ) ], but deteriorate and are unreliable in close vicinity of the front ( 1 x / x (t ) < 0.001). This yields changes in the third digit of and (0) , calculated by using x (t ) and w(0, t ) . The asymptotic behavior (4.4), as x x (t ) , is reproduced near the front except for its close vicinity. The results coincide with those given by Nordgren (1972) to the accuracy of two significant digits accepted in his work. In all the calculations, by no means could we have a correct third digit. Similar to self-similar solution, fine meshes did not improve the accuracy as compared with a rough mesh having the step x / x = 0.01. The results clearly show that the problem cannot be solved accurately without regularization. Comment 6. As mentioned in Sec. 3, the variable w has a singular partial derivative w / x at the liquid front. To remove the influence of the singularity, we also solved the problem by using w3 as an unknown function, because according to the SE (3.12) its spatial derivative is not singular. The conclusions when using w3 are the same as those above. Again, by no means could we have reliable a third digit, and results strongly deteriorated at a close vicinity of the liquid front. This shows that the inaccuracy is caused not by singularity of the derivative w / x at the liquid front. It is caused by the fact the BV problem, when solved by common finite differences, appears ill-posed. 6.2. Reformulation of PDE to form appropriate for using - regularization. Numerical results Extension of - regularization to solve PDE requires the combined use of the BC (3.11) on the front with the SE (3.12) to impose a BC at a small relative distance from the front. The distance being relative, we need to count it in the local system with the origin at the front. Hence it is reasonable to ( x x) / x from the front. The relative distance from the inlet is introduce the relative distance / t x const of a function ( x, t ) , which enters PDE 1 x / x (t ) . The partial time derivative and is evaluated under constant x, should be transformed into the partial time derivative / t const ( x (t ), t ) evaluated under constant . Omitting routine details of the change of the function ( , t ) of variables, we have the transformation: v (t ) , (6.1) t x const t const x (t ) where v (t ) dx / dt . When using the variable and transformation (6.1), equation (3.13) becomes: 13 2 Y 2 where Y ( , t ) A(Y , Y / w3 ( x (t ), t ) , A(Y , Y / Y ,x v ) ,x v ) B(Y , x ) Y/ Y t 0.75 x v 3Y 0, , B(Y , x ) (6.2) x2 . 4Y The BC (3.10), (3.11) in new variables read: 43Y Y 3 x ς Y ( ,t) q0 , (6.3) ς 0 1 0. (6.4) The SE (3.12) takes the form: Y 0.75 x v . (6.5) 1 Prove firstly, that the problem (6.2)-(6.4) is ill posed because like the self-similar formulation, the SE (6.5) is met identically by a solution of PDF (6.2) under the BC (6.4) of zero opening at the liquid front. Indeed, re-write (6.2) by using the expression for A(Y , Y / , x v ) as 2 1 Y Y Y (6.6) 0.75x v 0.75x 2 0. 3 t x ), for a solution, satisfying the BC (6.4), PDE (6.6) turns into the SE (6.5). In limit 1 (x Hence, for PDE (6.2), at the point 1 , we have imposed not only the BC (6.4) for unknown function Y , but also the BC (6.5) for its spatial derivative dY / d . Therefore, we have two rather than one BC at 1 and the problem appears ill-posed. Consequently, the starting problem (3.9)-(3.11) is ill-posed, as well, what explains the failure to solve it to the accuracy greater than two correct digits. Note that equations (6.4), (6.5) imply that the factor A(Y , Y / , x v ) in (6.2) is finite at the liquid front despite its denominator 3Y turns to zero: lim A(Y , dY / d , x*v* ) 1/ 3 . Y Y 2 The regularization of the problem (6.2)-(6.4) follows the line used for the self-similar formulation. Like the self-similar formulation, the BC (6.4) and the SE (6.5) yield the approximate equation near the liquid front 1: Y ( , t ) 0.75 x (t )v (t )(1 ) , (6.6) which defines the asymptotic behavior of the solution when 1 . As mentioned, the non-linear multiplier A(Y , Y / , x v ) in PDF (6.2) is non-singular at the liquid front 1 . The factor B (Y , x ) is finite except for the liquid front ( 1 ), because the opening is positive behind the front. Hence, similar to (5.1), we may impose the BC at the relative distance from the liquid front: (6.7) Y ( , t ) 0.75x (t )v (t ) , where 1 . In contrast with the problem (6.2)-(6.4), the problem (6.2), (6.3), (6.7) does not involve an additional BC. We may expect that it is well-posed and provides the needed regularization. Extensive numerical tests confirm the expectation. We solved the problem (6.2), (6.3), (6.7) by using the Crank-Nicolson scheme and iterations for non-linear multipliers A(Y , Y / , x v ) , B (Y , x ) at a time step. The velocity v (t ) is also iterated by using the equation following from (6.6): Y 3 xv . (6.8) 4 The condition (6.8) expresses (with an accepted tolerance) the continuity of the particle velocity at the point . After completing iterations, the final value of v (t ) shows the new coordinate of the t ) x (t ) v (t ) t . The value x (t t ) is used on the next time step. liquid front x (t 14 Initialization. In the considered problem there is no characteristic time and length. Thus, the starting equation x (0) 0 should be adjusted to this uncertainty. The adjustment concerns the initialization of time stepping. At t = 0 the liquid front coincides with the inlet. The opening is also zero. Then for „small‟ time, to have the factor A(Y , Y / , x v ) finite in the entire liquid, it should x v f ( ) with f ( ) be Y / 0.75 as 1 . Hence for „small‟ time, we have Y x v f ( ) with f (1) 0 . Substitution of Y and Y / into the BC (6.3), using v dx / dt and integration of 0.6 the resulting ODE yield x (t ) Ct 0.8 , where C 1.25 0.75q0 0.8 3 f (0) f (0) is a constant. Then v (t ) 0.8Ct 0.2 and Y 0.8C 2t 0.6 f ( ) . Insertion of these equations into PDF (6.2) gives ODE (3.14) / . Similarly, the BC (6.3), (6.4) and the SE (6.5) turn into the with y 0.8 f and corresponding equations (3.15), (3.16) and (3.17). The constant C actually represents , corresponding to the prescribed flux q0 . We see that when using time stepping, initialization requires solving the self-similar problem. The latter being ill-posed, the solution is obtained by - regularization as explained in Sec. 5.2. Having its solution, the initial data for an arbitrary chosen initialization time t 0 are found as x (t 0 ) t 00.8 , 2 0.6 0.8 t 0 0.2 , Y ( , t0 ) t0 ( * ) . For certainty, in the following calculations we set q0 = 1, t0 = 0.01. The same regularization parameter and the same spatial step were used for both the initialization and for each of the time steps. Numerical results. Two objectives were sought in numerical tests. First, we wanted to check the efficiency of -regularization, that is, its accuracy, stability and robustness. To this end, we used small relative distance , small spatial step , small time step t , and a large number of time steps. Secondly, we checked if the beneficial features of coarse meshes, observed for self-similar formulation, hold in time steps. Exploratory calculations have shown that fifty iterations in non-linear terms at a time step are sufficient to reproduce seven significant digits. Thus, in further tests the number of iterations was set equal to fifty. The benchmark solution served to evaluate the accuracy. In particular, for q0 = 1, the benchmark values of the front position and the front velocity are v (t 0 ) x (t ) 1.3208446 t 0.8 and v (t ) 1.0566757 t 0.2 , respectively. Results for fine meshes and small time steps. Obviously, the accuracy at the first time steps depends on the accuracy of the data obtained at the initialization stage. We could see that = 0.0001 and = 0.01 provide the accuracy of 0.026% for . To keep the accuracy on this level for x (t ) and v (t ) at first time steps, the step t should be notably less than t 0 . By taking t = 0.01 t 0 = 10-4, we had the accuracy of some 0.03% for both x (t ) and v (t ) at first hundred steps. With further growth of the time, the accuracy of v (t ) stayed on the same level, while the relative error of x (t ) decreased. In particular, at the step m = 1000 (t = 0.11), it is 0.0077%; at the step m = 20000 (t = 2.01) it is 0.0043%. The decrease of the relative error in x for large time is related to the growth of the absolute value of x (Recall that x (t ) t 0.8 .). The time expense on a conventional laptop for 20000 steps with 50 iterations at a step is near 15 s. Note that in view of the growth of x (t ) in time, there is no need to have the time step constant. By using exponentially growing time steps, the number of time steps may be drastically reduced without loss of the accuracy. For instance, with the same time expense we could reach time t = 36128 ( x = 5889.4) with no loss of the accuracy. The number of iterations could be reduced as well. There were no signs of instability in these and many other specially designed experiments. We 10 4 and could see that for 10 2 0.01, the results are accurate and stable in a wide range of 15 values of the time step and for very large number (up to 100000) of steps. Therefore - regularization (6.7) is quite efficient. Results for coarse mesh and large number of time steps. For the coarse mesh = 0.0(9) and = 0.1 at the initialization stage, we have one percent accuracy except for close vicinity of the liquid front. There is no need to take time steps notably less than the initial time. When taking t = 0.25 t0 = 0.0025, we had x with the error not exceeding 1.2% at any time instant for 20000 equal steps. Again, the relative error decreased with growing x . Using exponentially growing time step serves to obtain x (t ) and v (t ) with the accuracy of one percent even when x (t ) becomes of order 10000 (recall that the starting value is x (t0 ) 0.033170). As the number of nodes is small (only 11), the time expense is fractions of a second. Again, there were no signs of instability. Moreover, increasing the initial time step 3-10-fold, influenced the accuracy only of the first steps, while with growing time, the accuracy kept to the mentioned level of about 1%. Hence, use of a coarse mesh efficiently maintains this accuracy. 7. Extension of the regularization method to 2D fracture propagation According to the rationale presented in the preceding section, it appears that the strategy of using regularization when tracing 2-D hydrofracture propagation is as follows. At each point of the liquid front we introduce the local coordinate system moving with the front in the direction normal to the front. We transform field equations to this system. The prescribed BC at the front and the SE associated with front points are transformed accordingly. They are combined to formulate a new BC at a small distance behind the front. Below, we follow this path. We introduce the local Cartesian coordinates x1 ' O1 ' x2 ' with the origin O1 ' at a point x* at the liquid front and the axis x1 ' opposite to the direction of external normal to the front at this point. As the flux q* at the front is co-linear to the normal, its tangential component is zero. For the point x* , equation (2.11) written in the system x1 ' O1 ' x2 ' becomes a scalar equation qn* D ( w, p ) p x1' , x1 ' 0 and the SE (2.12) reads; 1 p D( w, p) w* x1' v where v obtain: , (7.1) x1 ' 0 dxn* / dt is the absolute value of the front velocity. Combining (7.1) with the BC (2.16) we p 1 D( w, p)dp v* x1 ' . p0 w Equation (7.2) serves to impose the BC at a small distance x1 ' p (7.2) from the front: 1 (7.3) D( w, p)dp v* , p0 w where p . The - regularization consists in using the BC p( ) is the pressure at the point (7.3) instead of the BC (2.16). In numerical calculations with - regularization, it is reasonable to use the aforementioned advantages of variables which express the particle velocity v q / w . In the local system x1 ' O1 ' x2 ' moving with the front, for points close to the front, the normal component of the velocity is: 1 p . Evaluating the partial time derivative under constant x1 ' rather than at constant v D(w, p) w x1 ' 16 global coordinate, the PDF (2.9) at points near the front takes the form (2.17). As noted in Sec. 2, it is reasonable to transform PDF (2.17) to (2.18) by introducing the variable y w1 / , where the exponent α accounts for asymptotic behavior of the opening at the front. As is small, the velocity v( ) is close to v* . This serves to employ the equation (7.4) v( ) v * 1 / 3 , p 4w , p0 4w( x* ) 0 , for iterations in v* at a time step. For the Nordgren problem, x* ; then equations (2.18), (7.3) and (7.4) are easily transformed to (6.2), (6.7) and (6.8), respectively. 8. Conclusions The paper presents the following conclusions. (i) The speed function for fluid flow in a thin channel is given by the ratio of the flux through a cross section to the channel width at the front. Its specification for hydraulic fracturing follows from the equation of Poiseuille type connecting the flux with gradient of pressure. The speed function, as the basis of the theory of propagating interfaces, facilitates employing such methods as level set methods and fast marching methods. The speed equation for hydraulic fracturing is a general condition at the liquid front not dependent on a particular BC defined by a physical situation ahead of the front, liquid properties, leak-off through the channel walls and/or presence of distributed sources. When there is no lag between the liquid front and the crack tip, both the flux and opening are zero at the liquid front; in this case, the SE is fulfilled in the limit when a point behind the front tends to the front. (ii) Using the SE gives a key to proper choice of unknown functions when solving a hydraulic fracture problem numerically. Specifically, the particle velocity, averaged over the opening, is a good choice, because it is non-singular at the front and non-zero in entire flow region. Using the velocity as an unknown may also serve to avoid the stiffness of the system of differential equations obtained in a conventional way. Another proper unknown is the opening to the degree 1/α, where α is non-negative exponent, characterizing the asymptotic behavior of the opening near the front. (iii) Existence of the SE also discloses the crucial feature of the problem: for zero or small lag, at the points of the front we actually have prescribed two rather than one BC; the BV problem appears ill-posed when solved numerically. It requires a proper regularization to have accurate and stable numerical results. (iv) Self-similar formulation of the Nordgren problem, reducing PDE to ODE, unambiguously demonstrates that the problem is ill-posed when considered as a BV problem. Numerical experiments confirm this theoretical conclusion: by no means could we obtain more than two correct digits when solving the problem without regularization, and the results always strongly deteriorate near the liquid front. (v) The solution of the self-similar problem, obtained when solving it as a Cauchy (initial value) problem by the Runge-Kutta method, provides benchmarks with at least five correct digits at entire liquid including its front. These numerical results may serve for testing methods of hydraulic fracture simulation. (vi) Studying of the Nordgren problem, accounting for the SE, suggests a means to overcoming the difficulty by - regularization. It consists in employing the SE together with a prescribed BC on the front to formulate a new BC at a small distance behind the front rather than on the front itself. Application of - regularization and comparing the results with the benchmarks shows that it is robust and provides highly accurate and stable results. (vii) Numerical experiments also disclose that using coarse spatial meshes is beneficial and serves as a specific regularization when high accuracy is not needed. Still, although 17 involving analytical work, - regularization is superior in the possibility to guarantee accurate and stable results and to evaluate the accuracy of calculations; it is competitive in time expense. (viii) The suggested - regularization may serve as a means to obtain accurate and stable results for simulation of hydrofracture with zero or small lag. When tracing 2-D hydrofracture propagation, - regularization is obtained by writing equations in the local coordinate system moving with the liquid front. Acknowledgment. The author appreciates the support of the EU Marie Curie IAPP program (Grant # 251475). References Adachi J.I. & Detournay E. 2002 Self-similar solution of plane-strain fracture driven by a power-law fluid. Int. J. Numer. Anal. Meth. Geomech., 26, 579-604. Adachi J., Siebrits E. et al. 2007 Computer simulation of hydraulic fractures. Int. J. Rock Mech. Mining Sci., 44, 739-757. Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press. Carter R.D. 1957 Derivation of the general equation for estimating the extent of the fractured area, Appendix to “Optimum fluid characteristics for fracture extension” by G.C. Howard and C.R. 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Interference Alignment with Power Splitting Relays in Multi-User Multi-Relay Networks Man Chu∗ , Biao He† , Xuewen Liao∗ , Zhenzhen Gao∗ , and Shihua Zhu∗ ∗ Department arXiv:1705.06791v2 [] 23 May 2017 † of Information and Communication Engineering, Xi’an Jiaotong University, China The Center for Pervasive Communications and Computing, University of California at Irvine, Irvine, CA 92697, USA Email: cmcc 1989414@stu.xjtu.edu.cn, biao.he@uci.edu, {yeplos, zhenzhen.gao, szhu}@mail.xjtu.edu.cn Abstract—In this paper, we study a multi-user multi-relay interference-channel network, where energy-constrained relays harvest energy from sources’ radio frequency (RF) signals and use the harvested energy to forward the information to destinations. We adopt the interference alignment (IA) technique to address the issue of interference, and propose a novel transmission scheme with the IA at sources and the power splitting (PS) at relays. A distributed and iterative algorithm to obtain the optimal PS ratios is further proposed, aiming at maximizing the sum rate of the network. The analysis is then validated by simulation results. Our results show that the proposed scheme with the optimal design significantly improves the performance of the network. I. I NTRODUCTION Energy harvesting has been envisioned as a promising technique to provide perpetual power supplies and prolong the lifetime of energy-constrained wireless networks, since wireline charging and battery replacement are not always feasible or unhazardous. In particular, simultaneous wireless information and power transfer (SWIPT) has drawn a significant amount of attention with its advantage of transporting both energy and information at the same time through radio frequency signals. The concept of SWIPT was first put forward by Varshney in his pioneer work [1], in which he investigated the tradeoff between information delivery and energy transfer. Recently developed practical receiver designs for SWIPT include power splitting (PS) and time switching (TS) [2]. For TS, the receiver switches over time between information decoding and energy harvesting. Differently, the receiver with PS splits the received signal power between information decoding and energy harvesting. It has been shown that the TS is theoretically a special case of the PS with binary PS ratios. Thus, the PS usually outperforms the TS in terms of the rate-energy tradeoffs. With the in-depth study on SWIPT, the research problem of how to combine SWIPT with other key technologies has gained much attention [3]. For the combination of the SWIPT and relays, both single-relay and multi-relay networks have been considered to acquire the efficient SWIPT scheme in [4][5]. In addition, the interference channel (IC) has been taken into account for the SWIPT with relays in, e.g., [6] and [7]. The authors in [6] analyzed the outage performance for SWIPT both with and without relay coordination in multi-user IC networks. In [7], the SWIPT with relays in IC networks was studied from a game-theoretic perspective. However, none of the aforementioned studies has directly addressed the issue of interference for SWIPT with relays in IC networks. In practice, interference alignment (IA) is recognized as an efficient technique to address the issue of interference for wireless transmissions. With the IA, the interferences are aligned into certain subspaces of the effective channel to the receiver, and the desired signals are aligned into interferencefree subspaces [8]. Without the consideration of relays, the authors in [9] studied the antenna selection with IA for the SWIPT in the IC network. While to the best of authors’ knowledge, few work has been investigated in the literature on SWIPT with the IA in multiple AF (amplify and forward) relay networks. In this paper, we study an IC network where multiple sources transmit messages to multiple users through multiple AF relays, where the energy-constrained relays harvest energy from the RF signals and use that harvested energy to forward the information to destinations. We propose a novel transmission scheme by adopting the IA and the PS design. We further propose a distributed and iterative algorithm to obtain the optimal PS ratios at relays that maximize the sum rate of the network. Our results show that the proposed scheme effectively improves the performance of the network, and it significantly outperforms benchmark schemes. II. S YSTEM M ODEL We consider a K × K × K dual-hop IC network consisting of K sources, K relays, and K destinations, as shown in Figure 1. Each source Si intends to transmit messages to its respective destination Di with the help of its dedicated AF relay Ri , i ∈ {1, · · · , K}. We assume that there are no direct links between the sources and the destinations. The numbers of antennas at each source, relay, and destination are denoted by M , N , and L, respectively. The sets of sources, relays and destinations are denoted by NS , NR , and ND , respectively. In addition, the set of all source-relay-destination (S-R-D) links, Si → Ri → Di , i = 1, · · · K, is denoted by K. We adopt the quasi-static block Rayleigh fading model and assume that all channels are independent of each other. The N × M normalized channel matrix from Si to Rj is denoted H11 S1 G11 R1 G12 H12 G1K H1K G21 H 21 H 22 S2 D1 G22 R2 H 2K D2 G2K GK 1 H K1 GK 2 HK2 SK H KK RK GKK DK Fig. 1. Illustration of multi-user multi-relay interference MIMO network. by Hij , and the L × N normalized channel matrix from Ri to Dj is denoted by Gij , i, j ∈ {1, · · · , K}. We assume that each node has the local channel state information (CSI) to enable the distributed IA. Moreover, the distance between Si and Rj is denoted by rij , and the distance between Ri and Dj is denoted by mij . The correspondingly path-loss effects are −τ modelled by rij and m−τ ij , respectively, where τ ≥ 2 denotes the path-loss exponent [10]. Half-duplex relays are considered in this work, and the two hop transmissions operate in two time slots. In the first slot, the sources simultaneously transmit the information and energy to the relays. In the second slot, the relays forward the received information to the destinations by the harvested power in the first slot. We assume that the wireless power transfer is the only energy access for the relays. Besides, every time slot is assumed to be unit interval for simplicity. III. I NTERFERENCE A LIGNMENT WITH P OWER -S PLITTING R ELAYS In this section, we proposed the transmission scheme for the dual-hop IC with the IA and the PS relays. A. IA Between Sources and Relays In the first time slot, the IA is employed for the transmissions from sources to relays. That is, all interference signals from unintended sources are aligned into certain interference subspaces at any relay Ri . As per the mechanism of IA, the following conditions should be satisfied to have the perfect IA: UH i Hji Vj = 0, ∀j 6= i, rank(UH i Hii Vi ) = di M + N ≥ (K + 1) × di , (1) (2) (3) where Vi ∈ CM ×di denotes the unitary precoding matrix at Si , Ui ∈ CN ×di denotes the unitary decoding matrix at Ri , and di denotes the number of data streams from Si . Here, the perfect IA means that the interferences are aligned into the null space of the effective channel to the relay and can be eliminated completely. We can interpret (1) as the condition of having the interference-free space of the desired dimensions, and (2) and (3) as the conditions that the desired signal is visible and resolvable within the interference-free space [11]. When the number of links is larger than 3, i.e., K > 3, the closed-form expressions for the precoding and decoding matrixes cannot be obtained. Hence, we use the channel reciprocity characteristic with the distributed and iterative IA to determine the IA matrices in this work. The received signal at Ri (without multiplexing the decoding matrix UH i ) in the first time slot is given by q −τ IA pi rii Hii Vi xi yR = i K q X −τ + pj rji Hji Vj xj + nAi , (4) j=1,j6=i where pi denotes the transmit power at Si , xi ∈ Cdi ×1 denotes the normalized desired signal for Di with E{||xi ||2 } = 1, and nAi denotes the additive white Gaussian noise (AWGN) at the receiver antennas of Ri . B. PS Relays In this work, the PS scheme is adopted at the relays for the SWIPT. The block diagram of the receiver at the relay is shown in Figure 2. For the received signal at Ri , a fraction 0 ≤ ρi ≤ 1 of the power is used for information processing and forwarding, and the remaining fraction 1−ρi of the power is used for energy harvesting. We refer to ρi as the PS ratio at Ri . We assume that the processing power required by the transmit/receive circuits at the relay is negligible compared with the power used for signal transmission [10]. Then, the signals received by the energy-harvesting unit at Ri is given by q p −τ EH yRi = (1 − ρi ) pi rii Hii Vi xi + K X q −τ pj rji Hji Vj xj + nAi  , (5) j=1,j6=i and the harvested energy at Ri is given by   K X 2 −τ QRi = η (1 − ρi )  pj rji ||Hji Vj ||  , (6) j=1 where η denotes the energy conversion efficiency. Note that the harvested energy due to the noise nAi is very small and thus ignored. At the information processing unit of Ri , the decoding matrix UH i is multiplexed to the received signal to eliminate (1 − ρi )Xi ρi pi Yii ρi pi b1ii +σ2 γDi (ρ) = PK j=1,j6=i (1 − ρj )Xj ρj pj Yji +cji σ 2 ρj pj bjj +σ 2 1  … x RF Unit and Antenna EH x IP IF . (12) j=1,j6=i nA =  σ2 η denotes the power normalization coefficient at Ri . The received signal at Di is given by q IP 2 QRi m−τ yDi = ii Gii βRi yRi K q X IP 2 + QRj m−τ ji Gji βRj yRj + nDi Power Splitting + 2 + (1 − ρi ) Xi cii ρi pi bσii +σ2 + U , nR q q −τ H QRi m−τ G β ρi pi rii Ui Hii Vi xi ii R i ii K q X q −τ H QRj m−τ ρj pj rjj Uj Hjj + ji Gji βRj j=1,j6=i Fig. 2. Power splitting model at the relay. Vj xj + K q X QRj m−τ ji Gji βRj nRj + nDi , (11) j=1 the interferences. The processed signals at the information processing unit of Ri is then given by IP 1 yR = i q −τ H ρi pi rii Ui Hii Vi xi + √ ρ i UH i nAi + nRi , (7) 2 where nRi ∼ CN (0, σR I ) denotes the AWGN introduced i N by the information processing unit at the relay. In practice, the power of antenna noise nAi is usually much smaller than the processing noise, and hence, the antenna noise has negligible impact on the signal processing. Then, the antenna noise nAi is ignored for simplicity in the rest of analysis in the paper [7], [12], and the processed signal at Ri is simplified as IP 2 yR = i q −τ H ρi pi rii Ui Hii Vi xi + nRi . (8) C. AF Signals to Destinations In the second time slot, the relays amplify and forward the processed signals to destinations by the harvested energy in the first time slot. Since the utilization of IA for the second hop would significantly increase the system overhead and the energy consumption at the energy-constrained relays in practice, the IA is not adopted for the second-hop transmission. The transmitted signal at Ri in the second time slot is then given by F yR = i p IP 2 QRi βRi yR , i (9) where QRi denotes the transmit power of Ri as given in (6) and βR i = q 1 2 −τ 2 ρi pi rii ||UH i Hii Vi || + σRi (10) 2 where nDi ∼ CN (0, σD I ) denotes the AWGN at Di . i N In order to simplify the expression, we define Xi = PK 2 2 −τ −τ −τ j=1 pj rji ||Hji Vj || , Yii = mii rii ||Gii H̃ii || , Yji = 2 2 2 −τ −τ −τ m−τ ji rii ||Gji H̃jj || , bii = rii ||H̃ii || , cji = mji ||Gji || , H and the effective channel matrix H̃ii = Ui Hii Vi . In addi2 2 tion, we assume that σD = σR = σ 2 for any i ∈ {1, · · · , K}, i i without loss of generality [7]. The signal to interference plus noise power ratio (SINR) at Di , denoted by γDi (ρ) is then given by (12) as shown at the top of this page, where T ρ = [ρ1 , ρ2 , · · · , ρK ] denotes the vector of all relays’ PS ratios. Finally, the rate of the transmission to Di is given by 21 B log2 (1 + γDi (ρ)), where B denotes the frequency bandwidth and 1/2 is due to the use of two time slots. IV. O PTIMIZATION OF P OWER S PLITTING R ATIO We note that the PS ratio ρ plays a pivotal in determining the network performance. Thus, we optimize the design of ρ in this section. A. Problem Formulation The objective of our design here is to determine a series of PS ratios, ρ1 , · · · , ρK , to maximize the total transmission rate of the network. The sum-rate maximization problem is formulated as: K X 1 (P1) max Rsum (ρ) = B log2 (1 + γDi (ρ)), ρ 2 i=1 s.t. ρ ∈ A, H Ui UH i = I, Vi Vi = I, E{||xi ||2 } = 1, ∀i ∈ {1, · · · , K}, (13) where A = {ρ | 0 ≤ ρi ≤ 1, ∀i ∈ {1, · · · , K}} is the feasible region of all PS ratios. We find that the optimization problem in (13) is not convex, and the closed-form solution of the globally optimal ρ cannot 

(z2 z3 + t) ρi 3 − z3 pi bii + z2 z3 σ 2 − 2t ρi 2 + 2z1 z3 − z2 z3 σ 2 + t + pi bii σ 2 ρ2 + z1 z3 ∂γDi (ρ¯i , ρi ) = 0. = i2 h ∂ρi −1 z2 + η (1 − ρi ) Xi cii σ 2 (ρi pi bii ) + σ 2 K X z1 = η (1 − ρj ) Xj pi bii + σ 2 j=1,j6=i K X z2 = η (1 − ρj ) Xj Yji K X σ2 ρj pj + . η (1 − ρ ) X c j j ji ρj pj bjj + σ 2 ρj pj bjj + σ 2 be obtained. Thus, we develop a distributed and iterative scheme, such that each Ri decides its own PS strategies aiming at maximizing the individual achievable transmission rate through the ith link, i.e., Si → Ri → Di , and the global optimal ρ that maximizes the sum rate of all links can be then obtained iteratively by the cooperation between the relays. The design problem is formulated as: K X i=1 max ρ 1 B log2 (1 + γDi (ρ¯i , ρi )), 2 s.t. ρ ∈ A, H Ui UH i = I, Vi Vi = I, E{||xi ||2 } = 1, ∀i ∈ {1, · · · , K}, (14) where ρ¯i = {ρj | 1 ≤ j ≤ K, ∀j 6= i} denotes the vector of all links’ PS ratios except for the ith link. B. Optimal Design We first determine the optimal ρi at Ri that maximizes the received SINR at Di for any given ρ¯i . The problem is formulated as: ρi s.t. ρ ∈ A, γ̃Di (ρ¯i , ρi ) η (1 − ρi ) Xi ρi pi Yii ρi pi b1ii +σ2 K P j=1,j6=i = I, Vi ViH (18) to find the optimal ρ. The main idea of our iterative algorithm is briefly summarized as follows. Assume that the result of nth iteration is ρ(n) = [ρ1 (n), ρ2 (n), · · · , ρK (n)]. The solution of (n + 1)th PS ratio ρi (n + 1) is then acquired by using Newton iteration method with the knowledge of {ρ1 (n + 1), ρ2 (n + 1), · · · , ρi−1 (n + 1), ρi+1 (n), · · · , ρK (n)}. We define a parameter ε to evaluate whether the algorithm is convergent or not. The iteration converges if and only if the condition of kρ(n + 1) − ρ(n)k2  ε is satisfied. In what follows, we determine the wise iterative initial values to reduce the number of the iterations to find the optimal ρ. To this end, we obtain the optimal PS ratios under the asymptotic scenario of high SNR, which in fact serve as the wise initial values of our iterative algorithm under general scenarios. We note that (12) can be simplified when the received SINRs at relays are high, i.e., ρi pi bii  σ 2 and ρj pj bjj  σ 2 . The second item and the third item of the denominator of γDi (ρ¯i , ρi ) approximate to zero under the high SINR consideration. Then, γDi (ρ¯i , ρi ) can be rewritten as = max γDi (ρ¯i , ρi ), Ui UH i (17) j=1,j6=i j=1,j6=i (P2)
ρj pj Yji + cji σ 2 − Xi cii σ 2 + pi bii σ 2 . ρj pj bji + σ 2 (16) = I, 2 E{||xi || } = 1, ∀i ∈ {1, · · · , K}. (15) We find that γDi |ρi =0 = 0 and γDi |ρi =1 = 0, and hence, γDi (ρ¯i , ρi ) is not a monotonous function of ρi in the range of [0, 1]. We will later show the uniqueness of the extreme value point of γDi (ρ¯i , ρi ) for 0 ≤ ρi ≤ 1, and verify that the extreme point is the optimal PS ratio maximizing the SINR. Let the first-order derivative of γDi (ρ¯i , ρi ) with respect to ρi be equal to zero, we have (16) shown at the top of this paper, where t = Xi cii σ 2 , z3 = Xi pi Yii , and z1 and z2 are given in (17) and (18), respectively. From (16), we find ∂γDi (ρ¯i ,ρi ) ∂γDi (ρ¯i ,ρi ) that |ρi =0 > 0 and |ρi =1 < 0. Thus, ∂ρi ∂ρi ∂γDi (ρ¯i ,ρi ) = 0. We ∂ρi ∂γDi (ρ¯i ,ρi ) ρi to = 0 ∂ρi there exists a ρi ∈ [0, 1] that satisfies note that the closed-form solution of cannot be obtained. Instead, we propose an iterative algorithm . (19) η (1 − ρj ) Xj ρj pj Yji ρi pi b1ii +σ2 + σ 2 The first-order derivative of γ̃Di (ρ¯i , ρi ) with respect to ρi is given by γ̃Di (ρ¯i , ρi ) ∂ρi

Xi Yii pi sji + σ 2 −ρi 2 pi bii − 2ρi σ 2 + σ 2 = , 2 [ρi pi bii (sji + σ 2 ) + σ 2 sji + σ 2 ] where sji = K P (20) η (1 − ρj )Xi ρj pj Yji ρj pi b1jj +σ2 . We find j=1,j6=i γ̃Di (ρ¯i ,ρi ) |ρi =0 > ∂ρi γ̃ (ρ¯ ,ρ ) that 0 and Di∂ρii i |ρi =1 < 0. Thus, there exists a ρi ∈ [0, 1] satisfying that γ̃Di (ρ¯i , ρi ) in (20) is equal to zero. In addition, we need to solve the following equation to acquire the initial PS ratios.

fˆ(ρi ) = Xi Yii pi sji + ρi 2 −ρi 2 pi bii − 2ρi ρi 2 + ρi 2 = 0. (21) −τ Since bii = rii ||H̃ii ||2 , it is always true that pi bii > 0. Note that the solution of (21) is uncorrelated with ρ¯i . Hence, each ρi can be calculated independently. We further find that (21) has two real roots, since ∆ = 4σ 2 (σ 2 + pi bi ) > 0. Moreover, according to the curve of the √ function as depicted in Figure 3, 2 − ∆ < 0 should be abandoned, and the solution of ρi = −2σ 2pi bii the optimal PS √ratio under the high SINR condition is given 2 + ∆ by ρoi = −2σ . 2pi bii TABLE I A LGORITHM TO SOLVE (P2) 1. Initialization: Set iteration number n = 0; Determine the initial PS ratios ρ(0) = {ρ1 (0), ρ2 (0) · · · ρK (0)} by (22). 2. Iteration: Obtain the nth iteration results ρ(n + 1) by solving (16) for each ρi with the Newton iteration method and {ρ1 (n + 1) · · · ρi−1 (n + 1), ρi+1 (n) · · · ρK (n)}. 3. Check if kρ(n + 1) − ρ(n)k2  ε is satisfied. If it is satisfied, go to Step 4. Otherwise, update the iteration fˆ ( ri ) number n = n + 1 and return to Step 2. 4. Obtain the optimal PS ratios as ρ∗ = ρ(n + 1). 0.8 0.75 0.7 r2 0 r1 1 ri 0.65 0.6 0.55 Fig. 3. The curve of fˆ(ρi ) versus ρi . 0.5 0.45 0.4 th Therefore, the initial PS ratio for the i link of our iterative algorithm is given by √ −2σ 2 + ∆ ρi (0) = ρoi = . (22) 2pi bii Different from the system of linear equations, it is difficult to provide a convergence condition for our proposed iterative algorithm. However, it can be found that ρi (n + 1) is the only root of equation γDi (ρ¯i , ρi ). Thus, the result of every iteration increases the objective function, i.e., γDi (ρi (n + 1)) > γDi (ρi (n)). Considering the monotonic increase of the objective function and the fact that the maximum achievable γDi is finite, the proposed algorithm converges to a certain point, at which the iteration stops. The detailed steps of the proposed algorithm are given in Table I. V. N UMERICAL R ESULTS In this section, we show the performance of the proposed scheme by simulations. Unless otherwise stated, we set the distance between each source and its dedicated relay rii = 1, the distance between each relay and its respective destination mii = 1, the path-loss exponent τ = 3, the energy conversion efficiency η = 0.5, the variance of AWGN at receivers σ 2 = 0.01, and the convergency criteria ε = 10−3 . We randomly generate 10,000 channel realizations to obtain the average sum rate of the network. 0.35 1 2 3 4 5 6 7 8 9 10 Fig. 4. Illustration of the convergence of the proposed algorithm. The system parameters are M = N = L = 4 and K = 3. We first demonstrate the convergence speed of the proposed iterative algorithm. Figure 4 plots the obtained PS ratio versus the iteration number for different initial values. As shown in the figure, the algorithm always converges to the same value of the PS ratio, i.e., the optimal PS ratio, for any given initial values. Comparing the results for different initial points, we find that our proposed initialized PS scheme outperforms all other schemes, in terms of the use of iterations to obtain the optimal PS ratio. We now present the performance comparison between our proposed scheme and benchmark schemes. We consider three benchmark schemes, which are the transmission with randomly select PS ratios, the transmission with fixed PS ratios of ρi = 0.5, and the transmission scheme without the IA as proposed in [7]. Figures 5 and 6 plot the average sum rate versus the transmit power at the source and the number of links, respectively. As shown in both figures, our proposed scheme always significantly outperforms all benchmark schemes. We note that the benchmark scheme without the IA always has 0.45 0.55 0.5 0.4 0.45 0.35 0.4 0.3 0.35 0.25 0.3 0.2 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0 5 10 15 20 25 30 35 40 0.05 3 Fig. 5. Sum rate of the network versus transmit power at the source. The system parameters are M = N = L = K = 4. the worst performance, which implies that the introduction of the IA improves the network performance even if the PS ratios are not optimally designed. The great advantage of our proposed scheme is due to not only the introduction of the IA to eliminate the interference to relays but also the welldesigned PS ratios that balance the information processing and energy harvesting to forward the information. We next discuss the impact of the transmit power at the source on the performance of the network. As depicted in Figure 5, the average sum rate increases as the transmit power increases. From all curves, we note that the speed of the increase of sum rate decreases as the transmit power increases, and it becomes very slow when the transmit power is high. Such an observation can be explained as follows. In this work, we have not employed the IA for the second hop due to the practical consideration of the system overhead and the high energy consumption at relays. Then, the increase of transmit power at relays may harm the network performance due to the interference. Thus, the relays actually cannot always increase the transmit power for the second hop, even if the harvested energy increases. Furthermore, we find that the performance gap between our proposed scheme and the other benchmark schemes increases as the transmit power at the source increases. This is because that the proposed scheme wisely takes advantage of the increase of the available energy resource at sources by the IA and the PS design, while the other schemes cannot take fully advantage of the increase of resource due to lack of either interference management or the PS design. Finally, we evaluate the impact of the number of links on the performance of the network. As illustrated in Figure 6, as the number of links increases, the sum rate of the network increases first until achieving a peak value, and then decreases. 3.5 4 4.5 5 5.5 6 6.5 7 Fig. 6. Sum rate of the network versus number of links. The system parameters are M = N = L = 4. We explain this observation as follows. When the number of links is small, the increase of the number of links can significantly increase the multiplexing gain of the network with the relatively small interference. When the number of links becomes large, further increasing the number of links would make the interference issue become serious, as the IA has not been employed at the relays for the second-hop transmission in the proposed scheme. From the discussions of Figures 5 and 6, we note that adopting the IA at the second hop has a considerable potential to further improve the performance of the network. However, as mentioned previously, the employment of the IA at relays would considerably increase the system overhead and the energy consumption of the energyconstrained relays. Thus, it is not always feasible to employ the IA at the wireless-powered relays in practice. VI. C ONCLUSION In this paper, we have introduced a novel transmission scheme for multi-user multi-relay IC networks, where the IA is employed at the sources and the PS is designed at the AF relays. We have proposed a distributed and iterative algorithm to determine the optimal PS ratios that maximize the sum rate of the network. In order to reduce the number of iterations to obtain the optimal PS ratios, we have further derived the closed-form expressions for the optimal PS ratios under the high SNR condition, which serve as the initial values of the iteration for the proposed algorithm. Our results have shown that the proposed scheme significantly outperforms the benchmark schemes to improve the network performance. ACKNOWLEDGMENT This work is supported by National Natural Science Foundation of China (NSFC) under grant 61461136001. R EFERENCES [1] L. R. Varshney, “Transporting information and energy simultaneously,” in Proc. IEEE ISIT, pp. 1612–1616, Jul. 2008. [2] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989–2001, Nov. 2013. [3] D. W. K. Ng, E. Lo, and R. Schober, “Wireless information and power transfer: Energy efficiency optimization in OFDMA systems,” IEEE Trans. Wireless Commun., vol. 12, no. 12, pp. 6352–6370, Dec. 2014. [4] Z. Chen, B. Xia, and H. Liu, “Wireless information and power transfer in two-way amplify-and-forward relaying channels,” in Proc. IEEE GLOBECOM, pp. 168–172, Dec. 2015. [5] L. Hu, C. Zhang, and Z. Ding, “Dynamic Power Splitting Policies for AF Relay Networks with Wireless Energy Harvesting,” in Proc. IEEE ICCW, pp. 2035–2039, Jun. 2015. [6] I. Krikidis, “Simultaneous information and energy transfer in large-scale networks with/without relaying,” IEEE Trans. Commun., vol. 62, no. 3, pp. 900–912, Mar. 2014. [7] H. Chen, Y. Jiang, Y. Li, Y. Ma, and B. Vucetic, “A game-theoretical model for wireless information and power transfer in relay interference channels,” in Proc. IEEE ISIT, pp. 1161–1165, Jun. 2014. [8] H. Zeng, F. Tian, Y. T. Hou, W. Lou, and S. F. Midkiff, “Interference alignment for multihop wireless networks: challenges and research directions,” IEEE Network, vol. 30, no. 2, pp. 74–80, Mar.–Apr. 2016. [9] X. Li, Y. Sun, F. R. Yu, and N. Zhao, “Antenna selection and power splitting for simultaneous wireless information and power transfer in interference alignment networks,” in Proc. IEEE GLOBECOM, pp. 2667–2672, Dec. 2014. [10] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and information processing,” IEEE Trans. Wireless Commun., vol. 12, no. 7, pp. 3622–3636, Jul. 2013. [11] K. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching the capacity of wireless networks through distributed interference alignment,” in Proc. IEEE GLOBECOM, pp. 1–6, Dec. 2008. [12] L. Liu, R. Zhang, and K. C. Chua, “Wireless information and power transfer: A dynamic power splitting approach,” IEEE Trans. Commun., vol. 61, no. 9, pp. 3990–4001, Sep. 2013. 

arXiv:1702.03152v1 [cs.CC] 10 Feb 2017 A Variation of Levin Search for All Well-Defined Problems Fouad B. Chedid A’Sharqiyah University, Ibra, Oman f.chedid@asu.edu.om February 13, 2017 Abstract In 1973, L.A. Levin published an algorithm that solves any inversion problem π as quickly as the fastest algorithm p∗ computing a solution for ∗ π in time bounded by 2l(p ) .t∗ , where l(p∗ ) is the length of the binary ∗ ∗ encoding of p , and t is the runtime of p∗ plus the time to verify its correctness. In 2002, M. Hutter published an algorithm that solves any well-defined problem π as quickly as the fastest algorithm p∗ computing a solution for π in time bounded by 5.tp (x) + dp .timetp (x) + cp , where dp = 40.2l(p)+l(tp ) and cp = 40.2l(f )+1 .O(l(f )2 ), where l(f ) is the length of the binary encoding of a proof f that produces a pair (p, tp ), where tp (x) is a provable time bound on the runtime of the fastest program p provably equivalent to p∗ . In this paper, we rewrite Levin Search using the ideas of Hutter so that we have a new simple algorithm that solves any well-defined problem π as quickly as the fastest algorithm p∗ computing a solution for π in time bounded by O(l(f )2 ).tp (x). keywords: Computational Complexity; Algorithmic Information Theory; Levin Search. 1 Introduction We recall that the class N P is the set of all decision problems that can be solved efficiently on a nondeterministic Turing Machine. Alternatively, N P is the set of all decision problems whose guessed solutions can be verified efficienlty on a deterministic Turing Machine, or equivalently today’s computers. The interesting part is that the set of all hardest problems in the class N P (also known as N P -complete problems) have so far defied any efficient solutions on real computers. This has been a very frustrating challenge given that many of the N P -complete problems are about real-world applications that we really would like to be able to solve efficiently. 1 Leonid Levin [6], and independently from Stephen Cook [1], defined the class N P , identified the class N P -complete, and gave a few reductions that show the N P -completeness of some N P problems (independent from the work of Richard Karp [3]). In that same paper [6], Levin gave an algorithm to deal with N P complete problems provided that someone develops a proof that shows P = N P . In this case, Levin’s algorithm can be used to develop a polynomial-time solution for every N P -complete problem. In particular, Levin’s algorithm works with a very broad class of mathematical problems that can be put in the form of inverting easily computable functions. For example, suppose we have a function y = f (x), where we know nothing about f except that it is easily computable. The challenge is to find x = f −1 (y) and to do so in a minimal amount of time. The idea of Levin’s algorithm is to go through the space of all algorithms in search for a fastest algorithm A that knows the secret of f . In general, let π be an inversion problem and let p∗ be a known fastest algorithm for π that runs in time t∗ , where t∗ is the runtime of p∗ plus the time to verify its correctness. The Universal Search algorithm of Levin, also known as Levin Search, is an effective ∗ procedure for findng p∗ in time bounded by 2l(p ) .t∗ , where l(p∗ ) is the length of the binary encoding of p∗ . Levin Search can also be used with some forms of optimization and prediction problems, and it is theoretically optimal. The following is a pseudocode for Levin Search: Pseudocode Levin Search t = 2. for all programs p, in parallel, do run p for at most t.2−l(p) steps if p is proved to generate a correct solution for π, then return p for p∗ and halt. endfor t = 2.t goto for all programs p End of Pseudocode The idea of Levin Search is to search the space of all programs p in increasing order of l(p) + log t, or equivalently in decreasing order of P rob.(p) , where t P rob.(p) = 2−l(p) is the probability contributed to p in the overall Solomonoff’s algorithmic probability [8] of finding a solution for π. By construction, P rob.(p) gives short programs a better chance of being successful on the assumption that short programs are more worthy (Occam’s razor). We mention that the main idea of Levin Search was behind the notion of Levin Complexity, which defines a computable version of Kolmogorov complexity based on time-bounded Turing machines. Let U be a fixed reference universal Turing machine, the Levin complexity of a finite binary string x is defined as Kt (x) = min .p {l(p) + log t : U (p) = x, in at most t steps.} Returning to Levin Search, during each iteration of the “for” loop, each program 2 is assigned a fraction of time proportional to its probability, which basically says that short programs get to run more often. The rest of this paper is organized as follows. Section 2 describes a sequential implementation of Levin Search and shows that it solves any inversion problem π as quickly as the fastest algorithm p∗ computing a solution for π, save for a ∗ factor of 2l(p ) . Section 3 describes an improvement on Levin Search for all welldefined problems. A problem is designated as well-defined if its solutions can be formally proved correct and have provably quickly computable time bounds. This improvement, known as Hutter’s algorithm [2], describes an algorithm that solves any well-defined problem π as quickly as the fastest algorithm computing a solution for π, save for a factor of 5. Section 4 contains our contribution. We rewrite Levin Search using the ideas of Hutter so that we have a new simple algorithm that solves any well-defined problem π as quickly as the fastest algorithm p∗ computing a solution for π, save for a factor of O(l(f )2 ), where l(f ) is the length of the binary encoding of a proof f that produces a pair (p, tp ), where tp is a provable time bound on the runtime of the fastest program p provably equivalent to p∗ . Section 5 contains some concluding remarks. 2 Sequential Levin Search The pseudocode of Levin Search from the previous section cannot be readily executed on today’s computers. A sequential version of Levin Search can simulate the parallel execution of all programs by running programs, each for a fraction of time, one after another, in increasing order of their length. Hereafter, we assume that all progams are encoded using prefix-free codes, using for example, Shannon-Fano coding [9]. Let π be an inversion problem and let p∗ be a known fastest algorithm for π that runs in time t∗ , where t∗ is the runtime of p∗ plus the time to verify its correctness. Also, let l(p) be the length of the binary encoding of program p. Algorithm Levin Search t = 2; for all programs p of length l(p) = 1, 2, . . . , log t do run p for at most t.2−l(p) steps. if p is proved to generate a correct solution for p, return p for p∗ and halt. endfor t = 2.t goto for all programs p End of Levin Search During each time step t, the algorithm runs all programs p of length ≤ log t in increasing order of their length and gives each program p a fraction of time proportional to 2−l(p) . Time Analysis. The following proof for the runtime complexity of Levin Search is due to Solomonoff [7]. The “for” loop in the algorithm takes time 3 P t.2−l(p) ≤ t. This is true by Kraft inequality [5]. Now, suppose that p:l(p)≤log t Levin Search stops at time t = T . Then, the total runtime of this algorithm is timeLevinSearch ≤ 2 + 4 + . . . + T T + + T = 2T − 2 < 2T. 4 2 Moreover, the runtime t∗ of p∗ is ∗ t∗ = T.2−l(p ) > timeLevinSearch −l(p∗ ). .2 2 Thus, ∗ timeLevinSearch < 2l(p )+1 ∗ .t = O(t∗ ) ∗ This is true because the multiplicative constant 2l(p )+1 is independent of the instance of the problem π. We mention that it is this huge multiplicative constant that limits the applicability of Levin Search in practice. It seems that a possible way to improve the runtime of Levin Search is to find a way, where not all programs of length ≤ log t get executed, for each value of t. Ideally, we would like a way that executes at most a single program for each value of t, preferably a shortest and fastest one, and this is exactly what we will present in Section 4. 3 Hutter’s Fastest and Shortest Algorithm for All Well-Defined Problems In [2], Hutter presented an improvement on Levin Search for problems that are well-defined. By this, Hutter means a problem whose solutions can be formally proved correct and have provably quickly computable time bounds. Let p∗ (x) be a known fastest algorithm for some well-defined problem π(x). Then, Hutter’s algorithm will construct a solution p(x) that is provably equivalent to p∗ (x), for all x, in time proportional to 5.tp (x) + d.timetp (x) + c, where tp (x) is a provable time bound on the runtime of p(x), timetp (x) is the time needed to compute tp (x), and d and c are constants which depend on p but not on π(x). This shows that Hutter’s agorithm runs in time O(tp ), save for a factor of 5. Our work in the next section is inspired by the ideas of Hutter, and so, for completeness, we have included below the details of Hutter’s algorithm as they appear in [2]. The main idea of Hutter’s algorithm is to search the space of all proofs1 in some formal axiomatic system, and not the space of all programs. In particular, the algorithm searches for those proofs that can tell us which programs are provably equivalent to p∗ and have provably quickly computable time bounds. This is doable since the set of all proofs in any formal sysem is enumerable. 1 Given a formal logic system F with a set of axioms and inference rules. A proof in F is a sequence of formulas, where each formula is either an axiom in F or is something that can be inferred using F ’s axioms and inference rules, and previoulsy inferred formulas. 4 Moreover, in [2], Hutter showed how to formalize the notions of provability, Turing Machines, and computation time. Let U be a fixed reference Universal Turing machine. For finite binary strings p and tp , we assume that • U (p, x) computes the output of program p on input x. • U (tp , x) returns a time bound on the runtime of program p on input x. Moreover, Hutter defined the following two terms: • A term u is defined such that the formula [∀y : u(p, y) = u(p∗ , y)] is true if and only if U (p, x) = U (p∗ , x), ∀x. • A term tm is defined such that the formula [tm(p, x) = n] is true if and only if U (p, x) takes n steps; that is, if timep (x) = n. Then, we say that programs p and p∗ are provably equivalent if the formula [∀y : u(p, y) = u(p∗ , y)] can be proved. The algorithm of Hutter is as follows. Algorithm Hutter(x) Let L = φ, tf ast = ∞, and pf ast = p∗ . Run A,B, and C concurrently with 10%, 10%, and 80% of computational resources, respectively. Algorithm A { This algorithm identifies programs p that are provably equivalent to p∗ and have provably computable time bounds tp } for i = 1, 2, 3, . . . do pick the ith proof in the list of all proofs and if the last formula in the proof is equal to [∀y, u(p∗ , x) = u(p, y) and u(tp , y) ≥ tm(p, y)], for some pair of strings (p, tp ) then Let L = L ∪ {(p, tp )}. End A Algorithm B { This algorithm finds the program with the shortest time bound in L.} for all (p, tp ) in L do run U on all (tp , x) in parallel for all tp with relative computational resources 2−l(p)−l(tp ) . if U halts for some tp and U (tp , x) < tf ast then tf ast = U (tp , x) and pf ast = p. endfor all (p, tp ) End B Algorithm C For k = 1, 2, 4, 8, . . . do pick the currenlty fastest program pf ast with time bound tf ast . run U on (p, x) for k steps. 5 if U halts in less than k steps, then print result U (p, x) and abort computation of A, B, and C. endfor k End C In [2], it was shown that the overall runtime of Hutter’s algorithm is timeHutter (x) ≤ 5.tpf ast (x) + dp .timetpf ast (x) + cp , where dp = 40.2l(pf ast )+l(tpf ast ) and cp = 40.2l(proof (pf ast ))+1 .O(l(proof (pf ast )2 ). 4 A Variation of Levin Search Inspired by the Fastest and Shortest Algorithm of Hutter We rewrite Levin Search so that it searches, in parallel, the space of all proofs, and not the space of all programs. Let p∗ be a known fastest algorithm for some well-defined problem π. Then, our algorithm will construct a solution p(x), for all x, that is provably equivalent to p∗ (x) in time O(tp (x)), save for a factor of O(l(proof (p))2 ), where tp (x) is a provable time bound on the runtime of the fastest program p provably equivalent to p∗ . We assume a formal axiomatic system F and the same terms u and tm defined in Hutter’s algorithm. Moreover, we assume all pairs (p, tp ) are encoded using P that prefix-free codes, and hence, we have 2−l(p)−l(tp ) ≤ 1, by Kraft inequality. (p,tp ) The following is our Modified Levin Search algorithm: Algorithm Modified Levin Search(x) t = 2; tf ast = ∞; pf ast = λ; { pf ast is initialized to the empty string } for all proofs f ∈ F of length l(f ) = 1, 2, . . . , log t do write down the first t.2−l(f ) characters of proof f . if the last formula in these characters is equal to ∀y, u(p∗ , x) = u(p, y) and u(tp , y) ≥ tm(p, y), for some pair of strings (p, tp ) then run tp (x) for at most t.2−l(p)−l(tp ) steps. if tp halts and tp (x) < tf ast then tf ast = tp (x) and pf ast = p. endfor if pf ast 6= λ then run pf ast for at most t steps with time bound tf ast . if pf ast (x) halts, then return pf ast for p∗ and halt. endif t = 2t. go to for all proofs f ∈ F End of Modified Levin Search 6 We note that during each time step t, Modified Levin Search runs at most a single provably fast program pf ast that is provably equivalent to p∗ . Running time calculation: Following [2], let naxioms be the finite number of axioms in F. Then, each proof f ∈ F is a sequence F1 F2 . . . Fn , for some positive integer n, where each Fi (1 ≤ i ≤ n) is either an axiom or a formula in F. It takes O(naxioms .l(Fi )) time to check if Fi is an axiom, and O(l(f )) time to check if Fi is a formula that can be inferred from other formulas Fj ∈ f , j < i. Thus, checking the validity of all formulas in f , and hence the validity of the proof f , takes O(l(f )2 ) time. The “for” loop in Modified Levin Search takes time X O(t.2−l(f ) ) + O((t.2−l(f ) )2 ) + t.2−l(p)−l(tp ) t1 = f :l(f )≤log t X ≤t+ O(l(f )) + O(l(f )2 ) f :l(f )≤log t This is true because P 2−l(p)−l(tp ) ≤ 1, by Kraft inequality. we next have (p,tp ) t1 ≤ t + X O(l(f )2 ) f :l(f )≤log t < t + 2log t+1 .O(l(f ′ )2 ) < t + 2t.O(l(f ′ )2 ) < O(l(f ′ )2 ).t, where f ′ is the proof that procuded the pair (p, tp ) with the shortest provable time bound tp (x), among all proofs of length ≤ log t. Thus, the runtime of Modified Levin Search is equal to X timeModif iedLevinSearch = O(l(f ′ )2 ).t + t t=2 = X < X O(l(f ′ )2 ).t t=2 O(l(f ∗ )2 ).t, t=2 where l(f ∗ ) is the length of the binary encoding of the proof that produced the resultant pair (pf ast , tpf ast ). 7 Suppose that Modified Levin Search stops at time t = T ≤ tpf ast (x).Then timeModif iedLevinSearch (x) = O(l(f ∗ )2 )}.(2 + 4 + . . . + T /2 + T ) = O(l(f ∗ )2 ).(2T − 2) < O(l(f ∗ )2 ).2T = O(l(f ∗ )2 ).T ≤ O(l(f ∗ )2 ).tpf ast (x). This shows that Modified Levin Search runs in time O(tpf ast ), save for a factor of O(l(f ∗ )2 ). 5 Concluding Remarks We recall that Hutter’s algorithm was designed for well-defined problems. These are problems the solutions of which can be formally proved correct and have provably quickly computable time bounds. For programs provably correct (they halt and compute the correct answer), but for which no quickly computable time bounds exist (For example, the traveling salesman problem), Hutter [2] explained that an obvious time bound for a progam p is its actual running time timep (.). By replacing tp (.) with timep (.) in the runtime of Hutter’s algorithm, and by noticing that, in this case, timetp (x) becomes timetimep (x) ≤ timep (x), we have timeHutter (x) ≤ 5.tp (x) + dp .timetp (x) + cp ≤ 5.timep (x) + dp .timep (x) + cp ≤ (5 + dp ).timep (x) + cp ≤ (5 + 40.2l(p) ).timep (x) + 40.2l(f )+1 .O(l(f )2 ) where l(p) is the length of the binary encoding of p, and l(f ) is the length of the binary encoding of a proof f that produces the program p that is provably equivalent to the fastest known algorithm p∗ for the problem in hand. Thus, for such programs, Hutter’s algorithm is optimal, save for a huge constant multiplicative term and a huge constant additive term. We next calculate the runtime of our algorithm in case we are dealing with programs that can be provably correct, but for which no quickly computable time bounds exist. Suppose that our algorithm stops at time t = T . Then, we have timeModif iedLevinSearch < O(l(f )2 ).T. We also have timep (x) = T.2−l(p) . Thus, timeModif iedLevinSearch < O(l(f )2 ).2l(p) .timep (x) 8 Thus, for such programs, the runtime of our algorithm also suffers from a huge constant multiplicative term, but with no additinoal huge additive terms. We conclude this paper by recalling an approach, proposed by Solomonoff in 1985 [7], to manage the huge multiplicative constant in the big-oh notation of the running time of Levin Search. We do so because we think that the importance of that work of Solomonoff is not widely appreciated. Solomonoff argued that if machines are going to have a problem solving capability similar to that of humans, then machines cannot start from scratch everytime they attempt to solve a new problem. We, humans, rely on our previous knolwedge of solutions to other problems to figure out a solution for a new problem. The basic idea of Solomonoff is that we should be able to construct p∗ via a list of references to previously discovered solutions for other related problems. We can imagine writing the code for p∗ as p1 p2 . . . pn , where pi is a reference to a solution for problem πi stored on a separate work tape of a Kolmogorov-Uspensky Machine [4]. This way, the solution p∗ is a sequence of calls to other solutions stored on the work tapes. This way, the length of p∗ would be made significantly smaller than the sum of the lengths of pi , and the saving in the length of p∗ would exponentially decrease the multiplicate constant in the big-oh notation of Levin Search. References [1] S. A. Cook, The complexity of theorem-proving procedures, Proceedings of the Third Annual ACM Symposium on Theory of Computing. (1971) 151–158. [2] M. Hutter, The fastest and shortest algorithm for all well-defined problems, International Journal of Foundations of Computer Science. 13:3 (2002) 431–443. [3] R. M. Karp, Reducibility among combinatorial problems, In R.E. Miller and J.W. Thatcher (editors). Complexity of Computer Computations. (1972) 85–103. [4] A.N. Kolmogorov and V.A. Uspenski, On the definition of an algorithm, Uspehi Mat. Nauk. 13 (1958), 3–28. AMS Transl. 2nd ser.29 (1963) 217– 245. [5] L.G. Kraft, A device for quantizing, grouping, and coding amplitude modulated pulses, MS Thesis, Electrical and Engineering Department, MIT. [6] L. A. Levin, Universal sequential search problems, Problemy Peredaci Informacii. 9 (1973) 115–116. Translated in Problems of Information Transmission. 9 (1973) 265–266. [7] R. J. Solomonoff, Optimum sequential searh. (1985) 1–13. 9 [8] R. J. Solomonoff, A formal theory of inductive inference: Part 1 and 2, Inform. Control. 7 (1964) 1–22, 224–254. [9] C.E. Shannon, A mathematical theory of communication, Bell System Technical Journal. 27 (1948) 379–423. 10 

arXiv:1510.00872v2 [math.NT] 9 Dec 2016 Compactifications of S-arithmetic quotients for the projective general linear group Takako Fukaya, Kazuya Kato, Romyar Sharifi December 12, 2016 Dedicated to Professor John Coates on the occasion of his 70th birthday Abstract Let F be a global field, let S be a nonempty finite set of places of F which contains Q the archimedean places of F , let d ≥ 1, and let X = v ∈S X v where X v is the symmetric space (resp., Bruhat-Tits building) associated to PGLd (Fv ) if v is archimedean (resp., nonarchimedean). In this paper, we construct compactifications Γ\X̄ of the quotient spaces Γ\X for S-arithmetic subgroups Γ of PGLd (F ). The constructions make delicate use of the maximal Satake compactification of X v (resp., the polyhedral compactification of X v of Gérardin and Landvogt) for v archimedean (resp., non-archimedean). We also consider a variant of X̄ in which we use the standard Satake compactification of X v (resp., the compactification of X v due to Werner). 1 Introduction 1.1. Let d ≥ 1, and let X = PGLd (R)/ POd (R) ∼ = SLd (R)/ SOd (R). The Borel-Serre space (resp., reductive Borel-Serre space) X̄ contains X as a dense open subspace [3] (resp., [26]). If Γ is a subgroup of PGLd (Z) of finite index, this gives rise to a compactification Γ\X̄ of Γ\X . 1.2. Let F be a global field, which is to say either a number field or a function field in one variable over a finite field. For a place v of F , let Fv be the local field of F at v . Fix d ≥ 1. In this paper, we will consider the space X v of all homothety classes of norms on Fvd and a certain space X̄ F,v which contains X v as a dense open subset. For F = Q and v the real place, X v is identified with PGLd (R)/ POd (R), and X̄ F,v is identified with the reductive Borel-Serre space associated to PGLd (Fv ). We have the following analogue of 1.1. 1 Theorem 1.3. Let F be a function field in one variable over a finite field, let v be a place of F , and let O be the subring of F consisting of all elements which are integral outside v . Then for any subgroup Γ of PGLd (O) of finite index, the quotient Γ\X̄ F,v is a compact Hausdorff space which contains Γ\X v as a dense open subset. 1.4. Our space X̄ F,v is not a very new object. In the case that v is non-archimedean, X v is identified as a topological space with the Bruhat-Tits building of PGLd (Fv ). In this case, X̄ F,v is similar to the polyhedral compactification of X v of Gérardin [7] and Landvogt [19], which we denote by X̄ v . To each element of X̄ v is associated a parabolic subgroup of PGLd ,Fv . We define X̄ F,v as the subset of X̄ v consisting of all elements for which the associated parabolic subgroup is F -rational. We endow X̄ F,v with a topology which is different from its topology as a subspace of X̄ v . In the case d = 2, the boundary X̄ v \ X v of X̄ v is P1 (Fv ), whereas the boundary X̄ F,v \ X v of X̄ F,v is P1 (F ). Unlike X̄ v , the space X̄ F,v is not compact, but the arithmetic quotient as in 1.1 and 1.3 is compact (see 1.6). 1.5. In §4, we derive the following generalization of 1.1 and 1.3. Let F be a global field. For a nonempty finite set S of places of F , let X̄ F,S be the subspace Q of v ∈S X̄ F,v consisting of all elements (x v )v ∈S such that the F -parabolic subgroup associated Q to x v is independent of v . Let X S denote the subspace v ∈S X v of X̄ F,S . Let S 1 be a nonempty finite set of places of F containing all archimedean places of F , let S 2 be a finite set of places of F which is disjoint from S 1 , and let S = S 1 ∪ S 2 . Let OS be the subring of F consisting of all elements which are integral outside S. Our main result is the following theorem (see Theorem 4.1.4). Theorem 1.6. Let Γ be a subgroup of PGLd (OS ) of finite index. Then the quotient Γ\(X̄ F,S 1 × X S 2 ) is a compact Hausdorff space which contains Γ\X S as a dense open subset. 1.7. If F is a number field and S 1 coincides with the set of archimedean places of F , then the space X̄ F,S 1 is the maximal Satake space of the Weil restriction of PGLd ,F from F to Q. In this case, the theorem is known for S = S 1 through the work of Satake [23] and in general through the work of Ji, Murty, Saper, and Scherk [14, 4.4]. ♭ ♭ 1.8. We also consider a variant X̄ F,v of X̄ F,v and a variant X̄ F,S of X̄ F,S with continuous surjections ♭ X̄ F,v → X̄ F,v , ♭ X̄ F,S → X̄ F,S . ♭ In the case v is non-archimedean (resp., archimedean), X̄ F,v is the part with “F -rational bound- ary” in Werner’s compactification (resp., the standard Satake compactification) X̄ v♭ of X v [24, 2 25] (resp., [22]), endowed with a new topology. We will obtain an analogue of 1.6 for this variant. To grasp the relationship with the Borel-Serre compactification [3], we also consider a vari♯ ♯ ant X̄ F,v of X̄ F,v which has a continuous surjection X̄ F,v → X̄ F,v , and we show that in the case ♯ that F = Q and v is the real place, X̄ Q,v coincides with the Borel-Serre space associated to ♯ PGLd ,Q (3.7.4). If v is non-archimedean, the space X̄ F,v is not Hausdorff (3.7.6) and does not seem useful. 1.9. What we do in this paper is closely related to what Satake did in [22] and [23]. In [22], he defined a compactification of a symmetric Riemannian space. In [23], he took the part of this compactification with “rational boundary” and endowed it with the Satake topology. Then he showed that the quotient of this part by an arithmetic group is compact. We take the part X̄ F,v of X̄ v with “F -rational boundary” to have a compact quotient by an arithmetic group. So, the main results and their proofs in this paper might be evident to the experts in the theory of Bruhat-Tits buildings, but we have not found them in the literature. 1.10. We intend to apply the compactification 1.3 to the construction of toroidal compactifications of the moduli space of Drinfeld modules of rank d in a forthcoming paper. In Section 4.7, we give a short explanation of this plan, along with two other potential applications, to asymptotic behavior of heights of motives and to modular symbols over function fields. 1.11. We plan to generalize the results of this paper from PGLd to general reductive groups in another forthcoming paper. The reason why we separate the PGLd -case from the general case is as follows. For PGLd , we can describe the space X̄ F,v via norms on finite-dimensional vector spaces over Fv (this method is not used for general reductive groups), and these norms play an important role in the analytic theory of toroidal compactifications. 1.12. In §2, we review the compactifications of Bruhat-Tits buildings in the non-archimedean setting and symmetric spaces in the archimedean setting. In §3 and §4, we discuss our compactifications. 1.13. We plan to apply the results of this paper to the study of Iwasawa theory over a function field F . We dedicate this paper to John Coates, who has played a leading role in the development of Iwasawa theory. Acknowledgments. The work of the first two authors was supported in part by the National Science Foundation under Grant No. 1001729. The work of the third author was partially supported by the National Science Foundation under Grant Nos. 1401122/1661568 and 1360583, and by a grant from the Simons Foundation (304824 to R.S.). 3 2 Spaces associated to local fields In this section, we briefly review the compactification of the symmetric space (resp., of the Bruhat-Tits building) associated to PGLd of an archimedean (resp., non-archimedean) local field. See the papers of Satake [22] and Borel-Serre [3] (resp., Gérardin [7], Landvogt [19], and Werner [24, 25]) for details. Let E be a local field. This means that E is a locally compact topological field with a nondiscrete topology. That is, E is isomorphic to R, C, a finite extension of Qp for some prime number p , or Fq ((T )) for a finite field Fq . Let | |: E → R≥0 be the normalized absolute value. If E ∼ = R, this is the usual absolute value. ∼ If E = C, this is the square of the usual absolute value. If E is non-archimedean, this is the unique multiplicative map E → R≥0 such that |a | = ♯(OE /aOE )−1 if a is a nonzero element of the valuation ring OE of E . Fix a positive integer d and a d -dimensional E -vector space V . 2.1 Norms 2.1.1. We recall the definitions of norms and semi-norms on V . A norm (resp., semi-norm) on V is a map µ: V → R≥0 for which there exist an E -basis (e i )1≤i ≤d of V and an element (ri )1≤i ≤d of Rd>0 (resp., Rd≥0 ) such that  (r 2 |a 1 |2 · · · + rd2 |a d |2 )1/2 if E ∼ = R,   1 µ(a 1 e 1 + · · · + a d e d ) = r1 |a 1 | + · · · + rd |a d | if E ∼ = C,   max(r1 |a 1 |, . . . , rd |a d |) otherwise. for all a 1 , . . . , a d ∈ E . 2.1.2. We will call the norm (resp., semi-norm) µ in the above, the norm (resp., semi-norm) given by the basis (e i )i and by (ri )i . 2.1.3. We have the following characterizations of norms and semi-norms. (1) If E ∼ = C), then there is a one-to-one correspondence between semi-norms = R (resp., E ∼ on V and symmetric bilinear (resp., Hermitian) forms ( , ) on V such that (x ,x ) ≥ 0 for all x ∈ V . The semi-norm µ corresponding to ( , ) is given by µ(x ) = (x ,x )1/2 (resp., µ(x ) = (x ,x )). This restricts to a correspondence between norms and forms that are positive definite. (2) If E is non-archimedean, then (as in [9]) a map µ: V → R≥0 is a norm (resp., semi-norm) if and only if µ satisfies the following (i)–(iii) (resp., (i) and (ii)): 4 (i) µ(a x ) = |a |µ(x ) for all a ∈ E and x ∈ V , (ii) µ(x + y ) ≤ max(µ(x ), µ(y )) for all x , y ∈ V , and (iii) µ(x ) > 0 if x ∈ V \ {0}. These well-known facts imply that if µ is a norm (resp., semi-norm) on V and V ′ is an E subspace of V , then the restriction of µ to V ′ is a norm (resp., semi-norm) on V ′ . 2.1.4. We say that two norms (resp., semi-norms) µ and µ′ on V are equivalent if µ′ = c µ for some c ∈ R>0 . 2.1.5. The group GLV (E ) acts on the set of all norms (resp., semi-norms) on V : for g ∈ GLV (E ) and a norm (resp., semi-norm) µ on V , g µ is defined as µ ◦ g −1 . This action preserves the equivalence in 2.1.4. 2.1.6. Let V ∗ be the dual space of V . Then there is a bijection between the set of norms on V and the set of norms on V ∗ . That is, for a norm µ on V , the corresponding norm µ∗ on V ∗ is given by ‚ |ϕ(x )| µ∗ (ϕ) = sup | x ∈ V \ {0} µ(x ) Œ for ϕ ∈ V ∗ . For a norm µ on V associated to a basis (e i )i of V and (ri )i ∈ Rd>0 , the norm µ∗ on V ∗ is associated to the dual basis (e i∗ )i of V ∗ and (ri−1 )i . This proves the bijectivity. 2.1.7. For a norm µ on V and for g ∈ GLV (E ), we have (µ ◦ g )∗ = µ∗ ◦ (g ∗ )−1 , which is to say (g µ)∗ = (g ∗ )−1 µ∗ , where g ∗ ∈ GLV ∗ (E ) is the transpose of g . 2.2 Definitions of the spaces 2.2.1. Let X V denote the set of all equivalence classes of norms on V (as in 2.1.4). We endow X V with the quotient topology of the subspace topology on the set of all norms on V inside RV . 2.2.2. In the case that E is archimedean, we have  PGLd (R)/ POd (R) ∼ = SLd (R)/ SOd (R) if E XV ∼ = PGL (C)/ PU(d ) ∼ if E = SLd (C)/ SU(d ) d ∼ =R ∼ = C. In the case E is non-archimedean, X V is identified with (a geometric realization of) the BruhatTits building associated to PGLV [4] (see also [5, Section 2]). 5 2.2.3. Recall that for a finite-dimensional vector space H 6= 0 over a field I , the following four objects are in one-to-one correspondence: (i) a parabolic subgroup of the algebraic group GLH over I , (ii) a parabolic subgroup of the algebraic group PGLH over I , (iii) a parabolic subgroup of the algebraic group SLH over I , and (iv) a flag of I -subspaces of H (i.e., a set of subspaces containing {0} and H and totally ordered under inclusion). The bijections (ii) 7→ (i) and (i) 7→ (iii) are the taking of inverse images. The bijection (i) 7→ (iv) sends a parabolic subgroup P to the set of all P-stable I -subspaces of H , and the converse map takes a flag to its isotropy subgroup in GLH . 2.2.4. Let X̄ V be the set of all pairs (P, µ), where P is a parabolic subgroup of the algebraic group PGLV over E and, if 0 = V−1 ( V0 ( · · · ( Vm = V denotes the flag corresponding to P (2.2.3), then µ is a family (µi )0≤i ≤m , where µi is an equivalence class of norms on Vi /Vi −1 . We have an embedding X V ,→ X̄ V which sends µ to (PGLV , µ). 2.2.5. Let X̄ V♭ be the set of all equivalence classes of nonzero semi-norms on the dual space V ∗ of V (2.1.4). We have an embedding X V ,→ X̄ V♭ which sends µ to µ∗ (2.1.6). This set X̄ V♭ is also identified with the set of pairs (W, µ) with W a nonzero E -subspace of V and µ an equivalence class of a norm on W . In fact, µ corresponds to an equivalence class µ∗ of a norm on the dual space W ∗ of W (2.1.6), and µ∗ is identified via the projection V ∗ → W ∗ with an equivalence class of semi-norms on V ∗ . We call the understanding of X̄ V♭ as the set of such pairs (W, µ) the definition of X̄ V♭ in the second style. In this interpretation of X̄ V♭ , the above embedding X V → X̄ V♭ is written as µ 7→ (V, µ). 2.2.6. In the case that E is non-archimedean, X̄ V is the polyhedral compactification of the Bruhat-Tits building X V by Gérardin [7] and Landvogt [19] (see also [11, Proposition 19]), and X̄ V♭ is the compactification of X V by Werner [24, 25]. In the case that E is archimedean, X̄ V is the maximal Satake compactification, and X̄ V♭ is the minimal Satake compactification for the standard projective representation of PGLV (E ), as constructed by Satake in [22] (see also [2, 1.4]). The topologies of X̄ V and X̄ V♭ are reviewed in Section 2.3 below. 6 2.2.7. We have a canonical surjection X̄ V → X̄ V♭ which sends (P, µ) to (V0 , µ0 ), where V0 is as in 2.2.4, and where we use the definition of X̄ V♭ of the second style in 2.2.5. This surjection is compatible with the inclusion maps from X V to these spaces. 2.2.8. We have the natural actions of PGLV (E ) on X V , X̄ V and X̄ V♭ by 2.1.5. These actions are compatible with the canonical maps between these spaces. 2.3 Topologies 2.3.1. We define a topology on X̄ V . Take a basis (e i )i of V . We have a commutative diagram PGLV (E ) × Rd>0−1 XV PGLV (E ) × Rd≥0−1 X̄ V . Here the upper arrow is (g , t ) 7→ g µ, where µ is the class of the norm on V associated to Q ((e i )i , (ri )i ) with ri = 1≤j <i t j−1 , and where g µ is defined by the action of PGLV (E ) on X V (2.2.8). The lower arrow is (g , t ) 7→ g (P, µ), where (P, µ) ∈ X̄ V is defined as follows, and g (P, µ) is then defined by the action of PGLV (E ) on X̄ V (2.2.8). Let I = {j | t j = 0} ⊂ {1, . . . , d − 1}, and write I = {c (i ) | 0 ≤ i ≤ m − 1}, where m = ♯I and 1 ≤ c (0) < · · · < c (m − 1) ≤ d − 1. If we also let c (−1) = 0 and c (m ) = d , then the set of c (i ) X Vi = Fej j =1 with −1 ≤ i ≤ m forms a flag in V , and P is defined to be the corresponding parabolic sub- group of PGLV (2.2.3). For 0 ≤ i ≤ m , we take µi to be the equivalence class of the norm on Q Vi /Vi −1 given by the basis (e j )c (i −1)<j ≤c (i ) and the sequence (r j )c (i −1)<j ≤c (i ) with r j = c (i −1)<k <j t k−1 . Both the upper and the lower horizontal arrows in the diagram are surjective, and the topology on X V coincides with the topology as a quotient space of PGLV (E ) × Rd>0−1 via the upper horizontal arrow. The topology on X̄ V is defined as the quotient topology of the topol- ogy on PGLV (E ) × Rd≥0−1 via the lower horizontal arrow. It is easily seen that this topology is independent of the choice of the basis (e i )i . 7 2.3.2. The space X̄ V♭ has the following topology: the space of all nonzero semi-norms on V ∗ ∗ has a topology as a subspace of the product RV , and X̄ V♭ has a topology as a quotient of it. 2.3.3. Both X̄ V and X̄ V♭ are compact Hausdorff spaces containing X V as a dense open subset. This is proved in [7, 19, 24, 25] in the case that E is non-archimedean and in [22, 2] in the archimedean case. 2.3.4. The topology on X̄ V♭ coincides with the image of the topology on X̄ V . In fact, it is easily seen that the canonical map X̄ V → X̄ V♭ is continuous (using, for instance, [25, Theorem 5.1]). Since both spaces are compact Hausdorff and this continuous map is surjective, the topology on X̄ V♭ is the image of that of X̄ V . 3 Spaces associated to global fields Let F be a global field, which is to say, either a number field or a function field in one variable over a finite field. We fix a finite-dimensional F -vector space V of dimension d ≥ 1. For a place v of F , let Vv = Fv ⊗F V . We set X v = X Vv and X v♭ = X V♭ v for brevity. If v is non-archimedean, we let Ov , k v , qv , ̟v denote the valuation ring of Fv , the residue field of Ov , the order of k v , and a fixed uniformizer in Ov , respectively. ⋆ In this section, we define sets X̄ F,v containing X v for ⋆ ∈ {♯, , ♭}, which serve as our rational ♭ partial compactifications. Here, X̄ F,v (resp., X̄ F,v ) is defined as a subset of X v (resp., X̄ v♭ ), and ♯ X̄ F,v has X̄ F,v as a quotient. In §3.2, by way of example, we describe these sets and various topologies on them in the case that d = 2, F = Q, and v is the real place. For ⋆ 6= ♯, we construct ⋆ ⋆ more generally sets X̄ F,S for a nonempty finite set S of places of F . In §3.1, we describe X̄ F,S as Q ⋆ a subset of v ∈S X̄ F,v . In §3.3 and §3.4, we define topologies on these sets. That is, in §3.3, we define the “Borel- Serre topology”, while in §3.4, we define the “Satake topology” on X̄ F,v and, assuming S con♭ tains all archimedean places of F , on X̄ F,S . In §3.5, we prove results on X̄ F,v . In §3.6, we com♭ pare the following topologies on X̄ F,v (resp., X̄ F,v ): the Borel-Serre topology, the Satake topol- ogy, and the topology as a subspace of X̄ v (resp., X̄ v♭ ). In §3.7, we describe the relationship between these spaces and Borel-Serre and reductive Borel-Serre spaces. 3.1 Definitions of the spaces 3.1.1. Let X̄ F,v = X̄ V,F,v be the subset of X̄ v consisting of all elements (P, µ) such that P is F rational. If P comes from a parabolic subgroup P ′ of PGLV over F , we also denote (P, µ) by (P ′ , µ). 8 ♭ 3.1.2. Let X̄ F,v be the subset of X̄ v♭ consisting of all elements (W, µ) such that W is F -rational (using the definition of X̄ v♭ in the second style in 2.2.5). If W comes from an F -subspace W ′ of V , we also denote (W, µ) by (W ′ , µ). ♯ 3.1.3. Let X̄ F,v be the set of all triples (P, µ, s ) such that (P, µ) ∈ X̄ F,v and s is a splitting s: m M i =0 ∼ (Vi /Vi −1 )v − → Vv over Fv of the filtration (Vi )−1≤i ≤m of V corresponding to P. ♯ We have an embedding X v ,→ X̄ F,v that sends µ to (PGLV , µ, s ), where s is the identity map of Vv . 3.1.4. We have a diagram with a commutative square ♯ X̄ F,v X̄ F,v ♭ X̄ F,v X̄ v X̄ v♭ . Here, the first arrow in the upper row forgets the splitting s , and the second arrow in the upper row is (P, µ) 7→ (V0 , µ0 ), as is the lower arrow (2.2.7). ♯ ♭ 3.1.5. The group PGLV (F ) acts on the sets X̄ F,v , X̄ F,v and X̄ F,v in the canonical manner. 3.1.6. Now let S be a nonempty finite set of places of F . Q • Let X̄ F,S be the subset of v ∈S X̄ F,v consisting of all elements (x v )v ∈S such that the parabolic subgroup of G = PGLV associated to x v is independent of v . Q ♭ ♭ • Let X̄ F,S be the subset of v ∈S X̄ F,v consisting of all elements (x v )v ∈S such that the F subspace of V associated to x v is independent of v . We will denote an element of X̄ F,S as (P, µ), where P is a parabolic subgroup of G and µ ∈ Q ♭ v ∈S,0≤i ≤m X (Vi /Vi −1 )v with (Vi )i the flag corresponding to P. We will denote an element of X̄ F,S Q as (W, µ), where W is a nonzero F -subspace of V and µ ∈ v ∈S X Wv . We have a canonical surjective map ♭ X̄ F,S → X̄ F,S which commutes with the inclusion maps from X S to these spaces. 9 3.2 Example: Upper half-plane 3.2.1. Suppose that F = Q, v is the real place, and d = 2. ♯ ♭ , and X̄ Q,v are described by using the upper In this case, the sets X v , X̄ v = X̄ v♭ , X̄ Q,v = X̄ Q,v half-plane. In §2, we discussed topologies on the first two spaces. The remaining spaces also ♯ have natural topologies, as will be discussed in §3.3 and §3.4: the space X̄ Q,v is endowed with the Borel-Serre topology, and X̄ Q,v has two topologies, the Borel-Serre topology and Satake topology, which are both different from its topology as a subspace of X̄ v . In this section, as a prelude to §3.3 and §3.4, we describe what the Borel-Serre and Satake topologies look like in this special case. 3.2.2. Let H = {x + y i | x , y ∈ R, y > 0} be the upper half-plane. Fix a basis (e i )i =1,2 of V . For z ∈ H, let µz denote the class of the norm on V corresponding to the class of the norm on V ∗ given by a e 1∗ + b e 2∗ 7→ |a z + b | for a ,b ∈ R. Here (e i∗ )1≤i ≤d is the dual basis of (e i )i , and | | denotes the usual absolute value (not the normalized absolute value) on C. We have a homeomorphism ∼ H −→ X v , z 7→ µz which is compatible with the actions of SL2 (R). For the square root i ∈ H of −1, the norm a e 1 +b e 2 7→ (a 2 +b 2)1/2 has class µi . For z = x +y i , we have µz = The action of ! −1 0 0 1 y x 0 1 ! µi . ∈ GL2 (R) on X v corresponds to x + y i 7→ −x + y i on H. 3.2.3. The inclusions X v ⊂ X̄ Q,v ⊂ X̄ v can be identified with H ⊂ H ∪ P1(Q) ⊂ H ∪ P1 (R). Here z ∈ P1 (R) = R ∪ {∞} corresponds to the class in X̄ v♭ = X̄ v of the semi-norm a e 1∗ + b e 2∗ 7→ |a z +b | (resp., a e 1∗ +b e 2∗ 7→ |a |) on V ∗ if z ∈ R (resp., z = ∞). These identifications are compatible with the actions of PGLV (Q). The topology on X̄ v of 2.3.1 is the topology as a subspace of P1 (C). 10 3.2.4. Let B be the Borel subgroup of PGLV consisting of all upper triangular matrices for the basis (e i )i , and let 0 = V−1 ( V0 = Qe 1 ( V1 = V be the corresponding flag. Then ∞ ∈ P1 (Q) is understood as the point (B, µ) of X̄ Q,v , where µ is the unique element of X (V0 )v × X (V /V0 )v . ♯ ♯ Let X̄ Q,v (B ) = H ∪ {∞} ⊂ X̄ Q,v and let X̄ Q,v (B ) be the inverse image of X̄ Q,v (B ) in X̄ Q,v . Then for the Borel-Serre topology defined in §3.3, we have a homeomorphism ♯ X̄ Q,v (B ) ∼ = {x + y i | x ∈ R, 0 < y ≤ ∞} ⊃ H. Here x + ∞i corresponds to (B, µ, s ) where s is the splitting of the filtration (Vi ,v )i given by the embedding (V /V0 )v → Vv that sends the class of e 2 to x e 1 + e 2 . ♯ The Borel-Serre topology on X̄ Q,v is characterized by the properties that ♯ (i) the action of the discrete group GLV (Q) on X̄ Q,v is continuous, ♯ (ii) the subset X̄ Q,v (B ) is open, and ♯ (iii) as a subspace, X̄ Q,v (B ) is homeomorphic to {x + y i | x ∈ R, 0 < y ≤ ∞} as above. 3.2.5. The Borel-Serre and Satake topologies on X̄ Q,v (defined in §3.3 and §3.4) are characterized by the following properties: (i) The subspace topology on X v ⊂ X̄ Q,v coincides with the topology on H. (ii) The action of the discrete group GLV (Q) on X̄ Q,v is continuous. (iii) The following sets (a) (resp., (b)) form a base of neighborhoods of ∞ for the Borel-Serre (resp., Satake) topology: (a) the sets U f = {x + y i ∈ H | y ≥ f (x )} ∪ {∞} for continuous f : R → R, (b) the sets Uc = {x + y i ∈ H | y ≥ c } ∪ {∞} with c ∈ R>0 . ♯ The Borel-Serre topology on X̄ Q,v is the image of the Borel-Serre topology on X̄ Q,v . 3.2.6. For example, the set {x +y i ∈ H | y > x }∪{∞} is a neighborhood of ∞ for the Borel-Serre topology, but it is not a neighborhood of ∞ for the Satake topology. 3.2.7. For any subgroup Γ of PGL2 (Z) of finite index, the Borel-Serre and Satake topologies induce the same topology on the quotient space X (Γ) = Γ\X̄ Q,v . Under this quotient topology, X (Γ) is compact Hausdorff. If Γ is the image of a congruence subgroup of SL2 (Z), then this is the usual topology on the modular curve X (Γ). 11 3.3 Borel-Serre topology ♯ 3.3.1. For a parabolic subgroup P of PGLV , let X̄ F,v (P) (resp., X̄ F,v (P)) be the subset of X̄ F,v ♯ (resp., X̄ F,v ) consisting of all elements (Q, µ) (resp., (Q, µ, s )) such that Q ⊃ P. The action of PGLV (Fv ) on X̄ v induces an action of P(Fv ) on X̄ F,v (P). We have also an action ♯ of P(Fv ) on X̄ F,v (P) given by g (α, s ) = (g α, g ◦ s ◦ g −1 ) for g ∈ P(Fv ), α ∈ X̄ F,v (P), and s a splitting of the filtration. 3.3.2. Fix a basis (e i )i of V . Let P be a parabolic subgroup of PGLV such that • if 0 = V−1 ( V0 ( · · · ( Vm = V denotes the flag of F -subspaces corresponding to P, then each Vi is generated by the e j with 1 ≤ j ≤ c (i ), where c (i ) = dim(Vi ). This condition on P is equivalent to the condition that P contains the Borel subgroup B of PGLV consisting of all upper triangular matrices with respect to (e i )i . Where useful, we will identify PGLV over F with PGLd over F via the basis (e i )i . Let ∆(P) = {dim(Vj ) | 0 ≤ j ≤ m − 1} ⊂ {1, . . . , d − 1}, and let ∆′ (P) be the complement of ∆(P) in {1, . . . , d − 1}. Let Rd≥0−1 (P) be the open subset of Rd≥0−1 given by Rd≥0−1 (P) = {(t i )1≤i ≤d −1 ∈ Rd≥0−1 | t i > 0 for all i ∈ ∆′ (P)}. In particular, we have ∆′ (P) Rd≥0−1 (P) ∼ = R>0 × R≥0 . ∆(P) 3.3.3. With P as in 3.3.2, the map PGLV (Fv ) × Rd≥0−1 → X̄ v in 2.3.1 induces a map π̄P,v : P(Fv ) × Rd≥0−1 (P) → X̄ F,v (P), which restricts to a map πP,v : P(Fv ) × Rd>0−1 → X F,v . The map π̄P,v is induced by a map ♯ ♯ π̄P,v : P(Fv ) × Rd≥0−1 (P) → X̄ F,v (P) defined as (g , t ) 7→ g (P, µ, s ) where (P, µ) is as in 2.3.1 and s is the splitting of the filtration (Vi )−1≤i ≤m defined by the basis (e i )i . For this splitting s , we set X V (i ) = s (Vi /Vi −1 ) = Fej c (i −1)<j ≤c (i ) 12 for 0 ≤ i ≤ m so that Vi = Vi −1 ⊕ V (i ) and V = Lm i =0 V (i ) . If P = B , then we will often omit the subscript B from our notation for these maps. 3.3.4. We review the Iwasawa decomposition. For v archimedean (resp., non-archimedean), let A v ≤ PGLd (Fv ) be the subgroup of elements of that lift to diagonal matrices in GLd (Fv ) with positive real entries (resp., with entries that are powers of ̟v ). Let K v denote the standard maximal compact subgroup of PGLd (Fv ) given by  POd (R)   K v = PUd   PGLd (Ov ) if v is real, if v is complex, otherwise. Let B u denote the upper-triangular unipotent matrices in the standard Borel B . The Iwasawa decomposition is given by the equality PGLd (Fv ) = B u (Fv )A v K v . 3.3.5. If v is archimedean, then the expression of a matrix in PGLd (Fv ) as a product in the Iwasawa decomposition is unique. 3.3.6. If v is non-archimedean, then the Bruhat decomposition is PGLd (k v ) = B (k v )S d B (k v ), where the symmetric group S d of degree d is viewed as a subgroup of PGLd over any field via the permutation representation on the standard basis. This implies that PGLd (Ov ) = B (Ov )S d Iw(Ov ), where Iw(Ov ) is the Iwahori subgroup consisting of those matrices in with upper triangular image in PGLd (k v ). Combining this with the Iwasawa decomposition (in the notation of 3.3.4), we have PGLd (Fv ) = B u (Fv )A v S d Iw(Ov ). This decomposition is not unique, since B u (Fv ) ∩ Iw(Ov ) = B u (Ov ). ∼ 3.3.7. If v is archimedean, then there is a bijection Rd>0−1 − → A v given by  t = (t k )1≤k ≤d −1 7→ a = where ri = Qi −1 t −1 k =1 k diag(r1 , . . . , rd )−1 if v is real, diag(r 1/2 , . . . , r 1/2 )−1 if v is complex, 1 as in 2.3.1. Proposition 3.3.8. 13 d (1) Let P be a parabolic subgroup of PGLV as in 3.3.2. Then the maps ♯ ♯ π̄P,v : P(Fv ) × Rd≥0−1 (P) → X̄ F,v (P) and π̄P,v : P(Fv ) × Rd≥0−1 (P) → X̄ F,v (P) of 3.3.3 are surjective. (2) For the Borel subgroup B of 3.3.2, the maps πv : B u (Fv ) × Rd>0−1 → X v , π̄v : B u (Fv ) × Rd≥0−1 → X̄ F,v (B ), ♯ ♯ and π̄v : B u (Fv ) × Rd≥0−1 → X̄ F,v (B ). of 3.3.3 are all surjective. ♯ (3) If v is archimedean, then πv and π̄v are bijective. (4) If v is non-archimedean, then π̄v induces a bijection (B u (Fv ) × Rd≥0−1 )/∼ → X̄ F,v (B ) where (g , (t i )i ) ∼ (g ′ , (t i′ )i ) if and only if (i) t i = t i′ for all i and Q (ii) |(g −1 g ′ )i j | ≤ ( i ≤k <j t k )−1 for all 1 ≤ i < j ≤ d , considering any c ∈ R to be less than 0−1 = ∞. ♯ ♯ Proof. If π̄v is surjective, then for any parabolic P containing B , the restriction of π̄v to B u (Fv )× ♯ ♯ Rd≥0−1 (P) has image X̄ F,v (P). Since B u (Fv ) ⊂ P(Fv ), this forces the surjectivity of π̄P,v , hence of π̄P,v as well. So, we turn our attention to (2)–(4). If r ∈ Rd>0 , we let µ(r ) ∈ X v denote the class of the norm attached to the basis (e i )i and r . Suppose first that v is archimedean. By the Iwasawa decomposition 3.3.4, and noting 3.3.5 and 2.2.2, we see that B u (Fv )A v → X v given by g 7→ g µ(1) is bijective, where µ(1) denotes the class of the norm attached to (e i )i and 1 = (1, . . . , 1) ∈ Rd>0 . For t ∈ Rd>0−1 , let a ∈ A v be its image under the bijection in 3.3.7. Since p a µ(1) = p µ(r ) , for p ∈ B u (Fv ) and r as in 2.3.1, we have the bijectivity of πv . Consider the third map in (2). For t ∈ Rd≥0−1 , let P be the parabolic that contains B and is determined by the set ∆(P) of k ∈ {1, . . . , d − 1} for which t k = 0. Let (Vi )−1≤i ≤m be the correQm sponding flag. Let M denote the Levi subgroup of P. (It is the quotient of i =0 GLV (i ) < GLV Lm by scalars, where V = i =0 V (i ) as in 3.3.3, and M ∩ B u is isomorphic to the product of the upper-triangular unipotent matrices in each PGLV (i ) .) The product of the first maps in (2) for the blocks of M is a bijection ∆′ (P) ∼ (M ∩ B u )(Fv ) × R>0 − → 14 m Y i =0 X Vv(i ) ⊂ X̄ F,v (P), such that (g , t ′ ) is sent to (P, g µ) in X̄ F,v , where µ is the sequence of classes of norms determined by t ′ and the standard basis. The stabilizer of µ in B u is the unipotent radical Pu of P, and this Pu acts simply transitively on the set of splittings for the graded quotients (Vi /Vi −1 )v . Since B u = (M ∩ B u )Pu , and this decomposition is unique, we have the desired bijectivity of ♯ π̄v , proving (3). Suppose next that v is non-archimedean. We prove the surjectivity of the first map in (2). Using the natural actions of A v and the symmetric group S d on Rd>0 , we see that any norm on Vv can be written as g µ(r ) , where g ∈ PGLd (Fv ) and r = (ri )i ∈ Rd>0 , with r satisfying r1 ≤ r2 ≤ · · · ≤ rd ≤ qv r1 . For such an r , the class µ(r ) is invariant under the action of Iw(Ov ). Hence for such an r , any ′ element of S d Iw(Ov )µ(r ) = S d µ(r ) is of the form µ(r ) , where r ′ = (rσ(i ) )i for some σ ∈ S d . Hence, ′ any element of A v S d Iw(Ov )µ(r ) for such an r is of the form µ(r ) for some r ′ = (ri′ )i ∈ Rd>0 . This proves the surjecivity of the first map of (2). The surjectivity of the other maps in (2) is then shown using this, similarly to the archimedean case. Finally, we prove (4). It is easy to see that the map π̄v factors through the quotient by the equivalence relation. We can deduce the bijectivity in question from the bijectivity of (B u (Fv )×Rd>0−1 )/∼ → X v , replacing V by Vi /Vi −1 as in the above arguments for the archimedean case. Suppose that πv (g , t ) = πv (1, t ′ ) for g ∈ B u (Fv ) and t , t ′ ∈ Rd>0−1 . We must show that ′ (g , t ) ∼ (1, t ′ ). Write πv (g , t ) = g µ(r ) and πv (1, t ′ ) = µ(r ) with r = (ri )i and r ′ = (ri′ )i ∈ Rd>0 such Q that r1 = 1 and r j /ri = ( i ≤k <j t k )−1 for all 1 ≤ i < j ≤ d , and similarly for r ′ and t ′ . It then ′ suffices to check that r ′ = r and ri |g i j | ≤ r j for all i < j . Since µ(r ) = g −1 µ(r ) , there exists c ∈ R>0 such that max{ri |x i | | 1 ≤ i ≤ d } = c max{ri′ |(g x )i | | 1 ≤ i ≤ d } for all x = (x i )i ∈ Fvd . Taking x = e 1 , we have g x = e 1 as well, so c = 1. Taking x = e i , we obtain ri ≥ ri′ , and taking x = g −1 e i , we obtain ri ≤ ri′ . Thus r = r ′ , and taking x = e j yields r j = max{ri |g i j | | 1 ≤ i ≤ j }, which tells us that r j ≥ ri |g i j | for i < j . ♯ Proposition 3.3.9. There is a unique topology on X̄ F,v (resp., X̄ F,v ) satisfying the following conditions (i) and (ii). ♯ (i) For every parabolic subgroup P of PGLV , the set X̄ F,v (P) (resp., X̄ F,v (P)) is open in X̄ F,v ♯ (resp., X̄ F,v ). (ii) For every parabolic subgroup P of PGLV and basis (e i )i of V such that P contains the Borel ♯ subgroup with respect to (e i )i , the topology on X̄ F,v (P) (resp., X̄ F,v (P)) is the topology as a quotient of P(Fv ) × Rd≥0−1(P) under the surjection of 3.3.8(1). 15 This topology is also characterized by (i) and the following (ii)’. (ii)’ If B is a Borel subgroup of PGLV consisting of upper triangular matrices with respect to a ♯ basis (e i )i of V , then the topology on X̄ F,v (B ) (resp., X̄ F,v (B )) is the topology as a quotient of B u (Fv ) × Rd≥0−1 under the surjection of 3.3.8(2). Proof. The uniqueness is clear if we have existence of a topology satisfying (i) and (ii). Let (e i )i be a basis of V , let B be the Borel subgroup of PGLV with respect to this basis, and let P be a parabolic subgroup of PGLV containing B . It suffices to prove that for the topology ♯ on X̄ F,v (B ) (resp., X̄ F,v (B )) as a quotient of B u (Fv ) × Rd≥0−1 (B ), the subspace topology on X̄ F,v (P) ♯ (resp., X̄ F,v (P)) coincides with the quotient topology from P(Fv )×Rd≥0−1 (P). For this, it is enough ♯ to show that the action of the topological group P(Fv ) on X̄ F,v (P) (resp., X̄ F,v (P)) is continuous ♯ with respect to the topology on X̄ F,v (P) (resp., X̄ F,v (P)) as a quotient of B u (Fv ) × Rd≥0−1 (P). We must demonstrate this continuity. Let (Vi )−1≤i ≤m be the flag corresponding to P, and let c (i ) = dim(Vi ). For 0 ≤ i ≤ m , we Lm regard GLV (i ) as a subgroup of GLV via the decomposition V = i =0 V (i ) of 3.3.3. Suppose first that v is archimedean. For 0 ≤ i ≤ m , let K i be the compact subgroup of GLV (i ) (Fv ) that is the isotropy group of the norm on V (i ) given by the basis (e j )c (i −1)<j ≤c (i ) and Q (1, . . . , 1) ∈ c (i −1)<j ≤c (i ) R>0 . We identify Rd>0−1 with A v as in 3.3.7. By the Iwasawa decomposition 3.3.4 and its uniqueness in 3.3.5, the product on P(Fv ) induces a homeomorphism ! m Y ∼ K i /{z ∈ Fv× | |z | = 1}. (a ,b, c ): P(Fv ) − → B u (Fv ) × Rd>0−1 × i =0 ∆′ (P) We also have a product map φ : P(Fv ) × B u (Fv ) × R>0 ∆′ (P) → P(Fv ), where we identify t ′ ∈ R>0 1/2 1/2 with the diagonal matrix diag(r1 , . . . , rd )−1 if v is real and diag(r1 , . . . , rd )−1 if v is complex, Q with r j−1 = c (i −1)<k <j t k′ for c (i − 1) < j ≤ c (i ) as in 2.3.1. These maps fit in a commutative diagram ∆(P) P(Fv ) × R≥0 (φ, id) ∆′ (P) ∆(P) P(Fv ) × B u (Fv ) × R>0 × R≥0 (id, π̄P,v ) π̄P,v B u (Fv ) × Rd≥0−1 (P) P(Fv ) × X̄ F,v (P) X̄ F,v (P) in which the right vertical arrow is the action of P(Fv ) on X̄ F,v (P), and the left vertical arrow is the continuous map (u , t ) 7→ (a (u ),b (u ) · (1, t )), ∆′ (P) ∆(P) (u , t ) ∈ P(Fv ) × R≥0 ∆(P) for (1, t ) the element of Rd≥0−1 (P) with R>0 -component 1 and R≥0 -component t . (To see the commutativity, note that c (u ) commutes with the block-scalar matrix determined by (1, t ).) 16 ♯ We also have a commutative diagram of the same form for X̄ F,v . Since the surjective horizontal arrows are quotient maps, we have the continuity of the action of P(Fv ). Next, we consider the case that v is non-archimedean. For 0 ≤ i ≤ m , let S (i ) be the group of permutations of the set I i = {j ∈ Z | c (i − 1) < j ≤ c (i )}, and regard it as a subgroup of GLV (i ) (F ). Let A v be the subgroup of the diagonal torus of PGLV (Fv ) with respect to the basis (e i )i with entries powers of a fixed uniformizer, as in 3.3.4. Qm Consider the action of A v i =0 S (i ) ⊂ P(Fv ) on Rd≥0−1 (P) that is compatible with the action of P(Fv ) on X̄ F,v (P) via the embedding Rd≥0−1 (P) → X̄ F,v (P). This action is described as follows. Any matrix a = diag(a 1 , . . . , a d ) ∈ A v sends t ∈ Rd≥0−1 (P) to (t j |a j +1 ||a j |−1 ) j ∈ Rd≥0−1 (P). The action Qm of i =0 S (i ) on Rd≥0−1 (P) is the unique continuous action which is compatible with the evident Qm action of i =0 S (i ) on Rd>0 via the map Rd>0 → Rd≥0−1 (P) that sends (ri )i to (t j ) j , where t j = r j /r j +1 . That is, for σ = (σi )0≤i ≤m ∈ m Y S (i ) , i =0 let f ∈ S d be the unique permutation with f |I i = σi−1 for all i . Then σ sends t ∈ Rd≥0−1 (P) to the element t ′ = (t j′ ) j given by  Q t j′ =  t f (j )≤k < f (j +1) k if f (j ) < f (j + 1), Q t −1 f (j +1)≤k < f (j ) k if f (j + 1) < f (j ). Let C be the compact subset of Rd≥0−1(P) given by ¨ « C = t = (t j ) j ∈ Rd≥0−1 (P) ∩ [0, 1]d −1 | Y c (i −1)<j <c (i ) t j ≥ qv−1 for all 0 ≤ i ≤ m . Qm We claim that for each x ∈ Rd≥0−1 (P), there is a finite family (h k )k of elements of A v i =0 S (i ) such S that the union k h k C is a neighborhood of x . This is quickly reduced to the following claim. Claim. Consider the natural action of H = A v S d ⊂ PGLV on the quotient space Rd>0 /R>0 , with the class of (a j ) j in A v acting as multiplication by (|a j |) j . Let C be the image of {r ∈ Rd>0 | r1 ≤ r2 ≤ · · · ≤ rd ≤ qv r1 } in Rd /R>0 . Then for each x ∈ Rd>0 /R>0 , there is a finite family (h k )k of elements of H such that S >0 h C is a neighborhood of x . k k 17 Proof of Claim. This is a well-known statement in the theory of Bruhat-Tits buildings: the quotient Rd>0 /R>0 is called the apartment of the Bruhat-Tits building X v of PGLV , and the set C is a (d − 1)-simplex in this apartment. Any (d − 1)-simplex in this apartment has the form hC for some h ∈ H , for any x ∈ Rd>0 /R>0 there are only finitely many (d − 1)-simplices in this apartment which contain x , and the union of these is a neighborhood of x in Rd>0 /R>0 . S By compactness of C , the topology on the neighborhood k h k C of x is the quotient topolQm ` ogy from k h k C . Thus, it is enough to show that for each h ∈ A v i =0 S (i ) , the composition (id,πP,v ) P(Fv ) × B u (Fv ) × hC −−−→ P(Fv ) × X̄ F,v (P) → X̄ F,v (P) ♯ (where the second map is the action) and its analogue for X̄ F,v are continuous. For 0 ≤ i ≤ m , let Iwi be the Iwahori subgroup of GLV (i ) (Fv ) for the basis (e j )c (i −1)<j ≤c (i ). By the the Iwasawa and Bruhat decompositions as in 3.3.6, the product on P(Fv ) induces a continuous surjection m Y B u (Fv ) × A v i =0 S (i ) × m Y i =0 Iwi → P(Fv ), and it admits continuous sections locally on P(Fv ). (Here, the middle group A v Qm i =0 S (i ) has the discrete topology.) Therefore, there exist an open covering (Uλ )λ of P(Fv ) and, for each λ, a subset Uλ of the above product mapping homeomorphically to Uλ , together with a contin- uous map (a λ ,b λ , c λ ): Uλ → B u (Fv ) × A v m Y S (i ) i =0 m Y × Iwi i =0 ∼ → Uλ . Let Uλ′ denote the such that its composition with the above product map is the map Uλ − inverse image of Uλ under P(Fv ) × B u (Fv ) → P(Fv ), (g , g ′ ) 7→ g g ′ h, so that (Uλ′ )λ is an open covering of P(Fv ) × B u (Fv ). For any γ in the indexing set of the cover, ′ ′ let Uλ,γ be the inverse image of Uλ′ in Uγ × B u (Fv ). Then the images of the Uλ,γ form an open ′ cover of P(Fv ) × B u (Fv ) as well. Let (a ′λ,γ ,b λ,γ ) be the composition (a λ ,b λ ) ′ Uλ,γ → Uλ −−−→ B u (Fv ) × A v As Qm i =0 m Y S (i ) . i =0 Iwi fixes every element of C under its embedding in X̄ F,v (P), we have a commutative 18 diagram ′ Uλ,γ × hC P(Fv ) × B u (Fv ) × hC B u (Fv ) × Rd≥0−1 (P) X̄ F,v (P) in which the left vertical arrow is ′ (u , hx ) 7→ (a ′λ,γ (u ),b λ,γ (u )x ) ♯ for x ∈ C . We also have a commutative diagram of the same form for X̄ F,v . This proves the continuity of the action of P(Fv ). ♯ 3.3.10. We call the topology on X̄ F,v (resp., X̄ F,v ) in 3.3.9 the Borel-Serre topology. The BorelSerre topology on X̄ F,v coincides with the quotient topology of the Borel-Serre topology on ♯ X̄ F,v . This topology on X̄ F,v is finer than the subspace topology from X̄ v . ♭ We define the Borel-Serre topology on X̄ F,v as the quotient topology of the Borel-Serre ♭ topology of X̄ F,v . This topology on X̄ F,v is finer than the subspace topology from X̄ v♭ . For a nonempty finite set S of places of F , we define the Borel-Serre topology on X̄ F,S (resp., Q Q ♭ ) for the as the subspace topology for the product topology on v ∈S X̄ F,v (resp., v ∈S X̄ F,v ♭ X̄ F,S ) ♭ Borel-Serre topology on each X̄ F,v (resp., X̄ F,v ). 3.4 Satake topology 3.4.1. For a nonempty finite set of places S of F , we define the Satake topology on X̄ F,S and, ♭ under the assumption S contains all archimedean places, on X̄ F,S . The Satake topology is coarser than the Borel-Serre topology of 3.3.10. On the other hand, the Satake topology and the Borel-Serre topology induce the same topology on the quotient space by an arithmetic group (4.1.8). Thus, the Hausdorff compactness of this quotient space can be formulated without using the Satake topology (i.e., using only the Borel-Serre topology). However, arguments involving the Satake topology appear naturally in the proof of this property. One nice aspect of the Satake topology is that each point has an explicit base of neighborhoods (3.2.5, 3.4.9, 4.4.9). 3.4.2. Let H be a finite-dimensional vector space over a local field E . Let H ′ and H ′′ be E subspaces of H such that H ′ ⊃ H ′′ . Then a norm µ on H induces a norm ν on H ′ /H ′′ as follows. Let µ′ be the restriction of µ to H ′ . Let (µ′ )∗ be the norm on (H ′ )∗ dual to µ′ . Let ν ∗ be the restriction of (µ′ )∗ to the subspace (H ′ /H ′′ )∗ of (H ′ )∗ . Let ν be the dual of ν ∗ . This norm ν is 19 given on x ∈ H ′ /H ′′ by ν(x ) = inf {µ(x̃ ) | x̃ ∈ H ′ such that x̃ + H ′′ = x }. 3.4.3. For a parabolic subgroup P of PGLV , let (Vi )−1≤i ≤m be the corresponding flag. Set X̄ F,S (P) = {(P ′ , µ) ∈ X̄ F,S | P ′ ⊃ P}. For a place v of F , let us set m Y ZF,v (P) = i =0 X (Vi /Vi −1)v and ZF,S (P) = Y ZF,v (P). v ∈S We let P(Fv ) act on ZF,v (P) through P(Fv )/Pu (Fv ), using the PGL(Vi /Vi −1 )v (Fv )-action on X (Vi /Vi −1)v for 0 ≤ i ≤ m . We define a P(Fv )-equivariant map φP,v : X̄ F,v (P) → ZF,v (P) with the product of these over v ∈ S giving rise to a map φP,S : X̄ F,S (P) → ZF,S (P). ′ ( V ′ ( · · · ( V ′ = V corresponding Let (P ′ , µ) ∈ X̄ F,v (P). Then the spaces in the flag 0 = V−1 m′ 0 to P ′ form a subset of {Vi | −1 ≤ i ≤ m }. The image ν = (νi )0≤i ≤m of (P ′ , µ) under φP,v is as follows: there is a unique j with 0 ≤ j ≤ m ′ such that Vj ′ ⊃ Vi ) Vi −1 ⊃ Vj ′−1 , and νi is the norm induced from µ j on the subquotient (Vi /Vi −1 )v of (Vj ′ /Vj ′−1 )v , in the sense of 3.4.2. The P(Fv )-equivariance of φP,v is easily seen using the actions on norms of 2.1.5 and 2.1.7. Though the following map is not used in this subsection, we introduce it here by way of ♭ comparison between X̄ F,S and X̄ F,S . 3.4.4. Let W be a nonzero F -subspace of V , and set ♭ ♭ X̄ F,S (W ) = {(W ′ , µ) ∈ X̄ F,S | W ′ ⊃ W }. For a place v of F , we have a map ♭ ♭ φW,v : X̄ F,v (W ) → X Wv ♭ ♭ which sends (W ′ , µ) ∈ X̄ F,v (W ) to the restriction of µ to Wv . The map φW,v is P(Fv )-equivariant, for P the parabolic subgroup of PGLV consisting of all elements that preserve W . Setting Q ♭ ♭ : X̄ F,S (W ) → Z♭F,S (W ) = v ∈S X Wv , the product of these maps over v ∈ S provides a map φW,S Z♭F,S (W ). 20 3.4.5. For a finite-dimensional vector space H over a local field E , a basis e = (e i )1≤i ≤d of H , and a norm µ on H , we define the absolute value |µ : e | ∈ R>0 of µ relative to e as follows. Suppose that µ is defined by a basis e ′ = (e i′ )1≤i ≤d and a tuple (ri )1≤i ≤d ∈ Rd>0 . Let h ∈ GLH (E ) be the element such that e ′ = he . We then define |µ : e | = | det(h)|−1 d Y ri . i =1 This is independent of the choice of e ′ and (ri )i . Note that we have |g µ : e | = | det(g )|−1 |µ : e | for all g ∈ GLH (E ). 3.4.6. Let P and (Vi )i be as in 3.4.3, and let v be a place of F . Fix a basis e (i ) of (Vi /Vi −1 )v for each 0 ≤ i ≤ m . Then we have a map ′ φP,v : X̄ F,v (P) → Rm , ≥0 (P ′ , µ) 7→ (t i )1≤i ≤m where (t i )1≤i ≤m is defined as follows. Let (Vj ′ )−1≤j ≤m ′ be the flag associated to P ′ . Let 1 ≤ i ≤ m . If Vi −1 belongs to (Vj ′ ) j , let t i = 0. If Vi −1 does not belong to the last flag, then there is a unique j such that Vj ′ ⊃ Vi ⊃ Vi −2 ⊃ Vj ′−1 . Let µ̃ j be a norm on (Vj ′ /Vj ′−1 )v which belongs to the class µ j , and let µ̃ j ,i and µ̃ j ,i −1 be the norms induced by µ j on the subquotients (Vi /Vi −1 )v and (Vi −1 /Vi −2 )v , respectively. We then let t i = |µ̃ j ,i −1 : e (i −1) |1/d i −1 · |µ̃ j ,i : e (i ) |−1/d i , where d i := dim(Vi /Vi −1 ). ′ is P(Fv )-equivariant for the following action of P(Fv ) on Rm The map φP,v ≥0 . For g ∈ P(Fv ), let g̃ ∈ GLV (Fv ) be a lift of g , and for 0 ≤ i ≤ m , let g i ∈ GLVi /Vi −1 (Fv ) be the element induced by m ′ g̃ . Then g ∈ P(Fv ) sends t ∈ Rm ≥0 to t ∈ R≥0 where t i′ = | det(g i )|1/d i · | det(g i −1 )|−1/d i −1 · t i . If we have two families e = (e (i ) )i and f = (f (i ) )i of bases e (i ) and f (i ) of (Vi /Vi −1 )v , and if the ′ map φP,v defined by e (resp., f ) sends an element to t (resp., t ′ ), then the same formula also describes the relationship between t and t ′ , in this case taking g i to be the element of GLVi /Vi −1 such that e (i ) = g i f (i ) . 3.4.7. Fix a basis e (i ) of Vi /Vi −1 for each 0 ≤ i ≤ m . Then we have a map ′ φP,S : X̄ F,S (P) → Rm , ≥0 where t i = Q t , v ∈S v,i (P ′ , µ) 7→ (t i )1≤i ≤m ′ with (t v,i )i the image of (P ′ , µv ) under the map φP,v of 3.4.6. 21 3.4.8. We define the Satake topology on X̄ F,S as follows. For a parabolic subgroup P of PGLV , consider the map ′ ψP,S := (φP,S , φP,S ): X̄ F,S (P) → ZF,S (P) × Rm ≥0 from 3.4.3 and 3.4.7, which we recall depends on a choice of bases of the Vi /Vi −1 . We say that a subset of X̄ F,S (P) is P-open if it is the inverse image of an open subset of ZF,S (P) × Rm ≥0 . By 3.4.6, the property of being P-open is independent of the choice of bases. We define the Satake topology on X̄ F,S to be the coarsest topology for which every P-open set for each parabolic subgroup P of PGLV is open. By this definition, we have: 3.4.9. Let a ∈ X̄ F,S be of the form (P, µ) for some µ. As U ranges over neighborhoods of the image (µ, 0) of a in ZF,S (P) × Rm ≥0 , the inverse images of the U in X̄ F,S (P) under ψP,S form a base of neighborhoods of a in X̄ F,S . 3.4.10. In §3.5 and §3.6, we explain that the Satake topology on X̄ F,S is strictly coarser than the Borel-Serre topology for d ≥ 2. 3.4.11. The Satake topology on X̄ F,S can differ from the subspace topology of the product topology for the Satake topology on each X̄ F,v with v ∈ S. Example. Let F be a real quadratic field, let V = F 2 , and let S = {v 1 , v 2 } be the set of real places of F . Consider the point (∞, ∞) ∈ (H ∪ {∞}) × (H ∪ {∞}) ⊂ X̄ v 1 × X̄ v 2 (see §3.2), which we regard as an element of X̄ F,S . Then the sets Uc := {(x 1 + y 1 i ,x 2 + y 2 i ) ∈ H × H | y 1 y 2 ≥ c } ∪ {(∞, ∞)} with c ∈ R>0 form a base of neighborhoods of (∞, ∞) in X̄ F,S for the Satake topology, whereas the sets Uc′ := {(x 1 + y 1 i ,x 2 + y 2 i ) ∈ H × H | y 1 ≥ c , y 2 ≥ c } ∪ {(∞, ∞)} for c ∈ R>0 form a base of neighborhoods of (∞, ∞) in X̄ F,S for the topology induced by the product of Satake topologies on X̄ F,v 1 and X̄ F,v 2 . 3.4.12. Let G = PGLV , and let Γ be a subgroup of G (F ). • For a parabolic subgroup P of G , let Γ(P) be the subgroup of Γ ∩ P(F ) consisting of all elements with image in the center of (P/Pu )(F ). • For a nonzero F -subspace W of V , let Γ(W ) denote the subgroup of elements of Γ that can be lifted to elements of GLV (F ) which fix every element of W . 22 3.4.13. We let AF denote the adeles of F , let ASF denote the adeles of F outside of S, and let Q AF,S = v ∈S Fv so that AF = ASF × AF,S . Assume that S contains all archimedean places of F . Let G = PGLV , let K be a compact open subgroup of G (ASF ), and let ΓK < G (F ) be the inverse image of K under G (F ) → G (ASF ). The following proposition will be proved in 3.5.15. Proposition 3.4.14. For S, G , K and ΓK as in 3.4.13, the Satake topology on X̄ F,S is the coarsest topology such that for every parabolic subgroup P of G , a subset U of X̄ F,S (P) is open if (i) it is open for Borel-Serre topology, and (ii) it is stable under the action of ΓK ,(P) (see 3.4.12). The following proposition follows easily from the fact that for any two compact open subgroups K and K ′ of G (ASF ), the intersection ΓK ∩ ΓK ′ is of finite index in both ΓK and ΓK ′ . ♭ Proposition 3.4.15. For S, K and ΓK as in 3.4.13, consider the coarsest topology on X̄ F,S such ♭ that for every nonzero F -subspace W , a subset U of X̄ F,S (W ) is open if (i) it is open for Borel-Serre topology, and (ii) it is stable under the action of ΓK ,(W ) (see 3.4.12). Then this topology is independent of the choice of K . ♭ 3.4.16. We call the topology in 3.4.15 the Satake topology on X̄ F,S . Proposition 3.4.17. (1) Let P be a parabolic subgroup of PGLV . For both the Borel-Serre and Satake topologies on X̄ F,S , the set X̄ F,S (P) is open in X̄ F,S , and the action of the topological group P(AF,S ) on X̄ F,S (P) is continuous. (2) The actions of the discrete group PGLV (F ) on the following spaces are continuous: X̄ F,S ♭ and X̄ F,S with their Borel-Serre topologies, X̄ F,S with the Satake topology, and assuming S ♭ contains all archimedean places, X̄ F,S with the Satake topology. Proposition 3.4.18. Let W be a nonzero F -subspace of V . Then for the Borel-Serre topology, ♭ and for the Satake topology if S contains all archimedean places of F , the subset X̄ F,S (W ) is open ♭ in X̄ F,S . Part (1) of 3.4.17 was shown in §3.3 for the Borel-Serre topology, and the result for the Satake topology on X̄ F,S follows from it. The rest of 3.4.17 and 3.4.18 is easily proven. 23 3.5 Properties of X̄ F,S Let S be a nonempty finite set of places of F . 3.5.1. Let P and (Vi )−1≤i ≤m be as before. Fix a basis e (i ) of Vi /Vi −1 for each i . Set Y0 = (RS>0 ∪ {(0)v ∈S })m ⊂ (RS≥0 )m . ′ The maps ψP,v := (φP,v , φP,v ): X̄ F,v (P) → ZF,v (P)×Rm ≥0 of 3.4.3 and 3.4.6 for v ∈ S combine to give the map ψP,S : X̄ F,S (P) → ZF,S (P) × Y0 . 3.5.2. In addition to the usual topology on Y0 , we consider the weak topology on Y0 that is the product topology for the topology on RS>0 ∪ {(0)v ∈S } which extends the usual topology on RS>0 by taking the sets ¨ « (t v )v ∈S ∈ RS>0 Y | v ∈S t v ≤ c ∪ {(0)v ∈S } for c ∈ R>0 as a base of neighborhoods of (0)v ∈S . In the case that S consists of a single place, we have Y0 = Rm ≥0 , and the natural topology and the weak topology on Y0 coincide. Proposition 3.5.3. The map ψP,S of 3.5.1 induces a homeomorphism ∼ Pu (AF,S )\X̄ F,S (P) − → ZF,S (P) × Y0 for the Borel-Serre topology (resp., Satake topology) on X̄ F,S and the usual (resp., weak) topology on Y0 . This homeomorphism is equivariant for the action of P(AF,S ), with the action of P(AF,S ) on Y0 being that of 3.4.6. This has the following corollary, which is also the main step in the proof. Corollary 3.5.4. For any place v of F , the map Pu (Fv )\X̄ F,v (P) → ZF,v (P) × Rm ≥0 is a homeomorphism for both the Borel-Serre and Satake topologies on X̄ F,v . We state and prove preliminary results towards the proof of 3.5.3. 3.5.5. Fix a basis (e i )i of V and a parabolic subgroup P of PGLV which satisfies the condition in 3.3.2 for this basis. Let (Vi )−1≤i ≤m be the flag corresponding to P, and for each i , set c (i ) = dim(Vi ). We define two maps ξ, ξ⋆ : Pu (Fv ) × ZF,v (P) × Rm → X̄ F,v (P). ≥0 24 3.5.6. First, we define the map ξ. Set ∆(P) = {c (0), . . . , c (m − 1)}. Let ∆i = {j ∈ Z | c (i − 1) < j < c (i )} for 0 ≤ i ≤ m . We then clearly have ‚ {1, . . . , d − 1} = ∆(P) ∐ For 0 ≤ i ≤ m , let V (i ) = P c (i −1)<j ≤c (i ) m a Œ ∆i . i =0 F e j , so Vi = Vi −1 ⊕ V (i ) . We have ∆(P) Rd≥0−1 (P) = R≥0 × m Y ∼ × = Rm ≥0 ∆ R>0i i =0 m Y ∆ R>0i . i =0 Let B be the Borel subgroup of PGLV consisting of all upper triangular matrices for the basis (e i )i . Fix a place v of F . We consider two surjections B u (Fv ) × Rd≥0−1(P) ։ X̄ F,v (P) and B u (Fv ) × Rd≥0−1(P) ։ Pu (Fv ) × ZF,v (P) × Rm . ≥0 The first is induced by the surjection π̄v of 3.3.8. The second map is obtained as follows. For 0 ≤ i ≤ m , let B i be the image of B in PGLV (i ) under P → PGLV /V ∼ = PGL (i ) . Then B i is a Borel subgroup of PGL (i ) , and we have a canonical i bijection i −1 V V Pu (Fv ) × m Y i =0 ∼ B i ,u (Fv ) − → B u (Fv ). ∆ By 3.3.8, we have surjections B i ,u (Fv )× R>0i ։ X (Vi /Vi −1 )v for 0 ≤ i ≤ m . The second (continuous) surjection is then the composite m Y ‚ ∼ B u (Fv ) × Rd≥0−1(P) − → Pu (Fv ) × ‚ ։ Pu (Fv ) × i =0 Œ B i ,u (Fv ) × m Y i =0 ‚ Rm × ≥0 m Y Œ ∆ R>0i i =0 Œ X (Vi /Vi −1 )v × Rm = Pu (Fv ) × ZF,v (P) × Rm . ≥0 ≥0 Proposition 3.5.7. There is a unique surjective continuous map ξ: Pu (Fv ) × ZF,v (P) × Rm ։ X̄ F,v (P) ≥0 for the Borel-Serre topology on X̄ F,v (P) that is compatible with the surjections from B u (Fv ) × Rd≥0−1 (P) to these sets. This map induces a homeomorphism ∼ − → Pu (Fv )\X̄ F,v (P) ZF,v (P) × Rm ≥0 that restricts to a homeomorphism of ZF,v (P) × Rm >0 with Pu (Fv )\X v . 25 This follows from 3.3.8. 3.5.8. Next, we define the map ξ⋆ . For g ∈ Pu (Fv ), (µi )i ∈ (X (Vi /Vi −1)v )0≤i ≤m , and (t i )1≤i ≤m ∈ Rm ≥0 , we let ξ⋆ (g , (µi )i , (t i )i ) = g (P ′ , ν), where P ′ and ν are as in (1) and (2) below, respectively. (1) Let J = {c (i − 1) | 1 ≤ i ≤ m , t i = 0}. Write J = {c ′ (0), . . . , c ′ (m ′ − 1)} with c ′ (0) < · · · < c ′ (m ′ − 1). Let c ′ (−1) = 0 and c ′ (m ′ ) = d . For −1 ≤ i ≤ m ′ , let c ′ (i ) X ′ Vi = j =1 F e j ⊂ V. Let P ′ ⊃ P be the parabolic subgroup of PGLV corresponding to the flag (Vi ′ )i . (2) For 0 ≤ i ≤ m ′ , set J i = {j | c ′ (i − 1) < c (j ) ≤ c ′ (i )} ⊂ {1, . . . , m }. L We identify Vi ′ /Vi ′−1 with j ∈ J i V (j ) via the basis (e k )c ′ (i −1)<k ≤c ′(i ) . We define a norm ν̃i on Vi ′ /Vi ′−1 as follows. Let µ̃ j be the unique norm on V (j ) which belongs to µ j and satisfies P |µ̃ j : (e k )c (j −1)<k ≤c (j )| = 1. For x = j ∈ J i x j with x j ∈ V (j ) , set P 2 2 1/2 (r µ̃ j (x j ) )  P j ∈ J i j ν̃i (x ) = j ∈ J i r j µ̃ j (x j )   maxj ∈ J i (r j µ̃ j (x j )) where for j ∈ J i , we set Y rj = if v is real, if v is complex, if v is non-archimedean, t ℓ−1 . ℓ∈ J i ℓ<j Let νi ∈ X (Vi ′ /Vi ′−1 )v be the class of the norm ν̃i . We omit the proofs of the following two lemmas. Lemma 3.5.9. The composition ξ⋆ ψP,v Pu (Fv ) × ZF,v (P) × Rm − → X̄ F,v (P) −→ ZF,v (P) × Rm ≥0 ≥0 coincides with the canonical projection. Here, the definition of the second arrow uses the basis (e j mod Vi −1 )c (i −1)<j ≤c (i ) of Vi /Vi −1 . 26 Lemma 3.5.10. We have a commutative diagram Pu (Fv ) × ZF,v (P) × Rm ≥0 Pu (Fv ) × ZF,v (P) × Rm ≥0 ξ X̄ F,v (P) ξ⋆ X̄ F,v (P) in which the left vertical arrow is (u , µ, t ) 7→ (u , µ, t ′ ), for t ′ defined as follows. Let I i : X (Vi /Vi −1 )v → ∆ R>0i be the unique continuous map for which the composition Ii ∆ ∆ B i ,u (Fv ) × R>0i → X (Vi /Vi −1 )v − → R>0i is projection onto the second factor, and for j ∈ ∆i , let I i ,j : X (Vi /Vi −1)v → R>0 denote the composi∆ tion of I i with projection onto the factor of R>0i corresponding to j . Then Y Y j −c (i −2) c (i )−j I i −1,j (µi ) c (i −1)−c (i −2) · t i′ = t i · I i ,j (µi ) c (i )−c (i −1) j ∈∆i −1 j ∈∆i for 1 ≤ i ≤ m . 3.5.11. Proposition 3.5.3 is quickly reduced to Corollary 3.5.4, which now follows from 3.5.7, 3.5.9 and 3.5.10. 3.5.12. For two topologies T1 , T2 on a set Z , we use T1 ≥ T2 to denote that the identity map of Z is a continuous map from Z with T1 to Z with T2 , and T1 > T2 to denote that T1 ≥ T2 and T1 6= T2 . In other words, T1 ≥ T2 if T1 is finer than T2 and T1 > T2 if T1 is strictly finer than T2 . By 3.5.3, the map ψP,S : X̄ F,S (P) → ZF,S (P) × Y0 is continuous for the Borel-Serre topology on X̄ F,S and usual topology on Y0 . On X̄ F,S , we therefore have Borel-Serre topology ≥ Satake topology. ♭ ♭ Corollary 3.5.13. For any nonempty finite set S of places of F , the map φW,S : X̄ F,S (W ) → Z♭F,S (W ) ♭ of 3.4.4 is continuous for the Borel-Serre topology on X̄ F,S . If S contains all archimedean places of F , it is continuous for the Satake topology. Proof. The continuity for the Borel-Serre topology follows from the continuity of ψP,S , noting ♭ that the Borel-Serre topology on X̄ F,S is the quotient topology of the Borel-Serre topology on ♭ X̄ F,S . Suppose that S contains all archimedean places. As φW,S is ΓK ,(W ) -equivariant, and ΓK ,(W ) acts trivially on Z♭F,S (W ), the continuity for the Satake topology is reduced to the continuity for the Borel-Serre topology. Remark 3.5.14. We remark that the map φP,v : X̄ F,v (P) → ZF,v (P) of 3.4.3 need not be continu♭ ♭ ous for the topology on X̄ F,v as a subspace of X̄ v . Similarly, the map φW,v : X̄ F,v (W ) → X Wv of ♭ 3.4.4 need not be continuous for the subspace topology on X̄ F,v ⊂ X̄ v♭ . See 3.6.6 and 3.6.7. 27 3.5.15. We prove Proposition 3.4.14. Proof. Let α = (P, µ) ∈ X̄ F,S . Let U be a neighborhood of α for the Borel-Serre topology which is stable under the action of ΓK ,(P). By 3.5.12, it is sufficient to prove that there is a neighborhood W of α for the Satake topology such that W ⊂ U . Let (Vi )−1≤i ≤m be the flag corresponding to P, and let V (i ) be as before. Let Γ1 = ΓK ∩ Pu (F ), and let Γ0 be the subgroup of ΓK consisting of the elements that preserve V (i ) and act on V (i ) as a scalar for all i . Then Γ1 is a normal subgroup of ΓK ,(P) and Γ1 Γ0 is a subgroup of ΓK ,(P) of finite index. Let )m ( Y1 = (a v )v ∈S ∈ RS>0 | Y av = 1 , v ∈S and set s = ♯S. We have a surjective continuous map 1/s ′ (t , t ′ ) 7→ (t i t v,i )v,i . Rm × Y1 ։ Y0 , ≥0 m The composition Rm ≥0 × Y1 → Y0 → R≥0 , where the second arrow is (t v,i )v,i 7→ ( cides with projection onto the first coordinate. Q t ), v ∈S v,i i coin- Let Φ = Pu (AF,S ) × Y1 and Ψ = ZF,S (P) × Rm . ≥0 Consider the composite map (ξ⋆v )v ∈S f : Φ × Ψ → Pu (AF,S ) × ZF,S (P) × Y0 −−−→ X̄ F,S (P). The map f is Γ1 Γ0 -equivariant for the trivial action on Ψ and the following action on Φ: for (g , t ) ∈ Φ, γ1 ∈ Γ1 and γ0 ∈ Γ0 , we have γ1 γ0 · (g , t ) = (γ1 γ0 g γ−1 , γ0 t ), 0 where γ0 acts on Y1 via the embedding ΓK → P(AF,S ) and the actions of the P(Fv ) described in 3.4.6. The composition ψP,S f Φ×Ψ− → X̄ F,S (P) −→ Ψ coincides with the canonical projection. There exists a compact subset C of Φ such that Φ = Γ1 Γ0C for the above action of Γ1 Γ0 on Φ. Let β = (µ, 0) ∈ Ψ be the image of α under ψP,S . For x ∈ Φ, we have f (x , β ) = α. Hence, there is an open neighborhood U ′ (x ) of x in Φ and an open neighborhood U ′′ (x ) of β in Ψ such that U ′ (x ) × U ′′ (x ) ⊂ f −1 (U ). Since C is compact, there is a finite subset R of C such that 28 C⊂ S W = U ′ (x ). x ∈R ′′ ψ−1 P,S (U ) ⊂ Let U ′′ be the open subset T x ∈R U ′′ (x ) of Ψ, which contains β . The P-open set X̄ F,S (P) is by definition an open neighborhood of α in the Satake topology on X̄ F,S . We show that W ⊂ f −1 (U ). Since the map Φ × Ψ → X̄ F,S (P) is surjective, it is sufficient to prove that the inverse image Φ × U ′′ of W in Φ × Ψ is contained f −1 (U ). For this, we note that ! [ U ′ (x ) × U ′′ ⊂ Γ1 Γ0 f −1 (U ) = f −1 (U ), Φ × U ′′ = Γ1 Γ0C × U ′′ = Γ1 Γ0 x ∈R the last equality by the stability of U under the action of ΓK ,(P) ⊃ Γ1 Γ0 and the Γ1 Γ0 -equivariance of f . ♭ 3.5.16. In the case d = 2, the canonical surjection X̄ F,S → X̄ F,S is bijective. It is a homeomor- phism for the Borel-Serre topology. If S contains all archimedean places of F , it is a homeomorphism for the Satake topology by 3.4.14. 3.6 Comparison of the topologies ♭ When considering X̄ F,v , we assume that all places of F other than v are non-archimedean. ♭ 3.6.1. For X̄ F,v (resp., X̄ F,v ), we have introduced several topologies: the Borel-Serre topology, the Satake topology, and the subspace topology from X̄ v (resp., X̄ v♭ ), which we call the weak topology. We compare these topologies below; note that we clearly have Borel-Serre topology ≥ Satake topology and Borel-Serre topology ≥ weak topology. ♭ 3.6.2. For both X̄ F,v and X̄ F,v , the following hold: (1) Borel-Serre topology > Satake topology if d ≥ 2, (2) Satake topology > weak topology if d = 2, (3) Satake topology 6≥ weak topology if d > 2. We do not give full proofs of these statements. Instead, we describe some special cases that give clear pictures of the differences between these topologies. The general cases can be proven in a similar manner to these special cases. ♭ are equal, their Borel-Serre Recall from 3.5.16 that in the case d = 2, the sets X̄ F,v and X̄ F,v topologies coincide, and their Satake topologies coincide. 3.6.3. We describe the case d = 2 of 3.6.2(1). Take a basis (e i )i =1,2 of V . Consider the point α = (B, µ) of X̄ F,v , where B is the Borel subgroup of upper triangular matrices with respect to (e i )i , and µ is the unique element of ZF,v (B ) = X Fv e 1 × X Vv /Fv e 1 . 29 Let π̄v be the surjection of 3.3.8(2), and identify B u (Fv ) with Fv in the canonical manner. The images of the sets {(x , t ) ∈ Fv × R≥0 | t ≤ c } ⊂ B u (Fv ) × R≥0 in X̄ F,v (B ) for c ∈ R>0 form a base of neighborhoods of α for the Satake topology. Thus, while the image of the set {(x , t ) ∈ Fv × R≥0 | t < |x |−1 } is a neighborhood of α for Borel-Serre topology, it is not a neighborhood of α for the Satake topology. 3.6.4. We prove 3.6.2(2) in the case that v is non-archimedean. The proof in the archimedean ♭ case is similar. Since all boundary points of X̄ F,v = X̄ F,v are PGLV (F )-conjugate, to show 3.6.2(2), it is sufficient to consider any one boundary point. We consider α of 3.6.3 for a fixed basis (e i )i =1,2 of V . For x ∈ Fv and y ∈ R>0 , let µy ,x be the norm on Vv defined by µy ,x (a e 1 + b e 2) = max(|a − x b |, y |b |). The class of µy ,x is the image of (x , y −1 ) ∈ B u (Fv ) × R>0 . Any element of X v is the class of the norm µy ,x for some x , y . If we vary x ∈ F∞ and y ∈ R>0 , the classes of µy ,x in X̄ F,v converge under the Satake topology to the point α if and only if y approaches ∞. In X̄ v , the point α is the class of the semi-norm ν on Vv∗ defined by ν(a e 1∗ + b e 2∗ ) = |a |. By 2.1.7, µ∗y ,x = µy ,0 ◦ 1 −x 0 !!∗ 1 = µ∗y ,0 ◦ ! 1 0 x 1 , from which we see that µ∗y ,x (a e 1∗ + b e 2∗) = max(|a |, y −1 |x a + b |). Then µ∗y ,x is equivalent to the norm νy ,x on Vv∗ defined by νy ,x (a e 1∗ + b e 2∗ ) = min(1, y |x |−1 ) max(|a |, y −1 |x a + b |), and the classes of the νy ,x converge in X̄ v to the class of the semi-norm ν as y → ∞. Therefore, the Satake topology is finer than the weak topology. Now, the norm µ∗1,x is equivalent to the norm ν1,x on Vv∗ defined above, which for sufficiently large x satisfies ν1,x (a e 1 + b e 2) = max(|a /x |, |a + (b /x )|). 30 Thus, as |x | → ∞, the sequence µ1,x converges in X̄ v = X̄ v♭ to the class of the semi-norm ν. ♭ However, the sequence of classes of the norms µ1,x does not converge to α in X̄ F,v = X̄ F,v for the Satake topology, so the Satake topology is strictly finer than the weak topology. 3.6.5. We explain the case d = 3 of 3.6.2(3) in the non-archimedean case. Take a basis (e i )1≤i ≤3 of V . For y ∈ R>0 , let µy be the norm on Vv defined by µy (a e 1 + b e 2 + c e 3 ) = max(|a |, y |b |, y 2 |c |). For x ∈ Fv , consider the norm µy ◦ g x , where  1 0 0    . gx =  0 1 x   0 0 1 If we vary x ∈ F∞ and let y ∈ R>0 approach ∞, then the class of µy ◦g x in X v converges under the Satake topology to the class α ∈ X̄ F,v of the pair that is the Borel subgroup of upper triangular Q2 matrices and the unique element of i =0 X (Vi /Vi −1 )v , where (Vi )−1≤i ≤2 is the corresponding flag. ♭ The quotient topology on X̄ F,v of the Satake topology on X̄ F,v is finer than the Satake topology ♭ on X̄ F,v by 3.4.14 and 3.4.15. Thus, if the Satake topology is finer than the weak topology on ♭ X̄ F,v or X̄ F,v , then the composite µy ◦ g x should converge in X̄ v♭ to the class of the semi-norm ν on Vv∗ that satisfies ν(a e 1∗ + b e 2∗ + c e 3∗ ) = |a |. However, if y → ∞ and y −2 |x | → ∞, then the class of µy ◦ g x in X v converges in X̄ v♭ to the class of the semi-norm a e 1∗ + b e 2∗ + c e 3∗ 7→ |b |. In fact, by 2.1.7 we have (µy ◦ g x )∗ (a e 1∗ + b e 2∗ + c e 3∗ ) = µ∗y ◦ (g x∗ )−1 (a e 1∗ + b e 2∗ + c e 3∗ ) = max(|a |, y −1 |b |, y −2 |−b x + c |) = y −2 |x |νy ,x where νy ,x is the norm a e 1∗ + b e 2∗ + c e 3∗ 7→ max(y 2 |x |−1 |a |, y |x |−1 |b |, |−b + x −1 c |) on Vv∗ . The norms νy ,x converge to the semi-norm a e 1∗ + b e 2∗ + c e 3∗ 7→ |b |. ♭ ♭ 3.6.6. Let W be a nonzero subspace of V . We demonstrate that the map φW,v : X̄ F,v (W ) → X Wv of 3.4.4 given by restriction to Wv need not be continuous for the weak topology, even though by 3.5.13, it is continuous for the Borel-Serre topology and (if all places other than v are nonarchimedean) for the Satake topology. For example, suppose that v is non-archimedean and d = 3. Fix a basis (e i )1≤i ≤3 of V , and let W = F e 1 + F e 2 . Let µ be the class of the norm a e 1 + b e 2 7→ max(|a |, |b |) 31 ♭ on Wv , and consider the element (W, µ) ∈ X̄ F,v . For x ∈ Fv and ε ∈ R>0 , let µx ,ε ∈ X v be the class of the norm a e 1 + b e 2 + c e 3 7→ max(|a |, |b |, ε−1 |c + b x |) on Vv . Then µx∗ ,ε is the class of the norm a e 1∗ + b e 2∗ + c e 3∗ 7→ max(|a |, |b − x c |, ε|c |) on Vv∗ . When x → 0 and ε → 0, the last norm converges to the semi-norm a e 1∗ + b e 2∗ + c e 3∗ 7→ max(|a |, |b |) on Vv∗ , and this implies that µx ,ε converges to (W, µ) for the weak topology. However, the restriction of µx ,ε to Wv is the class of the norm a e 1 + b e 2 7→ max(|a |, |b |, ε−1 |x ||b |). If x → 0 and ε = r −1 |x | → 0 for a fixed r > 1, then the latter norms converge to the norm a e 1 + b e 2 7→ max(|a |, r |b |), the class of which does not coincide with µ. 3.6.7. Let P be a parabolic subgroup of PGLV (F ). We demonstrate that the map φP,v : X̄ F,v (P) → ZF,v (P) of 3.4.3 is not necessarily continuous for the weak topology, though by 3.5.4, it is con- tinuous for the Borel-Serre topology and for the Satake topology. Let d = 3 and W be as in 3.6.6, and let P be the parabolic subgroup of PGLV corresponding to the flag 0 = V−1 ⊂ V0 = W ⊂ V1 = V. ♭ In this case, the canonical map X̄ F,v (P) → X̄ F,v (W ) is a homeomorphism for the weak topology on both spaces. It is also a homeomorphism for the Borel-Serre topology, and for the Satake topology if all places other than v are non-archimedean. Since ZF,v (P) = X (V ) × X (V /V ) ∼ = XW , 0 v 0 v v the argument of 3.6.6 shows that φP,v is not continuous for the weak topology. ♭ 3.6.8. For d ≥ 3, the Satake topology on X̄ F,v does not coincide with the quotient topology for the Satake topology on X̄ F,v , which is strictly finer. This is explained in 4.4.12. 3.7 Relations with Borel-Serre spaces and reductive Borel-Serre spaces 3.7.1. In this subsection, we describe the relationship between our work and the theory of ♯ Borel-Serre and reductive Borel-Serre spaces (see Proposition 3.7.4). We also show that X̄ F,v is not Hausdorff if v is a non-archimedean place. 32 3.7.2. Let G be a semisimple algebraic group over Q. We recall the definitions of the BorelSerre and reductive Borel-Serre spaces associated to G from [3] and [26, p. 190], respectively. Let Y be the space of all maximal compact subgroups of G (R). Recall from [3, Proposition 1.6] that for K ∈ Y , the Cartan involution θK of G R := R⊗Q G corresponding to K is the unique homomorphism G R → G R such that K = {g ∈ G (R) | θK (g ) = g }. Let P be a parabolic subgroup of G , let S P be the largest Q-split torus in the center of P/Pu , and let A P be the connected component of the topological group S P (R) containing the origin. We have AP ∼ = Rr>0 ⊂ S P (R) ∼ = (R× )r for some integer r . We define an action of A P on Y as follows (see [3, Section 3]). For K ∈ Y , we have a unique subtorus S P,K of PR = R ⊗Q P over R such that the projection P → P/Pu induces an isomorphism ∼ S P,K − → (S P )R := R ⊗Q S P and such that the Cartan involution θK : G R → G R of K satisfies θK (t ) = t −1 for all t ∈ S P,K (R). For t ∈ A P , let t K ∈ S P,K (R) be the inverse image of t . Then A P acts on Y by AP × Y → Y , (t , K ) 7→ t K K t K−1 . The Borel-Serre space is the set of pairs (P,Z ) such that P is a parabolic subgroup of G and Z is an A P -orbit in Y . The reductive Borel-Serre space is the quotient of the Borel-Serre space by the equivalence relation under which two elements (P,Z ) and (P ′ ,Z ′ ) are equivalent if (P ′ ,Z ′ ) = g (P,Z ) (that is, P = P ′ and Z ′ = g Z ) for some g ∈ Pu (R). 3.7.3. Now assume that F = Q and G = PGLV . Let v be the archimedean place of Q. We have a bijection between X v and the set Y of all maximal compact subgroups of G (R), whereby an element of X v corresponds to its isotropy group in G (R), which is a maximal compact subgroup. Suppose that K ∈ Y corresponds to µ ∈ X v , with µ the class of a norm that in turn cor- responds to a positive definite symmetric bilinear form ( , ) on Vv . The Cartan involution θK : G R → G R is induced by the unique homomorphism θK : GLVv → GLVv satisfying (g x , θK (g )y ) = (x , y ) for all g ∈ GLV (R) and x , y ∈ Vv . For a parabolic subgroup P of G corresponding to a flag (Vi )−1≤i ≤m , we have Œ ‚ m Y Gm,Q /Gm,Q , SP = i =0 33 where the i th term in the product is the group of scalars in GLVi /Vi −1 , and where the last Gm,Q is embedded diagonally in the product. The above description of θK shows that S P,K is the lifting of (S P )R to PR obtained through the orthogonal direct sum decomposition Vv ∼ = m M i =0 (Vi /Vi −1 )v with respect to ( , ). ♯ Proposition 3.7.4. If v is the archimedean place of Q, then X̄ Q,v (resp., X̄ Q,v ) is the Borel-Serre space (resp., reductive Borel-Serre space) associated to PGLV . ♯ Proof. Denote the Borel-Serre space by (X̄ Q,v )′ in this proof. We define a canonical map ♯ ♯ X̄ Q,v → (X̄ Q,v )′ , (P, µ, s ) 7→ (P,Z ), where Z is the subset of Y corresponding to the following subset Z ′ of X v . Let (Vi )−1≤i ≤m be the flag corresponding to P. Recall that s is an isomorphism s: m M i =0 ∼ (Vi /Vi −1 )v − → Vv . Then Z ′ is the subset of X v consisting of classes of the norms µ̃(s ) : x 7→ m X !1/2 µ̃i (s −1 (x )i )2 i =0 on Vv , where s −1 (x )i ∈ (Vi /Vi −1 )v denotes the i th component of s −1 (x ) for x ∈ Vv , and µ̃ = (µ̃i )0≤i ≤m ranges over all families of norms µ̃i on (Vi /Vi −1 )v with class equal to µi . It follows from the description of S P,K in 3.7.3 that Z is an A P -orbit. For a parabolic subgroup P of G , let ♯ ♯ (X̄ Q,v )′ (P) = {(Q,Z ) ∈ (X̄ Q,v )′ | Q ⊃ P}. ♯ ♯ By [3, 7.1], the subset (X̄ Q,v )′ (P) is open in (X Q,v )′ . Take a basis of V , and let B denote the Borel subgroup of PGLV of upper-triangular matrices for this basis. By 3.3.8(3), we have a homeomorphism ∼ ♯ → X̄ Q,v (B ). B u (R) × Rd≥0−1 − It follows from [3, 5.4] that the composition ♯ B u (R) × Rd≥0−1 → (X̄ Q,v )′ (B ) 34 induced by the above map ♯ ♯ X̄ Q,v (B ) → (X̄ Q,v )′ (B ), (P, µ, s ) 7→ (P,Z ) ♯ ♯ is also a homeomorphism. This proves that the map X̄ Q,v → (X̄ Q,v )′ restricts to a homeomor∼ ♯ ♯ ♯ ♯ phism X̄ Q,v (B ) − → (X̄ Q,v )′ (B ). Therefore, X̄ Q,v → (X̄ Q,v )′ is a homeomorphism as well. It then follows directly from the definitions that the reductive Borel-Serre space is identified with X̄ Q,v . 3.7.5. Suppose that F is a number field, let S be the set of all archimedean places of F , and let G be the Weil restriction ResF /Q PGLV , which is a semisimple algebraic group over Q. Then Y is identified with X F,S , and X̄ F,S is related to the reductive Borel-Serre space associated to G but does not always coincide with it. We explain this below. ♯ ′ Let (X̄ F,S )′ and X̄ F,S be the Borel-Serre space and the reductive Borel-Serre space associQ ♯ ♯ ated to G , respectively. Let X̄ F,S be the subspace of v ∈S X̄ F,v consisting of all elements (x v )v ∈S such that the parabolic subgroup of G associated to x v is independent of v . Then by similar arguments to the case F = Q, we see that Y is canonically homeomorphic to X F,S and this homeomorphism extends uniquely to surjective continuous maps ♯ ♯ (X̄ F,S )′ → X̄ F,S , ′ X̄ F,S → X̄ F,S . However, these maps are not bijective unless F is Q or imaginary quadratic. We illustrate the differences between the spaces in the case that F is a real quadratic field and d = 2. Fix a basis (e i )i =1,2 of V . Let P̃ be the Borel subgroup of upper triangular matrices in PGLV for this basis, and let P be the Borel subgroup Res F /Q P̃ of G . Then P/Pu ∼ = ResF /Q Gm ,F and S P = Gm ,Q ⊂ P/Pu . We have the natural identifications Y = X F,S = H × H. For a ∈ R>0 , the set Z a := {(y i , a y i ) ∈ H × H | y ∈ R>0 } is an A P -orbit. If a 6= b , the images of (P,Z a ) and (P,Z b ) in (X̄ F,S )′ do not coincide. On the other ♯ hand, both the images of (P,Z a ) and (P,Z b ) in X̄ F,S coincide with (x v )v ∈S , where x v = (P, µv , s v ) with µv the unique element of X Fv e 1 × X Vv /Fv e 1 and s v the splitting given by e 2. ♯ In the case that v is non-archimedean, the space X̄ F,v is not good in the following sense. ♯ Proposition 3.7.6. If v is non-archimedean, then X̄ F,v is not Hausdorff. Proof. Fix a ,b ∈ B u (Fv ) with a 6= b , for a Borel subgroup B of PGLV . When t ∈ Rd>0−1 is sufficiently near to 0 = (0, . . . , 0), the images of (a , t ) and (b, t ) in X v coincide by 3.3.8(4) applied to B u (Fv ) × Rd>0−1 → X v . We denote this element of X v by c (t ). The images f (a ) of (a , 0) and f (b ) ♯ of (b, 0) in X̄ F,v are different. However, c (t ) converges to both f (a ) and f (b ) as t tends to 0. ♯ Thus, X̄ F,v is not Hausdorff. 35 3.7.7. Let F be a number field, S its set of archimedean places, and G = ResF /Q PGLV , as in 3.7.5. Then X̄ F,S may be identified with the maximal Satake space for G of [23]. Its Satake topology was considered by Satake (see also [2, III.3]), and its Borel-Serre topology was con♭ sidered by Zucker [27] (see also [14, 2.5]). The space X̄ F,S is also a Satake space corresponding to the standard projective representation of G on V viewed as a Q-vector space. 4 Quotients by S-arithmetic groups As in §3, fix a global field F and a finite-dimensional vector space V over F . 4.1 Results on S-arithmetic quotients 4.1.1. Fix a nonempty finite set S 1 of places of F which contains all archimedean places of F , fix a finite set S 2 of places of F which is disjoint from S 1 , and let S = S 1 ∪ S 2 . 4.1.2. In the following, we take X̄ to be one of the following two spaces: (i) X̄ := X̄ F,S 1 , ♭ (ii) X̄ := X̄ F,S . 1 We endow X̄ with either the Borel-Serre or the Satake topology. 4.1.3. Let G = PGLV , and let K be a compact open subgroup of G (ASF ), with ASF as in 3.4.13. We consider the two situations in which (X, X̄) is taken to be one of the following pairs of spaces (for either choice of X̄ ): (I) X := X S × G (ASF )/K ⊂ X̄ := X̄ × X S 2 × G (ASF )/K , (II) X := X S ⊂ X̄ := X̄ × X S 2 . We now come to the main result of this paper. Theorem 4.1.4. Let the situations and notation be as in 4.1.1–4.1.3. (1) Assume we are in situation (I). Let Γ be a subgroup of G (F ). Then the quotient space Γ\X̄ is Hausdorff. It is compact if Γ = G (F ). (2) Assume we are in situation (II). Let ΓK ⊂ G (F ) be the inverse image of K under the canon- ical map G (F ) → G (ASF ), and let Γ be a subgroup of ΓK . Then the quotient space Γ\X̄ is Hausdorff. It is compact if Γ is of finite index in ΓK . ♭ 4.1.5. The case Γ = {1} of Theorem 4.1.4 shows that X̄ F,S and X̄ F,S are Hausdorff. 36 4.1.6. Let OS be the subring of F consisting of all elements which are integral outside S. Take an OS -lattice L in V . Then PGL L (OS ) coincides with ΓK for the compact open subgroup K = Q PGLL (Ov ) of G (ASF ). Hence Theorem 1.6 of the introduction follows from Theorem 4.1.4. v∈ /S 4.1.7. In the case that F is a number field and S (resp., S 1 ) is the set of all archimedean places of F , Theorem 4.1.4 in situation (II) is a special case of results of Satake [23] (resp., of Ji, Murty, Saper, and Scherk [14, Proposition 4.2]). 4.1.8. If in Theorem 4.1.4 we take Γ = G (F ) in part (1), or Γ of finite index in ΓK in part (2), then the Borel-Serre and Satake topologies on X̄ induce the same topology on the quotient space Γ\X̄. This can be proved directly, but it also follows from the compact Hausdorff property. 4.1.9. We show that some modifications of Theorem 4.1.4 are not good. Consider the case F = Q, S = {p, ∞} for a prime number p , and V = Q2 , and consider the S-arithmetic group PGL2 (Z[ p1 ]). Note that PGL2 (Z[ p1 ])\(X̄ Q,∞ × X p ) is compact Hausdorff, as is well known (and follows from Theorem 4.1.4). We show that some similar spaces are not Hausdorff. That is, we prove the following statements: (1) PGL2 (Z[ p1 ])\(X̄ Q,p × X ∞ ) is not Hausdorff. (2) PGL2 (Z[ p1 ])\(X̄ Q,∞ × PGL2 (Qp )) is not Hausdorff. (3) PGL2 (Q)\(X̄ Q,∞ × PGL2 (A∞ Q )) is not Hausdorff. Statement (1) shows that it is important to assume in 4.1.4 that S 1 , not only S, contains all archimedean places. Statement (3) shows that it is important to take the quotient G (ASF )/K in situation (I) of 4.1.3. Our proofs of these statements rely on the facts that the quotient spaces Z[ p1 ]\R, Z[ p1 ]\Qp , and Q\A∞ Q are not Hausdorff. Proof of statements (1)–(3). For an element x of a ring R, let ! 1 x gx = ∈ PGL2 (R). 0 1 In (1), for b ∈ R, let h b be the point i + b of the upper half plane H = X ∞ . In (2), for b ∈ Qp , ∞ let h b = g b ∈ PGL2 (Qp ). In (3), for b ∈ A∞ Q , let h b = g b ∈ PGL2 (AQ ). In (1) (resp., (2) and (3)), let ∞ ∈ X̄ Q,p (resp., X̄ Q,∞ ) be the boundary point corresponding to the the Borel subgroup of upper triangular matrices. In (1) (resp., (2), resp., (3)), take an element b of R (resp., Qp , resp., A∞ Q ) which does not belong to Z[ p1 ] (resp., Z[ p1 ], resp., Q). Then the images of (∞, h 0 ) and (∞, h b ) in the quotient space 37 are different, but they are not separated. Indeed, in (1) and (2) (resp., (3)), some sequence of elements x of Z[ p1 ] (resp., Q) will converge to b , in which case g x (∞, h 0 ) converges to (∞, h b ) since g x ∞ = ∞. 4.2 Review of reduction theory We review important results in the reduction theory of algebraic groups: 4.2.2, 4.2.4, and a variant 4.2.6 of 4.2.2. More details may be found in the work of Borel [1] and Godement [8] in the number field case and Harder [12, 13] in the function field case. Fix a basis (e i )1≤i ≤d of V . Let B be the Borel subgroup of G = PGLV consisting of all upper triangular matrices for this basis. Let S be a nonempty finite set of places of F containing all archimedean places. 4.2.1. For b = (b v ) ∈ A×F , set |b | = Q v |b v |. Let A v be as in 3.3.4. We let a ∈ image of a diagonal matrix diag(a 1 , . . . , a d ) in GLd (AF ). The ratios a i a −1 i +1 Q v A v denote the are independent of the choice. For c ∈ R>0 , we let B (c ) = B u (AF )A(c ), where ( ) Y A(c ) = a ∈ A v ∩ PGLd (AF ) | |a i a −1 | ≥ c for all 1 ≤ i ≤ d − 1 . i +1 v Let K 0 = Q v K v0 < G (AF ), where K v0 is identified via (e i )i with the standard maximal com- pact subgroup of PGLd (Fv ) of 3.3.4. Note that B u (Fv )A v K v0 = B (Fv )K v0 = G (Fv ) for all v . We recall the following known result in reduction theory: see [8, Theorem 7] and [12, Satz 2.1.1]. Lemma 4.2.2. For sufficiently small c ∈ R>0 , one has G (AF ) = G (F )B (c )K 0 . 4.2.3. Let the notation be as in 4.2.1. For a subset I of {1, . . . , d − 1}, let PI be the parabolic P subgroup of G corresponding to the flag consisting of 0, the F -subspaces 1≤j ≤i F e j for i ∈ I , and V . Hence PI ⊃ B for all I , with P∅ = G and P{1,...,d −1} = B . For c ′ ∈ R>0 , let B I (c , c ′ ) = B u (AF )A I (c , c ′ ), where A I (c , c ′ ) = {a ∈ A(c ) | |a i a −1 | ≥ c ′ for all i ∈ I }. i +1 Note that B I (c , c ′ ) = B (c ) if c ≥ c ′ . The following is also known [12, Satz 2.1.2] (see also [8, Lemma 3]): Lemma 4.2.4. Fix c ∈ R>0 and a subset I of {1, . . . , d − 1}. Then there exists c ′ ∈ R>0 such that {γ ∈ G (F ) | B I (c , c ′ )K 0 ∩ γ−1 B (c )K 0 6= ∅} ⊂ PI (F ). 38 4.2.5. We will use the following variant of 4.2.3. Q Let A S = v ∈S A v . For c ∈ R>0 , let A(c )S = A S ∩ A(c ) and B (c )S = B u (AF,S )A(c )S . For c 1 , c 2 ∈ R>0 , set A(c 1 , c 2 )S = {a ∈ A S | for all v ∈ S and 1 ≤ i ≤ d − 1, |a v,i a −1 | ≥ c 1 and |a v,i a −1 | ≥ c 2 |a w,i a −1 | for all w ∈ S}. v,i +1 v,i +1 w,i +1 Note that A(c 1 , c 2 )S is empty if c 2 > 1. For a compact subset C of B u (AF,S ), we then set B (C ; c 1, c 2 )S = C · A(c 1 , c 2 )S . Let DS = Q v ∈S D v , where D v = K v0 < G (Fv ) if v is archimedean, and D v < G (Fv ) is identified with S d Iw(Ov ) < PGLd (Fv ) using the basis (e i )i otherwise. Here, S d is the symmetric group of degree d and Iw(Ov ) is the Iwahori subgroup of PGLd (Ov ), as in 3.3.6. Lemma 4.2.6. Let K be a compact open subgroup of G (ASF ), let ΓK be the inverse image of K under G (F ) → G (ASF ), and let Γ be a subgroup of ΓK of finite index. Then there exist c 1 , c 2 ,C as above and a finite subset R of G (F ) such that G (AF,S ) = ΓR · B (C ; c 1 , c 2 )S DS . Proof. This can be deduced from 4.2.2 by standard arguments in the following manner. By the Iwasawa decomposition 3.3.4, we have G (ASF ) = B (ASF )K 0,S where K 0,S is the non-S-component of K 0 . Choose a set E of representatives in B (ASF ) of the finite set B (F )\B (ASF )/(B (ASF ) ∩ K 0,S ). Let D 0 = DS × K 0,S , and note that since A S ∩DS = 1, we can (by the Bruhat decomposition 3.3.6) replace K 0 by D 0 in Lemma 4.2.2. Using the facts that E is finite, |a | = 1 for all a ∈ F × , and D 0 is compact, we then have that there exists c ∈ R>0 such that G (AF ) = G (F )(B (c )S × E )D 0. For any finite subset R of G (F ) consisting of one element from each of those sets G (F )∩K 0,S e −1 with e ∈ E that are nonempty, we obtain from this that G (AF,S ) = ΓK R · B (c )S DS . 39 As ΓK is a finite union of right Γ-cosets, we may enlarge R and replace ΓK by Γ. Finally, we can replace B (c )S by C · A(c )S for some C by the compactness of the image of B u (AF,S ) → Γ\G (AF,S )/DS and then by B (C ; c 1 , c 2 )S for some c 1 , c 2 ∈ R>0 by the compactness of the cokernel of Γ ∩ B (AF,S ) → (B /B u )(AF,S )1 , where (B /B u )(AF,S )1 denotes the kernel of the homomorphism (B /B u )(AF,S ) → Rd>0−1 , Y a v,i a B u (AF,S ) 7→ a v,i +1 v ∈S ! . 1≤i ≤d −1 4.3 X̄ F,S and reduction theory 4.3.1. Let S be a nonempty finite set of places of F containing all archimedean places. We consider X̄ F,S . From the results 4.2.6 and 4.2.4 of reduction theory, we will deduce results 4.3.4 and 4.3.10 on X̄ F,S , respectively. We will also discuss other properties of X̄ F,S related to reduction theory. Let G , (e i )i , and B be as in §4.2. For c 1 , c 2 ∈ R>0 with c 2 ≥ 1, we define a subset T(c 1 , c 2 ) of (RS≥0 )d −1 by T(c 1 , c 2 ) = {t ∈ (RS≥0 )d −1 | t v,i ≤ c 1 , t v,i ≤ c 2 t w,i for all v, w ∈ S and 1 ≤ i ≤ d − 1}. Let Y0 = (RS>0 ∪ {(0)v ∈S })d −1 as in 3.5.1 (for the parabolic B ), and note that T(c 1 , c 2 ) ⊂ Y0 . Define the subset S(c 1 , c 2 ) of X̄ F,S (B ) as the image of B u (AF,S ) × T(c 1 , c 2 ) under the map πS = (πv )v ∈S : B u (AF,S ) × Y0 → X̄ F,S (B ), with πv as in 3.3.3. For a compact subset C of B u (AF,S ), we let S(C ; c 1 , c 2 ) ⊂ S(c 1 , c 2 ) denote the image of C × T(c 1 , c 2 ) under πS . 4.3.2. We give an example of the sets of 4.3.1. Example. Consider the case that F = Q, the set S contains only the real place, and d = 2, as in §3.2. Fix a basis (e i )1≤i ≤2 of V . Identify B u (R) with R in the natural manner. We have S(C ; c 1 , c 2 ) = {x + y i ∈ H | x ∈ C , y ≥ c 1−1 } ∪ {∞}, which is contained in S(c 1 , c 2 ) = {x + y i ∈ H | x ∈ R, y ≥ c 1−1 } ∪ {∞}. 40 4.3.3. Fix a compact open subgroup K of G (ASF ), and let ΓK ⊂ G (F ) be the inverse image of K under G (F ) → G (ASF ). Proposition 4.3.4. Let Γ be a subgroup of ΓK of finite index. Then there exist c 1 , c 2 ,C as in 4.3.1 and a finite subset R of G (F ) such that X̄ F,S = ΓR · S(C ; c 1 , c 2 ). Proof. It suffices to prove the weaker statement that there are c 1 , c 2 ,C and R such that X S = ΓR · (X S ∩ S(C ; c 1 , c 2 )). Indeed, we claim that the proposition follows from this weaker statement for the spaces in the Q product v ∈S X (Vi /Vi −1)v , where PI is as in 4.2.3 for a subset I of {1, . . . , d − 1} and (Vi )−1≤i ≤m is the corresponding flag. To see this, first note that there is a finite subset R ′ of G (F ) such that every parabolic subgroup of G has the form γPI γ−1 for some I and γ ∈ ΓR ′ . It then suffices to consider a = (P, µ) ∈ X̄ F,S , where P = PI for some I , and µ ∈ ZF,S (P). We use the notation of 3.5.6 and 3.5.1. By Proposition 3.5.7, the set X̄ F,S (P) ∩ S(C ; c 1 , c 2 ) is the image under ξ of the image of C × T(c 1 , c 2 ) in Pu (AF,S ) × ZF,S (P) × Y0 . Note that a has image (1, µ, 0) in the latter set (for 1 the identity matrix of Pu (AF,S )), and ξ(1, µ, 0) = a . Since the projection of T(c 1 , c 2 ) (resp., C ) to (RS>0 )∆i (resp., B i ,u (AF,S )) is the analogous set for c 1 and c 2 (resp., a compact subset), the claim follows. For v ∈ S, we define subsets Q v and Q v′ of X v as follows. If v is archimedean, let Q v = Q v′ be the one point set consisting of the element of X v given by the basis (e i )i and (ri )i with ri = 1 for all i . If v is non-archimedean, let Q v (resp., Q v′ ) be the subset of X v consisting of elements given by (e i )i and (ri )i such that 1 = r1 ≤ · · · ≤ rd ≤ qv (resp., r1 = 1 and 1 ≤ ri ≤ qv for 1 ≤ i ≤ d ). Then X v = G (Fv )Q v for each v ∈ S. Hence by 4.2.6, there exist c 1′ , c 2′ ,C as in 4.3.1 and a finite subset R of G (F ) such that X S = ΓR · B (C ; c 1′ , c 2′ )S · DSQ S , where Q S = Q v ∈S Qv . We have DSQ S = Q S′ for Q S′ = Q v ∈S Q v′ , noting for archimedean (resp., non-archimedean) v that K v0 (resp., Iw(Ov )) stabilizes all elements of Q v . We have B (C ; c 1′ , c 2′ )SQ S′ ⊂ S(C ; c 1 , c 2 ), where c 1 = max{qv | v ∈ S f }(c 1′ )−1 and c 2 = max{qv2 | v ∈ S f }(c 2′ )−1 , with S f the set of all non-archimedean places in S (and taking the maxima to be 1 if S f = ∅). 41 ′ 4.3.5. For v ∈ S and 1 ≤ i ≤ d −1, let t v,i : S(c 1 , c 2 ) → R≥0 be the map induced by φ B,v : X̄ F,v (B ) → Rd≥0−1 (see 3.4.6) and the i th projection Rd≥0−1 → R≥0 . Note that t v,i is continuous. 4.3.6. Fix a subset I of {1, . . . , d − 1}, and let PI be the parabolic subgroup of G defined in 4.2.3. For c 1 , c 2 , c 3 ∈ R>0 , let SI (c 1 , c 2 , c 3 ) = {x ∈ S(c 1 , c 2 ) | min{t v,i (x ) | v ∈ S} ≤ c 3 for each i ∈ I }. 4.3.7. For an element a ∈ X̄ F,S , we define the parabolic type of a to be the subset {dim(Vi ) | 0 ≤ i ≤ m − 1} of {1, . . . , d − 1}, where (Vi )−1≤i ≤m is the flag corresponding to the parabolic subgroup of G associated to a . Lemma 4.3.8. Let a ∈ X̄ F,S (B ), and let J be the parabolic type of a . Then the parabolic subgroup of G associated to a is PJ . This is easily proved. 4.3.9. In the following, we will often consider subsets of G (F ) of the form R 1 ΓK R 2 , ΓK R, or RΓK , where R 1 , R 2 , R are finite subsets of G (F ). These three types of cosets are essentially the same thing when we vary K . For finite subsets R 1, R 2 of G (F ), we have R 1 ΓK R 2 = R ′ ΓK ′ = ΓK ′′ R ′′ for some compact open subgroups K ′ and K ′′ of G (ASF ) contained in K and finite subsets R ′ and R ′′ of G (F ). Proposition 4.3.10. Given c 1 ∈ R>0 and finite subsets R 1 , R 2 of G (F ), there exists c 3 ∈ R>0 such that for all c 2 ∈ R>0 we have {γ ∈ R 1 ΓK R 2 | γSI (c 1 , c 2 , c 3 ) ∩ S(c 1, c 2 ) 6= ∅} ⊂ PI (F ). Proof. First we prove the weaker version that c 3 exists if the condition on γ ∈ R 1 ΓK R 2 is replaced by γSI (c 1 , c 2 , c 3 ) ∩ S(c 1 , c 2 ) ∩ X S 6= ∅. Let Q v′ for v ∈ S and Q S′ be as in the proof of 4.3.4. Claim 1. If c 1′ ∈ R>0 is sufficiently small (independent of c 2 ), then we have X S ∩ S(c 1 , c 2 ) ⊂ B (c 1′ )SQ S′ . Proof of Claim 1. Any x ∈ X S ∩ S(c 1 , c 2 ) satisfies t v,i (x ) ≤ c 1 for 1 ≤ i ≤ d − 1. Moreover, if Q ′ −1 for all such i , then x ∈ B (c ′ ) Q ′ . The claim v ∈S t v,i (x ) is sufficiently small relative to (c 1 ) 1 S S follows. 42 Let C v denote the compact set C v = {g ∈ G (Fv ) | gQ v′ ∩Q v′ 6= ∅}. If v is archimedean, then C v is the maximal compact open subgroup K v0 of 4.2.1. Set C S = Q S v ∈S C v . We use the decomposition G (A F ) = G (A F,S ) × G (A F ) to write elements of G (A F ) as pairs. Claim 2. Fix c 1′ ∈ R>0 . The subset B (c 1′ )S C S × R 1 K R 2 of G (AF ) is contained in B (c )K 0 for sufficiently small c ∈ R>0 . Proof of Claim 2. This follows from the compactness of the C v for v ∈ S and the Iwasawa decomposition G (AF ) = B (AF )K 0 . Claim 3. Let c 1′ be as in Claim 1, and let c ≤ c 1′ . Let c ′ ∈ R>0 . If c 3 ∈ R>0 is sufficiently small (independent of c 2 ), we have X S ∩ SI (c 1 , c 2 , c 3 ) ⊂ B I (c , c ′ )SQ S′ , where B I (c , c ′ )S = B (AF,S ) ∩ B I (c , c ′ ). Proof of Claim 3. An element x ∈ B (c )SQ S′ lies in B I (c , c ′ )SQ S′ if Q t (x ) ≤ (c v ∈S v,i ′ )−1 for all i ∈ I . An element x ∈ X S ∩ S(c 1 , c 2 ) lies in X S ∩ SI (c 1 , c 2 , c 3 ) if min{t v,i (x ) | v ∈ S} ≤ c 3 for all i ∈ I . In this case, x will lie in B I (c , c ′ )SQ S′ if c 3 ≤ (c ′ )−1 c 11−s , with s = ♯S. Let c 1′ be as in Claim 1, take c of Claim 2 for this c 1′ such that c ≤ c 1′ , and let c ′ ∈ R>0 . Take c 3 satisfying the condition of Claim 3 for these c 1′ , c , and c ′ . Claim 4. If X S ∩ SI (c 1 , c 2 , c 3 ) ∩ γ−1 S(c 1 , c 2 ) is nonempty for some γ ∈ R 1ΓR 2 ⊂ G (F ), then B I (c , c ′ ) ∩ γ−1 B (c )K 0 contains an element of G (AF,S ) × {1}. Proof of Claim 4. By Claim 3, any x ∈ X S ∩ SI (c 1 , c 2 , c 3 ) ∩ γ−1 S(c 1 , c 2 ) lies in gQ S′ for some g ∈ B I (c , c ′ )S . By Claim 1, we have γx ∈ g ′Q S′ for some g ′ ∈ B (c 1′ )S . Since γx ∈ γgQ S′ ∩ g ′Q S′ , we have (g ′ )−1 γg ∈ C S . Hence γg ∈ B (c 1′ )S C S , and therefore γ(g , 1) = (γg , γ) ∈ B (c )K 0 by Claim 2. We prove the weaker version of 4.3.10: let x ∈ X S ∩ SI (c 1 , c 2 , c 3 ) ∩ γ−1 S(c 1 , c 2 ) for some γ ∈ R 1 ΓR 2 . Then by Claim 4 and Lemma 4.2.4, with c ′ satisfying the condition of 4.2.4 for the given c , we have γ ∈ PI (F ). We next reduce the proposition to the weaker version, beginning with the following. Claim 5. Let c 1 , c 2 ∈ R>0 . If γ ∈ G (F ) and x ∈ S(c 1 , c 2 ) ∩ γ−1S(c 1 , c 2 ), then γ ∈ PJ (F ), where J is the parabolic type of x . 43 Proof of Claim 5. By Lemma 4.3.8, the parabolic subgroup associated to x is PJ and that associated to γx is PJ . Hence γPJ γ−1 = PJ . Since a parabolic subgroup coincides with its normalizer, we have γ ∈ PJ (F ). Fix J ⊂ {1, . . . , d − 1}, ξ ∈ R 1, and η ∈ R 2 . Claim 6. There exists c 3 ∈ R>0 such that if γ ∈ ΓK and x ∈ SI (c 1 , c 2 , c 3 ) ∩ (ξγη)−1S(c 1 , c 2 ) is of parabolic type J , then ξγη ∈ PI (F ). Proof of Claim 6. Let (Vi )−1≤i ≤m be the flag corresponding to PJ . Suppose that we have γ0 ∈ ΓK and x 0 ∈ S(c 1 , c 2 )∩(ξγ0η)−1 S(c 1 , c 2 ) of parabolic type J . By Claim 5, we have ξγ0 η, ξγη ∈ PJ (F ). Hence ξγη = ξ(γγ−1 )ξ−1 ξγ0η ∈ ΓK ′ η′ , 0 where K ′ is the compact open subgroup ξK ξ−1 ∩ PJ (ASF ) of PJ (ASF ), and η′ = ξγ0 η ∈ PJ (F ). The claim follows from the weaker version of the proposition in which V is replaced by Vi /Vi −1 (for 0 ≤ i ≤ m ), the group G is replaced by PGLVi /Vi −1 , the compact open subgroup K is replaced by the image of ξK ξ−1 ∩ PJ (ASF ) in PGLVi /Vi −1 (ASF ), the set R 1 is replaced by {1}, the set R 2 is replaced by the image of {η′ } in PGLVi /Vi −1 (F ), and PI (F ) is replaced by the image of PI (F )∩PJ (F ) in PGLVi /Vi −1 (F ). By Claim 6 for all J , ξ, and η, the result is proven. Pi Lemma 4.3.11. Let 1 ≤ i ≤ d − 1, and let V ′ = j =1 F e j . Let x ∈ X v for some v ∈ S, and let g ∈ GLV (Fv ) be such that g V ′ = V ′ . For 1 ≤ i ≤ d − 1, we have Œe (i ,j ) d −1 ‚ Y |det(g v : Vv′ → Vv′ )|(d −i )/i t v,j (g x ) = , t v,j (x ) |det(g v : Vv /Vv′ → Vv /Vv′ )| j =1 where  e (i , j ) =  j (d −i ) if j ≤ i , d − j if j ≥ i . i Proof. By the Iwasawa decomposition 3.3.4 and 3.4.5, it suffices to check this in the case that g is represented by a diagonal matrix diag(a 1 , . . . , a d ). It follows from the definitions that t v,j (g x )t v,j (x )−1 = |a j a −1 j +1 |, and the rest of the verification is a simple computation. Lemma 4.3.12. Let i and V ′ be as in 4.3.11. Let R 1 and R 2 be finite subsets of G (F ). Then there exist A, B ∈ R>0 such that for all γ ∈ GLV (F ) with image in R 1 ΓK R 2 ⊂ G (F ) and for which γV ′ = V ′ , we have A≤ Y |det(γ: V ′ → V ′ )|(d −i )/i v v v ∈S |det(γ: Vv /Vv′ → Vv /Vv′ )| 44 ≤ B. Proof. We may assume that R 1 and R 2 are one point sets {ξ} and {η}, respectively. Suppose that an element γ0 with the stated properties of γ exists. Then for any such γ, the image of −1 ∩ P (F ), and hence the image of γγ−1 in G (AS ) belongs to the γγ−1 {i } 0 in G (F ) belongs to ξΓ K ξ 0 F compact subgroup ξK ξ−1 ∩ P{i } (ASF ) of P{i } (ASF ). Hence |det(γγ−1 : Vv′ → Vv′ )| = |det(γγ−1 : Vv /Vv′ → Vv /Vv′ )| = 1 0 0 for every place v of F which does not belong to S. By Lemma 4.3.11 and the product formula, we have Y v ∈S (|det(γγ−1 : Vv′ → Vv′ )|(d −i )/i · |det(γγ−1 : Vv /Vv′ → Vv /Vv′ )|−1 ) = 1, 0 0 so the value of the product in the statement is constant under our assumptions, proving the result. Proposition 4.3.13. Fix c 1 , c 2 ∈ R>0 and finite subsets R 1 , R 2 of G (F ). Then there exists A > 1 such that if x ∈ S(c 1 , c 2 ) ∩ γ−1S(c 1 , c 2 ) for some γ ∈ R 1ΓK R 2, then A −1 t v,i (x ) ≤ t v,i (γx ) ≤ At v,i (x ) for all v ∈ S and 1 ≤ i ≤ d − 1. Proof. By a limit argument, it is enough to consider x ∈ X S ∩ S(c 1 , c 2 ). Fix v ∈ S. For x ,x ′ ∈ X S ∩ S(c 1 , c 2 ) and 1 ≤ i ≤ d − 1, let s i (x ,x ′ ) = t v,i (x ′ )t v,i (x )−1 . For each 1 ≤ i ≤ d − 1, take c 3 (i ) ∈ R>0 satisfying the condition in 4.3.10 for the set I = {i } and both pairs of finite subsets R 1 , R 2 and R 2−1 , R 1−1 of G (F ). Let c 3 = min{c 3 (i ) | 1 ≤ i ≤ d − 1}. For a subset I of {1, . . . , d − 1}, let Y (I ) be the subset of (X S ∩ S(c 1 , c 2 ))2 consisting of all pairs (x ,x ′ ) such that x ′ = γx for some γ ∈ R 1 ΓK R 2 and such that I = {1 ≤ i ≤ d − 1 | min(t v,i (x ), t v,i (x ′ )) ≤ c 3 }. For the proof of 4.3.13, it is sufficient to prove the following statement (S d ), fixing I . (S d ) There exists A > 1 such that A −1 ≤ s i (x ,x ′ ) ≤ A for all (x ,x ′ ) ∈ Y (I ) and 1 ≤ i ≤ d − 1. By Proposition 4.3.10, if γ ∈ R 1ΓK R 2 is such that there exists x ∈ X S with (x , γx ) ∈ Y (I ), then γ ∈ P{i } (F ) for all i ∈ I . Lemmas 4.3.11 and 4.3.12 then imply the following for all i ∈ I , noting that c 2−1 t w,i (y ) ≤ t v,i (y ) ≤ c 2 t w,i (y ) for all w ∈ S and y ∈ X S ∩ S(c 1 , c 2 ). 45 (Ti ) There exists B i > 1 such that for all (x ,x ′ ) ∈ Y (I ), we have B i−1 ≤ d −1 Y j =1 s j (x ,x ′ )e (i ,j ) ≤ B i , where e (i , j ) is as in 4.3.11. We prove the following statement (S i ) for 0 ≤ i ≤ d − 1 by induction on i . ′ ′ (S i ) There exists A i > 1 such that A −1 i ≤ s j (x ,x ) ≤ A i for all (x ,x ) ∈ Y (I ) and all j such that 1 ≤ j ≤ i and j is not the largest element of I ∩ {1, . . . , i } (if it is nonempty). That (S 0 ) holds is clear. Assume that (S i −1 ) holds for some i ≥ 1. If i ∈ / I , then since c 3 ≤ t v,i (x ) ≤ c 1 and c 3 ≤ t v,i (x ′ ) ≤ c 1 , we have c3 c1 ≤ s i (x ,x ′ ) ≤ , c1 c3 and hence (S i ) holds with A i := max(A i −1 , c 1 c 3−1 ). Assume that i ∈ I . If I ∩ {1, . . . , i − 1} = ∅, then (S i ) is evidently true with A i := A i −1 . If I ∩ {1, . . . , i − 1} 6= ∅, then let i ′ be the largest element of this intersection. We compare (Ti ) and (Ti ′ ). We have e (i , j ) = e (i ′ , j ) if j ≥ i and e (i , j ) < e (i ′ , j ) if j < i , so taking the quotient of the equations in (Ti ′ ) and (Ti ), we have (B i B i ′ ) −1 ≤ i −1 Y j =1 s j (x ,x ′ )e (i ,j )−e (i ,j ) ≤ B i B i ′ . ′ Since (S i −1 ) is assumed to hold, there then exists a ∈ R>0 such that (B i B i ′ )−1 A −a ≤ s i ′ (x ,x ′ )e (i ,i )−e (i ,i ) ≤ B i B i ′ A ai−1 i −1 ′ ′ ′ As the exponent e (i ′ , i ′ ) − e (i , i ′ ) is nonzero, this implies that (S i ) holds. By induction, we have (S d −1 ). To deduce (S d ) from it, we may assume that I is nonempty, and let i be the largest element of I . Then (S d −1 ) and (Ti ) imply (S d ). Proposition 4.3.14. Let c 1 , c 2 ∈ R>0 and a ∈ X̄ F,S . Let I be the parabolic type (4.3.7) of a . Fix a finite subset R of G (F ) and 1 ≤ i ≤ d − 1. (1) If i ∈ I , then for any ε > 0, there exists a neighborhood U of a in X̄ F,S for the Satake topology such that max{t v,i (x ) | v ∈ S} < ε for all x ∈ (ΓK R)−1U ∩ S(c 1 , c 2 ). (2) If i ∈ / I , then there exist a neighborhood U of a in X̄ F,S for the Satake topology and c ∈ R>0 such that min{t v,i (x ) | v ∈ S} ≥ c for all x ∈ (ΓK R)−1U ∩ S(c 1 , c 2 ). 46 Proof. The first statement is clear by continuity of t v,i and the fact that t v,i (γ−1 a ) = 0 for all γ ∈ G (F ), and the second follows from 4.3.13, noting 4.3.4. Proposition 4.3.15. Let a ∈ X̄ F,S , and let P be the parabolic subgroup of PGLV associated to a . Let ΓK ,(P) ⊂ ΓK be as in 3.4.12. Then there are c 1 , c 2 ∈ R>0 and ϕ ∈ G (F ) such that ΓK ,(P)ϕ S(c 1 , c 2 ) is a neighborhood of a in X̄ F,S for the Satake topology. Proof. This holds by definition of the Satake topology with ϕ = 1 if a ∈ X̄ F,S (B ). In general, let I be the parabolic type of a . Then the parabolic subgroup associated to a has the form ϕPI ϕ −1 for some ϕ ∈ G (F ). We have ϕ −1 a ∈ X̄ F,S (PI ) ⊂ X̄ F,S (B ). By that already proven case, there exists γ ∈ ΓK ,(PI ) such that ΓK ,(P) ϕγS(c 1 , c 2 ) is a neighborhood of a for the Satake topoology. ♭ The following result can be proved in the manner of 4.4.8 for X̄ F,S below, replacing R by {ϕ}, and ΓK ,(W ) by ΓK ,(P). Lemma 4.3.16. Let the notation be as in 4.3.15. Let U ′ be a neighborhood of ϕ −1 a in X̄ F,S for the Satake topology. Then there is a neighborhood U of a in X̄ F,S for the Satake topology such that U ⊂ ΓK ,(P)ϕ(S(c 1 , c 2 ) ∩ U ′ ). ♭ 4.4 X̄ F,S and reduction theory 4.4.1. Let S be a finite set of places of F containing the archimedean places. In this subsection, ♭ we consider X̄ F,S . Fix a basis (e i )1≤i ≤d of V . Let B ⊂ G = PGLV be the Borel subgroup of upper triangular matrices for (e i )i . Let K be a compact open subgroup of G (ASF ). ♭ 4.4.2. Let c 1 , c 2 ∈ R>0 . We let S♭ (c 1 , c 2 ) denote the image of S(c 1 , c 2 ) under X̄ F,S → X̄ F,S . For r ∈ {1, . . . , d − 1}, we then define S♭r (c 1 , c 2 ) = {(W, µ) ∈ S♭ (c 1 , c 2 ) | dim(W ) ≥ r }. Then the maps t v,i of 4.3.5 for v ∈ S and 1 ≤ i ≤ r induce maps t v,i : S♭r (c 1 , c 2 ) → R>0 (1 ≤ i ≤ r − 1) and t v,r : S♭r (c 1 , c 2 ) → R≥0 . For c 3 ∈ R>0 , we also set S♭r (c 1 , c 2 , c 3 ) = {x ∈ S♭r (c 1 , c 2 ) | min{t v,r (x ) | v ∈ S} ≤ c 3 }. Proposition 4.4.3. Fix c 1 ∈ R>0 and finite subsets R 1 , R 2 of G (F ). Then there exists c 3 ∈ R>0 such that for all c 2 ∈ R>0 , we have {γ ∈ R 1 ΓK R 2 | γS♭r (c 1 , c 2 , c 3 ) ∩ S♭r (c 1 , c 2 ) 6= ∅} ⊂ P{r } . 47 Proof. Take (W, µ) ∈ γS♭r (c 1 , c 2 , c 3 ) ∩ S♭r (c 1 , c 2 ), and let r ′ = dim W . Let P be the parabolic Pr ′ +i subgroup of V corresponding to the flag (Vi )−1≤i ≤d −r ′ with Vi = W + j =r ′+1 F e i for 0 ≤ i ≤ d − r ′ . Let µ′ ∈ ZF,S (P) be the unique element such that a = (P, µ′ ) ∈ X̄ F,S maps to (W, µ). Then a ∈ γS{r } (c 1 , c 2 , c 3 ) ∩ S(c 1 , c 2 ), so we can apply 4.3.10. Proposition 4.4.4. Fix c 1 , c 2 ∈ R>0 and finite subsets R 1 , R 2 of G (F ). Then there exists A > 1 such that if x ∈ S♭r (c 1 , c 2 ) ∩ γ−1S♭r (c 1 , c 2 ) for some γ ∈ R 1 ΓK R 2 , then A −1 t v,i (x ) ≤ t v,i (γx ) ≤ At v,i (x ) for all v ∈ S and 1 ≤ i ≤ r . Proof. This follows from 4.3.13. We also have the following easy consequence of Lemma 4.3.8. ♭ Lemma 4.4.5. Let a be in the image of X̄ F,S (B ) → X̄ F,S , and let r be the dimension of the F Pr subspace of V associated to a . Then the F -subspace of V associated to a is i =1 F e i . ♭ Proposition 4.4.6. Let a ∈ X̄ F,S and let r be the dimension of the F -subspace of V associated to a . Let c 1 , c 2 ∈ R>0 . Fix a finite subset R of G (F ). ♭ (1) For any ε > 0, there exists a neighborhood U of a in X̄ F,S for the Satake topology such that max{t v,r (x ) | v ∈ S} < ε for all x ∈ (ΓK R)−1U ∩ S♭r (c 1 , c 2 ). ♭ (2) If 1 ≤ i < r , then there exist a neighborhood U of a in X̄ F,S for the Satake topology and c ∈ R>0 such that min{t v,i (x ) | v ∈ S} ≥ c for all x ∈ (ΓK R)−1U ∩ S♭r (c 1 , c 2 ). Proof. This follows from 4.4.4, as in the proof of 4.3.14. Proposition 4.4.7. Let W be an F -subspace of V of dimension r ≥ 1. Let Φ be set of ϕ ∈ G (F ) Pr such that ϕ( i =1 F e i ) = W . ♭ (1) There exists a finite subset R of Φ such that for any a ∈ X̄ F,S (W ), there exist c 1 , c 2 ∈ R>0 for which the set ΓK ,(W ) R S♭r (c 1 , c 2 ) is a neighborhood of a in the Satake topology. ♭ (2) For any ϕ ∈ Φ and a ∈ X̄ F,S with associated subspace W , there exist c 1 , c 2 ∈ R>0 such that a ∈ ϕ S♭r (c 1 , c 2 ) and ΓK ,(W ) ϕ S♭r (c 1 , c 2 ) is a neighborhood of a in the Satake topology. Proof. We may suppose without loss of generality that W = Pr j =1 F e j , in which case Φ = G (F )(W ) (see 3.4.12). Consider the set Q of all parabolic subgroups Q of G such that W is contained in the smallest nonzero subspace of V preserved by Q. Any Q ∈ Q has the form 48 Q = ϕPI ϕ −1 for some ϕ ∈ G (F )(W ) and subset I of J := {i ∈ Z | r ≤ i ≤ d − 1}. There exists a finite subset R of G (F )(W ) such that we may always choose ϕ ∈ ΓK ,(W ) R. ♭ By 4.4.5, an element of X̄ F,S (B ) has image in X̄ F,S (W ) if and only if the parabolic subgroup associated to it has the form PI for some I ⊂ J . The intersection of the image of X̄ F,S (B ) → ♭ ♭ X̄ F,S with X̄ F,S (W ) is the union of the S♭r (c 1 , c 2 ) with c 1 , c 2 ∈ R>0 . By the above, for any a ∈ ♭ X̄ F,S (W ), we may choose ξ ∈ ΓK ,(W ) R such that ξ−1 a is in this intersection, and part (1) follows. ♭ Moreover, if W is the subspace associated to a , then ϕ −1 a ∈ X̄ F,S (W ) is in the image of X̄ F,S (B ) for all ϕ ∈ G (F )(W ) , from which (2) follows. ♭ Lemma 4.4.8. Let W , Φ, R be as in 4.4.7, fix a ∈ X̄ F,S (W ), and let c 1 , c 2 ∈ R>0 be as in 4.4.7(1) for ♭ this a . For each ϕ ∈ R, let Uϕ be a neighborhood of ϕ −1 a in X̄ F,S for the Satake topology. Then ♭ there is a neighborhood U of a in X̄ F,S for the Satake topology such that U⊂ [ ϕ∈R ΓK ,(W ) ϕ(S♭r (c 1 , c 2 ) ∩ Uϕ ). Proof. We may assume that each ϕ(Uϕ ) is stable under the action of ΓK ,(W ) . Let U = ΓK ,(W ) R S♭r (c 1 , c 2 ) ∩ \ ϕ(Uϕ ). ϕ∈R Then U is a neighborhood of a by 4.4.7(1). Let x ∈ U . Take γ ∈ ΓK ,(W ) and ϕ ∈ R such that x ∈ γϕ S♭r (c 1 , c 2 ). Since ϕ(Uϕ ) is ΓK ,(W ) -stable, γ−1x ∈ ϕ(Uϕ ) and hence ϕ −1 γ−1x ∈ S♭r (c 1 , c 2 ) ∩ Uϕ . ♭ Proposition 4.4.9. Let a = (W, µ) ∈ X̄ F,S , and let r = dim(W ). Take ϕ ∈ G (F ) and c 1 , c 2 ∈ R>0 as ♭ ♭ in 4.4.7(2) such that ΓK ,(W ) ϕ S♭r (c 1 , c 2 ) is a neighborhood of a . Let φW,S : X̄ F,S (W ) → Z♭F,S (W ) be as ♭ in 3.4.4. For any neighborhood U of µ = φW,S (a ) in Z♭F,S (W ) and any ε ∈ R>0 , set ♭ Φ(U , ε) = (φW,S )−1 (U ) ∩ ΓK ,(W ) ϕ{x ∈ S♭r (c 1 , c 2 ) | t v,r (x ) < ε for all v ∈ S}. ♭ Then the set of all Φ(U , ε) forms a base of neighborhoods of a in X̄ F,S under the Satake topology. Proof. We may suppose that W = Pr i =1 F e i without loss of generality, in which case ϕ ∈ G (F )(W ) . Let P be the smallest parabolic subgroup containing B with flag (Vi )−1≤i ≤m such that V0 = W and m = d − r . Let Q be the parabolic of all elements that preserve W . We then have G ⊃ Q ⊃ P ⊃ B . Let B ′ be the Borel subgroup of PGLV /W that is the image of P and which we regard as a subgroup of G using (e r +i )1≤i ≤m to split V → V /W . Let f v : Q u (Fv ) × X̄ V /W,F,v (B ′ ) × X Wv × R≥0 → X̄ F,v (P) 49 be the unique surjective continuous map such that ξ = f v ◦ h, where ξ is as in 3.5.6 and h is defined as the composition ∼ −1 Pu (Fv ) × X Wv × Rm − → Q u (Fv ) × B u′ (Fv ) × X Wv × R≥0 × Rm → Q u (Fv ) × X̄ V /W,F,v (B ′ ) × X Wv × R≥0 ≥0 ≥0 ∼ of the map induced by the isomorphism Pu (Fv ) − → Q u (Fv ) × B u′ (Fv ) and the map induced by −1 ′ the surjection π̄ B ′ ,v : B u′ (Fv ) × Rm ≥0 → X̄ V /W,F,v (B ) of 3.3.8(2). The existence of f v follows from 3.3.8(4). Set Y0 = RS>0 ∪ {(0)v ∈S }, and let f S : Q u (AF,S ) × X̄ V /W,F,S (B ′ ) × Z♭F,S (W ) × Y0 → X̄ F,S (P) be the product of the maps f v . Let t v,r : X̄ F,v (P) → R≥0 denote the composition ′ φB,v X̄ F,v (P) → X̄ F,v (B ) −→ Rd≥0−1 → R≥0 , where the last arrow is the r th projection. The composition of f S with (t v,r )v ∈S is projection onto Y0 by 3.5.7 and 3.4.6. ♭ Let X̄ F,S (W ) denote the inverse image of X̄ F,S (W ) under the canonical surjection ΠS : X̄ F,S → ♭ X̄ F,S . Combining f S with the action of G (F )(W ) , we obtain a surjective map f S′ : G (F )(W ) × (Q u (AF,S ) × X̄ V /W,F,S (B ′ ) × Z♭F,S (W ) × Y0 ) → X̄ F,S (W ), f S′ (g , z ) = g f S (z ). ♭ The composition of f S′ with φW,S ◦ ΠS is projection onto Z♭F,S (W ) by 3.5.9 and 3.5.10. Applying 4.3.4 with V /W in place of V , there exists a compact subset C of Q u (AF,S ) × X̄ V /W,F,S (B ′ ) and a finite subset R of G (F )(W ) such that f S′ (ΓK ,(W ) R × C × Z♭F,S (W ) × Y0 ) = X̄ F,S (W ). Consider the restriction of ΠS ◦ f S′ to a surjective map ♭ λS : ΓK ,(W ) R × C × Z♭F,S (W ) × Y0 → X̄ F,S (W ). We may suppose that R contains ϕ, since it lies in G (F )(W ) . ♭ Now, let U ′ be a neighborhood of a in X̄ F,S (W ) for the Satake topology. It is sufficient to prove that there exist an open neighborhood U of µ in Z♭F,S (W ) and ε ∈ R>0 such that Φ(U , ε) ⊂ U ′ . For ε ∈ R>0 , set Yε = {(t v )v ∈S ∈ Y0 | t v < ε for all v ∈ S}. For any x ∈ C , we have λS (α,x , µ, 0) = (W, µ) ∈ U ′ for all α ∈ R. By the continuity of λS , there exist a neighborhood D(x ) ⊂ Q u (AF,S ) × X̄ V /W,F,S (B ′ ) of x , a neighborhood U (x ) ⊂ Z♭F,v (W ) of µ, and ε(x ) ∈ R>0 such that λS (R × D(x ) × U (x ) × Yε(x )) ⊂ U ′ . 50 Since C is compact, some finite collection of the sets D(x ) cover C . Thus, there exist a neighborhood U of µ in Z♭F,v (W ) and ε ∈ R>0 such that λS (R ×C ×U ×Yε ) ⊂ U ′ . Since U ′ is ΓK ,(W ) -stable by 3.4.15, we have λS (ΓK ,(W ) R × C × U × Yε ) ⊂ U ′ . Let y ∈ Φ(U , ε), and write y = g x with g ∈ ΓK ,(W ) ϕ and x ∈ S♭r (c 1 , c 2 ) such that t v,r (x ) < ε ♭ for all v ∈ S. Since Φ(U , ε) ⊂ X̄ F,S (W ), we may by our above remarks write y = λS (g , c , ν, t ) = ♭ g ΠS (f S (c , ν, t )), where c ∈ C , ν = φW,S (y ), and t = (t v,r (x ))v ∈S . Since ν ∈ U and t ∈ Yε by definition, y is contained in U ′ . Therefore, we have Φ(U , ε) ⊂ U ′ . Example 4.4.10. Consider the case F = Q, S = {v } with v the archimedean place, and d = 3. ♭ We construct a base of neighborhoods of a point in X̄ Q,v for the Satake topology. ♭ Fix a basis (e i )1≤i ≤3 of V . Let a = (W, µ) ∈ X̄ Q,v , where W = Qe 1 , and µ is the unique element of X Wv . For c ∈ R>0 , let Uc be the subset of X v = PGL3 (R)/ PO3 (R) consisting of the elements    ! 1 x 12 x 13 y1y2 0 0   1 0  0 1 x 23   0  y 0 2   0 γ  0 0 1 0 0 1 such that γ ∈ PGL2 (Z), x i j ∈ R, y 1 ≥ c , and y 2 ≥ these elements converge to a in ♭ X̄ Q,v p 3 . 2 When γ, x i j and y 2 are fixed and y 1 → ∞, under the Satake topology. When γ, x i j , and y 1 are fixed and y 2 → ∞, they converge in the Satake topology to   ! 1 x 12 0  1 0  0 1 0 µ(y 1 ), µ(γ,x 12 , y 1 ) :=  0 γ  0 0 1 ♭ where µ(y 1 ) is the class in X̄ Q,v of the semi-norm a 1 e 1∗ + a 2 e 2∗ + a 3 e 3∗ 7→ (a 21 y 12 + a 22 )1/2 on Vv∗ . The set of Ūc := {a } ∪ {µ(γ,x , y ) | γ ∈ PGL2 (Z),x ∈ R, y ≥ c } ∪ Uc . ♭ is a base of neighborhoods for a in X̄ Q,v under the Satake topology. Note that H = SL2 (Z){z ∈ H | Im(z ) ≥ p 3 }, 2 which is the reason for the appearance of any b ∈ R>0 such that b ≤ p 3 . 2 p 3 . 2 It can of course be replaced by ♭ 4.4.11. We continue with Example 4.4.10. Under the canonical surjection X̄ Q,v → X̄ Q,v , the inverse image of a = (W, µ) in X̄ Q,v is canonically homeomorphic to X̄ (V /W )v = H ∪ P1 (Q) under the Satake topology on both spaces. This homeomorphism sends x + y 2 i ∈ H (x ∈ R, y 2 ∈ R>0 ) to the limit for the Satake topology of    1 0 0 y1 y2 0 0    0 1 x   0  ∈ PGL3 (R)/ PO3 (R) y 0 2    0 0 1 0 0 1 51 ♭ as y 1 → ∞. (This limit in X̄ Q,v depends on x and y 2 , but the limit in X̄ Q,v is a .) ♭ 4.4.12. In the example of 4.4.10, we explain that the quotient topology on X̄ Q,v of the Satake ♭ topology on X̄ Q,v is different from the Satake topology on X̄ Q,v . For a map 1 Z f : PGL2 (Z)/ 0 1 ! → R>0 , define a subset U f of X v as in the definition of Uc but replacing the condition on γ,x i j , y i by γ ∈ PGL2 (Z), x i j ∈ R, y 1 ≥ f (γ), and y 2 ≥ p 3 . 2 Let Ū f = {a } ∪ {µ(γ,x , y ) | γ ∈ PGL2 (Z),x ∈ R, y ≥ f (γ)} ∪ U f . ♭ When f varies, the Ū f form a base of neighborhoods of a in X̄ Q,v for the quotient topology of the Satake topology on X̄ Q,v . On the other hand, if inf{ f (γ) | γ ∈ PGL2 (Z)} = 0, then Ū f is not a ♭ neighborhood of a for the Satake topology on X̄ Q,v . 4.5 Proof of the main theorem In this subsection, we prove Theorem 4.1.4. We begin with the quasi-compactness asserted therein. Throughout this subsection, we set Z = X S 2 × G (ASF )/K in situation (I) and Z = X S 2 in situation (II), so X̄ = X̄ × Z . Proposition 4.5.1. In situation (I) of 4.1.3, the quotient G (F )\X̄ is quasi-compact. In situation (II), the quotient Γ\X̄ is quasi-compact for any subgroup Γ of ΓK of finite index. ♭ Proof. We may restrict to case (i) of 4.1.2 that X̄ = X̄ F,S 1 , as X̄ F,S of case (ii) is a quotient of 1 X̄ F,S 1 (under the Borel-Serre topology). In situation (I), we claim that there exist c 1 , c 2 ∈ R>0 , a compact subset C of B u (AF,S ), and a compact subset C ′ of Z such that X̄ = G (F )(S(C ; c 1 , c 2 ) × C ′ ). In situation (II), we claim that there exist c 1 , c 2 ,C ,C ′ as above and a finite subset R of G (F ) such that X̄ = ΓR(S(C ; c 1 , c 2 )×C ′ ). It follows that in situation (I) (resp., (II)), there is a surjective continuous map from the compact space C × T(c 1 , c 2 ) × C ′ (resp., R × C × T(c 1 , c 2 ) × C ′ ) onto the quotient space under consideration, which yields the proposition. For any compact open subgroup K ′ of G (ASF1 ), the set G (F )\G (ASF1 )/K ′ is finite. Each X v for v ∈ S 2 may be identified with the geometric realization of the Bruhat-Tits building for PGLVv , the set of i -simplices of which for a fixed i can be identified with G (Fv )/K v′ for some K ′ . So, we see that in situation (I) (resp., (II)), there is a compact subset D of Z such that Z = G (F )D (resp., Z = ΓD). 52 Now fix such a compact open subgroup K ′ of G (ASF1 ). By 4.3.4, there are c 1 , c 2 ∈ R>0 , a compact subset C of Pu (AF,S 1 ), and a finite subset R ′ of G (F ) such that X̄ F,S = ΓK ′ R ′ S(C ; c 1 , c 2 ). We consider the compact subset C ′ := (R ′ )−1 K ′ D of Z . Let (x , y ) ∈ X̄, where x ∈ X̄ F,S and y ∈ Z . Write y = γz for some z ∈ D and γ ∈ G (F ) (resp., γ ∈ Γ) in situation (I) (resp., (II)). In situation (II), we write ΓΓK ′ R ′ = ΓR for some finite subset R of G (F ). Write γ−1x = γ′ ϕs where γ′ ∈ ΓK ′ , ϕ ∈ R ′ , s ∈ S(C ; c 1 , c 2 ). We have (x , y ) = γ(γ−1x , z ) = γ(γ′ ϕs , z ) = (γγ′ ϕ)(s , ϕ −1 (γ′ )−1 z ). As γγ′ ϕ lies in G (F ) in situation (I) and in ΓR in situation (II), we have the claim. 4.5.2. To prove Theorem 4.1.4, it remains only to verify the Hausdorff property. For this, it is sufficient to prove the following. Proposition 4.5.3. Let Γ = G (F ) in situation (I) of 4.1.3, and let Γ = ΓK in situation (II). For every a , a ′ ∈ X̄, there exist neighborhoods U of a and U ′ of a ′ such that if γ ∈ Γ and γU ∩U ′ 6= ∅, then γa = a ′ . In the rest of this subsection, let the notation be as in 4.5.3. It is sufficient to prove 4.5.3 for the Satake topology on X̄. In 4.5.4–4.5.8, we prove 4.5.3 in situation (II) for S = S 1 . That is, we suppose that X̄ = X̄ . In 4.5.9 and 4.5.10, we deduce 4.5.3 in general from this case. Lemma 4.5.4. Assume that X̄ = X̄ F,S . Suppose that a , a ′ ∈ X̄ have distinct parabolic types (4.3.7). Then there exist neighborhoods U of a and U ′ of a ′ such that γU ∩ U ′ = ∅ for all γ ∈ Γ. Proof. Let I (resp., I ′ ) be the parabolic type of a (resp., a ′ ). We may assume that there exists an i ∈ I with i ∈ / I ′. By 4.3.15, there exist ϕ, ψ ∈ G (F ) and c 1 , c 2 ∈ R>0 such that ΓK ϕ S(c 1 , c 2 ) is a neighbor- hood of a and ΓK ψS(c 1 , c 2 ) is a neighborhood of a ′ . By 4.3.14(2), there exist a neighborhood U ′ ⊂ ΓK ψS(c 1 , c 2 ) of a ′ and c ∈ R>0 with the property that min{t v,i (x ) | v ∈ S} ≥ c for all x ∈ (ΓK ψ)−1U ′ ∩ S(c 1 , c 2 ). Let A ∈ R>1 be as in 4.3.13 for these c 1 , c 2 for R 1 = {ϕ −1 } and R 2 = {ψ}. Take ε ∈ R>0 such that Aε ≤ c . By 4.3.14(1), there exists a neighborhood U ⊂ ΓK ϕ S(c 1 , c 2 ) of a such that max{t v,i (x ) | v ∈ S} < ε for all x ∈ (ΓK ϕ)−1U ∩ S(c 1 , c 2 ). We prove that γU ∩ U ′ = ∅ for all γ ∈ ΓK . If x ∈ γU ∩ U ′ , then we may take δ, δ′ ∈ ΓK such that (δϕ)−1 γ−1x ∈ S(c 1 , c 2 ) and (δ′ ψ)−1x ∈ S(c 1 , c 2 ). Since (δϕ)−1 γ−1x = ϕ −1 (δ−1 γ−1 δ′ )ψ(δ′ ψ)−1x ∈ ϕ −1 ΓK ψ · (δ′ ψ)−1x , we have by 4.3.13 that c ≤ t v,i ((δ′ ψ)−1x ) ≤ At v,i ((δϕ)−1 γ−1x ) < Aε, for all v ∈ S and hence c < Aε, a contradiction. 53 ♭ Lemma 4.5.5. Assume that X̄ = X̄ F,S . Let a , a ′ ∈ X̄ and assume that the dimension of the F - subspace associated to a is different from that of a ′ . Then there exist neighborhoods U of a and U ′ of a ′ such that γU ∩ U ′ = ∅ for all γ ∈ Γ. Proof. The proof is similar to that of 4.5.4. In place of 4.3.13, 4.3.14, and 4.3.15, we use 4.4.4, 4.4.6, and 4.4.7, respectively. Lemma 4.5.6. Let P be a parabolic subgroup of G . Let a , a ′ ∈ ZF,S (P) (see 3.4.3), and let R 1 and R 2 be finite subsets of G (F ). Then there exist neighborhoods U of a and U ′ of a ′ in ZF,S (P) such that γa = a ′ for every γ ∈ R 1ΓK R 2 ∩ P(F ) for which γU ∩ U ′ 6= ∅. Proof. For each ξ ∈ R 1 and η ∈ R 2 , the set ξΓK η ∩ P(F ) is a ξΓK ξ−1 ∩ P(F )-orbit for the left Qm action of ξΓK ξ−1 . Hence its image in i =0 PGLVi /Vi −1 (AF,S ) is discrete, for (Vi )−1≤i ≤m the flag Qm corresponding to P, and thus the image of R 1 ΓK R 2 ∩ P(F ) in i =0 PGLVi /Vi −1 (AF,S ) is discrete as well. On the other hand, for any compact neighborhoods U of a and U ′ of a ′ , the set ) ( m Y ′ PGLVi /Vi −1 (AF,S ) | g U ∩ U 6= ∅ g∈ i =0 is compact. Hence the intersection M := {γ ∈ R 1 ΓK R 2 ∩P(F ) | γU ∩U ′ 6= ∅} is finite. If γ ∈ M and γa 6= a ′ , then replacing U and U ′ by smaller neighborhoods of a and a ′ , respectively, we have γU ∩ U ′ = ∅. Hence for sufficiently small neighborhoods U and U ′ of a and a ′ , respectively, we have that if γ ∈ M , then γa = a ′ . Lemma 4.5.7. Let W be an F -subspace of V . Let a , a ′ ∈ Z♭F,S (W ) (see 3.4.4), and let R 1 and R 2 be finite subsets of G (F ). Let P be the parabolic subgroup of G consisting of all elements which preserve W . Then there exist neighborhoods U of a and U ′ of a ′ in Z♭F,S (W ) such that γa = a ′ for every γ ∈ R 1ΓK R 2 ∩ P(F ) for which γU ∩ U ′ 6= ∅. Proof. This is proven in the same way as 4.5.6. 4.5.8. We prove 4.5.3 in situation (II), supposing that S = S 1 . In case (i) (that is, X̄ = X̄ = X̄ F,S ), we may assume by 4.5.4 that a and a ′ have the same ♭ parabolic type I . In case (ii) (that is, X̄ = X̄ = X̄ F,S ), we may assume by 4.5.5 that the dimension r of the F -subspace of V associated to a coincides with that of a ′ . In case (i) (resp., (ii)), take c 1 , c 2 ∈ R>0 and elements ϕ and ψ (resp., finite subsets R and R ′ ) of G (F ) such that c 1 , c 2 , ϕ (resp., c 1 , c 2 , R) satisfy the condition in 4.3.15 (resp., 4.4.7) for a and c 1 , c 2 , ψ (resp., c 1 , c 2 , R ′ ) satisfy the condition in 4.3.15 (resp., 4.4.7) for a ′ . In case (i), we set R = {ϕ} and R ′ = {ψ}. Fix a basis (e i )1≤i ≤d of V . In case (i) (resp., (ii)), denote S(c 1 , c 2 ) (resp., S♭r (c 1 , c 2 )) by S. In Pr case (i), let P = PI , and let (Vi )−1≤i ≤m be the associated flag. In case (ii), let W = i =1 F e i , and let P be the parabolic subgroup of G consisting of all elements which preserve W . 54 Note that in case (i) (resp., (ii)), for all ϕ ∈ R and ψ ∈ R ′ , the parabolic subgroup P is associated to ϕ −1 a and to ψ−1 a ′ (resp., W is associated to ϕ −1 a and to ψ−1 a ′ ) and hence these elements are determined by their images in ZF,S (P) (resp. Z♭F,S (W )). In case (i) (resp., case (ii)), apply 4.5.6 (resp., 4.5.7) to the images of ϕ −1 a and ψ−1 a ′ for ϕ ∈ R, ψ ∈ R ′ in ZF,S (P) (resp., Z♭F,S (W )). By this, and by 4.3.10 for case (i) and 4.4.3 for case (ii), we see that there exist neighborhoods Uϕ of ϕ −1 a for each ϕ ∈ R and Uψ′ of ψ−1 a ′ for each ψ ∈ R ′ for the Satake topology with the following two properties: (A) {γ ∈ (R ′ )−1 ΓK R | γ(S ∩ Uϕ ) ∩ (S ∩ Uψ′ ) 6= ∅ for some ϕ ∈ R, ψ ∈ R ′ } ⊂ P(F ), (B) if γ ∈ (R ′ )−1 ΓK R ∩ P(F ) and γUϕ ∩ Uψ′ 6= ∅ for ϕ ∈ R and ψ ∈ R ′ , then γϕ −1 a = ψ−1 a ′ . In case (i) (resp., (ii)), take a neighborhood U of a satisfying the condition in 4.3.16 (resp., 4.4.8) for (Uϕ )ϕ∈R , and take a neighborhood U ′ of a ′ satisfying the condition in 4.3.16 (resp., 4.4.8) for (Uψ′ )ψ∈R ′ . Let γ ∈ ΓK and assume γU ∩ U ′ 6= ∅. We prove γa = a ′ . Take x ∈ U and x ′ ∈ U ′ such that γx = x ′ . By 4.3.16 (resp., 4.4.8), there are ϕ ∈ R, ψ ∈ R ′ , and ε ∈ ΓK ,(ϕPϕ −1) and δ ∈ ΓK ,(ψPψ−1) in case (i) (resp., ε ∈ ΓK ,(ϕW ) and δ ∈ ΓK ,(ψW ) in case (ii)) such that ϕ −1 ε−1 x ∈ S ∩ Uϕ and ψ−1 δ−1x ′ ∈ S ∩ Uψ′ . Since (ψ−1 δ−1 γεϕ)ϕ −1 ε−1x = ψ−1 δ−1x ′ , we have ψ−1 δ−1 γεϕ ∈ P(F ) by property (A). By property (B), we have (ψ−1 δ−1 γεϕ)ϕ −1 a = ψ−1 a ′ . Since εa = a and δa ′ = a ′ , this proves γa = a ′ . We have proved 4.5.3 in situation (II) under the assumption S = S 1 . In the following 4.5.9 and 4.5.10, we reduce the general case to that case. Lemma 4.5.9. Let a , a ′ ∈ Z . In situation (I) (resp., (II)), let H = G (ASF1 ) (resp., H = G (AF,S 2 )). Then there exist neighborhoods U of a and U ′ of a ′ in Z such that g a = a ′ for all g ∈ H for which g U ∩ U ′ 6= ∅. Proof. For any compact neighborhoods U of a and U ′ of a ′ , the set M := {g ∈ H | g U ∩U ′ 6= ∅} is compact. By definition of Z , there exist a compact open subgroup N of H and a compact neighborhood U of a such that g x = x for all g ∈ N and x ∈ U . For such a choice of U , the set M is stable under the right translation by N , and M /N is finite because M is compact and N is an open subgroup of H . If g ∈ M and if g a 6= a ′ , then by shrinking the neighborhoods U and U ′ , we have that g U ∩U ′ = ∅. As M /N is finite, we have sufficiently small neighborhoods U and U ′ such that if g ∈ M and g U ∩ U ′ 6= ∅, then g a = a ′ . 55 4.5.10. We prove Proposition 4.5.3. Let H be as in Lemma 4.5.9. Write a = (a S 1 , a Z ) and a ′ = (a S′ 1 , a Z′ ) as elements of X̄ × Z . By 4.5.9, there exist neighborhoodsUZ of a Z and UZ′ of a Z′ in Z such that if g ∈ H and g UZ ∩UZ′ 6= ∅, then g a = a ′ . The set K ′ := {g ∈ H | g a Z = a Z } is a compact open subgroup of H . Let Γ′ be the inverse image of K ′ under Γ → H , where Γ = G (F ) in situation (I). In situation (II), the group Γ′ is of finite index in the inverse image of the compact open subgroup K ′ × K under S G (F ) → G (AF1 ). In both situations, the set M := {γ ∈ Γ | γa Z = a Z′ } is either empty or a Γ′ -torsor for the right action of Γ′ . Assume first that M 6= ∅, in which case we may choose θ ∈ Γ such that M = θ Γ′ . Since we have proven 4.5.3 in situation (II) for S 1 = S, there exist neighborhoods US 1 of a S 1 and US′ 1 of θ −1 a S′ 1 such that if γ ∈ Γ′ satisfies γUS 1 ∩ US′ 1 6= ∅, then γa S 1 = θ −1 a S′ 1 . Let U = US 1 × UZ and U ′ = θUS′ 1 × UZ′ , which are neighborhoods of a and a ′ in X̄, respectively. Suppose that γ ∈ Γ satisfies γU ∩U ′ 6= ∅. Then, since γUZ ∩UZ′ 6= ∅, we have γa Z = a Z′ and hence γ = θ γ′ for some γ′ ∈ Γ′ . Since θ γ′US 1 ∩ θUS′ 1 6= ∅, we have γ′US 1 ∩US′ 1 6= ∅, and hence γ′ a S 1 = θ −1 a S′ 1 . That is, we have γa S 1 = a S′ 1 , so γa = a ′ . In the case that M = ∅, take any neighborhoods US 1 of a S 1 and US′ 1 of a S′ 1 , and set U = US 1 × UZ and U ′ = US′ 1 × UZ′ . Any γ ∈ Γ such that γU ∩ U ′ 6= ∅ is contained in M , so no such γ exists. 4.6 Supplements to the main theorem We use the notation of §4.1 throughout this subsection. We suppose that Γ = G (F ) in situation (I), and we let Γ be a subgroup of ΓK of finite index in situation (II). For a ∈ X̄, let Γa < Γ denote the stabilizer of a . Theorem 4.6.1. For a ∈ X̄ (with either the Borel-Serre or the Satake topology), there is an open neighborhood U of the image of a in Γa \X̄ such that the image U ′ of U under the quotient map Γa \X̄ → Γ\X̄ is open and the map U → U ′ is a homeomorphism. Proof. By the case a = a ′ of Proposition 4.5.3, there is an open neighborhood U ′′ ⊂ X̄ of a such that if γ ∈ ΓK and γU ′′ ∩ U ′′ 6= ∅, then γa = a . Then the subset U := Γa \Γa U ′′ of Γa \X̄ is open and has the desired property. Proposition 4.6.2. Suppose that S = S 1 , and let a ∈ X̄. (1) Take X̄ = X̄ F,S , and let P be the parabolic subgroup associated to a . Then Γ(P) (as in 3.4.12) is a normal subgroup of Γa of finite index. 56 ♭ (2) Take X̄ = X̄ F,S , and let W be the F -subspace of V associated to a . Then Γ(W ) (as in 3.4.12) is a normal subgroup of Γa of finite index. Proof. We prove (1), the proof of (2) being similar. Let (Vi )−1≤i ≤m be the flag corresponding Qm to P. The image of Γ ∩ P(F ) in i =0 PGLVi /Vi −1 (AF,S ) is discrete. On the other hand, the stabiQm lizer in i =0 PGLVi /Vi −1 (AF,S ) of the image of a in ZF,S (P) is compact. Hence the image of Γa in Qm PGLVi /Vi −1 (F ), which is isomorphic to Γa /Γ(P) , is finite. i =0 Theorem 4.6.3. Assume that F is a function field and X̄ = X̄ F,S 1 , where S 1 consists of a single place v . Then the inclusion map Γ\X ,→ Γ\X̄ is a homotopy equivalence. Proof. Let a ∈ X̄. In situation (I) (resp., (II)), write a = (a v , a v ) with a v ∈ X̄ F,v and a v ∈ X S 2 × Q G (ASF )/K (resp., X S 2 ). Let K ′ be the isotropy subgroup of a v in G (AvF ) (resp., w ∈S 2 G (Fw )), and Q let Γ′ < Γ be the inverse image of K ′ under the map Γ → G (AvF ) (resp., Γ → w ∈S 2 G (Fw )). Let P be the parabolic subgroup associated to a . Let Γa be the isotropy subgroup of a in Γ, which is contained in P(F ) and equal to the isotropy subgroup Γ′a v of a v in Γ′ . In situation (I) (resp., (II)), take a Γa -stable open neighborhood D of a v in X S 2 × G (ASF )/K (resp., X S 2 ) that has compact closure. Claim 1. The subgroup ΓD := {γ ∈ Γa | γx = x for all x ∈ D} of Γa is normal of finite index. Proof of Claim 1. Normality follows from the Γa -stability of D. For any x in the closure D̄ of D, there exists an open neighborhood Vx of x and a compact open subgroup N x of G (AvF ) (resp., Q G (Fw )) in situation (I) (resp., (II)) such that g y = y for all g ∈ N x and y ∈ Vx . For a finite w ∈S 2 Tn subcover {Vx 1 , . . . , Vx n } of D̄, the group ΓD is the inverse image in Γa of i =1 N x i , so is of finite index. Claim 2. The subgroup H := ΓD ∩ Pu (F ) of Γa is normal of finite index. Proof of Claim 2. Normality is immediate from Claim 1 as Pu (F ) is normal in P(F ). Let H ′ = Γ′(P) ∩ Pu (F ), which has finite index in Γ′(P) and equals Γ′ ∩ Pu (F ) by definition of Γ′(P) . Since Γ′(P) ⊂ Γ′a v ⊂ Γ′ and Γ′a v = Γa , we have H ′ = Γa ∩ Pu (F ) as well. By Claim 1, we then have that H ′ contains H with finite index, so H has finite index in Γ′(P) . Proposition 4.6.2(1) tells us that Γ′(P) is of finite index in Γ′a v = Γa . Let (Vi )−1≤i ≤m be the flag corresponding to P. By Corollary 3.5.4, we have a homeomor- phism ∼ χ : Pu (Fv )\X̄ F,v (P) − → ZF,v (P) × Rm ≥0 ′ on quotient spaces arising from the P(Fv )-equivariant homeomorphism ψP,v = (φP,v , φP,v ) of 3.5.1 (see 3.4.3 and 3.4.6). 57 Claim 3. For a sufficiently small open neighborhood U of 0 = (0, . . . , 0) in Rm ≥0 , the map χ induces a homeomorphism ∼ χU : H \X̄ F,v (P)U − → ZF,v (P) × U , ′ where X̄ F,v (P)U denotes the inverse image of U under φP,v : X̄ F,v (P) → Rm ≥0 . Proof of Claim 3. By definition, χ restricts to a homeomorphism ∼ Pu (Fv )\X̄ F,v (P)U − → ZF,v (P) × U for any open neighborhood U of 0. For a sufficiently large compact open subset C of Pu (Fv ), we have Pu (Fv ) = HC . For U sufficiently small, every g ∈ C fixes all x ∈ X̄ F,v (P)U , which yields the claim. Claim 4. The map χU and the identity map on D induce a homeomorphism ∼ χU ,a : Γa \(X̄ F,v (P)U × D) − → (Γa \(ZF,v (P) × D)) × U . Proof of Claim 4. The quotient group Γa /H is finite by Claim 2. Since the determinant of an automorphism of Vi /Vi −1 of finite order has trivial absolute value at v , the Γa -action on Rm ≥0 is trivial. Since H acts trivially on D, the claim follows from Claim 3. m Now let c ∈ Rm >0 , and set U = {t ∈ R≥0 | t i < c for all 1 ≤ i ≤ m }. Set (X v )U = X v ∩ X̄ F,v (P)U . If c is sufficiently small, then (Γa \(ZF,v (P) × D)) × (U ∩ Rm ) ,→ (Γa \(ZF,v (P) × D)) × U >0 is a homotopy equivalence, and we can apply χU−1,a to both sides to see that the inclusion map Γa \((X v )U × D) ,→ Γa \(X̄ F,v (P)U × D) is also a homotopy equivalence. By Theorem 4.6.1, this proves Theorem 4.6.3. Remark 4.6.4. Theorem 4.6.3 is well-viewed as a function field analogue of the homotopy equivalence for Borel-Serre spaces of [3]. 4.6.5. Theorem 4.1.4 remains true if we replace G = PGLV by G = SLV in 4.1.3 and 4.1.4. It also remains true if we replace G = PGLV by G = GLV and we replace X̄ in 4.1.4 in situation (I) (resp., (II)) by X̄ × X S 2 × (RS>0 × G (ASF )/K )1 58 (resp., X̄ × X S 2 × (RS>0 )1 ), where ( )1 denotes the kernel of ((a v )v ∈S , g ) 7→ |det(g )| Y av v ∈S (resp., (a v )v ∈S 7→ Y a v ), v ∈S and γ ∈ GLV (F ) (resp., γ ∈ ΓK ) acts on this kernel by multiplication by ((| det(γ)|v )v ∈S , γ) (resp., (|det(γ)|v )v ∈S ). Theorems 4.6.1 and 4.6.3 also remain true under these modifications. These modified ver- sions of the results are easily reduced to the original case G = PGLV . 4.7 Subjects related to this paper 4.7.1. In this subsection, as possibilities of future applications of this paper, we describe connections with the study of • toroidal compactifications of moduli spaces of Drinfeld modules (4.7.2–4.7.5) • the asymptotic behavior of Hodge structures and p -adic Hodge structures associated to a degenerating family of motives over a number field (4.7.6, 4.7.7), and • modular symbols over function fields (4.7.8, 4.7.9). 4.7.2. In [21], Pink constructed a compactification of the moduli space of Drinfeld modules that is similar to the Satake-Baily-Borel compactification of the moduli space of polarized abalian varieties. In a special case, it had been previously constructed by Kapranov [15]. In [20], Pink, sketched a method for constructing a compactification of the moduli space of Drinfeld modules that is similar to the toroidal compactification of the moduli space of polarized abelian varieties (in part, using ideas of K. Fujiwara). However, the details of the construction have not been published. Our plan for constructing toroidal compactifications seems to be different from that of [20]. 4.7.3. We give a rough outline of the relationship that we envision between this paper and the analytic theory of toroidal compactifications. Suppose that F is a function field, and fix a place v of F . Let O be the ring of all elements of F which are integral outside v . In [6], the notion of a Drinfeld O-module of rank d is defined, and the moduli space of such Drinfeld modules is constructed. Let Cv be the completion of an algebraic closure of Fv and let | |: Cv → R≥0 be the absolute value which extends the normalized absolute value of Fv . Let Ω ⊂ Pd −1 (Cv ) be the (d − 1)dimensional Drinfeld upper half-space consisting of all points (z 1 : · · · : z d ) ∈ Pd −1 (Cv ) such that (z i )1≤i ≤d is linearly independent over Fv . 59 For a compact open subgroup K of GLd (AvF ), the set of Cv -points of the moduli space M K of Drinfeld O-modules of rank d with K -level structure is expressed as M K (Cv ) = GLd (F )\(Ω × GLd (AvF )/K ) (see [6]). Consider the case V = F d in §3 and §4. We have a map Ω → X v which sends (z 1 : · · · : z d ) ∈ Ω Pd to the class of the the norm Vv = Fvd → R≥0 given by (a 1 , . . . , a d ) 7→ | i =1 a i z i | for a i ∈ Fv . This map induces a canonical continuous map (1) M K (Cv ) = GLd (F )\(Ω × GLd (AvF )/K ) → GLd (F )\(X v × GLd (AvF )/K ). The map (1) extends to a canonical continuous map ♭ (2) M̄ KKP (Cv ) → GLd (F )\(X̄ F,v × GLd (AvF )/K ), where M̄ KKP denotes the compactification of Kapranov and Pink of M K . In particular, M̄ KKP is ♭ related to X̄ F,v . On the other hand, the toroidal compactifications of M K should be related to X̄ F,v . If we denote by M̄ Ktor the projective limit of all toroidal compactifications of M K , then the map of (1) should extend to a canonical continuous map (3) M̄ Ktor (Cv ) → GLd (F )\(X̄ F,v × GLd (AvF )/K ). that fits in a commutative diagram M̄ Ktor (Cv ) GLd (F )\(X̄ F,v × GL(AvF )/K ) M KKP (Cv ) ♭ GLd (F )\(X̄ F,v × GLd (AvF )/K ). 4.7.4. The expected map of 4.7.3(3) is the analogue of the canonical continuous map from the projective limit of all toroidal compactifications of the moduli space of polarized abelian varieties to the reductive Borel-Serre compactification (see [10, 16]). From the point of view of our study, the reductive Borel-Serre compactification and X̄ F,v are enlargements of spaces of norms. A polarized abelian variety A gives a norm on the polarized Hodge structure associated to A (the Hodge metric). This relationship between a polarized abelian variety and a norm forms the foundation of the relationship between the toroidal compactifications of a moduli space of polarized abelian varieties and the reductive BorelSerre compactification. This is similar to the relationship between M K and the space of norms X v given by the map of 4.7.3(1), as well as the relationship between M̄ Ktor and X̄ F,v given by 4.7.3(3). 4.7.5. In the usual theory of toroidal compactifications, cone decompositions play an important role. In the toroidal compactifications of 4.7.3, the simplices of Bruhat-Tits buildings 60 ♭ (more precisely, the simplices contained in the fibers of X̄ F,v → X̄ F,v ) should play the role of the cones in cone decompositions. 4.7.6. We are preparing a paper in which our space X̄ F,S with F a number field and with S containing a non-archimedean place appears in the following way. Let F be a number field, and let Y be a polarized projective smooth variety over F . Let m m ≥ 0, and let V = H dR (Y ) be the de Rham cohomology. For a place v of F , let Vv = Fv ⊗F V . For an archimedean place v of F , it is well known that Vv has a Hodge metric. For v non- archimedean, we can under certain assumptions define a Hodge metric on Vv by the method illustrated in the example of 4.7.7 below. The [Fv : Qv ]-powers of these Hodge metrics for v ∈ S Q are norms and therefore provide an element of v ∈S X Vv . When Y degenerates, this element Q of v ∈S X Vv can converge to a boundary point of X̄ F,S . 4.7.7. Let Y be an elliptic curve over a number field F , and take m = 1. Let v be a non-archimedean place of F , and assume that Fv ⊗F Y is a Tate elliptic curve of q -invariant qv ∈ Fv× with |qv | < 1. Then the first log-crystalline cohomology group D of the special fiber of this elliptic curve is a free module of rank 2 over the Witt vectors W (k v ) with a basis (e 1 , e 2 ) on which the Frobenius ϕ acts as ϕ(e 1 ) = e 1 and ϕ(e 2 ) = p e 2 , where p = char k v . The first ℓ-adic étale cohomology group of this elliptic curve is a free module of rank 2 over Zℓ with a basis (e 1,ℓ , e 2,ℓ ) such that the inertia subgroup of Gal(F̄v /Fv ) fixes e 1 . The monodromy operator N satisfies N e 2 = ξ′v e 1 , N e 1 = 0, N e 2,ℓ = ξ′v e 1,ℓ , N e 1,ℓ = 0 where ξ′v = ordFv (qv ) > 0. The standard polarization 〈 , 〉 of the elliptic curve satisfies 〈e 1 , e 2 〉 = 1 and hence 〈N e 2, e 2 〉 = ξ′v , 〈e 1 , N −1 e 1 〉 = 1/ξ′v 〈N e 2,ℓ, e 2,ℓ 〉 = ξ′v , 〈e 1,ℓ , N −1 e 1,ℓ 〉 = 1/ξ′v . 1 For V = H dR (Y ), we have an isomorphism Vv ∼ = Fv ⊗W (k v ) D. The Hodge metric | |v on Vv is defined by |a 1 e 1 + a 2 e 2 |v = max(ξ−1/2 |a 1 |p , ξv1/2 |a 2 |p ) v for a 1 , a 2 ∈ Fv , where | |p denotes the absolute value on Fv satisfying |p |p = p −1 and ξv := −ξ′v log(|̟v |p ) = − log(|qv |p ) > 0, where ̟v is a prime element of Fv . That is, to define the Hodge metric on Vv , we modify the naive metric (coming from the integral structure of the log-crystalline cohomology) by using 61 ξv which is determined by the polarization 〈 , 〉, the monodromy operator N , and the integral structures of the log-crystalline and ℓ-adic cohomology groups (for ℓ 6= p ). This is similar to what happens at an archimedean place v . We have Y (C) ∼ = C× /qvZ with qv ∈ Fv× . Assume for simplicity that we can take |qv | < e −2π where | | denotes the usual abso- lute value. Then qv is determined by Fv ⊗F Y uniquely. Let ξ := − log(|qv |) > 2π. If v is real, we further assume that qv > 0 and that we have an isomorphism Y (Fv ) ∼ = F × /q Z which is compatv v ible with Y (C) ∼ = C× /qvZ . Then in the case v is real (resp., complex), there is a basis (e 1 , e 2 ) of Vv such that (e 1 , (2πi )−1 e 2 ) is a Z-basis of H 1 (Y (C), Z) and such that the Hodge metric | |v on Vv −1/2 satisfies |e 1 |v = ξv 1/2 1/2 −1/2 and |e 2 |v = ξv (resp., ||e 2 |v − ξv | ≤ πξv Consider for example the elliptic curves y2 ). = x (x − 1)(x − t ) with t ∈ F = Q, t 6= 0, 1. As t approaches 1 ∈ Qv for all v ∈ S, the elliptic curves Fv ⊗F Y satisfy the above assumptions for all Q v ∈ S, and each qv approaches 0, so ξv tends to ∞. The corresponding elements of v ∈S X Vv defined by the classes of the | |v for v ∈ S converge to the unique boundary point of X̄ F,S with associated parabolic equal to the Borel subgroup of upper triangular matrices in PGLV for the basis (e 1 , e 2 ). We hope that this subject about X̄ F,S is an interesting direction to be studied. It may be related to the asymptotic behaviors of heights of motives in degeneration studied in [18]. 4.7.8. Suppose that F is a function field and let v be a place of F . Let Γ be as in 1.3. Kondo and Yasuda [17] proved that the image of H d −1(Γ\X v , Q) → H dBM −1 (Γ\X v , Q) is gener- ated by modular symbols, where H ∗BM denotes Borel-Moore homology. Our hope is that the compactification Γ\X̄ F,v of Γ\X v is useful in further studies of modular symbols. Let ∂ := X̄ F,v \ X v . Then we have an isomorphism H ∗BM (Γ\X v , Q) ∼ = H ∗ (Γ\X̄ F,v , Γ\∂ , Q) and an exact sequence · · · → H i (Γ\X̄ F,v , Q) → H i (Γ\X̄ F,v , Γ\∂ , Q) → H i −1 (Γ\∂ , Q) → H i −1 (Γ\X̄ F,v , Q) → . . . . Since Γ\X v → Γ\X̄ F,v is a homotopy equivalence by Thm. 4.6.3, we have ∼ H ∗ (Γ\X v , Q) − → H ∗ (Γ\X̄ F,v , Q). Hence the result of Kondo and Yasuda shows that the kernel of H dBM (Γ\X v , Q) ∼ = H d −1 (Γ\X̄ F,v , Γ\∂ , Q) → H d −2 (Γ\∂ , Q) −1 is generated by modular symbols. 62 If we want to prove that H dBM −1 (Γ\X v , Q) is generated by modular symbols, it is now sufficient to prove that the kernel of H d −2 (Γ\∂ , Q) → H d −2 (Γ\X̄ F,v , Q) is generated by the images (i.e., boundaries) of modular symbols. 4.7.9. In 4.7.8, assume d = 2. Then we can prove that H 1BM (Γ\X v , Q) is generated by modular symbols. In this case, the map H 0 (Γ\∂ , Q) = Map(Γ\∂ , Q) → H 0 (X̄ F,v , Q) = Q is just summation, and it is clear that the kernel of it is generated by the boundaries of modular symbols. References [1] BOREL, A., Some finiteness properties of adele groups over number fields, Publ. Math. Inst. Hautes Études Sci. 16 (1963), 5–30. [2] BOREL, A., JI, L., Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser, Boston, MA, 2006. [3] BOREL, A., SERRE J.P., Corners and arithmetic groups, Comment. Math. Helv. 4 (1973), 436– 491. [4] BRUHAT, F., TITS, J., Groupes réductifs sur un corps local: I. Données radicielles valuées, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251. [5] DELIGNE, P., HUSEMÖLLER, D., Survey of Drinfel’d modules, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987, 25–91. [6] DRINFELD, V.G., Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian), English translation: Math. USSR-Sb. 23 (1974), 561–592. [7] GÉRARDIN, P., Harmonic functions on buildings of reductive split groups, Operator algebras and group representations, Monogr. Stud. Math. 17, Pitman, Boston, MA, 1984, 208– 221. [8] GODEMENT, R., Domaines fondamentaux des groupes arithmétiques, Séminaire Bourbaki 8, no. 257 (1962–64), 201–205. [9] GOLDMAN, O., IWAHORI, N., The space of p -adic norms, Acta Math. 109 (1963), 137–177. [10] GORESKY, M., TAI, Y., Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces, Amer. J. Math. 121 (1999), 1095–1151. 63 [11] GUIVARC’H, Y., RÉMY, B., Group-theoretic compactifcation of Bruhat-Tits buildings, Ann. Sci. Éc. Norm. Supér. 39 (2006), 871–920. [12] HARDER, G. Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969) 33–54. [13] HARDER, G., Chevalley groups over function fields and automorphic forms, Ann. of Math. 100 (1974), 249–306. [14] JI, L., MURTY, V.K., SAPER, L., SCHERK, J. The fundamental group of reductive Borel-Serre and Satake compactifications, Asian J. Math. 19 (2015), 465–485. [15] KAPRANOV, M.M., Cuspidal divisors on the modular varieties of elliptic modules, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 568–583, 688, English translation: Math. USSR-Izv. 30 (1988), 533–547. [16] KATO, K., USUI, S., Classifying spaces of degenerating polarized Hodge structures, Ann. of Math. Stud., Princeton Univ. Press, 2009. [17] KONDO, S., YASUDA, S., The Borel-Moore homology of an arithmetic quotient of the BruhatTits building of PGL of a non-archimedean local field in positive characteristic and modular symbols, preprint, arXiv:1406.7047. [18] KOSHIKAWA, T., On heights of motives with semistable reduction, preprint, arXiv:1505.01873. [19] L ANDVOGT, E., A compactification of the Bruhat-Tits building, Lecture Notes in Math. 1619, Springer-Verlag, Berlin, 1996. [20] PINK, R., On compactification of Drinfeld moduli schemes, Sûrikaisekikenkyûsho Kôkyûroku 884 (1994), 178–183. [21] PINK, R., Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank, Manuscripta Math. 140 (2013), 333–361. [22] SATAKE, I., On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. 71 (1960), 77–110. [23] SATAKE, I., On compactifications of the quotient spaces for arithmetically defined discontinuous groups, Ann. of Math. 72 (1960), 555–580. 64 [24] WERNER, A., Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank, Doc. Math. 6 (2001), 315–341. [25] WERNER, A., Compactification of the Bruhat-Tits building of PGL by semi-norms, Math. Z. 248 (2004), 511–526. [26] ZUCKER, S., L 2 -cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982), 169–218. [27] ZUCKER, S., Satake compactifications, Comment. Math. Helv. 58 (1983), 312–343. 65 

Learning to learn by gradient descent by gradient descent arXiv:1606.04474v2 [] 30 Nov 2016 Marcin Andrychowicz1 , Misha Denil1 , Sergio Gómez Colmenarejo1 , Matthew W. Hoffman1 , David Pfau1 , Tom Schaul1 , Brendan Shillingford1,2 , Nando de Freitas1,2,3 1 Google DeepMind 2 University of Oxford 3 Canadian Institute for Advanced Research marcin.andrychowicz@gmail.com {mdenil,sergomez,mwhoffman,pfau,schaul}@google.com brendan.shillingford@cs.ox.ac.uk, nandodefreitas@google.com Abstract The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a number of tasks, including simple convex problems, training neural networks, and styling images with neural art. 1 Introduction Frequently, tasks in machine learning can be expressed as the problem of optimizing an objective function f (θ) defined over some domain θ ∈ Θ. The goal in this case is to find the minimizer θ∗ = arg minθ∈Θ f (θ). While any method capable of minimizing this objective function can be applied, the standard approach for differentiable functions is some form of gradient descent, resulting in a sequence of updates θt+1 = θt − αt ∇f (θt ) . The performance of vanilla gradient descent, however, is hampered by the fact that it only makes use of gradients and ignores second-order information. Classical optimization techniques correct this behavior by rescaling the gradient step using curvature information, typically via the Hessian matrix of second-order partial derivatives—although other choices such as the generalized Gauss-Newton matrix or Fisher information matrix are possible. Much of the modern work in optimization is based around designing update rules tailored to specific classes of problems, with the types of problems of interest differing between different research communities. For example, in the deep learning community we have seen a proliferation of optimization methods specialized for high-dimensional, non-convex optimization problems. These include momentum [Nesterov, 1983, Tseng, 1998], Rprop [Riedmiller and Braun, 1993], Adagrad [Duchi et al., 2011], RMSprop [Tieleman and Hinton, 2012], and ADAM [Kingma and Ba, 2015]. More focused methods can also be applied when more structure of the optimization problem is known [Martens and Grosse, 2015]. In contrast, communities who focus on sparsity tend to favor very different approaches [Donoho, 2006, Bach et al., 2012]. This is even more the case for combinatorial optimization for which relaxations are often the norm [Nemhauser and Wolsey, 1988]. 30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. This industry of optimizer design allows differameter update s par ent communities to create optimization methods which exploit structure in their problems of interest at the expense of potentially poor performance on problems outside of that scope. Moreover the No Free Lunch Theorems for Optimization [Wolpert and Macready, 1997] show that in the setting of combinatorial optimization, no algorithm is able to do better than a random optimizer optimizee strategy in expectation. This suggests that speerror signal cialization to a subclass of problems is in fact the only way that improved performance can be achieved in general. Figure 1: The optimizer (left) is provided with In this work we take a different tack and instead performance of the optimizee (right) and proposes propose to replace hand-designed update rules updates to increase the optimizee’s performance. with a learned update rule, which we call the op- [photos: Bobolas, 2009, Maley, 2011] timizer g, specified by its own set of parameters φ. This results in updates to the optimizee f of the form θt+1 = θt + gt (∇f (θt ), φ) . (1) A high level view of this process is shown in Figure 1. In what follows we will explicitly model the update rule g using a recurrent neural network (RNN) which maintains its own state and hence dynamically updates as a function of its iterates. 1.1 Transfer learning and generalization The goal of this work is to develop a procedure for constructing a learning algorithm which performs well on a particular class of optimization problems. Casting algorithm design as a learning problem allows us to specify the class of problems we are interested in through example problem instances. This is in contrast to the ordinary approach of characterizing properties of interesting problems analytically and using these analytical insights to design learning algorithms by hand. It is informative to consider the meaning of generalization in this framework. In ordinary statistical learning we have a particular function of interest, whose behavior is constrained through a data set of example function evaluations. In choosing a model we specify a set of inductive biases about how we think the function of interest should behave at points we have not observed, and generalization corresponds to the capacity to make predictions about the behavior of the target function at novel points. In our setting the examples are themselves problem instances, which means generalization corresponds to the ability to transfer knowledge between different problems. This reuse of problem structure is commonly known as transfer learning, and is often treated as a subject in its own right. However, by taking a meta-learning perspective, we can cast the problem of transfer learning as one of generalization, which is much better studied in the machine learning community. One of the great success stories of deep-learning is that we can rely on the ability of deep networks to generalize to new examples by learning interesting sub-structures. In this work we aim to leverage this generalization power, but also to lift it from simple supervised learning to the more general setting of optimization. 1.2 A brief history and related work The idea of using learning to learn or meta-learning to acquire knowledge or inductive biases has a long history [Thrun and Pratt, 1998]. More recently, Lake et al. [2016] have argued forcefully for its importance as a building block in artificial intelligence. Similarly, Santoro et al. [2016] frame multi-task learning as generalization, however unlike our approach they directly train a base learner rather than a training algorithm. In general these ideas involve learning which occurs at two different time scales: rapid learning within tasks and more gradual, meta learning across many different tasks. Perhaps the most general approach to meta-learning is that of Schmidhuber [1992, 1993]—building on work from [Schmidhuber, 1987]—which considers networks that are able to modify their own weights. Such a system is differentiable end-to-end, allowing both the network and the learning 2 algorithm to be trained jointly by gradient descent with few restrictions. However this generality comes at the expense of making the learning rules very difficult to train. Alternatively, the work of Schmidhuber et al. [1997] uses the Success Story Algorithm to modify its search strategy rather than gradient descent; a similar approach has been recently taken in Daniel et al. [2016] which uses reinforcement learning to train a controller for selecting step-sizes. Bengio et al. [1990, 1995] propose to learn updates which avoid back-propagation by using simple parametric rules. In relation to the focus of this paper the work of Bengio et al. could be characterized as learning to learn without gradient descent by gradient descent. The work of Runarsson and Jonsson [2000] builds upon this work by replacing the simple rule with a neural network. Cotter and Conwell [1990], and later Younger et al. [1999], also show fixed-weight recurrent neural networks can exhibit dynamic behavior without need to modify their network weights. Similarly this has been shown in a filtering context [e.g. Feldkamp and Puskorius, 1998], which is directly related to simple multi-timescale optimizers [Sutton, 1992, Schraudolph, 1999]. Finally, the work of Younger et al. [2001] and Hochreiter et al. [2001] connects these different threads of research by allowing for the output of backpropagation from one network to feed into an additional learning network, with both networks trained jointly. Our approach to meta-learning builds on this work by modifying the network architecture of the optimizer in order to scale this approach to larger neural-network optimization problems. 2 Learning to learn with recurrent neural networks In this work we consider directly parameterizing the optimizer. As a result, in a slight abuse of notation we will write the final optimizee parameters θ∗ (f, φ) as a function of the optimizer parameters φ and the function in question. We can then ask the question: What does it mean for an optimizer to be good? Given a distribution of functions f we will write the expected loss as h
i L(φ) = Ef f θ∗ (f, φ) . (2) As noted earlier, we will take the update steps gt to be the output of a recurrent neural network m, parameterized by φ, whose state we will denote explicitly with ht . Next, while the objective function in (2) depends only on the final parameter value, for training the optimizer it will be convenient to have an objective that depends on the entire trajectory of optimization, for some horizon T, L(φ) = Ef " T X # wt f (θt ) where t=1 θt+1 = θt + gt ,  gt = m(∇t , ht , φ) . ht+1 (3)  Here wt ∈ R≥0 are arbitrary weights associated with each time-step and we will also use the notation ∇t = ∇θ f (θt ). This formulation is equivalent to (2) when wt = 1[t = T ], but later we will describe why using different weights can prove useful. We can minimize the value of L(φ) using gradient descent on φ. The gradient estimate ∂L(φ)/∂φ can be computed by sampling a random function f and applying backpropagation to the computational graph in Figure 2. We allow gradients to flow along the solid edges in the graph, but gradients along the dashed edges are dropped. Ignoring gradients along the dashed edges amounts to making the assumption that the gradients of the optimizee do not depend on the optimizer parameters, i.e.  ∂∇t ∂φ = 0. This assumption allows us to avoid computing second derivatives of f . Examining the objective in (3) we see that the gradient is non-zero only for terms where wt 6= 0. If we use wt = 1[t = T ] to match the original problem, then gradients of trajectory prefixes are zero and only the final optimization step provides information for training the optimizer. This renders Backpropagation Through Time (BPTT) inefficient. We solve this problem by relaxing the objective such that wt > 0 at intermediate points along the trajectory. This changes the objective function, but allows us to train the optimizer on partial trajectories. For simplicity, in all our experiments we use wt = 1 for every t. 3 t-2 ft-2 Optimizee θt-2 t-1 ft-1 θt-1 + ht-2 gt ∇t-1 m θt+1 + gt-1 ∇t-2 Optimizer θt + gt-2 t ft ∇t m ht-1 m ht ht+1 Figure 2: Computational graph used for computing the gradient of the optimizer. 2.1 Coordinatewise LSTM optimizer One challenge in applying RNNs in our setting is that we want to be able to optimize at least tens of thousands of parameters. Optimizing at this scale with a fully connected RNN is not feasible as it would require a huge hidden state and an enormous number of parameters. To avoid this difficulty we will use an optimizer m which operates coordinatewise on the parameters of the objective function, similar to other common update rules like RMSprop and ADAM. This coordinatewise network architecture allows us to use a very small network that only looks at a single coordinate to define the optimizer and share optimizer parameters across different parameters of the optimizee. Different behavior on each coordinate is achieved by using separate activations for each objective function parameter. In addition to allowing us to use a small network for this optimizer, this setup has the nice effect of making the optimizer invariant to the order of parameters in the network, since the same update rule is used independently on each coordinate. ∇1 θ1 LSTMn + … ∇n LSTM1 … … … θn f … We implement the update rule for each coordinate using a two-layer Long Short Term Memory (LSTM) network [Hochreiter and Schmidhuber, 1997], using the now-standard forget gate architecture. The network takes as input the optimizee gradient for a single coordinate as well as the previous hidden state and outputs the update for the corresponding optimizee parameter. We will refer to this architecture, illustrated in Figure 3, as an LSTM optimizer. + The use of recurrence allows the LSTM to learn dynamic update rules which integrate informa- Figure 3: One step of an LSTM optimizer. All tion from the history of gradients, similar to LSTMs have shared parameters, but separate hidmomentum. This is known to have many desir- den states. able properties in convex optimization [see e.g. Nesterov, 1983] and in fact many recent learning procedures—such as ADAM—use momentum in their updates. Preprocessing and postprocessing Optimizer inputs and outputs can have very different magnitudes depending on the class of function being optimized, but neural networks usually work robustly only for inputs and outputs which are neither very small nor very large. In practice rescaling inputs and outputs of an LSTM optimizer using suitable constants (shared across all timesteps and functions f ) is sufficient to avoid this problem. In Appendix A we propose a different method of preprocessing inputs to the optimizer inputs which is more robust and gives slightly better performance. 4 Quadratics 10 1 Loss MNIST MNIST, 200 steps ADAM RMSprop 0 10 SGD NAG LSTM 10 0 10 -1 20 40 60 80 100 20 40 60 Step 80 100 120 140 160 180 200 Figure 4: Comparisons between learned and hand-crafted optimizers performance. Learned optimizers are shown with solid lines and hand-crafted optimizers are shown with dashed lines. Units for the y axis in the MNIST plots are logits. Left: Performance of different optimizers on randomly sampled 10-dimensional quadratic functions. Center: the LSTM optimizer outperforms standard methods training the base network on MNIST. Right: Learning curves for steps 100-200 by an optimizer trained to optimize for 100 steps (continuation of center plot). 3 Experiments In all experiments the trained optimizers use two-layer LSTMs with 20 hidden units in each layer. Each optimizer is trained by minimizing Equation 3 using truncated BPTT as described in Section 2. The minimization is performed using ADAM with a learning rate chosen by random search. We use early stopping when training the optimizer in order to avoid overfitting the optimizer. After each epoch (some fixed number of learning steps) we freeze the optimizer parameters and evaluate its performance. We pick the best optimizer (according to the final validation loss) and report its average performance on a number of freshly sampled test problems. We compare our trained optimizers with standard optimizers used in Deep Learning: SGD, RMSprop, ADAM, and Nesterov’s accelerated gradient (NAG). For each of these optimizer and each problem we tuned the learning rate, and report results with the rate that gives the best final error for each problem. When an optimizer has more parameters than just a learning rate (e.g. decay coefficients for ADAM) we use the default values from the optim package in Torch7. Initial values of all optimizee parameters were sampled from an IID Gaussian distribution. 3.1 Quadratic functions In this experiment we consider training an optimizer on a simple class of synthetic 10-dimensional quadratic functions. In particular we consider minimizing functions of the form f (θ) = kW θ − yk22 for different 10x10 matrices W and 10-dimensional vectors y whose elements are drawn from an IID Gaussian distribution. Optimizers were trained by optimizing random functions from this family and tested on newly sampled functions from the same distribution. Each function was optimized for 100 steps and the trained optimizers were unrolled for 20 steps. We have not used any preprocessing, nor postprocessing. Learning curves for different optimizers, averaged over many functions, are shown in the left plot of Figure 4. Each curve corresponds to the average performance of one optimization algorithm on many test functions; the solid curve shows the learned optimizer performance and dashed curves show the performance of the standard baseline optimizers. It is clear the learned optimizers substantially outperform the baselines in this setting. 3.2 Training a small neural network on MNIST In this experiment we test whether trainable optimizers can learn to optimize a small neural network on MNIST, and also explore how the trained optimizers generalize to functions beyond those they were trained on. To this end, we train the optimizer to optimize a base network and explore a series of modifications to the network architecture and training procedure at test time. 5 Loss MNIST, 40 units 10 0 20 40 60 MNIST, 2 layers ADAM RMSprop SGD NAG LSTM 80 100 20 40 60 Steps MNIST, ReLU 80 100 20 40 60 80 100 Figure 5: Comparisons between learned and hand-crafted optimizers performance. Units for the y axis are logits. Left: Generalization to the different number of hidden units (40 instead of 20). Center: Generalization to the different number of hidden layers (2 instead of 1). This optimization problem is very hard, because the hidden layers are very narrow. Right: Training curves for an MLP with 20 hidden units using ReLU activations. The LSTM optimizer was trained on an MLP with sigmoid activations. Final Loss LSTM ADAM NAG 10 0 Layers 10 -1 1 2 5 20 35 50 5 20 35 Hidden units 50 5 20 35 50 Figure 6: Systematic study of final MNIST performance as the optimizee architecture is varied, using sigmoid non-linearities. The vertical dashed line in the left-most plot denotes the architecture at which the LSTM is trained and the horizontal line shows the final performance of the trained optimizer in this setting. In this setting the objective function f (θ) is the cross entropy of a small MLP with parameters θ. The values of f as well as the gradients ∂f (θ)/∂θ are estimated using random minibatches of 128 examples. The base network is an MLP with one hidden layer of 20 units using a sigmoid activation function. The only source of variability between different runs is the initial value θ0 and randomness in minibatch selection. Each optimization was run for 100 steps and the trained optimizers were unrolled for 20 steps. We used input preprocessing described in Appendix A and rescaled the outputs of the LSTM by the factor 0.1. Learning curves for the base network using different optimizers are displayed in the center plot of Figure 4. In this experiment NAG, ADAM, and RMSprop exhibit roughly equivalent performance the LSTM optimizer outperforms them by a significant margin. The right plot in Figure 4 compares the performance of the LSTM optimizer if it is allowed to run for 200 steps, despite having been trained to optimize for 100 steps. In this comparison we re-used the LSTM optimizer from the previous experiment, and here we see that the LSTM optimizer continues to outperform the baseline optimizers on this task. Generalization to different architectures Figure 5 shows three examples of applying the LSTM optimizer to train networks with different architectures than the base network on which it was trained. The modifications are (from left to right) (1) an MLP with 40 hidden units instead of 20, (2) a network with two hidden layers instead of one, and (3) a network using ReLU activations instead of sigmoid. In the first two cases the LSTM optimizer generalizes well, and continues to outperform the hand-designed baselines despite operating outside of its training regime. However, changing the activation function to ReLU makes the dynamics of the learning procedure sufficiently different that the learned optimizer is no longer able to generalize. Finally, in Figure 6 we show the results of systematically varying the tested architecture; for the LSTM results we again used the optimizer trained using 1 layer of 20 units and sigmoid non-linearities. Note that in this setting where the 6 CIFAR-10 CIFAR-5 CIFAR-2 ADAM RMSprop SGD NAG LSTM LSTM-sub Loss 10 0 10 -1 10 0 200 400 600 800 1000 200 400 600 Step 800 1000 200 400 600 800 1000 Figure 7: Optimization performance on the CIFAR-10 dataset and subsets. Shown on the left is the LSTM optimizer versus various baselines trained on CIFAR-10 and tested on a held-out test set. The two plots on the right are the performance of these optimizers on subsets of the CIFAR labels. The additional optimizer LSTM-sub has been trained only on the heldout labels and is hence transferring to a completely novel dataset. test-set problems are similar enough to those in the training set we see even better generalization than the baseline optimizers. 3.3 Training a convolutional network on CIFAR-10 Next we test the performance of the trained neural optimizers on optimizing classification performance for the CIFAR-10 dataset [Krizhevsky, 2009]. In these experiments we used a model with both convolutional and feed-forward layers. In particular, the model used for these experiments includes three convolutional layers with max pooling followed by a fully-connected layer with 32 hidden units; all non-linearities were ReLU activations with batch normalization. The coordinatewise network decomposition introduced in Section 2.1—and used in the previous experiment—utilizes a single LSTM architecture with shared weights, but separate hidden states, for each optimizee parameter. We found that this decomposition was not sufficient for the model architecture introduced in this section due to the differences between the fully connected and convolutional layers. Instead we modify the optimizer by introducing two LSTMs: one proposes parameter updates for the fully connected layers and the other updates the convolutional layer parameters. Like the previous LSTM optimizer we still utilize a coordinatewise decomposition with shared weights and individual hidden states, however LSTM weights are now shared only between parameters of the same type (i.e. fully-connected vs. convolutional). The performance of this trained optimizer compared against the baseline techniques is shown in Figure 7. The left-most plot displays the results of using the optimizer to fit a classifier on a held-out test set. The additional two plots on the right display the performance of the trained optimizer on modified datasets which only contain a subset of the labels, i.e. the CIFAR-2 dataset only contains data corresponding to 2 of the 10 labels. Additionally we include an optimizer LSTM-sub which was only trained on the held-out labels. In all these examples we can see that the LSTM optimizer learns much more quickly than the baseline optimizers, with significant boosts in performance for the CIFAR-5 and especially CIFAR-2 datsets. We also see that the optimizers trained only on a disjoint subset of the data is hardly effected by this difference and transfers well to the additional dataset. 3.4 Neural Art The recent work on artistic style transfer using convolutional networks, or Neural Art [Gatys et al., 2015], gives a natural testbed for our method, since each content and style image pair gives rise to a different optimization problem. Each Neural Art problem starts from a content image, c, and a style image, s, and is given by f (θ) = αLcontent (c, θ) + βLstyle (s, θ) + γLreg (θ) The minimizer of f is the styled image. The first two terms try to match the content and style of the styled image to that of their first argument, and the third term is a regularizer that encourages smoothness in the styled image. Details can be found in [Gatys et al., 2015]. 7 Double resolution Loss Neural art, training resolution 20 40 60 Step 80 100 120 20 40 60 Step ADAM RMSprop SGD NAG LSTM 80 100 120 Figure 8: Optimization curves for Neural Art. Content images come from the test set, which was not used during the LSTM optimizer training. Note: the y-axis is in log scale and we zoom in on the interesting portion of this plot. Left: Applying the training style at the training resolution. Right: Applying the test style at double the training resolution. Figure 9: Examples of images styled using the LSTM optimizer. Each triple consists of the content image (left), style (right) and image generated by the LSTM optimizer (center). Left: The result of applying the training style at the training resolution to a test image. Right: The result of applying a new style to a test image at double the resolution on which the optimizer was trained. We train optimizers using only 1 style and 1800 content images taken from ImageNet [Deng et al., 2009]. We randomly select 100 content images for testing and 20 content images for validation of trained optimizers. We train the optimizer on 64x64 content images from ImageNet and one fixed style image. We then test how well it generalizes to a different style image and higher resolution (128x128). Each image was optimized for 128 steps and trained optimizers were unrolled for 32 steps. Figure 9 shows the result of styling two different images using the LSTM optimizer. The LSTM optimizer uses inputs preprocessing described in Appendix A and no postprocessing. See Appendix C for additional images. Figure 8 compares the performance of the LSTM optimizer to standard optimization algorithms. The LSTM optimizer outperforms all standard optimizers if the resolution and style image are the same as the ones on which it was trained. Moreover, it continues to perform very well when both the resolution and style are changed at test time. Finally, in Appendix B we qualitatively examine the behavior of the step directions generated by the learned optimizer. 4 Conclusion We have shown how to cast the design of optimization algorithms as a learning problem, which enables us to train optimizers that are specialized to particular classes of functions. Our experiments have confirmed that learned neural optimizers compare favorably against state-of-the-art optimization methods used in deep learning. We witnessed a remarkable degree of transfer, with for example the LSTM optimizer trained on 12,288 parameter neural art tasks being able to generalize to tasks with 49,152 parameters, different styles, and different content images all at the same time. We observed similar impressive results when transferring to different architectures in the MNIST task. The results on the CIFAR image labeling task show that the LSTM optimizers outperform handengineered optimizers when transferring to datasets drawn from the same data distribution. References F. Bach, R. Jenatton, J. Mairal, and G. Obozinski. Optimization with sparsity-inducing penalties. Foundations and Trends in Machine Learning, 4(1):1–106, 2012. 8 S. Bengio, Y. Bengio, and J. Cloutier. On the search for new learning rules for ANNs. Neural Processing Letters, 2(4):26–30, 1995. Y. Bengio, S. Bengio, and J. Cloutier. Learning a synaptic learning rule. 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This can make training an optimizer difficult, because neural networks naturally disregard small variations in input signals and concentrate on bigger input values. To this aim we propose to preprocess the optimizer’s inputs. One solution would be to give the optimizer (log(|∇|), sgn(∇)) as an input, where ∇ is the gradient in the current timestep. This has a problem that log(|∇|) diverges for ∇ → 0. Therefore, we use the following preprocessing formula  ( log(|∇|) , sgn(∇) if |∇| ≥ e−p p ∇k → (−1, ep ∇) otherwise where p > 0 is a parameter controlling how small gradients are disregarded (we use p = 10 in all our experiments). We noticed that just rescaling all inputs by an appropriate constant instead also works fine, but the proposed preprocessing seems to be more robust and gives slightly better results on some problems. B Visualizations Visualizing optimizers is inherently difficult because their proposed updates are functions of the full optimization trajectory. In this section we try to peek into the decisions made by the LSTM optimizer, trained on the neural art task. Histories of updates We select a single optimizee parameter (one color channel of one pixel in the styled image) and trace the updates proposed to this coordinate by the LSTM optimizer over a single trajectory of optimization. We also record the updates that would have been proposed by both SGD and ADAM if they followed the same trajectory of iterates. Figure 10 shows the trajectory of updates for two different optimizee parameters. From the plots it is clear that the trained optimizer makes bigger updates than SGD and ADAM. It is also visible that it uses some kind of momentum, but its updates are more noisy than those proposed by ADAM which may be interpreted as having a shorter time-scale momentum. Proposed update as a function of current gradient Another way to visualize the optimizer behavior is to look at the proposed update gt for a single coordinate as a function of the current gradient evaluation ∇t . We follow the same procedure as in the previous experiment, and visualize the proposed updates for a few selected time steps. These results are shown in Figures 11–13. In these plots, the x-axis is the current value of the gradient for the chosen coordinate, and the y-axis shows the update that each optimizer would propose should the corresponding gradient value be observed. The history of gradient observations is the same for all methods and follows the trajectory of the LSTM optimizer. 11 The shape of this function for the LSTM optimizer is often step-like, which is also the case for ADAM. Surprisingly the step is sometimes in the opposite direction as for ADAM, i.e. the bigger the gradient, the bigger the update. C Neural Art Shown below are additional examples of images styled using the LSTM optimizer. Each triple consists of the content image (left), style (right) and image generated by the LSTM optimizer (center). D Information sharing between coordinates In previous sections we considered a coordinatewise architecture, which corresponds by analogy to a learned version of RMSprop or ADAM. Although diagonal methods are quite effective in practice, we can also consider learning more sophisticated optimizers that take the correlations between coordinates into effect. To this end, we introduce a mechanism allowing different LSTMs to communicate with each other. Global averaging cells The simplest solution is to designate a subset of the cells in each LSTM layer for communication. These cells operate like normal LSTM cells, but their outgoing activations are averaged at each step across all coordinates. These global averaging cells (GACs) are sufficient to allow the networks to implement L2 gradient clipping [Bengio et al., 2013] assuming that each LSTM can compute the square of the gradient. This architecture is denoted as an LSTM+GAC optimizer. NTM-BFGS optimizer We also consider augmenting the LSTM+GAC architecture with an external memory that is shared between coordinates. Such a memory, if appropriately designed could allow the optimizer to learn algorithms similar to (low-memory) approximations to Newton’s method, e.g. (L-)BFGS [see Nocedal and Wright, 2006]. The reason for this interpretation is that such methods can be seen as a set of independent processes working coordinatewise, but communicating through the inverse Hessian approximation stored in the memory. We designed a memory architecture that, in theory, allows the 12 10 0 −10 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Step 13 Step 14 Step 15 Step 16 Step 17 Step 18 Step 19 Step 20 Step 21 Step 22 Step 23 Step 24 Step 25 Step 26 Step 27 Step 28 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 −400 0 400 Step 29 −400 0 400 Step 30 −400 0 400 Step 31 −400 0 400 Step 32 Figure 11: The proposed update direction for a single coordinate over 32 steps. 13 10 0 −10 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Step 13 Step 14 Step 15 Step 16 Step 17 Step 18 Step 19 Step 20 Step 21 Step 22 Step 23 Step 24 Step 25 Step 26 Step 27 Step 28 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 −400 0 400 Step 29 −400 0 400 Step 30 −400 0 400 Step 31 −400 0 400 Step 32 Figure 12: The proposed update direction for a single coordinate over 32 steps. 14 10 0 −10 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Step 13 Step 14 Step 15 Step 16 Step 17 Step 18 Step 19 Step 20 Step 21 Step 22 Step 23 Step 24 Step 25 Step 26 Step 27 Step 28 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 10 0 −10 −400 0 400 Step 29 −400 0 400 Step 30 −400 0 400 Step 31 −400 0 400 Step 32 Figure 13: The proposed update direction for a single coordinate over 32 steps. 15 LSTM r1 i1 a1 LSTM b1 LSTM r2 i2 a2 LSTM b2 LSTM r3 i3 a3 LSTM b3 LSTM … … … LSTM … LSTM … … … LSTM … LSTM … … … LSTM … LSTM … … … LSTM … LSTM rn in an LSTM bn x M = Mt + Outer Product ΔMt = Mt+1 Figure 14: Left: NTM-BFGS read operation. Right: NTM-BFGS write operation. network to simulate (L-)BFGS, motivated by the approximate Newton method BFGS, named for Broyden, Fletcher, Goldfarb, and Shanno. We call this architecture an NTM-BFGS optimizer, because its use of external memory is similar to the Neural Turing Machine [Graves et al., 2014]. The pivotal differences between our construction and the NTM are (1) our memory allows only low-rank updates; (2) the controller (including read/write heads) operates coordinatewise. In BFGS an explicit estimate of the full (inverse) Hessian is built up from the sequence of observed gradients. We can write a skeletonized version of the BFGS algorithm, using Mt to represent the inverse Hessian approximation at iteration t, as follows gt = read(Mt , θt ) θt+1 = θt + gt Mt+1 = write(Mt , θt , gt ) . Here we have packed up all of the details of the BFGS algorithm into the suggestively named read and write operations, which operate on the inverse Hessian approximation Mt . In BFGS these operations have specific forms, for example read(Mt , θt ) = −Mt ∇f (θt ) is a specific matrix-vector multiplication and the BFGS write operation corresponds to a particular low-rank update of Mt . In this work we preserve the structure of the BFGS updates, but discard their particular form. More specifically the read operation remains a matrix-vector multiplication but the form of the vector used is learned. Similarly, the write operation remains a low-rank update, but the vectors involved are also learned. Conveniently, this structure of interaction with a large dynamically updated state corresponds in a fairly direct way to the architecture of a Neural Turing Machine (NTM), where Mt corresponds to the NTM memory [Graves et al., 2014]. Our NTM-BFGS optimizer uses an LSTM+GAC as a controller; however, instead of producing the update directly we attach one or more read and write heads to the controller. Each read head produces a read vector rt which is combined with the memory to produce a read result it which is fed back into the controller at the following time step. Each write head produces two outputs, a left write vector at and a right write vector bt . The two write vectors are used to update the memory state by accumulating their outer product. The read and write operation for a single head is diagrammed in Figure 14 and the way read and write heads are attached to the controller is depicted in Figure 15. In can be shown that NTM-BFGS with one read head and 3 write heads can simulate inverse Hessian BFGS assuming that the controller can compute arbitrary (coordinatewise) functions and have access to 2 GACs. NTM-L-BFGS optimizer In cases where memory is constrained we can follow the example of L-BFGS and maintain a low rank approximation of the full memory (vis. inverse Hessian). The simplest way to do this is to store a sliding history of the left and right write vectors, allowing us to form the matrix vector multiplication required by the read operation efficiently. 16 gt ht Read it Controller … hk ht+1 LSTM1 LSTM2 ∇t rt gk rt+1 at bt Mt+1 Write k ∇ ik hk LSTMk rk ak bk Mt Figure 15: Left: Interaction between the controller and the external memory in NTM-BFGS. The controller is composed of replicated coordinatewise LSTMs (possibly with GACs), but the read and write operations are global across all coordinates. Right: A single LSTM for the kth coordinate in the NTM-BFGS controller. Note that here we have dropped the time index t to simplify notation. 17 

arXiv:1501.03411v4 [] 10 Dec 2015 PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS NEIL EPSTEIN AND JAY SHAPIRO Abstract. We introduce a new class of integral domains, the perinormal domains, which fall strictly between Krull domains and weakly normal domains. We establish basic properties of the class, and in the case of universally catenary domains we give equivalent characterizations of perinormality. (Later on, we point out some subtleties that occur only in the non-Noetherian context.) We also introduce and explore briefly the related concept of global perinormality, including a relationship with divisor class groups. Throughout, we provide illuminating examples from algebra, geometry, and number theory. 1. Introduction Motivated in part by the classical concept of a ring extension satisfying going-down from Cohen and Seidenberg [CS46], the concept of the goingdown domain has been fruitful in non-Noetherian commutative ring theory (see for example [Dob73, Dob74, DP76]); for Noetherian rings it merely coincides with domains of dimension ≤ 1 [Dob73, Proposition 7]. By definition, a ring extension R ⊆ S satisfies going-down if whenever p ⊂ q are prime ideals of R and Q ∈ Spec S with Q ∩ R = q, there is some prime ideal P ∈ Spec S with P ⊂ Q and P ∩ R = p (a condition that is satisfied whenever S is flat over R). Then an integral domain R is a going-down domain if for every (local) overring S of R, the inclusion R ⊆ S satisfies going-down. (In fact by [DP76, Theorem 1] it doesn’t matter whether one specifies ‘local’ or not.) It is natural to ask which overrings of an integral domain R satisfy goingdown over it. It is classical that any flat R-algebra (hence any flat overring) will satisfy going-down over R [Mat86, Theorem 9.5]. Moreover, the flat local overrings are precisely the rings Rp where p is a prime ideal of R [Ric65, Theorem 2]. In this context, since going-down domains have proven to be a useful concept, it makes sense to explore the orthogonal concept: • When does it happen that the only local overrings that satisfy goingdown over R are the localizations at prime ideals? Date: December 11, 2015. 2010 Mathematics Subject Classification. 13B21, 13F05, 13F45. Key words and phrases. Krull domain, going down, perinormal, globally perinormal, universally catenary. 1 2 NEIL EPSTEIN AND JAY SHAPIRO We call such a ring perinormal, and it is the subject of this paper. (The related concept of global perinormality stipulates that the only overrings, local or not, that satisfy going-down over the base are localizations of the base ring at multiplicative sets.) It turns out that the class of perinormal rings is closely related to Krull domains (and so Noetherian normal domains) and weakly normal (hence seminormal) domains in that Krull domain ⇒ perinormal ⇒ weakly normal and (R1 ), with neither implication reversible. Moreover, for universally catenary domains (and somewhat more generally), we can characterize perinormal domains as those domains R such that no prime localization Rp has an overring that induces a bijection on prime spectra. When R is smooth in codimension 1, we can restrict our attention to integral overrings of these Rp (cf. Theorem 4.7). On the other hand, the only perinormal going-down domains are Prüfer domains. The structure of the paper is as follows. We start by establishing some basic facts in Section 2, including Proposition 2.5 which shows that perinormality is a local property and Proposition 2.4, which reframes perinormality in terms of flatness. Section 3 explores the relationship of perinormality to (generalized) Krull domains, weakly normal domains, and (R1 ) domains. Theorem 3.10, Corollary 3.4, and Proposition 3.2 respectively show that perinormality is implied by the first and implies the latter two properties. We also exhibit some sharpening examples. Section 4 is dedicated to Theorem 4.7, which gives the two characterizations of perinormal domains among the Noetherian domains mentioned above. In Section 5, we find Theorem 5.2, which exhibits a method for producing perinormal domains that are not integrally closed. Section 6 is devoted to the related notion of global perinormality; in particular, we give a partial characterization (see Theorem 6.4) of which Krull domains may be globally perinormal, along with examples relevant to algebraic number theory. It turns out that the theory of perinormality is a bit different when one includes non-Noetherian rings; in Section 7, we point out the subtleties in a series of examples, including the fact that not every integrally closed domain is perinormal (unlike in the Noetherian case). We end with a list of interesting questions in Section 8. Conventions: All rings are commutative with identity, and ring homomorphisms and containments preserve the multiplicative identity. The term local means only that the ring has a unique maximal ideal. An overring of an integral domain R is a ring sitting between R and its fraction field. 2. First properties Definition 2.1. Let R be an integral domain. We say R is perinormal if whenever S is a local overring of R such that the inclusion R ⊆ S satisfies going-down, it follows that S is a localization of R (necessarily at a prime ideal). PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 3 We say that R is globally perinormal if the same conclusion holds when the condition on S being local is dropped (so that this time, the localization is just at a multiplicative set). Remark. The term perinormal is meant to reflect several aspects of the property: (1) It is closely related to the properties of seminormality and weak normality (cf. Corollary 3.4). (2) It is closely related to the concept of normality in the Noetherian case (cf. Theorem 3.10), which is usually the only situation where the word “normal” is used for integral closedness in the literature. (3) Perinormality is not a weakening of the property of integral closedness in general (cf. Example 7.1), whence the prefix peri- (unlike weak and semi-, which both imply weakenings). Lemma 2.2. A homomorphism of commutative rings R → S satisfies goingdown if and only if for all P ∈ Spec S, the induced map Spec (SP ) → Spec (RP ∩R ) is surjective. Proof. This follows immediately from the definition.  Recall the following theorem of Richman. Theorem 2.3 ([Ric65, Theorem 2]). Let A be an integral domain and B an overring of A. Then B is flat over A if and only if Bm = Am∩A for all maximal ideals m of B. This theorem allows us to characterize perinormality in terms of flatness: Proposition 2.4. A domain R is perinormal if and only if every overring of R that satisfies going-down is flat over R. Proof. Suppose R is perinormal. Let S be an overring of R that satisfies going-down over R. Let m be a maximal ideal of R. Clearly Sm satisfies going-down over R (since going-downness is transitive and Sm is going-down over S), so by perinormality, Sm is a localization of R, so that necessarily, Sm = Rm∩R . Then by Theorem 2.3, S is flat over R. Conversely, suppose every overring of R that satisfies going-down is flat over R. Let (S, m) be a local overring of R that satisfies going-down. Then it is flat over R, so again by Theorem 2.3, S = Rm∩R is a localization of R. Thus, R is perinormal.  Next, we show that perinormality is a local property. Proposition 2.5. If R is perinormal, so is RW for every multiplicative set W . Conversely, if Rm is perinormal for all maximal ideals m of R, then so is R. Proof. First suppose R is perinormal. Let S be a local overring of RW that satisfies going-down. Then S satisfies going-down over R (since no prime ideal of R lain over by a prime of S can intersect W ) and so S = RV for some multiplicatively closed subset V of R. But then V is also a multiplicatively closed subset of RW , and S = (RW )V . Therefore RW is perinormal. 4 NEIL EPSTEIN AND JAY SHAPIRO Conversely, suppose that Rm is perinormal for all maximal ideals m of R. Let (S, n) be a local overring of R such that the inclusion R ⊆ S satisfies going-down. Let m be a maximal ideal of R such that n∩R ⊆ m. Then Rm ⊆ S satisfies going-down, so that by perinormality of Rm , S = (Rm )n∩Rm = Rn∩R . Thus R is perinormal.  Example 2.6. Any valuation domain R is globally perinormal because every overring of R is a localization, as is easily shown. It then follows from Proposition 2.5 that every Prüfer domain is perinormal. 3. (R1 ) domains, weakly normal domains, and generalized Krull domains In this section, we fit perinormality into the context of three known important classes of integral domains. Namely, generalized Krull =⇒ Krull =⇒ perinormal =⇒ weakly normal and (R1 ), with neither arrow reversible. Definition 3.1. We say that a commutative ring R satisfies (R1 ) if RP is a valuation domain whenever P is a height one prime of R. Remark. It seems that in the literature, the term (R1 ) is only used for Noetherian rings (cf. [Mat86, p. 183]). Here we have extended it to arbitrary commutative rings in a way that both coincides with the established definition in the Noetherian case and suits our purpose in the general case. Proposition 3.2. Any perinormal domain R satisfies (R1 ). Proof. Let p be a height one prime of R. Let (V, m) be a valuation overring of R such that m ∩ R = p. (If R is Noetherian, we can choose V to be Noetherian as well.) Then the map R → V trivially satisfies going-down. Thus, V is a localization of R, whence V = Rm∩R = Rp , completing the proof that R satisfies (R1 ).  Proposition 3.3. If (R, m) is a local perinormal domain, then for any integral overring S of R such that Spec S → Spec R is a bijection, we have R = S. Proof. Let S be an integral overring of R such that Spec S → Spec R is bijective. By integrality of the extension, some prime ideal n of S lies over m; by bijectivity, there can be only one such prime; since fibers of Spec maps on integral extensions are antichains, n is maximal, and the unique maximal ideal of S. Now, let p ⊂ q be a chain of primes in R, and let Q ∈ Spec S with Q ∩ R = q. Since the Spec map is surjective, there is some P ∈ Spec S with P ∩ R = p. By the ‘going up’ property of integral extensions, there is some Q′ ∈ Spec S such that P ⊆ Q′ and Q′ ∩ R = q. But then by injectivity of the Spec map, Q′ = Q. This shows that the inclusion R ⊆ S satisfies going-down; hence S is a localization of R since R is perinormal. But since the map R → S is a local homomorphism of local rings, the only way S can be a localization of R is if R = S.  PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 5 Recall that an integral domain R is weakly normal 1 if for any integral overring S of R such that the map Spec S → Spec R is a bijection and for all P ∈ Spec S (where we set p := P ∩ R), the corresponding field extensions Rp /pRp → SP /P SP is purely inseparable, it follows that R = S. A domain R is seminormal if [Swa80] whenever x is an element of the fraction field with x2 , x3 ∈ R, we have x ∈ R. However, it is equivalent to say that for any integral overring S such that Spec S → Spec R is a bijection and the corresponding field extensions Rp /pRp → SP /P SP are isomorphisms, then R = S. From this, it is clear that every weakly normal domain is seminormal, and that for a domain that contains Q, the converse holds. Recall that both weak normality and seminormality are local properties in the sense of Proposition 2.5. Also every normal domain is weakly normal. For all this and more, cf. Vitulli’s survey article on weak normality and seminormality [Vit11]. Corollary 3.4. If R is perinormal, then it is weakly normal (hence seminormal). Proof. Since both perinormality and weak normality are local properties, we may assume R is local. Now let S be an integral overring of R where Spec S → Spec R is a bijection such that for any P ∈ Spec S, the field extension Rp /pRp → SP /P SP is purely inseparable (where p = P ∩ R). Then by Proposition 3.3, R = S. It follows that R is weakly normal.  We next present two examples to show that the converse to Corollary 3.4 is false, even under some additional restrictions. Example 3.5. Not all weakly normal (resp. seminormal) domains are perinormal, even in dimension 1. For example, A = R[x, ix] is seminormal, even weakly normal, without being perinormal. Failure of perinormality arises from the fact that C[x] is going-down over A (with the same fraction field C(x)) without being a localization of it. To see seminormality, merely observe that A consists of those polynomials whose constant term is real, and if f ∈ C[x] is such that its square and cube have real constant term, it follows that the constant term of f has its square and cube in R, whence the constant term of f is in R already. Example 3.6. (Thanks to Karl Schwede for this example.) Even for finitely generated algebras over algebraically closed fields, weakly normal (R1 ) domains are not necessarily perinormal. For an example, consider R = k[x, y, xz, yz, z 2 ] where k is any field of characteristic not equal to 2. Let A = k[x, y, z], and note that A is the integral closure of R. Hence every prime ideal of R is contracted from A. Let P ∈ Spec A. If P + (x, y), then z ∈ RP ∩R , whence RP ∩R = AP is regular. Therefore, RP ∩R is normal, weakly normal, and perinormal. This also shows that R satisfies (R1 ). 1This is not the original definition [AB69], but it is equivalent [Yan83, Remark 1]. 6 NEIL EPSTEIN AND JAY SHAPIRO Further, Yanagihara [Yan83, Proposition 1] has shown that an arbitrary pullback of a weakly normal inclusion is also a weakly normal inclusion. Hence, we may conclude that R is weakly normal, as R/I is a subring of A/I, where I = (x, y, xz, yz)R = (x, y)A, and the extension k[z 2 ] ֒→ k[z] is weakly normal, since char k 6= 2. (One may similarly show the ring is seminormal even when char k = 2 by using [GT80, 4.3] in place of [Yan83, Proposition 1], which shows the analogous fact for seminormal inclusions.) However, the ring R(x,y)∩R = k(z 2 )[x, y, xz, yz](x,y,xz,yz) is not perinormal. Its integral closure is A(x,y) = k(z)[x, y](x,y) . Then the map R(x,y)∩R → A(x,y) induces a bijection on spectra because for all other primes, we have an isomorphism, whereas the localness of the integral closure shows that we also have bijectivity at the maximal ideal. But the two rings are unequal because z ∈ / R(x,y)∩R . Then since R(x,y)∩R is not perinormal, neither is R. Lemma 3.7. Let R be an integral domain, S an overring of R, and p ∈ Spec S such that V := Rp∩R is a valuation domain of dimension 1. Then Rp∩R = Sp as subrings of the fraction field K of R, and ht p = 1. Proof. We have V = Rp∩R ⊆ Sp ⊆ K. But Sp 6= K, since p 6= 0. On the other hand, V is a valuation domain, so every overring is a localization at a prime ideal. Since V has only two primes, the only possibilities are V and K. Since Sp 6= K, it follows that Sp = V . Finally, ht p = dim Sp = dim V = 1.  Definition 3.8. For a commutative ring R, Spec 1 (R) denotes the set of all height one primes of R. Proposition 3.9. Let R be an (R1 ) domain and let S be an overring such that the extension R ⊆ S satisfies going-down. Then S satisfies (R1 ), and the map Spec S → Spec R induces an injective map Spec 1 (S) → Spec 1 (R) whose image consists of those height one primes p of R such that pS 6= S. Proof. First we need to show that given a height one prime Q of S, q := Q∩R is a height one prime of R. We have q 6= 0 because R ⊆ S is an essential extension of R-modules; hence ht q ≥ 1. On the other hand, suppose there is some p ∈ Spec R with 0 ( p ( q. Then by going-down, there is some P ∈ Spec S with P ∩ R = p. But then P 6= 0 (again by essentiality of the extension), whence 0 ( P ( Q is a chain of primes in S, so that ht Q ≥ 2, a contradiction. Then by Lemma 3.7, S satisfies (R1 ). Next, let p ∈ Spec 1 (R). If pS = S, then no prime of S can lie over p. On the other hand, if pS 6= S, then there is some maximal ideal Q of S with pS ⊆ Q. Then the going-down property implies that there is some P ∈ Spec S with P ∩ R = p. Moreover, Lemma 3.7 along with the (R1 )-ness of R implies that SP = Rp and ht P = 1. Finally, if there is some other prime ideal P ′ of S with P ′ ∩ S = p, then we have SP = Rp = SP ′ . But different prime ideals of a ring always give rise to different localizations, so P = P ′ , finishing the proof that the map of Spec 1 ’s is injective.  PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 7 Consider the following properties that an integral domain R may have: T (1) R = p∈Spec 1 (R) Rp . (2) For any nonzero element r ∈ R, the set {p ∈ Spec 1 (R) | r ∈ p} is finite. (3) Rp is a DVR for all p ∈ Spec 1 (R). One says R is a Krull domain (resp. generalized Krull domain) if it satisfies properties (1–3) (resp. properties (1), (2), and (R1 )). Recall the Mori-Nagata theorem (cf. [Fos73, Theorem 4.3]), which says that the integral closure of any Noetherian domain is Krull (though not necessarily Noetherian); hence every Noetherian normal domain is Krull. Theorem 3.10. If R is a generalized Krull domain (e.g. Noetherian normal), then R is perinormal. Proof. Let (S, m) be a local overring of R such that the inclusion R ⊆ S satisfies going-down. Let Q = m ∩ R; RQ is then also a generalized Krull domain [Gil72, Corollary 43.6]. Note that the going-down condition implies that the map Spec S → Spec RQ is surjective. Hence by Proposition 3.9, we get a bijective map Spec 1 (S) → Spec 1 (RQ ), and for each P ∈ Spec 1 (S) and corresponding p = P ∩ R ∈ Spec 1 (RQ ), we have (RQ )p = SP by Lemma 3.7. Therefore \ \ RQ ⊆ S ⊆ SP = (RQ )p = RQ . P ∈Spec 1 (S) p∈Spec 1 (RQ ) That is, S = RQ , so R is perinormal.  4. Local characterizations of perinormality In this section, after a preliminary exploration of how (R1 ) domains interact with overrings and the special relationship that occurs between two rings that share a nonzero ideal, we provide two surprising characterizations of perinormal domains within a large class of integral domains. Lemma 4.1. Let R be an (R1 ) integral domain whose integral closure R′ is a generalized Krull domain and such that for all P ∈ Spec 1 R′ , P ∩ R ∈ Spec 1 R. If there is a maximal ideal of R that contains all the height one primes of R, then R is local. Proof. Let m be a maximal ideal of R, and suppose that m contains all height one primes of R. By Lemma 3.7, since R is an (R 1 ) domain, if p ∈ Spec 1 R and P ∈ Spec 1 R′ lies over p, then Rp = RP′ . As height one primes of R′ contract to height one primes of R, we have \ \ \ Rm ⊆ (Rm )p = RP = RP = R ′ , p∈Spec 1 (Rm ) P ⊆m,ht P =1 P ∈Spec 1 R where the last equality follows since R′ is generalized Krull. We have shown that Rm is integral over R, which can only happen if R = Rm .  8 NEIL EPSTEIN AND JAY SHAPIRO Lemma 4.2. Let (R, m) be an (R1 ) local domain whose integral closure R′ is a generalized Krull domain and such that for all P ∈ Spec 1 R′ , P ∩ R ∈ Spec 1 R. Let S be an integral overring of R that satisfies going-down over R. Then S is local. Proof. By Proposition 3.9, S satisfies (R1 ) and the map Spec S → Spec R ∼ induces a bijection Spec 1 (S) → Spec 1 (R). Now let n be a maximal ideal of S that contains mS; it exists because S is integral over R. Then the extension R ⊆ Sn is going-down and mSn 6= Sn , so Proposition 3.9 applies again to ∼ produce a bijection Spec 1 (Sn ) → Spec 1 (R). This bijection composes with the inverse of the previous bijection to give a bijection of Spec 1 (Sn ) with Spec 1 (S). Hence, for all height one primes p of S, we have pSn 6= Sn – that is, p ⊆ n. Moreover, R and S have the same integral closure, which is generalized Krull by hypothesis. As height one primes of R′ contract to height one primes of R, one can show using the properties of integrality that the same holds for all intermediate rings. Thus by Lemma 4.1, S must be local.  Next we give conditions on a domain A that ensure that height one primes of A′ contract to height one primes of A. We first need to recall some definitions. A ring A is called catenary if given a pair p1 ⊂ p2 of prime ideals of A such that there exists a saturated chain of prime ideals between the two, then all such saturated chains have the same length. We say that A is universally catenary if it is Noetherian and every finitely generated A-algebra is catenary. It is clear that being catenary and hence being universally catenary is closed under localization. Let A ⊆ B be integral domains. Then tr.degA B denotes the transcendence degree of the fraction field of B over that of A. Recall that the ring extension is said to satisfy the dimension (or altitude) formula if the following equality holds for all P ∈ Spec B: htP + tr.degA/p B/P = ht p + tr.degA B where p = P ∩ A (see for example [Mat86, p. 119]). We note that if in addition B is integral over A, then tr.degA B = 0 = tr.degA/p B/P in which case the height of a prime of B is invariant under contraction to A. Lemma 4.3. Let R be a universally catenary integral domain with integral closure R′ . Then every height one prime ideal of R′ contracts to a height one prime ideal of R. Proof. Since R is Noetherian, by [Rat80, Corollary 2.5] it will suffice to show that if f ∈ R[x]′ , then the height of a prime ideal of R[x, f ] is invariant under contraction to R[x]. Since R[x] is also universally catenary and R[x, f ] is module finite over R[x] (in particular algbera finite), it follows that the extension R[x] ⊆ R[x, f ] satisfies the dimension formula (see for example [Mat86, Theorem 15.6]). Since it is also an integral extension, we have that the height is invariant under contraction as desired.  PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 9 Remark. The universal catenarity assumption is not particularly restrictive, as almost every Noetherian ring that arises in algebraic geometry, number theory, and everyday commutative algebra is universally catenary. Indeed, the class of universally catenary rings is closed under localization and finitely generated algebra extensions. Moreover, it includes CohenMacaulay rings (including Dedekind domains, fields, and regular local rings; see [Mat86, Theorem 17.9]) and complete Noetherian local rings (e.g. power series rings in finitely many variables over a field or over the p-adics; see [Mat86, Theorem 29.4]). The following three results are well-known to experts, and some of their statements appear (without proofs) in [Fon80]. However, we include them here for completeness and to make the paper self-contained. Lemma 4.4. Let R ⊆ T be an inclusion of commutative rings, and let I be an ideal that is common to R and T . (That is, I is an ideal of R and IT = I.) Let W be a multiplicatively closed subset of T , set V := W ∩R, and suppose that I ∩W 6= ∅. Then the natural map RV → TW is an isomorphism. Proof. Let z ∈ I ∩ W . To see injectivity, let vr ∈ RV (with r ∈ R, v ∈ V ) such that vr = 0 in TW . Then for some w ∈ W , we have wr = 0. Moreover, zw ∈ I ∩ W ⊆ R ∩ W = V and (zw)r = 0, whence vr = 0 in RV . To see surjectivity, let wt ∈ TW (with t ∈ T , w ∈ W ). Then zt ∈ IT ⊆ R zt and zw ∈ I ∩ W ⊆ R ∩ W = V , so that wt = zw ∈ RV .  Corollary 4.5. Let R ⊆ T be an inclusion of commutative rings, and let I be an ideal common to R and T . Let z ∈ I, and let P ∈ Spec T with I * P . Then the natural maps Rz → Tz and RP ∩R → TP are isomorphisms. Proof. In the first case, apply Lemma 4.4 with V = W = {z n | n ∈ N}. In the second case, apply the same lemma with W = T \ P .  Corollary 4.6. Let R ⊆ T be integral domains that share a common nonzero ideal I. Then the induced map of fraction fields is an isomorphism. Proof. Apply Lemma 4.4 with W = T \ {0}.  Theorem 4.7. Let R be a universally catenary integral domain with fraction field K. The following are equivalent. (a) R is perinormal. (b) For each p ∈ Spec R, Rp is the only ring S between Rp and K such that the induced map Spec S → Spec Rp is an order-reflecting bijection. (c) R satisfies (R1 ), and for each p ∈ Spec R, Rp is the only ring S between Rp and its integral closure such that the induced map Spec S → Spec Rp is an order-reflecting bijection. Proof. We note that we only need the universal catenarity condition for the implication (c) =⇒ (a). 10 NEIL EPSTEIN AND JAY SHAPIRO (a) =⇒ (b): Since perinormality localizes, we may assume that (R, p) is local. Now let S be a ring between R and K such that Spec S → Spec R is an order-reflecting bijection. Thus S satisfies going-down over R. Since S is local, the perinormality assumption on R implies that S is a localization of R. As the Spec map is onto, we conclude that R = S. (b) =⇒ (c): To see that R satisfies (R1 ), let p be a height one prime of R. Let V be a valuation ring centered on p. Then all nonzero prime ideals of V contract to p, and their intersection q is also a prime ideal of V . Since q contains no prime ideals other than itself and (0), we have ht q = 1. Now, the map Rp → Vq induces a bijection on Spec, so Rp = Vq , a valuation domain. On the other hand, the second condition in (c) follows trivially from (b). (c) =⇒ (a): Let (S, n) be a going-down local overring of R. Let p = n ∩ R. Note that Rp satisfies (R1 ), so that by Proposition 3.9, the map ∼ Spec S → Spec Rp induces a bijection Spec 1 (S) → Spec 1 (Rp ) where by Lemma 3.7, the corresponding localizations of S and Rp coincide. Hence S is (R1 ). Since Rp is also universally catenary it follows by integrality and Lemma 4.3 that the Spec map Spec (Rp )′ → Spec Rp induces a bijection ∼ Spec 1 (Rp )′ → Spec 1 Rp , where again the corresponding localizations of Rp and (Rp )′ coincide. Thus, \ \ Rp ⊆ S ⊆ SQ = (Rp )P = (Rp )′ , Q∈Spec 1 (S) P ∈Spec 1 (Rp ) where the last equality follows since (Rp )′ is a Krull domain. Hence S is integral over Rp . Next, we claim that the map Spec S → Spec Rp is injective. To see this, let Q be a prime ideal of Rp , and let W := Rp \ Q. Then the inclusion (Rp )Q ⊆ SW is integral, it satisfies going-down, and QSW 6= SW . Moreover, ′ the integral closure of Rp is RR\p , a Krull domain. Thus by Lemmas 4.2 and 4.3, SW is local. But this means that only one prime of S lies over Q, whence the map Spec S → Spec Rp is injective. However, the map is also surjective, since S is integral over Rp . Therefore the map is bijective. Finally, we must show that the map is order-reflecting – that is, if q1 ⊆ q2 in Rp , then the corresponding primes in S are also so ordered. So let Qj ∈ Spec S with Qj ∩ Rp = qj , j = 1, 2. By going-down, there is some P ∈ Spec S with P ⊆ Q2 and P ∩ Rp = q1 = Q1 ∩ Rp . But then by the injectivity of the Spec map, P = Q1 , whence Q1 ⊆ Q2 . Hence, condition (c) applies and Rp = S, whence R is perinormal.  Recall [FO70] that a ring extension A ⊆ B is called minimal if there are no rings properly between A and B. Corollary 4.8. Let (R, m) be a universally catenary local domain. Assume that dim R ≥ 2 and that the map R → S is a minimal ring extension, where PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 11 S is the integral closure of R. Then R is perinormal if and only if S is not local. Proof. By [FO70, Theorem 2.2], m is also an ideal of S. Now let p ∈ Spec R with p 6= m. Let P, P ′ ∈ Spec S with P ∩ R = P ′ ∩ R = p. Then by Corollary 4.5, SP = Rp = SP ′ , whence P = P ′ . Also, by integrality any maximal ideal of S must contract to m. Hence, there is a bijection between the nonmaximal primes of R and those of S. Suppose S is local. The only possibility of non-bijection of Spec happens at the maximal ideals, but it is clear that the unique maximal ideal of S contracts to m. Thus, R → S induces a bijection on Spec even though R 6= S. Then by the implication (a) =⇒ (b) of Theorem 4.7, R cannot be perinormal. On the other hand, if S is not local, then by minimality of the extension, there is no local integral overring of R = Rm other than R itself. Also, for any p ∈ Spec R \ {m}, Rp is integrally closed (because it equals SP , where P contracts to p), so again there is no local integral overring. The same observation shows that R satisfies (R1 ), since none of the height one primes of R are maximal. Then by the implication (c) =⇒ (a) of Theorem 4.7, R is perinormal.  We close this section by presenting an example that shows that Theorem 4.7 and Corollary 4.8 are in some sense best possible. Example 4.9. The fact that the last two results are false for arbitrary Noetherian rings can be demonstrated by [Nag62, Appendix, Example 2] with m = 0. This example consists of a Noetherian normal ring S with exactly two maximal ideals m1 and m2 where ht m1 = 1 and ht m2 = 2 and a field k ⊂ S such that the canonical map k → S/mi is an isomorphism for i = 1, 2. If R = k + J, where J = m1 ∩ m2 , then Nagata shows that S is the integral closure of R. We next claim that the set Spec 1 R is in natural bijection with the set X of height one primes contained in m2 , and that the corresponding localizations are equal. To see this, let p ∈ Spec 1 R. Then by integrality, there is some P ∈ Spec 1 S with P ∩ R = p. But all height one primes of S are in m2 except m1 , and m1 ∩ R = J ) p. Thus, P ⊂ m2 . Hence, contraction gives a surjective map X ։ Spec 1 R. Finally, if P, P ′ ∈ X with P ∩ R = P ′ ∩ R = p, then SP = Rp = SP ′ (by Corollary 4.5), whence P = P ′ . Hence R satisfies (R1 ) and the ring Sm2 satisfies going-down over R. However, the ring Sm2 cannot be a localization of R as the maximal ideal of the former contracts to the maximal ideal of R. Therefore R is not perinormal. On the other hand, S is a minimal ring extension of R by [PPL06, Theorem 3.3(b)]. Hence there are no local rings strictly between R and R′ = S and so condition (c) of Theorem 4.7 is satisfied. Thus both Theorem 4.7 and Corollary 4.8 are false without some assumption on R. 12 NEIL EPSTEIN AND JAY SHAPIRO 5. Gluing points of generalized Krull domains in high dimension In this section, we exhibit a method for constructing perinormal domains out of pre-existing generalized Krull domains, such that the new domains enjoy an arbitrary degree of branching-like behavior. We explain how to interpret this construction either in the algebraic context of pullbacks or the geometric context of gluing points. We begin with the following result, which may be known, but we include a proof for the convenience of the reader. Lemma 5.1. Let R ⊆ S ⊆ T be ring extensions. Let X := {P ∈ Spec S | P ∩ R is a maximal ideal}. Suppose that the induced map (Spec S \ X) → (Spec R\Max R) is injective and the inclusion R ⊆ S satisfies INC. If R ⊆ T satisfies going-down, so does S ⊆ T . Proof. Let P1 ⊂ P2 be a chain of two prime ideals of S such that there exists Q2 ∈ Spec T with Q2 ∩ S = P2 . Then setting pj := Pj ∩ R, j = 1, 2, we have p1 6= p2 (by INC) and Q2 ∩ R = p2 . Then by the going-down hypothesis on the extension R ⊆ T , there is a prime ideal Q1 of T with Q1 ⊆ Q2 and Q1 ∩ R = p1 . But then we have P1 ∩ R = p1 = Q1 ∩ R = (Q1 ∩ S) ∩ R, so by injectivity of the map in question (since p1 is a non-maximal ideal of R), we have P1 = Q1 ∩ S, completing the proof.  Theorem 5.2. Let S be a semilocal generalized Krull domain and let m1 , . . . , mn be its maximal ideals. Assume that n ≥ 2, and ht mj ≥ 2 for all 1 ≤ j ≤ n. Further suppose that the fields S/mi are all isomorphic to the same field k. For each i = 1, 2, . . . , n fix an isomorphism αi : k → S/mi . Let R be the pullback in the diagram R f /S p g  k h  / S/J Qn where J := m1 ∩ · · · ∩ mn = j=1 mj , p is the canonical projection, and h is Qn the composition of the maps k → Qn i=1 S/mi (given by λ 7→ (α1 (λ), . . . , αn (λ)) and the isomorphism between i=1 S/mi and S/J (given by the Chinese Remainder Theorem). Then R is local and perinormal. Also, R is globally perinormal if S is. But R is not integrally closed, because its integral closure is S. Proof. We first note that it follows from the properties of a pullback that as h is an injection (resp. p is a surjection), f is an injection (resp. g is a surjection). Thus we can view R as a subring of S where J = ker g is a common nonzero ideal of both rings. Then it follows from Corollary 4.6 that R and S have the same field of fractions. Next we show that S is integral over R (and hence equals the integral closure of R). To see this, let s ∈ S. Since J is a common ideal of R and S, PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 13 we have k∼ = R/J ֒→ S/J = S/(m1 ∩ . . . ∩ mn ) ∼ = n Y (S/mj ) ∼ = k × · · · × k, j=1 where the composite map is just the diagonal embedding. Now k × · · · × k is integral over k, which means that S/J is integral over R/J. In particular, there is some monic g ∈ (R/J)[X] such that g(s̄) = 0. But then g lifts to a monic polynomial G ∈ R[X] such that G(s) ∈ J. Say G(s) = j ∈ J. Then H(X) := G(X) − j is a monic polynomial over R such that H(s) = 0. It follows that the integral closure of R is S. Now we claim that R is local. This will follow if we can show that J is the Jacobson radical of R, since we already have that J is a maximal ideal of R. To this end, it suffices to show that for each j ∈ J, 1 − j is a unit of R. If not, then 1 − j ∈ p for some prime ideal of R, so that 1 − j ∈ pS. But since 1− j is a unit of S (since J is the Jacobson radical of S), it follows that pS = S, which contradicts the lying over property of the integral extension R ⊆ S. This contradiction proves the claim. Before showing that R is perinormal, we collect some observations about the relationship between Spec R and Spec S. Let P ∈ Spec S and p = P ∩ R. By integrality P is a nonmaximal ideal of S if and only if p is a nonmaximal ideal of R. Furthermore in this case by Corollary 4.5, we have Rp = SP , whence ht P = ht p. Since we are assuming that no maximal ideal of S has height 1, each height one prime of S must contract to a height one prime of R. Moreover by integrality each height one prime of R is lain over by a prime of S. Thus the Spec map induces a bijection Spec 1 (S) → Spec 1 (R) where the corresponding localizations coincide. In particular R satisfies (R1 ). Now let T be an overring of R such that R ⊆ T satisfies going-down. Case 1: Suppose JT = T . Then S ⊆ T . To see this, let n be a maximal ideal of T . Since J * n, we have n ∩ R = p ( J. Then there is some nonmaximal P ∈ Spec S with P ∩ R = p (since S is integral over R), whence we have Rp = SP . Hence, S ⊆ SP = R Tp ⊆ Tn . Since n was an arbitrary maximal ideal of T , it follows that S ⊆ n∈Max T Tn = T . Next, since the map Spec S → Spec R is injective on non-maximal ideals, S is integral over R, and R ⊆ T satisfies going-down, it follows from Lemma 5.1 that the extension S ⊆ T satisfies going-down. Thus, if T is local or S is globally perinormal, we have that T = SW for some multiplicative subset W of S. On the other hand, for any maximal ideal mi of S, we have SWQ= T = JT ⊆ mi T = mi SW , so mi ∩ W 6= ∅. Let zi ∈ mi ∩ W , and let z := ni=1 zi . Note that z ∈ J and that z is a unit in T . Thus by Corollary 4.5, T = SW = (Sz )V = (Rz )V = RV ′ for appropriate multiplicative sets V and V ′ , so that T is a localization of R. Case 2: On the other hand if JT 6= T , then by Proposition 3.9, the ∼ map Spec T → Spec R induces a bijection Spec 1 (T ) → Spec 1 (R), and by Lemma 3.7 the corresponding localizations are equal. Since we have a similar 14 NEIL EPSTEIN AND JAY SHAPIRO bijection between Spec 1 (S) and Spec 1 (R), we get \ \ R⊆T ⊆ TP = SQ = S. P ∈Spec 1 (T ) Q∈Spec 1 (S) where the last equality follows from S being a generalized Krull domain. It follows that T is integral over R and that J is a common ideal to R, S, and T , so we have k∼ = R/J ⊆ T /J ⊆ S/J ∼ = k × · · · × k. Thus, T /J must be isomorphic to a product of some finite number of copies of k. But by Lemma 4.2, T is local. Hence, T /J is local as well. Therefore, T /J ∼  = k, whence T = R. Example 5.3. For a geometrically relevant example of the above, let B = k[X, Y ], let pj = (xj , yj ) ∈ k2 be distinct ordered pairs (points of k2 ) for 1 ≤ j ≤ t, let nj := (X − xj , Y − yj ) (the maximal ideal corresponding to T T pj ), J := tj=1 nj , and A := k + tj=1 nj . Note that J is a maximal ideal of A. Then by the above theorem, the ring AJ is perinormal (even globally perinormal!), but it isn’t normal unless t = 1 (since there are t maximal ideals lying over JAJ in the integral closure of AJ .) By [Fer03, Théorème 5.1], Spec A can be seen, quite precisely, as the algebro-geometric result of gluing together the points p1 , . . . , pt of A2k together, and Spec AJ is the (geometric) localization at the resulting singular point. 6. Global perinormality Next, we explore the related but quite distinct concept of global perinormality. In particular, for Krull domains, there is a strong and surprising relationship to the divisor class group. We illustrate with examples from algebraic number theory. Proposition 6.1. Let R be a globally perinormal domain, and let W be a multiplicative subset of R. Then RW is globally perinormal as well. Proof. Let S be an overring of RW that satisfies going-down. Let Q ∈ Spec S. Then by Lemma 2.2, the map Spec SQ → Spec (RW )Q∩RW is surjective. But (RW )Q∩RW = RQ∩R canonically, so that the map Spec SQ → Spec RQ∩R is surjective. Since Q ∈ Spec S was arbitrarily chosen, Lemma 2.2 applies again to show that the map R → S satisfies going-down, whence since R is globally perinormal, S must be a localization of R. That is, S = RV for some multiplicative subset V of R. But since RW ⊆ S, we have W ⊆ V , so that V ′ := V RW is a multiplicative subset of RW , and S = RV = (RW )V ′ , finishing the proof that RW is globally perinormal.  We next give a result analogous to Proposition 2.4. Proposition 6.2. Let R be a perinormal domain. Then R is globally perinormal if and only if every flat overring of R is a localization of R. PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 15 Proof. Suppose R is globally perinormal. Let S be a flat overring of R. Then S satisfies going-down over R (by flatness), so by global perinormality, S = RW for some multiplicative set W ⊆ R. Conversely, suppose every flat overring of R is a localization of R. Let S be an overring that satisfies going-down over R. By perinormality and Proposition 2.4, S is flat over R, whence by assumption S is a localization of R. Hence, R is globally perinormal.  Proposition 6.3. Let R be a generalized Krull domain and let S be a goingdown overring of R. Then \ S= Rp =: R∆ , p∈∆ where ∆ := {p ∈ Spec 1 (R) | pS 6= S}. Proof. For any maximal ideal m of S, Sm is local overring of R such that R ⊆ Sm satisfies going-down. Hence by Theorem 3.10, Sm is a localization of R – i.e., Sm = Rm∩R . Now, for every p ∈ ∆, there is some such m ∈ Max S with pS ⊆ m, whereas when p ∈ Spec 1 (R) \ ∆, there is no such m. Also, every such Rm∩R is generalized Krull, by [Gil72, Corollary 43.6]. Thus:   \ \ \ \  Rm∩R = Sm = S= (Rm∩R )P  m∈Max S m∈Max S = \ m∈Max S   \ p∈Spec 1 (R), pS⊆m m∈Max S  Rp  = \ P ∈Spec 1 (Rm∩R ) Rp = R∆ . p∈∆  The next theorem involves the divisor class group; hence we restrict our attention to Krull domains (rather than generalized Krull domains), which is where the theory of the divisor class group is most well developed. Theorem 6.4. Let R be a Krull domain. (1) If the divisor class group Cl(R) of R is torsion, then R is globally perinormal. (2) The converse holds when dim R = 1. Proof. To prove part (1), it is enough to show that ever flat overring of R is a localization of R, due to Proposition 6.2 and Theorem 3.10. So let S be a flat overring of R. Then by Theorem 2.3, Sm = Rm∩R for all maximal ideals m of S. In particular, S is an intersection of localizations of R at prime ideals of R. But recall that Heinzer and Roitman prove [HR04, Corollary 2.9] that for a Krull domain with torsion divisor class group, any intersection of localizations of R at prime ideals is in fact a localization of R. Thus, S is a localization of R, whence R is globally perinormal. As for part (2), the following statement was proved independently in [Dav64, Theorem 2], [Gol64, Corollary (1)], and [GO64, Corollary 2.6]: 16 NEIL EPSTEIN AND JAY SHAPIRO Let R be a Dedekind domain. Then the class group Cl(R) of R is torsion if and only if every overring of R is a localization of R. But any 1-dimensional Krull domain is a Dedekind domain [Mat86, Theorem 12.5]. Hence, (2) follows.  Example 6.5. The ring of integers OK of any finite algebraic extension of K of Q is globally perinormal. This is because OK is a Dedekind domain (hence Krull) with finite (hence torsion) class group (cf. [FT93, Theorem 31]). The result then follows from Theorem 6.4. Example 6.6. If Rm is globally perinormal for all maximal ideals m, it does not follow that R is globally perinormal, even when R is a Dedekind domain finitely generated over a field. To see this, let E be any elliptic curve, with Weierstraß equation f = 0, considered as an affine curve in A2C . Then as a group, E = E(C) is analytically isomorphic (as an algebraic group) to C/Λ for some lattice Λ [Sil86, Corollary VI.5.1.1], which in turn is abstractly isomorphic (as a group) to R/Z × R/Z. The latter has uncountably many non-torsion elements (namely, whenever either of the two coordinates is irrational). On the other hand, E(C) is isomorphic to a particular subgroup (the so-called degree 0 part) of the divisor class group of the Dedekind domain R = C[X, Y ]/(f ) [Sil86, Proposition III.3.4], as the latter is the affine coordinate ring of E(C). Thus, Cl(R) contains (uncountably many) nontorsion elements, so by Theorem 6.4(2), R is not globally perinormal. But Rm is a DVR for any m ∈ Max R (since R is a Dedekind domain), so Rm is globally perinormal. On page 17, we have constructed a chart tracking many of the dependencies we have discussed so far. Note that none of the arrows are reversible, and that a crossed-out arrow indicates a specific non-implication. 7. Some subtleties of the non-Noetherian case As usual, nuances exist for general commutative rings that do not come up when one assumes all rings involved are Noetherian. We explore some of these in the current section. Example 7.1. There is a non-Noetherian one-dimensional local integrally closed domain that isn’t perinormal. In fact, any integrally closed onedimensional local domain that isn’t a valuation domain will suffice. For example, let K/F be a purely transcendental field extension, let X be an analytic indeterminate over K, let V := K[[X]], and then let R = F + XV . Then R is easily seen to be local with maximal ideal XV and integrally closed (but not completely integrally closed) in its fraction field K((X)). To see that R has dimension 1, let p ∈ Spec R with 0 ( p ( XV . Then by Lemma 4.4, the map Rp → VR\p is an isomorphism, so that VR\p is local and hence equals either V or K((X)). The former is impossible since every nonunit of R is a nonunit of V , and the latter means that p = 0, which PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 17 Dedekind Noetherian normal OK , where K is an algebraic number field Krull, with \ torsion Cl(R) globally perinormal \ Krull integrally closed generalized Krull weakly normal perinormal Prüfer (R1 ) seminormal contradicts the assumption. Hence, Spec R = {0, XV }, whence dim R = 1. But V is a going-down local overring of R that is not a localization. Example 7.2. There exist non-Prüfer, integrally closed integral domains (necessarily non-Noetherian) that are perinormal but not generalized Krull. For a concrete example, let k be a field, x, y indeterminates over k, and R = k[x, y, xy , xy2 , xy3 , . . .], considered as a subring of k(x, y). If m = xR, then m is a maximal ideal of R of height two (see below). If P is any other maximal ideal of R, then x 6∈ P . Thus R ⊆ k[x, y]p , where p = P ∩ k[x, y]. Hence RP = k[x, y]p , and so in particular RP is a Krull domain, whence perinormal. It is also now clear that R is not Prüfer. On the other hand, it is known (though apparently not written down) that Rm is a valuation ring. Specifically, Rm is the valuation ring associated to the valuation ν on k(x, y) with value group Z × Z (ordered lexicographically), where ν(x) = (0, 1) and ν(y) = (1, 0). We give a brief outline as to why Rm = V , where V is the valuation ring of ν. Clearly R ⊂ V , since y/x ∈ V . Moreover the maximal ideal of V is generated by x, and so Rm ⊆ V . For the reverse containment we can write an arbitrary element of V as f /g, where f, g ∈ k[x, y]. Evidently we can assume that g is not divisible by y. Thus ν(g) = (0, m) for some nonnegative integer m. We can then write g = xm h(x) + yp(x, y), where h(x) ∈ k[x] and p(x, y) ∈ k[x, y]. Since ν(f ) ≥ ν(g), v(f ) = (n, t) where either n > 0 or n = 0 and t ≥ m. In either case one can show that f /g can be written in the form F/G, where F, G ∈ k[x, y] and G ∈ / m. Thus f /g ∈ Rm , showing the two rings are equal. Hence, R is integrally closed. Finally to complete the example we must know that R is not a generalized Krull domain. However, m = xR is a 18 NEIL EPSTEIN AND JAY SHAPIRO T principal prime ideal of height two. Thus x−1 ∈ ( p∈Spec 1 (R) Rp ) \ R, contradicting the definition of a generalized Krull domain. Example 7.3. Let V be any rank 1 valuation ring and n ∈ N. Recall that generalized Krull domains are closed under finite polynomial extension [Gil72, Theorem 43.11(3)]. Thus, V [X1 , . . . , Xn ] is a generalized Krull domain (since V is obviously generalized Krull), hence perinormal (by Theorem 3.10). This provides a large class of examples of perinormal domains, of arbitrary Krull dimension, that are neither Krull nor Prüfer, even locally. 8. Questions We close with an incomplete but intriguing list of questions suitable for further research on perinormality and global perinormality. Question 1. Let k be a field, and let X, Y, Z, W be indeterminates over that field. Is the normal hypersurface R = k[X, Y, Z, W ]/(XW − Y Z) globally perinormal or not? Note that its divisor class group is well-known to be infinite cyclic [Fos73, Proposition 14.8]. If “yes”, this answer would mean that the converse to Theorem 6.4(1) is false in dimension 3. If “no”, this answer would provide evidence that the converse may be true. Question 2. Let R be a perinormal domain, K its fraction field, L a (finite?) extension field of K, and S the integral closure of R in L. Is S perinormal? In the non-Noetherian case, this question is interesting even when we further stipulate that R is integrally closed. Question 3. Let R be an integral domain and X an indeterminate over R. What can one say about the perinormality of R in relation to the perinormality of R[X]? Does one imply the other? Question 4. Let R be a Noetherian local domain whose completion R̂ is also a domain. If R is perinormal, is R̂ perinormal as well? What about the converse? Question 5. Is every completely integrally closed domain perinormal? Acknowledgments We wish to thank David Dobbs, Tiberiu Dumitrescu, Alan Loper, Karl Schwede, and Dana Weston for interesting and useful conversations at various stages of the project. We also wish to thank the referee for many useful comments and improvements, especially for Propositions 2.4 and 6.2, which are due to the referee. References [AB69] [CS46] Aldo Andreotti and Enrico Bombieri, Sugli omeomorfismi delle varietà algebriche, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 431–450. Irvin S. Cohen and Abraham Seidenberg, Prime ideals and integral dependence, Bull. Amer. Math. Soc. 52 (1946), 252–261. PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS 19 [Dav64] Edward D. Davis, Overrings of commutative rings. II. Integrally closed overrings, Trans. Amer. Math. Soc. 110 (1964), 196–212. [Dob73] David E. Dobbs, On going down for simple overrings, Proc. Amer. Math. Soc. 39 (1973), no. 3, 515–519. [Dob74] , On going down for simple overrings. II, Comm. Algebra 1 (1974), 439– 458. [DP76] David E. Dobbs and Ira J. Papick, On going down for simple overrings. III, Proc. Amer. Math. Soc. 54 (1976), 35–38. [Fer03] Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no. 4, 553–585. [FO70] Daniel Ferrand and Jean-Pierre Olivier, Homomorphisms minimaux d’anneaux, J. Algebra 16 (1970), 461–471. [Fon80] Marco Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. (4) 123 (1980), 331–355. [Fos73] Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 74, Springer-Verlag, New YorkHeidelberg, 1973. [FT93] Albrecht Fröhlich and Martin J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge Univ. Press, Cambridge, 1993. [Gil72] Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, no. 12, Dekker, New York, 1972. [GO64] Robert Gilmer and Jack Ohm, Integral domains with quotient overrings, Math. Ann. 153 (1964), 97–103. [Gol64] Oscar Goldman, On a special class of Dedekind domains, Topology 3 (1964), no. suppl. 1, 113–118. [GT80] Silvio Greco and Carlo Traverso, On seminormal schemes, Comp. Math. 40 (1980), no. 3, 325–365. [HR04] William Heinzer and Moshe Roitman, Well-centered overrings of an integral domain, J. Algebra 272 (2004), no. 2, 435–455. [Mat86] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, no. 8, Cambridge Univ. Press, Cambridge, 1986, Translated from the Japanese by M. Reid. [Nag62] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, 1962. [PPL06] Gabriel Picavet and Martine Picavet-L’Hermitte, About minimal morphisms, Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, pp. 369–386. [Rat80] Louis J. Ratliff, Jr., Notes on three integral dependence theorems, J. Algebra 66 (1980), no. 2, 600–619. [Ric65] Fred Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794–799. [Sil86] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. [Swa80] Richard G. Swan, On seminormality, J. Algebra 67 (1980), 210–229. [Vit11] Marie A. Vitulli, Weak normality and seminormality, Commutative algebra: Noetherian and non-Noetherian perspectives, Springer-Verlag, 2011, pp. 441– 480. [Yan83] Hiroshi Yanagihara, Some results on weakly normal ring extensions, J. Math. Soc. Japan 35 (1983), no. 4, 649–661. 20 NEIL EPSTEIN AND JAY SHAPIRO Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 E-mail address: nepstei2@gmu.edu Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 E-mail address: jshapiro@gmu.edu 

Submitted to the Annals of Statistics arXiv: arXiv:1610.03944 RANK VERIFICATION FOR EXPONENTIAL FAMILIES By Kenneth Hung and William Fithian arXiv:1610.03944v2 [stat.ME] 3 Jul 2017 University of California, Berkeley Many statistical experiments involve comparing multiple population groups. For example, a public opinion poll may ask which of several political candidates commands the most support; a social scientific survey may report the most common of several responses to a question; or, a clinical trial may compare binary patient outcomes under several treatment conditions to determine the most effective treatment. Having observed the “winner” (largest observed response) in a noisy experiment, it is natural to ask whether that candidate, survey response, or treatment is actually the “best” (stochastically largest response). This article concerns the problem of rank verification — post hoc significance tests of whether the orderings discovered in the data reflect the population ranks. For exponential family models, we show under mild conditions that an unadjusted two-tailed pairwise test comparing the first two order statistics (i.e., comparing the “winner” to the “runner-up”) is a valid test of whether the winner is truly the best. We extend our analysis to provide equally simple procedures to obtain lower confidence bounds on the gap between the winning population and the others, and to verify ranks beyond the first. 1. Introduction. 1.1. Motivating Example: Iowa Republican Caucus Poll. Table 1 shows the result of a Quinnipiac University poll asking 890 Iowa Republicans their preferred candidate for the Republican presidential nomination (Quinnipiac University Poll Institute, 2016). Donald Trump led with 31% of the vote, Ted Cruz came second with 24%, Marco Rubio third with 17%, and ten other candidates including “Don’t know” trailed behind. Rank 1* 2* 3* 4* 5 6 7 .. . Candidate Result Votes Trump 31% 276 Cruz 24% 214 Rubio 17% 151 Carson 8% 71 Paul 4% 36 Bush 4% 36 Huckabee 3% 27 .. .. .. . . . Table 1 Results from a February 1, 2016 Quinnipiac University poll of 890 Iowa Republicans. To compute the last column (Votes), we make the simplifying assumption that the reported percentages in the third column (Result) are raw vote shares among survey respondents. The asterisks indicate that the rank is verified at level 0.05 by a stepwise procedure. MSC 2010 subject classifications: Primary 62F07; secondary 62F03 Keywords and phrases: ranking, selective inference, exponential family 1 imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 2 K. HUNG AND W. FITHIAN Seeing that Trump leads this poll, several salient questions may occur to us: Is Trump really winning, and if so by how much? Furthermore, is Cruz really in second, is Rubio really in third, and so on? Note that there is implicitly a problem of multiple comparisons here, because if Cruz had led the poll instead, we would be asking a different set of questions (“Is Cruz really winning,” etc.). Indeed, the selection issue appears especially pernicious due to the so-called “winner’s curse”: given that Trump leads the poll, it more likely than not overestimates his support. Nevertheless, if we blithely ignore the selection issue, we might carry out the following analyses to answer the questions we posed before at significance level α = 0.05. We assume for simplicity that the poll represents a simple random sample of Iowa Republicans; i.e., that the data are a multinomial sample of size 890 and underlying probabilities (πTrump , πCruz , . . .). (The reality is a bit more complicated: before releasing the data, Quinnipiac has post-processed it to make the reported result more representative of likely caucus-goers. The raw data is proprietary.) 1. Is Trump really winning? If Trump and Cruz were in fact tied, then Trump’s share of their combined 490 votes would be distributed as Binomial (490, 0.5). Because the (two-tailed) p-value for this pairwise test is p = 0.006, we reject the null and conclude that Trump is really winning. 2. By how much? Using an exact 95% interval for the same binomial model, we conclude Trump has at least 7.5% more support than Cruz (i.e., πTrump ≥ 1.075 πCruz ) and also leads the other candidates by at least as much. 3. Is Cruz in second, Rubio in third, etc.? We can next compare Cruz to Rubio just as we compared Trump to Cruz (again rejecting because 214 is significantly more than half of 365), then Rubio to Carson, and so on, continuing until we fail to reject. The first four comparisons are all significant at level 0.05, but Paul and Bush are tied so we stop. Perhaps surprisingly, all of the three procedures described above are statistically valid despite their ostensibly ignoring the implicit multiple-comparisons issue. In other words, Procedures 1 and 2 control the Type I error rate at level α and Procedure 3 controls the familywise error rate (FWER) at level α. The remainder of this article is devoted to justifying these procedures for the multinomial family, and extending to analogous procedures in other exponential family settings. While methods analogous to Procedures 1 and 2 have been justified previously for balanced independent samples from log-concave location families (Gutmann and Maymin, 1987; Stefansson, Kim and Hsu, 1988), they have not been justified in exponential families before now. 1.2. Generic Problem Setting and Main Result. Generically, we will consider data drawn from an exponential family model with density
(1) X ∼ exp θ0 x − ψ (θ) g (x) , with respect to either the Lebesgue measure on Rn or counting measure on Zn . We assume further that g (x) is symmetric with respect to permutation, and Schur concave, a mild technical condition defined in Section 2. In addition to the multinomial family, model (1) also encompasses settings such as comparing independent binomial treatment outcomes in a clinical trial, competing sports teams under a Bradley–Terry model, entries of a Dirichlet distribution, and many more; see Section 2 for these and other examples. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 3 We will generically use the term population to refer to the treatment group, sports team, political candidate, etc. represented by a given random variable Xj . As we will see, θj ≥ θk if and only if Xj is stochastically larger than Xk ; thus, there is a well-defined stochastic ordering of the populations that matches the ordering of the entries of θ. We will refer to the population with maximal θj as the best, the population with second largest θj as the second best, the one with maximal Xj as the winner, and the one with the second-largest Xj as the runner-up, where ties between observations are broken randomly to obtain a full ordering. Following the convention in the ranking and selection literature, we assume that if there are multiple largest θj , then one is arbitrarily marked as the best. Note that in cases where it is more interesting to ask which is the smallest population (for example, if Xj is the number of patients on treatment j who suffer a heart attack during a trial) we can change the variables to −X and the parameters to −θ; this does not affect the Schur concavity assumption. Write the order statistics of X as X[1] ≥ X[2] ≥ · · · ≥ X[n] , where [j] will denote the random index for the j-th order statistic. Thus, θ[j] is the entry of θ corresponding to the j-th order statistic of X (so θ[1] might not equal maxj θj , for example). In each of the above examples, there is a natural exact test we could apply to test θj = θk for any two fixed populations j and k. In the multinomial case, we would apply the conditional binomial test based on the combined total Xj + Xk as discussed in the previous section. For the case of independent binomials we would apply Fisher’s exact test, again conditioning on Xj + Xk . These are both examples of a generic UMPU pairwise test in which we condition on the other n − 2 indices (notated X\{j,k} ) and Xj + Xk , and reject the null if Xj is outside the α/2 and 1 − α/2 quantiles of the conditional law Lθj =θk (Xj | Xj + Xk , X\{j,k} ). Crucially, this null distribution does not depend on the value of θ provided that θj = θk . We call this test the (two-tailed) unadjusted pairwise test since it makes no explicit adjustment for selection. Similarly, inverting this test for other values of θj − θk yields an unadjusted pairwise confidence interval. (To avoid trivialities in the discrete case, we assume these procedures are appropriately randomized at the rejection thresholds to give exact level-α control.) Generalizing the procedures described in Section 1.1 we obtain the following: 1. Is the winner really the best? To test the hypothesis H : θ[1] ≤ maxj6=[1] θj : Carry out the unadjusted pairwise test comparing the winner to the runner-up. If the test rejects at level α, reject H and declare that the winner is really the best. 2. By how much? To construct a lower confidence bound for θ[1] − maxj6=[1] θj : Construct the unadjusted pairwise confidence interval comparing the winner to the runner-up, and report the lower confidence bound obtained for θ[1] − θ[2] if it is nonnegative, report −∞ otherwise. 3. Is the runner-up really the second best, etc.? Continue by comparing the runner-up to the second runner-up, again using the unadjusted pairwise test, and so on down the list comparing adjacent values. Stop the first time the test does not reject; if there are j rejections, declare that θ[1] > θ[2] > · · · > θ[j] > max θ[j] k>j Procedures 2 and 3 are conservative stand-ins for exact, but slightly more involved, conditional inference procedures. In particular, as we will see, reporting −∞ in Procedure 2 is typically much more conservative than is necessary. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 4 K. HUNG AND W. FITHIAN We now state our main theorem: under a mild technical assumption, Procedures 1–3 described above are statistically valid, even accounting for the selection. Theorem 1. Assume the model (1) holds and g (x) is a Schur-concave function. Then: 1. Procedure lation not 2. Procedure 3. Procedure 1 has exact level α conditional on H being true (conditional on
the best popuwinning), and marginally has level α · P(H is true) ≤ α 1 − n1 . 2 gives a conservative 1 − α lower confidence bound for θ[1] − maxj6=[1] θj . 3 is a conservative stepwise procedure with FWER no larger than α. n Note that Theorem 1 implies that we could actually replace α with n−1 α to obtain a more powerful version of Procedure 1 when n is not too large. We define Schur-concavity and discuss its properties in Section 2. Because any log-concave and symmetric function is Schur-concave, Theorem 1 applies to all of the cases discussed above. The proof combines the conditional selective-inference framework of Fithian, Sun and Taylor (2014) with classical multiple-testing methods, as well as new technical tools involving majorization and Schur-concavity. Note that these procedures make an implicit adjustment for selection because they use twotailed, rather than one-tailed, unadjusted tests. If we instead based our tests on an independent realization X ∗ = (X1∗ , . . . , Xn∗ ) then, for example, Procedure 1 could use a right-tailed version of the unadjusted pairwise test. In the case n = 2, Procedure 1 amounts to a simple two-tailed test of the null hypothesis θ1 = θ2 , and it is intuitively clear that a one-tailed test would be too liberal. More surprising is that, no matter how large n is, Procedures 1–3 require no further adjustment beyond what is required when n = 2. 1.3. Related work. Rank verification has been studied extensively in the ranking and selection literature. See Gupta and Panchapakesan (1971, 1985) for surveys of the subset selection literature. The two main formulations of ranking and selection are closely related to procedures for multiple comparisons with the best treatment (Edwards and Hsu, 1983; Hsu, 1984), but more powerful methods are available in some cases for procedures involving only the first sample rank, the problem of comparisons with the sample best; see Hsu (1996) for an overview and discussion of the relationships between these problems. Comparisons with the sample best have been especially well-studied and the validity of Procedures 1 and 2 have been established in a different setting: balanced independent samples from log-concave location families. Gutmann and Maymin (1987) prove the validity of Procedure 1 in this setting, and Bofinger (1991); Maymin and Gutmann (1992); Karnnan and Panchapakesan (2009) give similar results for other models including scale and location-scale families. Stefansson, Kim and Hsu (1988) provide an alternative proof for the validity of Procedure 1 in the same setting, leading to a lower confidence bound analogous to that of Procedure 2; interestingly, the proof involves a very early application of the partitioning principle, later developed into fundamental technique in multiple comparisons (Finner and Strassburger, 2002). These results use very different technical tools than the ones we use here, require independence between the different groups (ruling out, for example, the multinomial family), and do not address the exponential family case. Because most exponential families are not location-scale families (the Gaussian being a notable exception), and because our results involve more general dependence structures, both our proof techniques and our technical results are complementary to the techniques and results in the above works. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 5 For the multinomial case, Gupta and Nagel (1967), discussed in Section 3.1, remain the state of the art in finite-sample tests; Gupta and Wong (1976) discuss related approaches for Poisson models. Berger (1980) mentions an alternative, simpler rule which performs a binomial test on each population, but its power does not necessarily increase as the size m of observations increases in cases like Multinomial(m; 2/3, 1/3, 0, . . . , 0). Nettleton (2009) proves validity for an asymptotic version of the winner-versus-runner-up test, and Gupta and Liang (1989) consider an empirical Bayes approach for selecting the best binomial population wherein a parametric prior distribution is assumed for the success probabilities for the different populations. Ng and Panchapakesan (2007) discuss an exact test for a modified problem in which the maximum count is fixed instead of the total count; that is, we sample until the leading candidate has at least m votes. As Section 3.1 shows, our test can be much more powerful than the one in Gupta and Nagel (1967), especially if there are many candidates, because of the way our critical rejection threshold for X[1] − X[2] adapts to the data. Thus, our work closes a significant gap in the ranking and selection literature, extending the result of Gutmann and Maymin (1987) and others to new families like the multinomial, independent binomials, and many others. 1.4. Outline. Section 2 defines Schur concavity, and gives several examples satisfying this condition. Section 3 justifies Procedure 1 and compares its power to that of Gupta and Nagel (1967). Sections 4 and 5 justify Procedures 2 and 3 respectively, and Section 6 concludes. 2. Majorization and Schur concavity. 2.1. Definitions and basic properties. We start by reviewing the notion of majorization, defined on both Rn and Zn . Definition 1. For two vectors a and b in Rn (or Zn ), suppose sorting the two vectors in descending order gives a(1) ≥ · · · ≥ a(n) and b(1) ≥ · · · ≥ b(n) . We say that a  b (a majorizes b) if for 1 ≤ i < n, a(1) + · · · + a(i) ≥ b(1) + · · · + b(i) , and a(1) + · · · + a(n) = b(1) + · · · + b(n) . This forms a partial order in Rn (or Zn ). Intuitively, majorization is a partial order that monitors the evenness of a vector: the more even a vector is, the “smaller” it is. There are two properties of majorization that we will use in the proofs. Lemma 2. 1. Suppose (x1 , x2 , x3 , . . .) and (x1 , y2 , y3 , . . .) are two vectors in Rn . Then (x1 , x2 , x3 , . . .)  (x1 , y2 , y3 , . . .) if and only if (x2 , x3 , . . .)  (y2 , y3 , . . .) . 2. (Principle of transfer) If x1 > x2 and t ≥ 0, then (x1 + t, x2 , x3 , . . .)  (x1 , x2 + t, x3 , . . .) . If t ≤ 0, the majorization is reversed. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 6 K. HUNG AND W. FITHIAN Proof. 1. The property follows from an equivalent formulation of majorization listed in Marshall, Olkin and Arnold (2010), where x  y if and only if n X j=1 xn = n X yn n X and j=1 (xj − a)+ ≥ j=1 n X (yj − a)+ for all a ∈ R. j=1 2. Proved in Marshall, Olkin and Arnold (2010). Definition 2. A function g is Schur-concave if x  y implies g (x) ≤ g (y). A Schur-concave function is symmetric by default since a  b and b  a if and only if b is a permutation of the coordinates of a. Conversely a symmetric and log-concave function is Schurconcave (Marshall, Olkin and Arnold, 2010). Interestingly, Gupta, Huang and Panchapakesan (1984) also show that, in the context of independent location families, Schur concavity of the probability density is equivalent to monotone likelihood ratio. 2.2. Examples. Many common exponential family models have Schur-concave carrier densities. Below we give a few examples: Example 1 (Independent binomial treatment outcomes in a clinical trial). If each of n different treatments are applied to m patients independently, the number of positive outcomes Xj for treatment j is Binomial (m, pj ). The best treatment would be the treatment with the highest success probability pj . The joint distribution of X is given by   X pj  1 p (x) ∝ exp  xj log 1 − pj x1 ! (m − x1 )! · · · xn ! (m − xn )! j The carrier measure above is Schur-concave. The unadjusted pairwise test in this family is Fisher’s exact test. Example 2 (Competitive sports under the Bradley–Terry model). Suppose n players compete in a round robin tournament, where player j has ability θj , and the probability of player j winning against player k is eθj −θk e(θj −θk )/2 = (θ −θ )/2 . θ −θ 1+e j k e j k + e(θk −θj )/2 Let Yjk be an indicator for the match between player j and k, where we take Yjk = 1 if j beats k and Yjk = 0 if k beats j. For symmetry, we will also adopt the convention that Yjk + Ykj = 1. Thus the joint distribution of Y = (Yjk )j6=k is   X X
p (y) ∝ exp  2θj yjk  = exp 2θ0 x , j k6=j P where xj = k6=j yjk . In other words, if Xj is the number of wins by player j, then X = (X1 , . . . , Xn ) is a sufficient statistic with distribution
p (x) = exp 2θ0 x g (x) , imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 7 where g (x) is a function that counts the number of possible tournament results giving the net win vector x. A bijection proof shows that x is indeed Schur-concave. Therefore, we can use Procedures 1–3 to compare player qualities. After conditioning on U (X) = (X1 + X2 , X3 , . . . , Xn ), and under the assumption θ1 = θ2 , every feasible configuration of Y is equally likely. If n is not too large (say, no more than 40 players), we can find the conditional distribution of X1 by enumerating over the configurations; for larger n, computation might pose a more serious problem, requiring us for example to compute the p-value using Markov Chain Monte Carlo techniques (Besag and Clifford, 1989). Example 3 (Comparing the variances of different normal populations). Suppose there are n normal populations with laws N (µj , σj2 ) and m independent observations from each of them. The sample variance for population j can be denoted as Rj . By Cochran’s theorem, (m − 1) Rj ∼ σj2 χ2m−1 , and thus the joint distribution of R is !(m−3)/2 2 (m − 1) rj r∼ e−(m−1)rj /2σj 1{rj >0} 2 σj j=1  Y n m−1 m−1 (m−3)/2 ∝ exp − r − · · · − r rj 1{r>0} . 1 n 2σn2 2σ12 j=1 n Y Q (m−3)/2 1{r>0} , which is Schur-concave. Thus, we can use ProceThe carrier measure is nj=1 rj dures 1–3 to find populations with the smallest or largest variances. In this example, the distribution of X1 /(X1 + X2 ) conditional on (X1 + X2 , X3 , . . . , X4 ) is distributed as Beta(m/2, m/2) under the null, or equivalently X1 /X2 is conditionally distributed as Fm,m ; hence a (two-tailed) F -test is valid for comparing the top two populations. 3. Verifying the Winner: Is the Winner Really the Best?. First, we justify the notion that the population with largest θj is also the largest population in stochastic order: Theorem 3. For a multivariate exponential family with a symmetric carrier distribution, X1 ≥ X2 in stochastic order if and only if θ1 ≥ θ2 . Proof. It suffices to prove the “if” part, as the “only if” part can be follows from swapping the role of θ1 and θ2 . For any fixed a, and x1 ≥ a and x2 < a, we have x1 > x2 and exp (θ1 x1 + θ2 x2 + · · · + θn xn − ψ (θ)) g (x) ≥ exp (θ1 x2 + θ2 x1 + · · · + θn xn − ψ (θ)) g (x) . Integrating both sides over the region {x : x1 ≥ a, x2 < a} gives P [X1 ≥ a, X2 < a] ≥ P [X1 < a, X2 ≥ a] . Now adding P [X1 ≥ a, X2 ≥ a] to both probabilities gives P [X1 ≥ a] ≥ P [X2 ≥ a] , meaning that X1 is greater than X2 in stochastic order. Before proving our main result for Procedure 1, we give the following lemmas, the first of which clarifies a key idea in the proof, and the second is needed for a sharper bound in (2). imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 8 K. HUNG AND W. FITHIAN Lemma 4 (Berger, 1982). If pj are valid p-values for testing null hypothesis H0j S, then p∗ = maxj pj is a valid p-value for the union null (i.e. disjunction null) hypothesis H0 = j H0j . Proof. Under H0 , one of the H0j is true; without loss of generality assume it is H01 . Then, P [p∗ ≤ α] ≤ P [p1 ≤ α] ≤ α. Therefore p∗ is a valid p-value for the union null hypothesis. Lemma 5. If θ1 ≥ maxj6=1 θj , then P [1 wins] ≥ n1 . Proof. We can prove so with a coupling argument: for any sequence x1 , x2 , . . . , xn , define τ (x) = {τ (xj )}j=1,...,n , obtained by swapping x1 with the largest value in the sequence x. Hence exp (θ1 τ (x1 ) + · · · + θn τ (xn ) − ψ (θ)) g (X) ≥ exp (θ1 x1 + · · · + θn xn − ψ (θ)) g (X) . If we integrating both sides over Rn (or Zn in the case of counting measure), the right hand side gives 1. Since τ is an n-to-1 mapping, the left hand side is n times the integral over {x1 ≥ maxj>1 xj }. In other words, nP [1 wins] ≥ 1 as desired. In the case of counting measure, the above argument follows if a subscript is attached to identical observations uniformly to ensure strict ordering. We are now ready to prove our result for Procedure 1, restated here for reference. Part 1 of Theorem 1. Assume the model (1) holds and g (x) is a Schur-concave function. Procedure 1 (the unadjusted pairwise test) has level α conditional on the best population not winning. Proof. Let j ∗ denote the (fixed) index of the best population, so θj ∗ ≥ maxj6=j ∗ θj . The type I error — the probability of incorrectly declaring any other j to be the best — is   [ X P declare j best ≤ P [declare j best | j wins] P [j wins] , j6=j ∗ j6=j ∗ recalling that ties are broken randomly, so there is only one winner in any realization. Thus, it is enough to bound Pθ [declare j best | j wins] ≤ α, for each j 6= j ∗ , and for all θ with j ∗ ∈ arg maxj θj . Then we will have   [ X n−1 α, (2) P declare j best ≤ α · P [j wins] = αP [j ∗ does not win] ≤ n ∗ ∗ j6=j j6=j where the last inequality follows from Lemma 5. We start by assuming that we are working with the Lebesgue measure rather than the counting measure (eliminating the possibility of ties). The necessary modification of the proof for the counting measure case is provided at the end of this proof. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 9 To minimize notational clutter, we consider only the case where the winner is 1, i.e. X1 ≥ maxj>1 Xj . Furthermore, we will denote the runner-up with 2. This is not necessarily true, but we will use it as a shorthand to simplify our notation. For other cases, the following proof remains valid under relabeling and can thus be applied. In this case, we will test the null hypothesis H01 : θ1 ≤ maxj>1 θj , which is the union of the null hypotheses H01j : θ1 ≤ θj for j ≥ 2. For each of these we can construct an exact p-value p1j , which is valid under H01j conditional on A1 , the event that X1 is the winner. Hence by Lemma 4, a test that rejects when p1∗ = maxj p1j ≤ α is valid for H01 conditional on A1 . Procedure 1 performs an unadjusted pairwise test comparing X1 to X2 . Hence it is sufficient to show that p12 = p1∗ and that rejecting when p12 ≤ α coincides with the unadjusted pairwise test. Our proof has three main parts: (1) deriving p1j for each j ≥ 2, (2) showing that p12 ≥ p1j for each j ≥ 2, and (3) showing that p12 is an unadjusted pairwise p-value. Derivation of p1j . Following the framework in Fithian, Sun and Taylor (2014), we first construct the p-values by conditioning on the selection event where the winner is 1:   A1 = X1 ≥ max Xj . j>1 For convenience, we let Djk = Xj − Xk 2 and Mjk = Xj + Xk . 2 We then re-parametrize to replace X1 and Xj with D1j and M1j . The distribution is now an exponential family with sufficient statistics D1j , M1j , X\{1,j} and corresponding natural parameters θ1 − θj , θ1 + θj , θ\{1,j} . We now consider
(3) Lθ1 −θj =0 D1j M1j , X\{1,j} , A1 . We can rewrite the selection event in terms of our new parameterization as   A1 = {X1 ≥ Xj } ∩ X1 ≥ max Xk k6=1,j   = {D1j ≥ 0} ∩ D1j ≥ max Xk − M1j . k6=1,j The conditional law of D1j in (3), in particular, is a truncated distribution.
p d1j | M1j , X\{1,j} , A1 ∝ exp ((θ1 − θj ) d1j + θ2 X2 + · · · + (θ1 + θj ) M1j + · · · + θn Xn ) g (M1j + dij , X2 , . . . , Mij − dij , . . . Xn ) 1A1 (a) ∝ g (M1j + d1j , X2 , . . . , M1j − d1j , . . . Xn ) 1A1 , where at step (a), conditioning on X\{1,j} and M1j removes dependence on θ\{1,j} and θ1 + θj respectively, while θ1 − θj is taken to be 0 under our null hypothesis. Note that we consider this as a one-dimensional distribution of D1j on R, where M1j and X\{1,j} are treated as fixed. The p-value for H01j is thus R∞ D1j g (M1j + z, X2 , . . . , M1j − z, . . . , Xn ) dz (4) p1j = R ∞ . max{X2 −M1j ,0} g (M1j + z, X2 , . . . , M1j − z, . . . , Xn ) dz imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 10 K. HUNG AND W. FITHIAN Finally, by construction, p1j satisfies   PH01j p1j < α M1j , X\{1,j} , A1 ≤ α a.s., Marginalizing over M1j , X\{1,j} , PH01j [p1j < α | A1 ] ≤ α. Therefore these p1j are indeed valid p-values. Demonstration that p1∗ = p12 . We now proceed to show that p12 , the p-value comparing the winner to the runner-up, is the largest of all p1j . Without loss of generality, it is sufficient to show that p12 ≥ p13 . From the first part of this proof, both p-values are constructed by conditioning on X\{1,2,3} . Upon conditioning these, (X1 , X2 , X3 ) follows an exponential family distribution, with carrier distribution gX4 ,...,Xn (X1 , X2 , X3 ) = g (X1 , . . . , Xn ) , here X4 , . . . , Xn are used in the subscript as they are conditioned on and no longer considered as variables. The first point in Lemma 2 says that the function gX4 ,...,Xn is Schur-concave as well. We have reduced the problem to the case when n = 3: we can apply the result for n = 3 to gX4 ,...,Xn to yield p12 ≥ p13 for n > 3. We have reduced to the case when n = 3. The p-values thus are R∞ g (M12 + z, M12 − z, X3 ) dz 12 p12 = RD∞ , 0 g (M12 + z, M12 − z, X3 ) dz R∞ g (M13 + z, X2 , M13 − z) dz p13 = R ∞ D13 max{X2 −M13 ,0} g (M13 + z, X2 , M13 − z) dz The maximum in the denominator of p13 prompts us to consider two separate cases. First, we suppose X2 < M13 . Changing variables such that the lower limits of both integrals in the numerator are 0, we can re-parametrize the integrals above to give R∞ g (X1 + z, X2 − z, X3 ) dz p12 = R ∞0 0 g (M12 + z, M12 − z, X3 ) dz R∞ g (X1 + z, X2 − z, X3 ) dz = R ∞0 , −D12 g (X1 + z, X2 − z, X3 ) dz R∞ g (X1 + z, X2 , X3 − z) dz p13 = R ∞0 0 g (M13 + z, X2 , M13 − z) dz R∞ g (X1 + z, X2 , X3 − z) dz = R ∞0 . −D13 g (X1 + z, X2 , X3 − z) dz To help see the re-parametrization, each of these integrals can be thought of in terms of integrals along segments and rays. For example p12 can be represented in terms of integrals A and B in Figure 1. Specifically, B p12 = A+B imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 11 RANK VERIFICATION FOR EXP. FAMILIES (M12 , M12 , X3 ) A (X1 , X2 , X3 ) + (0, 1, 0) X= X1 +X2 +X3 , X, X 3
B + (1, 0, 0) + (0, 0, 1) A1 Fig 1. The p-value p12 can be written in terms of integral A along the segment and B along the ray. The diagram is drawn a level set of x1 + x2 + x3 . The green region represents the selection event A1 . Figure 2 has both the p-values shown on the same diagram. Proving p12 ≥ p13 is the same as proving B D B D ≥ ⇐⇒ ≥ . A+B C +D A C We will prove so by extending A to include à on the diagram. We denote the sum A + à as A0 . Formally, Z 0 Z 0 0 (5) A = g (X1 + z, X2 − z, X3 ) dz ≥ g (X1 + z, X2 − z, X3 ) dz = A. −D13 −D12 It is thus sufficient to show that B ≥ D and C ≥ A0 . Indeed from the second point in Lemma 2 we have (X1 + z, X2 − z, X3 )  (X1 + z, X2 , X3 − z) for z ≤ 0 and the majorization reversed for z ≥ 0. This majorization relation is indicated as the dotted line in Figure 2. So Schur-concavity shows that g (X1 + z, X2 − z, X3 ) ≤ g (X1 + z, X2 , X3 − z) for z ≤ 0, and the inequality reversed for z ≥ 0. Taking integrals on both sides yields the desired inequality. For the second case where X2 ≥ M13 , the segment C will reach the line x1 = x2 first before it reaches x1 = x3 , ending at (X2 , X2 , X1 − X2 + X3 ) instead. But we can still extend A by à to (X2 , X1 , X3 ). The rest of the proof follows. In either cases, p12 ≥ p13 , or in generality, p12 ≥ p1j for j > 1. In other words, p12 = p1∗ . p12 is an unadjusted pairwise p-value. Before conditioning on A1 , the distribution in (3) is symmetric around 0 under θ1 = θj . Since the denominator of p12 integrates over half of this symmetric distribution, it is always equal to 1/2. Thus, the one-sided conditional test at level α is equivalent to the one-sided unadjusted test at level α/2, or equivalently the two-sided unadjusted pairwise test at level α. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 12 K. HUNG AND W. FITHIAN (M13 , X2 + D13 , X3 ) (M12 , M12 , X3 ) D à A C    (X1 , X2 , X3 ) B (M13 , X2 , M13 ) A1 Fig 2. The p-value p12 can be written in terms of integral A along the segment and B along the ray; and p13 in terms of C and D. A0 would refer to the sum of A with the dashed line portion labeled as Ã, formally explained in Equation (5). The majorization relation is indicated by the dotted line. Modification for counting measure. Now suppose the exponential family is defined on the counting measure instead. If ties are broken independently and randomly, the end points on the rays can be considered as “half an atom” if the coordinates are integers (or a smaller fraction of an atom in case of a multi-way tie). The number of atoms on each ray is the same (after the extension Ã) and the atoms on each ray can be paired up in exactly the same way as illustrated in Figure 2, with the inequalities above still holding for each pair of the atoms. Summing these inequalities yields our desired result. 3.1. Power Comparison in the Multinomial Case. As the construction of this test follows Fithian, Sun and Taylor (2014), it uses UMPU selective level-α tests for the pairwise p-values. This section compares the power of our procedure to the best previously known method for verifying multinomial ranks, by Gupta and Nagel (1967). They devise a rule to select a subset that includes the maximum πj . In other words, if the selected subset is J (X), it guarantees " # (6) P arg max πj ∈ J (X) ≥ 1 − α. j This is achieved by finding an integer d, as a function on m, n and α, and selecting the subset   J (X) = j : Xj ≥ max Xk − d . k We take d(m, n, α) to be the smallest integer such that (6) holds for any π; Gupta and Nagel (1967) provide an algorithm for determining d. Subset selection is closely related to testing whether the winner is the best. In particular, we can define a test that declares j the best whenever J (X) = {j}. If J (X) satisfies (6), this test is valid at level α. We next compare the power of the resulting test against the power of imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 13
our Procedure 1 in a multinomial example with π ∝ eδ , 1, . . . , 1 , for several combinations of m and n. Figure 3 gives the power curves for Multinomial (m, π) and   π ∝ eδ , 1, . . . , 1 , for various combinations of m and n. For their method, we use α = 0.05; but in light of the extra n factor of n−1 n in (2), we will apply the selective procedure with n−1 α such that the marginal type I error rate of both procedures are controlled at α. Their test coincides with our test at n = 2; however as n grows, the selective test shows significantly more power than Gupta and Nagel’s test. Fig 3. Power curves as a function of δ. The plots in the first row all have m = 50 and the second row m = 250. The solid line and the dashed line are the power for the selective test and Gupta and Nagel’s test, respectively. To interpret, e.g., the upper right panel of Figure 3, suppose that in a poll of m = 50 respondents, one candidate enjoys 30% support and the other n − 1 = 9 split the remainder 0.3 (δ = log 0.7/9 ≈ 1.35). Then our procedure has power approximately 0.3 to detect the best candidate, while Gupta and Nagel’s procedure has power around 0.1. To understand why our method is more powerful, note that both procedures operate by comparing X[1] − X[2] to some threshold, but the two methods differ in how that threshold is determined. The threshold from Gupta and Nagel (1967) is fixed and depends on m and n alone, whereas in our procedure the threshold depends on X[1] + X[2] , a data-adaptive choice. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 14 K. HUNG AND W. FITHIAN The difference between the two methods is amplified when n is large and π(1)  1/2. In that case, d from
Gupta and Nagel is usually computed based on the worst-case scenario π = 12 , 12 , 0, . . . , 0 ; i.e. d is the upper α quantile of   1 ≈ Normal (0, m) . X1 − X2 ∼ m − 2 · Binomial m, 2 √ Thus d ≈ mzα , where zα is the upper α quantile of a standard Gaussian. On the other hand, n our method defines a threshold based on the upper n−1 · α2 quantile of   1 X1 − X2 | X1 + X2 ∼ X1 + X2 − 2 · Binomial X1 + X2 , , 2 √ which is approximately X1 + X2 zα/2 . If π(1)  1/2 then with high probability X1 + X2  m, making our test much more liberal. 4. Confidence Bounds on Differences: By How Much?. By generalizing the above, we can construct a lower confidence bound for θ[1] − maxj6=[1] θj . Here we provide a more powerful Procedure 2’ first. We will proceed by inverting a statistical test of the hypothesis δ H0[1] : θ[1] − maxj6=[1] θj ≤ δ, which can be written as a union of null hypotheses: δ H0[1] = [ H0[1]j : θ[1] − θj ≤ δ. j6=[1] By Lemma 4, we can construct selective exact one-tailed p-values pδ[1]j for each of these by conditioning on A[1] , M[1]j and X\{[1],j} , giving us an exact test for H0[1] by rejecting whenever maxj6=[1] pδ[1]j < α. Theorem 6. The p-values constructed above satisfy pδ[1][2] ≥ pδ[1]j for any j 6= [1]. Proof. Again we start with assuming X1 ≥ X2 ≥ maxj>2 Xj for convenience. The p-values in question are derived from the conditional law Lθ1 −θj =δ (D1j | M1j , X2 , . . . , Xn , A) , which is the truncated distribution p (d1j ) ∝ exp ((θ1 − θj ) d1j + θ2 X2 + · · · + (θ1 + θj ) M1j + · · · + θn Xn ) g (M1j + d1j , X2 , . . . , M1j − d1j , . . . Xn ) 1A1 ∝ exp (δd1j ) g (M1j + d1j , X2 , . . . , M1j − d1j , . . . Xn ) 1A1 . The p-values thus are R∞ pδ1j = R ∞ D1j exp (δz) g (M1j + z, X2 , . . . , M1j − z, . . . , Xn ) dz max{X2 −M1j ,0} exp (δz) g (M1j + z, X2 , . . . , M1j − z, . . . , Xn ) dz . imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 15 As before in Part 1 of Theorem 1, the conditioning reduces to the case where n = 3. Once again it is sufficient to show that p12 ≥ p13 . We have the same two cases. If X2 < M13 , then R∞ exp (δ (z + D12 )) g (X1 + z, X2 − z, X3 ) dz δ p12 = R ∞0 −D12 exp (δ (z + D12 )) g (X1 + z, X2 − z, X3 ) dz R∞ exp (δz) g (X1 + z, X2 − z, X3 ) dz = R ∞0 −D12 exp (δz) g (X1 + z, X2 − z, X3 ) dz R∞ exp (δ (z + D13 )) g (X1 + z, X2 , X3 − z) dz δ p13 = R ∞0 −D13 exp (δ (z + D13 )) g (X1 + z, X2 , X3 − z) dz R∞ exp (δz) g (X1 + z, X2 , X3 − z) dz = R ∞0 . −D13 exp (δz) g (X1 + z, X2 , X3 − z) dz The same argument in Figure 2 shows that pδ12 ≥ pδ13 . This is again true for the case where X2 ≥ M13 as well. In other words, Procedure 2’ can be summarized as: Find the minimum δ such that pδ[1][2] ≤ α. And by construction, Procedure 2’ gives exact 1 − α confidence bound for θ[1] − maxj6=[1] θj . Part 2 of Theorem 1. Assume the model (1) holds and g (x) is a Schur-concave function. Procedure 2 (the lower bound of unadjusted pairwise confidence interval) gives a conservative 1 − α lower confidence bound for θ[1] − maxj6=[1] θj . Proof. When Procedure 2 reports −∞ as a confidence lower bound, it is definitely valid and conservative. It remains to show that when Procedure 2 reports a finite confidence lower bound, it is smaller than the confidence lower bound reported by Procedure 2’. If Procedure 2 reports a finite confidence lower bound δ ∗ , then δ ∗ ≥ 0. Also R∞ exp (δ ∗ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz α (7) = RD∞12 ∗ 2 −∞ exp (δ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz as Procedure 2 is constructed from an unadjusted two-tail pairwise confidence interval. However, as δ ∗ ≥ 0, we have R0 exp (δ ∗ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz R−∞ ≤1 ∞ ∗ 0 exp (δ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz R∞ exp (δ ∗ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz R−∞ ≤ 2. ∞ ∗ 0 exp (δ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz Multiplying this to (7), we have R∞ exp (δ ∗ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz 12 α ≥ RD∞ , ∗ 0 exp (δ z) g (M12 + z, X2 , . . . , M12 − z, . . . , Xn ) dz indicating that δ ∗ is smaller than the confidence bound that Procedure 2’ would report. Hence δ ∗ is a valid and conservative. Note that Procedure 2 reporting −∞ in case of δ ∗ ≤ 0 is rather extreme. In reality, we can always just adopt Procedure 2’ in the case when Procedure 1 rejects. In fact, by Procedure 2’, the multinomial example for polling in Section 1.1 can give a stronger lower confidence bound, that πTrump / maxj6=Trump πj ≥ 1.108 (Trump leads the field by at least 10.8%). imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 16 K. HUNG AND W. FITHIAN 5. Verifying Other Ranks: Is the Runner-Up Really the Second Best, etc.?. Often we will be interested in verifying ranks beyond the winner. More generally, we could imagine declaring that the first j populations are all in the correct order, that is θ[1] > · · · > θ[j] > max θ[k] . (8) k>j Let j0 denote the largest j for which (8) is true. Note that j0 is both random and unknown, because it depends on both the data and population ranks. Procedure 3 declares that j0 ≥ j if the unadjusted pairwise tests between X[k] and X[k+1] , reject at level α for all of k = 1, . . . , j. In terms of the Iowa polling example of Section 1, we would like to produce a statement of the form “Trump has the most support, Cruz has the second-most, and Rubio has the third-most.” Procedure 3 performs unadjusted pairwise tests to ask if Cruz is really the runner-up upon verifying that Trump is the best, and if Rubio is really the second runner-up upon verifying that Cruz is the runner-up, etc., until we can no longer infer that a certain population really holds its rank. While we aim to declare more populations to be in the correct order, declaring too many populations, i.e. out-of-place populations, to be in the right order is undesirable. It is possible to consider false discovery rate (the expected portion of out-of-place populations declared) here, but we restrict our derivation to FWER (the probability of having any out-of-place populations declared). Formally, let ĵ0 denote the number of ranks validated by a procedure (the number of rejections). h iThen the FWER of ĵ0 is the probability that too many rejections are made; i.e. P ĵ0 > j0 . For example, suppose that the top three data ranks and population ranks coincide, but not the fourth (j0 = 3). Then we will have made a Type I error if we declare that the top five ranks are correct (ĵ0 = 5), but not if we declare that the top two are correct (ĵ0 = 2). In other words, ĵ0 is a lower confidence bound for j0 . To show that Procedure 3 is valid, we will prove the validity of a more liberal Procedure 3’, described in Algorithm 1. Procedure 3 is equivalent to Procedure 3’ for the most part, except  that Procedure 3 conditions on a larger event X[j] ≥ maxk>j X[k] in Line 7. Theorem 7. Procedure 3’ is a stepwise procedure that an estimate ĵ0 of j0 at the FWER controlled at α, where j0 is given by   j0 = max θ[1] > · · · > θ[j] > max θ[k] . j k>j Proof. We will first show that Procedure 3’ falls into the sequential goodness-of-fit testing framework proposed by Fithian, Taylor and Tibshirani (2015). We thus analyze Procedure 3’ as a special case of the BasicStop procedure on random hypothesis, described in the same paper. This enables us to construct valid selective p-values and derive Procedure 3’. Application of the sequential goodness-of-fit testing framework. X[n] , we can set up a sequence of nested models M1 (X) ⊆ · · · ⊆ Mn (X) , Upon observing X[1] ≥ · · · ≥   where Mj (X) = θ : θ[1] > · · · > θ[j] > max θ[k] . c k>j imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 17 Algorithm 1. Procedure 3’, a more liberal version of Procedure 3 1 2 3 4 5 6 7 8 input : X1 , . . . , Xn output: ĵ0 , an estimate for j0 # Initialization τj ← [j]; # Consider τj as part of the observation and the fixed realization of the random index [j] Xτ0 ← ∞; j ← 0; rejected ← true; while rejected do j ← j + 1; Dτj ← Xτj − Xτj+1 ; Set up the distribution of Dτj τj+1 , conditioned on • the variables Xτ1 , . . . , Xτj−1 , Xτj+2 , . . . , Xτn ] , and  • the event Xτj−1 ≥ Xτj ≥ maxk>j Xτk ) ; # The distribution of Dτj τj+1 depends only on θτj − θτj+1 now test H0 : θτj − θτj+1 ≤ 0 against H1 : θτj − θτj+1 > 0 according to the distribution of Dτj τj+1 ; Set rejected as the output of the test; 9 10 11 end ĵ0 ← j − 1; If we define the j-th null hypothesis as e 0j : θ[j] ≤ max θ[k] , H k>j e 01 , . . . , H e 0j are all false if and only if θ ∈ then H / Mj (X). In other words, Mj (X) is a family of distributions that does not have all first j ranks correct. e 0j , stating that without As we will see later, each step in Procedure 3’ is similar to testing H the first j ranks correct, it is hard to explain the observations. Thus, returning ĵ0 = j amounts e 01 , . . . , H e 0j , or equivalently determining that the models M1 (X), . . . , Mj (X) do to rejecting H not fit the data. e 0j provided intuition in the setting up the nested models, they While the null hypotheses H are rather cumbersome to work with. Inspired by Fithian, Taylor and Tibshirani (2015), we will instead consider another sequence of random hypothesis that are more closely related to the nest models, H0j : θ ∈ Mj (X) , or equivalently, that θ[1] , . . . , θ[j] are not the best j parameters in order.   Adapting this notation, the FWER can be viewed as P reject H0(j0 +1) . Special case of the BasicStop procedure. While impractical, Procedure 3’ can be thought of as performing all n tests first, producing a sequence of p-values pj , and returning (9) ĵ0 = min {j : pj > α} − 1. This is a special case of the BasicStop procedure. Instead of simply checking that Procedure 3’ fits all the requirement for FWER control in BasicStop, we will give the construction of Procedure 3’, assuming that we are to estimate j0 with BasicStop. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 18 K. HUNG AND W. FITHIAN In general, the FWER for BasicStop can be rewritten as P [pj0 +1 ≤ α]. This is however difficult to analyze, as j0 itself is random and dependent on X, thus we break the FWER down as follows: X P [pj0 +1 ≤ α] = P [pj0 +1 ≤ α | j0 = j] P [j0 = j] j = X = X P [pj+1 ≤ α | j0 = j] P [j0 = j] j P [pj+1 ≤ α | θ ∈ Mj+1 (X) \ Mj (X)] P [j0 = j] . j We emphasize here that θ is not random, but Mj+1 is. Thus it suffices to construct the p-values such that (10) P [pj ≤ α | θ ∈ Mj (X) \ Mj−1 (X)] ≤ α for all j. Considerations for conditioning. By smoothing, we are free to condition on additional variables in (10). A logical choice that simplified (10) is conditioning on the variables Mj−1 (X) and Mj (X). Note that the choice of the model Mj (X), once again, based solely on the random indices [1], . . . , [j], so conditioning on both Mj−1 (X) and Mj (X) is equivalent to conditioning on the random indices [1], . . . , [j], which in turns is equivalent to conditioning on the σ-field generated by the partition of the observation space X    Xτ1 ≥ · · · ≥ Xτj ≥ max Xτk : τ is any permutation of (1, . . . , n) , k>j or colloquially, the set of all possible choices  of [1], . . . , [j]. Within each set in this partition, the event {θ ∈ Mj (X) \ Mj−1 (X)} is simply θτ1 > · · · > θτj and θτj ≤ maxk>j θτk , a trivial event. As a brief summary, we want to construct p-values pj such that   pj ≤ α Xτ1 ≥ · · · ≥ Xτj ≥ max Xτk . P θτ1 >···>θτj θτj ≤maxk>j θτk k>j Construction of the p-values. To avoid the clutter in the subscripts, we will drop the τ in the subscript. Hence our goal is now   P θ1 >···>θj pj ≤ α X1 ≥ · · · ≥ Xj ≥ max Xk θj ≤maxk>j θk k>j Construction of pj for other permutations τ can be obtained similarly. There are many valid options for pj (such as constant α). We will follow the idea in the proof of Part 1 of Theorem 1 here. pj is intended to test H0j : θ ∈ Mj (X), which is equivalent to the union of the null hypotheses: 1. θk ≤ θk+1 for k = 1, . . . , j − 1, and e 0j .) 2. θj ≤ θk for k = j + 1, . . . , n. (The union of these null hypotheses is H imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 19 RANK VERIFICATION FOR EXP. FAMILIES Since the joint distribution of X, restricted to {X1 ≥ · · · ≥ Xj ≥ maxk>j Xk }, remains in the exponential family, we can construct the p-values for each of the hypotheses above by conditioning on the variables corresponding to the nuisance parameters here, similar to the proof of Part 1 of Theorem 1. Then we can take pj as the maximum of such p-values. For the hypothesis H0jk : θj ≤ θk , we can construct pjk , by considering the survival function of the conditional law     X1 ≥ · · · ≥ Xj ≥ max X` , X\{j,k} , Mjk Lθj =θk Djk `>j       = Lθj =θk Djk X ≥ Xj ≥ max X` and Xj ≥ Mjk , X\{j,k} , Mjk    j−1 `>j `6=k Once again, Xj+1 = max`>j X` is simply shorthand for simplifying our notation. Now the p-values are similar to the ones in Equation (4), for k > j: R Xj−1 Djk g (X1 , . . . , Mjk + z, . . . , Mjk − z, . . . , Xn ) dz pjk = R X . j−1 g (X , . . . , M + z, . . . , M − z, . . . , X ) dz 1 n jk jk max{Xj+1 −Mjk ,0} We can graphically represent pjk in Figure 4, a diagram analogous to Figure 2. D C B    (Xj , Xj+1 , Xk ) Truncation for Xj ≤ Xj−1 (Mjk , Xj+1 + Djk , Xk )
Mj(j+1) , Mj(j+1) , Xk à A (Mjk , Xj+1 , Mjk ) A1 Fig 4. The two p-values constructed corresponds to taking integrals of g along these segments, that lie on a level set of xj + xj+1 + xk . The dashed line corresponds to extension in (5). The dotted line on the far right is the truncation that enforces Xj < Xj−1 . We have pj(j+1) ≥ maxk>j pjk by Section 3: the upper truncation for Xj can be represented by cropping Figure 2 along a vertical line, shown in Figure 4. Considering pj(j+1) is sufficient in rejecting all the H0jk . We will take pj∗ = pj(j+1) , noting that this is the p-value that Procedure 3’ would produce. In fact, pj∗ is also the p-value we would have constructed if we were to reject e 0j . only H Upon constructing pj , one should realize that the p-values for testing θk ≤ θk+1 would have been constructed in earlier iterations of BasicStop, as pk∗ . In other words, pj = maxk≤j pk∗ is imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 20 K. HUNG AND W. FITHIAN the sequence of p-values that works with BasicStop. However, from (9),   ĵ0 = min j : max pk∗ > α − 1 = min {j : pj∗ > α} − 1, k≤j so it is safe to apply BasicStop to pj∗ directly, yielding Procedure 3’. Part 3 of Theorem 1. Assume the model (1) holds and g (x) is a Schur-concave function. Procedure 3 is a conservative stepwise procedure with FWER no larger than α. Proof. The p-values pj(j+1) obtained in Procedure 3’ are always smaller than their counterpart in Procedure 3, as the upper truncation at Xj−1 is on the upper tail. Therefore Procedure 3 is conservative and definitely valid. 6. Discussion. Combining ideas from conditional inference and multiple testing, we have proven the validity of several very simple and seemingly “naive” procedures for significance testing of sample ranks. In particular, we have shown that an unadjusted pairwise test comparing the winner with the runner-up is a valid significance test for the first rank. Our result complements and extends pre-exisiting analogous results for location and location-scale families with independence between observations. Our approach is considerably more powerful than previously known solutions. We provide similarly straightforward conservative methods for producing a lower confidence bound for the difference between the winner and runner up, and for verifying ranks beyond the first. Claims reporting the “winner” are commonly made in the scientific literature, usually with no significance level reported or an incorrect method applied. For example, Uhls and Greenfield (2012) asked n = 20 elementary and middle school students which of seven personal values they most hoped to embody as adults, with “Fame” (8 responses) being the most commonly selected, with “Benevolence” (5 responses) second. The authors’ main finding — which appeared in the abstract, the first paragraph of the article, and later a CNN.com headline (Alikhani, 2011) — was that “Fame” was the most likely response, accompanied by a significance level of 0.006, which the authors computed by testing whether the probability of selecting “Fame” was larger than 1/7. The obvious error in the authors’ reasoning could have been avoided if they had performed an equally straightforward two-tailed binomial test of “Fame” vs. “Benevolence,” which would have produced a p-value of 0.58. Reproducibility. A git repository containing with the code generating the image in this paper is available at https://github.com/kenhungkk/verifying-winner. References. Alikhani, L. (2011). Study: Tween TV today is all about fame. Berger, R. L. (1980). Minimax subset selection for the multinomial distribution. Journal of Statistical Planning and Inference 4 391–402. Berger, R. L. (1982). Multiparameter hypothesis testing and acceptance sampling. Technometrics. Besag, J. and Clifford, P. (1989). Generalized monte carlo significance tests. Biometrika 76 633–642. Bofinger, E. (1991). Selecting “Demonstrably best” or “Demonstrably worst” exponential population. Australian Journal of Statistics 33 183–190. Edwards, D. G. and Hsu, J. C. (1983). Multiple comparisons with the best treatment. Journal of the American Statistical Association 78 965–971. Finner, H. and Strassburger, K. (2002). The partitioning principle: a powerful tool in multiple decision theory. The Annals of Statistics 30 1194–1213. imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 RANK VERIFICATION FOR EXP. FAMILIES 21 Fithian, W., Sun, D. L. and Taylor, J. E. (2014). Optimal Inference After Model Selection. arXiv.org. Fithian, W., Taylor, J. E. and Tibshirani, R. J. (2015). Selective Sequential Model Selection. arXiv.org. Gupta, S. S., Huang, D.-Y. and Panchapakesan, S. (1984). On some inequalities and monotonicity results in selection and ranking theory. In Inequalities in statistics and probability (Lincoln, Neb., 1982) 211–227. Inst. Math. Statist., Hayward, CA, Hayward, CA. Gupta, S. S. and Liang, T. (1989). Selecting the best binomial population: parametric empirical Bayes approach. Journal of Statistical Planning and Inference 23 21–31. Gupta, S. S. and Nagel, K. (1967). On selection and ranking procedures and order statistics from the multinomial distribution. Sankhyā: The Indian Journal of Statistics 29. Gupta, S. S. and Panchapakesan, S. (1971). On Multiple Decision (Subset Selection) Procedures Technical Report, Purdue University. Gupta, S. S. and Panchapakesan, S. (1985). Subset Selection Procedures: Review and Assessment. American Journal of Mathematical and Management Sciences 5 235–311. Gupta, S. S. and Wong, W.-Y. (1976). On Subset Selection Procedures for Poisson Processes and Some Applications to the Binomial and Multinomial Problems Technical Report. Gutmann, S. and Maymin, Z. (1987). Is the selected population the best? The Annals of Statistics 15 456–461. Hsu, J. C. (1984). Constrained Simultaneous Confidence Intervals for Multiple Comparisons with the Best. The Annals of Statistics 12 1136–1144. Hsu, J. (1996). Multiple comparisons: theory and methods. CRC Press. Quinnipiac University Poll Institute (2016). First-Timers Put Trump Ahead In Iowa GOP Caucus, Quinnipiac University Poll Finds; Sanders Needs First-Timers To Tie Clinton In Dem Caucus. Karnnan, N. and Panchapakesan, S. (2009). Does the Selected Normal Population Have the Smallest Variance? American Journal of Mathematical and Management Sciences 29 109–123. Marshall, A. W., Olkin, I. and Arnold, B. (2010). Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics. Springer, New York, NY. Maymin, Z. and Gutmann, S. (1992). Testing retrospective hypotheses. The Canadian Journal of Statistics. La Revue Canadienne de Statistique 20 335–345. Nettleton, D. (2009). Testing for the supremacy of a multinomial cell probability. Journal of the American Statistical Association 104 1052–1059. Ng, H. K. T. and Panchapakesan, S. (2007). Is the selected multinomial cell the best? Sequential Analysis 26 415–423. Stefansson, G., Kim, W.-C. and Hsu, J. C. (1988). On confidence sets in multiple comparisons. In Statistical Decision Theory and Related Topics IV 89–104. . . . Decision Theory and . . . . Uhls, Y. T. and Greenfield, P. M. (2012). The value of fame: preadolescent perceptions of popular media and their relationship to future aspirations. Developmental psychology. Department of Mathematics 951 Evans Hall, Suite 3840 Berkeley, CA 94720-3840 E-mail: kenhung@berkeley.edu Department of Statistics 301 Evans Hall Berkeley, CA 94720 E-mail: wfithian@berkeley.edu imsart-aos ver. 2014/10/16 file: verifying-winner.tex date: July 4, 2017 

arXiv:1704.06398v2 [] 24 Jul 2017 Tail sums of Wishart and GUE eigenvalues beyond the bulk edge. Iain M. Johnstone∗ Stanford University and Australian National University July 26, 2017 Abstract Consider the classical Gaussian unitary ensemble of size N and the real Wishart ensemble WN (n, I). In the limits as N → ∞ and N/n → γ > 0, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of γ. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation. 1 Introduction This paper develops some tail sum bounds on eigenvalues outside the bulk that are needed for results on estimation of covariance matrices in the spiked model, Donoho et al. [2017]. This application is described briefly in Section 4. It depends on properties of the eigenvalues of real white Wishart matrices, distributed as WN (n, I), which are the main focus of this note. Specifically, suppose that A ∼ WN (n, I), and that λ1 ≥ . . . ≥ λN are eigenvalues of the sample covariance matrix n−1 A. In the limit N/n → γ > 0, it is well known that the empirical distribution of {λi } converges to the Marcenko-Pastur law (see e.g. Pastur and Shcherbina [2011, Corollary 7.2.5]), which is supported on an interval Iγ — √ augmented with 0 if γ > 1 — having upper endpoint λ(γ) = (1 + γ)2 . We focus on the eigenvalues λi that exit this “bulk” interval Iγ on the upper side. In statistical application, such exiting eigenvalues might be mistaken for “signal” and so it is useful to have some bounds on what can happen under the null hypothesis of no signal. Section 3 studies the mean value behavior of quantities such as N X TN = [λi − λ(γ)]q+ , i=1 q≥0 Thanks to David Donoho and Matan Gavish for discussions and help. Peter Forrester provided an important reference. Work on this paper was completed during a visit to the Research School of Finance, Actuarial Studies and Statistics, A. N. U., whose hospitality and support is gratefully acknowledged. Work supported in part by the U.S. National Science Foundation and the National Institutes of Health. ∗ 1 which for q = 0 reduces to the number TN0 of exiting eigenvalues. a.s. It is well known that the largest eigenvalue λ1 → λ(γ) [Geman, 1980], and that closed intervals outside the bulk support contain no eigenvalues for N large with probability one [Bai and Silverstein, 1998]. However these and even large deviation results for λ1 [Majumdar and Vergassola, 2009] and TN0 [Majumdar and Vivo, 2012] seem not to directly yield the information on E(TN ) that we need. Marino et al. [2014] looked at the variance of TN0 using methods related to those of this note. Recently, Chiani [2017] has studied the probability that all eigenvalues of Gaussian, Wishart and double Wishart random matrices lie within the bulk, and derived universal limiting values of 0.6921 and 0.9397 in the real and complex cases respectively. In summary, the motivation for this note is high-dimensional covariance estimation, but there are noteworthy byproducts: the asymptotic values of E(TN0 ) are perhaps suprisingly small, and numerically for the Gaussian Unitary Ensemble (GUE), it is found that the chance of even two exiting eigenvalues is very small, of order 10−6 . 2 The Gaussian Unitary Ensemble Case (GUE) We begin with GUE to illustrate the methods in the simplest setting, and to note an error in the literature. Recall that the Gaussian Unitary ensemble GUE(N) is the Gaussian probability measure on the space of N × N Hermitian matrices with density proportional to exp{− 21 NtrA2 }. Theorem 1. Let λ1 , . . . , λN be eigenvalues of an N-by-N matrix from the GUE. Denote by λ+ = 2 the upper edge of the Wigner semicircle, namely, the asymptotic density of the eigenvalues. For q ≥ 0, let TN = N X i=1 (λi − λ+ )q+ . (1) Then, with a constant cq specified at (3) below, E(TN ) = cq N −2q/3 (1 + o(1)). In particular, for q = 0 and TN = #{i : λi > λ+ }, 1 E(TN ) → c0 = √ ≈ 0.030629. 6 3π (2) Proof. We use the so-called one-point function and bounds due to Tracy and Widom [1994, 1996]. To adapt to their let (yi )N 1 be the eigenvalues of GUE with joint density Pnnotation, 2 2 proportional to exp(− 1 yi )∆ (y), where ∆(y) is the usual Vandermonde. In √ this scaling the eigenvalue bulk p concentrates as the psemi-circle on [−µN , µN ] with µN = 2N. We have yi = N/2 λi and µN = N/2 λ+ , for λ+ = 2, so that TN = N X 1 (λi − λ+ )q+ = N  2 q/2 X N 2 1 (yi − µN )q+ . From the determinantal structure of GUE, the marginal density of a single (unordered) eigenvalue yi is given by the one-point function N −1 SN (y, y) = N −1 N −1 X φ2k (y), k=0 where φk (y) are the (Hermite) functions obtained by orthonormalizing y k e−y  2 q/2 Z ∞ E(TN ) = (y − µN )q SN (y, y)dy. N µN 2 /2 . Thus Now introduce the TW scaling τN = √ y = µN + τN x, 1 , 2N 1/6 and let Ai denote the Airy function. Tracy and Widom [1996, p 745-6] show that SτN (x, x) = τN SN (µN + τN x, µN + τN x) Z ∞ → KA (x, x) = Ai2 (x + z)dz, 0 with the convergence being dominated: SτN (x, x) ≤ M 2 e−2x . Consequently,  2τ 2 q/2 Z ∞ N xq SτN (x, x)dx E(TN ) = N 0 Z ∞ −2q/3 ∼N xq KA (x, x)dx. 0 In particular, E(TN ) = O(N −2q/3 ), and if q = 0, then E(TN ) converges to a positive constant. Integration by parts and Olver et al. [2010, 9.11.15] yield Z ∞ Z ∞ Z ∞ q cq = x KA (x, x)dx = xq Ai2 (z)dzdx 0 0 x Z ∞ (3) 1 2Γ(q + 1) q+1 2 = x Ai (x)dx = √ q+1 0 π12(2q+9)/6 Γ((2q + 9)/6) √ For q = 0 the constant becomes c0 = 1/(6 3π). Remarks. 1. Ullah [1983] states, in our notation, that the expected number of eigenvalues above the bulk edge, E(TN ) ∼ 0.25N −1/2 . This claim cannot be correct: a counterexample uses the limiting law F2 for y(1) = maxi yi of Tracy and Widom [1994]: √ (4) E(TN ) ≥ Pr(y(1) > 2N ) → 1 − F2 (0) = 0.030627. We evaluated numerically in Mathematica the formulas (U3), (U6) and (U7) for p = (2/N)E(TN ) given in Ullah [1983]. While numerical results from intermediate formula (U3) 3 Table 1: For √ GUE(N), the probabilities pN (k) of exactly k eigenvalues exceeding the upper bulk edge 2N, along with the expected number E(TN ), to be compared with limiting value (2). N pN (1) pN (2) pN (3) E(TN ) 10 25 50 100 250 500 2.868 · 10−2 2.955 · 10−2 2.994 · 10−2 3.019 · 10−2 3.039 · 10−2 3.048 · 10−2 1.36 · 10−6 1.70 · 10−6 1.88 · 10−6 2.00 · 10−6 2.09 · 10−6 2.14 · 10−6 6.9 · 10−14 1.4 · 10−13 1.9 · 10−13 2.3 · 10−13 2.6 · 10−13 2.8 · 10−13 0.028681 0.029551 0.029944 0.030195 0.030392 0.030480 are consistent with our (2), neither those from (U6) nor those from the final result (U7) are consistent with (U3), or indeed with each other! 2. The striking closeness of the right side of (4) to (2) led us to use the Matlab toolbox of Bornemann [2010] to evaluate numerically √ (n) pN (k) = Pr( exactly k of {yi } > 2N) = E2 (k, J) √ with J = ( 2N, ∞), in the notation of Bornemann [2010]. The results, in Table 1, confirm that the probability of 2 or more eigenvalues exiting the bulk is very small, of order 10−6 , for all N. This is also suggested by the plots of the densities of y(1) , y(2) , . . . in the scaling limit in Figure 4 of Bornemann [2010], which itself extends Figure 2 of Tracy and Widom [1994]. 3 The real Wishart case Suppose λi are eigenvalues of n−1 XX ⊤ for X a N × n matrix with i.i.d. N(0, 1) entries. √ Assume that γN = N/n → γ ∈ (0, 1]. Set λ(γ) = (1 + γ)2 . We recall the scaling for the Tracy-Widom law from the largest eigenvalue λ1 : λ1 = λ(γN ) + N −2/3 τ (γN )WN where WN converges in distribution to W ∼ T W1 and τ (γ) = √ √ γ( γ + 1)4/3 . Theorem 2. (a) Suppose η(λ, c) ≥ 0 is jointly continuous in λ and c, and satisfies η(λ, c) = 1 η(λ, c) ≤ Mλ for λ ≤ λ(c) for some M and all λ. Suppose also that cN − γN = O(N −2/3 ). Then for q > 0, E X N i=1 [η(λi , cN ) − 1] 4 q  → 0. (5) (b) Suppose cN − γN ∼ sσ(γ)N −2/3 , where σ(γ) = τ (γ)/λ′ (γ) = γ(1 + X  Z N q q −2q/3 E [λi − λ(cN )]+ ∼ τ (γ)N i=1 √ γ)1/3 . Then ∞ s (x − s)q+ K1 (x, x)dx. (6) where K1 is defined at (9) below. (c) In particular, let Nn = #{i : λi ≥ λ(cN )} and suppose that cN − γN = o(N −2/3 ). Then Z ∞ ENn → c0 = K1 (x, x)dx ≈ 0.17. 0 Remarks. 1. Part (b) represents a sharpening of (5) that is relevant when η(λ) = η(λ, γ) is Hölder continuous in λ near the bulk edge λ(γ), η(λ) − η(λ(γ)) ∼ (λ − λ(γ))q+ . The example q = 1/2 occurs commonly for optimal shrinkage rules η ∗ (λ) in Donoho et al. [2017]. 2. Section 4 explains why we allow cN to differ from γN . Proof. Define TN = N X ( [η(λ, c) − 1]q F (λ, c) = [λ − λ(c)]q+ F (λi , cN ), i=1 (a) (b). We adapt the discussion here to the notation used in Tracy and Widom [1998] and Johnstone [2001]. Let (yi )N 1 = nλi be the eigenvalues of WN (n, I) with joint density function PN (y1 , . . . , yN ) with explicit form given, for example, in [Johnstone, 2001, eq. (4.1)]. We obtain Z ∞ E(TN ) = F (y/n, cN )R1 (y)dy, 0 R where R1 (y1 ) = N (0,∞)N−1 PN (y1 , . . . , yN )dy2 · · · dyN is the one-point (correlation) function. It follows from Tracy and Widom [1998, p814–16] that R1 (y) = T1 (y) = 21 tr KN (y, y) (7) where KN (x, y) is the 2 × 2 matrix kernel associated with PN , see e.g. [Tracy and Widom, 1998, eq. (3.1)]. It follows from Widom [1999] that 1 tr 2 KN (y, y) = S(y, y) + ψ(y)(ǫφ)(y) = S1 (y, y), (8) where the functions S(y, y ′), ψ(y) and φ(y) are defined in terms of orthonormalized Laguerre polynomials in Widom [1999] and studied further in Johnstone [2001]. The function ǫ(x) = 1 sgnx and the operator ǫ denotes convolution with the kernel ǫ(x − y). 2 5 For convergence, introduce the Tracy-Widom scaling y = µN + σN x, 1 2 and nh = n + 21 and define p √ 1/3 σN = c(Nh /nh )Nh , µN = ( Nh + nh )2 , √ √ √ where c(γ) = (1 + γ)1/3 (1 + 1/ γ) = (1 + γ)1/3 λ′ (γ) We now rescale the scalar-valued function (8): S1τ (x, x) = σN S1 (µN + σN x, µN + σN x). where we set Nh = N + We can rewrite our target E(TN ) using (7), (8) and this rescaling in the form Z ∞ E(TN ) = F (ℓN (x), cN )S1τ (x, x)dx, δN where ℓN (x) = (µN +σN x)/n, δN = (nλ(cN )−µN )/σN and we used the fact that F (λ, c) = 0 for λ ≤ λ(c). It follows from [Johnstone, 2001, eq. (3.9)] that   Z ∞ Z ∞ S1τ (x, x) = 2 φτ (x + u)ψτ (x + u)du + ψτ (x) cφ − φτ (u)du . 0 x It is shown in equations (3.7), 3.8) and Sec. 5 of that paper that 1 φτ (x), ψτ (x) → √ Ai(x) 2 and, uniformly in N and in intervals of x that are bounded below, that φτ (x), ψτ (x) = O(e−x ). √ Along with cφ → 1/ 2 (cf. App. A7 of same paper), this shows that   Z ∞ Z ∞ 2 1 S1τ (x, x) → K1 (x, x) = Ai (x + z)dz + 2 Ai(x) 1 − Ai(z)dz > 0 0 (9) x with the convergence being dominated S1τ (x, x) ≤ M 2 e−2x + M ′ e−x . (10) Before completing the argument for (a) – (c), we note it is easily checked that n−1 µN = λ(γN ) + O(N −1 ), so that δN = n [λ(cN ) − λ(γN )] + O(N −1/3 ). σN If cN − γN = θN N −2/3 for θN = O(1) then δN ∼ n −2/3 N θN λ′ (γ) ∼ θN /σ(γ), σN 6 (11) since we have N 2/3 σN /n ∼ σ(γ)λ′ (γ) = τ (γ). (12) In case (a), then, δN ≥ −A for some A. We then have ℓN (x) → λ(γ) for all x ≥ −A, and so from joint continuity η(ℓN (x), cN ) → η(λ(γ), γ) = 1, and hence for all x ≥ −A, F (ℓN (x), cN ) = [η(ℓN (x), cN ) − 1]q → 0 (13) The convergence is dominated since the assumption η(λ, c) ≤ Mλ implies that |F (ℓN (x), cN )| ≤ C(1 + |x|q ). Hence the convergence (13) along with (10) and the dominated convergence theorem implies (5). For case (b), Z ∞ 2q/3 N E(TN ) = [N 2/3 (ℓN (x) − λ(cN ))]q+ S1τ (x, x)dx. δN Observe that N 2/3 (λ(γN ) − λ(cN )) ∼ N 2/3 λ′ (γ)(γN − cN ) ∼ −sτ (γ), and so from (11) and (12), we have N 2/3 (ℓN (x)−λ(cN )) = O(N −1/3 )+N 2/3 (λ(γN )−λ(cN ))+N 2/3 n−1 σN x ∼ τ (γ)(x−s). (14) In addition, from (14), we have N 2/3 |ℓN (x) − λ(cN )| ≤ M(1 + |x|), so that the convergence is dominated and (6) is proven. For case (c), we have only to evaluate Z ∞ Z ∞ Z 1 c0 = K1 (x, x)dx = KA (x, x)dx + 4 0 0 ∞ G′ (x)dx = I1 + I2 , 0 I1 was evaluated in the previous section and G(x) = [1 − Rwhere ∞ Ai(z)dz = 1/3, from Olver et al. [2010, 9.10.11], we obtain 0 4I2 = G(∞) − G(0) = 1 − (2/3)2 = 5/9, with the result 5 1 ≈ 0.031 + 0.139 = 0.16952. c0 = √ + 6 3π 36 7 R∞ x Ai(z)dz]2 . Since 4 Application to covariance estimation We indicate how Theorem 2 is applied to covariance estimation in the spiked model studied in Donoho et al. [2017]. Consider a sequence of statistical problems indexed by dimension p and sample size n. In the nth problem X̌ ∼ Np (0, Σ) where p = pn sastisfies pn /n → γ ∈ (0, 1] and the population covariance matrix Σ = Σp has fixed ordered eigenvalues ℓ1 ≥ . . . ≥ ℓr > 1 for all n, and then ℓr+1 = . . . = ℓpn = 1. Suppose that the sample covariance matrix Š = Šn,pn has eigenvalues λ̌1 ≥ . . . ≥ λ̌p and corresponding eigenvectors v1 , . . . , vp . Consider shrinkage estimators of the form Σ̂η = p X η(λ̌j , cp )vj vj⊤ , (15) j=1 where η(λ, c) is a continuous bulk shrinker, that is, satisfies the conditions (a) of Theorem 2. Without loss of generality, as explained in the reference cited, we may also assume that λ → η(λ, c) is non-decreasing. In the spiked model, the typical choice for cp in practice would be to set cp = p/n, and we adopt this choice below. It is useful to analyse an “oracle” or “rank-aware” variant of (15) which takes advantage of the assumed structure of Σp , especially the fixed rank r of Σp − I: Σ̂η,r = r X η(λ̌j , cp )vj vj⊤ + p X vj vj⊤ . j=r+1 j=1 The error in estimation of Σ using Σ̂ is measured by a loss function Lp (Σ, Σ̂). One seeks conditions under which the losses Lp (Σ, Σ̂η ) and Lp (Σ, Σ̂η,r ) are asymptotically equivalent. They consider a large class of loss functions which satisfy a Lipschitz condition which implies that, for some q, |Lp (Σ, Σ̂η ) − Lp (Σ, Σ̂η,r )| ≤ C(ℓ1 , η(λ̌1)) p X [η(λ̌j , cp ) − 1]q . j=r+1 Suppose now that Π : Rp → Rp−r is a projection on the span of the p − r unit eigenvectors of Σ. Let X = ΠX̌ and let λ1 ≥ · · · ≥ λp−r denote the eigenvalues of n−1 XX ⊤ . By the Cauchy interlacing Theorem (e.g. [Bhatia, 1997, p. 59]), we have λ̌j ≤ λj−r for r + 1 ≤ j ≤ p, (16) where the (λi )p−r i=1 are the eigenvalues of a white Wishart matrix Wp−r (n, I). From the monotonicity of η, p−r p X X q [η(λi , cp ) − 1]q . (17) [η(λ̌j , cp ) − 1] ≤ i=1 j=r+1 Now apply part (a) of Theorem 2 with the identifications N ← p − r, cN ← cp . 8 Clearly γN = N/n → γ and cN − γ N = N +r N − = O(N −2/3 ), n n since r is fixed. We conclude that the right side of (17) and hence |Lp (Σ, Σ̂η ) − Lp (Σ, Σ̂η,r )| converge to 0 in L1 and in probability. Part (c) of Theorem 2 helps to give an example where the losses Lp (Σ, Σ̂η ) and Lp (Σ, Σ̂η,r ) −1 are not asymptotically equivalent. Indeed, let Lp (Σ, Σ̂η ) = kΣ̂−1 η − Σ k, with k · k denoting matrix operator norm. Here the shrinkage rule η = η ∗ (λ, c) is discontinuous at the √ optimal upper bulk edge λ(c) = (1 + c)2 : η ∗ (λ, c) = 1 ∗ η (λ, c) → 1 + √ c for λ ≤ λ(c) for λ ↓ λ(c). Proposition 3 of Donoho et al. [2017] shows that D −1 −1 −1 kΣ̂−1 η − Σ k − kΣ̂η,r − Σ k → W, (18) where W has a two point distribution (1 − π)δ0 + πδw with non-zero √ probability π = Pr(T W1 > 0) at location w = f (ℓ+ ) − f (ℓr ), where ℓ+ = 1 + c and the function  1/2 c(ℓ − 1) f (ℓ) = ℓ(ℓ − 1 + γ) is strictly decreasing for ℓ ≥ ℓ+ . Part (c) of Theorem 2, along with interlacing inequality (16), is used in the proof to establish that Nn = #{i ≥ r + 1 : λin > λ+ (cn )}, the number of noise eigenvalues exiting the bulk, is bounded in probability. 5 Final Remarks It is apparent that the same methods will show that the value of c0 for the Gaussian Orthogonal Ensemble will be the same as for the real Wishart (Laguerre Orthogonal Ensemble), and similarly that the value of c0 for the white complex Wishart (Laguerre Unitary Ensemble) will agree with that for GUE. Some natural questions are left for further work. First, the evaluation of c0 for values of β other than 1 and 2, and secondly universality, i.e. that the limiting constants do not require the assumption of Gaussian matrix entries. Finally, this article appears in a special issue dedicated to the memory of Peter Hall. Hall’s many contributions to high dimensional data have been reviewed by Samworth [2016]. However, it seems that Peter did not publish specifically on problems connected with the application of random matrix theory to statistics — the exception that proves the rule of his extraordinary breadth and depth of interests. Nevertheless the present author’s work on this specific topic, as well as on many others, has been notably advanced by Peter’s support — academic, collegial and financial – in promoting research visits to Australia and contact with specialists there in random matrix theory, particularly at the University of Melbourne, Peter’s academic home since 2006. 9 References Z. D. Bai and Jack W. Silverstein. 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Event excitation for event-driven control and optimization of multi-agent systems Yasaman Khazaeni and Christos G. Cassandras Division of Systems Engineering and Center for Information and Systems Engineering Boston University, MA 02446 arXiv:1604.00691v1 [math.OC] 3 Apr 2016 yas@bu.edu,cgc@bu.edu Abstract— We consider event-driven methods in a general framework for the control and optimization of multi-agent systems, viewing them as stochastic hybrid systems. Such systems often have feasible realizations in which the events needed to excite an on-line event-driven controller cannot occur, rendering the use of such controllers ineffective. We show that this commonly happens in environments which contain discrete points of interest which the agents must visit. To address this problem in event-driven gradient-based optimization problems, we propose a new metric for the objective function which creates a potential field guaranteeing that gradient values are non-zero when no events are present and which results in eventual event excitation. We apply this approach to the class of cooperative multi-agent data collection problems using the event-driven Infinitesimal Perturbation Analysis (IPA) methodology and include numerical examples illustrating its effectiveness. I. I NTRODUCTION The modeling and analysis of dynamic systems has historically been founded on the time-driven paradigm provided by a theoretical framework based on differential (or difference) equations: we postulate the existence of an underlying “clock” and with every “clock tick” a state update is performed which synchronizes all components of the system. As systems have become increasingly networked, wireless, and distributed, the universal value of this paradigm has come to question, since it may not be feasible to guarantee the synchronization of all components of a distributed system, nor is it efficient to trigger actions with every time step when such actions may be unnecessary. The event-driven paradigm offers an alternative to the modeling, control, communication, and optimization of dynamic systems. The main idea in event-driven methods is that actions affecting the system state need not be taken at each clock tick. Instead, one can identify appropriate events that trigger control actions. This approach includes the traditional time-driven view if a clocktick is considered a system “event”. Defining the right events is a crucial modeling step and has to be carried out with a good understanding of the system dynamics. The importance of event-driven behavior in dynamic systems was recognized in the development of Discrete Event The authors work is supported in part by NSF under grants CNS1239021, ECCS-1509084, and IIP-1430145, by AFOSR under grant FA9550-15-1-0471, by ONR under grant N00014-09-1-1051, and by the Cyprus Research Promotion Foundation under Grant New Infrastructure Project/Strategic/0308/26. Systems (DES) and later Hybrid Systems (HS) [1]. More recently there have been significant advances in applying event-driven methods (also referred to as “event-based” and “event-triggered”) to classical feedback control systems; e.g., see [2], [3], [4], as well as [5] and [6] and references therein. Event-driven approaches are also attractive in receding horizon control, where it is computationally inefficient to re-evaluate an optimal control value over small time increments as opposed to event occurrences defining appropriate planning horizons for the controller (e.g., see [7]). In distributed networked systems, event-driven mechanisms have the advantage of significantly reducing communication among networked components which cooperate to optimize a given objective. Maintaining such cooperation normally requires frequent communication among them; it was shown in [8] that we can limit ourselves to event-driven communication and still achieve optimization objectives while drastically reducing communication costs (hence, prolonging the lifetime of a wireless network), even when delays are present (as long as they are bounded). Clearly, the premise of these methods is that the events involved are observable so as to “excite” the underlying event-driven controller. However, it is not always obvious that these events actually take place under every feasible control: it is possible that under some control no such events are excited, in which case the controller may be useless. In such cases, one can resort to artificial “timeout events” so as to eventually take actions, but this is obviously inefficient. Moreover, in event-driven optimization mechanisms this problem results in very slow convergence to an optimum or in an algorithm failing to generate any improvement in the decision variables being updated. In this work, we address this issue of event excitation in the context of multi-agent systems. In this case, the events required are often defined by an agent “visiting” a region or a single point in a mission space S ⊂ R2 . Clearly, it is possible that such events never occur for a large number of feasible agent trajectories. This is a serious problem in trajectory planning and optimization tasks which are common in multi-agent systems seeking to optimize different objectives associated with these tasks, including coverage, persistent monitoring or formation control [9], [10], [11], [12], [13], [14], [15], [16]. At the heart of this problem is the fact that objective functions for such tasks rely on a non- zero reward (or cost) metric associated with a subset S + ⊂ S of points, while all other points in S have a reward (or cost) which is zero since they are not “points of interest” in the mission space. We propose a novel metric which allows all points in S to acquire generally non-zero reward (or cost), thus ensuring that all events are ultimately excited. This leads to a new method allowing us to apply event-based control and optimization to a large class of multi-agent problems. We will illustrate the use of this method by considering a general trajectory optimization problem in which Infinitesimal Perturbation Analysis (IPA) [1] is used as an event-driven gradient estimation method to seek optimal trajectories for a class of multi-agent problems where the agents must cooperatively visit a set of target points to collect associated rewards (e.g., to collect data that are buffered at these points.) This defines a family within the class of Traveling Salesman Problems (TSPs) [17] for which most solutions are based on techniques typically seeking a shortest path in the underlying graph. These methods have several drawbacks: (i) they are generally combinatorially complex, (ii) they treat agents as particles (hence, not accounting for limitations in motion dynamics which should not, for instance, allow an agent to form a trajectory consisting of straight lines), and (iii) they become computationally infeasible as on-line methods in the presence of stochastic effects such as random target rewards or failing agents. As an alternative we seek solutions in terms of parameterized agent trajectories which can be adjusted on line as a result of random effects and which are scalable, hence computationally efficient, especially in problems with large numbers of targets and/or agents. This approach was successfully used in [18], [19]. In section II we present the general framework for multiagent problems and address the event excitation issue. In section III we overview the event-driven IPA methodology and how it is applied to a general hybrid system optimization problem. In section IV we introduce a data collection problem as an application of the general framework introduced in section II and will show simulation results of applying the new methodology to this example in section V. II. E VENT-D RIVEN O PTIMIZATION IN M ULTI -AGENT S YSTEMS Multi-agent systems are commonly modeled as hybrid systems with time-driven dynamics describing the motion of the agents or the evolution of physical processes in a given environment, while event-driven behavior characterizes events that may occur randomly (e.g., an agent failure) or in accordance to control policies (e.g., an agent stopping to sense the environment or to change directions). In some cases, the solution of a multi-agent dynamic optimization problem is reduced to a policy that is naturally parametric. As such, a multi-agent system can be studied with parameterized controllers aiming to meet certain specifications or to optimize a given performance metric. Moreover, in cases where such a dynamic optimization problem cannot be shown to be reduced to a parametric policy, using such a policy is still near-optimal or at least offers an alternative. Fig. 1. Multi-agent system in a dynamic setting, blue areas are obstacles In order to build a general framework for multi-agent optimization problems, assuming S as the mission space, we introduce the function R(w) : S → R as a “property” of point w ∈ S. For instance, R(w) could be a weight that gives relative importance to one point in S compared to another. Setting R(w) > 0 for only a finite number of points implies that we limit ourselves to a finite set of points of interest while the rest of S has no significant value. Assuming F to be the set of all feasible agent states, We define P (w, s) : S × F → R to capture the cost/reward resulting from how agents with state s ∈ F interact with w ∈ S. For instance, in coverage problems if an “event” occurs at w, then P (w, s) is the probability of agents jointly detecting such events based on the relative distance of each agent from w. In general settings, the objective is to find the best state vector s1 , · · · , sN so that N agents achieve a maximal reward (minimal cost) from interacting with the mission space S: Z min J = P (w, s)R(w)dw (1) s∈F S This static problem can be extended to a dynamic version where the agents determine optimal trajectories si (t), t ∈ [0, T ], rather than static states: Z TZ min J = P (w, s(u(t)))R(w, t)dwdt (2) u(t)∈U 0 S subject to motion dynamics: ṡj (t) = fj (sj , uj , t), j = 1, · · · , N (3) In Fig. 1, such a dynamic multi agent system is illustrated. As an example, consensus problems are just a special case of (1). Suppose that we consider a finite set of points w ∈ S which coincide with the agents states s1 , ..., sN (which are not necessarily their locations). Then we can set P (w, s) = ksi − sj k2 and, therefore, replace the integral in (1) by a sum. In this case, R(w) = Ri is just the weight that an agent carries in the consensus algorithm. An optimum occurs when ksi − sj k2 = 0 for all i, j, i.e., all agents “agree” and consensus is reached. This is a special case because of the simplicity in P (w, s) making the problem convex so that a global optimum can be achieved, in contrast to most problems we are interested in. As for the formulation in (2), consider a trajectory planning problem where N mobile agents are tasked to visit M stationary targets in the mission space S. Target behavior is described through state variables xi (t) which may model reward functions, the amount of data present at i, or other problem-dependent target properties. More formally, let (Ω, F, P) be an appropriately defined probability space and ω ∈ Ω a realization of the system where target dynamics are subject to random effects: ẋi (t) = gi (xi (t), ω) (4) gi (·) is as such that xi (t) is monotonically increasing by t and it resets to zero each time a target is completely emptied by an agent. In the context of (2), we assume the M targets are located at pointswi , i = 1, · · · , M and define R(xi (t), w) if w ∈ C(wi ) R(w, t) = (5) 0 otherwise to be the value of point w, where C(wi ) is a compact 2manifold in R2 containing wi which can be considered to be a region defined by the sensing range of that target relative to agents (e.g., a disk centered at wi ). Note that R(w, t) is also a random variable defined on the same probability space above. Given that only points w ∈ C(wi ) have value for the agents, there is an infinite number of points w ∈ / C(wi ) such that R(w, t) = 0 provided the following condition holds: Condition 1: If ∃i such that w ∈ C(wi ) then w ∈ / C(wj ) holds ∀j 6= i. This condition is to ensure that two targets do not share any point w in their respective sensing ranges. Also it ensures that the set {C(wi ) | i = 1 : · · · , M } does not create a compact partitioning of the mission space and there exist points w which do not belong to any of the C(wi ). Viewed as a stochastic hybrid system, we may define different modes depending on the states of agents or targets and events that cause transitions between these modes. Relative to a target i, any agent has at least two modes: being at a point w ∈ C(wi ), i.e., visiting this target or not visiting it. Within each mode, agent j’s dynamics, dictated by (3), and target i’s dynamics in (4) may vary. Accordingly, there are 0 at least two types of events in such a system: (i) δij events + occur when agent j initiates a visit at target i, and (ii) δij events occur when agent j ends a visit at target i. Additional event types may be included depending on the specifics of a problem, e.g., mode switches in the target dynamics or agents encountering obstacles. An example is shown in Fig. 2, where target sensing ranges are shown with green circles and agent trajectories are shown in dashed lines starting at a base shown by a red triangle. In the blue trajectory, agent 1 moves along the trajectory that passes through points A → B → C → D. It is easy to see that when passing through points A and C we have 0 δi1 and δi00 1 events, while passing through B and D we + have δi1 and δi+0 1 events. The red trajectory is an example where none of the events is excited. Suppose we consider an on-line trajectory adjustment process in which the agent improves its trajectory based on its performance measured through (5). In this case, R(w, t) = 0 over all t, as long as the agent keeps using the red trajectory, i.e., no event ever occurs. Therefore, if an event-driven approach is used to control the trajectory adjustment process, no action is ever triggered and the approach is ineffective. In contrast, in the blue trajectory the controller can extract useful information from every observed event; such information (e.g., a gradient of J with respect to controllable parameters as described in the next section) can be used to adjust the current trajectory Fig. 2. Sample trajectories so as to improve the objective function J in (1) or (2). Therefore, if we are to build an optimization framework for this class of stochastic hybrid systems to allow the application of event-driven methods by calculating a performance measure gradient, then a fundamental property required is the occurrence of at least some events in a sample realization. In particular, the IPA method [20] is based on a single sample realization of the system over which events are observed along with their occurrence times and associated system states. Suppose that the trajectories can be controlled through a set of parameters forming a vector θ. Then, IPA provides an unbiased estimate of the gradient of a performance metric J(θ) with respect to θ. This gradient is then used to improve the trajectory and ultimately seek an optimal one when appropriate conditions hold. As in the example of Fig. 2, it is possible to encounter trajectory realizations where no events occur in the system. In the above example, this can easily happen if the trajectory does not pass through any target. The existence of such undesirable trajectories is the direct consequence of Condition 1. This lack of event excitation results in event-based controllers being unsuitable. New Metric: In order to overcome this issue we propose a new definition for R(w, t) in (5) as follows: M X R(w, t) = hi (xi (t), di (w)) (6) i=1 where w ∈ S, hi (·) is a function of the target’s state xi (t) and di (w) = kwi −wk. Note that, if hi (·) is properly defined, (6) yields R(w, t) > 0 at all points. While the exact form of hi (·) depends on the problem, we impose the condition that hi (·) is monotonically decreasing in di (w). We can think of hi (·) as a value function associated with point wi . Using the definition of R(w, t), this value is spread out over all points w ∈ S rather than being concentrated at the single point wi . This creates a continuous potential field for the agents leading to a non-zero gradient of the performance measure even when the trajectories do not excite any events. This non-zero gradient will then induce trajectory adjustments that naturally bring them toward ones with observable events. Finally, recalling the definition in (2), we also define: N X P (w, s) = ksj (t) − wk2 (7) j=1 the total quadratic travel cost for agents to visit point w. In Section IV, we will show how to apply R(w, t) and P (w, s) defined as above in order to determine optimal agent trajectories for a class of multi-agent problems of the form (2). First, however, we review in the next section the eventdriven IPA calculus which allows us to estimate performance gradients with respect to controllable parameters. III. E VENT-D RIVEN IPA C ALCULUS Let us fix a particular value of the parameter θ ∈ Θ and study a resulting sample path of a general SHS. Over such a sample path, let τk (θ), k = 1, 2, · · · denote the occurrence times of the discrete events in increasing order, and define τ0 (θ) = 0 for convenience. We will use the notation τk instead of τk (θ) when no confusion arises. The continuous state is also generally a function of θ, as well as of t, and is thus denoted by x(θ, t). Over an interval [τk (θ), τk+1 (θ)), the system is at some mode during which the time-driven state satisfies ẋ = fk (x, θ, t), in which x is any of the continuous state variables of the system and ẋ denotes ∂x ∂t . Note that we suppress the dependence of fk on the inputs u ∈ U and d ∈ D and stress instead its dependence on the parameter θ which may generally affect either u or d or both. The purpose of perturbation analysis is to study how changes in θ influence the state x(θ, t) and the event times τk (θ) and, ultimately, how they influence interesting performance metrics that are generally expressed in terms of these variables. An event occurring at time τk+1 (θ) triggers a change in the mode of the system, which may also result in new dynamics represented by fk+1 . The event times τk (θ) play an important role in defining the interactions between the time-driven and event-driven dynamics of the system. Following the framework in [20], consider a general performance function J of the control parameter θ: J(θ; x(θ, 0), T ) = E[L(θ; x(θ, 0), T )] (8) where L(θ; x(θ, 0), T ) is a sample function of interest evaluated in the interval [0, T ] with initial conditions x(θ, 0). For simplicity, we write J(θ) and L(θ). Suppose that there are K events, with occurrence times generally dependent on θ, during the time interval [0, T ] and define τ0 = 0 and τN +1 = T . Let Lk : Rn × Θ × R+ → R be a function and define L(θ) by K Z τk+1 X L(θ) = Lk (x, θ, t)dt (9) k=0 τk where we reiterate that x = x(θ, t) is a function of θ and t. We also point out that the restriction of the definition of J(θ) to a finite horizon T which is independent of θ is made merely for the sake of simplicity. Returning to the stochastic setting, the ultimate goal of the iterative process shown is to maximize Eω [L(θ, ω)], where we use ω to emphasize dependence on a sample path ω of a SHS (clearly, this is reduced to L(θ) in the deterministic case). Achieving such optimality is possible under standard ergodicity conditions imposed on the underlying stochastic processes, as well as the assumption that a single global optimum exists; otherwise, the gradientbased approach is simply continuously attempting to improve the observed performance L(θ, ω). Thus, we are interested in estimating the gradient dEω [L(θ, ω)] dJ(θ) = (10) dθ dθ based on directly observed data. We by evaluating dL(θ,ω) dθ obtain θ ∗ by optimizing J(θ) through an iterative scheme of the form θn+1 = θn − ηn Hn (θn ; x(θ, 0), T, ωn ), n = 0, 1, · · · (11) where ηn is a step size sequence and Hn (θn ; x(θ, 0), T, ωn ) at θ = θn . In using IPA, is the estimate of dJ(θ) dθ , Hn (θn ; x(θ, 0), T, ωn ) is the sample derivative dL(θ,ω) dθ which is an unbiased estimate of dJ(θ) if the condition dθ (dropping the symbol ω for simplicity)  dL(θ)  dE[L(θ)] dJ(θ) E = = (12) dθ dθ dθ is satisfied, which turns out to be the case under mild technical conditions. The conditions under which algorithms of the form (11) converge are well-known (e.g., see [21]). Moreover, in addition to being unbiased, it can be shown that such gradient estimates are independent of the probability laws of the stochastic processes involved and require minimal information from the observed sample path. The process is based on analyzing through which IPA evaluates dL(θ) dθ how changes in θ influence the state x(θ, t) and the event times τk (θ). In turn, this provides information on how L(θ) is affected, because it is generally expressed in terms of these variables. Given θ = [θ1 , ..., θl ]T , we use the Jacobian matrix notation: ∂x(θ, t) ∂τk (θ) x0 (θ, t) = , τk 0 = , k = 1, · · · , K (13) ∂θ ∂θ for all state and event time derivatives. For simplicity of notation, we omit θ from the arguments of the functions above unless it is essential to stress this dependence. It is shown in [20] that x0 (t) satisfies: dx0 (t) ∂fk (t) 0 ∂fk (t) = x (t) + (14) dt ∂x ∂θ for t ∈ [τk (θ), τk+1| (θ)) with boundary condition x0 (τk+ ) = x0 (τk− ) + [fk−1 (τk− ) − fk (τk+ )]τk0 (15) for k = 0, · · · , K. We note that whereas x(t) is often continuous in t, x0 (t) may be discontinuous in t at the event times τk ; hence, the left and right limits above are generally different. If x(t) is not continuous in t at t = τk (θ), the value of x(τk+ ) is determined by the reset function r(q, q 0 , x, ν, δ) and dr(q, q 0 , x, ν, δ) x0 (τk+ ) = (16) dθ 0 + Furthermore, once the initial condition x (τk ) is given, the linearized state trajectory x0 (t) can be computed in the interval t ∈ [τk (θ), τk+1 (θ)) by solving (14) to obtain R t ∂fk (u) h Z t ∂f (v) R t ∂fk (u) i du du − k 0 ∂x τk x (t) = e e τk ∂x dv + ξk ∂θ τk (17) with the constant ξk determined from x0 (τk+ ). In order to complete the evaluation of x0 (τk+ ) we need to also determine τk0 . If the event at τk (θ) is exogenous τk0 = 0 and if the event at τk (θ) is endogenous: h ∂g i ∂g ∂gk 0 −
k k τk0 = − fk (τk− ) + x (τk ) (18) ∂x ∂θ ∂x ∂gk + where gk (x, θ) = 0 and it is defined as long as ∂x fk (τk ) 6= 0 (details may be found in [20].) The derivative evaluation process involves using the IPA calculus in order to evaluate the IPA derivative dL dθ . This is accomplished by taking derivatives in (9) with respect to θ: Z τk+1 K dL(θ) X d Lk (x, θ, t)dt (19) = dθ dθ τk k=0 Applying the Leibnitz rule, we obtain, for every k = 0, · · · , ZK, τk+1 d Lk (x, θ, t)dt dθ τk Z τk+1 h ∂Lk (x, θ, t) i ∂Lk (x, θ, t) 0 dt x (t) + = ∂x ∂θ τk 0 + Lk (x(τk+1 ), θ, τk+1 )τk+1 − Lk (x(τk ), θ, τk )τk0 (20) In summary the three equations (15), (17) and (18) form the basis of the IPA calculus and allow us to calculate the final derivative in (20). In the next section IPA is applied to a data collection problem in a multi-agent system. connected to a target i even if there are other agents l with pil (t) > 0; this is not the only possible model, but we adopt it based on the premise that simultaneous downloading of packets from a common source creates problems of proper data reconstruction. This means that j in (22) is the index of the agent that is connected to target i at time t. The dynamics of xi (t) in (22) results in two new event types added to what was defined earlier, (i) ξi0 events occur when xi (t) reaches zero, and (ii) ξi+ events occur when xi (t) leaves zero. The performance measure is the total content of data left at targets at the end of a finite mission time T . Thus, we define J1 (t) to be the following (recalling that {σi (t)} are random processes): M X J1 (t) = αi E[xi (t)] (23) i=1 IV. T HE DATA C OLLECTION P ROBLEM We consider a class of multi-agent problems where the agents must cooperatively visit a set of target points to collect associated rewards (e.g., to collect data that are buffered at these points.). The mission space is S ⊂ R2 . This class of problems falls within the general formulation introduced in (2). The state of the system is the position of agent j time t, sj (t) = [sxj (t), syj (t)] and the state of the target i, xi (t). The agent’s dynamics (3) follow a single integrator: ṡxj (t) = uj (t) cos θj (t), ṡyj (t) = uj (t) sin θj (t) (21) where uj (t) is the scalar speed of the agent (normalized so that 0 ≤ uj (t) ≤ 1) and θj (t) is the angle relative to the positive direction, 0 ≤ θj (t) < 2π. Thus, we assume that each agent controls its speed and heading. We assume the state of the target xi (t) represents the amount of data that is currently available at target i (this can be modified to different state interpretations). The dynamics of xi (t) in (4) for this problem are: 0 if xi (t) = 0 and σi (t) ≤ µij p(sj (t), wi ) ẋi (t) = σi (t) − µij p(sj (t), wi ) otherwise (22) i.e., we model the data at the target as satisfying simple flow dynamics with an exogenous (generally stochastic) inflow σi (t) and a controllable rate with which an agent empties the data queue given by µij p(sj (t), wi ). For brevity we set p(sj (t), wi ) = pij (t) which is the normalized data collection rate from target i by agent j and µij is a nominal rate corresponding to target i and agent j. Assuming M targets are located at wi ∈ S, i = 1, . . . , M, and have a finite range of ri , then agent j can collect data from wi only if dij (t) = kwi − sj (t)k ≤ ri . We then assume that: (A1) pij (t) ∈ [0, 1] is monotonically nonincreasing in the value of dij (t) = kwi − sj (t)k, and (A2) it satisfies pij (t) = 0 if dij (t) > ri . Thus, pij (t) can model communication power constraints which depend on the distance between a data source and an agent equipped with a receiver (similar to the model used in [22]) or sensing range constraints if an agent collects data using on-board sensors. For simplicity, we will also assume that: (A3) pij (t) is continuous in dij (t) and (A4) only one agent at a time is where αi is a weight factor for target i. We can now formulate a stochastic optimization problem P1 where the control variables are the agent speeds and headings denoted by the vectors u(t) = [u1 (t), . . . , uN (t)] and θ(t) = [θ1 (t), . . . , θN (t)] respectively (omitting their dependence on the full system state at t). Z 1 T J1 (t)dt (24) P1 : min J(T ) = T 0 u(t),θ(t) where 0 ≤ uj (t) ≤ 1, 0 ≤ θj (t) < 2π, and T is a given finite mission time. This problem can be readily placed into the general framework (2). In particular, the right hand side of (24) is:" # Z T XZ 1 αi E 2 xi (t)dwdt T 0 C(wi ) πri i "Z Z # (25) T X αi 1{w ∈ C(wi )} 1 xi (t)dwdt = E T πri2 0 S i This is now in the form of the general framework in (2) with X αi 1{w ∈ C(wi )} R(w, t) = xi (t) (26) πri2 i and P (sj (t), w) = 1 (27) Recalling the definition in (5), only points within the sensing range of each target have non-zero values, while all other point value are zero, which is the case in (26) above. In addition, (27) simply shows that there is no meaningful dynamic interaction between an agent and the environment. Problem P1 is a finite time optimal control problem. In order to solve this, following previous work in [19] we proceed with a standard Hamiltonian analysis leading to a Two Point Boundary Value Problem (TPBVP) [23]. We omit this, since the details are the same as the analysis in [19]. The main result of the Hamiltonian analysis is that the optimal speed is always the maximum value, i.e., u∗j (t) = 1 (28) Hence, we only need to calculate the optimal θj (t). This TPBVP is computationally expensive and easily becomes intractable when problem size grows. The ultimate solution of the TPBVP is a set of agent trajectories that can be put in a parametric form defined by a parameter vector θ and then optimized over θ. If the parametric trajectory family is broad enough, we can recover the true optimal trajectories; otherwise, we can approximate them within some acceptable accuracy. Moreover, adopting a parametric family of trajectories and seeking an optimal one within it has additional benefits: it allows trajectories to be periodic, often a desirable property, and it allows one to restrict solutions to trajectories with desired features that the true optimal may not have, e.g., smoothness properties to achieve physically feasible agent motion. Parameterizing the trajectories and using gradient based optimization methods, in light of the discussions from the previous sections, enables us to make use of Infinitesimal Perturbation Analysis (IPA) [20] to carry out the trajectory optimization process. We represent each agent’s trajectory through general parametric equations sxj (t) = fx (θj , ρj (t)), syj (t) = fy (θj , ρj (t)) (29) where the function ρj (t) controls the position of the agent on its trajectory at time t and θj is a vector of parameters controlling the shape and location of the trajectory. Let θ = [θ1 , . . . , θN ]. We now revisit problem P1 in (24): Z 1 T (30) J1 (θ, t)dt min J(θ, T ) = θ∈Θ T 0 and will bring in the equations that were introduced in the previous section in order to calculate an estimate of dJ(θ) dθ as in (10). For this problem due to the continuity of xi (t) the last two terms in (20) vanish. From (23) we have: Z τk+1 X Z τk+1 X M M d αi xi (θ, t)dt = αi x0i (θ, t)dt (31) dθ τk τk i=1 i=1 In summary, the evaluation of (31) requires the state derivatives x0i (t) explicitly and s0j (t) implicitly, (dropping the dependence on θ for brevity). The latter are easily obtained for any specific choice of f and g in (29). The former require a rather laborious use of (15),(17),(18) which, reduces to a simple set of state derivative dynamics as shown next. Proposition 1. After an event occurrence at t = τk , the state derivatives x0i (τk+ ) with respect to the controllable parameter θ satisfy the following:  if e(τk ) = ξi0  0 0 − + 0 + xi (τk ) = x0 (τ ) − µil (t)pil (τk )τk if e(τk ) = δij  i0 k− xi (τk ) otherwise where l 6= j with pil (τk ) > 0 if such l exists and 0 τk = ∂dij (sj ) 0 ∂sj sj ∂dij (sj ) ∂sj ṡj (τk ) −1 . Proof: The proof is omitted due to space limitations, but it is very similar to the proofs of Propositions 1-3 in [24]. As is obvious from Proposition 1, the evaluation of x0i (t) is entirely dependent on the occurrence of events ξi0 and + + δij in a sample realization, i.e., ξi0 and δij cause jumps in this derivative which carry useful information. Otherwise, x0i (τk+ ) = x0i (τk− ) is in effect and these gradients remain unchanged. However, we can easily have realizations where 0 no events occur in the system (specifically, events of type δij + and δij ) if the trajectory of agents in the sample realization does not pass through any target. This lack of event excitation results in the algorithm in (11) to stall. In the next section we overcome the problem of no event excitation using the definitions in (6) and (7). We accomplish this by adding a new metric to the objective function that generates a non-zero sensitivity with respect to θ. A. Event Excitation Our goal here is to select a function hi (·) in (6) with the property of “spreading” the value of xi (t) over all w ∈ S. We begin by determining the convex hull produced by the targets, since the trajectories need not go outside this convex hull. Let T = {w1 , w2 , · · · , wM } be the set of all target points. Then, the convex hull of these points is X  M X C= βi w i | βi = 1, ∀i, βi ≥ 0 (32) i=1 i Given that C ⊂ S, we seek some R(w, t) that satisfies the following property for constants ci > 0: Z M X R(w, t)dw = ci xi (t) (33) C i=1 so that R(w, t) can be viewed as a continuous density defined for all points w ∈ C which results in a total value equivalent to a weighted sum of the target states xi (t), i = 1, . . . , M . In order to select an appropriate h(xi (t), di (w)) in (6), we first define d+ i (w) = max(kw − wi k, ri ) where ri is the target’s sensing range. We then define: M X αi xi (t) (34) R(w, t) = d+ (w) i=1 i Here, we are spreading a target’s reward (numerator) over all w so as to obtain the “total weighted reward density” at w. Note that d+ i (w) = max(kw − wi k, ri ) > 0 to ensure that the target reward remains positive and fixed for points w ∈ C(wi ). Moreover, following (7), N X P (w, s(t)) = ksj (t) − wk2 (35) j=1 Using these definitions we introduce a new objective function metric which is added to the objective function in (24): hZ i J2 (t) = E P (w, s(t))R(w, t)dw (36) C The expectation is a result of P (w, s(t)) and R(w, t) being random variables defined on the same probability space as xi (t). Proposition 2. For R(w, t) in (34), there exist ci > 0, i = 1, . . . , M , such that: Z M X R(w, t)dw = ci xi (t) (37) C i=1 Proof: We have Z Z X M αi xi (t) R(w, t) = dw + C C i=1 di (w) (38) Z M X xi (t) = αi dw + C di (w) i=1 R (t) We now need to find the value of C dx+i(w) for each target i i. To do this we first look at the case of one target in a 2D space and for now we assume C is just a disk with radius Λ around the target (black circle with radius Λ in Fig. 3). We can now calculate the above integral for this target using the polar coordinates: Z Z 2π Z Λ xi (t) xi (t) dw = drdθ + max(r i , r) 0 0 C di (w) Z 2π Z ri Z 2π Z Λ xi (t) xi (t) (39) = drdθ + drdθ ri r 0 ri 0 0  Λ
 = xi (t) 2π 1 + log( ) ri In our case C is the convex hull of all targets. We will use the idea to calculate the R xsame i (t) dw for the actual con+ C di (w) vex hull. We do this for an interior target i.e., a target inside the convex hull. Extending the same to targets on the edge is straightforward. Using the same Fig. 3. One Target R(w, t) polar coordinate for each θ we Calculation define Λ(θ) to be the distance of the target to the edge of C in the direction of θ. (C shown by a red polygon in Fig. 3). Z 2π Z Λ Z xi (t) xi (t) dw = drdθ + + 0 0 di (r, θ) C di (w) Z 2π Z ri Z 2π Z Λ(θ) xi (t) xi (t) = drdθ (40) drdθ + r r i 0 0 0 ri Z 2π   Λ(θ) = xi (t) 2π + log( )dθ r i 0 The second part in (40) has to be calculated knowing Λ(θ) but since we assumed the target is inside the convex hull we know Λ(θ) ≥ ri . This means log( Λ(θ) ri ) > 0 and the xi (t)’s multiplier is a positive value. We can define ci in (37) as: Z 2π   Λ(θ) ci = αi 2π + )dθ log( (41) r i 0 The significance of J2 (t) is that it accounts for the movement of agents through P (w, s(t)) and captures the target state values through R(w, t). Introducing this term in the objective function in the following creates a non-zero gradient even if the agent trajectories are not passing through any targets. We now combine the two metrics in (24) and (36) and define problem P2: Z  1 T P2 : min J(T ) = J1 (t) + J2 (t) dt (42) T 0 u(t),θ(t) In this problem, the second term is responsible for adjusting the trajectories towards the targets by creating a potential field, while the first term is the original performance metric which is responsible for adjusting the trajectories so as to maximize the data collected once an agent is within a target’s sensing range. It can be easily shown that the results in (28) hold for problem P2 as well, through the same Hamiltonian analysis presented in [19]. When sj (t) follows the parametric functions in (29), the new metric simply becomes a function of the parameter vector θ and we have: Z  1 T J1 (θ, t) + J2 (θ, t) dt (43) θ∈Θ T 0 The new objective function’s derivative follows the same procedure that was described previously. The first part’s derivative can be calculated from (31). For the second part we have: Z τk+1 Z min J(θ, T ) = d dθ P (w, θ, t)R(w, θ, t)dw τk Z C τk+1 Z = τk C h dP (w, θ, t) dθ R(w, θ, t) + P (w, θ, t) dR(w, θ, t) i dw dθ (44) In the previous section, we raised the problem of no events being excited in a sample realization, in which case the total derivative in (31) is zero and the algorithm in (11) stalls. Now, looking at (44) we can see that if no events occur the second part in the integration which involves dR(w,θ,t) dθ PM will be zero, since i=1 x0i (t) = 0 at all t. However, the first part in the integral does not depend on the events, but calculates the sensitivity of P (w, s(t)) in (35) with respect to the parameter θ. Note that the dependence on θ comes through the parametric description of s(t) through (29). This term ensures that the algorithm in (11) does not stall and adjusts trajectories so as to excite the desired events. V. S IMULATION R ESULTS We provide some simulation results based on an elliptical parametric description for the trajectories in (29). The elliptical trajectory formulation is: sxj (t) = Aj + aj cos ρj (t) cos φj − bj sin ρj (t) sin φj syj (t) = Bj + aj cos ρj (t) sin φj + bj sin ρj (t) cos φj (45) Here, θj = [Aj , Bj , aj , bj , φj ] where Aj , Bj are the coordinates of the center, aj and bj are the major and minor axis respectively while φj ∈ [0, π) is the ellipse orientation which is defined as the angle between the x axis and the major axis of the ellipse. The time-dependent parameter ρj (t) is the eccentric anomaly of the ellipse. Since an agent is moving with constant speed of 1 on this trajectory, based on (28), we have ṡxj (t)2 + ṡyj (t)2 = 1, which gives h 2 ρ̇j (t) = a sin ρj (t) cos φj + bj cos ρj (t) sin φj  2 i− 21 + a sin ρj (t) sin φj − bj cos ρj (t) cos φj (46) The first case we consider is a problem with one agent and seven targets located on a circle, as shown in Fig. 4. We consider a deterministic case with σi (t) = 0.5 for all i. The other problem parameters are T = 50, µij = 100, ri = 0.2 and αi = 1. A target’s sensing range is denoted with solid black circles with the target location at the center. The blue polygon indicates the convex hull produced by the targets. The direction of motion on a trajectory is shown with the small arrow. Starting with an initial trajectory shown in light blue, the on-line trajectory optimization process converges to the trajectory passing through all targets in an efficient manner (shown in dark solid blue). In contrast, starting with this trajectory - which does not pass through any targets using the event-based IPA calculus to estimate the objective function gradient. R EFERENCES Fig. 4. One agent and seven target scenario Fig. 5. Two agent and seven targets scenario - problem P1 does not converge and the initial trajectory remains unchanged. At the final trajectory, J1∗ = 0.0859 and J ∗ = 0.2128. Using the obvious shortest path solution, the actual optimal value for J1 is 0.0739 that results from moving on the edges of the convex hull (which allows for shorter agent travel times). In the second case, 7 targets are randomly distributed and two agents are cooperatively collecting the data. The problem parameters are σi = 0.5, µij = 10, ri = 0.5, αi = 1, T = 50. The initial trajectories for both agents are shown in light green and blue respectively. We can see that both agent trajectories converge so as to cover all targets, shown in dark green and blue ellipses. At the final trajectories, J1∗ = 0.1004 and J ∗ = 0.2979. Note that we may use these trajectories to initialize the corresponding TPBVP, another potential benefit of this approach. This is a much slower process which ultimately converges to J1∗ = 0.0991 and J ∗ = 0.2776. VI. C ONCLUSIONS We have addressed the issue of event excitation in a class of multi-agent systems with discrete points of interest. We proposed a new metric for such systems that spreads the point-wise values throughout the mission space and generates a potential field. This metric allows us to use eventdriven trajectory optimization for multi-agent systems. The methodology is applied to a class of data collection problems [1] C. G. Cassandras and S. Lafortune, Introduction to Discrete Event Systems. 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Cassandras, “A new event-driven cooperative receding horizon controller for multi-agent systems in uncertain environments,” In Proceedings of IEEE 53rd Annual Conference on Decision and Control, pp. 2770–2775, Dec 2014. [8] M. Zhong and C. G. Cassandras, “Asynchronous distributed optimization with event-driven communication,” Automatic Control, IEEE Trans. on, vol. 55, no. 12, pp. 2735–2750, 2010. [9] M. Schwager, D. Rus, and J.-J. Slotine, “Decentralized, adaptive coverage control for networked robots,” The International Journal of Robotics Research, vol. 28, no. 3, pp. 357–375, 2009. [10] C. G. Cassandras, X. Lin, and X. Ding, “An optimal control approach to the multi-agent persistent monitoring problem,” IEEE Trans. on Aut. Cont., vol. 58, pp. 947–961, April 2013. [11] M. Cao, A. Morse, C. Yu, B. Anderson, and S. Dasgupta, “Maintaining a directed, triangular formation of mobile autonomous agents,” Communications in Information and Systems, vol. 11, no. 1, p. 1, 2011. [12] K.-K. Oh and H.-S. Ahn, “Formation control and network localization via orientation alignment,” IEEE Trans. on Automatic Control, vol. 59, pp. 540–545, Feb 2014. [13] H. Yamaguchi and T. Arai, “Distributed and autonomous control method for generating shape of multiple mobile robot group,” in Proc. of the IEEE International Conf. on Intelligent Robots and Systems, vol. 2, pp. 800–807 vol.2, Sep 1994. [14] J. Desai, V. Kumar, and J. Ostrowski, “Control of changes in formation for a team of mobile robots,” in Proc. of the IEEE International Conf. on Robotics and Automation, vol. 2, pp. 1556–1561, 1999. [15] M. Ji and M. B. Egerstedt, “Distributed coordination control of multi-agent systems while preserving connectedness.,” IEEE Trans. on Robotics, vol. 23, no. 4, pp. 693–703, 2007. [16] J. Wang and M. Xin, “Integrated optimal formation control of multiple unmanned aerial vehicles,” IEEE Trans. on Control Systems Technology, vol. 21, pp. 1731–1744, Sept 2013. [17] D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook, The traveling salesman problem: a computational study. Princeton University Press, 2011. [18] X. Lin and C. G. Cassandras, “An optimal control approach to the multi-agent persistent monitoring problem in two-dimensional spaces,” IEEE Trans. on Automatic Control, vol. 60, pp. 1659–1664, June 2015. [19] Y. Khazaeni and C. G. Cassandras, “An optimal control approach for the data harvesting problem,” in 54th IEEE Conf. on Decision and Cont., pp. 5136–5141, 2015. [20] C. G. Cassandras, Y. Wardi, C. G. Panayiotou, and C. Yao, “Perturbation analysis and optimization of stochastic hybrid systems,” European Journal of Cont., vol. 16, no. 6, pp. 642 – 661, 2010. [21] H. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications. Springer, 2003. [22] J. L. Ny, M. a. Dahleh, E. Feron, and E. Frazzoli, “Continuous path planning for a data harvesting mobile server,” Proc. of the IEEE Conf. on Decision and Cont., pp. 1489–1494, 2008. [23] A. E. Bryson and Y. C. Ho, Applied optimal control: optimization, estimation and control. CRC Press, 1975. [24] Y. Khazaeni and C. G. Cassandras, “An optimal control approach for the data harvesting problem,” arXiv:1503.06133. 

arXiv:1610.03625v2 [] 8 Apr 2017 THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS MARC KEILBERG Abstract. We investigate a possible connection between the F SZ properties of a group and its Sylow subgroups. We show that the simple groups G2 (5) and S6 (5), as well as all sporadic simple groups with order divisible by 56 are not F SZ, and that neither are their Sylow 5-subgroups. The groups G2 (5) and HN were previously established as non-F SZ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be F SZ. We conclude by considering all perfect groups available through GAP with order at most 106 , and show they are non-F SZ if and only if their Sylow 5-subgroups are non-F SZ. Introduction The F SZ properties for groups, as introduced by Iovanov et al. [4], arise from considerations of certain invariants of the representation categories of semisimple Hopf algebras known as higher Frobenius-Schur indicators [5, 10, 11]. See [9] for a detailed discussion of the many important uses and generalizations of these invariants. When applied to Drinfeld doubles of finite groups, these invariants are described entirely in group theoretical terms, and are in particular invariants of the group itself. The F SZ property is then concerned with whether or not these invariants are always integers—which gives the Z in F SZ. While the F SZ and non-F SZ group properties are well-behaved with respect to direct products [4, Example 4.5], there is currently little reason to suspect a particularly strong connection to proper subgroups which are not direct factors. Indeed, by [2, 4] the symmetric groups Sn are F SZ, while there exist non-F SZ groups of order 56 . Therefore, Sn is F SZ but contains non-F SZ subgroups for all sufficiently large n. On the other hand, non-F SZ groups can have every proper subquotient be F SZ. Even the known connection to the one element centralizers—see the comment following Definition 1.1—is relatively weak. In this paper we will establish a few simple improvements to this situation, and then proceed to establish a number of examples of F SZ and non-F SZ groups that support a potential connection to Sylow subgroups. We propose this connection as Conjecture 2.7. We will make extensive use of GAP [3] and the AtlasRep[15] package. Most of the calculations were designed to be completed with only 2GB of memory or (much) less available—in particular, using only a 32-bit implementation of GAP—, though in a few cases a larger workspace was necessary. In all cases the calculations can 2010 Mathematics Subject Classification. Primary: 20D08; Secondary: 20F99, 16T05, 18D10. Key words and phrases. sporadic groups, simple groups, Monster group, Baby Monster group, Harada-Norton group, Lyons group, projective symplectic group, higher Frobenius-Schur indicators, FSZ groups, Sylow subgroups. This work is in part an outgrowth of an extended e-mail discussion between Geoff Mason, Susan Montgomery, Peter Schauenburg, Miodrag Iovanov, and the author. The author thanks everyone involved for their contributions, feedback, and encouragement. 1 2 MARC KEILBERG be completed in workspaces with no more than 10GB of memory available. The author ran the code on an Intel(R) Core(TM) i7-4770 CPU @ 3.40GHz machine with 12GB of memory. All statements about runtime are made with respect to this computer. Most of the calculations dealing with a particular group were completed in a matter of minutes or less, though calculations that involve checking large numbers of groups can take several days or more across multiple processors. The structure of the paper is as follows. We introduce the relevant notation, definitions, and background information in Section 1. In Section 2 we present a few simple results which offer some connections between the F SZ (or non-F SZ) property of G and certain of its subgroups. This motivates the principle investigation of the rest of the paper: comparing the F SZ properties for certain groups and their Sylow subgroups. In Section 3 we introduce the core functions we will need to perform our calculations in GAP. We also show that all groups of order less than 2016 (except possibly those of order 1024) are F SZ. The remainder of the paper will be dedicated to exhibiting a number of examples that support Conjecture 2.7. In Section 4 we show that the simple groups G2 (5), HN , Ly, B, and M , as well as their Sylow 5-subgroups, are all non-F SZ5 . In Section 5 we show that all other sporadic simple groups (including the Tits group) and their Sylow subgroups are F SZ. This is summarized in Theorem 5.4. The case of the simple projective symplectic group S6 (5) is handled in Section 6, which establishes S6 (5) as the second smallest non-F SZ simple group after G2 (5). It follows from the investigations of Schauenburg [13] that HN is then the third smallest non-F SZ simple group. S6 (5) was not susceptible to the methods of Schauenburg [13], and requires further modifications to our own methods to complete in reasonable time. We finish our examples in Section 7 by examining those perfect groups available through GAP, and show that they are F SZ if and only if their Sylow subgroups are F SZ. Indeed, they are non-F SZ if and only if their Sylow 5-subgroup is non-F SZ5 . Of necessity, these results also establish that various centralizers and maximal subgroups in the groups in question are also non-F SZ5 , which can be taken as additional examples. If the reader is interested in F SZ properties for other simple groups, we note that Schauenburg [13] has checked all simple groups of order at most |HN | = 273,030,912,000,000 = 214 ·36 ·56 ·7·11·19, except for S6 (5) (which we resolve here); and that several families of simple groups were established as F SZ by Iovanov et al. [4]. We caution the reader that the constant recurrence of the number 5 and Sylow 5-subgroups of order 56 in this paper is currently more of a computationally convenient coincidence than anything else. The reasons for this will be mentioned during the course of the paper. 1. Background and Notation Let N be the set of positive integers. The study of F SZ groups is connected to the following sets. Definition 1.1. Let G be a group, u, g ∈ G, and m ∈ N. Then we define Gm (u, g) = {a ∈ G : am = (au−1 )m = g}. Note that Gm (u, g) = ∅ if u 6∈ CG (g), and that in all cases Gm (u, g) ⊆ CG (g). In particular, letting H = CG (g), then when u ∈ H we have Gm (u, g) = Hm (u, g). THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 3 The following will then serve as our definition of the F SZm property. It’s equivalence to other definitions follows easily from [4, Corollary 3.2] and applications of the Chinese remainder theorem. Definition 1.2. A group G is F SZm if and only if for all g ∈ G, u ∈ CG (g), and n ∈ N coprime to the order of g, we have |Gm (u, g)| = |Gm (u, g n )|. We say a group is F SZ if it is F SZm for all m. The following result is useful for reducing the investigation of the F SZ properties to the level of conjugacy classes or even rational classes. Lemma 1.3. For any group G and u, g, x ∈ G we have a bijection Gm (u, g) → Gm (ux , g x ) given by a 7→ ax . If n ∈ N is coprime to |G| and r ∈ N is such that rn ≡ 1 mod |G|, we also have a bijection Gm (u, g n ) → Gm (ur , g). Proof. The first part is [5, Proposition 7.2] in slightly different notation. The second part is [14, Corollary 5.5].  All expressions of the form Gm (u, g n ) will implicitly assume that n is coprime to the order of g. We are free to replace n with an equivalent value which is coprime to |G| whenever necessary. Moreover, when computing cardinalities |Gm (u, g)| it suffices to compute the cardinalities |Hm (u, g)| for H = CG (g), instead. This latter fact is very useful when attempting to work with groups of large order, or groups with centralizers that are easy to compute in, especially when the group is suspected of being non-F SZ. + , the union of which yields Remark 1.4. There are stronger conditions called F SZm + the F SZ + condition, which are also introduced by Iovanov et al. [4]. The F SZm condition is equivalent to the centralizer of every non-identity element with order not in {1, 2, 3, 4, 6} being F SZm , which is in turn equivalent to the sets Gm (u, g) and Gm (u, g n ) being isomorphic permutation modules for the two element centralizer CG (u, g) [4, Theorem 3.8], with u, g, n satisfying the same constraints as for the F SZm property. Here the action is by conjugation. We note that while the F SZ property is concerned with certain invariants being in Z, the F SZ + property is not concerned with these invariants being non-negative integers. When the invariants are guaranteed to be non-negative is another area of research, and will also not be considered here. Example 1.5. The author has shown that quaternion groups and certain semidirect products defined from cyclic groups are always F SZ [7, 8]. This includes the dihedral groups, semidihedral groups, and quasidihedral groups, among many others. Example 1.6. Iovanov et al. [4] showed that several groups and families of groups are F SZ, including: • All regular p-groups. • Zp ≀r Zp , the Sylow p-subgroup of Sp2 , which is an irregular F SZ p-group. • P SL2 (q) for a prime power q. 4 MARC KEILBERG • Any direct product of F SZ groups. Indeed, any direct product of F SZm groups is also F SZm , as the cardinalities of the sets in Definition 1.1 split over the direct product in an obvious fashion. • The Mathieu groups M11 and M12 . • Symmetric and alternating groups. See also [2]. Because of the first item, Susan Montgomery has proposed that we use the term F S-regular instead of F SZ, and F S-irregular for non-F SZ. Similarly for F Sm regular and F Sm -irregular. These seem reasonable choices, but for this paper the author will stick with the existing terminology. Example 1.7. On the other hand, Iovanov et al. [4] also established that non-F SZ groups exist by using GAP [3] to show that there are exactly 32 isomorphism classes of groups of order 56 which are not F SZ5 . Example 1.8. The author has constructed examples of non-F SZpj p-groups for all primes p > 3 and j ∈ N in [6]. For j = 1 these groups have order pp+1 , which is the minimum possible order for any non-F SZ p-group. Combined, [4, 6, 13] show, among other things, that the minimum order of non-F SZ 2-groups is at least 210 , and the minimum order for non-F SZ 3-groups is at least 38 . It is unknown if any non-F SZ 2-groups or 3-groups exist, however. Example 1.9. Schauenburg [13] provides several equivalent formulations of the F SZm properties, and uses them to construct GAP [3] functions which are useful for testing the property. Using these functions, it was shown that the Chevalley group G2 (5) and the sporadic simple group HN are not F SZ5 . These groups were attacked directly, using advanced computing resources for HN , often with an eye on computing the values of the indicators explicitly. We will later present an alternative way of using GAP to prove that these groups, and their Sylow 5subgroups, are not F SZ5 . We will not attempt to compute the actual values of the indicators, however. One consequence of these examples is that the smallest known order for a nonF SZ group is 56 = 15,625. The groups with order divisible by pp+1 for p > 5 that are readily available through GAP are small in number, problematically large, and frequently do not have convenient representations. Matrix groups have so far proven too memory intensive for what we need to do, so we need permutation or polycyclic presentations for accessible calculations. For these reasons, all of the examples we pursue in the following sections will hone in on the non-F SZ5 property for groups with order divisible by 56 , and which admit known or reasonably computable permutation representations. In most of the examples, 56 is the largest power of 5 dividing the order, with the Monster group, the projective symplectic group S6 (5), and the perfect groups of order 12 · 57 being the exceptions. 2. Obtaining the non-F SZ property from certain subgroups Our first elementary result offers a starting point for investigating non-F SZm groups of minimal order. Lemma 2.1. Let G be a group with minimal order in the class of non-F SZm groups. Then |Gm (u, g)| 6= |Gm (u, g n )| for some (n, |G|) = 1 implies g ∈ Z(G). Proof. If not then CG (g) is a smaller non-F SZm group, a contradiction.  THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 5 The result applies to non-F SZm groups in a class that is suitably closed under the taking of centralizers. For example, we have the following version for p-groups. Corollary 2.2. Let P be a p-group with minimal order in the class of non-F SZpj p-groups. Then |Ppj (u, g)| 6= |Ppj (u, g n )| for some p ∤ n implies g ∈ Z(P ). Example 2.3. From the examples in the previous section, we know the minimum possible order for a non-F SZp p-group for p > 3 is pp+1 . It remains unknown if the examples of non-F SZpj p-groups from [6] for j > 1 have minimal order among non-F SZpj p-groups. We also know that to check if a group of order 210 or 38 is F SZ it suffices to assume that g is central. Next, we determine a condition for when the non-F SZ property for a normal subgroup implies the non-F SZ property for the full group. Lemma 2.4. Let G be a group and suppose H is a non-F SZm normal subgroup with m coprime to [G : H]. Then G is non-F SZm . Proof. Let u, g ∈ H and (n, |g|) = 1 be such that |Hm (u, g)| 6= |Hm (u, g n )|. By the index assumption, for all x ∈ G we have xm ∈ H ⇔ x ∈ H, so by definitions Gm (u, g) = Hm (u, g) and Gm (u, g n ) = Hm (u, g n ), which gives the desired result.  Corollary 2.5. Let G be a finite group and suppose P is a normal non-F SZpj Sylow p-subgroup of G for some prime p. Then G is non-F SZpj . Corollary 2.6. Let G be a finite group and P a non-F SZpj Sylow p-subgroup of G. Then the normalizer NG (P ) is non-F SZpj . Sadly, we will find no actual use for Corollary 2.5 in the examples we consider in this paper. However, this result, [13, Lemma 8.7], and the examples we collect in the remainder of this paper do suggest the following conjectural relation for the F SZ property. Conjecture 2.7. A group is F SZ if and only if all of its Sylow subgroups are F SZ. Some remarks on why this conjecture may involve some deep results to establish affirmatively seems in order. Consider a group G and let u, g ∈ G and n ∈ N with (n, |G|) = 1. Suppose that g has order a power of p, for some prime p. Then [ Gpj (u, g) = Gpj (u, g) ∩ P x , where the union runs over all distinct conjugates P x in CG (g) of a fixed Sylow p-subgroup P of CG (g). Let Ppxj (u, g) = Gpj (u, g) ∩ P x . Then |Gpj (u, g)| = S S |Gpj (u, g n )| if and only if there is a bijection P x (u, g) → P x (u, g n ). In the special case u ∈ P , if P was F SZpj we would have a bijection P (u, g) → P (u, g n ), x x but this does not obviously guarantee a bijection g n ) for all conS x P (u, g) S →xP (u, n jugates. Attempting to get a bijection P (u, g) → P (u, g ) amounts, via the Inclusion-Exclusion Principle, to controlling the intersections of any number of conjugates and how many elements those intersections contribute to Gpj (u, g) and Gpj (u, g n ). There is no easy or known way to predict the intersections of a collection of Sylow p-subgroups for a completely arbitrary G, so any positive affirmation of the conjecture will impose a certain constraint on these intersections. 6 MARC KEILBERG Moreover, we have not considered the case of the sets Gm (u, g) where m has more than one prime divisor, nor those cases where u, g do not order a power of a fixed prime, so a positive affirmation of the conjecture is also expected to show that the F SZm properties are all derived from the F SZpj properties for all prime powers dividing m. On the other hand, a counterexample seems likely to involve constructing a large group which exhibits a complex pattern of intersections in its Sylow p-subgroups for some prime p, or otherwise exhibits the first example of a group which is F SZpj for all prime powers but is nevertheless not F SZ. Example 2.8. All currently known non-F SZ groups are either p-groups (for which the conjecture is trivial), are nilpotent (so are just direct products of their Sylow subgroups), or come from perfect groups (though the relevant centralizers need not be perfect). The examples of both F SZ and non-F SZ groups we establish here will also all come from perfect groups and p-groups. In the process we obtain, via the centralizers and maximal subgroups considered, an example of a solvable, non-nilpotent, non-F SZ group; as well as an example of a non-F SZ group which is neither perfect nor solvable. All of these examples, of course, conform to the conjecture. 3. GAP functions and groups of small order The current gold standard for general purpose testing of the F SZ properties in GAP [3] is the FSZtest function of Schauenburg [13]. In certain specific situations, the function FSInd from [4] can also be useful for showing a group is non-F SZ. However, with most of the groups we will consider in this paper both of these functions are impractical to apply directly. The principle obstruction for FSZtest is that this function needs to compute both conjugacy classes and character tables of centralizers, and this can be a memory intensive if not wholly inaccessible task. For FSInd the primary obstruction, beyond its specialized usage case, is that it must completely enumerate, store, and sort the entire group (or centralizer). This, too, can quickly run into issues with memory consumption. We therefore need alternatives for testing (the failure of) the FSZ properties which can sidestep such memory consumption issues. For Section 7 we will also desire functions which can help us detect and eliminate the more ”obviously” FSZ groups. We will further need to make various alterations to FSZtest to incorporate these things, and to return a more useful value when the group is not F SZ. The first function we need, FSZtestZ, is identical to FSZtest—and uses several of the helper functions found in [13]—except that instead of calculating and iterating over all rational classes of the group it iterates only over those of the center. It needs only a single input, which is the group to be checked. If it finds that the group is non-F SZ, rather than return false it returns the data that established the non-F SZ property. Of particular importance are the values m and z. If the group is not shown to be non-F SZ by this test, then it returns fail to indicate that the test is typically inconclusive. FSZtestZ := function (G) l o c a l CT, zz , z , c l , div , d , c h i , m, b ; c l := R a t i o n a l C l a s s e s ( Center (G) ) ; c l := F i l t e r e d ( c l , c−>not Order ( R e p r e s e n t a t i v e ( c ) ) THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 7 in [ 1 , 2 , 3 , 4 , 6 ] ) ; for z z in c l do z := R e p r e s e n t a t i v e ( z z ) ; d i v := F i l t e r e d ( D i v i s o r s I n t ( Exponent (G) / Order ( z ) ) , m−>not Gcd (m, Order ( z ) ) in [ 1 , 2 , 3 , 4 , 6 ] ) ; i f Length ( d i v ) < 1 then continue ; f i ; CT := O r d i n a r y C h a r a c t er T a b le (G) ; for c h i in I r r (CT) do for m in d i v do i f not I s R a t ( beta (CT, z , m, c h i ) ) then return [ z ,m, c h i ,CT ] ; fi ; od ; od ; od ; #t h e t e s t i s i n c o n c l u s i v e i n g e n e r a l return f a i l ; end ; This function is primarily useful for testing groups with minimal order in a class closed under centralizers, such as in Lemma 2.1 and Corollary 2.2. Or for any group with non-trivial center that is suspected of failing the F SZ property at a central value. We next desire a function which can quickly eliminate certain types of groups as automatically being F SZ. For this, the following result on groups of small order is helpful. Theorem 3.1. Let G be a group with |G| < 2016 and |G| 6= 1024. Then G is F SZ. Proof. By Lemma 2.1 it suffices to run FSZtestZ over all groups in the SmallGroups library of GAP. This library includes all groups with |G| < 2016, except those of order 210 = 1024. In practice, the author also used the function IMMtests introduced below, but where the check on the size of the group is constrained initially to 100 by [4, Corollary 5.5], and can be increased whenever desired to eliminate all groups of orders already completely tested. This boils down to quickly eliminating p-groups and groups with relatively small exponent. By using the closure of the F SZ properties with respect to direct products, one need only consider a certain subset of the orders in question rather than every single one in turn, so as to avoid essentially double-checking groups. We note that the groups of order 1536 take the longest to check. The entire process takes several days over multiple processors, but is otherwise straightforward.  8 MARC KEILBERG We now define the function IMMtests. This function implements most of the more easily checked conditions found in [4] that guarantee the F SZ property, and calls FSZtestZ when it encounters a suitable p-group. The function returns true if the test conclusively establishes that the group is F SZ; the return value of FSZtestZ if it conclusively determines the group is non-F SZ; and fail otherwise. Note that whenever this function calls FSZtestZ that test is conclusive by Corollary 2.2, so it must adjust a return value of fail to true. IMMtests := function (G) l o c a l sz , b , l , p2 , p3 , po ; i f I s A b e l i a n (G) then return true ; fi ; s z := S i z e (G) ; i f ( s z < 2 0 1 6 ) and ( not s z =1024) then return true ; fi ; i f IsPGroup (G) then #R e g u l a r p−g r o u p s are a l w a y s FSZ . l := C o l l e c t e d ( F a c t o r s I n t ( s z ) ) [ 1 ] ; i f l [1] >= l [ 2 ] or Exponent (G) = l [ 1 ] then return true ; fi ; s z := Length ( U p p e r C e n t r a l S e r i e s (G) ) ; i f l [ 1 ] = 2 then i f l [2 ] < 1 0 or s z < 3 or Exponent (G)<64 then return true ; e l i f l [ 2 ] = 1 0 and s z >= 3 then b := FSZtestZ (G) ; i f I s L i s t ( b ) then return b ; e l s e return true ; fi ; fi ; e l i f l [ 1 ] = 3 then i f l [2 ] < 8 or s z < 4 or Exponent (G)<27 THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 9 then return true ; e l i f l [ 2 ] = 8 and sz >=4 then b := FSZtestZ (G) ; i f I s L i s t ( b ) then return b ; e l s e return true ; fi ; fi ; e l i f sz < l [1]+1 then return true ; e l i f s z = l [ 1 ] + 1 and s z=l [ 2 ] then b := FSZtestZ (G) ; i f I s L i s t ( b ) then return b ; e l s e return true ; fi ; fi ; else #c h e c k t h e e x p o n e n t f o r non−p−g r o u p s l := F a c t o r s I n t ( Exponent (G) ) ; p2 := Length ( P o s i t i o n s ( l , 2 ) ) ; p3 := Length ( P o s i t i o n s ( l , 3 ) ) ; po := F i l t e r e d ( l , x−>x > 3 ); i f F o r A l l ( C o l l e c t e d ( po ) , x−>x [ 2 ] < 2 ) and ( ( p2 < 4 and p3 < 4 ) or ( p2 < 6 and p3 < 2 ) ) then return true ; fi ; fi ; #t e s t s were i n c o n c l u s i v e return f a i l ; end ; We then incorporate these changes into a modified version of FSZtest, which we give the same name. Note that this function also uses the function beta and its corresponding helper functions from [13]. It has the same inputs and outputs as FSZtestZ, except that the test is definitive, and so returns true when the group is F SZ. FSZtest := function (G) l o c a l C, CT, zz , z , c l , div , d , c h i , m, b ; b := IMMtests (G ) ; ; 10 MARC KEILBERG i f not b= f a i l then return b ; fi ; c l := R a t i o n a l C l a s s e s (G) ; c l := F i l t e r e d ( c l , c−>not Order ( R e p r e s e n t a t i v e ( c ) ) in [ 1 , 2 , 3 , 4 , 6 ] ) ; for z z in c l do z := R e p r e s e n t a t i v e ( z z ) ; C := C e n t r a l i z e r (G, z ) ; d i v := F i l t e r e d ( D i v i s o r s I n t ( Exponent (C) / Order ( z ) ) , m−>not Gcd (m, Order ( z ) ) in [ 1 , 2 , 3 , 4 , 6 ] ) ; i f Length ( d i v ) < 1 then continue ; f i ; # Check f o r t h e e a s y c a s e s b := IMMtests (C ) ; i f b=true then continue ; e l i f I s L i s t ( b ) then i f R a t i o n a l C l a s s (C, z)= R a t i o n a l C l a s s (C, b [ 2 ] ) then return b ; fi ; fi ; CT := O r d i n a r y C h a r a c t er T a b le (C) ; for c h i in I r r (CT) do for m in d i v do i f not I s R a t ( beta (CT, z , m, c h i ) ) then return [m, z , c h i ,CT ] ; fi ; od ; od ; od ; return true ; end ; Our typical procedure will be as follows: given a group G, take its Sylow 5subgroup P and find u, g ∈ P such that |P5 (u, g)| 6= |P5 (u, g n )| for 5 ∤ n, and then show that |G5 (u, g)| 6= |G5 (u, g n )|. The second entry in the list returned by THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 11 FSZtest gives precisely the g value we need. But it does not provide the u value directly, nor the n. As it turns out, we can always take n = 2 when o(g) = 5, but for other orders this need not necessarily hold. In order to acquire these values we introduce the function FSIndPt below, which is a variation on FSInd [4]. This function has the same essential limitation that FSInd does, in that it needs to completely enumerate, store, and sort the elements of the group. This could in principle be avoided, at the cost of increased run-time. However our main use for the function is to apply it to Sylow 5-subgroups which have small enough order that this issue does not pop up. The inputs are a group G, m ∈ N and g ∈ G. It is best if one in fact passes in CG (g) for G, but the function will compute the centralizer regardless. The function looks for an element u ∈ CG (g) and an integer j coprime to the order of g such that |Gm (u, g)| 6= |Gm (u, g j )|. The output is the two element list [u,j] if such data exists, otherwise it returns fail to indicate that the test is normally inconclusive. Note that by Lemma 1.3 and centrality of g in C = CG (g) we need only consider the rational classes in C to find such a u. FSIndPt:= function (G,m, g ) l o c a l GG, C, Cl , g u c o e f f , elG , Gm, a l i s t , a u l i s t , u m l i s t , npos , j , n , u , pr ; C := C e n t r a l i z e r (G, g ) ; GG := EnumeratorSorted (C ) ; ; elG := S i z e (C ) ; Gm := L i s t (GG, x−>P o s i t i o n (GG, xˆm) ) ; pr := P r imeResidues ( Order ( g ) ) ; for Cl in R a t i o n a l C l a s s e s (C) do u := R e p r e s e n t a t i v e ( Cl ) ; npos := [ ] ; a l i s t := [ ] ; a u l i s t := [ ] ; u m l i s t := [ ] ; g u c o e f f := [ ] ; u m l i s t := L i s t (GG, a−>P o s i t i o n (GG, ( a∗ I n v e r s e ( u ) ) ˆm) ) ; ; #The f o l l o w i n g computes t h e c a r d i n a l i t i e s # o f G m( u , g ˆn ) . for n in pr 12 MARC KEILBERG do npos := P o s i t i o n (GG, g ˆn ) ; a l i s t := P o s i t i o n s (Gm, npos ) ; a u l i s t := P o s i t i o n s ( u m l i s t , npos ) ; g u c o e f f [ n ] := S i z e ( Intersection ( alist , aulist )); #Check i f we ’ ve fou n d our u i f not g u c o e f f [ n ] = g u c o e f f [ 1 ] then return [ u , n ] ; fi ; od ; od ; #No u was fou n d f o r t h i s G,m, g return f a i l ; end ; Lastly, we introduce the function FSZSetCards, which is the most naive and straightforward way of computing both |Gm (u, g)| and |Gm (u, g n )|. The inputs are a set C of group elements—normally this would be CG (g), but could be a conjugacy class or some other subset or subgroup—; group elements u, g; and integers m, n such that g 6= g n . The output is a two element list, which counts the number of elements of C in Gm (u, g) in the first entry and the number of elements of C in Gm (u, g n ) in the second entry. It is left to the user to check that the inputs satisfy whatever relations are needed, and to then properly interpret the output. FSZSetCards := function (C, u , g ,m, n ) l o c a l c o n t r i b s , apow , aupow , a ; c o n t r i b s := [ 0 , 0 ] ; for a in C do apow := a ˆm; aupow := ( a ∗ I n v e r s e ( u ) ) ˆm; i f ( apow = g and aupow = g ) then c o n t r i b s [ 1 ] := c o n t r i b s [ 1 ] + 1 ; e l i f ( apow=g ˆn and aupow=gˆn ) then c o n t r i b s [ 2 ] := c o n t r i b s [ 2 ] + 1 ; fi ; od ; return ( c o n t r i b s ) ; end ; THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 13 As long as C admits a reasonable iterator in GAP then this function can compute these cardinalities with a very minimal consumption of memory. Any polycyclic or permutation group satisfies this, as well as any conjugacy class therein. However, for a matrix group GAP will attempt to convert to a permutation representation, which is usually very costly. The trade-off, as it often is, is in the speed of execution. For permutation groups the run-time can be heavily impacted by the degree, such that it is almost always worthwhile to apply SmallerDegreePermutationRepresentation whenever possible. If the reader wishes to use this function on some group that hasn’t been tested before, the author would advise adding in code that would give you some ability to gauge how far along the function is. By default there is nothing in the above code, even if you interrupt the execution to check the local variables, to tell you if the calculation is close to completion. Due to a variety of technical matters it is difficult to precisely benchmark the function, but when checking a large group it is advisable to acquire at least some sense of whether the calculation may require substantial amounts of time. Remark 3.2. Should the reader opt to run our code to see the results for themselves, they may occasionally find that the outputs of FSZSetCards occur in the opposite order we list here. This is due to certain isomorphisms and presentations for groups calculated in GAP not always being guaranteed to be identical every single time you run the code. As a result, the values for u or g may sometimes be a coprime power (often the inverse) of what they are in other executions of the code. Nevertheless, there are no issues with the function proving the non-F SZ property thanks to Lemma 1.3, and there is sufficient predictability to make the order of the output the only variation. While very naive, FSZSetCards will suffice for most of our purposes, with all uses of it completing in an hour or less. However, in Section 6 we will find an example where the expected run-time for this function is measured in weeks, and for which FSZtest requires immense amounts of memory—Schauenburg [13] says that FSZtest for this group consumed 128 GB of memory without completing! We therefore need a slightly less naive approach to achieve a more palatable runtime in this case. We leave this to Section 6, but note to the reader that the method this section uses can also be applied to all of the other groups for which FSZSetCards suffices. The reason we bother to introduce and use FSZSetCards is that the method of Section 6 relies on being able to compute conjugacy classes, which can hit memory consumption issues that FSZSetCards will not encounter. It is not our goal with these functions to find the most efficient, general-purpose procedure. Instead we seek to highlight some of the ways in which computationally problematic groups may be rendered tractable by altering the approach one takes, and to show that the non-F SZ property of these groups can be demonstrated in a (perhaps surprisingly) short amount of time and with very little memory consumption. 4. The non-F SZ sporadic simple groups The goal for this section is to show that the Chevalley group G2 (5), and all sporadic simple groups with order divisible by 56 , as well as their Sylow 5-subgroups, are non-F SZ5 . We begin with a discussion of the general idea for the approach. Our first point of observation is that the only primes p such that pp+1 divides the order of any of these groups have p ≤ 5. Indeed, a careful analysis of the non-F SZ 14 MARC KEILBERG groups of order 56 found in [4] shows that several of them are non-split extensions with a normal extra-special group of order 55 , which can be denoted in AtlasRep notation as 51+4 .5. Consulting the known maximal subgroups for these groups we can easily infer that the Sylow 5-subgroups of HN , G2 (5), B, and Ly have this same form, and that the Monster has such a p-subgroup. Indeed, G2 (5) is a maximal subgroup of HN , and B and Ly have maximal subgroups containing a copy of HN , so these Sylow subgroups are all isomorphic. Furthermore, the Monster’s Sylow 5-subgroup has the form 51+6 .52 , a non-split extension of the elementary abelian group of order 25 by an extra special group of order 57 . Given this, we suspect that these Sylow 5-subgroups are all non-F SZ5 , and that this will cause the groups themselves to be non-F SZ5 . We can then exploit the fact that non-trivial p-groups all have non-trivial centers to obtain centralizers in the parent group that contain a Sylow 5-subgroup. In the case of G = HN or G = G2 (5), we can quickly find u, g ∈ P , with P a Sylow 5-subgroup of G, such that |P5 (u, g)| 6= |P5 (u, g 2 )|, and show that for H = CG (g) we have |H5 (u, g)| 6= |H5 (u, g 2 )|. Since necessarily |H5 (u, g)| = |G5 (u, g)| and |H5 (u, g 2 )| = |G5 (u, g 2 )|, this will show that HN and G2 (5) are non-F SZ5 . Unfortunately, it turns out that P is not normal in H in either case, so the cardinalities of these sets in H must be checked directly, rather than simply applying Corollary 2.5. The remaining groups require a little more work, for various reasons. In the case of the Monster, there is a unique non-identity conjugacy class yielding a centralizer with order divisible by 59 . So we are free to pick any subgroup G of M that contains a centralizer with this same order. Fortunately, not only is such a (maximal) subgroup known, but Bray and Wilson [1] have also computed a permutation representation for it. This is available in GAP via the AtlasRep package. This makes all necessary calculations for the Monster accessible. The Sylow 5-subgroup is fairly easily shown to be non-F SZ5 directly. However, the centralizer we get in this way has large order, and its Sylow 5-subgroup is not normal, making it impractical to work with on a personal computer. However, further consultation of character tables shows that the Monster group has a unique conjugacy class of an element of order 10 whose centralizer is divisible by 56 . So we may again pick any convenient (maximal) subgroup with such a centralizer, and it turns out the same maximal subgroup works. We construct the appropriate element of order 10 by using suitable elements from Sylow subgroups of the larger centralizer, and similarly to get the element u. Again it turns out that the Sylow 5-subgroup of this smaller subgroup is not normal, so we must compute the set cardinalities over the entire centralizer in question. However, this centralizer is about 1/8000-th the size of the initial one, and we are subsequently able to calculate the appropriate cardinalities in under an hour. The Baby Monster can then be handled by using the fact that the Monster contains the double cover of B as the centralizer of an involution to obtain the centralizer we need in B from a centralizer in M . The author thanks Robert Wilson for reminding them of this fact. For the Lyons group, the idea is much the same as for HN and G2 (5), with the additional complication that the AtlasRep package does not currently contain any permutation representations for Ly. To resolve this, we obtain a permutation representation for Ly, either computed directly in GAP THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 15 or downloaded [12]. This is then used to construct a suitable permutation representation of the maximal subgroup in question. Once this is done the calculations proceed without difficulties. These calculations all make extensive use of the functions given in Section 3. 4.1. Chevalley group G2 (5). We now show that G2 (5) and its Sylow 5-subgroups are not F SZ5 . This was independently verified in [13]. Since G2 (5) is of relatively small order, it can be attacked quickly and easily. Theorem 4.1. The simple Chevalley group G2 (5) and its Sylow 5-subgroup are non-F SZ5 . Proof. The claims follow from running the following GAP code. G := AtlasGroup ( ”G2 ( 5 ) ” ) ; ; P := SylowSubgroup (G, 5 ) ; ; # The f o l l o w i n g shows P i s n ot FSZ 5 g := FSZtestZ (P ) [ 2 ] ; # Find u u := FSIndPt (P , 5 , g ) [ 1 ] ; ; C := C e n t r a l i z e r (G, g ) ; ; #Check t h e c a r d i n a l i t i e s FSZSetCards (C, u , g , 5 , 2 ) ; The output is [0,625], so it follows that G and P are both non-F SZ5 as desired.  We note that P is not normal in C, and indeed C is a perfect group of order 375,000 = 23 · 3 · 56 . The call to FSZSetCards above runs in approximately 11 seconds, which is approximately the amount of time necessary to run FSZtest on G2 (5) directly. In this case, the use of FSZSetCards is not particularly efficient, as the groups in question are of reasonably small sizes and permutation degree. Nevertheless, this demonstrates the basic method we will employ for all subsequent groups. 4.2. The Harada-Norton group. For the group HN the idea proceeds similarly as for G2 (5). Theorem 4.2. The Harada-Norton simple group HN and its Sylow 5-subgroup are not F SZ5 . Proof. To establish the claims it suffices to run the following GAP code. G := AtlasGroup ( ”HN” ) ; ; P := SylowSubgroup (G, 5 ) ; ; # G, t h u s P, has v e r y l a r g e d e g r e e . # P o l y c y c l i c g r o u p s are e a s i e r t o work w i t h . i s o P := IsomorphismPcGroup (P ) ; ; P := Image ( i s o P ) ; ; 16 MARC KEILBERG #Find u , g f o r P g := FSZtestZ (P ) [ 2 ] ; u := FSIndPt (P , 5 , g ) [ 1 ] ; g := Image ( Inver seGener a lMa pping ( i s o P ) , g ) ; ; u := Image ( Inver seGener a lMa pping ( i s o P ) , u ) ; ; C := C e n t r a l i z e r (G, g ) ; ; iso C := IsomorphismPcGroup (C ) ; ; C := Image ( iso C ) ; ; FSZSetCards (C, Image ( isoC , u ) , Image ( isoC , g ) , 5 , 2 ) ; This code executes in approximately 42 minutes, with approximately 40 of that spent finding P . The final output is [3125,0], so we conclude that both P and HN are non-F SZ5 , as desired.  P is again not a normal subgroup of C, so we again must test the entire centralizer rather than just P . We note that |C| = 25 56 = 500,000. Indeed, C is itself nonF SZ5 of necessity, and the fact that the call to IsomorphismPcGroup did not fail means that C is solvable, and in particular not perfect and not a p-group. 4.3. The Monster group. We will now consider the Monster group M . The full Monster group is famously difficult to compute in. But, as detailed in the beginning of the section, by consulting character tables of M and its known maximal subgroups, we can find a maximal subgroup which contains a suitable centralizer (indeed, two suitable centralizers) and also admits a known permutation representation [1]. Theorem 4.3. The Monster group M and its Sylow 5-subgroup are not F SZ5 . Proof. The Sylow 5-subgroup of M has order 59 . Consulting the character table of M , we see that M has a unique conjugacy class yielding a proper centralizer with order divisible by 59 , and a unique conjugacy class of an element of order 10 whose centralizer has order divisible by 56 ; moreover, the order of the latter centralizer is precisely 12 million, and in particular is not divisible by 57 . It suffices to consider any maximal subgroups containing such centralizers. The maximal subgroup of shape 51+6 : 2.J2 .4, which is the normalizer associated to a 5B class, is one such + choice. We first show that the Sylow 5-subgroup of M is not F SZ5 . G := P := isoP P := AtlasGroup ( ” 5 ˆ ( 1 + 6 ) : 2 . J2 . 4 ” ) ; ; SylowSubgroup (G, 5 ) ; ; := IsomorphismPcGroup (P ) ; ; Image ( i s o P ) ; ; ex := FSZtestZ (P ) ; The proper centralizer with order divisible by 59 is still impractical to work with. So we will use the data for P to construct the element of order 10 mentioned above. THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 17 zp := ex [ 2 ] ; ; zp := Image ( Inver seGener a lMa pping ( i s o P ) , zp ) ; ; C := C e n t r a l i z e r (G, zp ) ; ; Q := SylowSubgroup (C , 2 ) ; ; zq := F i r s t ( Center (Q) , q−>Order ( q)>1 and S i z e ( C e n t r a l i z e r (G, zp ∗q ) ) = 1 2 0 0 0 0 0 0 ) ; ; #This g i v e s us t h e g and c e n t r a l i z e r we want . g := zp ∗ zq ; ; C := C e n t r a l i z e r (G, g ) ; ; #Reducing t h e p e r m u t a t i o n d e g r e e w i l l #s a v e a l o t o f compu t at ion t ime l a t e r . iso C := S m a l l e r D e g r e e P e r m u t a t i o n R e p r e s e n t a t i o n (C ) ; ; C := Image ( iso C ) ; ; g := Image ( isoC , g ) ; ; zp := Image ( isoC , zp ) ; ; zq := Image ( isoC , zq ) ; ; #Now P := isoP P := proceed to con st ru ct a choice of u . SylowSubgroup (C , 5 ) ; ; := IsomorphismPcGroup (P ) ; ; Image ( i s o P ) ; ; ex := FSIndPt (P, 5 , Image ( iso P , zp ) ) ; up := Image ( Inver seGener a lMa pping ( i s o P ) , ex [ 1 ] ) ; ; #D e f i n e our c h o i c e o f u . #In t h i s case , u has o r d e r 5 0 . u := up∗ zq ; ; #F i n a l l y , we compute t h e c a r d i n a l i t i e s # of the relevant s e t s . FSZSetCards (C, u , g , 5 , 7 ) ; This final function yields [0,15000], which proves that M is not F SZ5 , as desired.  This final function call takes approximately 53 minutes to complete, while all preceding operations can complete in about 5 minutes combined—though the conversion of C to a lower degree may take more than this, depending. The lower degree C has degree 18, 125, but requires (slightly) more than 2 GB of memory to acquire. This conversion can be skipped to keep the memory demands well under 2GB, but the execution time for FSZSetCards will inflate to approximately a day and a half. 18 MARC KEILBERG Remark 4.4. In the first definition of C above, containing the full Sylow 5-subgroup of M , we have |C| = 9.45 × 1010 = 28 · 33 · 59 · 7. For the second definition of C, corresponding to the centralizer of an element of order 10, we have |C| = 1.2×107 = 28 · 3 · 56 . The first centralizer is thus 7875 = 32 · 53 · 7 times larger than the second one. Either one is many orders of magnitude smaller than |M | ≈ 8.1 × 1053 , but the larger one was still too large to work with for practical purposes. 4.4. The Baby Monster. We can now consider the Baby Monster B. Theorem 4.5. The Baby Monster B and its Sylow 5-subgroup are both non-F SZ5 . Proof. The Baby Monster is well known to have a maximal subgroup of the form HN.2, so it follows that B and HN have isomorphic Sylow 5-subgroups. By Theorem 4.2 HN has a non-F SZ5 Sylow 5-subgroup, so this immediately gives the claim about the Sylow 5-subgroup of B. From the character table of B we see that there is a unique non-identity conjugacy class whose centralizer has order divisible by 56 . This corresponds to an element of order 5 from the 5B class, and the centralizer has order 6,000,000 = 27 ·3·56 . In the double cover 2.B of B, this centralizer is covered by the centralizer of an element of order 10. This centralizer necessarily has order 12, 000, 000. Since M contains 2.B as a maximal subgroup, and there is a unique centralizer of an element of order 10 in M with order divisible by 12,000,000, these centralizers in 2.B and M are isomorphic. We have already computed this centralizer in M in Theorem 4.3. To obtain the centralizer in B, we need only quotient by an appropriate central involution. In the notation of the proof of Theorem 4.3, this involution is precisely zq. GAP will automatically convert this quotient group D into a lower degree representation, yielding a permutation representation of degree 3125 for the centralizer. This will require as much as 8GB of memory to complete. Moreover, the image of zp from Theorem 4.3 in this quotient group yields the representative of the 5B class we desire, denoted here by g. Using the image of up in the quotient for u, we can then easily run FSZSetCards(C,u,g,5,2) to get a result of [15000,3125], which shows that B is non-F SZ5 as desired. This final call completes in about 4 minutes.  Note that in M the final return values summed to 15,000, with one of the values 0, whereas in B they sum to 18,125 and neither is zero. This reflects how there is no clear relationship between the F SZ properties of a group and its quotients, even when the quotient is by a (cyclic) central subgroup. In particular, it does not immediately follow that the quotient centralizer would yield the non-F SZ property simply because the centralizer in M did, or vice versa. Moreover, we also observe that the cardinalities computed in Theorem 4.2 implies that for a Sylow 5-subgroup P of B we have P5 (u, g 2 ) = ∅, so the 3125 ”extra” elements obtained in B5 (u, g 2 ) come from non-trivial conjugates of P . This underscores the expected difficulties in a potential proof (or disproof) of Conjecture 2.7. 4.5. The Lyons group. There is exactly one other sporadic group with order divisible by 56 (or pp+1 for p > 3): the Lyons group Ly. Theorem 4.6. The maximal subgroup of Ly of the form 51+4 : 4.S6 has a faithful permutation representation on 3,125 points, given by the action on the cosets of THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 19 4.S6 . Moreover, this maximal subgroup, Ly, and their Sylow 5-subgroups are all non-F SZ5 . Proof. It is well-known that Ly contains a copy of G2 (5) as a maximal subgroup, and that the order of Ly is not divisible by 57 . Therefore Ly and G2 (5) have isomorphic Sylow 5-subgroups, and by Theorem 4.1 this Sylow subgroup is not F SZ5 . Checking the character table for Ly as before, we find there is a unique nonidentity conjugacy class whose corresponding centralizer has order divisible by 56 . In particular, the order of this centralizer is 2,250,000 = 24 ·32 ·56 , and it comes from an element of order 5. So any maximal subgroup containing an element of order 5 whose centralizer has this order will suffice. The maximal subgroup 51+4 : 4.S6 is the unique such choice. The new difficulty here is that, by default, there are only matrix group representations available though the AtlasRep package for Ly and 51+4 : 4.S6 , which are ill-suited for our purposes. However, faithful permutation representations for Ly are known, and they can be constructed through GAP with sufficient memory available provided one uses a well-chosen method. A detailed description of how to acquire the permutation representation on 8,835,156 points, as well as downloads for the generators (including MeatAxe versions courtesy of Thomas Breuer) can be found on the web, courtesy Pfeiffer [12]. Using this, we can then obtain a permutation representation for the maximal subgroup 51+4 : 4.S6 on 8,835,156 points using the programs available on the online ATLAS [16]. This in turn is fairly easily converted into a permutation representation on a much smaller number of points, provided one has up to 8 GB of memory available, via SmallerDegreePermutationRepresentation. The author obtained a permutation representation on 3125 points, corresponding to the action on the cosets of 4.S6 . The exact description of the generators is fairly long, so we will not reproduce them here. The author is happy to provide them upon request. One can also proceed in a fashion similar to some of the cases handled in [1] to find such a permutation representation. Once this smaller degree representation is obtained, it is then easy to apply the same methods as before to show the desired claims about the F SZ5 properties. We can directly compute the Sylow 5-subgroup, then find u, g through FSZtestZ and FSIndPt irrespectively, set C to be the centralizer of g, then run FSZSetCards(C,u,g,5,2). This returns [5000,625], which gives the desired nonF SZ5 claims.  Indeed, FSZtest can be applied to (both) the centralizer and the maximal subgroup once this permutation representation is obtained. This will complete quickly, thanks to the relatively low orders and degrees involved. We also note that the centralizer C so obtained will not have a normal Sylow 5-subgroup, and is a perfect group. The maximal subgroup in question is neither perfect nor solvable, and does not have a normal Sylow 5-subgroup. 5. The F SZ sporadic simple groups We can now show that all other sporadic simple groups and their Sylow subgroups are F SZ. 20 MARC KEILBERG Example 5.1. Any group which is necessarily F SZ (indeed, F SZ + ) by [4, Corollary 5.3] necessarily has all of its Sylow subgroups F SZ, and so satisfies the conjecture. This implies that all of the following sporadic groups, as well as their Sylow p-subgroups, are F SZ (indeed, F SZ + ). • The Mathieu groups M11 , M12 , M22 , M23 , M24 . • The Janko groups J1 , J2 , J3 , J4 . • The Higman-Simms group HS. • The McLaughlin group M cL. • The Held group He. • The Rudvalis group Ru. • The Suzuki group Suz. • The O’Nan group O′ N . • The Conway group Co3 . • The Thompson group T h. • The Tits group 2 F4 (2)′ . Example 5.2. Continuing the last example, it follows that the following are the only sporadic simple groups not immediately in compliance with the conjecture thanks to [4, Corollary 5.3]. • The Conway groups Co1 , Co2 . • The Fischer groups F i22 , F i23 , F i′24 . • The Monster M . • The Baby Monster B. • The Lyons group Ly. • The Harada-Norton group HN . The previous section showed that the last four groups were all non-F SZ5 and have non-F SZ5 Sylow 5-subgroups, and so conform to the conjecture. By exponent considerations the Sylow subgroups of the Conway and Fischer groups are all F SZ + . The function FSZtest can be used to quickly show that Co1 , Co2 , F i22 , and F i23 are F SZ, and so conform to the conjecture. This leaves just the largest Fischer group F i′24 . Theorem 5.3. The sporadic simple group F i′24 and its Sylow subgroups are all F SZ. Proof. The exponent of F i′24 can be calculated from its character table and shown to be 24,516,732,240 = 24 · 33 · 5 · 7 · 11 · 13 · 17 · 23 · 29. As previously remarked, this automatically implies that the Sylow subgroups are all F SZ (indeed, F SZ + ). By [4, Corollary 5.3] it suffices to show that every centralizer of an element with order not in {1, 2, 3, 4, 6} in F i′24 that contains an element of order 16 is F SZ. There is a unique conjugacy class in F i′24 for an element with order (divisible by) 16. The centralizer of such an element has order 32, and is isomorphic to Z16 × Z2 . So it suffices to consider the elements of order 8 in this centralizer, and show that their centralizers (in F i′24 ) are F SZ. Every such element has a centralizer of order 1536 = 29 · 3. So by Theorem 3.1 the result follows. The following is GAP code verifying these claims. G := AtlasGroup ( ” Fi2 4 ’ ” ) ; ; THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 21 GT := C h a r a c t e r T a b l e ( ” Fi2 4 ’ ” ) ; ; P o s i t i o n s ( O r d e r s C l a s s R e p r e s e n t a t i v e s (GT) mod 1 6 , 0 ) ; exp := Lcm( O r d e r s C l a s s R e p r e s e n t a t i v e s (GT) ) ; C o l l e c t e d ( F a c t o r s I n t ( exp ) ) ; SetExponent (G, exp ) ; ; P := SylowSubgroup (G, 2 ) ; ; #There are many ways t o g e t an e l e m e n t o f o r d e r 1 6 . #Here ’ s a v e r y crude , i f non−d e t e r m i n i s t i c , one . x := Random(P ) ; ; while not Order ( x ) = 16 do x:=Random(P ) ; od ; C := C e n t r a l i z e r (G, x ) ; ; c e n t s := F i l t e r e d (C, y−>Order ( y ) = 8 ) ; ; c e n t s := L i s t ( c e n t s , y−>C e n t r a l i z e r (G, y ) ) ; ; L i s t ( cents , Size ) ;  The following then summarizes our results on sporadic simple groups. Theorem 5.4. The following are equivalent for a sporadic simple group G. (1) G is not F SZ. (2) G is not F SZ5 . (3) The order of G is divisible by 56 . (4) G has a non-F SZ Sylow subgroup. (5) The Sylow 5-subgroup of G is not F SZ5 . Proof. Combine the results of this section and the previous one.  6. The symplectic group S6 (5) In [13] it was mentioned that the symplectic group S6 (5) was likely to be the second smallest non-F SZ simple group, after G2 (5). Computer calculations there ran into issues when checking a particular centralizer, as the character table needed excessive amounts of memory to compute. Our methods so far also place this group at the extreme end of what’s reasonable. In principle the procedure and functions we’ve introduced so far can decide that this group is non-F SZ in an estimated two weeks of uninterrupted computations, and with nominal memory usage. However, we can achieve a substantial improvement that completes the task in about 8 hours (on two processes; 16 hours for a single process), while maintaining nominal memory usage. The simple yet critical observation comes from [13, Definition 3.3]. In particular, if a ∈ Gm (u, g), then am = g implies that for all b ∈ classCG (g) (a) we have bm = g. So while FSZSetCards acts as naively as possible and iterates over all elements of 22 MARC KEILBERG C = CG (g), we in fact need to only iterate over the elements of those conjugacy classes of C whose m-th power is g (or g n ). GAP can often compute the conjugacy classes of a finite permutation or polycyclic group quickly and efficiently. So while it is plausible that finding these conjugacy classes can be too memory intensive for certain centralizers, there will nevertheless be centralizers for which all other methods are too impractical for either time or memory reasons, but for which this reduction to conjugacy classes makes both time and memory consumption a nonissue. The otherwise problematic centralizer of S6 (5) is precisely such a case, as we will now see. Theorem 6.1. The projective symplectic group S6 (5) and its Sylow 5-subgroup are both non-F SZ5 . Proof. As usual, our first task is to show that the Sylow 5-subgroup is non-F SZ5 , and then use the data obtained from that to attack S6 (5). G := P := isoP P := AtlasGroup ( ” S6 ( 5 ) ” ) ; ; SylowSubgroup (G, 5 ) ; ; := IsomorphismPcGroup (P ) ; ; Image ( i s o P ) ; ; #Show P i s non−FSZ 5 , and #g e t t h e g we need v i a FS Zt est Z g := FSZtestZ (P ) [ 2 ] ; #Get t h e u we need v i a FSIndPt u := FSIndPt (P , 5 , g ) [ 1 ] ; One can of course store the results of FSZtestZ and FSIndPt directly to see the complete data returned, and then extract the specific data need. We can then show that G = S6 (5) is itself non-F SZ5 by computing G5 (u, g) and G5 (u, g 2 ) with the following code. G := isoG G := g := u := uinv C e n t r a l i z e r (G, g ) ; ; := S m a l l e r D e g r e e P e r m u t a t i o n R e p r e s e n t a t i o n (G ) ; ; Image ( isoG ) ; ; Image ( isoG , g ) ; ; Image ( isoG , u ) ; ; := I n v e r s e ( u ) ; ; #Now we compute t h e c o n j u g a c y c l a s s e s # of the c e n t r a l i z e r . c l := C o n j u g a c y C l a s s e s (G ) ; ; #We t h e n need o n l y c o n s i d e r t h o s e # c l a s s e s w i t h a s u i t a b l e 5− t h power cand1 := F i l t e r e d ( c l , x−>R e p r e s e n t a t i v e ( x)ˆ5=g ) ; ; cand2 := F i l t e r e d ( c l , x−>R e p r e s e n t a t i v e ( x)ˆ5=g ˆ 2 ) ; ; THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 23 #There i s i n f a c t o n l y one c o n j u g a c y # c l a s s in both cases . Length ( cand1 ) ; Length ( cand2 ) ; cand1 := cand1 [ 1 ] ; ; cand2 := cand2 [ 1 ] ; ; #The f o l l o w i n g computes | G 5 ( u , g ) | Number ( cand1 , x−>(x∗ uinv )ˆ5=g ) ; #The f o l l o w i n g computes | G 5 ( u , g ˆ 2 ) | Number ( cand2 , x−>(x∗ uinv )ˆ5=g ˆ 2 ) ; This code shows that |G5 (u, g)| = 1,875,000; |G5 (u, g 2 )| = 375,000. Therefore S6 (5) is non-F SZ5 , as desired.  The calculation of |G5 (u, g)| takes approximately 8.1 hours, and the calculation of |G5 (u, g 2 )| takes approximately 7.45 hours. The remaining calculations are done in significantly less combined time. We note that the calculations of these two cardinalities can be done independently, allowing each one to be calculated simultaneously on separate GAP processes. We also note that the centralizer in S6 (5) under consideration in the above is itself a perfect group; is a permutation group of degree 3125 and order 29.25 billion; and has a non-normal Sylow 5-subgroup. Moreover, it can be shown that the g we found yields the only rational class of P at which P fails to be F SZ. One consequence of this, combined with the character table of S6 (5), is that, unlike in the case of the Monster group, we are unable to switch to any other centralizer with a smaller Sylow 5-subgroup to demonstrate the non-F SZ5 property. Similarly as with the Baby Monster group, it is interesting to note that |P5 (u, g)| = 62,500 and |P5 (u, g 2 )| = 0 for P, u, g as in the proof. These cardinalities can be quickly computed exactly as they were for S6 (5), simply restricted to P , or using the slower FSZSetCards, with the primary difference being that now there are multiple conjugacy classes to check and sum over. Before continuing on to the next section, where we consider small order perfect groups available in GAP, we wish to note a curious dead-end, of sorts. Lemma 6.2. Given u, g ∈ G with [u, g] = 1, let C = CG (g), D = CC (u), and m ∈ N. Then a ∈ Gm (u, g) if and only if ad ∈ Gm (u, g) for some/any d ∈ D. Proof. This is noted by Iovanov et al. [4] when introducing the concept of an F SZ + group. It is an elementary consequence of the fact that D = CG (u, g) centralizes both g and u by definition.  So suppose we have calculated those conjugacy classes in C whose m-th power is g. As in the above code, we can iterate over all elements of these conjugacy classes in order to compute |Gm (u, g)|. However, the preceding lemma shows that we could instead partition each such conjugacy class into orbits under the D action. The 24 MARC KEILBERG practical upshot then being that we need only consider a single element of each orbit in order to compute |Gm (u, g)|. In the specific case of the preceding theorem, we can show that the single conjugacy classes cand1 and cand2 both have precisely 234 million elements, and that D is a non-abelian group of order 75,000, and is in fact the full centralizer of u in S6 (5). Moreover the center of C is generated by g, and so has order 5. Thus in the best-case scenario partitioning these conjugacy classes into D orbits can result in orbits with |D/Z(C)| = 15,000 elements each. The cardinalities we computed can also be observed to be multiples 15,000. That would constitute a reduction of more than four orders of magnitude on the total number of elements we would need to check. While this is a best-case scenario, since D also has index 390,000 in C it seems very plausible that such a partition would produce a substantial reduction in the number of elements to be checked. So provided that calculating these orbits can be done reasonably quickly, we would expect a significant reduction in run-time. There is a practical problem, however. The problem being that, as far as the author can tell, there is no efficient way for GAP to actually compute this partition. Doing so evidently requires that GAP fully enumerate and store the conjugacy class in question. In our particular case, a conjugacy class of 234 million elements in a permutation group of degree 3125 simply requires far too much memory—in excess of 1.5 terabytes. As such, while the lemma sounds promising, it seems to be lacking in significant practical use for computer calculations. It seems likely, in the author’s mind, that any situation in which it is useful could have been handled in reasonable time and memory by other methods. Nevertheless, the author cannot rule out the idea as a useful tool. 7. Perfect groups of order less than 106 We now look for examples of additional non-F SZ perfect groups. The library of perfect groups stored by GAP has most perfect groups of order less than 106 , with a few exceptions noted in the documentation. So we can iterate through the available groups, of which there are 1097 at the time this paper was written. We can use the function IMMtests from Section 3 to show that most of them are F SZ. #Get a l l a v a i l a b l e s i z e s G l i s t := F i l t e r e d ( S i z e s P e r f e c t G r o u p s ( ) , n−>N r P e r f e c t L i b r a r y G r o u p s ( n ) > 0 ) ; ; #Get a l l a v a i l a b l e p e r f e c t g r o u p s G l i s t := L i s t ( G l i s t , n−>L i s t ( [ 1 . . N r P e r f e c t L i b r a r y G r o u p s ( n ) ] , k−>P er fectGr o up ( IsPermGroup , n , k ) ) ) ; ; G l i s t := F l a t ( G l i s t ) ; ; #Remove t h e o b v i o u s l y FSZ on es F l i s t := F i l t e r e d ( G l i s t , G−>not IMMtests (G)=true ) ; ; This gives a list of 63 perfect groups which are not immediately dismissed as being F SZ. THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 25 Theorem 7.1. Of the 1097 perfect groups of order less than 106 available through the GAP perfect groups library, exactly 7 of them are not F SZ, all of which are extensions of A5 . All seven of them are non-F SZ5 . Four of them have order 375,000 = 23 · 3 · 56 , and three of them have order 937,500 = 22 · 3 · 57 . Their perfect group ids in the library are: [375000, 2], [375000, 8], [375000, 9], [937500, 3], [937500, 4], [937500, 5] [375000, 11], Proof. Continuing the preceding discussion, we can apply FSZtest to the 63 groups in Flist to obtain the desired result. This calculation takes approximately two days of total calculation time on the author’s computer, but can be easily split across multiple GAP instances. Most of the time is spent on the F SZ groups of orders 375,000 and 937,500.  On the other hand, we can also consider the Sylow subgroups of all 1097 available perfect groups, and test them for the F SZ property. Theorem 7.2. If G is one of the 1097 perfect groups of order less than 106 available through the GAP perfect groups library, then the following are equivalent. (1) (2) (3) (4) G G G G is not F SZ. has a non-F SZ Sylow subgroup. has a non-F SZ5 Sylow 5-subgroup. is not F SZ5 . Proof. Most of the GAP calculations we need to perform now are quick, and the problem is easily broken up into pieces, should it prove difficult to compute everything at once. The most memory intensive case requires about 1.7 GB to test. With significantly more memory available than this, the cases can simply be tested by FSZtest en masse, which will establish the result relatively quickly—a matter of hours. We sketch the details here and leave it to the interested reader to construct the relevant code. Recall that it is generally worthwhile to convert p-groups into polycyclic groups in GAP via IsomorphismPcGroup. Let Glist be constructed in GAP as before. Running over each perfect group, we can easily construct their Sylow subgroups. We can then use IMMtests from Section 3 to eliminate most cases. There are 256 Sylow subgroups, each from a distinct perfect group, for which IMMtests is inconclusive; and there are exactly 4 cases where IMMtests definitively shows the non-F SZ property, which are precisely the Sylow 5-subgroups of each of the non-F SZ perfect groups of order 375,000. These 4 Sylow subgroups are all non-F SZ5 . We can also apply FSZtestZ to the Sylow 5-subgroups of the non-F SZ perfect groups of order 937,500 to conclude that they are all non-F SZ5 . All other Sylow subgroups remaining that come from a perfect group of order less than 937,500 can be shown to be F SZ by applying FSZtest without difficulty. Of the three remaining Sylow subgroups, one has a direct factor of Z5 , and the other factor is easily tested and shown to be F SZ, whence this Sylow subgroup is F SZ. This leaves two other cases, which are the Sylow 5-subgroups of the perfect groups with ids [937500,7] and [937500,8]. The second of these is easily shown to be FSZ by FSZtest. The first can also be 26 MARC KEILBERG tested by FSZtest, but this is the case that requires the most memory and time— approximately 15 minutes and the indicated 1.7 GB. In this case as well the Sylow subgroups are F SZ. This completes the proof.  References [1] John N. Bray and Robert A. Wilson. Explicit representations of maximal subgroups of the monster. Journal of Algebra, 300(2):834 – 857, 2006. ISSN 0021-8693. doi: http://dx.doi.org/10.1016/j.jalgebra.2005.12.017. URL http://www.sciencedirect.com/science/article/pii/S0021869305007313. [2] Pavel Etingof. On some properties of quantum doubles of finite groups. Journal of Algebra, 394:1 – 6, 2013. ISSN 00218693. doi: http://dx.doi.org/10.1016/j.jalgebra.2013.07.004. URL http://www.sciencedirect.com/science/article/pii/S0021869313003529. [3] GAP. GAP – Groups, Algorithms, and Programming, Version 4.8.4. http://www.gap-system.org, Jun 2016. [4] M. Iovanov, G. Mason, and S. Montgomery. 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ArXiv e-prints, April 2016. in preparation. THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS 27 [14] Peter Schauenburg. Some quasitensor autoequivalences of drinfeld doubles of finite groups. Journal of Noncommutative Geometry, 11:51–70, 2017. doi: 10.4171/JNCG/11-1-2. URL http://dx.doi.org/10.4171/JNCG/11-1-2. [15] R. A. Wilson, R. A. Parker, S. Nickerson, J. N. Bray, and T. Breuer. AtlasRep, a gap interface to the atlas of group representations, Version 1.5.1. http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep, Mar 2016. Refereed GAP package. [16] Robert Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel Abbott. ATLAS of Finite Group Representations - Version 3. URL http://brauer.maths.qmul.ac.uk/Atlas/v3/. Accessed: October 3, 2016. E-mail address: keilberg@usc.edu 

1 Multiple Right-Hand Side Techniques in Semi-Explicit Time Integration Methods for Transient Eddy Current Problems arXiv:1611.06721v2 [] 23 Sep 2017 Jennifer Dutiné∗ , Markus Clemens∗ , and Sebastian Schöps† ∗ Chair of Electromagnetic Theory, University of Wuppertal, 42119 Wuppertal, Germany † Graduate School CE, Technische Universität Darmstadt, 64293 Darmstadt, Germany Abstract—The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential algebraic equation system into a finitely stiff ordinary differential equation (ODE) system by application of a generalized Schur complement for nonconducting parts. The ODE can be integrated in time using explicit time integration schemes, e.g. the explicit Euler method. This requires the repeated evaluation of a pseudo-inverse of the discrete curlcurl matrix in nonconducting material by the preconditioned conjugate gradient (PCG) method which forms a multiple right-hand side problem. The subspace projection extrapolation method and proper orthogonal decomposition are compared for the computation of suitable start vectors in each time step for the PCG method which reduce the number of iterations and the overall computational costs. ∂Ω Ωc Ωs Ωn Fig. 1. Computational domain Ω split into three regions: conductive and nonlinearly permeable (Ωc ), nonconductive with constant permeability (Ωn ) and nonconductive with excitation (Ωs ). Index Terms—Differential equations, eddy currents. I. I NTRODUCTION I N the design process of transformers, electric machines, etc., simulations of magnetoquasistatic field problems are an important tool. In particular in multi-query scenarios, as needed e.g. in the case of uncertainty quantification or optimization, using efficient and fast algorithms is important. The spatial discretization of the magnetic vector potential formulation of eddy current problems yields an infinitely stiff differential-algebraic equation system of index 1 (DAE). It can only be integrated in time using implicit time integration schemes, as e.g. the implicit Euler method, or singly diagonal implicit Runge-Kutta schemes [1], [2]. Due to the nonlinear B-H-characteristic of ferromagnetic materials large nonlinear equation system have to be linearized, e.g. by the NewtonRaphson method, and resolved in every implicit time step. At least one Newton-Raphson iteration is required per time step. The Jacobian and the stiffness matrix have to be updated in every iteration. A linearization within each time step is avoided if explicit time integration methods are used. First approaches for this were published in [3] and [4], where different methods are used in the conductive and nonconductive regions respectively. In [3], the Finite Difference Time Domain (FDTD) method is applied in the conductive regions, while the solution in the nonconductive regions is computed using the Boundary Element Method (BEM) [3]. In [4] an explicit time integration method and the discontinuous Galerkin finite element method Corresponding author: J. Dutiné (email: dutine@uni-wuppertal.de) (DG-FEM) are applied in conductive materials, while the finite element method based on continuous shape functions and an implicit time integration scheme are used in nonconductive domains [4]. In another recent approach, a similar DG-FEM explicit time stepping approach is used for an H − Φ formulation of the magnetoquasistatic field problem [5]. This work is based on an approach originally presented in [6], where the magnetoquasistatic DAE based on an ~ ∗ −field formulation is transformed into a finitely stiff orA dinary differential equation (ODE) system by applying a generalized Schur complement. The structure of this paper is as follows: Section II introduces the mathematical formulation of the eddy current problem and the transformation to an ordinary differential equation. In Section III the time stepping and the resulting multiple right-hand side problem are discussed. Here, also the use of the subspace projection extrapolation method and of the proper orthogonal decomposition method as multiple righthand side techniques is described. In Section IV the simulation results for validating the presented approach and the effect of subspace projection extrapolation method and of the proper orthogonal decomposition method on a nonlinear test problem are presented. The main results of this paper are summarized in Section V. II. M ATHEMATICAL F ORMULATION ~ ∗ −formulation is given The eddy current problem in the A by the partial differential equation     ~ ∂A ~ ∇×A ~ = ~JS , κ +∇× ν ∇×A (1) ∂t c 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 2 ~ is the time-dependent where κ is the electrical conductivity, A magnetic vector potential, ν is the reluctivity that can be ~ S iS (t), where nonlinear in ferromagnetic materials and ~JS = X ~ S distributes iS (t) is the time-dependent source current and X the current density spatially. Furthermore, initial values and boundary conditions are needed. The weak formulation of (1) ~ leads to the variational problem: find A Z ~ ∂A w ~ ·κ dΩ + ∂t Ω Z Z   ~ ∇×A ~ dΩ = ∇×w ~ ·ν ∇×A w ~ · J~s dΩ Ω Ω for all w ~ ∈ H0 (curl, Ω) where we denote the spatial domain with Ω and assume Dirichlet conditions at the boundary ∂Ω, see Fig. 1. Discretization and choosing test and ansatz functions from the same space according to Ritz-Galerkin, i.e., ~ r, t) ≈ A(~ N dof X w ~ i (~r)ai (t) where Mc is the conductivity matrix, Kc is the nonlinear curl-curl reluctivity matrix in conducting regions, Kn is the typically constant curl-curl matrix in nonconducting regions, Kcn is a coupling matrix, and jSn is the source current typically defined in the nonconducting domain only. The conductivity matrix in (3) is not invertible and therefore the problems consists of differential-algebraic equations (DAEs). The numerical solution of these systems is more difficult than in the case of ordinary differential equations (ODEs). The level of difficulty is measured by the DAE index, which can be roughly interpreted as the number of differentiations needed to obtain an ODE from the DAE [1]. System (3) is essentially an index-1 DAE with the speciality that the algebraic constraint, i.e., the second equation in (3), is formally not uniquely solvable for an without defining a gauge condition due to the nullspace of the discrete curl-curl operator Kn . However, it is well known that many iterative solvers have a weak gauging property, e.g. [7], such that a formal regularization can be avoided. Relying on this weak gauging property, the generalized Schur complement (4) where K+ n represents a pseudo-inverse of Kn in matrix form, is applied to (3) and transforms the DAE into d ac + KS (ac )ac dt an = −Kcn K+ n js,n , (7) are computed in the m-th time step, where ∆t is the time step size. The Courant-Friedrich-Levy (CFL) criterion determines the maximum stable time step size of explicit time integration methods [1]. For the explicit Euler method ∆t ≤ 2
λmax M−1 c KS (ac ) (8) is an estimation for the maximum stable time step size, where λmax is the maximum eigenvalue [8]. The maximum eigenvalue can be estimated using the power method [10]. III. M ULTIPLE R IGHT-H AND S IDE P ROBLEM leads to a spatially discretized symmetric equation system in time domain. Separation of the degrees of freedom (dofs) into two vectors ac storing the dofs allocated in conducting regions (if ~r ∈ Ωc ) and an holding the dofs allocated in nonconducting regions (if ~r ∈ Ωn ∪ Ωs ) yields the DAE system          Kc (ac ) Kcn ac 0 Mc 0 d ac + = , (3) jSn 0 0 dt an Kn an K> cn Mc + m + > m am n = Kn js,n − Kn Kcn ac (2) i=0 > KS (ac ) := Kc (ac ) − Kcn K+ n Kcn , evaluated using the preconditioned conjugate gradient method (PCG) [9]. The finitely stiff ODE (5a) can be integrated explicitly in time, e.g. by using the explicit Euler method. Using this time integration method, the expressions   m−1 m m−1 m−1 am +∆tM−1 Kcn K+ )ac , (6) c = ac c n js,n −KS (ac (5a) + > = K+ n js,n − Kn Kcn ac . (5b) A regularization of Kn by a grad-div or tree/cotree gauging can be used alternatively [6], [8]. Here, the pseudo-inverse is As the matrix Kn remains constant within each explicit time step, the repeated evaluation of a pseudo-inverse K+ n in (6), (7) forms a multiple right-hand side (mrhs) problem of the form Kn ap = jp ⇔ ap = K+ (9) n jp . Here, jp represents one of the right-hand side vectors > m−1 > m . Instead of computing the matrix jm s,n , Kcn ac , and Kcn ac + Kn explicitly, a vector ap is computed according to (9) using the preconditioned conjugate gradient (PCG) method [9]. Improved start vectors for the PCG method can be obtained by the subspace projection extrapolation (SPE) method or the proper orthogonal decomposition (POD) method. In the SPE method, the linearly independent column vectors of a matrix USPE are formed by a linear combination of an orthonormalized basis of the subspace spanned by solutions ap from previous time steps. The modified Gram-Schmidt method is used for this orthonormalization procedure [11]. The improved start vector x0,SPE is then computed by [12]
−1 > x0,SPE := USPE U> USPE K> (10) SPE Kn USPE cn jp . As only the last column vector in the matrix USPE changes in every time step, all other matrix-column-vector products in computing Kn USPE in (10) are reused from previous time steps in a modification of the procedure in [12] referred to as the ”Cascaded SPE” (CSPE) [9]. When using the POD method for the PCG start vector generation, NPOD solution vectors from previous time steps form the column vectors of a snapshot matrix X which is decomposed into X = UΣV> (11) using the singular value decomposition (SVD) [13], [14], [15]. Here, U and V are orthonormal matrices and Σ is a diagonal matrix of the singular values ordered by magnitude (σi ≥ σj for i < j). The index k is chosen such that the information of the largest singular values is kept σk ≤ εPOD . (12) σ1 3 The threshold value εPOD is here chosen as εPOD := 10−4 . A measure how much information is kept can be computed by the relative information criterion [16] k P σi i=1 NP POD ! ≈ 1. (13) σi i=1 Defining UPOD = [U:,1 , ... , U:,k ] as the first k columns of U allows to compute an improved start vector x0,POD by  −1 > x0,POD := UPOD U> UPOD K> POD Kn UPOD cn jp . (14) The repeated evaluation of M−1 in (6) also forms a mrhs c problem, and both the POD and the CSPE method can be used for computing improved start vectors for the PCG method. In the case of small matrix dimensions of the regular matrix Mc , the inverse can also be computed directly using GPUacceleration. IV. N UMERICAL VALIDATION The ferromagnetic TEAM 10 benchmark problem is used for numerical validation of the presented explicit time integration scheme for magnetoquasistatic fields [17]. The domain consists of two square-bracket-shaped steel plates opposite of each other and a rectangular steel plate between them, resulting in two 0.5 mm wide air gaps. The model geometry is shown in Fig. 2. The position where the magnetic field is evaluated is marked as S1. The excitation current iS = (1 − exp(−t/τ )), where τ = 0.5 s, is applied for a time interval of 120 ms starting at t = 0 s [17]. The resulting magnetic flux density is computed for this time interval. The finite element method (FEM) using 1st order edge elements is used for the spatial discretization [18]. All simulations are computed on a workstation with an Intel Xeon E5 processor and an NVIDIA TESLA K80 GPU. The conjugate gradient method is preconditioned by an algebraic multigrid method [19]. The matrix Mc is inverted using the Magmalibrary and GPU-acceleration [20]. A fine mesh resulting in about 700,000 dofs and the implicit Euler method are used to validate the simulation code. A good agreement between the measured results published in [17] and the simulation of this fine spatial discretization is shown in Fig. 2. The required simulation time of this simulation using the implicit Euler method is 5.38 days using an in-house implicit time integration magnetoquasistatic code. For benchmarking the proposed mrhs techniques for the (semi-)explicit time integration scheme, a model with a coarse spatial discretization yielding about 30,000 dofs and the explicit Euler method is used. For this spatial discretization, the resulting maximum stable time step size according to (8) is ∆tCFL = 1.2 µs. Both meshes are presented in Fig. 3 The results for the average magnetic flux density are compared with the results obtained using the same discretization in space and the implicit Euler method for time integration and show good agreement, depicted in Fig. 2. The resulting field plots for both spatial discretizations are shown in Fig. 4. The simulation time for the implicit time integration method is still 2.58 h. Fig. 2. Comparison of results for the average magnetic flux density evaluated at position S1 and model geometry as inset. TABLE I S IMULATION T IME AND AVERAGE N UMBER OF PCG I TERATIONS U SING D IFFERENT S TART V ECTORS x0 Start vector Avg. Number of PCG Iterations x0 := am−1 p 3.16 Simulation Time 2.35 h x0 := x0,POD 2.18 17.35 h x0 := x0,CSPE 1.02 1.62 h The effect of computing improved start vectors using POD or CSPE on the average number of PCG iterations and on the simulation time is compared to using the solution from as start vector for the PCG the previous time step am−1 p method. An overview is presented in Table I and shows that both the CSPE and the POD start vector generation methods significantly reduce the number of PCG iterations. When using CSPE the number of column vectors in the operator USPE in (10) is increased during the simulation to improve the spectral information content of USPE . This number remains below 20. Thus, only small systems have to be solved for the inversion in (10) and the effort to perform all computations of the CSPE method is low. This is also confirmed by the simulation time which is shortest when using CSPE. The simulation time resulting from using explicit time integration and CSPE for start vector generation is 63 % of the simulation time of the implicit reference simulation. A bar plot showing the reduced simulation time by using the explicit Euler scheme and CSPE compared to using the standard formulation and the implicit Euler method for time integration is depicted in Fig. 5. In case of the POD, the amount of information kept according to (13) is > 0.99 during the entire simulation. However, the computational effort for performing the SVD and the computations in (14) is higher than the effort for CSPE. Although the number of PCG iterations is further decreased, the simulation time resulting from using POD for start vector generation is higher than when using apm−1 as start vector for the PCG method due to the costs of the repeated SVD. 4 using the POD and the CSPE method. Although both reduce the number of PCG iterations needed, the computational effort of the CSPE is significantly lower than for the POD method. Reducing the computational effort of the POD, e.g. by accelerating the computation of the SVD is subject to further investigations. Using the CSPE method, the overall simulation time was reduced by 37 % compared to the simulation time of the implicit reference simulation. Fig. 3. Meshes resulting in about 700,000 dofs (left) and in about 30,000 dofs (right). ACKNOWLEDGMENT This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant numbers CL143/11-1 and SCHO1562/1-1. The third author is supported by the Excellence Initiative of the German Federal and State Governments and The Graduate School of CE at TU Darmstadt. R EFERENCES | B |/ T 0.00 0.35 0.70 1.05 1.50 ~ for the spatial discretization Fig. 4. Field plots of the magnetic flux density B with about 700,000 dofs (left) and with about 30,000 dofs (right). Fig. 5. Comparison of simulation times. V. C ONCLUSION The magnetic vector potential formulation of eddy current problems was transformed into an ODE system of finite stiffness using a generalized Schur complement. The resulting ODE system was integrated in time using the explicit Euler method. A pseudo-inverse of the singular curl-curl matrix in nonconducting material was evaluated using the PCG method. Improved start vectors for the PCG method were calculated [1] E. Hairer and G. Wanner, Solving ordinary differential equations II: stiff and differential-algebraic problems, 2nd rev. ed., Springer, Berlin, 1996. [2] A. 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Magn., vol. 35, pp. 1163-1166, May 1999. [8] S. Schöps, A. Bartel, and M. Clemens, ”Higher order half-explicit time integration of eddy current problems,” IEEE Trans. Magn., vol. 48, pp. 623-626, Feb. 2012. [9] J. Dutiné, M. Clemens, S. Schöps, and G. Wimmer, ”Explicit time integration of transient eddy current problems,” presented at the 10th International Symposium on Electric and Magnetic Fields (EMF) 2016. Full paper to appear in J. Num. Mod.: ENDF. [10] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., The Johns Hopkins University Press, Baltimore, 2013. [11] L. N. Trefethen and D. Bau, Numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, 1997. [12] M. Clemens, M. Wilke, R. Schuhmann, and T. Weiland, ”Subspace projection extrapolation scheme for transient field simulations,” IEEE Trans. Magn., vol. 40, pp. 934–937, Apr. 2004. [13] T. Henneron and S. Clénet, ”Model order reduction of non-linear magnetostatic problems based on POD and DEI methods,” IEEE Trans. Magn., vol. 50, pp. 33–36, Feb. 2014. [14] T. Henneron and S. Clénet, ”Model-order reduction of multiple-input non-linear systems based on POD and DEI methods,” IEEE Trans. Magn., vol. 51, pp. 1–4, Mar. 2015. [15] Y. Sato, M. Clemens, and H. Igarashi, ”Adaptive subdomain model order reduction with discrete empirical interpolation method for nonlinear magneto-quasi-static problems,” IEEE Trans. Magn., vol. 52, pp. 1–4, Mar. 2016. [16] D. Schmidthäusler, S. Schöps, and M. Clemens, ”Linear subspace reduction for quasistatic field simulations to accelerate repeated computations,” IEEE Trans. Magn., vol. 50, pp. 421–424, Feb. 2014. [17] T. Nakata and K. Fujiwara, ”Results for benchmark problem 10 (steel plates around a coil),” Compel, vol. 9, pp. 181–192, 1990. [18] A. Kameari, ”Calculation of transient 3d eddy current using edgeelements,” IEEE Trans. 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1 Deep Local Binary Patterns arXiv:1711.06597v1 [] 17 Nov 2017 Kelwin Fernandes and Jaime S. Cardoso Abstract—Local Binary Pattern (LBP) is a traditional descriptor for texture analysis that gained attention in the last decade. Being robust to several properties such as invariance to illumination translation and scaling, LBPs achieved stateof-the-art results in several applications. However, LBPs are not able to capture high-level features from the image, merely encoding features with low abstraction levels. In this work, we propose Deep LBP, which borrow ideas from the deep learning community to improve LBP expressiveness. By using parametrized data-driven LBP, we enable successive applications of the LBP operators with increasing abstraction levels. We validate the relevance of the proposed idea in several datasets from a wide range of applications. Deep LBP improved the performance of traditional and multiscale LBP in all cases. xref xref xref (a) r=1, n=8 (b) r=2, n=16 (c) r=2, n=8 Figure 1: LBP neighborhoods with radius (r) and angular resolution (n). The first two cases use Euclidean distance to define the neighborhood, the last case use Manhattan distance. Index Terms—Local Binary Patterns, Deep Learning, Image Processing. • I. I NTRODUCTION I N recent years, computer vision community has moved towards the usage of deep learning strategies to solve a wide variety of traditional problems, from image enhancement [1] to scene recognition [2]. Deep learning concepts emerged from traditional shallow concepts from the early years of computer vision (e.g. filters, convolution, pooling, thresholding, etc.). Although these techniques have achieved state-of-the-art performance in several of tasks, the deep learning hype has overshadowed research on other fundamental ideas. Narrowing the spectrum of methods to a single class will eventually saturate, creating a monotonous environment where the same basic idea is being replicated over and over, and missing the opportunity to develop other paradigms with the potential to lead to complementary solutions. As deep strategies have benefited from traditional -shallowmethods in the past, some classical methods started to take advantage of key deep learning concepts. That is the case of deep Kernels [3], which explores the successive application of nonlinear mappings within the kernel umbrella. In this work, we incorporate deep concepts into Local Binary Patterns [4], [5], a traditional descriptor for texture analysis. Local Binary Patterns (LBP) is a robust descriptor that briefly summarizes texture information, being invariant to illumination translation and scaling. LBP has been successfully used in a wide variety of applications, including texture classification [6]–[9], face/gender recognition [10]–[15], among others [16]–[18]. LBP has two main ingredients: • The neighborhood (N ), usually defined by an angular resolution (typically 8 sampling angles) and radius r of the neighborhoods. Fig. 1 illustrates several possible neighborhoods. Kelwin Fernandes (kafc@inesctec.pt) and Jaime S. Cardoso (jaime.s.cardoso@inesctec.pt) are with Faculdade de Engenharia da Universidade do Porto and with INESC TEC, Porto, Portugal. The binarization function b(xref , xi ) ∈ {0, 1}, which allows the comparison between the reference point (central pixel) and each one of the points xi in the neighborhood. Classical LBP is applicable when xref (and xi ) are in an ordered set (e.g., R and Z), with b(xref , xi ) defined as b(xref , xi ) = (xref ≺ xi ), (1) where ≺ is the order relation on the set (interpolation is used to compute xi when a neighbor location does not coincide with the center of a pixel). The output of the LBP at each position ref is the code resulting from the comparison (binarization function) of the value xref with each of the xi in the neighborhood, with i ∈ N (ref ), see Figure 2. The LBP codes can be represented by their numerical value as formally defined in (2). X LBP (xref ) = 2i · b(xref , xi ) (2) i∈|N (ref )| LBP codes can take 2|N | different values. In predictive tasks, for typical choices of angular resolution, LBP codes are compactly summarized into a histogram with 2|N | bins, being this the feature vector representing the object/region/image (see Fig. 3). Also, it is typical to compute the histograms in sub-regions and to build a predictive model by using as features the concatenation of the region histograms, being nonoverlapping and overlapping [19] blocks traditional choices (see Figure 4). In the last decade, several variations of the LBP have been proposed to attain different properties. The two best-known variations were proposed by Ojala et. al, the original authors of the LBP methodology: rotation invariant and uniform LBP [5]. Rotation invariance can be achieved by assigning a unique identifier to all the patterns that can be obtained by applying circular shifts. The new encoding is defined in (3), where 2 0 22 110 < 20 > 1 1 < > 42 < 80 80 > ⇒ 1 0 0 ⇒ 10011010 > < 0 0 1 60 1 (a) LBP code with 8 neighbors. 94 1 < 30 0 222 > ⇒ 65 > 15 1 < ⇒ 100001 > > 40 0 0 0 22 (b) LBP code with 6 neighbors. Figure 2: Cylinder and linear representation of the codes at some pixel positions. Encodings are built in a clockwise manner from the starting point indicated in the middle section of both figures. LBP encodings ⇒ Histogram e1 e5 ... e7 e5 e8 e1 ... e7 e3 e10 e1 ... e6 .. . .. . .. . .. .. . e2 ... e2 e9 e10 . ⇒ Frequency Original image ⇒ Classifier e1 e2 e3 . . . en e10 Encodings Frequency Figure 3: Traditional pipeline for image classification using LBP. e1 e2 e3 . . . en Frequency Encodings e1 e2 e3 . . . en Classifier Frequency Encodings e1 e2 e3 . . . en Frequency Encodings e1 e2 e3 . . . en Encodings Figure 4: Multi-block LBP with 2 × 2 non-overlapping blocks. ROR(e, i) applies a circular-wrapped bit-wise shift of i positions to the encoding e. LBP (xref ) = min{ROR(LBP (xref ), i) | i ∈ [0, . . . |N |)} (3) In the same work, Ojala et. al [5] identified that uniform patterns (i.e. patterns with two or less circular transitions) are responsible for the vast majority of the histogram frequencies, leaving low discriminative power to the remaining ones. Then, the relatively small proportion of non-uniform patterns limits the reliability of their probabilities, all the non-uniform patterns are assigned to a single bin in the histogram construction, while uniform patterns are assigned to individual bins. [5]. Heikkila et. al proposed Center-Symmetric LBP [20], which increases robustness on flat image areas and improves computational efficiency. This is achieved by comparing pairs of neighbors located in centered symmetric directions instead of comparing each neighbor with the reference pixel. Thus, an encoding with four bits is generated from a neighborhood of 8 pixels. Also, the binarization function incorporates an activation tolerance given by b(xi , xj ) = xi > xj + T . Further extensions of this idea can be found in [21]–[23]. Local ternary patterns (LTP) are an extension of LBP that use a 3-valued mapping instead of a binarization function. The function used by LTP is formalized at (4). LTP are less sensitive to noise in uniform regions at the cost of losing invariance to illumination scaling. Moreover, LTP induce an additional complexity in the number of encodings, producing histograms with up to 3N bins.   −1 bT (xref , xi ) = 0   +1 xi < xref − T xref − T ≤ xi ≤ xref + T xref + T < xi (4) So far, we have mainly described methods that rely on redefining the construction of the LBP encodings. A different line of research focuses on improving LBP by modifying the texture summarization when building the frequency histograms. Two examples of this idea were presented in this work: uniform LBP [5] and multi-block LBP (see Fig. 4). Since different scales may bring complementary information, one can concatenate the histograms of LBP values at different scales. Berkan et al. proposed this idea in the Over-Complete LBP (OCLBP) [19]. Besides computing the encoding at multiple scales, OCLBP computes the histograms on several spatially overlapped blocks. An alternative to this way of modeling multiscale patterns is to, at each point, compute the LPB code at different scales, concatenate the codes and summarize (i.e., compute the histogram) of the concatenated feature vector. This latter option has difficulties concerning the dimensionality of the data (potentially tackled with a bag of words approach) and the nature of the codes (making unsuitable standard k-means to find the bins-centers for a bag of words approach). Multi-channel data (e.g. RGB) has been handled in a similar way, by 1) computing and summarizing the LBP codes in each channel independently and then concatenating the histograms [24] and by 2) computing a joint code for the three channels [25]. As LBP have been successfully used to describe spatial relations between pixels, some works explored embedding temporal information on LBP for object detection and background removal [10], [26]–[30]. It is trivial to prove that the only options for the binarization function that hold Eq. (5) are the constant functions (always zeros or always ones), the one given by Eq. (1) and its reciprocal. Model Histogram ... LBP LBP LBP Image 3 Figure 5: Recursive application of Local Binary Patterns. Finally, Local Binary Pattern Network (LBPNet) was introduced by Xi et al. [31] as a preliminary attempt to embed Deep Learning concepts in LBP. Their proposal consists on using a pyramidal approach on the neighborhood scales and histogram sampling. Then, Principal Component Analysis (PCA) is used on the frequency histograms to reduce the dimensionality of the feature space. Xi et al. analogise the pyramidal construction of LBP neighborhoods and histogram sampling as a convolutional layer, where multiple filters operate at different resolutions, and the dimensionality reduction as a Pooling layer. However, LBPNets aren’t capable of aggregating information from a single resolution into higher levels of abstraction which is the main advantage of deep neural networks. In the next sections, we will bring some ideas from the Deep Learning community to traditional LBP. In this sense, we intend to build LBP blocks that can be applied recursively in order to build features with higher-level of abstraction. II. D EEP L OCAL B INARY PATTERNS The ability to build features with increasing abstraction level using a recursive combination of relatively simple processing modules is one of the reasons that made Convolutional Neural Networks – and in general Neural Networks – so successful. In this work, we propose to represent “higher order” information about texture by applying LBP recursively, i.e., cascading LBP computing blocks one after the other (see Fig 5). In this sense, while an LBP encoding describes the local texture, a second order LBP encoding describes the texture of textures. However, while it is trivial to recursively apply convolutions – and many other filters – in a cascade fashion, traditional LBP are not able to digest their own output. Traditional LBP rely on receiving as input an image with domain in an ordered set (e.g. grayscale intensities). However, LPB codes are not in an ordered set, dismissing the direct recursive application of standard LBP. As such we will first generalize the main operations supporting LBP and discuss next how to assemble together a deep/recursive LBP feature extractor. We start the discussion with the binarization function b(xref , xi ). It is instructive to think, in the conventional LBP, if nontrivial alternative functions exist to the adopted one, Eq. (1). What is(are) the main property(ies) required by the binarization function? Does it need to make use of a (potentially implicit) order relationship? A main property of the binarization function is to be independent of scaling and translation of the input data, that is, b(k1 xref + k2 , k1 xi + k2 ) = b(xref , xi ), k1 > 0. (5) Proof. Assume xi , xj > xref and b(xref , xi ) 6= b(xref , xj ). Under the independence to translation and scaling (Eq. (5)), b(xref , xi ) = b(xref , xj ) as shown below, which is a contradiction. b(xref , xi ) = h Identity of multiplication i   xj − xref xj − xref b xref , xi xj − xref xj − xref = h Identity of addition i    xj − xref xref xj − xref xj xj − xref + (xi − xi ) xref , xi + b xj − xref xj − xref xj − xref = hArithmetici   xi − xref xj − xi xi − xref xj − xi b xref + xref , xj + xref x − xref xj − xref xj − xref x − xref  j  j xi − xref xj − xi = Eq. (5), where k1 = , k2 = xref xj − xref xj − xref b(xref , xj ) Therefore, b(xref , xi ) must be equal to all xi above xref . Similarly, b(xref , xi ) must be equal to all xi below xref . Among our options, the constant binarization function is not a viable option, since the information (in information theory perspective) in the output is zero. Since the recursive application of functions can be understood as a composition, invariance to scaling and translation is trivially ensured by using a traditional LBP in the first transformation. Given that transitivity is a relevant property held by the natural ordering of real numbers, we argue that such property should be guaranteed by our binarization function. In this sense, we will focus on strict partial orders of encodings. Following, we show how to build such binarization functions for the i-th application of the LBP operator, where i > 1. We will consider both predefined/expert driven solutions and data driven solutions (and therefore, application specific). Hereafter, we will refer to the binarization function as the partial ordering of LBP encodings. Despite the existence of other types of functions may be of general interest, narrowing the search space to those that can be represented as a partial ordering induce efficient learning mechanisms. A. Preliminaries Let us formalize the deep binarization function as the order relation b+ ∈ P(EN ×EN ), where EN is the set of encodings induced by the neighborhood N . Let Φ be an oracle Φ :: P(EN ×EN ) → R that assesses the performance of a binarization function. For example, among other options, the oracle can be defined as the performance of the traditional LBP pipeline (see Fig. 3) on a dataset for a given predictive task. 4 Table I: Lower bound of the number of combinations for deciding the best LBP binarization function as partial orders. # Neighbors 2 3 4 5 6 7 8 Rotational Inv. 2 · 101 5 · 102 3 · 106 7 · 1011 1 · 1036 2 · 1072 1 · 10225 Uniform 2 · 104 1 · 1015 2 · 1041 6 · 1094 2 · 10190 3 · 10346 1 · 10585 Traditional 5 · 102 7 · 1011 8 · 1046 2 · 10179 1 · 10685 3 · 102640 3 · 1010288 2) Learning b+ from a High-dimensional Space: A second option is to map the code space to a new (higher-dimensional) space that characterizes LBP encodings. Then, an ordering or preference relationship can be learned in the extended space, for instance resorting to preference learning algorithms [36]– [38]. Some examples of properties that characterize LBP encodings are: • • B. Deep Binarization Function • From the entire space of binarization functions, we restrict our analysis to those induced by strict partial orders. Within this context, it is easy to see that learning the best binarization function by exhaustive exploration is intractable since the number of combinations equals the number of directed acyclic graph (DAG) with 2|N | = |EN | nodes. The DAG counting problem was studied by Robinson [32] and is given by the recurrence relation Eqs. (6)-(7). • a0 = 1   n X k−1 n an>1 = (−1) 2k(n−k) an−k k (6) (7) k=1 Table I illustrates the size of the search space for several numbers of neighbors. For instance, for the traditional setting with 8 neighbors, the number of combinations has more than 10,000 digits. Thereby, a heuristic approximation must be carried out. 1) Learning b+ from a User-defined dissimilarity function: The definition of a dissimilarity function between two codes seems reasonably accessible. For instance, an immediate option is to adopt the hamming distance between codes, dH . With rotation invariance in mind, one can adopt the minimum hamming distance between all circularly shifted versions of xref and xi , dri H . The circular invariant hamming distance between xref and xi can be computed as dri H = min s∈0,··· ,N −1 dH (ROR(xref , s), xi ) (8) Having defined such a dissimilarity function between pairs of codes, one can know proceed with the definition of the binarization function. Given the dissimilarity function, we can learn a mapping of the codes to an ordered set. Resorting to Spectral Embedding [33], one can obtain such a mapping. The conventional binarization function, Eq. (1), can then be applied. Other alternatives for building the mappings can be found in the manifold learning literature: Isomaps [34], Multidimensional scaling (MDS) [35], among others. In this case, the oracle function can be defined as the intrinsic loss functions used in the optimization process of such algorithms. Preserving a desired property P such as rotational invariance and sign invariance (i.e. interchangeability between ones and zeros) can be achieved by considering P -aware dissimilarities. • Number of transitions of size 1 (e.g. 101, 010). Number of groups/transitions. Size of the smallest/largest group. Diversity on the group sizes. Number of ones. Techniques to learn the final ordering based on the new high-dimensional space include: • • Dimensionality reduction techniques, including Spectral embeddings, PCA, MDS and other manifold techniques. Preference learning strategies for learning rankings [36]– [38]. A case of special interest that will be used in the experimental section of this work are Lexicographic Rankers (LR) [36], [38]. In this work, we will focus on the simplest type of LR, linear LR. Let us assume that features in the new high-dimensional space are scoring rankers (e.g. monotonic functions) on the texture complexity of the codes. Thus, for each code ei and feature sj , the complexity associated to ei by sj is denoted as sj (ei ). We assume sj (ei ) to lie in a discrete domain with a well-known order relation. Thus, each feature is grouping the codes into equivalence classes. For example, the codes with 0 transitions (i.e. flat textures), 2 transitions (i.e. uniform textures) and so on. If we concatenate the output of the scoring rankers in a linear manner (s0 (ei ), s1 (ei ), · · · , sn (ei )), a natural arrangement is their lexicographic order (see Eq. (9)), where each sj (ei ) is subordering the equivalence class obtained by the previous prefix of rankers (s0 (ei ), · · · , sj−1 (ei )).  a=b    a ≺ b LexRank(a, b) =  ab    LexRank(t(a), t(b)) , |a| = 0 ∨ |b| = 0 , a0 ≺ b0 , a0  b0 , a0 = b0 (9) where t(a) returns the tail of the sequence. Namely, the order between two encodings is decided by the first scoring ranker in the hierarchy that assigns different values to the encodings. Therefore, the learning process is reduced to find the best feature arrangement. A heuristic approximation to this problem can be achieved by iteratively appending to the sequence of features the one that maximizes the performance of the oracle Φ. Similarly to property-aware dissimilarity functions, if the features in the new feature vector V(x) are invariant to P , the P -invariance of the learned binarization function is automatically guaranteed. 5 Image LBP LBP-b+ 1 ··· LBP-b+ n Histogram Model Figure 6: Deep LBP. (a) Input image (b) LBP (c) DeepLBP(1) (d) DeepLBP(2) (e) DeepLBP(3) (f) DeepLBP(4) III. D EEP A RCHITECTURES Given the closed form of the LBP with deep binarization functions, their recursive combination seems feasible. In this section, several alternatives for the aggregation of deep LBP operators are proposed. A. Deep LBP (DLBP) The simplest way of aggregating Deep LBP operators is by applying them recursively and computing the final encoding histograms. Figure 6 shows this architecture. The first transformation is done by a traditional shallow LBP while the remaining transformations are performed using deep binarization functions. Figure 7 illustrates the patterns detected by several deep levels on a cracker image from the Brodatz database. In this case, the ordering between LBP encodings was learned by using a lexicographic ordering of encodings on the number of groups, the size of the largest group and imbalance ratio between 0’s and 1’s. We can observe that the initial layers of the architecture extract information from local textures while the later layers have higher levels of abstraction. B. Multi-Deep LBP (MDLBP) Despite it may be a good idea to extract higher-order information from images, for the usual applications of LBP, it is important to be able to detect features at different levels of abstraction. For instance, if the image has textures with several complexity levels, it may be relevant to keep the patterns extracted at different abstraction levels. Resorting to the techniques employed in the analysis of multimodal data [40], we can aggregate this information in two ways: feature and decision-level fusion. 1) Feature-level fusion: one histogram is computed at each layer and the model is built using the concatenation of all the histograms as features. 2) Decision-level fusion: one histogram and decision model is computed at each layer. The final model uses the probabilities estimated by each individual model to produce the final decision. Figures 8a and 8b show Multi-Deep LBP architectures with feature-level and decision-level fusion respectively. In our experimental setting, feature-level fusion was used. Figure 7: Visualization of LBP encodings from a Brodatz database [39] image. The results obtained by applying n layers of Deep LBP operators are denoted as DeepLBP(n). A neighborhood of size 8, radius 10 and Euclidean distance was used. The grayscale intensity is defined by the order of the equivalence classes. Image LBP LBP-b+ 1 Histogram Histogram-1 ... LBP-b+ n Histogram-n .. . Model (a) Multi-Deep LBP with feature-level fusion. Image LBP LBP-b+ 1 Histogram Histogram-1 ... LBP-b+ n Histogram-n .. . Model-0 Model-1 Model Model-n Global Model (b) Multi-Deep LBP with decision-level fusion. Figure 8: Deep LBP architectures. C. Multiscale Deep LBP (Multiscale DLBP) In the last few years, deep learning approaches have benefited from multi-scale architectures that are able to aggregate information from different image scales [41]–[43]. Despite being able to induce higher abstraction levels, deep networks 6 Scale 1 Image Deep LBP1 Scale 2 Image Deep LBP2 .. . ... Image Deep LBPn (a) KTH TIPS (b) FMD (c) Virus (d) Brodatz (e) Kylberg (f) 102 Flowers Model Scale n Figure 9: Multi-scale Deep LBP. Table II: Summary of the datasets used in the experiments Dataset KTH TIPS FMD Virus Brodatz* Kylberg 102 Flowers Caltech 101 Reference [44] [45] [46] [39] [47] [48] [49] Task Texture Texture Texture Texture Texture Object Object Images 810 1000 1500 1776 4480 8189 9144 Classes 10 10 15 111 28 102 102 are restrained to the size of the individual operators. Thereby, aggregating multi-scale information in deep architectures may exploit their capability to detect traits that appear at different scales in the images in addition to turning the decision process scale invariant. In this work, we consider the stacking of independent deep architectures at several scales. The final decision is done by concatenating the individual information produced at each scale factor (cf. Figure 9). Depending on the fusion approach, the final model operates in different spaces (i.e. feature or decision level). In an LBP context, we can define the scale operator of an image by resizing the image or by increasing the neighborhood radius. IV. E XPERIMENTS In this section, we compare the performance of the proposed deep LBP architectures against shallow LBP versions. Several datasets were chosen from the LBP literature covering a wide range of applications, from texture categorization to object recognition. Table II summarizes the datasets used in this work. Also, Fig. 10 shows an image per dataset in order to understand the task diversity. We used a 10-fold stratified cross-validation strategy and the average performance of each method was measured in terms of: • Accuracy. (g) Caltech 101 Figure 10: Sample images from each dataset • Class rank: Position (%) of the ground truth label in the ranking of classes ordered by confidence. The ranking was induced using probabilistic classifiers. While high values are preferred when using accuracy, low values are preferred for class rank. All the images were resized to have a maximum size of 100 × 100 and the neighborhood used for all the LBP operators was formed by 8 neighbors on a radius of size 3, which proved to be a good configuration for the baseline LBP. The final features were built using a global histogram, without resorting to image blocks. Further improvements in each application can be achieved by finetuning the LBP neighborhoods and by using other spatial sampling techniques on the histogram construction. Since the objective of this work was to objectively compare the performance of each strategy, we decided to fix these parameters. The final decision model is a Random Forest with 1000 trees. In the last two datasets, which contain more than 100 classes, the maximum depth of the decision trees was bounded to 20 in order to limit the required memory. In all our experiments, training data was augmented by including vertical and horizontal flips. 7 Table III: Class rank (%) of the ground-truth label and accuracy with single-scale strategies Dataset KTH TIPS FMD Virus Brodatz Kylberg 102 Flowers Caltech 101 Strategy Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim LBP Similarity High Dim 1 1.55 26.96 8.08 0.25 0.23 13.46 13.05 - Class Rank Layers 2 3 4 19.18 18.87 18.26 18.64 18.78 18.79 1.60 1.68 1.82 0.94 0.91 1.09 25.77 25.79 25.61 23.20 23.36 23.30 6.61 6.78 6.72 6.65 6.50 6.53 0.22 0.22 0.23 0.21 0.21 0.22 0.18 0.16 0.14 0.07 0.07 0.07 13.34 13.56 13.99 13.10 12.99 13.15 12.37 12.23 12.32 11.98 12.19 12.16 A. Single-scale First, we validated the performance of the proposed deep architectures on single scale settings with increasing number of deep layers. Information from each layer was merged at a feature level by concatenating the layerwise histogram (c.f. Section III-B). Table III summarizes the results of this setting. In all the datasets, the proposed models surpassed the results achieved by traditional LBP. Furthermore, even when the accuracy gains are small, the large gains in terms of class rank suggest that the deep architectures induce more stable models, which assign a high probability on the ground-truth level, even on misclassified cases. For instance, in the Kylberg dataset, a small relative accuracy gain of 3.23% was achieved by the High Dimensional rule, the relative gain on the class rank was 69.56%. With a few exceptions (e.g. JAFFE dataset), the data-driven deep operator based on a high dimensional projection achieved the best performance. Despite the possibility to induce encoding orderings using user-defined similarity functions, the final orderings are static and domain independent. In this sense, more flexible data-driven approaches as the one suggested in Section II-B2 are able to take advantage of the dataset-specific properties. Despite the capability of the proposed deep architectures to achieve large gain margins, the deep LBP operators saturate rapidly. For instance, most of the best results were found on architectures with up to three deep layers. Further research on aggregation techniques to achieve higher levels of abstraction should be conducted. For instance, it would be interesting to explore efficient non-parametric approaches for building encoding orderings that allow more flexible data-driven optimization. 5 18.80 19.10 1.86 1.11 25.59 23.50 6.73 6.55 0.25 0.23 0.14 0.07 14.29 13.32 12.38 12.34 1 89.22 29.20 56.80 89.23 95.29 23.18 39.71 - 2 53.28 53.28 88.96 92.96 28.90 33.40 61.00 61.53 89.73 90.72 96.14 98.37 25.59 24.56 40.35 41.45 Accuracy Layers 3 53.28 53.28 88.57 93.58 30.00 32.60 61.33 61.27 90.23 90.72 96.52 98.35 24.46 23.76 40.07 40.78 4 53.28 53.28 86.99 92.72 30.90 33.30 60.93 61.47 90.50 90.09 96.72 98.26 24.92 22.81 39.74 40.56 5 53.28 53.28 87.97 92.36 30.80 33.00 61.93 62.27 90.36 89.59 96.81 98.24 24.58 22.36 39.81 40.43 B. Multi-Scale A relevant question in this context is if the observed gains are due to the higher abstraction levels of the deep LBP encodings or to the aggregation of information from larger neighborhoods. Namely, when applying a second order operator, the neighbors of the reference pixel include information from their own neighborhood which was initially out of the scope of the local LBP operator. Thereby, we compare the performance of the Deep LBP and multiscale LBP. In order to simplify the model assessment, we fixed the number of layers to 3 in the deep architectures. A scaling factor of 0.5 was used on each layer of the pyramidal multiscale operator. Guided by the results achieved in the single-scale experiments, the deep operator based on the lexicographic sorting of the high-dimensional feature space was used in all cases. Table IV summarizes the results on the multiscale settings. In most cases, all the deep LBP architectures surpassed the performance of the best multiscale shallow architecture. Thereby, the aggregation level achieved by deep LBP operators goes beyond a multiscale analysis, being able to address meta-texture information. Furthermore, when combined with a multiscale approach, deep LBP achieved the best results in all the cases. C. LBPNet Finally, we compare the performance of our deep LBP architecture against the state of the art LBPNet [31]. As referred in the introduction, LBPNet uses LBP encodings at different neighborhood radius and histogram sampling in order to simulate the process of learning a bag of convolutional filters in deep networks. Then, the dimensionality of the descriptors are reduced by means of PCA, resorting to the idea of pooling layers from Convolutional Neural Networks 8 Table IV: Class rank (%) of the ground-truth label and accuracy with multi-scale strategies Dataset KTH TIPS FMD Virus Brodatz Kylberg 102 Flowers Caltech 101 Strategy Shallow Deep Shallow Deep Shallow Deep Shallow Deep Shallow Deep Shallow Deep Shallow Deep Class Rank Scales 1 2 3 1.55 1.17 1.22 0.91 0.79 0.62 26.96 26.31 26.32 23.36 23.54 23.77 8.08 7.51 7.97 6.50 5.92 6.04 0.25 0.20 0.23 0.21 0.13 0.13 0.23 0.13 0.12 0.07 0.05 0.04 13.46 13.10 12.79 12.99 12.68 12.71 12.92 12.46 12.28 12.21 11.74 11.60 Accuracy Scales 1 2 3 89.22 90.94 90.93 93.58 94.21 94.96 29.20 29.60 29.80 32.60 33.20 33.00 56.80 60.60 58.60 61.27 66.13 64.87 89.23 90.77 90.00 90.72 92.97 93.11 95.29 97.34 97.57 98.35 98.84 98.95 23.18 25.10 26.40 23.76 26.02 26.87 40.07 40.84 41.03 40.68 41.67 42.01 Table V: Class rank (%) of the ground-truth label and accuracy Dataset KTH TIPS FMD Virus Brodatz Kylberg (CNN). However, the output of a LBPNet cannot be used by itself in successive calls of the same function. Thereby, it is uncapable of building features with higher expresiveness than the individual operators. In our experiments, we considered the best LBPNet with up to three scales and histogram computations with nonoverlapping histograms that divide the image in 1 × 1, 2 × 2 and 3 × 3 blocks. The number of components kept in the PCA transformation was chosen in order to retain 95% of the variance for most datasets with the exception of 102 Flowers and Caltech, where a value of 99% was chosen due to poor performance of the previous value. A global histogram was used in our deep LBP architecture Table V summarizes the results obtained by multiscale LBP (shallow), LBPNet and our proposed deep LBP. In order to understand if the gains achieved by the LBPNet are due to the overcomplete sampling or to the PCA transformation preceeding the final classifier, we validated the performance of our deep architecture with a PCA transformation on the global descriptor before applying the Random Forest classifier. Despite being able to surpass the performance of our deep LBP without dimensionality reduction, LBPNet did not improve the results obtained by our deep architecture with PCA in most cases. In this sense, even whithout resorting to local descriptors on the histogram sampling, our model was able to achieve the best results within the family of LBP methods. The only exception was observed in the 102 Flowers dataset (see Fig. 10f), where the spatial information can be relevant. It is important to note that our model can also benefit from using spatial sampling of the LBP activations. Moreover, deep learning concepts such as dropout and pooling layers can be introduced within the Deep LBP architectures in a straightforward manner. V. C ONCLUSIONS Local Binary Patterns have achieved competitive performance in several computer vision tasks, being a robust and easy to compute descriptor with high discriminative power on a wide spectrum of tasks. In this work, we proposed Deep Local Binary Patterns, an extension of the traditional LBP that 102 Flowers Caltech 101 Strategy Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Shallow LBPNet Deep LBP Deep LBP (PCA) Class Rank 1.17 0.43 0.62 0.16 26.31 25.71 23.36 23.20 7.51 7.18 5.92 5.91 0.20 0.20 0.13 0.12 0.12 0.19 0.04 0.02 12.79 9.61 12.68 22.30 12.46 12.11 11.60 10.87 Accuracy 90.94 96.29 94.96 98.39 29.80 30.00 33.20 32.30 60.60 60.73 66.13 65.60 90.77 91.49 93.11 94.46 97.57 95.80 98.95 99.55 26.40 35.56 26.87 8.80 41.03 42.69 42.01 45.14 allow successive applications of the operator. By applying LBP in a recursive way, features with higher level of abstraction are computed that improve the descriptor discriminability. The key aspect of our proposal is the introduction of flexible binarization rules that define an order relation between LBP encodings. This was achieved with two main learning paradigms. First, learning the ordering based on a userdefined encoding similarity metric. Second, allowing the user to describe LBP encodings on a high-dimensional space and learning the ordering on the extended space directly. Both ideas improved the performance of traditional LBP in a diverse set of datasets, covering various applications such as face analysis, texture categorization and object detection. As expected, the paradigm based on a projection to a high-dimensional space achieved the best performance, given its capability of using application specific knowledge in an efficient way. The proposed deep LBP are able to aggregate information from local neighborhoods into higher abstraction levels, being able to surpass the performance obtained by multiscale LBP as well. While the advantages of the proposed approach were demonstrated in the experimental section, further research can be conducted on several areas. For instance, it would be interesting to find the minimal properties of interest that should be guaranteed by the binarization function. In this work, since we are dealing with intensity-based image, we restricted our analysis to partial orderings. However, under the presence of other types of data such as directional (i.e. periodic, angular) data, cycling or local orderings could be more suitable. In the most extreme case, the binarization function may be arbitrarily complex without being restricted to strict orders. On the other hand, constraining the shape of the binarization 9 function allows more efficient ways to find suitable candidates. In this sense, it is relevant to explore ways to improve the performance of the similarity-based deep LBP. Two possible options would be to refine the final embedding by using training data and allowing the user to specify incomplete similarity information. In this work, each layer was learned in a local fashion, without space for further refinement. While this idea was commonly used in the deep learning community when training stacked networks, later improvements take advantage of refining locally trained architectures [50]. Therefore, we plan to explore global optimization techniques to refine the layerwise binarization functions. Deep learning imposed a new era in computer vision and machine learning, achieving outstanding results on applications where previous state-of-the-art methods performed poorly. While the foundations of deep learning rely on very simple image processing operators, relevant properties held by traditional methods, such as illumination and rotational invariance, are not guaranteed. 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International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 STUDY OF Ε-SMOOTH SUPPORT VECTOR REGRESSION AND COMPARISON WITH Ε- SUPPORT VECTOR REGRESSION AND POTENTIAL SUPPORT VECTOR MACHINES FOR PREDICTION FOR THE ANTITUBERCULAR ACTIVITY OF OXAZOLINES AND OXAZOLES DERIVATIVES Doreswamy1 and Chanabasayya .M. Vastrad2 1 Department of Computer Science, MangaloreUniversity, Mangalagangotri-574 199, Karnataka,INDIA Doreswamyh@yahoo.com 2 Department of Computer Science, MangaloreUniversity, Mangalagangotri-574 199, Karnataka, INDIA channu.vastrad@gmail.com ABSTRACT A new smoothing method for solving ε -support vector regression (ε-SVR), tolerating a small error in fitting a given data sets nonlinearly is proposed in this study. Which is a smooth unconstrained optimization reformulation of the traditional linear programming associated with a ε-insensitive support vector regression. We term this redeveloped problem as ε-smooth support vector regression (ε-SSVR). The performance and predictive ability of ε-SSVR are investigated and compared with other methods such as LIBSVM (ε-SVR) and P-SVM methods. In the present study, two Oxazolines and Oxazoles molecular descriptor data sets were evaluated. We demonstrate the merits of our algorithm in a series of experiments. Primary experimental results illustrate that our proposed approach improves the regression performance and the learning efficiency. In both studied cases, the predictive ability of the εSSVR model is comparable or superior to those obtained by LIBSVM and P-SVM. The results indicate that ε-SSVR can be used as an alternative powerful modeling method for regression studies. The experimental results show that the presented algorithm ε-SSVR, , plays better precisely and effectively than LIBSVMand P-SVM in predicting antitubercular activity. KEYWORDS ε-SSVR , Newton-Armijo, LIBSVM, P-SVM 1.INTRODUCTION The aim of this paper is supervised learning of real-valued functions. We study a sequence S = x , y , . . . , x , y of descriptor-target pairs, where the descriptors are vectors in ℝ and the targets are real-valued scalars, yi ∈ ℝ.Our aim is to learn a function f: ℝ → ℝ which serves a good closeness of the target values from their corresponding descriptor vectors. Such a function is usually mentioned to as a regression function or a regressor for short.The main aimof DOI : 10.5121/ijscai.2013.2204 49 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 regression problems is to find a function fx that can rightly predict the target values,y of new input descriptor data points, x, by learning from the given training data set, S.Here, learning from a given training dataset means finding a linear surface that accepts a small error in fitting this training data set. Ignoring thevery smallerrors that fall within some acceptance, say εthat maylead to a improvedgeneralization ability is performed bymake use of an ε -insensitive loss function. As well as applying purpose of support vector machines (SVMs) [1-4], the function fx is made as flat as achievable, in fitting the training data. This issue is called ε -support vector regression (ε-SVR) and a descriptor data pointx ∈ R is called a support vector if"f#x $ − y " ≥ ε.Generally, ε-SVR is developed as a constrained minimization problem [5-6], especially, a convex quadratic programming problem or a linear programming problem[7-9].Suchcreations presents 2m more nonnegative variablesand 2m inequality constraints that increase the problem sizeand could increase computational complexity for solvingthe problem. In our way, we change the model marginally and apply the smooth methods that have been widely used for solving important mathematical programming problems[10-14] and the support vector machine for classification[15]to deal with the problem as an unconstrained minimizationproblemstraightly. We name this reformulated problem as ε – smooth support vector regression(ε-SSVR). Because ofthe limit less arrangement of distinguishability of the objectivefunction of our unconstrained minimization problem, weuse a fast Newton-Armijo technique to deal with this reformulation. It has been shown that the sequence achieved by the Newton-Armijo technique combines to the unique solutionglobally and quadratically[15]. Taking benefit of ε-SSVR generation, we only need to solve a system of linear equations iteratively instead of solving a convex quadratic program or a linear program, as is the case with a conventionalε-SVR. Thus, we do not need to use anysophisticated optimization package tosolve ε-SSVR. In order to deal with the case of linear regression with aOxazolines and Oxazoles molecular descriptor dataset. The proposed ε-SSVR model has strong mathematical properties, such as strong convexity and infinitely often differentiability. To demonstrate the proposed ε-SSVR’s capability in solving regression problems, we employ ε-SSVR to predict ant tuberculosis activity for Oxazolines and Oxazoles agents. We also compared our ε-SSVR model with P-SVM[16-17] and LIBSVM [18] in the aspect of prediction accuracies. The proposed ε-SSVR algorithm is implemented in MATLAB. A word about our representation and background material is given below. Entire vectors will be column vectors by way of this paper.For a vector xin the n-dimensional real descriptor space R , the plus functionx( is denoted as x( = max0, x , i = 1, … . . , n. The scalar(inner) product of two vectors x and y in the n-dimensional real descriptor space R will be reprsented by x, y and the p-norm of x will be represnted by ‖x‖. . For a matrix A ∈ R ⨯ , A is the iTh row of A which is a row vector inR ? A column vector of ones of arbitrary dimension will be reprsented by 1. For A ∈ R ⨯ and B ∈ R⨯2 , the kernel KA, B maps R ⨯ ⨯ R⨯2 intoR ⨯2 . In exact, if x andy are column vectors in R , then Kx , , y is a real number , , KA, x = K#x , , A, $ is a column vector in R . andKA, A, is an m ⨯ m matrix . If f is a real valued function interpreted on the n-dimensional real descriptor spaceR , the gradient of f at x is represented by ∇fx which is a row vector in R and n ⨯ n Hessian matrixof second partial derivatives of f at x is represented by∇5 fx . The base of the natural logarithm will be represented bye. 50 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 2. MATERIALS AND ALGORITHAMS 2.1 The Data Set The molecular descriptors of 100Oxazolines and Oxazoles derivatives [19-20] based H37Rv inhibitors analyzed. These molecular descriptors are generated using Padel-Descriptor tool [21]. The dataset covers a diverse set of molecular descriptors with a wide range of inhibitory activities against H37Rv. The pIC50 (observed biological activity) values range from -1 to 3. The dataset can be arranged in data matrix. This data matrix x contains m samples (molecule structures) in rows and n descriptors in columns. Vector y with order m × 1 denotes the measured activity of interest i.e. pIC50. Before modeling, the dataset is scaled. 2.2 The Smooth ε –support vector regression(ε-SSVR) We allow a given dataset Swhich consists of m points in n-dimensional real descriptor space R denoted by the matrix A ∈ R ⨯ and m observations of real value associated with each descriptor. That is,S = A , y | A ∈ R , y ∈ R, for i = 1, … … , m we would like to search a nonlinear regression function,fx , accepting a small error in fitting this given data set. This can be performed by make use of the ε- insensitive loss function that sets ε- insensitive “tube” around the data, within which errors are rejected. Also, put into using the idea of support vector machines (SVMs) [1-4],thefunction fx is made as 8lat as possible in fitting thetraining data set. We start with the regression function f(x) and it is expressed as f(x) = x , w + b. This problem can be formulated as an unconstrained minimization problem given as follows: 1 min>?@ w , w + C1, |ξ|D (1) (;,<)∈= 2 Where |ξ| ∈ R , (|ξ|D ) = max{0, |A w + b + y | − ε } that denotes the fitting errors and positive control parameter C here weights the agreement between the fitting errors and the flatnessof the regression functionf(x). To handle ε-insensitive loss function in the objective function of the above minimization problem,traditionallyit is reformulated as a constrained minimization problem expressed as follows: 1 min ∗ w , w + C1, (ξ + ξ∗ )Aw − 1b − y ≤ 1ε + ξ − Aw − 1b + y ≤ 1ε + ξ∗ ξ, ξ∗ (;,<,E,E ) 2 ≥ 0. (2) This equation (2), which is equivalent to the equation (1), is a convex quadratic minimization problem with n + 1 free variables, 2m nonnegative variables, and 2m imparity constraints.However, presenting morevariables (and constraints in the formulation increases theproblem size and could increase computational complexityfor dealing with the regression problem. In our smooth way, we change the model marginally and solve it as an unconstrained minimization problem preciselyapart from adding any new variable and constraint. 51 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure1. (a) |x|5D and (b) p5D (x, α) with α = 5,ε=1. That is, the squares of 2-norm ε- insensitive loss, ‖|Aw − 1b + y|D ‖55 is minimized with weight J  in place of the 1-norm of ε- insensitive loss as in Eqn (1). Additional, we add the term 5 b5 5 in the objective function to induce strong convexity and to certainty that the problem has a only global optimal solution. These produce the following unconstrained minimization problem: 1 C min>?@ (w , w + b5 ) + K|A w + b − y |5D (;,<)∈= 2 2 L (3) This formulation has been projected in active set support vector regression [22] and determined in its dual form. Motivated by smooth support vector machine for classification (SSVM) [15] the squares of ε- insensitive loss function in the above formulation can be correctly approximated by a smooth function which is extremely differentiable and described below. Thus, we are admitted to use a fast Newton-Armijo algorithm to determine the approximation problem. Before we make out the smooth approximation function, we exhibit some interesting observations: |x|D = max {0, |x| − ε } = max{0, x − ε } + max{0, −x − ε } (4) = (x − ε)( + (−x − ε)( . In addition, (x − ε)( . (−x − ε)( = 0 for all x ∈ R and ε > 0 . Thus, we have (5) |x|5D = (x − ε)5( + (−x − ε)5( . In SSVM [15], the plus function x( approximated by a smooth p-function,p(x, α) = x +  log(1 + eSQT ), α > 0. It is straightforward to put in place of |x|5D by a very correct smooth Q approximation is given by: 5 5 p5D (x, α) = #p(x − ε, α)$ + #p(−x − ε, α)$ . (6) 52 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure1.exemplifies the square of ε- insensitive loss function and its smooth approximation in the case of α = 5 and ε = 1. We call this approximation p5D -function with smoothing parameterα. This p5D -function is used hereto put in place of the squares of ε- insensitive loss function of Eqn. (3) to get our smooth support vector regression(ε-SSVR): min>?@ ΦD,Q (w, b) (;,<)∈= ∶= 1 min>?@ #w , w + b5 $ (;,<)∈= 2 C + K p5D (A w + b − y , α) (7) 2 L 1 = min>?@ #w , w + b5 $ (;,<)∈= 2 C , 5 + 1 pD (A w + b − y, α) , 2 Where p5D (A w + b − y, α) ∈ R is expressed by p5D (A w + b − y, α) = p5D (A w + b − y , α). This problem is a powerfully convex minimization problem without any restriction. It is not difficult to show that it has a one and only solution. Additionally, the objective function in Eqn. (7)is extremelydifferentiable, thus we can use a fast Newton-Armijo technique to deal with the problem. Before we deal with the problem in Eqn. (7) we have to show that the result of the equation (3) can be got by analyzing Eqn. (7) with α nearing infinity. We begin with a simple heading thatlimits the difference betweenthe squares of ε- insensitive loss function,|x|5D and its smooth approximation p5D (x, α). Heading 2.2.1.For x ∈ Rand |x| < Z + [: log 2 5 2σ p5D (x, α) − |x|5D ≤ 2 \ ] + log 2, α α where p5D x, α is expressed in Eqn. (6). 8 Proof. We allow for three cases. For −ε ≤ x ≤ ε, |x|D = 0 and p(x, α)5 are a continuity increasing function, so we have p5D (x, α) – |x|5D = p(x − ε, α)5 + p(−x − ε, α)5 log 2 5 5 ≤ 2p(0, α) = 2 \ ] , α sincex − ε ≤ 0 and – x − ε ≤ 0. 2cd 5 5 For ε < a < [ + Z , using the result in SSVM[15] that px, α 5 − x( 5 ≤ b Q e + for |x| < Z , we have p5D (x, α) – (|x|D )5 = (p(x − ε, α))5 + (p(−x − ε, α))5 − (x − ε)5( 5f log 2 Q 53 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 5 ≤ #p(x − ε, α)$ − (x − ε)5( + (p(0, α))5 log 2 5 2σ ≤ 2\ ] + log 2. α α Likewise, for the case of – ε − σ < a < – [ , we have log 2 5 2σ ] + log 2. p5D x, α – |x|D )5 ≤ 2 \ α α 2cd 5 5 e Q Hence, p5D x, α – |x|5D ≤ 2 b + 5f log 2. Q By Heading 2.1, we have that as the smoothing parameter α reaches infinity, the one and only solution of Equation (7) reaches, the one and only solution of Equation (3). We shall do this for a function fD (x) given in Eqn. (9) below that includes the objective function of Eqn. (3) and for a function g D (x, α) given in Eqn. (10) below which includes the SSVR function of Eqn. (7). Axiom 2.2.2. Let A ∈ R ⨯ andb ∈ R ⨯ . Explain the real valued functions fD (x) and g D (x, α) in the n-dimensional real molecular descriptor spaceR : 1 1 5 fD (x) = K"Ag x − b" + ‖x‖55 D 2 2 (9) L And g D (x, α) = Withε,α > 0.  ∑ L 5  p5D ( Ag x − b, α) + 5 ‖x‖55 , (10) 1. There exists a one and only solution x of minT∈=> fD (x) and one and only solution xQ of minT∈=> g D (x, α). 2. For all α > 0 , we have the following inequality: log 2 5 log 2 ‖xQ − x‖55 ≤ m j\ ] +ξ k, α α (11) Whereξ is expressed as follows: ξ = max |(Ax − b) |. l l (12) Thus xQ gathers to xas α goes to endlessness with an upper limit given by Eqn. (11). The proof can be adapted from the results in SSVM [15] and, thus, excluded here. We now express a Newton-Armijo algorithm for solving the smooth equation (7). 54 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 2.2.1A NEWTON-ARMIJO ALGORITHM FOR m-SSVR By utilizing the results of the preceding section and taking benefitof the twice differentiability of the objectivefunction in Eqn. (7), we determine a globally and quadratically convergent Newton-Armijo algorithm for solving Eqn. (7). Algorithm 2.3.1 Newton-ArmijoAlgorithm For n-SSVR Start with any choice of initial point (w o , bo ) ∈ R( . Having (w , b ), terminate if the gradient of the objective function of Eqn. (7) is zero, that is, ∇ΦD,Q (w , b )=0. Else calculate (w ( ,b ( ) as follows: 1. Newton Direction:Decide the directiond ∈ R( by allocatingequal to zero the Linearization of∇ΦD,Q (w, b) all over(w , b ), which results inn + 1 Linear equations with n + 1 variables: ∇5 ΦD,Q (w , b )d = −∇ΦD,Q (w , b ), . 2. (13) Armijo Step size [1]: Choose a stepsize λ ∈ R such that: #w ( , b ( $ = (w , b )+λ d , (14)  5r whereλ = max{1, , , … … }such that: ΦD,Q #w , b $ − ΦD,Q ((w , b ) +λ d ≥ −δλ ΦD,Q (w , b )d , (15)  where δ ∈ b0, 5e . Note that animportant difference between our smoothingapproach and that of the traditional SVR [7-9] is that we are solving a linear system of equations (13) here, rather solving a quadratic program, as is the case with the conventional SVR. 2.3LIBSVM LIBSVM [18] is a library for support vector machines. LIBSVM is currentlyone of the most widely used SVM software. This software contains C-support vector classification (C-SVC), vsupport vector classification (v-SVC), ε-support vector regression (ε-SVR), v-support vector regression (v-SVR). All SVM formulations supported in LIBSVM are quadratic minimization problems 2.4Potential-Support Vector Machines(P-SVM) P-SVM [16-17] is a supervised learning method used for classification and regression. As well as standard Support Vector Machines, it is based on kernels. Kernel Methods approach the problem by mapping the data into a high dimensional feature space, where each coordinate corresponds to one feature of the data items, transforming the data into a set of points in a Euclidean space. In that space, a variety of methods can be used to find relations between the data. 55 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 2.5Experimental Evaluation In order to evaluate how well each method generalizedto unseen data, we split the entire data set into two parts,the training set and testing set. The training data was usedto generate the regression function that is learning fromtraining data; the testing set, which is not involved in thetraining procedure, was used to evaluate the predictionability of the resulting regression function.We also used a tabular structure scheme in splitting the entire data set to keep the “similarity” between training and testing data sets [23]. That is, we tried to make the training set and the testing set have the similar observation distributions. A smaller testing error indicates better prediction ability. We performed tenfold cross-validation on each data set [24] and reported the average testing error in our numerical results. Table 1 gives features of two descriptor datasets. Table 1: Features of two descriptor datasets Data set(Molecular Descriptors of Oxazolines and Oxazoles Derivatives) Train Size Test Size Full 75 X 254 25 X 254 254 Reduced 75 X 71 25 X 71 71 Attributes In all experiments, 2-norm relative error was chosen to evaluate the tolerance between the predicted values and the observations. For an observation vector y and the predicted vector yv , the 2-norm relative error (SRE) of two vectors y and yv was defined as follows. SRE = ‖y − yv‖5 ‖y‖5 (16) In statistics, the mean absolute error is a quantity used to measure how close predictions are to the eventual outcomes. The mean absolute error (MAE) is given by   1 1 MAE = K|yv − y | = K|e | (17) n n L L As the name suggests, the mean absolute error is an average of the absolute errors e =yv − y , where yv is the prediction and y the observed value. In statistics, the coefficient of determination, denoted R5 and pronounced R squared, is used in the context of statistical models whose main purpose is the prediction of future outcomes on the basis of other related information. R5 is most often seen as a number between 0 and 1, used to describe how well a regression line fits a set of data. A R5 near 1 indicates that a regression line fits the data well, while a R5 close to 0 indicates a regression line does not fit the data very well. It is the proportion of variability in a data set that is accounted for by the statistical model. It provides a measure of how well future outcomes are likely to be predicted by the model. 56 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 R5 = 1 − ∑L(y − yv )5 ∑L(y − y)5 (18) The predictive power of the models developed on the calculated statistical parameters standard error of prediction (SEP) and relative error of prediction (REP)% as follows: SEP = z o.| ∑L(yv − y )5 { n  100 1 REP(%) = ~ K(yv − y )5  y n L o.| (19) (20) The performancesof models were evaluated in terms of root mean square error (RMSE), which was defined as below: ∑ˆ†L# „ €‚ƒ = † 5 − ‡† $ (21) ‰ Whereyv ,y and y are the predicted, observed and mean activity property, respectively. 3.RESULTS AND DISCUSSION In this section, we demonstrate the effectiveness of our proposed approachε-SSVR by comparing it to LIBSVM (ε-SVR) and P-SVM. In the following experiments, training is done 5 with Gaussian kernel function k(x1, x2) = exp b−ϒ‹x − xg ‹ e , where ϒis the is the width of the Gaussian kernel, i, j = 1, … . . , l. We perform tenfold cross-validation on each dataset and record the average testing error in our numerical results. The performances of ε-SSVR for regression depend on the combination of several parameters They are capacity parameter , ε of ε- insensitive loss function and ϒparameter.  is a regularization parameter that controls the tradeoff between maximizing the margin and minimizing the training error. In practice the parameter  is varied through a wide range of values and the optimal performance assessed using a separate test set. Regularization parameter , whose effect on the RMSE is shown in Figure 1a for full descriptor datasetandFigure 1b for reduced descriptor dataset. 57 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure 1a.The selection of the optimal capacity factor (8350) for ε-SSVR(ε=0.1, ϒ=0.0217) For the Full descriptordataset, The RMSE valuefor ε-SSVRmodel 0.3563 is small for selected optimal parameter C, compared to RMSE values for other two models i.e. LIBSVM (ε-SVR) and P-SVM are 0.3665 and 0.5237.Similarly,for the reduced descriptor dataset,The RMSE value for ε-SSVR model 0.3339 is small for selected optimal parameter C, compared to RMSE values for other two models i.e. LIBSVM (ε-SVR) and P-SVM are 0.3791 and 0.5237. The optimal value for ε depends on the type of noise present in the data, which is usually unknown. Even if enough knowledge of the noise is available to select an optimal value for ε, there is the practical consideration of the number of resulting support vectors. Ε insensitivity prevents the entire training set meeting boundary conditions and so allows for the possibility of sparsely in the dual formulations solution. 58 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure 1b. The selection of the optimal capacity factor(1000000) for ε-SSVR(ε=0.1, ϒ=0.02) So, choosing the appropriate value of ε is critical from theory. To find an optimal ε, the root mean squares error (RMSE) on LOO cross-validation on different ε was calculated. The curves of RMSE versus the epsilon (ε) is shown in Figure 2a and Figure 2b. Figure 2a. The selection of the optimal epsilon (0.1) for ε-SSVR( = 1000, ϒ=0.02) For the Full descriptor dataset , The RMSE value for ε-SSVR model 0.3605 is small for selected optimal epsilon(ε), compared to RMSE value for LIBSVM(ε-SVR) model is closer i.e. 0.3665 but comparable to the proposed model and bigRMSE value for P-SVM model is 0.5237. 59 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure 2b.The selection of the optimal epsilon (0.1) for ε-SSVR(= 10000000, ϒ=0.01) Similarly , for the Reduced descriptor dataset , The RMSE value for ε-SSVR model 0.3216 is small for selected optimal epsilon(ε) , compared to RMSE values for other two models i.e. LIBSVM(ε-SVR) and P-SVM are 0.3386 and 0.4579. Figure 3a. The selection of the optimal ϒ(0.02) for ε-SSVR(C =1000, ε=0.1) Parameter tuning was conducted in ε-SSVR, where the ϒ parameter in the Gaussian kernel function was varied from 0.01 to 0.09 in steps 0.01 to select optimal parameter. The value of ϒ is updated based on the minimization LOO tuning error rather than directly minimizing the training error. The curves of RMSE versus the gamma(ϒ) is shown in Figure 3a and Figure 3b. 60 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure 3b. The selection of the optimal ϒ(0.01) for ε-SSVR(C =1000000, ε=0.1) For the Full descriptordataset , The RMSE value for ε-SSVR model 0.3607 is small for selected optimal parameter ϒ , compared to RMSE values for other two models i.e. LIBSVM(ε-SVR) and P-SVM are 0.3675 and 0.5224. Similarly , for the Reduced descriptor dataset , The RMSE value for ε-SSVR model 0.3161 is small for selected optimal parameterϒ, compared to RMSE values for other two models i.e.LIBSVM(ε-SVR) and P-SVM are 0.3386 and 0.4579. The statistical parameters calculated for the ε-SSVR, LIBSVM(ε-SVR) and P-SVM models are represented in Table 2 and Table 3. Table 2. Performance Comparison between ε-SSVR,ε-SVR and P-SVM for Full descriptor dataset Algorithm (ε, C,ϒ) Train Error(Ž ) 0.9790 0.9825 0.8248 Test Error(Ž ) 0.8183 0.8122 0.6166 MAE SRE SEP REP(%) ε-SSVR ε-SVR P-SVM (0.1,1000,0.0217) 0.0994 0.0918 0.2510 0.1071 0.0979 0.3093 0.3679 0.3741 0.5345 53.7758 54.6693 78.1207 ε-SSVR ε-SVR P-SVM (0.1,8350,0.0217) 0.9839 0.9825 0.8248 0.8226 0.8122 0.6166 0.0900 0.0918 0.2510 0.0939 0.0979 0.3093 0.3636 0.3741 0.5345 53.1465 54.6693 78.1207 ε-SSVR ε-SVR P-SVM (0.1,1000,0.02) 0.9778 0.9823 0.8248 0.8181 0.8113 0.6186 0.1019 0.0922 0.2506 0.1100 0.0984 0.3093 0.3681 0.3750 0.5332 53.8052 54.8121 77.9205 61 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 In these tables, statistical parameters R-square (R5 ) ,Mean absolute error (MAE),2-N Normalization(SRE), standard error of prediction (SEP) and relative error of prediction (REP%) obtained by applying the ε-SSVR, ε-SVR and P-SVM methods to the test set indicate a good external predictability of the models. Table 3. Performance Comparison between ε-SSVR,ε-SVR and P-SVM for Reduced descriptor dataset Algorithm ( ε, C,ϒ) ε-SSVR ε-SVR P-SVM (0.1,1000000,0.02) ε-SSVR ε-SVR P-SVM ε-SSVR ε-SVR P-SVM (0.1,10000000,0.01) (0.1,1000000,0.01) Train Error(Ž ) 0.9841 0.9847 0.8001 Test Error(Ž ) 0.8441 0.7991 0.7053 MAE SRE SEP REP(%) 0.0881 0.0827 0.2612 0.0931 0.0914 0.3304 0.3408 0.3870 0.4687 49.8084 56.5533 68.4937 0.9849 0.9829 0.8002 0.9796 0.9829 0.8002 0.8555 0.8397 0.7069 0.8603 0.8397 0.7069 0.0851 0.0892 0.2611 0.0964 0.0892 0.2611 0.0908 0.0967 0.3303 0.1056 0.0967 0.3303 0.3282 0.3456 0.4673 0.3226 0.3456 0.4673 47.9642 50.5103 68.3036 47.1515 50.5103 68.3036 An experimental results show that experiments carried out from reduced descriptor datasets shows good results rather than full descriptor dataset. As from can be seen from table 4 , the results of ε-SSVR models are better than those obtainedby ε-SVR and P-SVM models for Reduced descriptor data set. Figure 4. Correlation between observed and predicted values for training set and test set generated by ε-SSVR Figure4, 5and 6 are the scatter plot of the three models, which shows a correlation between observed value and ant tuberculosisactivity prediction in the training and test set. 62 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Figure 5. Correlation between observed and predicted values for training set and test set generated by ε-SVR Figure 6. Correlation between observed and predicted values for training set and test set generated by P-SVM algorithm Our numerical results have demonstrated that ε-SSVR is a powerful tool for solving regressionProblems handle the massive data sets without scarifying any prediction accuracy. In the tuning process of these experiments, we found out that LIBSVM and P-SVM become very slow when the control parameter  becomes bigger, while ε-SSVR is quite robust to the control parameter . Although we solved the ε-insensitive regression problem is an unconstrained minimization problem. 63 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 4CONCLUSION In the present work, ε-SSVR, which is a smooth unconstrained optimization reformulation of the traditional quadratic program associated with a ε-insensitive support vector regression.We have compared the performance of, ε-SSVR, LIBSVM and P-SVM models with two datasets. The obtained results show that ε-SSVR can be used to derive statistical model with better qualities and better generalization capabilities than linear regression methods. ΕSSVRalgorithm exhibits the better overall performance and a better predictive ability than the LIBSVM and P-SVM models. The experimental results indicate ε-SSVR has high precision and good generalization ability. ACKNOLDGEMENTS We gratefully thank to the Department of Computer Science Mangalore University, Mangalore India for technical support of this research. REFERENCES [1] Jan Luts,,Fabian Ojeda, Raf Van de Plasa, Bart De Moor, Sabine Van Huffel, Johan A.K. Suykens ,“A tutorialon support vector machine-based methods for classification problems in chemometrics”, AnalyticaChimicaActa665 (2010) 129–145 [2] HongdongLi ,Yizeng Liang, QingsongXu ,”Support vector machines and its applications in chemistry”,Chemometrics and Intelligent Laboratory Systems 95 (2009) 188–198 [3] Jason Weston,”Support Vector Machines and Stasitical Learning Theory”, NEC Labs America 4 IndependenceWay, Princeton, USA.http://www.cs.columbia.edu/~kathy/cs4701/documents/jason_svm_tutorial.pdf [4] AlyFaragandRefaat M Mohamed ,“Regression Using Support Vector Machines: Basic Foundations” ,http://www.cvip.uofl.edu/wwwcvip/research/publications/TechReport/SVMRegressionTR.pdf [5] Chih-Jen Lin ,“Optimization, Support Vector http://www.csie.ntu.edu.tw/~cjlin/talks/rome.pdf [6] Max Welling ,“Support Vector Regression” http://www.ics.uci.edu/~welling/teaching/KernelsICS273B/SVregression.pdf [7] ALEX J. SMOLA and BERNHARD SCHO¨ LKOPF,, “Tutorial on support vector regression” ,Statistics and Computing 14: 199–222, 2004 [8] Qiang Wu Ding-Xuan Zhou,” SVM Soft Margin Classifiers: Linear Programming versus Quadratic Programming” ,www6.cityu.edu.hk/ma/doc/people/zhoudx/LPSVMfinal.pdf [9] Laurent El Ghaoui,” Convex Optimization www.stanford.edu/class/ee392o/mit022702.pdf Machines, in and Machine Classifcation Learning” Problems” , , , [10] DONGHUI LI MASAO FUKUSHIMA, “Smoothing Newton and Quasi-Newton methods for mixed Complementarity problems” ,Computational Optimization and Applications ,17,203230,2000 [11] C. Chen and O.L. Mangasarian ,“Smoothing Methods for Convex Inequalities and Linear Complementarity problems”, Math. Programming, vol. 71, no. 1, pp. 51-69, 1995 64 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 [12] X Chen,L. Qi and D. Sun,“Globalandsuperlinear convergence of the Smoothing Newton methodapplication to general box constrained variational inequalities“, Mathematics of Computation Volume 67,Number 222, April 1998, Pages 519-540 [13] X. Chen and Y. Ye, SIAM J. , “ On Homotopy-Smoothing Methods For Variational Inequalities” ,Control andOptimization, vol. 37, pp. 589-616, 1999. [14] Peter W. ,“A semi-smooth Newton method for elasto-plastic contact problems”, Christensen International Journal of Solids and Structures 39 (2002) 2323–2341 [15] Y. J. Lee and O. L. Mangasarian ,” SSVM: Asmooth support vector machine forclassification”, Computational Optimization and Applications, Vol. 22, No. 1, 2001, pp. 5-21. [16] SeppHochreiter and Klaus Obermanyer ,”Support Vector Machines for Dyadic Data” ,Neural Computation,18, 1472-1510, 2006. http://ni.cs.tu-berlin.de/software/psvm/index.html [17] Ismael F. Aymerich, JaumePiera and AureliSoria-Frisch ,“Potential Suport Vector Machines and Self-Organizing Maps for phytoplankton discrimination”, In proceeding of: International Joint Conference onNeural Networks, IJCNN 2010, Barcelona, Spain, 18-23 July, 2010 [18] C.-C. Chang and C.-J. Lin, 2010 , “LIBSVM: A Library for Support Vector Machines” ,http://www.csie.ntu.edu.tw/~cjlin/libsvm [19] Andrew J. Phillips, Yoshikazu Uto, Peter Wipf, Michael J. Reno, and David R. Williams, “Synthesis of Functionalized Oxazolines and Oxazoles with DAST and Deoxo-Fluor” Organic Letters 2000 Vol 2 ,No.81165-1168 [20] Moraski GC, Chang M, Villegas-Estrada A, Franzblau SG, Möllmann U, Miller MJ.,”Structureactivityrelationship of new anti-tuberculosis agents derived from oxazoline and oxazole benzyl esters” ,Eur J Med Chem. 2010 May;45(5):1703-16. doi: 10.1016/j.ejmech.2009.12.074. Epub 2010 Jan 14. [21] “Padel-Descriptor” http://padel.nus.edu.sg/software/padeldescriptor/ [22] David R. Musicant and Alexander Feinberg ,“Active Set Support Vector Regression” , IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO.2, MARCH 2004 [23] Ian H.Witten&Eibe Frank, “Data mining: Practical Machine Learning Tools ,SecondEditionElseveir. andtechinques” [24] PayamRefaeilzadeh, Lei Tang, Huanliu ,“Cross-Validation” http://www.cse.iitb.ac.in/~tarung/smt/papers_ppt/ency-cross-validation.pdf , 65 International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.2, No.2, April 2013 Authors Doreswamyreceived B.Sc degree in Computer Science andM.Sc Degree inComputer Science from University of Mysore in 1993 and 1995 respectively. Ph.Ddegree in Computer Science from Mangalore University in the year 2007. Aftercompletion of his Post-Graduation Degree, he subsequently joined and served asLecturer in Computer Science at St.Joseph’s College, Bangalore from 19961999.Then he has elevated to the position Reader in Computer Science at Mangalore Universityin year 2003. He was the Chairman of the Department of Post-Graduate Studies and researchin computer science from 2003-2005 and from 2009-2008 and served at varies capacitiesin Mangalore University at present he is the Chairman of Board of Studies and AssociateProfessor in Computer Science of Mangalore University. His areas of Researchinterestsinclude Data Mining and Knowledge Discovery,ArtificialIntelligence and Expert Systems, Bioinformatics ,Molecular modelling and simulation ,Computational Intelligence ,Nanotechnology, ImageProcessing and Pattern recognition. He has been granted a Major Research project entitled “Scientific Knowledge DiscoverySystems(SKDS) for Advanced Engineering Materials Design Applications” fromthe funding agency University Grant Commission, New Delhi, India. Hehas been published about 30 contributedpeer reviewed Papers at national/International Journal and Conferences.Hereceived SHIKSHA RATTAN PURASKAR for his outstanding achievementsin the year 2009 and RASTRIYA VIDYA SARASWATHI AWARD for outstanding achievement in chosenfield of activityin the year 2010. ChanabasayyaM. Vastradreceived B.E. degree and M.Tech.degree in the year2001 and 2006 respectively. Currently working towards his Ph.D Degree in Computer Science andTechnology under the guidance of Dr. Doreswamyin the Department of Post-Graduate Studies and Research in Computer Science, Mangalore University. 66 

Provable and practical approximations for the degree distribution using sublinear graph samples∗ Talya Eden School of Computer Science, Tel Aviv University Tel Aviv, Israel talyaa01@gmail.com Shweta Jain University of California, Santa Cruz Santa Cruz, CA, USA sjain12@ucsc.edu arXiv:1710.08607v2 [cs.SI] 19 Jan 2018 Dana Ron School of Computer Science, Tel Aviv University Tel Aviv, Israel danaron@tau.ac.il Ali Pinar Sandia National Laboratories Livermore, CA apinar@sandia.gov C. Seshadhri University of California, Santa Cruz Santa Cruz, CA sesh@ucsc.edu ABSTRACT 1 The degree distribution is one of the most fundamental properties used in the analysis of massive graphs. There is a large literature on graph sampling, where the goal is to estimate properties (especially the degree distribution) of a large graph through a small, random sample. The degree distribution estimation poses a significant challenge, due to its heavy-tailed nature and the large variance in degrees. We design a new algorithm, SADDLES, for this problem, using recent mathematical techniques from the field of sublinear algorithms. The SADDLES algorithm gives provably accurate outputs for all values of the degree distribution. For the analysis, we define two fatness measures of the degree distribution, called the h-index and the z-index. We prove that SADDLES is sublinear in the graph size when these indices are large. A corollary of this result is a provably sublinear algorithm for any degree distribution bounded below by a power law. We deploy our new algorithm on a variety of real datasets and demonstrate its excellent empirical behavior. In all instances, we get extremely accurate approximations for all values in the degree distribution by observing at most 1% of the vertices. This is a major improvement over the state-of-the-art sampling algorithms, which typically sample more than 10% of the vertices to give comparable results. We also observe that the h and z-indices of real graphs are large, validating our theoretical analysis. In domains as diverse as social sciences, biology, physics, cybersecurity, graphs are used to represent entities and the relationships between them. This has led to the explosive growth of network science as a discipline over the past decade. One of the hallmarks of network science is the occurrence of specific graph properties that are common to varying domains, such as heavy tailed degree distributions, large clustering coefficients, and small-world behavior. Arguably, the most significant among these properties is the degree distribution, whose study led to the foundation of network science [7, 8, 20]. Given an undirected graph G, the degree distribution (or technically, histogram) is the sequence of numbers n(1), n(2), . . ., where n(d) is the number of vertices of degree d. In almost all real-world scenarios, the average degree is small, but the variance (and higher moments) is large. Even for relatively large d, n(d) is still non-zero, and n(d) typically has a smooth non-increasing behavior. In Fig. 1, we see the typical degree distribution behavior. The average degree in a Google web network is less than 10, but the maximum degree is more than 5000. There are also numerous vertices with all intermediate degrees. This is referred to as a “heavy tailed” distribution. The degree distribution, especially the tail, is of significant relevance to modeling networks, determining their resilience, spread of information, and for algorithmics [6, 9, 13, 16, 33–36, 42]. With full access to G, the degree distribution can be computed in linear time, by simply determining the degree of each vertex. Yet in many scenarios, we only have partial access to the graph, provided through some graph samples. A naive extrapolation of the degree distribution can result in biased results. The seminal research paper of Faloutsos et al. claimed a power law in the degree distribution on the Internet [20]. This degree distribution was deduced by measuring a power law distribution in the graph sample generated by a collection of traceroute queries on a set of routers. Unfortunately, it was mathematically and empirically proven that traceroute responses can have a power law even if the true network does not [1, 11, 27, 37]. In general, a direct extrapolation of the degree distribution from a graph subsample is not valid for the ACM Reference format: Talya Eden, Shweta Jain, Ali Pinar, Dana Ron, and C. Seshadhri. 2016. Provable and practical approximations for the degree distribution using sublinear graph samples. In Proceedings of , , , 12 pages. DOI: ∗ Both Talya Eden and Shweta Jain contributed equally to this work, and are joint first authors of this work. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). , © 2016 Copyright held by the owner/author(s). . DOI: INTRODUCTION The Hidden Degrees Model (HDM) Vertex and neighbor queries allowed, not degree queries: This is a substantially weaker model. In numerous cybersecurity and network monitoring settings, an algorithm cannot query for degrees, and has to infer them indirectly. Observe that this model is significantly harder than the standard model. It takes O((m + n) log n) to determine all the degrees, since one has to at least visit all the edges to find degrees exactly. In this model, we express the number of queries as a fraction of m. We stress that other query models are possible. Recent work of Dasgupta, Kumar, and Sarlos argues that vertex queries are too powerful, especially in social network contexts [14]. In their query model, a small set of random seeds are provided, but only neighbor and degree queries are subsequently allowed. underlying graph. This leads to the primary question behind our work. How can we provably and practically estimate the degree distribution without seeing the entire graph? There is a rich literature in statistics, data mining, and physics on estimating graph properties (especially the degree distribution) using a small subsample [2, 3, 5, 17, 28, 30, 31, 39, 46, 47]. Nonetheless, there is no provable algorithm for the entire degree distribution, with a formal analysis on when it is sublinear in the number of vertices. Furthermore, most empirical studies typically sample 10-30% of the vertices for reasonable estimates. 1.1 Problem description We focus on the complementary cumulative degree histogram (often called the cumulative degree distribution) or ccdh of G. This is Í the sequence {N (d)}, where N (d) = r ≥d n(r ). The ccdh is typically used for fitting distributions, since it averages out noise and is monotonic [12]. Our aim is to get an accurate bicriteria approximation to the ccdh of G, at all values of d. 1.2 Our contributions Our main theoretical result is a new sampling algorithm, the Sublinear Approximations for Degree Distributions Leveraging Edge Samples, or SADDLES. This algorithm provably provides (ε, δ )-approximations for the ccdh. We show how to design SADDLES under both the Standard Model and the Hidden Degrees Model. We apply SADDLES on a variety of real datasets and demonstrate its ability to accurately approximate the ccdh with a tiny sample of the graph. • Sampling algorithm for estimating ccdh: Our algorithm combines a number of techniques in random sampling to get (ε, δ )-estimates for the ccdh. A crucial component is an application of an edge simulation technique, first devised by Eden et al. in the context of triangle counting [18, 19]. This (theoretical) technique shows how to get a collection of weakly correlated uniform random edges from independent uniform vertices. SADDLES employs a weighting scheme on top of this method to estimate the ccdh. • Heavy tails leads to sublinear algorithms: The challenge in analyzing SADDLES is in finding parameters of the ccdh that allow for sublinear query complexity. To that end, we discuss two parameters that measure “heaviness” of the distribution tail: the classic h-index and a newly defined z-index. We prove that the running time of SADDLES is sublinear (for both models) whenever these indices are large. This yields a provably sublinear time algorithm to get accurate (ε, δ )-estimates for the ccdh. • Excellent empirical behavior: We deploy an implementation of SADDLES on a collection of large real-world graphs. In all instances, we achieve extremely accurate estimates for the entire ccdh by sampling at most 1% of the vertices of the graph. Refer to Fig. 1. Observe how SADDLES tracks various jumps in the ccdh, for both graphs in Fig. 1. • Comparison with existing sampling methods: A number of graph sampling methods have been proposed in practice, such as vertex sampling, snowball sampling, forest-fire sampling, induced graph sampling, random walk, edge sampling [5, 17, 28, 30, 38, 39, 47]. A recent work of Zhang et al. explicitly addresses biases in these sampling methods, and fixes them using optimization techniques [47]. We run head-to-head comparisons with all these sampling methods, and demonstrate the SADDLES gives significantly better practical performance. Fig. 1 shows the output of all these sampling methods with a total sample size of 1% of e(d)} is an (ε, δ )-estimate of the Definition 1.1. The sequence {N e(d) ≤ (1 + δ )N ((1 − ε)d). ccdh if ∀d, (1 − δ )N ((1 + ε)d) ≤ N Computing an (ε, δ )-estimate is significantly harder than approximating the ccdh using standard distribution measures. Statistical measures, such as the KS-distance, χ 2 , `p -norms, etc. tend to ignore the tail, since (in terms of probability mass) it is a negligible portion of the distribution. From a network science standpoint, the few vertices that are of extremely high degree are essential for graph analysis. An (ε, δ )-estimate is accurate for all d. The query model: A formal approach requires specifying a query model for accessing G. We look to the subfields of property testing and sublinear algorithms within theoretical computer science for such models [22, 23]. Consider the following three kinds of queries. • Vertex queries: acquire a uniform random vertex v ∈ V . • Neighbor queries: given v ∈ V , acquire a uniform random neighbor u of V . • Degree queries: given v ∈ V , acquire the degree dv . An algorithm is only allowed to make these queries to process the input. It has to make some number of queries, and finally produce an output. We discuss two query models, and give results for both. The Standard Model (SM) All queries allowed: This is the standard model in numerous sublinear algorithms results [18, 19, 22– 24]. Furthermore, most papers on graph sampling implicitly use this model for generating subsamples. Indeed, any method involving crawling from a random set of vertices and collecting degrees is in the Standard Model. This model is the primary setting for our work, and allows for comparison with rich body of graph sampling algorithms. It is worth noting that in the Standard Model, one can determine the entire degree distribution in O(n log n) queries (the extra log n factor comes from the coupon collector bound of finding all the vertices through uniform sampling). Thus, it makes sense to express the number of queries made by an algorithm as a fraction of n. Alternately, the number of queries is basically the number of vertices encountered by the algorithm. Thus, a sublinear algorithm makes o(n) queries. 2 (a) amazon0601 copurchase network (b) web-Google web network (c) cit-Patents citation network (d) com-orkut social network Figure 1: The output of SADDLES on a collection of networks: amazon0601 (403K vertices, 4.9M edges), web-Google (870K vertices, 4.3M edges), cit-Patents (3.8M vertices, 16M edges), com-orkut social network (3M vertices, 117M edges). SADDLES samples 1% of the vertices and gives accurate results for the entire (cumulative) degree distribution. For comparison, we show the output of a number of sampling algorithms from past work, each run with the same number of samples. (Because of the size of com-Orkut, methods involving optimization [47] fail to produce an estimate in reasonable time.) the vertices. Observe how across the board, the methods make erroneous estimates for most of the degree distribution. The errors are also very large, for all the methods. This is consistent with previous work, where methods sample more than 10% of the number of vertices. 1.3 Our main result is more nuanced, and holds for all degree distributions. If the ccdh has a heavy tail, we expect N (d) to be reasonably large even for large values of d. We describe two formalisms of this notion, through fatness indices. Definition 1.4. The h-index of the degree distribution is the largest d such that there are at least d vertices of degree at least d. Theoretical results in detail This is the exact analogy of the bibliometric h-index [26]. As we show in the §2.1, h can be approximated by mind (d + N (d))/2. A more stringent index is obtained by replacing the arithmetic mean by the (smaller) geometric mean. Our main theoretical result is a new sampling algorithm, the Sublinear Approximations for Degree Distributions Leveraging Edge Samples, or SADDLES. We first demonstrate our results for power law degree distributions [7, 8, 20]. Statistical fitting procedures suggest they occur to some extent in the real-world, albeit with much noise [12]. The classic power law degree distribution sets n(d) ∝ 1/d γ , where γ is typically in [2, 3]. We build on this to define a power law lower bound. Definition p1.5. The z-index of the degree distribution is z = mind :N (d )>0 d · N (d). Our main theorem asserts that large h and z indices lead to a sublinear algorithm for degree distribution estimation. Theorem 1.3 is a direct corollary obtained by plugging in values of the indices for power laws. Definition 1.2. Fix γ > 2. A degree distribution is bounded below by a power law with exponent γ , if the ccdh satisfies the following property. There exists a constant τ > 0 such that for all d, N (d) ≥ bτn/d γ −1 c. Theorem 1.6. For any ε > 0, the SADDLES algorithm outputs (with high probability) an (ε, ε)-approximation to the ccdh, and makes the following number of queries. e • SM: O(n/h + m/z 2 ) e • HDM: O(m/z) The following is a corollary of our main result. For convenience, we will suppress query complexity dependencies on ε and log n e factors, using O(·). 1.4 Theorem 1.3. Suppose the degree distribution of G is bounded below by a power law with exponent γ . Let the average degree be denoted by d. For any ε > 0, the SADDLES algorithm outputs (with high probability) an (ε, ε)-approximation to the ccdh and makes the following number of queries. 1 1 e 1− γ + n1− γ −1 d) • SM: O(n Challenges The heavy-tailed behavior of the real degree distribution poses the primary challenge to computing (ε, δ )-estimates to the ccdh. As d increases, there are fewer and fewer vertices of that degree. Sampling uniform random vertices is inefficient when N (d) is small. A natural idea to find high degree vertices to pick a random neighbor of a random vertex. Such a sample is more likely to be a high degree vertex. This is the idea behind methods like snowball sampling, forest fire sampling, random walk sampling, graph sample-and-hold, etc. [5, 17, 28, 30, 38, 39, 47]. But these lead to biased samples, since vertices with the same degree may be picked with differing probabilities. 1 e 1− 2(γ −1) d) • HDM: O(n In most real-world instances, the average degree d is typically constant. Thus, the complexities above are strongly sublinear. For e 1/2 ) for both models. When γ = 3, example, when γ = 2, we get O(n e 2/3 ) and O(n e 3/4 ). we get O(n 3 A direct extrapolation/scaling of the degrees in the observed graph does not provide an accurate estimate. Our experiments show that existing methods always miss the head or the tail. A more principled approach was proposed recently by Zhang et al. [47], by casting the estimation of the unseen portion of the distribution as an optimization problem. From a mathematical standpoint, the vast majority of existing results tend to analyze the KS-statistic, or some `p -norm. As we mentioned earlier, this does not work well for measuring the quality of the estimate at all scales. As shown by our experiments, none of these methods give accurate estimate for the entire ccdh with less than 5% of the vertices. The main innovation in SADDLES comes through the use of a recent theoretical technique to simulate edge samples through vertex samples [18, 19]. The sampling of edges occurs through two stages. In the first stage, the algorithm samples a set of r vertices and sets up a distribution over the sampled vertices such that any edge adjacent to a sampled vertex may be sampled with uniform probability. In the second stage, it samples q edges from this distribution. While a single edge is uniform random, the set of edges are correlated. For a given d, we define a weight function on the edges, such that the total weight is exactly N (d). SADDLES estimates the total weight by scaling up the average weight on a random sample of edges, generated as discussed above. The difficulty in the analysis is the correlation between the edges. Our main insight is that if the degree distribution has a fat tail, this correlation can be contained even for sublinear r and q. Formally, this is achieved by relating the concentration behavior of the average weight of the sample to the h and z-indices. The final algorithm combines this area with vertex sampling to get accurate estimates for all d. The hidden degrees model is dealt with using birthday paradox techniques formalized by Ron and √ Tsur [41]. It is possible to estimate the degree dv using O( dv ) queries neighbor queries. But this adds overhead to the algorithm, especially for estimating the ccdh at the tail. As discussed earlier, we need methods that bias towards higher degrees, but this significantly adds to the query cost of actually estimating the degrees. 1.5 Zhang et al. observe that the degree distribution of numerous sampling methods is a random linear projection of the true distribution [47]. They attempt to invert this (ill-conditioned) linear problem, to correct the biases. This leads to improvement in the estimate, but the empirical studies typically sample more than 10% of the vertices for good estimates. A recent line of work by Soundarajan et al. on active probing also has flavors of graph sampling [44, 45]. In this setting, we start with a small, arbitrary subgraph and try to grow this subgraph to achieve some coverage objective (like discover the maximum new vertices, find new edges, etc.). The probing schemes devised in these papers outperform uniform random sampling methods for coverage objectives. All these results aim to capture numerous properties of the graph, using a single graph sample. Nonetheless, the degree distribution is typically considered as the most important, and empirical analyses always focus on estimating it accurately. Ribiero and Towsley [39] and Stumpf and Wiuf [46] specifically study degree distributions. Ribiero and Towsley [39] do detailed analysis on degree distribution estimates (they also look at the ccdh) for a variety of these sampling methods. Their empirical results show significant errors either at the head or the tail. We note that almost all these results end up sampling up to 20% of the graph to estimate the degree distribution. Some methods try to match the shape/family of the distribution, rather than estimate it as a whole [46]. Thus, statistical methods can be used to estimate parameters of the distribution. But it is reasonably well-established that real-world degree distributions are rarely pure power laws in most instances [12]. Indeed, fitting a power law is rather challenging and naive regression fits on log-log plots are erroneous, as results of Clauset-Shalizi-Newman showed [12]. The subfield of property testing and sublinear algorithms for sparse graphs within theoretical computer science can be thought of as a formalization of graph sampling to estimate properties. Indeed, our description of the main problem follows this language. There is a very rich body of mathematical work in this area (refer to Ron’s survey [40]). Practical applications of graph property testing are quite rare, and we are only aware of one previous work on applications for finding dense cores in router networks [25]. The specific problem of estimating the average degree (or the total number of edges) was studied by Feige [21] and Goldreich-Ron [23]. Gonen et al. and Eden et al. focus on the problem of estimating higher moments of the degree distribution [19, 24]. One of the main techniques we use of simulating edge queries was developed in sublinear algorithms results of Eden et al. [18, 19] in the context of triangle counting and degree moment estimation. We stress that all these results are purely theoretical, and their practicality is by no means obvious. On the practical side, Dasgupta, Kumar, and Sarlos study average degree estimation in real graphs, and develop alternate algorithms [14]. They require the graph to have low mixing time and demonstrate that the algorithm has excellent behavior in practice (compared to implementations of Feige’s and the Goldreich-Ron algorithm [21, 23]). Dasgupta et al. note that sampling uniform random vertices is not possible in many settings, and thus they consider a significantly weaker setting than Model 1. Chierichetti Related Work There is a rich body of literature on generating a graph sample that reveals graph properties of the larger “true” graph. We do not attempt to fully survey this literature, and only refer to results directly related to our work. The works of Leskovec & Faloutsos [30], Maiya & Berger-Wolf [31], and Ahmed, Neville, & Kompella [2, 5] provide excellent surveys of multiple sampling methods. There are a number of sampling methods based on random crawls: forest-fire [30], snowball sampling [31], and expansion sampling [30]. As has been detailed in previous work, these methods tend to bias certain parts of the network, which can be exploited for more accurate estimates of various properties [30, 31, 39]. A series of papers by Ahmed, Neville, and Kompella [2–5] have proposed alternate sampling methods that combine random vertices and edges to get better representative samples. Notably, this yields one of the best streaming algorithms for triangle counting [4]. 4 Proof. Let s = mind max(d, N (d)) and let the minimum be attained at d ∗ . If there are multiple minima, let d ∗ be the largest among them. We consider two cases. (Note that N (d) is a monotonically non-increasing sequence.) Case 1: N (d ∗ ) ≥ d ∗ . So s = N (d ∗ ). Since d ∗ is the largest minimum, for any d > d ∗ , d > N (d ∗ ). (If not, then the minimum is also attained at d > d ∗ .) Thus, d > N (d ∗ ) ≥ N (d). For any d < d ∗ , N (d) ≥ N (d ∗ ) ≥ d ∗ > d. We conclude that d ∗ is largest d such that N (d) ≥ d. Thus, h = d ∗ . If s , h, then d ∗ < N (d ∗ ). Then, N (d ∗ + 1) < N (d ∗ ), otherwise the minimum would be attained at d ∗ + 1. Furthermore, max(d ∗ + 1, N (d ∗ + 1)) > N (d ∗ ), implying d ∗ + 1 > N (d ∗ ). This proves that h + 1 > s. Case 2: d ∗ > N (d ∗ ). So s = d ∗ . For d > d ∗ , N (d) ≤ N (d ∗ ) < ∗ d < d. For d < d ∗ , N (d) ≥ d ∗ > d (if N (d) < d ∗ , then d ∗ would not be the minimizer). Thus, d ∗ − 1 is the largest d such that N (d) ≥ d, and h = d ∗ − 1 = s − 1.  et al. focus on sampling uniform random vertices, using only a small set of seed vertices and neighbor queries [10]. We note that there is a large body of work on sampling graphs from a stream [32]. This is quite different from our setting, since a streaming algorithm observes every edge at least once. The specific problem of estimating the degree distribution at all scales was considered by Simpson et al. [43]. They observe many of the challenges we mentioned earlier: the difficulty of estimating the tail accurately, finding vertices at all degree scales, and combining estimates from the head and the tail. 2 PRELIMINARIES We set some notation. The input graph G has n vertices and m edges. For any vertex v, let Γ(v) be the neighborhood of v, and dv be the degree. As mentioned earlier, n(d) is the number of vertices Í of degree d and N (d) = r ≥d n(r ) is the ccdh at d. We use “u.a.r.” as a shorthand for “uniform at random”. We stress that the all mention of probability and error is with respect to the randomness of the sampling algorithm. There is no stochastic assumption on the input graph G. We use the shorthand A ∈ (1 ± α)B for A ∈ [(1 − α)B, (1 + α)B]. We will apply the following (rescaled) Chernoff bound. The h-index does not measure d vs N (d) at different scales, and a large h-index only ensures that there are ”enough” high-degree vertices. For instance, the h-index does not distinguish between N 1 and N 2 when N 1 (100) = 100 and N 1 (d) = 0 for d > 100 and N 2 (100, 000) = 100 and N 2 (d) = 0 for all other values of d ≥ 100. The h and z-indices are related to each other. √ Claim 2.4. h ≤ z ≤ h. Theorem 2.1. [Theorem 1 in [15]] Let X 1 , X 2 , . . . , X k be a sequence of iid random variables with expectation µ. Furthermore, X i ∈ [0, B]. Í • For ε < 1, Pr[| ki=1 X i − µk | ≥ εµk] ≤ 2 exp(−ε 2 µk/3B). Í • For t ≥ 2eµ, Pr[ ki=1 X i ≥ tk] ≤ 2−t k/B . Proof. Since Np(d) is integral, if N (d) > 0, then N (d) ≥ 1. Thus, for all N (d) > 0, max(d, N (d)) ≤ d · N (d) ≤ max(d, N (d)). We take the minimum over all d to complete the proof.  We will require the following “boosting through medians” lemma, which is a routine application of the Chernoff bound. To give some intuition about these indices, we compute the h and z index for power laws. The classic power law degree distribution sets n(d) ∝ 1/d γ , where γ is typically in [2, 3]. Theorem 2.2. Consider two quantities A < B. Suppose there exists a randomized algorithm (that does not know A or B) of expected running time T that outputs a value in [A, B] with probability at least 2/3. Then, for any 0 < δ < 1/3, there exists a randomized algorithm with expected running time O(T log(1/δ )) that outputs a value in (1 ± ε)A with probability at least 1 − δ . Claim 2.5. If a degree distribution is bounded below by a power 1 Proof. Consider d ≤ τn 1/γ , where τ is defined according to Definition 1.2. Then, N (d) ≥ bτn/(τ 1/γ n (γ −1)/γ )c = Ω(n 1/γ ). This proves the h-index bound. Proof. We simply run k = 200 log(1/δ ) independent invocations of the algorithm and output the median output. Let X i be the indicator random variable for the ith invocation outputting a number in [A, B]. Observe that µ = E[X i ] ≥ 2/3. By Í Theorem 2.1, Pr[| ki=1 X i − µk | ≥ µk/5] ≤ 2 exp(−µk/75) ≤ 2 exp(−(2/3)(200/75) log(1/δ )) ≤ δ . Thus, with probability at least 1 − δ , at least (4/5)(2/3)k > k/2 outputs lie in [A, B]. This implies that the median also lies in [A, B].  1 Set d ∗ = (τn) γ −1 . For d ≤ d ∗ , N (d) ≥ 1 and d · N (d) ≥ 1 (τ /2)n/d γ −2 = Ω(n γ −1 ). If there exists no d > d ∗ such that 1 2(γ −1) ). If there does exist some such d, N (d) > 0, then √ z = Ω(n then z = Ω( d ∗ ) which yields the same value.  √ Plugging in values, for γ = 2, both h and z are Ω( n). For γ = 3, h = Θ(n1/3 ) and z = Θ(n1/4 ). It will be convenient to fix the approximation parameter ε > 0 at the very outset. So we will not pass ε as a parameter to our various subroutines. 2.1 1 law with exponent γ , then h = Ω(n γ ) and z = Ω(n 2(γ −1) ). 2.2 Simulating degree queries for HDM The Hidden Degrees Model does not allow for querying the degree dv of a vertex v. Nonetheless, it is possible to get accurate estimates of dv by sampling u.a.r. neighbors (with replacement) of v. This can be done by using the birthday paradox argument, as formalized by Ron and Tsur [41]. Roughly speaking, one repeatedly samples neighbors until the same vertex is seen twice. If this happens after t samples, t 2 is a constant factor approximation for dv . This argument √ can be refined to get accurate approximations for dv using O( dv ) random edge queries. More on Fatness indices The following characterization of the h-index will be useful for analysis. Since (d + N (d))/2 ≤ max(d, N (d)) ≤ d + N (d), this proves that mind (d + N (d))/2 is a 2-factor approximation to the h-index. Lemma 2.3. mind max(d, N (d)) ∈ {h, h + 1} 5 • HDM: O((m/z)(ε −3 log(n/εδ ))). Theorem 2.6. [Theorem 3.1 of [41], restated] Fix any α > 0. There is an algorithm that outputs a value √ in (1 ± α)dv with probability > 2/3, and makes an expected O( dv /α 2 ) u.a.r. neighbor samples. Observe how a larger h and z-index lead to smaller running times. Ignoring constant factors and assuming m = O(n), asymptotically increasing h and z-indices lead to sublinear algorithms. Suppose the degree distribution was a power law with exponent γ > 1. The average degree is constant, so m = O(n). Using the index calculations in §2.1, for SM, the running time is O(n 1−1/γ ). For √ HDM, it is O(n1−1/2(γ −1) ). For γ = 2, the running times are O( n) for both models. For γ = 3, the running times are O(n2/3 ) and O(n3/4 ) respectively. We now describe the algorithm itself. The main innovation in SADDLES comes through the use of a recent theoretical technique to simulate edge samples through vertex samples [18, 19]. The sampling of edges occurs through two stages. In the first stage, the algorithm samples a set of r vertices and sets up a distribution over the sampled vertices such that any edge adjacent to a sampled vertex may be sampled with uniform probability. In the second stage, it samples q edges from this distribution. For each edge, we compute a weight based on the degrees of its vertices and generate our final ccdh estimate by averaging these weights. Additionally, we use vertex sampling to estimate the head of the distribution. Straightforward Chernoff bound arguments can be used to determine when to use the vertex sampling over the edge sampling method. In the following description, we use c to denote a sufficiently large constant. The same algorithmic structure is used for the Standard Model and the Hidden Degrees Model. The only difference is the use the algorithm of Corollary 2.7 to estimate degrees in the HDM, while the degrees are directly available in the Standard Model. We abuse notation somewhat, and use SADDLES to denote the core sampling procedure. As described, this works for a single choice of d to estimate N (d). The final algorithm simply invokes this procedure for various degrees. For the sake of the theoretical analysis, we will simply assume this theorem. In the actual implementation of SADDLES, we will discuss the specific parameters used. It will be helpful to abstract out the estimation of degrees through the following corollary. The procedure DEG(v) will be repeatedly invoked by SADDLES. This is a direct consequence of setting α − ε/10 and applying Theorem 2.2 with δ = 1/n3 . Corollary 2.7. There is an algorithm DEG that takes as input a vertex v, and has the following properties: • For all v: with probability > 1 − 1/n 3 , the output DEG(v) is in (1 ± ε/10)dv . √ • The expected running time of DEG(v) is O(ε −2 dv log n). We will assume that invocations to DEG with the same arguments use the same sequence of random bits. Alternately, imagine that a call to DEG(v, ε) stores the output, so subsequent calls output the same value. Definition 2.8. The output DEG(v) is denoted by dˆv . The random bits used in all calls to DEG is collectively denoted Λ. (Thus, Λ completely specifies all the values {dˆv }.) We say Λ is good if ∀v ∈ V , dˆv ∈ (1 ± ε/10)dv . The following is a consequence of conditional probabilities. Claim 2.9. Consider any event A, such that for any good Λ, Pr[A|Λ] ≥ p. Then Pr[A] ≥ p − 1/n 2 . Proof. The probability that Λ is not good is at most the probability that for some v, DEG(v) < (1 ± ε/10). By the union bound and Corollary 2.7, the probability is at most 1/n 2 . Note that Í Pr[A] ≥ Λgood Pr[Λ] Pr[A|Λ] ≥ p Pr[Λ is good]. Since Λ is good with probability at least 1−1/n2 , Pr[A] ≥ (1−1/n2 )p ≥ p−1/n 2 .  cΛ (d) to be |{v |dˆv ≥ d}|. We will For any fixed Λ, we set N cΛ -values. perform the analysis of SADDLES with respect to the N 4 cΛ (v) ∈ [N ((1 + Claim 2.10. Suppose Λ is good. For all v, N ε/9)d), N ((1 − ε/9)d)]. Claim 4.1. The following holds with probability > 9/10. If e(d) ∈ (1 ± SADDLES(d, r , q) outputs an estimate in Step 6, then N cΛ (d). If it does not output in Step 6, then N cΛ (d) < (2c/ε 2 )(n/r ). ε/10)N Proof. Since Λ is good, ∀u, dˆu ∈ (1 ± ε/10)du , Furthermore, if du ≥ (1 + ε/9)d, then dˆu ≥ (1 − ε/10)(1 + ε/9)d ≥ d. Analogously, if du ≤ (1 − ε/9)d, then dˆu ≤ (1 + ε/10)(1 − ε/9)d ≤ d. Thus, {u|du ≥ d(1 + ε/9)} ⊆ {u|dˆu ≥ d} ⊆ {u|du ≥ d(1 − ε/9)}.  3 ANALYSIS OF SADDLES The estimate of Step 6 can be analyzed with a direct Chernoff bound. Proof. Each X i is an iid Bernoulli random variable, with success cΛ (d)/n. We split into two cases. probability precisely N c Case 1: N Λ (d) ≥ (c/10ε 2 )(n/r ). By the Chernoff bound of Í cΛ (d)/n| ≥ (ε/10)(r N cΛ (d)/n)] ≤ Theorem 2.1, Pr[| i ≤r X i − r N 2 c 2 exp(−(ε /100)(r N Λ (d)/n) ≤ 1/100. cΛ (d) ≤ (c/10ε 2 )(n/r ). Note that E[Íi ≤r X i ] ≤ c/10ε 2 Case 2: N Í 2 ≤ (c/ε )/2e. By the upper tail bound of Theorem 2.1, Pr[ i ≤r X i ≥ 2 c/ε ] < 1/100. Thus, with probability at least 99/100, if an estimate is output cΛ (d) > (c/10ε 2 )(n/r ). By the first case, with probability in Step 6, N e(d) is a (1 + ε/10)-estimate for N cΛ (d). A union at least 99/100, N bound completes the first part. THE MAIN RESULT AND SADDLES We begin by stating the main result, and explaining how heavy tails lead to sublinear algorithms. Theorem 3.1. There exists an algorithm SADDLES with the following properties. For any ε > 0, β > 0, it outputs an (ε, ε)-approximation of the ccdh with probability > 1 − β. The total representation size is O((log n)/ε). The expected running time depends on the model. • SM: O((n/h + m/z 2 )(ε −3 log(n/εδ ))). 6 Lemma 4.4. Fix any good Λ and d. Suppose r ≥ cε −2n/d. With probability at least 9/10, Õ cΛ (d) wtΛ,d (v) ∈ (1 ± ε/8)(r /n)N Algorithm 1: SADDLES(d, r , q) Inputs: d: degree for which N(d) is to be computed r: budget for vertex samples q: budget for edge samples Output: e(d): estimated N (d) N 1 2 3 4 5 6 7 8 9 10 11 12 13 . Í Proof. Let wt(R) denote v ∈R wtΛ,d (v). By linearity of Í cΛ (d). To expectation, E[wt(R)] = (r /n)· v ∈V wtΛ,d (v) ≥ (r /2n)N apply the Chernoff bound, we need to bound the maximum weight of a vertex. For good Λ, the weight wtΛ,d of any ordered pair is at most 1/(1 − ε/10)d ≤ 2/d. The number of neighbors of v such that cΛ (d). Thus, wtΛ,d (v) ≤ 2N cΛ (d)/d. dˆu ≥ d is at most N By the Chernoff bound of Theorem 2.1 and setting r ≥ cε −2n/d, Repeat for i = 1, . . . , r ; Select u.a.r. vertex v and add it to multiset R ; In HDM, call DEG(v) to get estimate dˆv . In SM, set dˆv to dv . ; If dˆ ≥ d, set X i = 1. Else, X i = 0. ; Í v If i ≤r X i ≥ c/ε 2 ; e(d) = n Íi ≤k X i ; Return N r Í Let dˆR = v ∈R dˆv and D denote the distribution over R where v ∈ R is selected with probability dˆv /dˆR ; Repeat for i = 1, . . . , q ; Sample v ∼ D ; Pick u.a.r. neighbor u of v ; Call DEG(u) to get dˆu ; If dˆu ≥ d, set Yi = 1/dˆu . Else, set Yi = 0 ; q e(d) = n · dˆR Í Yi . Return N r q Pr [|wt(R) − E[wt(R)]| > (ε/20)E[wt(R)]] ! cΛ (d)/2n) ε 2 · (cε −2n/d) · (N < 2 exp − ≤ 1/10 cΛ (d)/d 3 · 202 · 2N With probability at least 9/10, wt(R) ∈ (1 ± ε/20)E[wt(R)]. By cΛ (d). We the arguments given above, E[wt(R)] ∈ (1 ± ε/9)(r /n)N combine to complete the proof.  Now, we determine the number of edge samples required to estimate the weight wtΛ,d (R). e(d) be as defined in Step 13 of SADDLES. Assume Lemma 4.5. Let N cΛ (d)). Then, with Λ is good, r ≥ cε −2n/d, and q ≥ cε −2m/(d N e(d) ∈ (1 ± ε/4)N cΛ (d). probability > 7/8, N i=1 cΛ (d) ≥ (2c/ε 2 )(n/r ), then with probability at Furthermore, if N Í cΛ (d)/n ≥ c/ε 2 . A union boun least 99/100, i ≤r X i ≥ (1 − ε/10)r N proves (the contrapositive of) the second part.  Proof. We define the random set R selected in Step 2 to be Í sound if the following hold. (1) wt(R) = v ∈R wtΛ,d (v) ∈ (1 ± Í cΛ (d) and (2) v ∈R dv ≤ 100r (2m/n). By Lemma 4.4, the ε/8)(r /n)N Í first holds with probability > 9/10. Observe that E[ v ∈R dv ] = r (2m/n), since 2m/n is the average degree. By the Markov bound, the second holds with probability > 99/100. By the union bound, R is sound with probability at least 1 − (1/10 + 1/100) > 8/9. Fix a sound R. Recall Yi from Step 12. The expectation of Yi |R Í Í is v ∈R Pr[v is selected]· u ∈Γ(v) Pr[u is selected]wtΛ,d (hv, ui). We plug in the probability values, and observe that for good Λ, for all v, dˆv /dv ∈ (1 ± ε/10). Õ Õ (1/dv )wtΛ,d (hv, ui) E[Yi |R] = (dˆv /dˆR ) We define weights of ordered edges. The weight only depends on the second member in the pair, but allows for a more convenient analysis. The weight of hv, ui is the random variable Yi of Step 12. Definition 4.2. The d-weight of an ordered edge hv, ui for a given Λ (the randomness of DEG) is defined as follows. We set wtΛ,d (hv, ui) to be 1/dˆu if dˆu ≥ d, and zero otherwise. Í For vertex v, wtΛ,d (v) = u ∈Γ(v) wtΛ,d (hv, ui). The utility of the weight definition is captured by the following e(d), and thus, we claim. The total weight is an approximation of N can analyze how well SADDLES approximates the total weight. Í cΛ (d). Claim 4.3. If Λ is good, v ∈V wtΛ,d (v) ∈ (1 ± ε/9)N Proof. Õ wtΛ,d (v) = v ∈V Õ Õ v ∈V u ∈Γ(v) = Õ v ∈R = u:dˆu ≥d v ∈Γ(u) 1/dˆu = (1/dˆR ) u ∈Γ(v) Õ (dˆv /dv ) v ∈R ∈ 1dˆu ≥d /dˆu Õ v ∈R (1 ± ε/10)(1/dˆR ) Õ wtΛ,d (hv, ui) u ∈Γ(v) Õ Õ wtΛ,d (hv, ui) v ∈R u ∈Γ(v) Õ du /dˆu (1 ± ε/10)(wt(R)/dˆR ) (2) Í ˆ e N (d) = (n/r )(d R /q) i ≤q Yi Note that and Í (n/r )(dˆR /q)E[ i ≤q Yi |R] ∈ (1 ± ε/10)(n/r )wt(R). Since R is cΛ (d). Also, note that sound, the latter is in (1 ± ε/4)N ∈ (1) u:dˆu ≥d Since Λ is good, ∀u, dˆu ∈ (1 ± ε/10)du , and du /dˆu ∈ (1 ± ε/9). Í cΛ (d). Applying in (1), v ∈V wtΛ,d (v) ∈ (1 ± ε/9)N  cΛ (d) c (d) qwt(R) (r /n)N N ≥ = Λ (3) ˆ 4(100r (2m/n) 800m 2d R Í By linearity of expectation, E[ i ≤q Yi |R] = qE[X 1 |R]. Observe that Yi ≤ 1/d. We can apply the Chernoff bound of Theorem 2.1 to E[Yi |R] = E[X 1 |R] ≥ We come to an important lemma, that shows that the weight of the random subset R (chosen in Step 2) is well-concentrated. This is proven using a Chernoff bound, but we need to bound the maximum possible weight to get a good bound on r = |R|. 7 Proof. (of Theorem 3.1) The overall algorithm is the same for both models, involving multiple invocations to SADDLES. The only difference is in DEG, which is trivial when degree queries are allowed. We first argue about correctness. Consider the set D = {b(1 + ε/10)i c|0 ≤ i ≤ 10ε −1 log n}. We will run a boosted version of SADDLES for each degree in D. The e(d 0 ), where d 0 is the largest output for an arbitrary d will be N power of (1 + ε/10) smaller than d (rounding down). The boosting is done through Theorem 2.2, which ensures we can get the desired estimate for each d with probability > 1 − εβ/100n. A union bound over all d ∈ D yields a total error probability of at most β. Observe that the query complexity and running time of SADDLES are within constant factors of each other. Hence, we only focus on the number of queries made. For the Standard Model, the bound for a single invocation of SADDLES is simply O(r + q) = O(ε −2 (n/h + m/z 2 )). For the Hidden Degrees Model, we have to account for the overhead of using Ron-Tsur birthday paradox algorithm of Corollary 2.7 for each degree estimated. The number of queries √ for a single call to DEG(d) is O(ε −2 d log n). The total overhead v √ Í of all calls in Step 3 is E[ v ∈R d√v (ε −2 log n)]. By linearity of expectation, this is O((ε −2 log n)r E[ dv ], where the√expectation is over a uniform random vertex. We can bound r E[ dv ] ≤ r E[dv ] = O(ε −2n(m/n)/h) = O(ε −2n/h). The total overhead of all calls in Step 11 requires more care. Note that when DEG(v) is called multiple times for a fixed v, the subsequent calls require no further queries. (This is because the output of the first call can be stored.) We partition the vertices into two sets S 0 = {v |dv ≤ z 2 } and S 1 = {v |dv > z 2 }. The total query cost of queries to S 0 is at most O(q f ) = O((ε −2 log n)m/z). For the total cost we directly√bound by (ignoring the ε −2 log n √ to S 1 ,Í Í Í factor) v ∈S 1 dv = v ∈S 1 dv / dv ≤ z −1 v dv = O(m/z). All −4 in all, the total query complexity is O((ε log2 )(n/h + m/z)). Since m ≥ n and z ≤ h, we can simplify to O((ε −4 log2 n)(m/z)).  the iid random variables (Yi |R). Õ Õ Õ Pr[| Yi − E[ Yi ]| > (ε/100)E[ Yi ]|R] i  i  · d · qE[X 1 |R] (4) 3 · 1002 We use (3) to bound the (positive) term in the exponent is at least ≤ 2 exp − i ε2 c (d) cε −2m N ε2 · · Λ ≥ 10. 2 cΛ (d) 800m 3 · 100 N Thus, if R is sound, the following bound holds with probability at least 0.99. We also apply (2). cΛ (d) N = (n/r )(dˆR /q) q Õ Yi i=1 ∈ (1 ± ε/100)(n/r )(dˆR /q)qE[Yi |R] ∈ e(d) (1 ± ε/100)(1 ± ε/10)(n/r )wt(R) ∈ (1 ± ε/4)N The probability that R is sound is at least 8/9. A union bound completes the proof.  The bounds on r and q in Lemma 4.5 depend on the degree d. We now bring in the h and z-indices to derive bounds that hold for all d. We also remove the conditioning over a good Λ. Theorem 4.6. Let c be a sufficiently large constant. Suppose r ≥ cε −2n/h and q ≥ cε −2m/z 2 . Then, for all d, with probability ≥ 5/6, e(d) ∈ [(1 − ε/2)N ((1 + ε/2)d), (1 + ε/2)N ((1 − ε/2)d]. N Proof. We will first assume that Λ is good. By Claim 2.10, cΛ (d) ∈ [N ((1 + ε/9)d, N ((1 − ε/9)d)]. N cΛ (d) = 0, so there are no vertices with dˆv ≥ d. By the Suppose N bound above, N ((1 + ε/9)d) = 0, implying that N ((1 + ε/2)d) = 0. e(d) = 0, since the random variables X i and Yi in Furthermore N e(d) = N ((1 + ε/2)d), SADDLES can never be non-zero. Thus, N completing the proof. cΛ (d) > 0. We split into two cases, We now assume that N depending on whether Step 6 outputs or not. By Claim 4.1, with e(d) ∈ (1 ± ε/9)N cΛ (d). probability > 9/10, if Step 6 outputs, then N e(d) holds with By combining these bounds, the desired bound on N probability > 9/10, conditioned on a good Λ. Henceforth, we focus on the case that Step 6 does not output. By cΛ (d) < 2cε −2 (n/r ). By the choice of r and Claim 2.10, Claim 4.1, N cΛ ((1 + ε/9)d) < h. By the characterization of h of Lemma 2.3, N cΛ ((1 + ε/9)d), (1 + ε/9)d) = (1 + ε/9)d. z 2 ≤ max(N cε −2n/d. 5 EXPERIMENTAL RESULTS We implemented our algorithm in C++ and performed our experiments on a MacBook Pro laptop with 2.7 GHz Intel Core i5 with 8 GB RAM. We performed our experiments on a collection of graphs from SNAP [29], including social networks, web networks, and infrastructure networks. The graphs typically have millions of edges, with the largest with more than 100M edges. Basic properties of these graphs are presented in Table 1. We ignore direction and treat all edges as undirected edges. (5) z2 This implies that r ≥ By the definition of z, ≤ N (min(dmax , (1 +ε/9)d)) · min(dmax , (1 +ε/9)d). By the Claim 2.10 cΛ (d) ≥ N ((1+ε/9)d). Since N cΛ (d) > bound in the first paragraph, N 2 c c c 0, N Λ (d) ≥ N Λ (dmax ). Plugging into (5), z ≤ N Λ (d) · (1 + ε/9)d. cΛ (d)). The parameters satisfy the conditions Thus, m ≤ cε −2m/(d N e(d) ∈ (1 ± ε/4)N cΛ (d), and in Lemma 4.5. With probability > 7/8, N e(d) has the desired accuracy. by Claim 2.10, N e(d) All in all, assuming Λ is good, with probability at least 7/8, N has the desired accuracy. The conditioning on a good Λ is removed by Claim 2.9 to complete the proof.  5.1 Implementation Details For the Hidden Degrees Model, we explicitly describe the procedure DEG, which estimates the degree of a given vertex. In the algorithm Algorithm 2: DEG(v) 1 2 3 We finally prove Theorem 3.1. 8 Initialize S = ∅ ; Repeatedly add u.a.r. vertex to S, until the number of pair-wise collisions is at least k = 50 ;
Output |S | /k as estimate dˆv 2 size of n/h + m/z 2 (as given by Theorem 3.1, ignoring constants) is significantly sublinear. This is consistent with our choice of r + q = n/100 leading to accurate estimates for the ccdh. DEG, a “pair-wise collision” refers to a pair of neighbor samples that yield the same vertex.
If S has size t, the expected number of pair-wise collisions is t2 /dv . We simply reverse engineer that inequality to get the estimate dˆv . Ron√and Tsur essentially prove that with high probability, |S | = Θ( dv ) and furthermore, this suffices to bound the variance of the estimate [41]. Our implementation of SADDLES is identical to the pseudo-code given in Alg. 1. The only constant to be set is c/ε 2 in Step 5, which our implementation fixes at 25. There are two parameters r and q that are chosen to be typically around 0.005n. To get the entire degree distribution, we run SADDLES on all degrees d = b1.1i c. 5.2 5.3 Comparison with previous work There are several graph sampling algorithms that have been discussed in [2, 17, 28, 30, 38, 39, 47]. We describe these methods below in more detail, and discuss our implementation of the method. • Vertex Sampling (VS, also called egocentric sampling) [5, 17, 28, 30, 38, 39]: In this algorithm, we sample vertices u.a.r. and scale the ccdh obtained appropriately, to get an estimate for the ccdh of the entire graph. • Edge Sampling (ES) [5, 17, 28, 30, 38, 39]: This algorithm samples edges u.a.r. and includes one or both end points in the sampled network. Note that this does not fall into the standard model. In our implementation we pick a random end point. • Random walk with jump (RWJ) [5, 17, 30, 38, 39]: We start a random walk at a vertex selected u.a.r. and collect all vertices encountered on the path in our sampled network. At any point, with a constant probability (0.15 in our implementation, based on previous results) we jump to another u.a.r. vertex. • One Wave Snowball (OWS) [5, 17, 28]: Snowball sampling starts with some vertices selected u.a.r. and crawls the network until a network of the desired size is sampled. In our implementation, we typically stop at the one level since that accumulates enough vertices. • Forest fire (FF) [5, 17, 30]: This method generates random sub-crawls of the network, and is related to snowball sampling. A vertex is picked u.a.r. and randomly selects a subset of its neighbors. In previous work, this is done by choosing x such neighbors, where x is a geometric random variable with mean 0.2. The process is repeated from every selected vertex until it ends. It is then repeated from another u.a.r. vertex. We run all these algorithms on the amazon0601, web-Google, cit-Patents, and com-orkut networks. To make fair comparisons, we run each method until it selects 1% of the vertices. The comparisons are shown in Fig. 1. Observe how none of the methods come close to accurately measuring the ccdh. (This is consistent with previous work, where typically 10-20% of the vertices are sampled for results.) Naive vertex sampling is accurate at the head of the distribution, but completely misses the tail. Except for vertex sampling, all other algorithms are biased towards the tail. Crawls find high degree vertices with disproportionately higher probability, and overestimate the tail. Evaluation of SADDLES The sample size of SADDLES in the Standard Model is exactly r + q. We will typically fix this to be 1% of the number of vertices in our runs, unless otherwise stated. Accuracy over all graphs. We show results of running SADDLES with the parameters discussed above for a variety of graphs. Fig. 1, and Fig. 2 show the results for the Standard Model on all graphs in Tab. 1. For all these runs, we set r + q to be 1% of the number of vertices in the graph. For the Hidden Degrees Model, we show results in Fig. 3. For space reasons, we only show results on HDM for the graphs in Fig. 2, though results are consistent over all our experiments. Again, we set r + q to be 1%, though the number of edges sampled varies quite a bit. The required number of samples are provided in Tab. 1. Note that the number of edges sampled is well within 10% of the total, except for the com-youtube graph. Visually, we can see that the estimates are accurate for all degrees, in all graphs, for both models. This is despite there being sufficient irregular behavior in N (d). For example, the web-BerkStan ccdh (Fig. 1) is quite “bumpy” between degree 102 and 104 , and the extreme tail has sudden jumps. Note that the shape of the various ccdhs are different and none of them form an obvious straight line. Nonetheless, SADDLES captures the distribution almost perfectly in all cases by observing 1% of the vertices. Convergence. To demonstrate convergence, we use the following setup. In the figures, we fix the graph com-orkut, and run SADDLES only for the degrees 10, 100, and 1000. For each choice of degree, we vary the total number of samples r + q. (We set r = q in all runs.) Finally, for each setting of r + q and each degree, we perform 100 independent runs of SADDLES. For each such run, we compute an error parameter α. Suppose the output of a run is M, for degree d. The value of α is the smallest value of ϵ, such that M ∈ [(1 − ϵ)N ((1 + ϵ)d), (1 + ϵ)N ((1 − ϵ)d)]. (It is the smallest ϵ such that M is an (ϵ, ϵ)-approximation of N (d).) Fig. 4 shows the spread of α, for the 100 runs, for each choice of r + q. Observe how the spread decreases as r + q goes to 10%. In all cases, the values of α decay to less than 0.05. We notice that convergence is much faster for d = 10. This is because N (10) is quite large, and SADDLES is using vertex sampling to estimate the value. Inverse method of Zhang et al [47]. An important result of estimating degree distributions is that of Zhang et al [47], that explicitly points out the bias problems in various sampling methods. They propose a bias correction method by solving an ill-conditioned linear system. Essentially, given one of the above sampled networks, it applies a constrained, penalized weighted least-squares approach to solving the problem of debiasing the estimated degree distribution. We apply this method for the sampling methods demonstrated in their paper, namely vertex sampling (VS), one-wave snowball (OWS), and induced sampling (IN) (sample vertices u.a.r. and only retain edges between sampled vertices). We show results in Fig. 1, again with a sample size of 1% of Large value of h and z-index on real graphs. The h and z-index of all graphs is given in Tab. 1. Observe how they are typically in the hundreds. Note that the average degree is typically an order of magnitude smaller than these indices. Thus, a sample 9 Table 1: Graph properties: #vertices (n), #edges (m), maximum degree, h-index and z-index. The last column indicates the median number of samples over 100 runs (as a percentage of m) required by SADDLES under HDM to estimate the ccdh for r + q = 0.01n. For all graphs except one, the number of samples required is < 0.1m. graph loc-gowalla web-Stanford com-youtube web-Google web-BerkStan wiki-Talk as-skitter cit-Patents com-lj soc-LiveJournal1 com-orkut #vertices 1.97E+05 2.82E+05 1.13E+06 8.76E+05 6.85E+05 2.39E+06 1.70E+06 3.77E+06 4.00E+06 4.85E+06 3.07E+06 #edges 9.50E+05 1.99E+06 2.99E+06 4.32E+06 6.65E+06 9.32E+06 1.11E+07 1.65E+07 3.47E+07 8.57E+07 1.17E+08 max. degree 14730 38625 28754 6332 84230 100029 35455 793 14815 20333 33313 avg. degree 4.8 7.0 2.6 4.9 9.7 3.9 6.5 4.3 8.6 17.7 38.1 H-index 275 427 547 419 707 1055 982 237 810 989 1638 Z-index 101 148 121 73 220 180 184 28 114 124 172 Perc. edge samples for HDM) 7.0 6.4 11.7 6.2 5.5 8.5 6.7 5.6 4.7 2.4 2.0 (a) as-skitter (b) loc-gowalla (c) web-Google (d) wiki-Talk (e) soc-LiveJournal (f) com-lj (g) web-BerkStan (h) com-youtube Figure 2: The result of runs of SADDLES on a variety of graphs, for the Standard Model. We set r + q to be 1% of the number of vertices, for all graphs. Observe the close match at all degrees between the true degree distribution and output of SADDLES. the vertices. Observe that no method get even close to estimating the ccdh accurately, even after debiasing. Fundamentally, these methods require significantly more samples to generate accurate estimates. The running time and memory requirements of this method grow superlinearly with the maximum degree in the graph. The maximum degree is not known in advance, but the algorithm needs to know this value , so it uses an upper bound. The largest graph processed by [47] has a few hundred thousand edges, which is on the smaller side of graphs in Tab. 1. SADDLES processes a graph with more than 100M edges in less than a minute, while our attempts to run the [47] algorithm on this graph did not terminate in hours. 6 ACKNOWLEDGEMENTS Ali Pinar’s work is supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia 10 (a) as-skitter (b) loc-gowalla (c) web-Google (d) wiki-Talk Figure 3: The result of runs of SADDLES on a variety of graphs, for the Hidden Degrees Model. We set r + q to be 1% of the number of vertices, for all graphs. The actual number of edges sampled varies, and is given in Tab. 1. (a) d = 10 (b) d = 100 (c) d = 1000 (d) d = 10000 Figure 4: Convergence of SADDLES: We plot the values of the error parameter α (as defined in §5.2) for 100 runs at increasing values of r + q. We have a different plot for d = 10, 100, 1000, 10000 to show the convergence at varying portions of the ccdh. National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. Both Shweta Jain and C. Seshadhri are grateful to the support of the Sandia National Laboratories LDRD program for funding this research. C. Seshadhri also acknowledges the support of NSF TRIPODS grant. This research was partially supported by the Israel Science Foundation grant No. 671/13 and by a grant from the Blavatnik fund. Talya Eden is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. Both Talya Eden and C. Seshadhri are grateful to the support of the Simons Institute, where this work was initiated during the Algorithms and Uncertainty Semester. [5] Nesreen K Ahmed, Jennifer Neville, and Ramana Kompella. 2014. 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arXiv:1704.00699v2 [math.DS] 14 Jan 2018 FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS CLINTON T. CONLEY, STEVE C. JACKSON, DAVID KERR, ANDREW S. MARKS, BRANDON SEWARD, AND ROBIN D. TUCKER-DROB Abstract. We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G (“shapes”) with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz–Huczek– Zhang tiling theorem for countable amenable groups and strengthens the Ornstein–Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable. 1. Introduction A discrete group G is said to be amenable if it admits a finitely additive probability measure which is invariant under the action of G on itself by left translation, or equivalently if there exists a unital positive linear functional ℓ∞ (G) → C which is invariant under the action of G on ℓ∞ (G) induced by left translation (such a functional is called a left invariant mean). This definition was introduced by von Neumann in connection with the Banach–Tarski paradox and shown by Tarski to be equivalent to the absence of paradoxical decompositions of the group. Amenability has come to be most usefully leveraged through its combinatorial expression as the Følner property, which asks that for every finite set K ⊆ G and δ > 0 there exists a nonempty finite set F ⊆ G which is (K, δ)-invariant in the sense that |KF ∆F | < δ|F |. The concept of amenability appears as a common thread throughout much of ergodic theory as well as the related subject of operator algebras, where it is known via a number of avatars like injectivity, hyperfiniteness, and nuclearity. It forms the cornerstone of the theory of orbit equivalence, and also underpins both Kolmogorov–Sinai entropy and the classical ergodic theorems, whether explicitly in their most general formulations or implicitly in the original setting of single transformations (see Chapters 4 and 9 of [11]). A key tool in applying amenability to dynamics is the Rokhlin lemma of Ornstein and Weiss, which in one of its simpler forms says that for every free probability-measurepreserving action G y (X, µ) of a countably infinite amenable group and every finite set K ⊆ G and δ > 0 there exist (K, δ)-invariant finite sets T1 , . . . , Tn ⊆ G and measurable sets A1 , . . . , An ⊆ X such that the sets sAi for i = 1, . . . , n and s ∈ Ti are pairwise disjoint and have union of measure at least 1 − δ [17]. The proportionality in terms of which approximate invariance is expressed in the Følner condition makes it clear that amenability is a measure-theoretic property, and it is not Date: November 9, 2017. 1 2 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB surprising that the most influential and definitive applications of these ideas in dynamics (e.g., the Connes–Feldman–Weiss theorem) occur in the presence of an invariant or quasi-invariant measure. Nevertheless, amenability also has significant ramifications for topological dynamics, for instance in guaranteeing the existence of invariant probability measures when the space is compact and in providing the basis for the theory of topological entropy. In the realm of operator algebras, similar comments can be made concerning the relative significance of amenability for von Neumann algebras (measure) and C∗ -algebras (topology). While the subjects of von Neumann algebras and C∗ -algebras have long enjoyed a symbiotic relationship sustained in large part through the lens of analogy, and a similar relationship has historically bound together ergodic theory and topological dynamics, the last few years have witnessed the emergence of a new and structurally more direct kind of rapport between topology and measure in these domains, beginning on the operator algebra side with the groundbreaking work of Matui and Sato on strict comparison, Z-stability, and decomposition rank [14, 15]. On the side of groups and dynamics, Downarowicz, Huczek, and Zhang recently showed that if G is a countable amenable group then for every finite set K ⊆ G and δ > 0 one can partition (or “tile”) G by left translates of finitely many (K, δ)-invariant finite sets [3]. The consequences that they derive from this tileability are topological and include the existence, for every such G, of a free minimal action with zero entropy. One of the aims of the present paper is to provide some insight into how these advances in operator algebras and dynamics, while seemingly unrelated at first glance, actually fit together as part of a common circle of ideas that we expect, among other things, to lead to further progress in the structure and classification theory of crossed product C∗ -algebras. Our main theorem is a version of the Downarowicz–Huczek–Zhang tiling result for free p.m.p. (probability-measure-preserving) actions of countable amenable groups which strengthens the Ornstein–Weiss Rokhlin lemma in the form recalled above by shrinking the leftover piece down to a null set (Theorem 3.6). As in the case of groups, one does not expect the utility of this dynamical tileability to be found in the measure setting, where the Ornstein–Weiss machinery generally suffices, but rather in the derivation of topological consequences. Indeed we will apply our tiling result to show that, for every countably infinite amenable group G, the crossed product C(X) ⋊ G of a generic free minimal action G y X on the Cantor set possesses the regularity property of Z-stability (Theorem 5.4). The strategy is to first prove that such an action admits clopen tower decompositions with arbitrarily good Følner shapes (Theorem 4.2), and then to demonstrate that the existence of such tower decompositions implies that the crossed product is Z-stable (Theorem 5.3). The significance of Z-stability within the classification program for simple separable nuclear C∗ -algebras is explained at the beginning of Section 5. It is a curious irony in the theory of amenability that the Hall–Rado matching theorem can be used not only to show that the failure of the Følner property for a discrete group implies the formally stronger Tarski characterization of nonamenability in terms of the existence of paradoxical decompositions [2] but also to show, in the opposite direction, that the Følner property itself implies the formally stronger Downarowicz–Huczek–Zhang characterization of amenability which guarantees the existence of tilings of the group by translates of finitely many Følner sets [3]. This Janus-like scenario will be reprised FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 3 here in the dynamical context through the use of a measurable matching argument of Lyons and Nazarov that was originally developed to prove that for every simple bipartite nonamenable Cayley graph of a discrete group G there is a factor of a Bernoulli action of G which is an a.e. perfect matching of the graph [13]. Accordingly the basic scheme for proving Theorem 3.6 will be the same as that of Downarowicz, Huczek, and Zhang and divides into two parts: (i) using an Ornstein–Weiss-type argument to show that a subset of the space of lower Banach density close to one can be tiled by dynamical translates of Følner sets, and (ii) using a Lyons–Nazarov-type measurable matching to distribute almost all remaining points to existing tiles with only a small proportional increase in the size of the Følner sets, so that the approximate invariance is preserved. We begin in Section 2 with the measurable matching result (Lemma 2.6), which is a variation on the Lyons–Nazarov theorem from [13] and is established along similar lines. In Section 3 we establish the appropriate variant of the Ornstein–Weiss Rokhlin lemma (Lemma 3.4) and put everything together in Theorem 3.6. Section 4 contains the genericity result for free minimal actions on the Cantor set, while Section 5 is devoted to the material on Z-stability. Acknowledgements. C.C. was partially supported by NSF grant DMS-1500906. D.K. was partially supported by NSF grant DMS-1500593. Part of this work was carried out while he was visiting the Erwin Schrödinger Institute (January–February 2016) and the Mittag– Leffler Institute (February–March 2016). A.M. was partially supported by NSF grant DMS-1500974. B.S. was partially supported by ERC grant 306494. R.T.D. was partially supported by NSF grant DMS-1600904. Part of this work was carried out during the AIM SQuaRE: Measurable Graph Theory. 2. Measurable matchings Given sets X and Y and a subset R ⊆ X × Y , with each x ∈ X we associate its vertical section Rx = {y ∈ Y : (x, y) ∈ R} and with each y ∈ Y we associate S its horizontal section Ry = {x ∈ X : (x, y) ∈ R}. Analogously, for A ⊆ X we put RA = x∈A Rx = {y ∈ Y : ∃x ∈ A (x, y) ∈ R}. We say that R is locally finite if for all x ∈ X and y ∈ Y the sets Rx and Ry are finite. If now X and Y are standard Borel spaces equipped with respective Borel measures µ and ν, we say that R ⊆ X × Y is (µ, ν)-preserving if whenever f : A → B is a Borel bijection between subsets A ⊆ X and B ⊆ Y with graph(f ) ⊆ R we have µ(A) = ν(B). We say that R is expansive if there is some c > 1 such that for all Borel A ⊆ X we have ν(RA ) ≥ cµ(A). We use the notation f : X ⇀ Y to denote a partial function from X to Y . We say that such a partial function f is compatible with R ⊆ X × Y if graph(f ) ⊆ R. Proposition 2.1 (ess. Lyons–Nazarov [13, Theorem 1.1]). Suppose that X and Y are standard Borel spaces, that µ is a Borel probability measure on X, and that ν is a Borel measure on Y . Suppose that R ⊆ X × Y is Borel, locally finite, (µ, ν)-preserving, and 4 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB expansive. Then there is a µ-conull X ′ ⊆ X and a Borel injection f : X ′ → Y compatible with R. Proof. Fix a constant of expansivity c > 1 for R. We construct a sequence (fn )n∈N of Borel partial injections from X to Y which are compatible with R. Moreover, we will guarantee that the set X ′ = {x ∈ X : ∃m ∈ N ∀n ≥ m x ∈ dom(fn ) and fn (x) = fm (x)} is µ-conull, establishing that the limiting function satisfies the conclusion of the lemma. Given a Borel partial injection g : X ⇀ Y we say that a sequence (x0 , y0 , . . . , xn , yn ) ∈ X × Y × · · · × X × Y is a g-augmenting path if • x0 ∈ X is not in the domain of g, • for all distinct i, j < n, yi 6= yj , • for all i < n, (xi , yi ) ∈ R, • for all i < n, yi = g(xi+1 ), • yn ∈ Y is not in the image of g. We call n the length of such a g-augmenting path and x0 the origin of the path. Note that the sequence (x0 , y0 , y1 , . . . , yn ) in fact determines the entire g-augmenting path, and moreover that xi 6= xj for distinct i, j < n. In order to proceed we require a few lemmas. Lemma 2.2. Suppose that n ∈ N and g : X ⇀ Y is a Borel partial injection compatible with R admitting no augmenting paths of length less than n. Then µ(X \ dom(g)) ≤ c−n . Proof. Put A0 = X \ dom(g). Define recursively for i < n sets Bi = RAi and Ai+1 = Ai ∪ g −1 (Bi ). Note that the assumption that there are no augmenting paths of length less than n implies that each Bi is contained in the image of g. Expansivity of R yields ν(Bi ) ≥ cµ(Ai ) and (µ, ν)-preservation of R then implies that µ(Ai+1 ) ≥ ν(Bi ) ≥ cµ(Ai ). Consequently, 1 ≥ µ(An ) ≥ cn µ(A0 ), and hence µ(A0 ) ≤ c−n .  We say that a graph G on a standard Borel space X has a Borel N-coloring if there is a Borel function c : X → N such that if x and y are G-adjacent then c(x) 6= c(y). Lemma 2.3 (Kechris–Solecki–Todorcevic [12, Proposition 4.5]). Every locally finite Borel graph on a standard Borel space has a Borel N-coloring. Proof. Fix a countable algebra {Bn : n ∈ N} of Borel sets which separates points (for example, the algebra generated by the basic open sets of a compatible Polish topology), and color each vertex x by the least n ∈ N such that Bn contains x and none of its neighbors.  Analogously, for k ∈ N, we say that a graph on a standard Borel X has a Borel kcoloring if there is a Borel function c : X → {1, . . . , k} giving adjacent points distinct colors. Lemma 2.4 (Kechris–Solecki–Todorcevic [12, Proposition 4.6]). If a Borel graph on a standard Borel X has degree bounded by d ∈ N, then it has a Borel (d + 1)-coloring. Proof. By Lemma 2.3, the graph has a Borel N-coloring c : X → N. We recursively build sets An for n ∈ N by A0 = {x ∈ X : c(x) = 0} and An+1 = An ∪ {x ∈ X : FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 5 S c(x) = n + 1 and no neighbor of x is in An }. Then A = n An is a Borel set which is G-independent, and moreover is maximal with this property. So the restriction of G to X \ A has degree less than d, and the result follows by induction.  Lemma 2.5 (ess. Elek–Lippner [5, Proposition 1.1]). Suppose that g : X ⇀ Y is a Borel partial injection compatible with R, and let n ≥ 1. Then there is a Borel partial injection g′ : X ⇀ Y compatible with R such that • dom(g ′ ) ⊇ dom(g), • g′ admits no augmenting paths of length less than n, • µ({x ∈ X : g ′ (x) 6= g(x)} ≤ nµ(X \ dom(g)). Proof. Consider the set Z of injective sequences (x0 , y0 , x1 , y1 , . . . , xm , ym ), where m < n, such that for all i ≤ m we have (xi , yi ) ∈ R and for all i < m we have (xi+1 , yi ) ∈ R. Equip Z with the standard Borel structure it inherits as a Borel subset of (X × Y )≤n . Consider also the locally finite Borel graph G on Z rendering adjacent two distinct sequences in F Z if they share any entries. By Lemma 2.3 there is a partition Z = k∈N Zk of Z into Borel sets such that for all k, no two elements of Zk are G-adjacent. In other words, we partition potential augmenting paths into countably many colors, where no two paths of the same color intersect. Thus we may flip paths of the same color simultaneously without risk of causing conflicts. Towards that end, fix a bookkeeping function s : N → N such that s−1 (k) is infinite for all k ∈ N in order to consider each color class infinitely often. Given a g-augmenting path z = (x0 , y0 , . . . , xm , ym ), define the flip along z to be the Borel partial function gz : X ⇀ Y given by ( yi if ∃i ≤ m x = xi , gz (x) = g(x) otherwise. The fact that z is g-augmenting ensures that gz is injective. More generally, for any Borel G-independent set Z aug ⊆ Z of g-augmenting paths, we may simultaneously flip g along all paths in Z aug to obtain another Borel partial injection (g)Z aug . We iterate this construction. Put g0 = g. Recursively assuming that gk : X ⇀ Y has been defined, let Zkaug be the set of gk -augmenting paths in Zs(k) , and let gk+1 = (gk )Z aug k be the result of flipping gk along all paths in Zkaug . As each x ∈ X is contained in only finitely many elements of Z, and since each path in Z can be flipped at most once (after the first flip its origin is always in the domain of the subsequent partial injections), it follows that the sequence (gk (x))k∈N is eventually constant. Defining g′ (x) to be the limiting value, it is routine to check that there are no g′ -augmenting paths of length less than n. Finally, to verify the third item of the lemma, put A = {x ∈ X : g′ (x) 6= g(x)}. With each x ∈ A associate the origin of the first augmenting path along which it was flipped. This is an at most n-to-1 Borel function from A to X \ dom(g), and since R is (µ, ν)-preserving the bound follows.  We are now in position to follow the strategy outlined at the beginning of the proof. Let f0 : X ⇀ Y be the empty function. Recursively assuming the Borel partial injection fn : X ⇀ Y has been defined to have no augmenting paths of length less than n, let fn+1 be the Borel partial injection (fn )′ granted by applying Lemma 2.5 to fn . Thus fn+1 has no augmenting paths of length less than n + 1 and the recursive construction continues. 6 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB Lemma 2.2 ensures that µ(X \ dom(fn )) ≤ c−n , and thus the third item of Lemma 2.5 ensures that µ({x ∈ X : fn+1 (x) 6= fn (x)}) ≤ (n + 1)c−n . As the sequence (n + 1)c−n is summable, the Borel–Cantelli lemma implies that X ′ = {x ∈ X : ∃m ∈ N ∀n ≥ m x ∈ dom(fn ) and fn (x) = fm (x)} is µ-conull. Finally, f = limn→∞ fn ↾ X ′ is as desired.  Lemma 2.6. Suppose X and Y are standard Borel spaces, that µ is a Borel measure on X, and that ν is a Borel measure on Y . Suppose R ⊆ X × Y is Borel, locally finite, (µ, ν)-preserving graph. Assume that there exist numbers a, b > 0 such that |Rx | ≥ a for µ-a.e. x ∈ X and |Ry | ≤ b for ν-a.e. y ∈ Y . Then ν(RA ) ≥ ab µ(A) for all Borel subsets A ⊆ X. R R Proof. Since R is (µ, ν)-preserving we have A |Rx | dµ = RA |Ry ∩ A| dν. Hence Z Z Z Z y  b dν = bν(RA ). |R ∩ A| dν ≤ |Rx | dµ = a dµ ≤ aµ(A) = A A RA RA 3. Følner tilings Fix a countable group G. For finite sets K, F ⊆ G and δ > 0, we say that F is (K, δ)invariant if |KF △F | < δ|F |. Note this condition implies |KF | < (1 + δ)|F |. Recall that G is amenable if for every finite K ⊆ G and δ > 0 there exists a (K, δ)-invariant set F . A Følner sequence is a sequence of finite sets Fn ⊆ G with the property that for every finite K ⊆ G and δ > 0 the set Fn is (K, δ)-invariant for all but finitely many n. Below, we always assume that G is amenable. Fix a free action G y X. For A ⊆ X we define the lower and upper Banach densities of A to be |A ∩ F x| |A ∩ F x| D (A) = sup inf and D̄(A) = inf sup . F ⊆G x∈X ¯ |F | |F | F ⊆G x∈X F finite F finite Equivalently [3, Lemma 2.9], if (Fn )n∈N is a Følner sequence then D (A) = lim inf n→∞ x∈X ¯ |A ∩ Fn x| |Fn | and D̄(A) = lim sup n→∞ x∈X |A ∩ Fn x| . |Fn | We now define an analogue of ‘(K, δ)-invariant’ for infinite subsets of X. A set A ⊆ X (possibly infinite) is (K, δ)∗ -invariant if there is a finite set F ⊆ G such that |(KA△A) ∩ F x| < δ|A ∩ F x| for all x ∈ X. Equivalently, A is (K, δ)∗ -invariant if and only if for every Følner sequence (Fn )n∈N we have limn supx |(KA△A) ∩ Fn x|/|A ∩ Fn x| < δ. A collection {Fi : i ∈ I} of finite subsets of X is called ǫ-disjoint if for each i there is an Fi′ ⊆ Fi such that |Fi′ | > (1 − ǫ)|Fi | and such that the sets {Fi′ : i ∈ I} are pairwise disjoint. Lemma 3.1. Let K, W ⊆ G be finite, let ǫ, δ > 0, let C ⊆ X, and for c ∈ CS let Fc ⊆ W be (K, δ(1 − ǫ))-invariant. If the collection {Fc c : c ∈ C} is ǫ-disjoint and c∈C Fc c has S positive lower Banach density, then c∈C Fc c is (K, δ)∗ -invariant. S Proof. Set A = c∈C Fc c and set T = W W −1 ({1G } ∪ K)−1 . Since W is finite and each Fc ⊆ W , there is 0 < δ0 < δ such that each Fc is (K, δ0 (1 − ǫ))-invariant. Fix a finite set D (A) (A) U ⊆ G which is (T, D¯2|T | (δ − δ0 ))-invariant and satisfies inf x∈X |A ∩ U x| > ¯ 2 |U |. Now FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 7 fix x ∈ X. Let B be the set of b ∈ U x such that T b 6⊆ U x. Note that B ⊆ T −1 (T U x△U x) and thus |U | |T U △U | · · |A ∩ U x| < (δ − δ0 )|A ∩ U x|. |B| ≤ |T | · |U | |A ∩ U x| P Set C ′ = {c ∈ C : Fc c ⊆ U x}. Note that the ǫ-disjoint assumption gives (1−ǫ) c∈C ′ |Fc | ≤ |A∩U x|. Also, our definitions of C ′ , T , and B imply that if c ∈ C \C ′ and S ({1G }∪K)Fc c∩ U x 6= ∅ then (({1G }∪K)Fc c)∩U x ⊆ B. Therefore (KA△A)∩U x ⊆ B∪ c∈C ′ (KFc c△Fc c). Combining this with the fact that each set Fc is (K, δ0 (1 − ǫ))-invariant, we obtain X |(KA△A) ∩ U x| ≤ |B| + |KFc c△Fc c| c∈C ′ < (δ − δ0 )|A ∩ U x| + X δ0 (1 − ǫ)|Fc | c∈C ′ ≤ (δ − δ0 )|A ∩ U x| + δ0 |A ∩ U x| = δ|A ∩ U x|. Since x was arbitrary, we conclude that A is (K, δ)∗ -invariant.  Lemma 3.2. Let T ⊆ G be finite and let ǫ, δ > 0 with ǫ(1 + δ) < 1. Suppose that A ⊆ X is (T −1 , δ)∗ -invariant. If B ⊇ A and |B ∩ T x| ≥ ǫ|T | for all x ∈ X, then D (B) ≥ (1 − ǫ(1 + δ)) · D (A) + ǫ. ¯ ¯ Proof. This is implicitly demonstrated in [3, Proof of Lemma 4.1]. As a convenience to the reader, we include a proof here. Fix θ > 0. Since A is (T −1 , δ)∗ -invariant, we can pick a finite set U ⊆ G which is (T, θ)-invariant and satisfies inf x∈X |A ∩ U x| > D(A) − θ ¯ |U | and sup x∈X |T −1 A ∩ U x| < 1 + δ. |A ∩ U x| |A∩U x| |U | > D (A) − θ, and set U ′ = {u ∈ U : A ∩ T ux = ∅}. Notice that ¯ |U | − |T −1 A ∩ U x| |T −1 A ∩ U x| |A ∩ U x| |U ′ | = =1− · > 1 − (1 + δ)α. |U | |U | |A ∩ U x| |U | Fix x ∈ X, set α = Since A ∩ T U ′ x = ∅ and |B ∩ T y| ≥ ǫ|T | for all y ∈ X, it follows that |(B \ A) ∩ T ux| ≥ ǫ|T | for all u ∈ U ′ . Thus there are ǫ|T ||U ′ | many pairs (t, u) ∈ T × U ′ with tux ∈ B \ A. It follows there is t∗ ∈ T with |(B \ A) ∩ t∗ U ′ x| ≥ ǫ · |U ′ |. Therefore   |A ∩ U x| |(B \ A) ∩ t∗ U ′ x| |U ′ | |U | |B ∩ T U x| ≥ + · · ′ |T U | |U | |U | |U | |T U |   −1 > α + ǫ(1 − (1 + δ)α) · (1 + θ)   = (1 − ǫ(1 + δ))α + ǫ · (1 + θ)−1   > (1 − ǫ(1 + δ))(D (A) − θ) + ǫ · (1 + θ)−1 . ¯ Letting θ tend to 0 completes the proof.  8 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB Lemma 3.3. Let X be a standard Borel space and let G y X be a free Borel action. Let Y ⊆ X be Borel, let T ⊆ G be finite, and let ǫ ∈ (0, 1/2). Then there is a Borel set C ⊆ X and a Borel function c ∈ C 7→ Tc ⊆ T suchSthat |Tc | > (1− ǫ)|T |, the sets {Tc c : c ∈ C} are pairwise disjoint and disjoint with Y , Y ∪ c∈C Tc c = Y ∪ T C, and |(Y ∪ T C) ∩ T x| ≥ ǫ|T | for all x ∈ X. Proof. Using Lemma 2.4, fix a Borel partition P = {P1 , . . . , Pm } of X such that T x∩T x′ = ∅ for S all x 6= x′ ∈ Pi and all 1 ≤ i ≤ m. We will pick Borel sets Ci ⊆ Pi and set C = 1≤i≤m Ci . Set Y0 = Y . Let 1 ≤ i ≤ m and inductively assume that Yi−1 has been defined. Define Ci = {c ∈ Pi : |Yi−1 ∩ T c| < ǫ|T |}, defineSYi = Yi−1 ∪ T Ci , and for c ∈ Ci set Tc = {t ∈ T : tc 6∈ Yi−1 }. It is easily seen that C = 1≤i≤m Ci has the desired properties.  The following lemma is mainly due to Ornstein–Weiss [17], who proved it with an invariant probability measure taking the place of Banach density. Ornstein and Weiss also established a purely group-theoretic counterpart of this result which was later adapted to the Banach density setting by Downarowicz–Huczek–Zhang in [3] and will be heavily used in Section 5, where it is recorded as Theorem 5.2. The only difference between this lemma and prior versions is that we simultaneously work in the Borel setting and use Banach density. Lemma 3.4. [17, II.§2. Theorem 5] [3, Lemma 4.1] Let X be a standard Borel space and let G y X be a free Borel action. Let K ⊆ G be finite, let ǫ ∈ (0, 1/2), and let n satisfy (1 − ǫ)n < ǫ. Then there exist (K, ǫ)-invariant sets F1 , . . . , Fn , a Borel set C ⊆ X, and a Borel function c ∈ C 7→ Fc ⊆ G such that: (i) for every c ∈ C there is 1 ≤ i ≤ n with Fc ⊆ Fi and |Fc | > (1 − ǫ)|Fi |; (ii) theSsets Fc c, c ∈ C, are pairwise disjoint; and (iii) D ( c∈C Fc c) > 1 − ǫ. ¯ Proof. Fix δ > 0 satisfying (1 + δ)−1 (1 − (1 + δ)ǫ)n < ǫ − 1 + (1 + δ)−1 . Fix a sequence of (K, ǫ)-invariant sets F1 , . . . , Fn such that Fi is (Fj−1 , δ(1 − ǫ))-invariant for all 1 ≤ j < i ≤ n. The set C will be the disjoint union of sets Ci , 1 ≤ i ≤ n. The construction S S will be such that Fc ⊆ Fi and |Fc | > (1 − ǫ)|Fi | for c ∈ Ci . We will define Ai = i≤k≤n c∈Ck Fc c S and arrange that Ai+1 ∪ Fi Ci = Ai+1 ∪ c∈Ci Fc c and D (Ai ) ≥ (1 + δ)−1 − (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i . ¯ S In particular, we will have Ai = i≤k≤n Fk Ck . To begin, apply Lemma 3.3 with Y = ∅ and T = Fn to get a Borel set Cn and a Borel map c ∈ S Cn 7→ Fc ⊆ Fn such that |Fc | > (1 − ǫ)|Fn |, the sets {Fc c : c ∈ Cn } are pairwise disjoint, c∈Cn Fc c = Fn Cn , and |Fn Cn ∩ Fn x| ≥ ǫ|Fn | for all x ∈ X. Applying Lemma 3.2 with A = ∅ and B = Fn Cn we find that the set An = Fn Cn satisfies D (An ) ≥ ǫ. ¯ Inductively assume that Cn through Ci+1 have been defined and An through Ai+1 are defined as above and satisfy (3.1). Using Y = Ai+1 and T = Fi , apply Lemma 3.3 to get a Borel set Ci and a Borel map c ∈ Ci 7→ Fc ⊆ Fi such that |FcS| > (1 − ǫ)|Fi |, the sets {Fc c : c ∈ Ci } are pairwise disjoint and disjoint with Ai+1 , Ai+1 ∪ c∈Ci Fc c = Ai+1 ∪ Fi Ci , and |(Ai+1 ∪ Fi Ci ) ∩ Fi x| ≥ ǫ|Fi | for all x ∈ X. The set Ai+1 is the union of an ǫ-disjoint (3.1) FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 9 collection of (Fi−1 , δ(1 − ǫ))-invariant sets and has positive lower Banach density. So by Lemma 3.1 Ai+1 is (Fi−1 , δ)∗ -invariant. Applying Lemma 3.2 with A = Ai+1 , we find that Ai = Ai+1 ∪ Fi Ci satisfies D (Ai ) ≥ (1 − ǫ(1 + δ)) · D (Ai+1 ) + ǫ ¯ ¯ ǫ(1 + δ) (1 − ǫ(1 + δ)) − (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i + ≥ (1 + δ) 1+δ = (1 + δ)−1 − (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i . This completes the inductive step and completes the definition of C. It is immediate from the construction that (i) and (ii) are satisfied. Clause (iii) also follows by noting that (3.1) is greater than 1 − ǫ when i = 1.  We recall the following simple fact. Lemma 3.5. [3, Lemma 2.3] If F ⊆ G is (K, δ)-invariant and F ′ satisfies |F ′ △F | < ǫ|F | then F ′ is (K, (|K|+1)ǫ+δ )-invariant. 1−ǫ Now we present the main theorem. Theorem 3.6. Let G be a countable amenable group, let (X, µ) be a standard probability space, and let G y (X, µ) be a free p.m.p. action. For every finite K ⊆ G and every δ > 0 there exist a µ-conull G-invariant Borel set X ′ ⊆ X, a collection {Ci : 0 ≤ i ≤ m} of Borel subsets of X ′ , and a collection {Fi : 0 ≤ i ≤ m} of (K, δ)-invariant sets such that {Fi c : 0 ≤ i ≤ m, c ∈ Ci } partitions X ′ . < δ. Apply Lemma 3.4 to get (K, ǫ)-invariant Proof. Fix ǫ ∈ (0, 1/2) satisfying (|K|+1)6ǫ+ǫ 1−6ǫ ′ ′ sets F1 , . . . , Fn , a Borel set C ⊆ X, and a Borel function c ∈ C 7→ Fc ⊆ G satisfying (i) for every c ∈ C there is 1 ≤ i ≤ n with Fc ⊆ Fi′ and |Fc | > (1 − ǫ)|Fi′ |; (ii) theSsets Fc c, c ∈ C, are pairwise disjoint; and (iii) D ( c∈C Fc c) > 1 − ǫ. ¯ S Set Y = X \ c∈C Fc c. If µ(Y ) = 0 then we are done. So we assume µ(Y ) > 0 and we let ν denote the restriction of µ to Y . Fix a Borel map c ∈ C 7→ Zc ⊆ Fc satisfying 4ǫ|Fc | < |Zc | < 5ǫ|Fc | for all c ∈ C (it’s clear from the proof of Lemma S 3.4 that we may choose the sets Fi′ so that ǫ|Fc | > mini ǫ(1 − ǫ)|Fi′ | > 1). Set Z = c∈C Zc c and let ζ denote the restriction of µ to Z (note that µ(Z) > 0). S Set W = ni=1 Fi′ and W ′ = W W −1 . Fix a finite set U ⊆ G which is (W ′ , (1/2−ǫ)/|W ′ |)invariant and satisfies inf x∈X |(X \ Y ) ∩ U x| > (1 − ǫ)|U |. Since every amenable group admits a Følner sequence consisting of symmetric sets, we may assume that U = U −1 [16, Corollary 5.3]. Define R ⊆ Y ×Z by declaring (y, z) ∈ R if and only if y ∈ U z (equivalently z ∈ U y). Then R is Borel, locally finite, and (ν, ζ)-preserving. We now check that R is expansive. We automatically have |Rz | = |Y ∩ U z| < ǫ|U | for all z ∈ Z. By Lemma 2.6 it suffices to show that |Ry | = |Z ∩ U y| ≥ 2ǫ|U | for all y ∈ Y . Fix y ∈ Y . Let B be the set of b ∈ U y such that W ′ b 6⊆ U y. Then B ⊆ W ′ (W ′ U y△U y) and thus |W ′ U △U | |B| ≤ |W ′ | · < 1/2 − ǫ. |U | |U | 10 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB Let A be the union of those sets Fc c, c ∈ C, which are contained in U y. Notice that (X \ Y ) ∩ U y ⊆ B ∪ A. Therefore |A| |(B ∪ A) ∩ U y| |(X \ Y ) ∩ U y| 1 −ǫ+ > ≥ > 1 − ǫ, 2 |U | |U | |U | hence |A| > |U |/2. By construction |Z ∩ A| > 4ǫ|A|. So |Z ∩ U y| ≥ |Z ∩ A| > 2ǫ|U |. We conclude that R is expansive. Apply Proposition 2.1 to obtain a G-invariant µ-conull set X ′ ⊆ X and a Borel injection ρ : Y ∩X ′ → Z with graph(ρ) ⊆ R. Consider the sets Fc ∪{g ∈ U : gc ∈ Y and ρ(gc) ∈ Fc c} as c ∈ C varies. These are subsets of W ∪ U and thus there are only finitely many such sets which we can enumerate as F1 , . . . , Fm . We partition C ∩ X ′ into Borel sets C1 , . . . , Cm with c ∈ Ci if and only if c ∈ X ′ and Fc ∪ {g ∈ U : gc ∈ Y and ρ(gc) ∈ Fc c} = Fi . Since ρ is defined on all of Y ∩ X ′ , we see that the sets {Fi c : 1 ≤ i ≤ m, c ∈ Ci } partition X ′ . Finally, for c ∈ Ci ∩ X ′ , if we let Fj′ be such that |Fc △Fj′ | < ǫ|Fj′ |, then |Fi △Fj′ | ≤ |Fi △Fc | + |Fc △Fj′ | ≤ |ρ−1 (Fc c)| + ǫ|Fj′ | ≤ |Zc | + ǫ|Fj′ | < 5ǫ|Fc | + ǫ|Fj′ | ≤ 6ǫ|Fj′ |. Using Lemma 3.5 and our choice of ǫ, this implies that each set Fi is (K, δ)-invariant.  4. Clopen tower decompositions with Følner shapes Let G y X be an action of a group on a compact space. By a clopen tower we mean a pair (B, S) where B is a clopen subset of X (the base of the tower) and S is a finite subset of G (the shape of the tower) such that the sets sB for s ∈ S are pairwise disjoint. By a clopen tower decomposition of X we mean a finite collection {(Bi , Si )}ni=1 of clopen towers such that the sets S1 B1 , . . . , Sn Bn form a partition of X. We also similarly speak of measurable towers and measurable tower decompositions for an action G y (X, µ) on a measure space, with the bases now being measurable sets instead of clopen sets. In this terminology, Theorem 3.6 says that if G y (X, µ) is a free p.m.p. action of a countable amenable group on a standard probability space then for every finite set K ⊆ G and δ > 0 there exists, modulo a null set, a measurable tower decomposition of X with (K, δ)invariant shapes. Lemma 4.1. Let G be a countably infinite amenable group and G y X a free minimal action on the Cantor set. Then this action has a free minimal extension G y Y on the Cantor set such that for every finite set F ⊆ G and δ > 0 there is a clopen tower decomposition of Y with (F, δ)-invariant shapes. Proof. Let F1 ⊆ F2 ⊆ . . . be an increasing sequence of finite subsets of G whose union is equal to G. Fix a G-invariant Borel probability measure µ on X (such a measure exists by amenability). The freeness of the action G y X means that for each n ∈ N we can apply Theorem 3.6 to produce, modulo a null set, a measurable tower decomposition Un for the p.m.p. action G y (X, µ) such that each shape is (Fn , 1/n)-invariant. Let A be the unital G-invariant C∗ -algebra of L∞ (X, µ) generated by C(X) and the indicator functions of the levels of each of the tower decompositions Un . Since there are countably many such indicator functions and the group G is countable, the C∗ -algebra A is separable. Therefore by the Gelfand–Naimark theorem we have A = C(Z) for some zero-dimensional FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 11 metrizable space Z and a G-factor map ϕ : Z → X. By a standard fact which can be established using Zorn’s lemma, there exists a nonempty closed G-invariant set Y ⊆ Z such that the restriction action G y Y is minimal. Note that Y is necessarily a Cantor set, since G is infinite. Also, the action G y Y is free, since it is an extension of a free action. Since the action on X is minimal, the restriction ϕ|Y : Y → X is surjective and hence a G-factor map. For each n we get from Un a clopen tower decomposition Vn of Y with (Fn , 1/n)-invariant shapes, and by intersecting the levels of the towers in Vn with Y we obtain a clopen tower decomposition of Y with (Fn , 1/n)-invariant shapes, showing that the extension G y Y has the desired property.  Let X be the Cantor set and let G be a countable infinite amenable group. The set Act(G, X) is a Polish space under the topology which has as a basis the sets Uα,P,F = {β ∈ Act(G, X) : αs A = βs A for all A ∈ P and s ∈ F } where α ∈ Act(G, X), P is a clopen partition of X, and F is a finite subset of G. Write FrMin(G, X) for the set of actions in Act(G, X) which are free and minimal. Then FrMin(G, X) is a Gδ set. To see this, fix an enumeration s1 , s2 , s3 , . . . of G \ {e} (where e denotes the identity element of the group) and for every n ∈ N and S nonempty clopen set A ⊆ X define the set Wn,A of all α ∈ Act(G, X) such that (i) s∈F αs A = X for some finite set F ⊆ G, and (ii) there exists a clopen partition {A1 , . . . , Ak } of A such that αsn Ai ∩ Ai = ∅ for all i = 1, . . . , k. Then each Wn,A is open, which means, with A ranging T over T the countable collection of nonempty clopen subsets of X, that the intersection n∈N A Wn,A , which is equal to FrMin(G, X), is a Gδ set. It follows that FrMin(G, X) is a Polish space. Theorem 4.2. Let G be a countably infinite amenable group. Let C be the collection of actions in FrMin(G, X) with the property that for every finite set F ⊆ G and δ > 0 there is a clopen tower decomposition of X with (F, δ)-invariant shapes. Then C is a dense Gδ subset of FrMin(G, X). α Proof. That C is a Gδ set is a simple exercise. Let G y X be any action in FrMin(G, X). β By Lemma 4.1 this action has a free minimal extension G y Y with the property in the theorem statement, where Y is the Cantor set. Let P be a clopen partition of X and F a nonempty finite subset of G. Write A1 , . . . , An for the members of the clopen partition W −1 −1 s∈F s P. Then for each i = 1, . . . , n the set Ai and its inverse image ϕ (Ai ) under the extension map ϕ : Y → X are Cantor sets, and so we can find a homeomorphism ψi : Ai → ϕ−1 (Ai ). Let ψ : X → Y be the homeomorphism which is equal to ψi on Ai for γ each i. Then the action G y X defined by γs = ψ −1 ◦ βs ◦ ψ for s ∈ G belongs to C as well as to the basic open neighborhood Uα,P,F of α, establishing the density of C.  5. Applications to Z-stability A C∗ -algebra A is said to be Z-stable if A ⊗ Z ∼ = A where Z is the Jiang–Su algebra [10], with the C∗ -tensor product being unique in this case because Z is nuclear. Z-stability has become an important regularity property in the classification program for simple separable nuclear C∗ -algebras, which has recently witnessed some spectacular advances. Thanks to recent work of Gong–Lin–Niu [6], Elliott–Gong–Lin–Niu [4], and Tikuisis–White–Winter 12 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB [22], it is now known that simple separable unital C∗ -algebras satisfying the universal coefficient theorem and having finite nuclear dimension are classified by ordered K-theory paired with tracial states. Although Z-stability does not appear in the hypotheses of this classification theorem, it does play an important technical role in the proof. Moreover, it is a conjecture of Toms and Winter that for simple separable infinite-dimensional unital nuclear C∗ -algebras the following properties are equivalent: (i) Z-stability, (ii) finite nuclear dimension, (iii) strict comparison. Implications between (i), (ii), and (iii) are known to hold in various degrees of generality. In particular, the implication (ii)⇒(i) was established in [23] while the converse is known to hold when the extreme boundary of the convex set of tracial states is compact [1]. It remains a problem to determine whether any of the crossed products of the actions in Theorem 5.4 falls within the purview of these positive results on the Toms–Winter conjecture, and in particular whether any of them has finite nuclear dimension (see Question 5.5). By now there exist highly effectively methods for establishing finite nuclear dimension for crossed products of free actions on compact metrizable spaces of finite covering dimension [20, 21, 7], but their utility is structurally restricted to groups with finite asymptotic dimension and hence excludes many amenable examples like the Grigorchuk group. One can show using the technology from [7] that, for a countably infinite amenable group with finite asymptotic dimension, the crossed product of a generic free minimal action on the Cantor set has finite nuclear dimension. Our intention here has been to remove the restriction of finite asymptotic dimension by means of a different approach that establishes instead the conjecturally equivalent property of Z-stability but for arbitrary countably infinite amenable groups. To verify Z-stability in the proof of Theorem 5.3 we will use the following result of Hirshberg and Orovitz [8]. Recall that a linear map ϕ : A → B between C∗ -algebras is said to be complete positive if its tensor product id ⊗ ϕ : Mn ⊗ A → Mn ⊗ B with the identity map on the n × n matrix algebra Mn is positive for every n ∈ N. It is of order zero if ϕ(a)ϕ(b) = 0 for all a, b ∈ A satisfying ab = 0. One can show that ϕ is an order-zero completely positive map if and only if there is an embedding B ⊆ D of B into a larger C∗ -algebra, a ∗ -homomorphism π : A → D, and a positive element h ∈ D commuting with the image of π such that ϕ(a) = hπ(a) for all a ∈ A [24]. Below - denotes the relation of Cuntz subequivalence, so that a - b for positive elements a, b in a C∗ -algebra A means that there is a sequence (vn ) in A such that limn→∞ ka − vn bvn∗ k = 0. Theorem 5.1. Let A be a simple separable unital nuclear C∗ -algebra not isomorphic to C. Suppose that for every n ∈ N, finite set Ω ⊆ A, ε > 0, and nonzero positive element a ∈ A there exists an order-zero complete positive contractive linear map ϕ : Mn → A such that (i) 1 − ϕ(1) - a, (ii) k[b, ϕ(z)]k < ε for all b ∈ Ω and norm-one z ∈ Mn . Then A is Z-stable. FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 13 The following is the Ornstein–Weiss quasitiling theorem [17] as formulated in Theorem 3.36 of [11]. For finite sets A, F ⊆ G we write ∂F A = {s ∈ A : F s ∩ A 6= ∅ and F s ∩ (G \ A) 6= ∅}. For λS≤ 1, a collection C of finite subsets of G is said to λ-cover a finite subset A of G if |A ∩ C| ≥ λ|A|. For β ≥ 0, a collection C of finite subsets of G is said to be β-disjoint if for each C ∈ C there is a set C ′ ⊆ C with |C ′ | ≥ (1 − β)|C| so that the sets C ′ for C ∈ C are pairwise disjoint. Theorem 5.2. Let 0 < β < 12 and let n be a positive integer such that (1 − β/2)n < β. Then whenever e ∈ T1 ⊆ T2 ⊆ · · · ⊆ Tn are finite subsets of a group G such that |∂Ti−1 Ti | ≤ (η/8)|Ti | for i = 2, . . . , n, for every (Tn , β/4)-invariant nonempty finite set E ⊆ G there exist C1 , . . . , Cn ⊆ G such that S (i) ni=1 Ti Ci ⊆ E, and S (ii) the collection of right translates ni=1 {Ti c : c ∈ Ci } is β-disjoint and (1−β)-covers E. Theorem 5.3. Let G be a countably infinite amenable group and let G y X be a free minimal action on the Cantor set such that for every finite set F ⊆ G and δ > 0 there is a clopen tower decomposition of X with (F, δ)-invariant shapes. Then C(X) ⋊ G is Z-stable. Proof. Let n ∈ N. Let Υ be a finite subset of the unit ball of C(X), F a symmetric finite subset of G containing the identity element e, and ε > 0. Let a be a nonzero positive element of C(X) ⋊ G. We will show the existence of an order-zero completely positive contractive linear map ϕ : Mn → C(X) ⋊ G satisfying (i) and (ii) in Theorem 5.1 where the finite set Ω there is taken to be Υ ∪ {us : s ∈ F }. Since C(X) ⋊ G is generated as a C∗ -algebra by the unit ball of C(X) and the unitaries us for s ∈ G, we will thereafter be able to conclude by Theorem 5.1 that C(X) ⋊ G is Z-stable. By Lemma 7.9 in [18] we may assume that a is a function in C(X). Taking a clopen set A ⊆ X on which a is nonzero, we may furthermore assume that a is equal to the indicator function 1A . Minimality implies that the clopen sets sA for s ∈ G cover X, and so by compactness there is a finite set D ⊆ G such that D −1 A = X. Equip X with a compatible metric d. Choose an integer Q > n2 /ε. Let γ > 0, to be determined. Take a 0 < β < 1/n which is small enough so that if T is a nonempty finite subset of G which is sufficiently invariant under left translation by F Q T and T ′ is a subset of T with |T ′ | ≥ (1 − nβ)|T | then | s∈F Q s−1 T ′ | ≥ (1 − γ)|T |. Choose an L ∈ N large enough so that (1 − β/2)L < β. By amenability there exist finite subsets e ∈ T1 ⊆ T2 ⊆ · · · ⊆ TL of G such that |∂Tl−1 Tl | ≤ (β/8)|Tl | for l = 2, . . . , L. By the previous paragraph, we may also assume that for each l the set Tl is sufficiently invariant under left translation by F Q so that for all T ⊆ Tl satisfying |T | ≥ (1 − nβ)|Tl | one has \ s−1 T ≥ (1 − γ)|Tl |. (5.1) s∈F Q By uniform continuity there is a η > 0 such that |f (x)−f (y)| < ε/(3n2 ) for all f ∈ Υ∪Υ2 and all x, y ∈ X satisfying d(x, y) < η. Again by uniform continuity there is an η ′ > 0 14 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB S such that d(tx, ty) < η for all x, y ∈ X satisfying d(x, y) < η ′ and all t ∈ L l=1 Tl . Fix a clopen partition {A1 , . . . , AM } of X whose members all have diameter less that η ′ . Let E be a finite subset of G containing TL and let δ > 0 be such that δ ≤ β/4. We will further specify E and δ below. By hypothesis there is a collection {(Vk , Sk )}K k=1 of clopen towers such that the shapes S1 , . . . , SK are (E, δ)-invariant and the sets S1 V1 , . . . , SK VK partition X. We may assume that for each k = 1, . . . , K the set Sk is large enough so that X  L Mn (5.2) |Tl | ≤ β|Sk |. l=1 By a simple procedure we can construct, for each k, a clopen partition Pk of Vk such that each level of every one of the towers (V, Sk ) for V ∈ Pk is contained in one of the sets A1 , . . . , AM as well as in one of the sets A and X \ A. By replacing (Vk , Sk ) with these thinner towers for each k, we may therefore assume that each level in every one of the towers (V1 , S1 ), . . . , (VK , SK ) is contained in one of the sets A1 , . . . , AM and in one of the sets A and X \ A. Let 1 ≤ k ≤ K. Since Sk is (TL , β/4)-invariant, by Theorem 5.2 and our choice of the sets T1 , . . . , TL we can find Ck,1 , . . . , Ck,L ⊆ Sk such that the collection {Tl c : l = 1, . . . , L, c ∈ Ck,l } is β-disjoint, has union contained in Sk , and (1 − β)-covers Sk . By β-disjointness, for every l = 1, . . . , L and c ∈ Ck,l we can find a Tk,l,c ⊆ Tl satisfying |Tk,l,c | ≥ (1 − β)|Tl | so that the collection of sets Tk,l,cc for l = 1, . . . , L and c ∈ Ck,l is disjoint and has the same union as the sets Tl c for l = 1, . . . , L and c ∈ Ck,l , so that it (1 − β)-covers Sk . For each l = 1, . . . , L and m = 1, . . . , M write Ck,l,m for the set of all c ∈ Ck,l such that (n) (1) cVk ⊆ Am , and choose pairwise disjoint subsets Ck,l,m, . . . , Ck,l,m of Ck,l,m such that each has cardinality ⌊|Ck,l,m |/n⌋. For each i = 2, . . . , n choose a bijection G (1) G (i) Λk,i : Ck,l,m → Ck,l,m l,m (1) l,m (i) which sends Ck,l,m to Ck,l,m for all l, m. Also, define Λk,1 to be the identity map from F (1) l,m Ck,l,m to itself. T F (1) ′ = ni=1 Tk,l,Λk,i (c) , which satisfies Let 1 ≤ l ≤ L and c ∈ m Ck,l,m. Define the set Tk,l,c (5.3) ′ | ≥ (1 − nβ)|Tl | ≥ (1 − nβ)|Tk,l,c | |Tk,l,c since each Tk,l,Λk,i (c) is a subset of Tl of cardinality at least (1 − β)|Tl |. Set \ ′ ′ Bk,l,c,Q = sTk,l,c , Bk,l,c,0 = Tk,l,c \ F Q−1 Bk,l,c,Q, s∈F Q and, for q = 1, . . . , Q − 1, using the convention F 0 = {e}, Bk,l,c,q = F Q−q Bk,l,c,Q \ F Q−q−1 Bk,l,c,Q. ′ . For s ∈ F we have Then the sets Bk,l,c,0, . . . , Bk,l,c,Q partition Tk,l,c (5.4) sBk,l,c,Q ⊆ Bk,l,c,Q−1 ∪ Bk,l,c,Q, FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 15 while for q = 1, . . . , Q − 1 we have sBk,l,c,q ⊆ Bk,l,c,q−1 ∪ Bk,l,c,q ∪ Bk,l,c,q+1, (5.5) for if we are given a t ∈ Bk,l,c,q then st ∈ F Q−q+1 Bk,l,c,Q, while if st ∈ F Q−q−2 Bk,l,c,Q then t ∈ F Q−q−1 Bk,l,c,Q since F is symmetric, contradicting the membership of t in Bk,l,c,q . Also, from (5.1) and (5.3) we get |Bk,l,c,Q| ≥ (1 − γ)|Tl |. (5.6) For i = 2, . . . , n, c ∈ F (i) m Ck,l,m , and q = 0, . . . , Q we set Bk,l,c,q = Bk,l,λ−1(c),q . k,i Write Λk,i,j for the composition Λk,i ◦ Λ−1 k,j . Define a linear map ψ : Mn → C(X) ⋊ G by declaring it on the standard matrix units {eij }ni,j=1 of Mn to be given by X X X utΛk,i,j (c)c−1 t−1 1tcVk ψ(eij ) = k,l,m c∈C (j) k,l,m ′ t∈Tk,l,c and extending linearly. Then ψ(eij )∗ = ψ(eji ) for all i, j and the product ψ(eij )ψ(ei′ j ′ ) is 1 or 0 depending on whether i = i′ , so that ψ is a ∗ -homomorphism. F (j) For all k and l, all 1 ≤ i, j ≤ n, and all c ∈ m Ck,l,m we set hk,l,c,i,j = Q X X q=1 t∈Bk,l,c,q q u −1 −1 1tcV k Q tΛk,i,j (c)c t and put h= n X X X hk,l,c,i,i. k,l,m i=1 c∈C (i) k,l,m Then h is a norm-one function which commutes with the image of ψ, and so we can define an order-zero completely positive contractive linear map ϕ : Mn → C(X) ⋊ G by setting ϕ(z) = hψ(z). Note that ϕ(eij ) = P k,l,m P (j) c∈Ck,l,m hk,l,c,i,j . We now verify condition (ii) in Theorem 5.1 for the elements of the set {us : s ∈ F }. F (j) Let 1 ≤ i, j ≤ n. For all k and l, all c ∈ m Ck,l,m, and all s ∈ F we have us hk,l,c,i,j u−1 s − hk,l,c,i,j = Q X X q=1 t∈Bk,l,c,q q u −1 −1 1stcV k Q stΛk,i,j (c)c (st) − Q X X q=1 t∈Bk,l,c,q q u −1 −1 1tcV , k Q tΛk,i,j (c)c t and so in view of (5.4) and (5.5) we obtain kus hk,l,c,i,j u−1 s − hk,l,c,i,j k ≤ 1 ε < 2. Q n 16 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB ∗ ∗ Since each of the elements b = us hk,l,c,i,j u−1 s −hk,l,c,i,j is such that b b and bb are dominated ′ ′ by twice the indicator functions of Tk,l,Λ−1 (c) cVk and Tk,l,Λ−1 (c) Λk,i,j (c)Vk , respectively, and j j F (i′ ) ′ ′ the sets Tk,l,Λ−1(c) cVk over all k, l, all i = 1, . . . , n, and all c ∈ m Ck,l,m are pairwise i′ disjoint, this yields −1 kus ϕ(eij )u−1 s − ϕ(eij )k = max max kus hk,l,c,i,j us − hk,l,c,i,j k < k,l,m c∈C (j) k,l,m ε n2 and hence, for every norm-one element z = (zij ) ∈ Mn , k[us , ϕ(z)]k = kus ϕ(z)u−1 s − ϕ(z)k ≤ n X |zij |kus ϕ(eij )u−1 s − ϕ(eij )k i,j=1 ε = ε. n2 Next we verify condition (ii) in Theorem 5.1 for the functions in Υ. Let 1 ≤ i, j ≤ n. (j) Let g ∈ Υ ∪ Υ2 . Let 1 ≤ k ≤ K, 1 ≤ l ≤ L, 1 ≤ m ≤ M , and c ∈ Ck,l,m. Then < n2 · (5.7) h∗k,l,c,i,j ghk,l,c,i,j = Q X X q=1 t∈Bk,l,c,q q2 (tcΛk,i,j (c)−1 t−1 g)1tcVk Q2 and (5.8) gh∗k,l,c,i,j hk,l,c,i,j = Q X X q=1 t∈Bk,l,c,q q2 g1tcVk . Q2 Now let x ∈ Vk . Since Λk,i,j (c)x and cx both belong to Am , we have d(Λk,i,j (c)x, cx) < η ′ . It follows that for every t ∈ Tl we have d(tΛk,i,j (c)x, tcx) < η by our choice of η ′ , so that |g(tΛk,i,j (c)x) − g(tcx)| < ε/(3n2 ) by our choice of η, in which case k(tcΛk,i,j (c)−1 t−1 g − g)1tcVk k = kc−1 t−1 ((tcΛk,i,j (c)−1 t−1 g − g)1tcVk )k = k(Λk,i,j (c)−1 t−1 g − c−1 t−1 g)1Vk k = sup |g(tΛk,i,j (c)x) − g(tcx)| x∈Vk < ε . 3n2 Using (5.7) and (5.8) this gives us (5.9) kh∗k,l,c,i,j ghk,l,c,i,j − gh∗k,l,c,i,j hk,l,c,i,j k = max max q=1,...,Q t∈Bk,l,c,q q2 ε k(tcΛk,i,j (c)−1 t−1 g − g)1tcVk k < 2 . Q2 3n Set w = ϕ(eij ) for brevity. Let f ∈ Υ. For g ∈ {f, f 2 } the functions h∗k,l,c,i,j ghk,l,c,i,j − (j) gh∗k,l,c,i,j hk,l,c,i,j over all k, l, and m and all c ∈ Ck,l,m have pairwise disjoint supports, so that (5.9) yields ε kw∗ gw − gw∗ wk < 2 . 3n FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 17 It follows that kw∗ f 2 w − f w∗ f wk ≤ kw∗ f 2 w − f 2 w∗ wk + kf (f w∗ w − w∗ f w)k < 2ε 3n2 and so kf w − wf k2 = k(f w − wf )∗ (f w − wf )k = kw∗ f 2 w − f w∗ f w + f w∗ wf − w∗ f wf k ≤ kw∗ f 2 w − f w∗ f wk + k(f w∗ w − w∗ f w)f k 2ε ε ε < 2 + 2 = 2. 3n 3n n Therefore for every norm-one element z = (zij ) ∈ Mn we have k[f, ϕ(z)]k ≤ n X |zij |k[f, ϕ(eij )]k < n2 · i,j=1 ε = ε. n2 Finally, we verify that the parameters in the construction of ϕ can be chosen so that 1 − ϕ(1) - 1A . By taking the sets S1 , . . . , SK to be sufficiently left invariant (by enlarging E and shrinking δ if necessary) we may assume that for every k = 1, . . . , K there is an Sk′ ⊆ Sk such that the set {s ∈ Sk′ : Ds ⊆ Sk } has cardinality at least |Sk |/2. Let 1 ≤ k ≤ K. Take a maximal set Sk′′ ⊆ Sk′ such that the sets Ds for s ∈ Sk′′ are pairwise disjoint, and note that |Sk′′ | ≥ |Sk′ |/|D −1 D| ≥ |Sk |/(2|D|2 ). Since D −1 A = X, each of the sets DsVk for s ∈ Sk′′ intersects A, and so the set Sk♯ of all s ∈ Sk such that sVk ⊆ A has F F F cardinality at least |Sk |/(2|D|2 ). Define Sk,1 = l,m ni=1 c∈C (i) Bk,l,c,Qc, which is the k,l,m set of all s ∈ Sk such that the function ϕ(1) takes the value 1 on sVk . Set Sk,0 = Sk \ Sk,1 . P (i) Since ni=1 |Ck,l,m | ≥ |Ck,l,m | − n for every l and m, by (5.2) we have n XX (i) |Tl ||Ck,l,m | ≥ l,m i=1 X |Tl ||Ck,l,m | − M n l,m ≥ [ X |Tl | l Tl Ck,l − β|Sk | l ≥ (1 − 2β)|Sk |. (i) Since for all l and i and all c ∈ Ck,l,m we have |Bk,l,c,Q| ≥ (1 − γ)|Tl | by (5.6), it follows, putting λ = (1 − γ)(1 − 2β), that |Sk,1 | ≥ (1 − γ) n XX (i) |Tl ||Ck,l,m | ≥ λ|Sk |. l,m i=1 By taking γ and β small enough we can guarantee that 1 − λ ≤ 1/(2|D|2 ) and hence |Sk,0 | = |Sk | − |Sk,1 | ≤ (1 − λ)|Sk | ≤ |Sk♯ |, 18 CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB so that there exists an injection θk : Sk,0 → Sk♯ . Define z= K X X uθk (s)s−1 1sVk . k=1 s∈Sk,0 A simple computation shows that z ∗ 1A z is the indicator function of is the support of 1 − ϕ(1), and so putting v = (1 − ϕ(1))1/2 z ∗ we get FK k=1 Sk,0 Vk , which v1A v ∗ = (1 − ϕ(1))1/2 z ∗ 1A z(1 − ϕ(1))1/2 = 1 − ϕ(1). This demonstrates that 1 − ϕ(1) - 1A , as desired.  Combining Theorems 5.1 and 4.2 yields the following. Theorem 5.4. Let G be a countably infinite amenable group and X the Cantor set. Then the set of all actions in FrMin(G, X) whose crossed product is Z-stable is comeager, and in particular nonempty. Question 5.5. Do any of the crossed products in Theorem 5.4 have tracial state space with compact extreme boundary (from which we would be able to conclude finite nuclear dimension by [1] and hence classifiability)? For G = Z a generic action in FrMin(G, X) is uniquely ergodic, so that the crossed product has a unique tracial state [9]. However, already for Z2 nothing of this nature seems to be known. On the other hand, it is known that the crossed products of free minimal actions of finitely generated nilpotent groups on compact metrizable spaces of finite covering dimension have finite nuclear dimension, and in particular are Z-stable [21]. References [1] J. Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White and W. Winter. Covering dimension of C∗ -algebras and 2-coloured classification. To appear in Mem. Amer. Math. Soc.. [2] T. Ceccherini-Silberstein, P. de la Harpe, and R. I. Grigorchuk. Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. (Russian) Tr. Mat. Inst. Steklova 224 (1999), 68–111; translation in Proc. Steklov Inst. Math. 224 (1999), 57–97. [3] T. Downarowicz, D. Huczek, and G. Zhang. Tilings of amenable groups. To appear in J. Reine Angew. Math. [4] G. Elliott, G. Gong, H. Lin, and Z. Niu. On the classification of simple amenable C∗ -algebras with finite decomposition rank, II. arXiv:1507.03437. [5] G. Elek and G. Lippner. Borel oracles. An analytic approach to constant time algorithms. Proc. Amer. Math. Soc. 138 (2010), 2939–2947. [6] G. Gong, H. Lin, and Z. Niu. Classification of finite simple amenable Z-stable C∗ -algebras. arXiv:1501.00135. [7] E. Guentner, R. Willett, and G. Yu. Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and C∗ -algebras. Math. Ann. 367 (2017), 785–829. [8] I. Hirshberg and J. Orovitz. Tracially Z–absorbing C∗ -algebras. J. Funct. Anal. 265 (2013), 765–785. [9] M. Hochman. Genericity in topological dynamics. Ergodic Theory Dynam. Systems 28 (2008), 125– 165. [10] X. Jiang and H. Su. On a simple unital projectionless C∗ -algebra. Amer. J. Math. 121 (1999), 359–413. [11] D. Kerr and H. Li. Ergodic Theory: Independence and Dichotomies. Springer, Cham, 2016. [12] A. Kechris, S. Solecki, and S. Todorcevic. Borel chromatic numbers, Adv. in Math. 141 (1999), 1–44. FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS 19 [13] R. Lyons and F. Nazarov. Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32 (2011), 1115–1125. [14] H. Matui and Y. Sato. Strict comparison and Z-absorption of nuclear C∗ -algebras. Acta Math. 209 (2012), 179–196. [15] H. Matui and Y. Sato. Decomposition rank of UHF-absorbing C∗ -algebras. Duke Math. J. 163 (2014), 2687–2708. [16] I. Namioka, Følner’s conditions for amenable semi-groups. Math. Scand. 15 (1964), 18–28. [17] D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48 (1987), 1–141. [18] N. C. Phillips. Large subalgebras. arXiv:1408.5546. [19] M. Rørdam and W. Winter. The Jiang–Su algebra revisited. J. Reine Angew. Math. 642 (2010), 129–155. [20] G. Szabó. The Rokhlin dimension of topological Zm -actions. Proc. Lond. Math. Soc. (3) 110 (2015), 673–694. [21] G. Szabo, J. Wu, and J. Zacharias. Rokhlin dimension for actions of residually finite groups. arXiv:1408.6096. [22] A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear C∗ -algebras. Ann. of Math. (2) 185 (2017), 229–284. [23] W. Winter. Nuclear dimension and Z-stability of pure C∗ -algebras. Invent. Math. 187 (2012), 259– 342. [24] W. Winter and J. Zacharias. Completely positive maps of order zero. Münster J. Math. 2 (2009), 311–324. Clinton T. Conley, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. E-mail address: clintonc@andrew.cmu.edu Steve Jackson, Department of Mathematics, University of North Texas, Denton, TX 762035017, U.S.A. E-mail address: jackson@unt.edu David Kerr, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. E-mail address: kerr@math.tamu.edu Andrew Marks, UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A. E-mail address: marks@math.ucla.edu Brandon Seward, Courant Institute of Mathematical Sciences, New York, NY 10012, U.S.A. E-mail address: bseward@cims.nyu.edu Robin Tucker-Drob, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. E-mail address: rtuckerd@math.tamu.edu 

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 1 Capacity Scaling in MIMO Systems with General Unitarily Invariant Random Matrices arXiv:1306.2595v4 [] 5 Mar 2018 Burak Çakmak, Ralf R. Müller, Senior Member, IEEE, Bernard H. Fleury, Senior Member, IEEE Abstract—We investigate the capacity scaling of MIMO systems with the system dimensions. To that end we quantify how the mutual information varies when the number of antennas (at either the receiver or transmitter side) is altered. For a system comprising R receive and T transmit antennas with R > T , we find the following: By removing as many receive antennas as needed to obtain a square system (provided the channel matrices before and after the removal have full rank) the maximum resulting loss of mutual information over all signal-to-noise ratios (SNRs) depends only on R, T and the matrix of left-singular vectors of the initial channel matrix, but not on its singular values. In particular, if the latter is Haar distributed the P matrix P 1 ergodic rate loss is given by Tt=1 R r=T +1 r−t nats. Under the same assumption, if T, R → ∞ with the ratio φ , T /R fixed, the rate loss normalized by R converges almost surely to H(φ) bits with H(·) denoting the binary entropy function. We also quantify and study how the mutual information as a function of the system dimensions deviates from the traditionally assumed linear growth in the minimum of the system dimensions at high SNR. Index Terms—multiple-input–multiple-output, mutual information, high SNR, multiplexing gain, unitary invariance, binary entropy function, Haar random matrix, S-transform I. I NTRODUCTION HE capacity of a multiple-input–multiple-output (MIMO) system with perfect channel state information at the receiver can be expressed as [1] T min(T, R) log2 SNR + O(1) (1) whenever the channel matrix has full rank almost surely. Here T and R denote the number of receive and transmit antennas, respectively, and O(1) is a bounded function of the signal-to-noise ratio (SNR) that does depend on T and R, in general. The scaling term min(T, R) is often referred to as the multiplexing gain. The explicit expression for the capacity scaling when the number of transmit or receive antennas varies, is difficult to calculate. Closed-form expressions can be obtained only in few particular cases, e.g. for a channel matrix of asymptotically large size with independent identically distributed (iid) zero-mean entries [2]. Burak Çakmak and Bernard H. Fleury were supported by the research project VIRTUOSO funded by Intel Mobile Communications, Keysight, Telenor, Aalborg University, and the Danish National Advanced Technology Foundation. Ralf R. Müller was supported by the Alexander von Humboldt Foundation. Burak Çakmak is with the Department of Computer Science, Technical University of Berlin, 10587 Berlin, Germany (e-mail: burak.cakmak@tuberlin.de). Ralf R. Müller is with the Institute for Digital Communications, FriedrichAlexander Universität Erlangen-Nürnberg, 91058 Erlangen, Germany (e-mail: mueller@lnt.de). Bernard H. Fleury is with the Department of Electronic Systems, Aalborg University, 9220 Aalborg, Denmark (e-mail: fleury@es.aau.dk). In order to better understand capacity scaling in MIMO channels with more complicated structures, such as correlation at transmit and/or receive antennas, related works use either implicit solutions, e.g. [3], or consider asymptotically high SNR and express the capacity in terms of the multiplexing gain, e.g. [4]. However, implicit solutions provide limited intuitive insight into the capacity scaling and the multiplexing gain is a crude measure of capacity. In this article, we consider an affine approximation to the mutual information at high SNR. In particular, we investigate how mutual information varies when the numbers of antennas (at either the receiver or transmitter side) is altered. Our affine approximation to the mutual information leads to a generalization of the multiplexing gain which we call the multiplexing rate. Such an approximation was formerly addressed in [1], which was the baseline of many published works, e.g. [5]–[7]. We study the variation of the multiplexing rate when the number of antennas either at the transmit or receive side varies. More specifically, we formulate the reduction of the number of antennas by means of a convenient linear projection operator. This formulation allows us to asses the mutual information at high SNR in insightful and explicit closed form. We consider unitarily invariant matrix ensembles [8] which model a broad class of MIMO channels [9]. Specifically, our sole restriction is that the matrix of left (right) singular vectors of the initial channel matrix, i.e. before the reduction, is Haar distributed. Informally speaking, this implies that the channel matrix involves some symmetry with respect to the antennas. An individual antenna contributes in a “democratic fashion” to the mutual information. There is no preferred antenna in the system. In fact, such an invariance seems a natural property for the mutual information to depend on T and R only, but not on the specific antennas in the system. Since the term O(1) in (1) is a bounded function of SNR, the expression (1) has more than once led to misinterpretations in the wireless communications community: (i) when the number of antennas at either the transmit or receive side varies, while the minimum of the system dimensions (i.e. the numbers of transmit and receive antennas) is kept fixed, the mutual information does not vary at high SNR; (ii) the mutual information scales linearly with the minimum of the system dimensions at high SNR. It is the goal of this paper to debunk these misinterpretations. We summarize our main contributions as follows: 1) As regards misinterpretation (i) we find the following: For a system comprising R receive and T transmit antennas IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 with R > T (T > R), let some of the receive (transmit) antennas be removed from the system to obtain a system with R̃ ≥ T receive (T̃ ≥ R transmit) antennas. Note that min(T, R̃) = T (min(T̃ , R) = R). Then, the loss of mutual information in the high SNR limit depends only on R, T and R̃ (T̃ ) and the matrix of left (right)singular vectors of the initial R × T channel matrix, but not on its singular values. Assuming the matrix of left-(right-)singular vectors to bePHaarPdistributed, T R 1 the ergodic rate loss is given by t=1 r=R̃+1 r−t PR PT 1 ( r=1 t=T̃ +1 t−r ) nats. 2) As regards misinterpretation (ii), we quantify how the mutual information as a function of the number of antennas deviates from the approximate linear growth (versus the minimum of the system dimensions) in the high SNR limit. This deviation does depend on the singular values of the channel matrix. We show that in the large system limit the deviation is additive for compound unitarily invariant channels and can be easily expressed in terms of the S-transform (in free probability) of the limiting eigenvalue distribution (LED) of the Gramian of the channel matrix. 3) We show that the aforementioned results on the variation of mutual information in the high SNR limit provide least upper bounds on said variation over all SNRs. Thus, these results have a universal character related to the SNR. 4) We derive novel formulations of the mutual information and the multiplexing rate in terms of the S-transform of the empirical eigenvalue distribution of the Gramian of the channel matrix. These formulations establish a fundamental relationship between the mutual information and the multiplexing rate. A. Related Work The work presented in paper [5] is related to contribution 1). Specifically, in [5, Section 3] the authors unveiled misinterpretation (i) for iid Gaussian unitarily invariant channel matrices. We elucidate misinterpretation (i) by considering arbitrary unitarily invariant matrices that need neither be Gaussian nor iid. In particular, our results and/or statements do not require any assumptions on the singular values of the channel matrix. They solely depend on the singular vectors of the channel matrix, e.g. see contribution 1). Our proof technique - which is based on an algebraic manipulation of the projection operator that we introduce - is different from any related work we are aware of. B. Organization The paper is organized as follows. In Section II, we introduce the preliminary notations and definitions. In Section III, we present the system model. In Section IV, we introduce new formulations of the mutual information and the multiplexing rate in terms of the S-transform. Section V and VI are dedicated to lift misinterpretations (i) and (ii), respectively. Conclusions are outlined in Section VII. The technical lemmas and the proofs are located in the Appendix. 2 II. N OTATIONS & D EFINITIONS N OTATION 1 We denote the binary entropy function as ( (p − 1) log2 (1 − p) − p log2 p p ∈ (0, 1) . (2) H(p) , 0 p ∈ {0, 1} N OTATION 2 For an N × K matrix X, FK X denotes the empirical eigenvalue distribution function of X † X, i.e. FK X (x) = 1 | {λi ∈ L : λi ≤x} | K (3) with L and |·| denoting the set of eigenvalues of X † X and the cardinality of a set, respectively. Here, (·)† denotes conjugate transposition. Moreover, for N, K → ∞ with φ = K/N fixed, if FK X converges weakly and almost surely to a LED function, this limit is denoted by FX . D EFINITION 1 A K-dimensional projector P β with β ≤ 1 is a βK × K matrix with entries (P β )ij = δij , ∀i, j, where δij denotes the Kronecker delta. D EFINITION 2 For an N × K matrix X 6= 0, we define the normalized rank of X † X as K αX , 1 − FK X (0) (4) and the distribution function of non-zero eigenvalues of X † X as
1  K αX − 1 u(x) + FK (5) F̃K X (x) X (x) , K αX with u(x) denoting the unit-step function. The S-transform introduced by Voiculescu in the context of free probability is defined as follows: D EFINITION 3 [10] Let F be a probability distribution function with support in [0, ∞). Moreover, let α , 1 − F(0) 6= 0. Define Z zx Ψ(z) , dF(x), −∞ < z < 0. (6) 1 − zx Then, the S-transform of F is defined as S(z) , z + 1 −1 Ψ (z), z −α < z < 0 (7) where Ψ−1 denotes the composition inverse of Ψ. N OTATION 3 For an N × K matrix X 6= 0, the S-transform K of FK X is denoted by SX . For N, K → ∞ with φ = K/N † fixed, if X X has a LED function FX almost surely, the Stransform of FX is denoted by SX . Similarly, we define ΨK X and ΨX . All large-system limits are assumed to hold in the almost sure sense, unless explicitly stated otherwise. Where obvious, limit operators indicating the large-system limit are omitted for the sake of compactness and readability. IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 3 III. S YSTEM M ODEL B. Unitary Invariance Consider the MIMO system y = Hx + n (8) where H ∈ CR×T , x ∈ CT ×1 , y ∈ CR×1 , n ∈ CR×1 are respectively the channel matrix, the input vector, the output vector, and the noise vector. The entries of x and n are assumed to be independent (circularly symmetric) complex Gaussian distributed with zero mean and variances σx2 and σn2 , respectively. The transmit SNR is defined as γ, σx2 , σn2 0 < γ < ∞. (9) The mutual information per transmit antenna of the communication link (8) is given by [11] Z I(γ; FTH ) , log2 (1 + γx) dFTH (x). (10) Similarly, I(γ; FR ) is the mutual information per receive H† antenna of (8). A. Antenna Removal Via Projector I(γ; FTV HU ) = I(γ; FTH ) In the sequel, we formulate the variation of mutual information when the number of antennas either at the transmit or receive side of reference system (8) changes. This variation is achieved by removing a certain fraction of antennas at the corresponding side of the system. We formulate this removal process via a multiplication of the channel matrix with a rectangular projector matrix. We distinguish between two cases: the removal of receive antennas and the removal of transmit antennas. In the first case, the system model resulting after removing a fraction 1 − β of receive antennas in (8) reads y β = P β (Hx + n) (11) = P β Hx + nβ . (12) The βR × R matrix P β is an R-dimensional projector which removes a fraction 1 − β of receive antennas in reference system (8) and nβ = P β n. The mutual information of the MIMO system (12) is equal to T I(γ; FTP β H ). (13) Similarly, removing a fraction 1 − β of transmit antennas in (8) yields the R × βT system ỹ = HP †β xβ + n. (14) Here, xβ is the vector obtained by removing from x the (1 − β)T entries fed to the removed transmit antennas, i.e. xβ = P β x with P β being a T -dimensional projector. The mutual information of system (14) reads βT I(γ; FβT HP †β For channel matrices that are unitarily invariant from right, i.e. H and HU admit the same distribution for any unitary matrix U independent of H, it does not matter which transmit antennas are removed. Only their number counts. The same applies to channel matrices that are unitarily invariant from left for the removal of receive antennas. For channel matrices that involve an asymmetry with respect to the antennas, i.e. some antennas contribute more to the mutual information than others, it must be specified which antennas are to be removed and the mutual information will depend (typically in a complicated manner) on the choice of the removed antennas. In this paper, we restrict the considerations to cases where only the number of removed antennas matters, since this leads to explicit closed-form expressions. For asymmetric channel matrices, one could obtain antennaindependent scaling laws if all antennas with equal contributions to mutual information are grouped together and all those groups are decimated proportionally. Doing so would heavily complicate the formulation of the antenna removal by means of multiplication with projector matrices. However, we can utilize the fact that for the channel in (8), mutual information is invariant to multiplication with unitary matrices, i.e. ). (15) (16) for all unitary matrices U and V . Since the channel matrix U HV is bi-unitarily invariant for all random unitary matrices U and V independent of H, and has the same mutual information as H, we can assume without loss of generality that H is unitarily invariant from left for receive and from right for transmit antenna removal, respectively, and keep the projector formulation of Section III-A as it is. The multiplication with a random unitary matrix followed by a fixed selection of antennas has statistically the same effect as a random selection of antennas. It provides the symmetry required to make mutual information only depend on the number of removed antennas and not on which antennas are removed. Equivalence to the ergodic capacity variation: The ergodic capacity of channel (8) is [12] i h ¯ FTH ) , max E I(γ; FT √ ) . (17) C(γ, H Q Q≥0 tr(Q)=T Conceptually, we relax the iid assumption on the entries of x in (8) and assume arbitrary correlation between these entries described by the covariance matrix σx2 Q where Q is nonnegative definite with unit trace and σx2 , T1 E[x† x]. It is shown in [12] that for channel matrices that are unitarily invariant from right the ergodic capacity in (17) is attained with Q = I, i.e.   ¯ FT ) = E I(γ; FT ) . C(γ, (18) H H In particular, the unitary invariance property of the channel is not broken by removing some of the transmit or receive antennas. For example, if H is invariant from right, then HP †β is invariant from right too. In summary, for bi-unitarily invariant channel matrices the variation of ergodic mutual informations that results from removing some number of IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 4 transmit or receive antennas does actually coincide with the corresponding variation of ergodic capacities. IV. M UTUAL I NFORMATION AND M ULTIPLEXING R ATE The normalized mutual information in (10) can be decomposed as Z T T I(γ; FH ) = αH log2 (γx) dF̃TH (x) {z } | I0 (γ;FT H) + T αH | Z   1 log2 1 + dF̃TH (x) . xγ {z } (19) ∆I(γ;FT H) We refer to the first term I0 (γ; FTH ) as the multiplexing T rate per transmit antenna. The factor αH is the multiplexing gain normalized by the number of transmit antennas. The second term ∆I(γ; FTH ) is the difference between the mutual information per transmit antenna and the multiplexing rate per transmit antenna. To alleviate the terminology, in the sequel we skip the explicit reference to the normalization by the number of transmit (or receive, see later) antennas when we refer to quantities such as those arising in (19). Whether the quantities considered are absolute or normalized will be clear from the context. We have lim ∆I(γ; FTH ) = 0. (20) γ→∞ If H † H is invertible we have   1 (21) I0 (γ; FTH ) = log2 det γH † H T   1 ∆I(γ; FTH ) = log2 det I + (γH † H)−1 (22) T with I denoting the identity matrix. The affine approximation of the ergodic mutual information at high SNR introduced in [1], see also [5, Eq. (9)] for a compact formulation of it, coincides with the ergodic formulation of our definition of the multiplexing rate. We next uncover a fundamental link between the mutual information and the multiplexing rate. This result makes use of the minimum-mean-square-error (MMSE) achieved by the optimal receiver for (8) normalized by the number of transmit antennas Z dFTH (x) T . (23) ηH (γ) , 1 + γx T Clearly, ηH (γ) is a strictly decreasing function of γ with range T (1 − αH , 1) [9]. T HEOREM 1 Define Z fH (x) , H(x)− x log2 STH (−z) dz, T 0 ≤ x ≤ αH . (24) 0 Then, we have T T I(γ; FTH ) = fH (1 − ηH ) + (1 − ηH ) log2 γ T T I0 (γ; FTH ) = fH (αH ) + αH log2 γ. T T For short we write ηH for ηH (γ) in (25). (25) (26) P ROOF 1 See Appendix B. Note that by definition the function fH (x) in (24) may T involve αH via STH (z). We have the following implications of Theorem 1: i) the mutual information can be directly expressed as a function of the (normalized) MMSE; ii) for any expression T of the mutual information as a function of the MMSE ηH the T multiplexing rate results immediately by substituting ηH for T 1−αH , e.g. see Examples 1 and 2; iii) the converse of ii) is not always true: given an expression of the multiplexing rate as a T T T function of αH , substituting αH for 1 − ηH does not always yield the mutual information. An intermediate step is required here to guarantee that the converse holds: the expression needs T first to be recast as a function of fH . Then substituting αH T for 1−ηH in the latter function yields the mutual information. If any probability distribution function with support in [0, ∞), say F, is substituted for FTH in (19) the formulas (25) and (26) remain valid provided I(γ; F) is finite and log(x) is absolutely integrable over F̃, respectively1 . The absolute integrability condition holds if, and only if, I(γ; F) and ∆I(γ; F) are finite, see (171)-(173). In the sequel we substitute FH for FTH to calculate I(γ; FH ) and I0 (γ; FH ). In Appendix C, we provide some sufficient conditions that guarantee the almost sure convergence of I(γ; FTH ) and I0 (γ; FTH ) to I(γ; FH ) and I0 (γ; FH ), respectively. We conclude that these asymptotic convergence are reasonable assumptions in practice, for the details see Appendix C. It is well-known that the S-transform of the LED of the product of asymptotically free matrices is the product of the respective S-transforms of the LEDs of these matrices. Therefore, for MIMO channel matrices that involve a compound structure, Theorem 1 provides a means to analytically calculate the large-system limits of the mutual information and multiplexing rate in terms of the large-system limits of the MMSE and the multiplexing gain. We next address two relevant random matrix ensembles that share this structure. E XAMPLE 1 We consider the concatenation of vector-valued fading channels described in [13]. Specifically, we assume that the channel matrix H factorizes according to H = X N X N −1 · · · X 2 X 1 (27) where the entries of the Kn × Kn−1 matrix X n are iid with zero mean and variance 1/Kn for n ∈ [1, N ]. Furthermore, the ratios ρn , Kn /K0 n ∈ [1, N ] are fixed as Kn → ∞. T Moreover, let ηH denote the large-system limit of MMSE ηH . By invoking Theorem 1 we obtain an analytical expression of the large-system limit of the mutual information in terms of (the large-system limit of) the MMSE2 as I(γ; FH ) = H(ηH ) + (1 − ηH )(log2 γ − N log2 e) # "N   X ρn 1 − ηH 1 − ηH H + log2 . + (1 − ηH ) 1 − ηH ρn ρn n=1 (28) F̃ is defined by substituting F̃T H for F in (5). explicit expression of the MMSE as a function of SNR is difficult to obtain. However, ηH (γ) can be solved numerically from the fixed point ηH (γ) QN ηH (γ)+ρn −1 equation γ = 1−η [13, Eq. (21)]. n=1 (γ) ρ 1 Here 2 An H n IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 5 Furthermore, as regards the multiplexing rate, we have I0 (γ; FH ) =H(αH ) + αH (log2 γ − N log2 e) # "N X ρn  αH  αH H + log2 (29) + αH α ρn ρn n=1 H with αH = min(1, ρ1 , · · · , ρN ). Normalizing this loss with the number of transmit antennas yields I(γ; FTH ) − I(γ; FTP β H ). (34) Assume that H and P β H have both full rank almost surely. Then, we define the rate loss χTH (R, βR) , lim I(γ; FTH ) − I(γ; FTP β H ), β ≥ φ. (35) γ→∞ = I0 (γ; FTH ) − I0 (γ; FTP β H ) P ROOF 2 See Appendix D. (36) † E XAMPLE 2 We consider a Jacobi matrix ensemble, see e.g. [14], [15], which find application in the context of optical MIMO communications [16], [17]. Accordingly, the channel matrix factorizes as H = P β2 U P †β1 (30) where U is an N × N Haar unitary matrix. From Theorem 1 we obtain I(γ; FH ) =H(ηH ) + (1 − ηH ) log2 γ   β1 H(β1 (1 − ηH )) β2 + H (1 − ηH ) − β1 β1 β2 (31) where ηH = ηH (γ) is given by p −(1 + κγ) + (1 + κγ)2 − 4β1 β2 γ(1 + γ) ηH (γ) = 1 + 2β1 (1 + γ) (32) with κ , β1 + β2 . Moreover, we have I0 (γ; FH ) =H(αH ) + αH log2 γ   H(β1 αH ) β2 β1 − + H αH β1 β1 β2 (33) with αH = min(1, β2 /β1 ). P ROOF 3 See Appendix E. V. T HE U NIVERSAL R ATE L OSS In Section 1 we underlined the following misinterpretation of mutual information: when the number of antennas (at either the transmit or receive side) varies, with the minimum of the system dimensions kept fixed, the mutual information does not vary at high SNR. It is the goal of this section to elucidate this misinterpretation. To do so we need to distinguish between two cases as to reference system (8): (i) T ≤ R; (ii) T ≥ R. In the former (latter) case we consider the removal of receive (transmit) antennas. In both cases the reduction of antennas is constrained in a way that keeps the minimum of the numbers of antennas at both sides fixed. A. Case (i) – Removing receive antennas We remove a fraction (1 − β) of receive antennas in system (8) to obtain system (12). We constrain the reduction with the condition β ≥ φ , T /R to ensure that min(T, βR) = T . This reduction of the number of receive antennas causes a loss in mutual information given by T I(γ; FTH ) − T I(γ; FTP β H ). = det H H 1 log2 . T det H † P †β P β H (37) T T The full-rank assumption implies αH = αP which is βH essential in the definition (35). Otherwise the difference in (35) diverges as γ → ∞. Next, we present some general important properties of the rate loss χTH (R, βR). 1) Universality related to SNR: Note that both quantities in (34) increase with the SNR. It is shown in Appendix F that their difference, i.e. (34), increases with the SNR too. Hence, the rate loss χTH (R, βR) provides the least upper bound on the mutual information loss over the entire SNR range. R EMARK 1 Let H and P β H have both full rank almost surely. Then, we have χTH (R, βR) = sup{I(γ; FTH ) − I(γ; FTP β H )}. (38) γ P ROOF 4 See Appendix F 2) Equivalence to capacity loss: Let us denote the capacity of channel (12) as C(γ; FTP β H ) , max I(γ; FTP β H √Q ). Q≥0 tr(Q)=T (39) It turns out that (35) also holds when the mutual informations in (35) are replaced by the respective capacities. R EMARK 2 Let H and P β H have both full rank almost surely. Then, we have χTH (R, βR) = lim C(γ; FTH ) − C(γ; FTP β H ). γ→∞ (40) P ROOF 5 See Appendix G. 3) The invariance related to singular values: Though χTH (R, βR) is defined through the distribution functions FTH and FTP β H in (35), it actually depends solely on the matrix of left singular vectors of H: T HEOREM 2 Let H and P β H have both full rank almost surely. Consider the spectral decomposition H = LSR (41) where L is a R × R unitary matrix whose columns are the left singular vectors of H, R is a T × T unitary matrix whose IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 6 columns are the right singular vectors of H and the diagonal entries of S are the singular values of H. Then, we have χTH (R, βR) = − 1 log2 det P φ L† P †β P β LP †φ . T (42) P ROOF 6 See Appendix H. 4) Statistical properties resulting from unitarily invariance: Let H † H have full rank almost surely and be unitarily invariant3 . Thereby, the matrix of left singular vectors of H, i.e. L, is Haar, see [9, Lemma 2.6]. Thus, P φ L† P †β P β LP †φ belongs to the Jacobi matrix ensemble, see Example 2. In other words, the rate loss χTH (R, βR) becomes nothing but minus the log det of the Jacobi matrix ensemble normalized by T . We also refer the reader to [14] for a detailed study of the determinant of the Jacobi matrix ensemble. In particular, from [14, Proposition 2.4], the rate loss admits the explicit statistical characterization χTH (R, βR) ∼ − T 1X log2 ρt T t=1 (43) where {ρ1 , · · · , ρT } are independent random variables and ρt ∼ Be ((βR + 1 − t), (1 − β)R). Here, X ∼ Y indicates that random variables X and Y are identically distributed. For a > 0 and b > 0, Be(a, b) denotes the Beta distribution with density Be(x; a, b) = Γ(a + b) a−1 x (1 − x)b−1 , Γ(a)Γ(b) x>0 (44) C OROLLARY 1 (U NIVERSAL R ATE L OSS ) Let H † H have full rank almost surely and be unitarily invariant. Define4 R T 1 X X T ln 2 t=1 0 r=R +1 R EMARK 3 The function T χT (R, R0 ) (see (45)) satisfies the symmetry property 0 T χT (R, T ) = T 0 χT (R, T 0 ) , T <R (49) where T 0 , R − T . where Γ is the gamma function. χT (R, R0 ) , Note that χT (R, T ) equals to the ergodic rate loss when we remove as many antennas as needed to obtain a square system. Furthermore, if R, T → ∞ with the ratio φ = T /R fixed, the first and the second terms of (48) converge to respectively the first and the second terms of (47). We coin the limit (47) the binary entropy loss as it only involves the binary entropy function evaluated at the aspect ratios φ and β/φ of two channel matrices – the one before and the one after the removal of the antennas. In particular, for β = φ, i.e. we remove as many receive antennas as needed to obtain a square system, the binary entropy loss has the compact expression H(φ)/φ. 5) A symmetry property of the universal rate loss: We show a symmetry property of the universal rate loss in the case when the end system after (completion of the antenna removal) is square, i.e. β = φ. Let us start with an illustrative example. Consider two separate MIMO systems one of dimensions 3×2 and one of dimensions 3×1. Let the antenna removal processes be 3 × 1 → 1 × 1 for the former system and 3 × 2 → 2 × 2 for the latter. Thus, in both cases two communication links are removed from the reference systems. Let the channel matrices of the reference systems fulfill the conditions stated in Corollary 1 (i.e. full-rank and unitary invariance). Both removal process lead to the same the binary entropy loss equal to 3H(1/3) = 3H(2/3) = 2.75 bit. 1 , r−t T ≤ R0 ≤ R. (45) Then, we have E[χTH (R, βR)] = χT (R, βR). (46) Moreover, if R, T → ∞ with φ = T /R fixed, we have almost surely   H (φ) β φ T χH (R, βR) → − H . (47) φ φ β P ROOF 7 See Appendix I. The name Universal Rate Loss refers to the fact that the results in Corollary 1 solely refer to the number of transmit and receive antennas before and after the variation. The ergodic rate loss has the additive property χT (R, R0 ) = χT (R, T ) − χT (R0 , T ) , T ≤ R0 ≤ R. (48) 3 Provided H † H is unitarily invariant, when H has almost surely full rank, so does P β H too, see Appendix I. 4 The sum over an empty index set is by definition zero. P ROOF 8 See Appendix J 0 Note that the expressions T χT (R, T ) and T 0 χT (R, T 0 ) corresponds to the ergodic rate losses for the antenna removal processes R×T → T ×T and R×T 0 → T 0 ×T 0 , respectively. In both cases T × (R − T ) communications links are removed from the reference systems. In other words, for R being fixed the ergodic rate loss T χT (R, T ) is a symmetric function of T with respect to T = R/2 (see Figure 2). Since χφR (R, βR) ≤ χφR (R, φR), the symmetry property (49) implies that the maximum ergodic rate loss is attained when φ = β = 1/2. R EMARK 4 Let H † H have full rank almost surely and be unitarily invariant. Then, for φ ≤ β ≤ 1 we have   1 1 , = arg max E[χφR (50) H (R, βR)]. φ,β 2 2 B. Case (ii) – Removing transmit antennas We remove a fraction (1 − β) of transmit antennas in (8) to obtain system (14). We constrain the reduction of receive antennas with β ≥ 1/φ (φ = T /R) to ensure min(βT, R) = T . Reducing the number of transmit antennas results in a loss of mutual information equal to T I(γ; FTH ) − βT I(γ; FβT † ). HP β IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 7 Normalizing this loss with the number of transmit antennas of the reference system gives I(γ; FTH ) − βI(γ; FβT HP †β ). (51) Let H and HP †β have both full rank almost surely. Then, we define the large SNR limit 1 . φ (52) Again the full rank assumption is important here. Otherwise the difference (52) may diverge as γ → ∞. βT T χ̃R H (T, βT ) , lim I(γ; FH ) − βI(γ; F HP †β γ→∞ ), β≥ C OROLLARY 2 Let H and HP †β have both full rank almost surely. Then, we have 1 (53) log2 det P φ1 RP †β P β R† P †1 φ T where R is a T × T unitary matrix whose columns are the right singular vectors of H, see (41). χ̃R H (T, βT ) = − Note that the right-hand side in (53) is obtained by formally replacing φ with φ−1 in the right-hand side of (47). This follows from the identity βI(γ; FβT HP †β )= 1 I(γ; FR P β H † ). φ (54) This substitution is valid for any result that refers to mutual information, e.g. as in Corollary 1. However, it does not apply in general to capacity related results, such as in Remark 2, due to the placement of the projection operator on the transmitter side. C. The rate loss with antenna power profile In this subsection we address the rate loss χTH for a channel model that takes into consideration the power imbalance at the transmitter and receiver sides: H = ΛR H̃ΛT . (55) Here, the matrices ΛR ∈ CR×R and ΛT ∈ CT ×T are diagonal, full-rank, and deterministic. The matrix ΛR (ΛT ) represents the power imbalance at receive (transmit) side. We generalize Theorem 2 for the model (55) as (see Appendix H) † χTH (R, βR) det P φ L̃ Θ1 L̃P †φ 1 = log2 † T det P φ L̃ Θβ L̃P †φ Moreover, let X β , P β X where X is a R × T matrix with iid zero-mean complex Gaussian entries. Let D β be the βR × βR diagonal matrix whose diagonal entries are the non-zero eigenvalues of Θβ , ΛR † P †β P β ΛR . Then, we have " # det X †1 D 1 X 1 1 T . (57) E[χH (R, βR)] = E log2 T det X †β D β X β (56) where Θβ , ΛR † P †β P β ΛR for β ≤ 1 and L̃ is a R × R unitary matrix whose columns are the left singular vectors of H̃, see (41). Note that the rate loss does not depend on the singular values of H̃. This property allows for obtaining a convenient expression for the ergodic rate loss E[χTH (R, βR)] † when H̃ H̃ is unitarily invariant, i.e. L̃ in (56) is Haar distributed. C OROLLARY 3 Let H be defined as in (55). Furthermore, let † H̃ H̃ have full rank almost surely and be unitarily invariant. P ROOF 9 See Appendix K The expectation in (57) can be simply computed by using the following result. L EMMA 1 [5, Lemma 2] Let X be an n × m matrix with iid zero-mean complex Gaussian entries such that n > m. Let D be an n × n deterministic Hermitian positive-definite matrix whose jth eigenvalue is denoted by λj . Moreover, let Ω be the n × n Vandermonde matrix with (Ω)ij = λij−1 and Γ be the (n − m) × (n − m) principal submatrix of Ω. Then, we have m E[ln det X † DX] = det Γ X det Ψi det Ω i=1 (58) where Ψi is m × m matrix whose entries are n−m−1+l (Ψi )k,l =νn−m+k λn−m+k − n−m X d−1 d−1 νq (Γ−1 )d,q λn−m+k λqn−m−1+1 . (59) d=1,q=1 In this expression, νq = ψ(l) + ln λq if l = i else νq = 1 with ψ(·) denoting the digamma function. D. Further discussions based on numerical results As a warm up example, consider a 4 × 2 MIMO system that is stripped off two of its four receive antennas. For fullrank channel matrices that are unitarily invariant from left Theorem 1 gives the exact high SNR limit of the ergodic loss equal to 4χ2 (4, 2) = 3.37 bit. The asymptotic loss (47) is 4H(2/4) = 4 bit. Note also that 4χ2 (4, 2) is the supremum of the mutual information loss over all SNRs. This is depicted in Figure 1 for a Gaussian channel. We illustrate the universal rate loss and the tightness of the approximation provided by the binary entropy loss, i.e. RH(T /R), already for small system dimensions. To this end we consider three different channel models that are unitarily invariant from the left: (i) the channel matrix H = U Λ where U ∈ CR×T is uniformly distributed over the manifold of complex R×T matrices such that U † U = I and Λ ∈ RT ×T is a positive diagonal matrix that represents the power imbalance at the transmitter. This is a typical channel model in the context of massive MIMO, i.e. in the regime of T  R. Here we point out that Λ does not affect the rate loss. Therefore for convenience we set Λ = I. (ii) the channel matrix H = X with the entries of X being zero-mean iid complex-valued Gaussian with finite variance; (iii) the channel matrix H = X 2 DX 1 . Here X 1 ∈ CS×T and X 2 ∈ CR×S represent the propagation channel from the transmit antennas IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 8 15 VI. D EVIATION FROM L INEAR G ROWTH T E[I(.; FTH )] T E[I0 (.; FTH )] T E[I(.; FTP- H )] T E[I0 (.; FTP- H )] [bits] 10 3:37 5 0 -10 -5 0 5 10 15 20 .(dB) Fig. 1: Ergodic mutual information (continuous lines) and ergodic multiplexing rate (dashed lines) versus the SNR of a zero-mean iid complex Gaussian MIMO channel with T = 2 transmit antennas and the number of receive antennas decreased from R = 4 (blue curves) to R = 2 (red curves). 4H(T =4) H=U H=X H = X2 DX1 4 4E[@TH (4; T )] 3 2 In this section we clarify the second misinterpretation underlined in Section 1. Specifically, we analyze the variation of the multiplexing rate when either the number of receive or the number transmit antennas varies while their maximum is kept fixed. For a channel matrix having orthogonal columns when the number of transmit or receive antennas varies, the linear growth of mutual information is obvious. However, for a channel matrix with e.g. iid entries, a substantial crosstalk arises due to the lack of orthogonality of its columns. The effect of this crosstalk onto mutual information is non-linear in the number of antennas. The mutual information scales approximately linearly in the minimum of the numbers of transmit and receive antennas. For a tall rectangular channel matrix that becomes wider and wider, the mutual information can only grow approximately linearly until the matrix becomes square. The same holds for a wide rectangular channel matrix growing taller and taller. Therefore, we have to distinguish between two cases: (i) the number of receive antennas is smaller than the number of transmit antennas, i.e. a wide channel matrix, and (ii) the converse of (i), i.e. a tall channel matrix. Since case (ii) can be easily treated by replacing the channel matrix with its conjugate transpose, we restrict our investigations to case (i). The linear growth cannot continue once the channel matrix has grown square. Thus, it makes sense to constrain the matrix of reference system (8) to be square; i.e. we assume that the channel matrix H in (8) is N × N i.e. N = R = T . The exact mutual information of the (rectangular) system (14) of size βN × N , β ≤ 1 is N I(γ; FN P β H ). (60) The mutual information (60) scales approximately linearly with the number of receive antennas, if it is close to 1 βN I(γ; FN H ). 0 0 1 2 3 4 T Fig. 2: The maximal ergodic mutual information loss over the SNR range: The entries of X ∈ C4×T , X 1 ∈ CS=4×T and X 2 ∈ C4×S=4 are zero mean iid complex Gaussian. The matrix U ∈ C4×T is uniformly distributed over the manifold of complex 4 × T matrices. The S × S matrix D is positive diagonal. Its diagonal entries are iid and uniformly distributed. to the scatterers and from the scatterers to the receive antennas respectively, while the diagonal entries in the diagonal matrix D are the individual scattering coefficients of the scatterers. This random matrix ensemble models the channel under the assumption of propagation via one-bounce scattering only [18]. To fulfill the full-rank condition we restrict to the case S ≥ T . From Figure 2, we conclude that the binary entropy loss yields an accurate approximation even for small system dimensions. (61) Thus, in the high SNR limit, the deviation from the linear growth normalized to N (the deviation from linear growth for short) is given by N N ∆L(β; FN H ) , lim I(γ; FP β H ) − βI(γ; FH ) γ→∞ N = I0 (γ; FN P β H ) − βI0 (γ; FH ) (62) (63) where H is assumed to have full rank almost surely. The fullT T rank assumption implies αH = αP which is necessary in βH the definition (62). Otherwise, (62) is divergent. E XAMPLE 3 Let H be unitary. Then, we have ∆L(β; FN H ) = 0. (64) A. The large-system limit consideration The deviation from linear growth (63) differs from the quantity χTH defined in (35) only by the factor β scaling the second term. Unlike χTH , ∆L does depend on the singular values of channel matrix. This makes the analysis somehow IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 9 A SSUMPTION 1 The channel matrix H has full rank almost surely. Furthermore, HH † is unitarily invariant, has a uniformly bounded spectral norm, and its empirical eigenvalue distribution converges almost surely as N → ∞. Moreover, ∆I(1; FH ) is finite. We carry out the analysis on the basis of the LED function FP β H . Specifically, we consider ∆L(β; FH ) = I0 (γ; FP β H ) − βI0 (γ; FH ). (65) When we interpret the asymptotic results in the numerical investigations we assume that A 40 N E[I0 (.; FN P- H )] -N E[I0 (.; FN H )] [bits] 30 20 10 0 0 1 2 3 4 5 3 4 5 -N B 3 N E["L(-; FN H )] N (- ! 1) log2 (1 ! -) 2 [bits] intractable. On the other hand, it is well-known that asymptotic results when the numbers of antennas grow large provide very good approximations already for systems with a dozen (or even less) of antennas in practice. Thus, we can resort to the asymptotic regime in the number of antennas to study the deviation from linear growth. To that end, in this section we make use of the following underlying assumption: 1 0 0 1 2 -N lim E[I0 (γ; FN P β H )] = I0 (γ; FP β H ) , N →∞ β ≤ 1. (66) It is easy to show that the convergence (66) is a mild assumption for β < 1: as FH is assumed to have a compact support, FP β H has a compact support too, see [19, Corollary 1.14]. Note that a compactly supported probability distribution can be uniquely characterized by its moments. This fact allow us to use the machinery provided in Proposition 1 in Appendix C. R Specifically, supN E[ x−1 dF̃N P β H (x)] < ∞ is sufficient for (66) to hold. Indeed this is a reasonable condition for β < 1 since Z 1 dF̃P β H (x) , 0 < β < 1 (67) x Fig. 3: Ergodic multiplexing rate and corresponding linear growth (A) and (ergodic) deviation from linear growth (B) versus number of receive antennas βN . The entries of H ∈ C5×5 are iid Gaussian with zero mean and variance 1/5. The SNR is γ = 20 dB. L EMMA 2 Let H fulfill Assumption 1. Then, we have Z 1 SH (−βz) dz. ∆L(β; FH ) = −β log2 SH (−z) 0 (69) is strictly increasing with β, see Remark 5. P ROOF 11 See Appendix M. E XAMPLE 4 Let the entries of H be iid with zero mean and variance σ 2 /N . Then, we have Alternatively, we may bypass the need for using the Stransform by invoking the following result: ∆L(β; FH ) = (β − 1) log2 (1 − β) R EMARK 5 Let H fulfill Assumption 1. Furthermore, let P t be an N -dimensional projector with 0 < t < 1. Then, we have Z 1 SH (−t) = dF̃P t H (x) , 0 < t < 1. (70) x (68) where by convention 0 log2 0 = 0. P ROOF 10 See Appendix L. In other words, at high SNR the normalized mutual information of a MIMO system of sufficiently large dimensions with zero-mean iid channel entries grows approximately linearly with the minimum of the numbers of transmit and receive antennas up to 1st order and the deviation from the linear growth is close to (β − 1) log2 (1 − β). Figure 3 illustrates this behavior. The result in (70) also provides a convenient means to calculate the deviation from linear growth in the large-system limit. The right-hand side of (70) is nothing but the asymptotic inverse spectral mean of the channel matrix P t H. B. The S-transform formulation C. The universality related to the SNR range The result in Example 4 can be obtained from previous capacity results, e.g. [9, Eq. (2.63)]. We obtained it as a special case of the following lemma. Note that the difference I(γ; FP β H ) − βI(γ; FH ) converges to ∆L(β; FH ) as the SNR tends to infinity, see (63). In Appendix O we show that this difference actually increases with SNR unless FH is a Dirac distribution function. Thus, we P ROOF 12 See Appendix M. IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 10 VII. C ONCLUSIONS have the following universal characterization over the whole SNR range. R EMARK 6 Let H fulfill Assumption 1. Then, we have  ∆L(β, FH ) = sup I(γ; FP β H ) − βI(γ; FH ) . (71) γ P ROOF 13 See Appendix O. D. The additive property We now draw the attention to another important property of the deviation from linear growth: T HEOREM 3 Let X and Y be independent CN ×N random matrices. Moreover, let X and Y fulfill Assumption 1. Then, we have ∆L(β; FXY ) = ∆L(β; FX ) + ∆L(β; FY ). (72) P ROOF 14 See Appendix P. E XAMPLE 5 Consider a random matrix defined as H= M Y Am (73) m=1 where the N × N matrices Am , m = 1, . . . , M , are independent, have iid entries with zero mean and variance σ 2 /N . Then we have almost surely ∆L(β; FH ) = M ∆L(β; FA1 ) (74) = M (β − 1) log2 (1 − β). A variation of the number of antennas in a MIMO system affects the mutual information at asymptotically large SNR in following way: If the minimum number of antennas at transmitter and receiver side stays unaltered, the change of mutual information depends only on the system dimensions and the matrix of left (or right) singular vectors of initial channel matrix but not on its singular values. For channel matrices that are unitarily invariant from left (or right) this change of mutual information in the ergodic sense can be expressed with a simple analytic function of the system dimensions. Moreover, the large system limit of this expression involves only the binary entropy functions of the aspect ratios of two varying channel matrices – the one before and the one after altering the number of antennas. Mutual information grows only approximately linear with the minimum of the system dimensions even at high SNR. This deviation from that linear growth, i.e. the error of the linear approximation, does depend on the singular values of the channel matrix. It can be quantified and has the following remarkable property in the large system limit: For certain factorizable MIMO channel matrices, the deviation is the sum of the deviations of the individual factors. The results derived in this work for asymptoticly large SNR are least upper bounds over the whole SNR range. This gives them a universal character. Finally, a fundamental relation between mutual information and its affine approximation (the multiplexing rate) was unveiled. This relation can be conveniently described via the S-transform of free probability. (75) As mentioned previously, the crosstalk due to non-orthogonal columns in H affects the mutual information in a way that is non-linear in the number of antennas. Thus, it causes the deviation from linear growth. Let us be more precise here and (inspired from [20, Eq. (1)]) introduce the concept of crosstalk ratio: PN P † 2 i=1 j<i |hi hj | . (76) CTH , lim PN † 2 N →∞ i=1 |hi hi | Here hi denotes the ith column of H. For example, for an unitary matrix H, we have CTH = 0. As a second example, let the entries of H be iid complex Gaussian with zero mean and variance 1/N . Then, from (196) we get 1 CTH = . (77) 2 We next show that the crosstalk ratio has the same additive property as the deviation from linear growth. A PPENDIX A P RELIMINARIES L EMMA 3 Let p ∈ [0, 1]. Then, we have Zp log2 P ROOF 16 We first recast (79) into the equivalent identity Zx lim log2 x→p 1−z dz = H(p). x−z (80) 0 To prove (80), we first apply a variable substitution Zx 1−t log2 dt = x x−t 0 Z1 log2 x−1 − z dz 1−z (81) 0 and decompose the right hand side of (81) as Z1 x P ROOF 15 See Appendix Q. (79) 0 N ×N R EMARK 7 Let X and Y be independent C random matrices. Moreover let X and Y fulfill Assumption 1. Then we have CTXY = CTX + CTY . (78) 1−z dz = H(p). p−z log2 (x 0 −1 Z1 − z) dz − x log2 (1 − z) dz. 0 (82) IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 11 Define u , log2 (x−1 −z) and v = z. Applying the integration by part rule, we obtain for the first integral: Z1 log2 x −1
− z dz 1 uv|0 = 0 Z1 − v du (83) lim (Ψn )−1 (z) = Ψ−1 (z), n→∞ 0 = x−1 H(x) − log2 e. Furthermore, Ψn (z) is a strictly increasing homeomorphism of (−∞, 0) onto (−α, 0) [22]. This implies that (see e.g. [23, Proposition 0.1]) (84) −α < z < 0. (93) This completes the proof. Using (84), we compute the second integral: Z1 Z1 log2 (1 − z) dz = lim x→1 0 0 L EMMA 6 Consider a random matrix X and a projector P β . Assume that X † X and P †β P β are asymptotically free. Then, log2 (x−1 − z) dz = − log2 e. (85) we have SXP † (z) = SX (βz). (94) β This completes the proof. L EMMA 4 [21] Let A and A + B be invertible and B have rank 1. Furthermore let g , tr(BA−1 ) 6= −1. Then, we have (A + B) −1 1 A−1 BA−1 . = A−1 − g+1 (86) L EMMA 5 [22, Lemma 2 & Lemma 4] Let F be a probability distribution function with support in [0, ∞) and S its Stransform. Moreover, let F be not a Dirac distribution function. Then, S is strictly decreasing on (−α, 0) with α , 1 − F(0). In particular, we have Z −1 lim− S(z) = x dF(x) (87) z→0 Z 1 lim + S(z) = dF(x) (88) x z→−α P ROOF 18 The S-transform of P β reads [9, Example 2.32] SP β (z) = z+1 . z+β (95) By invoking the identity [9, Theorem 2.32] and the asymptotic freeness between X † X and P †β P β , we obtain SXP † (z) = β = z+1 SP (βz)SX (βz) z + 1/β β SX (βz). (96) (97) R EMARK 8 Let H = P β2 U P †β1 with U an N -dimensional Haar unitary. Then, we have almost surely SH (z) = 1 + β1 z . β2 + β1 z (98) where we use the convention 1/0 = ∞ in (88) when F(0) > 0. T HEOREM 4 [22, Proposition 1] Let F be a probability distribution function R with support in (0, ∞) and S its Stransform. Then | log x| dF(x) is finite if, and only if, R1 | log S(−z)| dz is finite. If either of these integrals is finite, 0 Z1 Z log(x) dF(x) = − log S(−z) dz. (89) 0 T HEOREM 5 For n ∈ N+ , {1, 2, ...} let Fn be probability distribution functions on [0, ∞). Furthermore let 1 − Fn (0) = α > 0, ∀n ∈ N+ . Moreover let Sn denote the S-transform of Fn . Then if Fn converges weakly to a probability distribution function F as n → ∞, we have lim Sn (z) = S(z), n→∞ −α < z < 0 (90) P ROOF 19 By invoking Lemma 6 to (95) we obtain (98). A PPENDIX B P ROOF OF T HEOREM 1 A. Proof of (25) By definition, I(γ; FTH ) < ∞. Then, from identity [22, Eq. (5)] we write Z 1 I(γ; FTH ) = − log2 (s)∂ΨT√γH (−s) ds (99) 0 dΨT H (x) where ∂ΨTH (ω) , dx . x=ω At this stage we point out two identities: T ΨT√γH (−1) + 1 = ΨTH (−γ) + 1 = ηH (γ) where S is the S-transform of F. lim x→0− P ROOF 17 Let us consider the function, (see (6)) Z zx dFn (x), −∞ < z < 0. Ψn (z) , 1 − zx (91) zx For z ∈ (−∞, 0), z → 1−zx is bounded and continuous. Hence, the weak convergence of Fn implies that n lim Ψ (z) = Ψ(z), n→∞ −∞ < z < 0. (92) ΨT√ γH (x) + 1 = 1. (100) (101) Now we apply the variable substitution z , ΨT√γH (−s) + 1 in the integral in (99). Notice that with this substitution the upper and lower limits of this integral read (100) and (101), respectively. As a result (99) is recast in the form I(γ; FTH ) = Z 1 T ηH   <−1> log2 −ΨT√γH (z − 1) dz (102) IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 12 <−1> with ΨTH denoting the inverse of ΨTH . Then, by the definition of the S-transform, see (7), we obtain T T Z ηH Z ηH 1−z dz + log2 ST√γH (z − 1) dz I(γ; FTH ) = log2 z 1 1 (103) T Z ηH T = H(ηH )+ (104) log2 ST√γH (z − 1) dz 1 T = H(ηH )− T 1−ηH Z log2 ST√γH (−z) dz. (105) 0 Finally, we obtain (25) by using the scaling property of the S-transform [24, Lemma 4.2]. B. Proof of (26) Let S̃TH be the S-transform of F̃TH . By using [9, Theorem 2.32] we write z+1 T STH (αH z), −1 < z < 0. (106) S̃TH (z) = T z + 1/αH Note that F̃TH is an empirical distribution function. Thus, log2 (x) is absolutely integrable over it. We use Theorem 4 and Lemma 3 to complete the proof: I0 (γ; FTH ) = T αH log2 γ − T αH Z1 log2 S̃TH (−x) dx (107) we have almost surely lim I0 (γ; FTH ) = I0 (γ; FH ). T →∞ Condition (111) is reasonable in practice. Otherwise the power amplification per dimension of the MIMO system explodes as its dimensions grow to infinity. One can show that for rectangular and unitarily invariant channel matrices, the condition (113) is reasonable too due to the strict decreasing property of the function of β in (67). However, it might not hold when the channel matrix is square. As an example, consider a channel matrix H whose entries are iid with zero mean and variance σ 2 /T . Then, condition (113) holds if φ 6= 1, but is violated if φ = 1. Indeed the latter case turns out critical for the “log det” convergence of the zero-mean iid matrix ensemble, e.g. see [9], [25]. Nevertheless, both [26, Proposition 2.2] and numerical evidence lead us to conjecture that (114) holds when φ = 1 as well. Thus, we conclude that the asymptotic convergence of the multiplexing rate, i.e. (114), is a mild assumption in practice. Proof of Proposition 1 For the sake of readability of the proof, whenever we use the limit operator indicating that T tends to infinity, we implicitly assume that the ratio φ = T /R is fixed. For convenience we define Y , I + γH † H. 0 T T = αH log2 γ − αH Z1 log2 0 1−x T ST (−αH x) dx T −x H 1/αH T = αH log2 γ − Z log2 T αH −x T S (−x) dx 1−x H I(γ; FTH ) (109) T T T = αH log2 γ + H(αH )− log2 STH (−x) dx. (110) 0 A PPENDIX C O N THE CONVERGENCE OF M UTUAL INFORMATION AND M ULTIPLEXING R ATE In this section we provide some sufficient conditions that guarantee the convergence of the mutual information (10) and multiplexing rate (see (19)) in the large system limit. P ROPOSITION 1 As R, T → ∞ with the ratio φ , T /R fixed let H † H have a LED FH . Furthermore, let Z sup x dFTH (x) < ∞ a.s.. (111) T Then we have almost surely lim I(γ; FTH ) = I(γ; FH ). T →∞ Moreover if in addition Z 1 dF̃TH (x) < ∞ a.s. sup x T Z0 =− log2 ST√Y (z) dz. (116) −1 0 ZαH (115) By Theorem 4 we have (108) αT H (114) (112) (113) The function ST√Y is strictly decreasing on (−1, 0) if, and only if, FTH is not a Dirac distribution function, see Lemma 5. If FTH is a Dirac distribution function then ST√Y is a constant function. Without loss of generality, we can assume that FTH is not a Dirac distribution function. Then, by invoking Lemma 5 again we have −1  1 1 tr(Y ) < ST√Y (z) < tr(Y −1 ), −1 < z < 0. T T (117) For convenience we define the random variable Z T M , sup x dFTH (x) s.t. φ = . (118) R T Since the upper bound in (117) is smaller than one we have | log2 ST√Y (z)| = − log2 ST√Y (z), 1 < log2 tr(Y ) T ≤ log2 (1 + γM ). −1 < z < 0 (119) (120) (121) Because of (121), we can apply Lebesgue’s dominated convergence theorem [27, Theorem 10.21]: lim I(γ; FTH ) = − Z0 T →∞ −1 lim log2 ST√Y (z) dz. T →∞ (122) IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 13 By invoking Theorem 5 we complete the proof of (111): lim log2 ST√Y (z) = log2 lim ST√Y (z) (123) = (124) T →∞ T →∞ log2 S√Y (z). I0 (γ, FTH ) where the result (131) follows from the identity (84). We obtain (31) from (25) with (131) inserted in (24). Moreover, by the definition of the S-transform we have β1 (1 − z)Ψ2H (z) + (1 − (β1 + β2 )z)ΨH (z) − β2 z = 0. (132) To prove (114), we use the same arguments for as for I(γ, FTH ). In particular, by invoking Lemma 5 again we can write Z −1 Z 1 x dF̃TH (x) < S̃TH (z) < dF̃TH (x), −1 < z < 0 x (125) with S̃TH denoting the S-transform of F̃TH . Unlike (117), the right-most integral is not bounded in general, so we need the additional assumption (113). This completes the proof. Note that 1 + ΨH (−γ) = ηH (γ). Thus, (132) has two solutions for ηH (γ). Only one fulfills the properties of ηH (γ) in [9, pp. 41]. Specifically, from the property ηH (γ) → 1 as γ → 0 we concludeR that (32) is this solution. Finally it is also 1 easy to show that R 0 | log2 S̃H (−z)| dz is finite in this case. This implies that | log(x)|dF̃H (x) is finite too. Thus, the multiplexing rate is obtained by replacing the term (1 − ηH ) in (31) with αH , which leads to (33). A PPENDIX D P ROOF OF E XAMPLE 1 A PPENDIX F P ROOF OF R EMARK 1 With a convenient re-parameterization of [13, Eq. (19)] we write N Y ρn . (126) SH (z) = z + ρn n=1 We can write the integral terms in (127) as Z 1−ηH z log2 (1 − ) dz = ρn 0 Z 1 1 − ηH ρn log2 + − z) dz log2 ( ρn 1 − ηH 0 d{I(γ; FTH ) − I(γ; FTP β H )} dγ = T T ηP (γ) − ηH (γ) βH γ ln 2 (127) where the equality holds when β = 1. To prove (134) it is sufficient to consider the removal of a single receive antenna, i.e. β = (R − 1)/R. It is immediate that H † H = H † P †β P β H + h†R hR (128) for n ∈ [1, N ]. By invoking the result in (84) we obtain (28). From the linearity R 1 property of the Lebesgue integral, it is easy Rto show that 0 | log2 S̃H (−z)dz| is finite, which implies that | log(x)|dF̃H (x) is finite too, due to Theorem 4. Thus, the multiplexing rate is obtained by replacing the term (1−ηH ) in (28) with αH (due to Theorem 1). This leads to (29). Finally, we note that if αH < 1 the S-transform SH (z) diverges as z → (−αH ), see Lemma 5. Thus, from (126) the unique solution of αH is αH = min(1, ρ1 , ρ2 , . . . , ρN ). (135) with hR ∈ C1×T representing the Rth row of H. Then (134) follows directly from Lemma 4 in Appendix A. A PPENDIX G P ROOF OF R EMARK 2 We decompose the capacity expression in (39) as C(γ, FTP β H ) = I0 (γ; FTP β H √Q∗ ) + ∆I(γ; FTP β H √Q∗ ) (136) with Q? denoting the capacity achieving covariance matrix. We define C0 (γ, FTP β H ) , max I0 (γ; FTP β H √Q ). Recall (98): 1 + β1 z . β2 + β1 z (137) Q≥0 tr(Q)=T A PPENDIX E P ROOF OF E XAMPLE 2 SH (z) = . (133) Hence, to prove Remark 1 we simply need to show that n o tr (I + γH † P †β P β H)−1 − (I + γH † H)−1 ≥ 0 (134) From Theorem 1 we have I(γ; FH ) = H(ηH ) + (1 − ηH ) log2 γ N Z 1−ηH X z ) dz. + log2 (1 − ρ n n=1 0 We first point out the relationship [9] (129) Moreover, notice that αH = 1 − FH (0) = min(1, β2 /β1 ). For convenience let a , 1 − ηH (γ) < αH . Then, we have Z a Z 1 1 − β1 at log2 log2 SH (−z) dz = a dt (130) β2 − β1 at 0  0 β1 H(β1 a) β2 = − H a (131) β1 β1 β2 In particular, by the definitions in (39) and (137) we have C0 (γ, FTP β H ) ≥ I0 (γ; FTP β H √Q∗ ). Hence, we have lim C0 (γ; FTP β H ) − C(γ; FTP β H ) ≥ 0. (138) γ→∞ Since H † P †β P β H has almost surely full rank, we have T = α√ and thereby Q T √ αP βH Q T T √ I0 (γ; FTP β H √Q ) = α√ Q log2 γ+α Q Z T log2 x dF̃P β H √Q (x). (139) IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 14 For a sufficiently large SNR a full-rank matrix Q maximizes (139). Therefore, to prove the result we can assume without loss of generality that Q has full rank. Doing so, we have I0 (γ; FTP β H √Q ) 1 = log2 γ + log2 det H † P †β P β H T 1 + log2 det Q. (140) T Due to the constraint tr(Q) = T , the identity operator maximizes (140). Hence, from (138) we have lim I0 (γ; FTP β H ) − C(γ; FTP β H ) ≥ 0. γ→∞ (141) On the other hand we have I0 (γ; FTP β H ) < I(γ; FTP β H ) ≤ C(γ; FTP β H ). (142) Thus (141) must be zero. This completes the proof. A PPENDIX I P ROOF OF C OROLLARY 1 We first show that provided H † H is unitarily invariant, when H † H has full rank almost surely, so does H † P †β P β H too for φ ≤ β: From (147) we have det H † P †β P β H = det Σ2 det P φ L† P †β P β LP †φ (149) where Σ is a T ×T diagonal matrix whose diagonal entries are the positive singular values of H. By the unitary invariance assumption, P φ L† P †β P β LP †φ is a Jacobi matrix ensemble with a positive determinant for φ ≤ β [14]. Thereby, (149) is positive. Given x ∼ Be(a, b) we have E[ln x] = ψ(a) − ψ(a + b) where ψ(·) denotes the digamma function. For natural arguments, the digamma function can be expressed as n−1 X 1 . l A PPENDIX H P ROOF OF T HEOREM 2 ψ(n) = ψ(1) + We prove (56) which is a generalization of Theorem 2. We make use of (55) to write Hence, from (43) we can write the ergodic rate loss as l=1 (150) T † det H † P †β P β H = det Λ†T ΛT det H̃ Θβ H̃ (143) E[χTH (R, βR)] = − where Θβ , ΛR † P †β P β ΛR for β ≤ 1. Hence, from (37) the rate loss reads as det H̃ Θ1 H̃ 1 log2 . † T det H̃ Θβ H̃ T 1 X [ψ(R + 1 − t) − ψ(βR + 1 − t)] T ln 2 t=1 (152) " # βR−t T R−t X 1 1 X X1 − (153) = T ln 2 t=1 r=1 r r r=1 (144) To simplify this expression, we consider the singular value decomposition of H̃ † H̃ = L̃[Σ|0] R̃ H̃ =L̃P †φ ΣR̃. (146) For notational compactness, let us define Z β , † † P φ L̃ Θβ L̃P φ and A , ΣR̃. Thereby, we can write † H̃ Θβ H̃ = A† Z β A. Note that A† Z β A and Z β AA† have the same eigenvalues. Thus, we have † det H̃ Θβ H̃ = det Σ2 det Z β . T = (145) where L̃ and R̃ are respectively R × R and T × T unitary matrices, Σ is a T × T positive diagonal matrix and 0 is a (R − T ) × T zero matrix. Remark that we can actually write (145) as (147) (151) = † χTH (R, βR) = 1 X E[ln ρt ] T ln 2 t=1 1 X T ln 2 t=1 R−t X r=βR−t+1 1 . r (154) This completes the derivation of (46). As regards to derivation of (47), we first note the almost sure convergence of the limit [14, Theorem 3.6 and Eq. (4.23)] 1 log2 det P φ L† P †β P β LP †φ = T →∞ T Z lim log2 (x) dFP β U P † (x). φ (155) Using (33) we express this limit in terms of binary entropy function:   Z 1 β φ log2 (x) dFP β U P † (x) = − H(φ) + H . (156) φ φ φ β This completes the derivation of (47). We complete the derivation of (56) by plugging (147) in (144): χTH (R, βR) = det Z 1 1 log2 . T det Z β (148) Note also that Z 1 = I for ΛR = I. This completes the proof of Theorem 2. A PPENDIX J P ROOF OF R EMARK 3 It is sufficient to prove the result for (R − T ) <PT . For the p sake of notational compactness, we define hp , l=1 1l and IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 15 g(R, T ) , ln(2)T χT (R, T ). Then, from (153) we write g(R, T ) = = T X hR−t − t=1 R−T X hT −t (157) t=1 T X hR−t + t=1 − T X A PPENDIX L S OLUTION OF E XAMPLE 4 hR−t t=(R−T )+1 2T −R X T X hT −t − hT −t . (158) t=2T −R+1 t=1 Notice that T X hR−t = hT −t (159) A PPENDIX M P ROOF OF L EMMA 2 t=1 t=(R−T )+1 T X 2T −R X Note that we do not assume that H has Gaussian entries. However it is well known that for any distribution of the entries of H, the distribution function FN P β H converges weakly and almost surely to the Marc̆enko-Pastur law. In other words, we get the same asymtotic results regardless of whether we restrict the entries of H to Gaussian or not. Thus, without loss of generality we can assume that the entries of H are Gaussian, so that HH † is unitarily invariant. Doing so we have SH (z) = (1 + z)−1 [9]. Then, we immediately obtain (68) from (33). hT −t = t=2T −R+1 R−T X h(R−T )−t . (160) t=1 We have αP β H = β. Thus I0 (γ; FP β H ) = βI0 (γ; F̃P β H ) = βI0 (γ; FH † P † ). Thereby, we get (168) β g(R, T ) = R−T X hR−t − R−T X t=1 Furthermore, with Lemma 6 we have h(R−T )−t (161) SH † P † (z) = SH † (βz) = SH (βz). t=1 = g(R, R − T ). (162) In the sequel we first show that This completes the proof. (170) where S̃P β H is the S-transform of F̃P β H . To do so, it is R1 sufficient to show that |log2 SH (−z)| dz < ∞. Since FH 0 (163) has a compact support, I(γ; FH ) is finite. Now we show that log x is absolutely integrable over FH if, and only if, I(1; FH ) and ∆I(1; FH ) are finite [22]: where U is a R × R unitary matrix whose columns are the left singular vectors of the Gaussian random matrix X. Since X † X is unitarily invariant U is Haar distributed. The matrix of the left singular vectors of H̃, i.e. L̃, is Haar distributed too † as H̃ H̃ is unitarily invariant. Thereby, from (56) and (163) we have (164) Thereby, we have which completes the proof. Z1 |log2 (x)| dFH (x) = 0   1 dFH (x) log2 x 0 Z∞ + log2 (x) dFH (x). (171) Thus, we have Z1   1 dFH (x) < ∞ ⇐⇒ ∆I(1; FH ) < ∞, (172) log2 x 0 (165) where U β is a R × R unitary matrix. Since X ∼ U β X, we have X † Θβ X ∼ X † P †β D β P β X. (166) # " 1 det X † D 1 X T E[χH (R, βR)] = E log2 T det X † P †β D β P β X Z∞ 1 Note that the rank of Θβ is βR. Thus, we can consider the eigenvalue decomposition Θβ = U †β P †β D β P β U β 0 0 Following the same line of argumentation as used to obtain (148) we get 1 det X † Θ1 X log2 . T det X † Θβ X |log2 SH (−βz)| dz < ∞ log2 S̃P β H (−z) dz = A PPENDIX K P ROOF OF C OROLLARY 3 χTH (R, βR) ∼ Z1 Z1 det P φ U † Θ1 U P †φ det X † Θ1 X = log log2 2 det X † Θβ X det P φ U † Θβ U P †φ (169) β Z∞ log2 (x) dFH (x) < ∞ ⇐⇒ I(1; FH ) < ∞. 1 with ⇐⇒ implying ‘’if, and only if”. Hence (171) is finite. R1 Due to Theorem 4 this implies that |log2 SH (−z)| dz is 0 (167) (173) finite too. By invoking Theorem 4, (168) and (169) we obtain Z 1 I0 (γ; FP β H ) = β log2 γ − β log2 SH (−βz) dz. (174) 0 IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 16 Due to (170), it follows from the linearity property of the Lebesgue integral that ∆L(β; FH ) = I0 (γ; FP β H ) − βI0 (γ; FH ) Z 1 SH (−βz) dz. = −β log2 SH (−z) 0 Then, by using (178) and (182) we obtain ηH † P † = lim S√Y β (z) (175) z→−1+ β (176) (187) = lim + S√Y 1 (βz) (188) = (189) z→−1 S√Y 1 (−β) , 0<β<1 This completes the proof. which is strictly increasing with β, see Lemma 5. This completes the proof. A PPENDIX N P ROOF OF R EMARK 5 Invoking Lemma 6 we can write SH (−t) = SH † (−t) = lim SH † P † (z). z→−1+ t A PPENDIX P P ROOF OF T HEOREM 3 (177) Since H has almost surely full rank, αHP † = 1, so that t F̃P t H = FH † P † . Then from Lemma 5 we have t Z 1 lim + SH † P † (z) = dFH † P † (x). (178) t t x z→−1 The matrices XX † , Y Y † and P †β P β are asymptotically free [28]. Then, from Lemma 2 and the linearity property of the Lebesgue integral we have Z1 ∆L(β; FXY ) = −β This completes the proof. log2 SX (−βz)SY (−βz) dz (190) SX (−z)SY (−z) 0 = ∆L(β; FX ) + ∆L(β; FY ). (191) A PPENDIX O P ROOF OF R EMARK 6 A PPENDIX Q P ROOF OF R EMARK 7 For the sake of notational simplicity we introduce Y β , I + γP β HH † P †β (179) = P β (I + γHH † )P †β (180) = P β Y 1 P †β . (181) It follows that Y 1 is unitarily invariant since HH † is. Furthermore, since HH † has a compactly support LED so does Y 1 . Thus Y 1 is asymptotically free of P †β P β [28]. Then, with Lemma 6 we have in the limit N → ∞ S√Y β (z) = S√Y 1 (βz). For an N × N matrix A, we define φ(A) , lim N →∞ 1 tr(A) N whenever the limit exists. Since XX † and Y Y † are asymptotically free, we have (see [29, Eq. (120)]) φ(X † Y † Y X) = φ(X † X)φ(Y † Y ) (182) φ((X † Y † Y X)2 ) = φ(X † X)2 φ((Y † Y )2 ) Here we note that S√Y β (z) is strictly decreasing on (−1, 0) if, and only if, F√Y β is not a Dirac distribution function, see Lemma 5. We recall the following property of ηH (γ) see (23) [9]: + φ(Y † Y )2 φ((X † X)2 ) 1 − ηP β H − β(1 − ηH ) d{I(γ; FP β H ) − βI(γ; FH )} = , dγ γ ln 2 (183) where for convenience ηH is short for ηH (γ). Hence, in order to prove the remark it is sufficient to show that (1 − β) + βηH − ηP β H ≥ 0 − φ(X † X)2 φ(Y † Y )2 . lim (A† A)ii → φ(A† A), ∀i. N →∞ (185) Thus, the right-hand side of (184) is equal to β(ηH −ηH † P † ). β Therefore we are left with proving ηH ≥ ηH † P † . Firstly, β remark that Z 1 dF√Y β (x). (186) ηH † P † = β x (194) (195) Inserting (195) in the definition of the crosstalk ratio in (76), we get for A ∈ {X, Y , XY } that (184) β (193) Furthermore, from [30, Theorem 2.1] for A ∈ {X, Y , XY } we have almost surely CTA = where the equality holds when β = 1. Furthermore, by using [9, Lemma 2.26] we have ηP β H = (1 − β) + βηH † P † . (192) φ((A† A)2 ) 1 − . 2 2φ(A† A)2 (196) We complete the proof by plugging (193) and (194) in (196) for A = XY : CTXY = φ(X † X)2 φ((Y † Y )2 ) + φ(Y † Y )2 φ((X † X)2 ) −1 2φ(X † X)2 φ(Y † Y )2 (197) = CTX + CTY . (198) IEEE TRANSACTIONS ON INFORMATION THEORY, 2018 R EFERENCES [1] S. Shamai (Shitz) and S. Verdú, “The effect of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Transactions on Information Theory, vol. 47, no. 4, pp. 1302–1327, May 2001. [2] S. Verdú and S. Shamai (Shitz), “Spectral efficiency of CDMA with random spreading,” IEEE Transactions on Information Theory, vol. 45, no. 2, pp. 622–640, March 1999. [3] A. M. 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New York: Springer-Verlag, 1996. [28] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy (Mathematical Surveys & Monographs). Boston, MA, USA: American Mathematical Society, 2006. 17 [29] R. R. Müller, G. Alfano, B. M. Zaidel, and R. de Miguel, “Applications of large random matrices in communications engineering,” arXiv preprint arXiv:1310.5479, 2013. [30] B. Cakmak, “Random matrices for information processing–a democratic vision,” Ph.D. dissertation, Aalborg Universitetsforlag, 2016. Burak Çakmak was born in Istanbul, Turkey, 1986. He received the B.Eng. degree from Uludağ University, Turkey in 2009, M.Sc. degree from Norwegian University of Science and Technology, Norway in 2012 and Ph.D degree from Aalborg University, Denmark in 2017. Dr. Çakmak is a postdoctoral researcher at the Department of Computer Science, Technical University of Berlin, Germany. His research interests include random matrix theory, communication theory, statistical physics of disorder systems, machine learning and Bayesian inference. Ralf R. Müller (S’96–M’03–SM’05) was born in Schwabach, Germany, 1970. He received the Dipl.-Ing. and Dr.-Ing. degree with distinction from Friedrich-Alexander-Universität (FAU) Erlangen-Nürnberg in 1996 and 1999, respectively. From 2000 to 2004, he directed a research group at The Telecommunications Research Center Vienna in Austria and taught as an adjunct professor at TU Wien. In 2005, he was appointed full professor at the Department of Electronics and Telecommunications at the Norwegian University of Science and Technology in Trondheim, Norway. In 2013, he joined the Institute for Digital Communications at FAU Erlangen-Nürnberg in Erlangen, Germany. He held visiting appointments at Princeton University, US, Institute Eurcom, France, University of Melbourne, Australia, University of Oulu, Finland, National University of Singapore, Babes-Bolyai University, Cluj-Napoca, Romania, Kyoto University, Japan, FAU Erlangen-Nürnberg, Germany, and TU München, Germany. Prof. Müller received the Leonard G. Abraham Prize (jointly with Sergio Verdú) for the paper “Design and analysis of low-complexity interference mitigation on vector channels” from the IEEE Communications Society. He was presented awards for his dissertation “Power and bandwidth efficiency of multiuser systems with random spreading” by the Vodafone Foundation for Mobile Communications and the German Information Technology Society (ITG). Moreover, he received the ITG award for the paper “A random matrix model for communication via antenna arrays” as well as the Philipp-Reis Award (jointly with Robert Fischer). Prof. Müller served as an associate editor for the IEEE TRANSACTIONS ON INFORMATION THEORY from 2003 to 2006 and as an executive editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 2014 to 2016. Bernard H. Fleury (M’97–SM’99) received the Diplomas in Electrical Engineering and in Mathematics in 1978 and 1990 respectively and the Ph.D. Degree in Electrical Engineering in 1990 from the Swiss Federal Institute of Technology Zurich (ETHZ), Switzerland. Since 1997, he has been with the Department of Electronic Systems, Aalborg University, Denmark, as a Professor of Communication Theory. From 2000 till 2014 he was Head of Section, first of the Digital Signal Processing Section and later of the Navigation and Communications Section. From 2006 to 2009, he was partly affiliated as a Key Researcher with the Telecommunications Research Center Vienna (ftw.), Austria. During 19781985 and 19921996, he was a Teaching Assistant and a Senior Research Associate, respectively, with the Communication Technology Laboratory, ETHZ. Between 1988 and 1992, he was a Research Assistant with the Statistical Seminar at ETHZ. Prof. Fleurys research interests cover numerous aspects within communication theory, signal processing, and machine learning, mainly for wireless communication systems and networks. His current scientific activities include stochastic modeling and estimation of the radio channel, especially for large systems (operating in large bandwidths, equipped with large antenna arrays, etc.) deployed in harsh conditions (e.g. in highly time-varying environments); iterative messagepassing processing (with focus on the design of efficient feasible architectures for wireless receivers); localization techniques in wireless terrestrial systems; and radar signal processing. Prof. Fleury has authored and coauthored more than 150 publications and is co-inventor of 6 filed or published patents in these areas. He has developed, with his staff, a high-resolution method for the estimation of radio channel parameters that has found a wide application and has inspired similar estimation techniques both in academia and in industry. 

1 One-pass Person Re-identification by Sketch Online Discriminant Analysis arXiv:1711.03368v1 [] 9 Nov 2017 Wei-Hong Li, Zhuowei Zhong, and Wei-Shi Zheng∗ Abstract—Person re-identification (re-id) is to match people across disjoint camera views in a multi-camera system, and re-id has been an important technology applied in smart city in recent years. However, the majority of existing person re-id methods are not designed for processing sequential data in an online way. This ignores the real-world scenario that person images detected from multi-cameras system are coming sequentially. While there is a few work on discussing online re-id, most of them require considerable storage of all passed data samples that have been ever observed, and this could be unrealistic for processing data from a large camera network. In this work, we present an onepass person re-id model that adapts the re-id model based on each newly observed data and no passed data are directly used for each update. More specifically, we develop an Sketch online Discriminant Analysis (SoDA) by embedding sketch processing into Fisher discriminant analysis (FDA). SoDA can efficiently keep the main data variations of all passed samples in a low rank matrix when processing sequential data samples, and estimate the approximate within-class variance (i.e. within-class covariance matrix) from the sketch data information. We provide theoretical analysis on the effect of the estimated approximate within-class covariance matrix. In particular, we derive upper and lower bounds on the Fisher discriminant score (i.e. the quotient between between-class variation and within-class variation after feature transformation) in order to investigate how the optimal feature transformation learned by SoDA sequentially approximates the offline FDA that is learned on all observed data. Extensive experimental results have shown the effectiveness of our SoDA and empirically support our theoretical analysis. Index Terms—Online learning, Person re-identification, Discriminant feature extraction I. I NTRODUCTION Person re-identification (re-id) [51], [1], [13], [22], [31], [20], [54] is crucially important for successfully tracking people in a large camera network. It is to match the same person’s images captured at non-overlapping camera views at different time. Person re-id by visual matching is inherently challenging because of the existence of many visually similar persons and dramatic appearance changes of the same person caused by the serious cross-camera-view variations such as illumination, viewpoint, occlusions and background clutter. Recently, a large number of works [22], [23], [3], [16], [27], [30], [35], [44], [53] have been reported to solve this challenge. Wei-Hong Li is with the School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, China. E-mail: liweih3@mail2.sysu.edu.cn Zhuowei Zhong is with the School of Data and Computer Science, Sun Yat-sen University, Guangzhou, China. E-mail: zhongzhw6@gmail.com Wei-Shi Zheng is with the School of Data and Computer Science, Sun Yat-sen University, Guangzhou, China. E-mail: wszheng@ieee.org/zhwshi@mail.sysu.edu.cn * Corresponding author. However, it is largely unsolved to perform online learning for person re-identification, since most person re-id models except [25], [39], [29], [37] are only suitable for offline learning. On one hand, the offline learning mode cannot enable a real-time update of person re-id model when a large amount of persons are detected in a camera network. An online update is important to keep the cross-view matching system work on recent mostly interested persons, that is to make the whole reid system work on sequential data. On the other hand, online learning is helpful to alleviate the large scale learning problem (either with high-dimensional feature, or on large-scale data set, or both) nowadays. By using online learning, especially the one-pass online learning, it is not necessary to always store (all) observed/passed data samples. In this paper, we overcome the limitation of offline person re-id methods by developing an effective online person reid model. We proposed to embed the sketch processing into Fisher discriminant analysis (FDA), and the new model is called Sketch online Discriminant Analysis (SoDA). In SoDA, the sketch processing preserves the main variations of all passed data samples in a low-rank sketch matrix, and thus SoDA enables selecting data variation for acquring discriminant features during online learning. SoDA enables the newly learned discriminant model to embrace information from a new coming data sample in the current round and meanwhile retain important information learned in previous rounds in a light and fast manner without directly saving any passed observed data samples and keeping large-scale covariance matrices, so that SoDA is formed as an one-pass online adaptation model. While no passed data samples are saved in SoDA, we propose to estimate the within-class variation from the sketch information (i.e. a low-rank sketch matrix), and thus in SoDA an approximate within-class covariance matrix can be derived. We have provided in-depth theoretical analysis on how sketch affects the discriminant feature extraction in an online way. The rigorous upper and lower bounds on how SoDA approaches its offline model (i.e. the classical Fisher Discriminant Analysis [41]) are presented and proved. Compared to existing online models for person re-id [25], [39], [29], [37], SoDA is succinct, but it is theoretically guaranteed and effective. While most existing online re-id models have to retain all observed passed data samples, the proposed SoDA relies on the sketch information from historical data without any explicit storage of passed data samples, and sketch information will assist our online model in preventing one-pass online model from being biased by a new coming data. While a more conventional way for online learning of FDA is to update both within-class and between- 2 class covariance matrices directly [33], [48], [38], [26], [34], [15], we introduce a novel approach to realize online FDA by mining any within-class information from a sketch data matrix, and this provides a lighter, more effecient and effective online learning for FDA. We also find that an extra benefit of embedding sketch processing in SoDA is to simultaneously embed dimension reduction as well, so that no extra learning task on learning dimension reduction technology (e.g. PCA) is required and SoDA is more flexible when learning on some high dimensional data [22], [5] in an online manner. We have conducted extensive experiments on three largest scale person re-identification datasets in order to evaluate the effectiveness of SoDA for learning person re-identification model in an online way. Extensive experiments are also included for comparing SoDA with related online learning models, even though they were not applied to person reidentification before. The rest of the paper is organized as follows. In Sec. II, the related literatures are first reviewed. We elaborate our online algorithm and analyze the space and time complexity of SoDA in Sec. III. Then we present theoretical analysis on the relationship between our SoDA and the offline FDA in Sec. IV. Experimental results for evaluation and verification of our theoretical analysis are reported in Sec. V and finally we conclude the work in Sec. VI. II. R ELATED W ORK Online Person re-identification. While person reidentification has been investigated in a large number of works [51], [1], [13], [31], [20], [54], [22], [23], [3], [16], [27], [30], [35], [44], [53], [32], [28], [47], the majority of them only address by offline learning. That is person re-id model is learned on a fixed training dataset. This ignores the increase demand of data from a visual surveillance system, since thousands of person images are captured day by day and it is demanded to train a person re-id system on streaming data so as to keep the system update to date. Recently, only a few works [37], [25], [39], [29] have been developed towards online processing for person reidentification. The most related work is the incremental distance metric based online learning mechanism (OL-IDM) proposed in [37]. For updating the KISSME metric [17], the OL-IDM utilizes the modified Self-Organizing Incremental Neural Network (SOINN) [8] to produce two pairwise sets: a similar pairs set and a dissimilar pairs set. Although SOINN enables learning KISSME [17] on sequential data, SOINN has to compare the newly observed sample with all the preserved nodes and adds the newly observed sample as a new node if it does not appear in the network. This would be costly as sequential data increase and when feature dimension is high. Another related work is the human-in-the-loop ones [39], [25], [29], which proposed incremental method learned with the involvement of humans’ feedback. Wang et al. [39] assumes that an operator is available to scan the rank list provided by the proposed algorithm when matching a new probe sample with existing observed gallery ones, and this operator will select the true match, strong-negative match, and weak-negative match for the probe. After having the human feedback, the algorithm is able to be update. Martinel et al. presented a graph-based approach to exploit the most informative probe-gallery pairs for reducing human efforts and developed an incremental and iterative approach based on the feedback [29]. Unlike these models, we design a sketch FDA model called SoDA for one-pass online learning, without any storage of passed observed samples, maintaining a small size sketch matrix on handling streaming data so that the discriminant projections can be updated efficiently for extracting discriminative features for identifying different individuals. Thanks to the sketch matrix, our SoDA is capable of obtaining comparable performance with offline FDA models on streaming data or large and high dimensional datasets with very low cost on space and time. Compared to the related online person re-id models, SoDA is theoretically sounded since the bounds on approximating the offline model is provided. In particular, compared to Wang et al.’s and Martinel et al.’s work, our work has the following distinct aspects: Firstly, the proposed SoDA is developed for the one-pass online learning, while Wang et al.’s and Martinel et al.’s work cannot work for one-pass online learning, because the former one requires human feedback between probe sample and all preserved gallery samples, and the latter one needs to store all sample pairs during interative learning. Secondly, the proposed SoDA could be orthogonal to the human-in-the-loop work, since we discuss how to automatically update a person re-identification model on streaming data without elaborated human interaction (feedback), and thus our work and the idea of incorporating more human interaction in human-in-the-loop work can accompany each other. SoDA vs. Incremental Fisher Discriminant Learning. SoDA is related to existing incremental/online Fisher Discriminant Analysis (FDA) methods, which aim to update within-class and between-class covariance matrix sequentially. Pang et al. proposed to directly update the between-class and within-class scatter matrices [33]. However, Pang et al.’s method has to preserve the whole scatter matrices in the memory, which becomes impractical for high dimensional data. Ye et al. [48] and Uray et al. [38] performed online learning by updating PCA components to derive an approximate update of scatter matrices. Compared to Pang’s method, Ye’s and Uray’s can only perform online learning sample by sample, which can be time consuming for large scale data. Also, Ye’s method is based on QR decomposition of between-class covariance matrix, and therefore it would increase computational cost when the number of class is large. Since, Ye’s method is limited to learning discriminant projections in the range space of between-class covariance matrix but not the range space of total-class covariance matrix [46], which may lose discriminant information. Lu et al. proposed a complete model that picks up the lost discriminant information [26]. But Lu’s method only can update the model sample by sample. Peng et al. alternately proposed a chuck version of Ye’s method in order to process multiple data points at a time [34]. Kim et 3 2𝑙 𝑡=0 2𝑙 𝑡=1 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 − 𝑐𝑙𝑎𝑠𝑠 𝑆𝑐𝑎𝑡𝑡𝑒𝑟 𝑀𝑎𝑡𝑟𝑖𝑥 …… …… 𝑆𝑘𝑒𝑡𝑐ℎ 𝒘1 𝒘𝑘 𝒘2 𝒘3 𝒘4 𝑙 𝐴 𝑆𝑒𝑡 𝑜𝑓 𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑡=𝑇 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑊𝑖𝑡ℎ𝑖𝑛 − 𝑐𝑙𝑎𝑠𝑠 𝑆𝑐𝑎𝑡𝑡𝑒𝑟 𝑀𝑎𝑡𝑟𝑖𝑥 Collecting Images and Labels (𝑎) Feature Representation (𝑏) Sketching Main Data Variations (𝑐) Constructing Covariance Matrices (𝑑) Learning Discriminant Components (𝑒) Fig. 1. Illustration of our proposed Sketch online Discriminant Analysis (SoDA) (Best viewed in color). (a) In real-world application, images are generated endlessly from visual surveillance camera network. (b) (t = 0, 1, · · · , T ), every presented image is represented by a d−dimensional row feature vector. (c) We maintain a low rank sketch matrix to summarize all passed data by matrix sketch: 1) At the begining, we set B ∈ R2`×d , the sketch matrix, to be a zero matrix. 2) All rows of B would be filled by 2` samples from top to bottom one by one. 3) we maintain the main data variations in the upper half of B by sketch. 4) Each row of the lower half of B is set to be all zero and will be replaced by a new sample. (d) After sketch, the between-class and within-class covariance matrices are constructed. (e) Due to the sketch, we can update a set of discriminant components efficiently only using limited space and time. al. proposed a sufficient spanning set based incremental FDA [15] to overcome the limitations in the previous works. Since it is hard to directly update the discriminant components in FDA, Yan et al. [45] and Hiraoka et al. [10] modified FDA in order to get the discriminant components updated. They proposed iterative methods for directly updating discriminant projections. Compared to the above mentioned incremental/online FDA methods, our proposed SoDA embeds sketch processing into FDA and therefore mines the within-class scatter information from a sketch data matrix rather than directly from samples. This gives the benefit that while the passed data samples are not necessary to be saved, SoDA is still able to extract useful within-class information from the compressed data information contained in the sketch matrix. In general, SoDA is an online version of FDA, and SoDA can not only approximate the FDA, which optimizes discriminant components on whole data directly, but also run faster with limited memory. Also, dimension reduction is naturally embedded into SoDA and no extra online model for dimension reduction is required. Indepth theoretical investigation is provided in Sec. IV to explain its rationale and to guarantee its effectiveness. Although the proposed SoDA can be seen as embedding sketch processing into FDA, we contribute solid theoretical analysis on how SoDA will approximate the Batch mode FDA when estimating the within-class variations from sketch information, where the lower bound and upper bound are provided. The theoretical analysis guarantees SoDA to be an effective and efficient online learning method. Online Learning. SoDA is an online learning methods. In literatures, online learning [2], [6], [12], [40], [11] is known as a light and rapid means to process streaming data or large-scale datasets, and it has been widely exploited in many real-world tasks such as Face Recognition [14], [36], Images Retrieval [21], [42] and Object Tracking [19], [18]. It enables learning a up-to-date model based on streaming data. However, most of these online leaning based models [6], [18], [19] are not suitable for person re-identification, since they are incapable of predicting labels of data samples from unseen classes which do not appear in the training stage. III. S KETCH ONLINE D ISCRIMINANT A NALYSIS (S O DA) In this section, we start to present the Sketch online Discriminant Analysis (SoDA) for Person re-identification. In real-world scenario, samples come endlessly and sequentially from vision system (Figure 1). The number of samples received in each round is random, and the individual sample obtained is also stochastic. Suppose the tth (t = 1, 2, · · · ) new coming sample represented as a d−dimensional feature vector xi ∈ Rd is labelled with class label yi . For convenience, at the tth round, we denote all passed data (i.e. N training samples collected in the current and previous rounds) as a training T sample matrix X = [x1 , x2 , · · · , xN ] ∈ RN ×d , and denote all the corresponding labels as y = [y1 , y2 , · · · , yN ]T ∈ RN where yi is the class label of xi and yi ∈ {1, 2, ..., C}. At each round (t = 1, 2, · · · ), the proposed SoDA maintains the main variations of all passed data (X ∈ RN ×d ) in a low rank matrix, which is named as the “sketch matrix”. 4 B. Estimating Approximate Within-class covariance matrix Algorithm 1: Sketch online Discriminant Analysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Input: X = [x1 , x2 , · · · , xN ]T ∈ RN ×d , y ∈ RN , λ > 0 B ←− zero matrix ∈ R2`×d ; for each data xi ∈ Rd and label yi do using xT i to replace one zero row of B; if all samples in X are processed then deleting all zero rows of B; end if B has no zero rows then [U, Σ, V] = SVD(B); settingpξ as the (` + 1)th largest element Σ`+1 of Σ; Σ̂ = max(Σ2 − I2` ξ 2 , O); B = Σ̂VT (B contains ` rows non-zero values); end mc ←− (Nc mc + xi )/(Nc + 1) ( c = 0, yi ); Nc ←− Nc + 1 ( c = 0, yi ); end B ←− B+ , P = V+ ; P Nc T Sb = C c=1 N0 (mc − m0 )(mc − m0 ) ; T T S̃t = B B/N0 − m0 m0 ; S̃w = S̃t − Sb ; Ŝb = PT Sb P; Ŝw = PT S̃w P; [W, Λ] = EVD(Ŝb , Ŝw ); Output: B, W, Λ The sketch matrix keeps a small number of selected frequent directions, which are obtained and updated by a matrix sketch technique during the whole online learning process. While sketching main data variations, the population mean and the one of each class are also updated. We further utilize these updated means and the low rank sketch matrix to estimate between-class covariance matrix and derive the approximate within-class covariance matrix after all new coming samples are compressed into the sketch matrix. Finally, we generate discriminant components by eigenvalue decomposition for simultaneously minimizing the approximate within-class variance and maximizing the between-class variance. The whole procedure of SoDA is illustrated in Figure 1 and presented in Algorithm 1. The in-depth theoretical investigation to explain why SoDA can approximate the offline FDA model by sketch and guarantee its effectiveness on extracting discriminant components is provided in Sec. IV. During online learning, we keep updating the population mean m0 and mean of each class mc (c = 1, 2, . . ., C) so as to construct the between-class covariance matrix Sb . When having a new coming sample xi with class label yi , the population mean and mean of class yi are updated by (1) and the population number and the number of samples for class yi are also updated by: Nc = Nc + 1, c = 0, yi . In the above, S̃w is not always the exact within-class covariance matrix but it is an approximate one. In Sec. IV, we will provide in-depth theoretical analysis of the bias of this approximation on discriminant feature component extraction. C. Dimension Reduction and Extraction of Discriminant Components A. Estimating Between-class covariance matrix mc = (Nc mc + xi )/(Nc + 1), c = 0, yi , For realizing one-pass online learning, we aim to update/form the within-class covariance matrix which describes the within-class variation without using any passed observed data samples. Different from previous online FDA approaches, we embed sketch processing into FDA and derive a novel approximate within-class covariance matrix efficiently and effectively. For this purpose, we first employ the sketch technique [24] to compress the passed data samples into a sketch matrix so as to maintain the main variations of passed data. More specifically, we maintain the main variations of all passed data X in a small size matrix B ∈ R2`×d , called a sketch matrix, where B is initialized by a zero matrix. Each new coming sample xTi (i.e. the i-th row of X) replaces a zero row of B from top to bottom until B is full without any all zero rows. When B is full, we apply Singular Value Decomposition (SVD) on B such that UΣVT = B, where Σ is a diagonal matrix with singular values on the diagonal in decreasing order. Each row in VT corresponds to a singular value in Σ, and let vectors {vj } of VT corresponding to the first half singular values denoted as f requent directions and the ones corresponding to lower half singular values denoted as unf requent directions. By employingpthe sketch algorithm, the frequent directions vj are scaled by λ2i − ξ 2 and retained in B, where ξ is the (` + 1)th largeast singular value in Σ`+1 of Σ. In thisp way, the sketch matrix B is obtained by Σ̂VT , where Σ̂ = max(Σ2 − I2` ξ 2 , O) and O is a zero matrix. Therefore, the sketch matrix B is a 2` × d matrix, where B+ , the upper half of B, retains the main variations of passed data samples, and B− , the lower half of B, is reset to zero. Although no passed observed data samples are saved, we propose to derive an approximate within-class covariance matrix using the sketch matrix B below: S̃w = S̃t − Sb , (4) where S̃t = BT B/N0 − m0 mT0 . (5) (2) We then use the updated means to estimate the betweenclass covariance matrix as follows: C X Nc Sb = (mc − m0 )(mc − m0 )T . (3) N 0 c=1 Normally, after updating the two covariance matrices Sb and S̃w , it is only necessary to compute the generalized eigen-vectors of ΛS̃w W = Sb W. However, in person reidentification, some kinds of features are of high dimensionality such as HIPHOP [5], LOMO [22] and etc, and the size of the two covariance matrices Sb and S̃w was determined by the feature dimensionality. Thus the above eigen-decomposition remains costly when the size of both Sb and S̃w are large. An intuitive solution is to conduct another online learning for dimension reduction, which spends extra time and space. However, SoDA does not require such an extra learning. Due to sketch, SoDA actually maintains a set of frequent directions that describe main data variations. And thus we take these 5 frequent directions as basis vectors and the span of them can + approximate the data space. Hence, we set P = VT , the upper half of matrix VT (Line 16 in Algorithm 1), and the dimension reduction is performed by: Ŝb = PT Sb P, (6) Ŝw = PT S̃w P, where P = [v1 , v2 , . . . , vk ] consists of k frequent directions. In this way, Ŝb and Ŝw become matrices in Rk×k , and computing generalized eigen-vectors will become much faster. Finally, the generalized eigen-vectors (Line 22 in Algorithm 1) are computed by ΛŜw W = Ŝb W, and they are the discriminant components we pursuit. D. Computational Complexity 𝑏 𝑆𝑤 − 𝑆ሚ𝑤 𝑎 𝑇ℎ𝑒 𝑆𝑘𝑒𝑡𝑐ℎ 𝑀𝑎𝑡𝑟𝑖𝑥 (𝑩) 𝑐 𝑆ሚ𝑤 As presented above, after processing all observed samples, we maintain B ∈ R`×d , P ∈ Rd×k , mc ∈ Rd and Nc (c = 0, 1, 2, . . . , C). The time and space cost of the rest procedure is O(d`2 ) (After the whole processing, N0 is equal to N ) and O((` + C)d), respectively. Therefore, the cost of time and space is O(d`2 ) and O((` + k + C)d), respectively, almost the same as the cost of sketch algorithm [24]. 𝑑 𝑆𝑤 Fig. 2. (a) is the sketch matrix (B). (c) is the approximate within-class covariance matrix (S̃w ) generated by SoDA while (d) is the groundtruth one (Sw ) produced by FDA. (b) is the difference (Sw − S̃w ) of the groundtruth within-class covariance matrix and the approximate one. It is noteworthy that the distinction between Sw and S̃w is less than 1 × 10−12 , which indicates that S̃w estimated by SoDA can approximate the groundtruth one (Best viewed in color). In this section, we theoretically show that SoDA approximates FDA in a principled way, although SoDA is formed based on the approximate within-class covariance matrix mined from sketch data information. eigenvectors, that is ΛSw W = Sb W and Λ is a diagonal matrix with generalized eigenvalues on the diagonal. Here, the eigenvectors corresponding to the largest eigenvalues are used to compress a high dimensional data vector to a low dimensional feature representation. A. Fisher Discriminant Analysis B. Relation between SoDA and FDA Fisher discriminant analysis (FDA) aims to seek discriminant projections for minimizing within-class variance and maximizing between-class variance, which are estimated over the data matrix X and its label set y in an offline way. There are several equivalent criteria JF for the multi-class case. For analysis, we consider the one that maxmizes the following criterion: T Before presenting the theoretical analysis, we first define the following notations. Let tr(WT Sb W) J1F (W) = , tr(WT Sw W) (11) tr(WT Sb W) J2F (W) = , tr(WT S̃w W) 1 where JF (W) is the conventional FDA criterion and J2F (W) is SoDA criterion by replacing Sw with S̃w that is mined from sketch data information. Let the largest Fisher scores in the above equations be J1F (W1 ) = max J1F (W) = µ1 , W∈Rd×k (12) 2 2 JF (W ) = max J2F (W) = µ2 . IV. T HEORETICAL A NALYSIS JF (W) = W Sb W , WT Sw W (7) where Sb is the between-class covariance matrix and Sw is the within-class covariance matrix. They are given by Sb = C X Nc (mc − m0 )(mc − m0 )T , N c=1 C X Nc X 1 Sw = (xi − mc )(xi − mc )T , N N c c=1 y =c (8) W∈Rd×k (10) Since for optimizing Eqs. (12), we can form a Lagrangian function by imposing the constraint tr(WT Sb W) = 1 for both criteria [41] We define D = {W = [w1 , · · · , wk ] ∈ Rd×k |tr(WT Sb W) = 1}, and thus we can reform Eqs. (12) by: µ−1 {J1F (W1 )}−1 = tr(WT Sw W), 1 = min W1 ∈D (13) µ−1 {J2F (W2 )}−1 = tr(WT S̃w W). 2 = min 2 Generally, the analysis seeks a set of feature vectors {wj } that maximize the criterion subject to the normalization constraint tr(WT Sb W) = 1, where W is the matrix whose columns are {wj }. This leads to the computation of generalized In the following sections, we first discuss the relationship between µ1 and µ2 . And then this relationship will be used to present a bound for J1F (W2 ). Note that J1F (W2 ) is to measure how well the optimal projection learned by our SoDA approximates the optimal solution that maximizes J1F (W). Note (9) i where mc and Nc are the data mean and the number of samples of the cth class, respectively, and N and m0 are the population number and population mean, respectively. And the total covariance matrix is St = Sw + Sb = C 1 XX (xi − m0 )(xi − m0 )T . N c=1 y =c i W ∈D 6 that our analysis will not take any dimension reduction before extracting discriminant components below for discussion. Our analysis can be extended if the same dimension reduction is applied to all methods discussed below. wT Sw w and wT S̃w w when constraining wT Sb w = 1. In T addition, since w2 Sb w2 = 1, we have s0 rb ||w2 ||22 ≤ 1, i.e. 1 ||w2 ||2 ≤ (s0 rb )− 2 . Therefore, based on Theorem 1, we have T T T T T 2 µ−1 S̃w w2 ≤ w1 S̃w w1 ≤ w1 Sw w1 = µ−1 2 =w 1 , C. Relationship Between the Maximum Fisher Score of FDA and that of SoDA We first present the relationship between the maximum Fisher score of FDA and the one of SoDA, i.e. the relationship between µ1 and µ2 . Suppose that matrix X ∈ RN ×d is the totally training sample set consisting of samples acquired at each time step. However, it is not intuitive to obtain the relationship between the maximum Fisher score of FDA and the one of SoDA based on the covariance matrices inferred in Eq. (5). In order to exploit such a relationship, we first investigate the Fisher score obtained by Sb and the approximate within-class covariance matrix S̃w as follows: S̃w = S̃t − Sb = BT B/N − m0 mT0 − Sb . (14) Let Sw be the within-class covariance matrix computed in batch mode (i.e. for offline FDA). Since it is known that Sw = St − Sb = XT X/N − m0 mT0 − Sb , it can be verified that Sw − S̃w =(XT X/N − m0 mT0 − Sb ) − (BT B/N − m0 mT0 − Sb ) (15) =(XT X − BT B)/N . By combining Eq. (25) as stated in the Appendix, it is not hard to have the following theorem about the relation between Sw and S̃w , and we visualize the approximation between the groundtruth within-class covaraince matrix and our approximate one in Figure 2. We assume that Sw , S̃w and Ŝw are not singular in the following analysis 1 . 2 Theorem 1. S̃w  Sw , and ||Sw − S̃w || ≤ 2||X||f /(N `), where || ∗ || is the induced norm of a matrix and || ∗ ||f is the Frobenius norm. Based on the above theorem, we particularly consider the two-class classification case. Theorem 2. Considering the two criteria in Eq. (13) when the discriminant feature transformation is a one-dimensional vector, i.e. W1 = w1 ∈ Rd and W2 = w2 ∈ Rd , the relationship between µ1 and µ2 is as follow: 2 −1 − 21 ||X||f /(N `) ≤ µ−1 (16) µ−1 2 ≤ µ1 , 1 − 2(s0 rb ) where s0 is the smallest (non-zero) singular value of matrix Sb and rb = rank(Sb ). 2 Proof. Let D = 2||X||f /(N `). From the Theorem 1, we have wT (Sw −S̃w )w ||w||2 wT Sw w, wT Sw w ≤ for any nonzero w ∈ Rd , 0 ≤ ∀w ∈ Rd , wT S̃w w ≤ ≤ D. That is 1 µ−1 Sw w1 ≤ w2 Sw w2 1 =w 2T (18) − 21 − 21 = µ−1 . ≤ w S̃w w2 + D(s0 rb ) 2 + D(s0 rb ) 2 −1 −1 − 12 Then µ1 − 2(s0 rb ) ||X||f /(N `) ≤ µ−1 ≤ µ . 2 1 From the theorem above, we can claim that the largest Fisher score J2F (w2 ) is always greater than or equal to the original one J1F (w1 ) after sketch. From another aspect, the 2 − 21 ||X||f /(N `) ≤ µ−1 ≤ µ−1 inequalities “µ−1 1 − 2(s0 rb ) 2 1 ” means when more rows are set in the sketch matrix B, (i.e. much larger ` is set), µ2 becomes µ1 , and thus SoDA becomes exactly the FDA. For the multi-class case, we can generalize the above proof below. Theorem 3. Considering the two criteria in Eq. (13), when the discriminant feature transformation is a d-dimensional transformation where d > 1, we have µ1 ≤ µ2 . Proof. Note that W1 and W2 (∈ Rd×k ) make the two criteria minimized in Eq. (13), respectively. Let W1 = [w11 , · · · , wk1 ] and W2 = [w12 , · · · , wk2 ]. Since for any w ∈ Rd , wT Sw w ≥ wT S̃w w by Theorem 1, we have T T 2 S̃w W2 ) ≤ tr(W1 S̃w W1 ) µ−1 2 = tr(W = k X T wi1 S̃w wi1 ≤ k X T wi1 Sw wi1 i=1 i=1 T = tr(W1 Sw W1 ) = µ−1 1 . Hence, the theorem is proved. D. How Does the Projection Learned by SoDA Optimize the Original Fisher Criterion Approximately? In the above, we analyze the quotient values between tr(WT Sb W) and tr(W . However, in SoDA, our withinT S̃ W) w class covariance matrix is estimated by sketch and is not the exact within-class covariance matrix. In the following, we will present the effect of the learned discriminant component using SoDA on minimizing the grouth-truth within-class covariance. For this purpose, the following theorems are presented. tr(WT Sb W) tr(WT Sw W) Theorem 4. For any w ∈ Q = {w ∈ Rd |wT w = 1}, we have wT S̃w w ≤ wT Sw w ≤ wT S̃w w + 2 ||X||2f /`. N (19) wT S̃w w+D||w||2 . Proof. While the inequality wT S̃w w ≤ wT Sw w is obvious (17) by using Theorem 1, we focus on the latter one. Since Sw = Let w1 and w2 be the discriminant vectors that minimize S̃w + (XT X − BT B)/N in Eq. (15), by applying Eq. (25), T the Criterion in Eq. (13) under the constraints w1 Sb w1 = 1 we have wT Sw w = wT S̃w w + N1 wT (XT X − BT B)w ≤ 2T 2 1T and w Sb w = 1, respectively. That is w Sw w1 = µ−1 wT S̃w w + N2 ||X||2f /`. 1 −1 2T 2 1 2 and w S̃w w = µ2 , i.e. w and w would minimize Theorem 5. Considering the two criteria in Eq. (13), we de1 The analysis can be generalized to the case when S̃ is not invertible if fine D = {W = [w1 , · · · , wk ] ∈ Rd×k |tr(WT Sb W) = 1}, w the same regularization is imposed on both Sw , S̃w and Ŝw denote the smallest non-zero singular value of Sb as s0 , and 7 (a) (b) Fig. 3. Fisher Score comparison on three datasets using JLH feature. (Best viewed in color). (c) TABLE I C OMPARISION AMONG DIFFERENT ONLINE / INCREMENTAL APPROACHES Approaches IFDA [15] Pang’s IFDA [33] IDR/QR [48] OL-IDM [37] Wang Save within-class -scatter matrix? Save between-class -scatter matrix? Is an one-pass algorithm? Human feedback Can the model be trained on streaming data? Is the model embedded with dimension reduction? time O(ndc) -O(d3 ) O(nd2 ) complexity space 2 2 2 -O(d ) O(d ) O(d ) complexity ! ! % % ! % ! ! ! % ! % ! % ! % ! % ! % ! % et al. [39] Martinel et al. [29] SoDA (Ours) -- -- -- -- % ! % % % ! % % % % ! % ! ! -- -- O(min(`, d)2 max(`, d)) -- -- O((` + k + C)d) T let rb = rank(Sb ). Suppose the norm of each data vector xi (i.e. each row of the data matrix X ∈ RN ×d ) is bounded by M , that is ||xi ||22 ≤ M. Then we have 1 ≤ J1F (W2 ) ≤ µ1 . (20) 2k µ−1 + M/` 1 s0 rb 2 Proof. First, given W ∈ D that minimize {J2F (W)}−1 . T {J1F (W2 )}−1 =tr(W2 Sw W2 ) = k X T wi2 Sw wi2 i=1 = k X T ||wi2 ||22 wi2 wi2 Sw 2 ||wi ||2 ||wi2 ||2 ||wi2 ||22 wi2 wi2 S̃w 2 ||wi ||2 ||wi2 ||2 i=1 ≤ k X T i=1 + ≤ k X (21) k X 2 ||wi2 ||22 ||X||2f /` N i=1 {J1F (W2 )}−1 =tr(W2 Sw W2 ) ≤ k X T wi2 S̃w wi2 + i=1 =µ−1 2 + 2k (s0 rb )−1 ||X||2f /` N (22) 2k (s0 rb )−1 ||X||2f /`. N Pk 2T 2 Note that µ−1 = 2 i=1 wi S̃w wi since it is assumed that −1 2 2 W ∈ D minimizes {JF (W)} . Thus, under the constraint T tr(W2 Sb W2 ) = 1, we have 1 ≤ J1F (W2 ) ≤ µ1 , (23) −1 2k µ2 + N (s0 rb )−1 ||X||2f /` where the latter equation is obvious since W2 may not be the optimal projection for mamixizing JF1 (W). Finally, since µ1 ≤ µ2 and ||xi ||22 ≤ M that means the norm of any data vector xi (i.e. each row of the data matrix X ∈ RN ×d ) is bounded by M , we have 1 ≤ J1F (W2 ) ≤ µ1 . (24) −1 µ1 + s2k M/` r 0 b T wi2 S̃w wi2 i=1 + T 2k ||wi2 ||22 ||X||2f /`. N T Since tr(W2 Sb W2 ) = 1, we have wi2 Sb wi2 ≤ 1. Here, T for convenience, one can further assume wi2 Sb wi2 > 0, otherwise a much tighter bound can be inferred. And thus s0 rb ||wi2 ||22 ≤ 1. So we have E. Discussion 1) SoDA vs. FDA: The above theorem indicates that 1) the learned transformation by SoDA may not be the optimal one for the FDA directly learned on all observed data since J1F (W2 ) ≤ µ1 , which is obvious and reasonable; 2) however, there is a lower bound on J1F (W2 ), since µ−1 + 12k M/` ≤ 1 s0 r b 8 (a) (b) Fig. 4. Comparison on three datasets using JSTL feature. (Best viewed in color). 𝑀𝑎𝑟𝑘𝑒𝑡 − 1501 𝑆𝑌𝑆𝑈 𝐸𝑥𝑀𝑎𝑟𝑘𝑒𝑡 (c) V. E XPERIMENTS A. Datasets and Evaluation Settings Fig. 5. Example images from different person re-id datasets. For each dataset, two images in a column correspond to the same person. J1F (W2 ); 3) as long as more and more rows are set in the sketch matrix B used in SoDA, i.e. ` is larger and larger, 2k 2 1 s0 M/` → 0 and so that JF (W ) ≈ µ1 in such a case. The latter case is reasonable because although the sketch in SoDA enables selecting data variation during the online learning, more data information is kept when a much larger sketch matrix B is used, and this will be verified in the experiments (see Figure 3 for example). 2) SoDA vs. Incremental/online models: In Table I, we compare SoDA with related incremental/online FDA models in details. A distinct and important characteristic of SoDA is that it is able to perform one-pass online learning directly only relying on sketch data information. SoDA does not have to keep within-class covariance matrix and betweenclass covariance matrix in memory during online learning, due to embedding sketch processing, which has not been considered for online learning of FDA before. Moreover, as compared to the others, SoDA does not need any extra online learning progress on dimension reduction, which is naturally embedded. Thus the training cost of SoDA is much lighter. When applied SoDA to person re-id, we perform the comparison with related online person re-id models. An important distinction is that no extra human feedback is required, and SoDA is able to be applied on streaming data in an onepass learning manner. In comparison with OL-IDM, SoDA has its merits: 1) dimension reduction is naturally embedded in SoDA; 2) embedding sketch into person re-id model learning is a more efficient and effective way to maintain the main variations of data, which has been verified by our experimental results. 1) Datasets: We extensively evaluated the proposed approach on three large person re-id benchmarks: Market-1501, SYSU, and ExMarket. • Market-1501 dataset [51] contains person images collected in front of a campus supermarket at a University. It consists of 32,643 person images of 1,501 identities. • SYSU dataset contains totally 48,892 images of 502 pedestrians captured by two cameras. Similar to [4], we randomly selected 251 identities from two views as training set which contains 12308 images. And we randomly selected three images of each person from the rest 251 identities of both cameras to form the testing set, where the 753 images of the first camera were used as query images. • ExMarket dataset was formed by combining the MARS dataset [50] and Market-1501 dataset. MARS was formed as a video dataset for person re-identification. All the identities from MARS are of a subset of those from Market. More specifically, for each identity, we extracted one frame for each five consecutive frames firstly and combined images extracted from MARS and the ones from Market-1501 of the same person. Therefore, ExMarket contains 237147 images of 1501 identities, the largest population size among the three benchmark datasets tested. 2) Features: In this work, we conducted the evaluation based on four types of feature for evaluation: 1) JSTL, 2) LOMO, 3) HIPHOP, 4) JSTL + LOMO + HIPHOP (JLH). • JSTL is a kind of low-dimensional deep feature representation (R256 ) extracted by a deep convolutional network [43]; • LOMO is an effective handcraft feature proposed for person re-id in [22], and it is a 26960-dimensional vector; • HIPHOP is another recently proposed person re-id feature (R84096 ) [5] that extracts more view invariant histogram features from shallow layers of a convolution network. In addition, since person re-id can benefit from using multiple different types of appearance features as shown in [5], [7], [9], [49], [52]. we concatenated JSTL, LOMO and HIPHOP as a high dimensional feature (R111312 ), named JLH 9 (a) (b) Fig. 6. Comparison on three datasets using LOMO feature. (Best viewed in color). (c) Market-1501 100 Matching Rate (%) 90 80 70 60 50 OL-IDM IDR/QR IFDA Pang's IFDA FDA SoDA (Ours) 40 30 20 10 5 10 15 20 Rank (a) (b) Fig. 7. Comparison on three datasets using HIPHOP feature. (Best viewed in color). SYSU 100 Matching Rate (%) (c) 80 60 40 OL-IDM IDR/QR IFDA Pang's IFDA FDA SoDA (Ours) 20 0 5 10 15 Rank (a) (b) Fig. 8. Comparison on three datasets using JLH feature. (Best viewed in color). in this work for convenience of description. On all datasets, we report experimental results of SoDA using the concatenated feature in Table VI. Since LOMO, HIPHOP, and JLH are of high dimension, for all methods except SoDA, we first reduced their feature dimension of the three types of feature to 2000, 2000 and 2500, respectively, on all datasets. For SoDA, we set the sketch size (`) to the (reduced) feature dimension menthioned above on all datasets. 3) Evaluation protocol: On all datasets, we followed the standard evaluation settings on person re-identification, i.e. images of half of the persons were used for training and images of the rest half were used for testing, so that there is no overlap in persons between training and testing sets. More specifically, on Market-1501 dataset, we used the standard training (12936 images of 750 people) and testing (19732 images of 751 people) sets provided in [51]. On SYSU dataset, 20 (c) similar to [4], we randomly picked all images of the selected 251 identities from two views to form the training set which contains 12308 images, and we randomly picked 3 images of each pedestrian of the rest 251 identities in each view for forming the gallery and query sets for testing. On ExMarket dataset, we conducted the same identity split as the Market1501 dataset. The training set contains 112351 images, and the testing set contains 124796 images, among which 3363 images are considered as query images and the rest are considered as gallery images. On all datasets, the cumulative matching characteristic (CMC) curves is shown to measure the performance of the compared methods on re-identifying individuals across different camera views under online setting. In addition to this, we also report results using another two performance metrics: 1) rank-1 Matching Rate, and 2) mean Average Precision 10 TABLE II C OMPARISON WITH FDA ON ALL BENCHMARKS . Feature Dataset Method Market FDA -1501 SoDA FDA SYSU SoDA ExFDA Market SoDA JSTL rank-1 rank-5 rank-10 rank-20 mAP 57.30 75.53 81.38 86.49 28.57 57.13 74.79 81.18 85.90 28.25 31.21 52.99 61.49 71.85 25.86 31.74 52.86 62.15 71.31 26.04 53.89 68.11 73.13 77.97 22.71 54.93 68.79 73.13 77.46 22.87 LOMO rank-1 rank-5 rank-10 rank-20 mAP 51.90 74.26 81.12 87.14 23.60 52.41 73.37 81.38 87.17 23.58 46.61 70.78 79.42 86.19 41.81 47.81 70.39 78.75 86.72 41.69 45.64 60.42 66.86 72.89 17.98 46.08 61.31 67.81 73.63 17.77 HIPHOP rank-1 rank-5 rank-10 rank-20 mAP 60.27 80.52 87.05 91.18 31.45 61.88 81.41 86.70 91.60 33.39 52.86 73.84 81.67 87.78 48.20 53.12 73.97 80.88 87.25 48.48 57.24 71.38 77.11 81.74 27.20 55.76 70.40 76.10 81.59 24.97 JLH rank-1 rank-5 rank-10 rank-20 mAP 74.20 88.75 92.19 94.80 49.01 75.27 89.28 92.70 95.22 49.82 63.08 80.35 86.32 91.50 56.82 64.81 80.74 87.25 91.77 59.82 66.86 78.18 82.63 86.70 39.00 66.18 78.36 82.48 86.64 37.11 TABLE III C OMPARISON WITH INCREMENTAL FDA MODELS AND ONLINE METHOD USING JSTL. Dataset Market-1501 SYSU ExMarket rank-1 Accumulative rank-1 Accumulative rank-1 Accumulative Method matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) OL-IDM 31.50 10.48 3706.84 12.08 10.29 10588.15 50.24 18.93 1646433.70 IDR/QR 41.15 13.20 803.59 12.88 10.24 247.17 42.70 11.20 6172.79 IFDA 51.45 21.21 38.22 22.97 18.12 12.40 49.91 16.58 394.31 Pang’s IFDA 57.36 28.58 13.68 31.08 25.28 7.65 55.46 22.97 120.94 SoDA 57.13 28.25 7.84 31.74 26.04 4.68 54.93 22.87 50.52 TABLE IV C OMPARISON WITH INCREMENTAL FDA MODELS AND ONLINE METHOD USING LOMO. Dataset Market-1501 SYSU ExMarket rank-1 Accumulative rank-1 Accumulative rank-1 Accumulative Method matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (min) OL-IDM 3.95 0.73 736707.11 1.06 1.59 743335.02 3.86 0.33 > 1 week IDR/QR 19.36 5.09 345181.63 6.37 5.16 83903.98 19.92 3.58 74393.24 IFDA 38.75 13.32 314470.08 26.83 22.59 67003.60 35.63 10.43 69668.26 Pang’s IFDA 44.80 18.64 314461.09 35.99 31.82 66646.88 43.50 15.42 69625.84 SoDA 52.41 23.53 2127.47 47.81 41.69 3345.30 46.08 17.77 359.28 TABLE V C OMPARISON WITH INCREMENTAL FDA MODELS AND ONLINE METHOD USING HIPHOP. Dataset Market-1501 SYSU ExMarket rank-1 Accumulative rank-1 Accumulative rank-1 Accumulative Method matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) OL-IDM 11.97 2.22 277104.72 1.46 2.00 252626.33 7.24 0.54 > 1 week IDR/QR 19.98 6.00 225226.32 10.49 9.34 86513.64 21.97 5.32 2392922.71 IFDA 52.14 21.30 185390.31 25.50 22.32 66202.88 46.08 15.50 2133499.12 Pang’s IFDA 60.42 31.30 185174.97 51.79 47.51 65593.56 54.84 25.11 2135671.23 SoDA 61.88 33.39 3620.00 53.12 48.48 13849.61 55.76 24.97 83319.79 (mAP). mAP first computes the area under the PrecisionRecall curve for each query and then calculates the mean of Average Precision over all query persons. All experiments were implemented using MATLAB on a machine with CPU E5 2686 2.3 GHz and 256 GB RAM, and the accumulative time of all compared methods were also computed and reported for measuring efficiency. B. SoDA vs. FDA In Sec. IV, we provide theoretical analysis on the relation between SoDA and FDA. In this section, we provide empirical evaluation on three datasets by the comparison on Fisher Score between SoDA and FDA in Figure 3. The figure indicates that by keeping more rows in the sketch matrix, SoDA can acquire more similar Fisher Score as the one of FDA, and this is supported by Theorem 5. We also compared SoDA with FDA on the three datasets in Table II, and the comparison shows that they work comparably. Therefore the results reported here have validated that our sketch approach approximates FDA (i.e. the offline model) for extracting discriminant information very well, and thus the effectiveness of our model is verfied both theoretically and empirically. C. SoDA vs. Incremental FDA Model There are existing works that are related to incremental learning of FDA, which also process sequential data and update the models online. We compared extensively our method SoDA with three related online/incremental FDA methods, including IFDA [15], IDR/QR [48] and Pang’s IFDA [33]. We show CMC curve of all methods using different types of features in Figure 4, Figure 6, Figure 7 and Figure 8. The results illustrate that the proposed SoDA outperformed the compared incremental FDA. For instance, when using JLH, SoDA outperformed Pang’s IFDA and achieved 75.27%, 64.81% and 66.18% rank-1 matching rate on Market, SYSU and ExMarket, respectively. We further report mAP and accumulative time in Table III, Table IV, Table V and Table VI. It suggests that SoDA has a better mAP values especially on 11 (a) (b) (c) Fig. 9. Effect of the sketch size on accumulative time consumption. (Best viewed in color). TABLE VI C OMPARISON WITH INCREMENTAL FDA MODELS AND ONLINE METHOD USING JLH. Dataset Market-1501 SYSU ExMarket rank-1 Accumulative rank-1 Accumulative rank-1 Accumulative Method matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) matching rate (%) mAP (%) Time (s) OL-IDM 14.43 2.48 356136.53 3.32 4.91 554908.70 10.84 0.70 > 1 week IDR/QR 36.70 13.73 251934.68 15.80 12.82 220962.28 39.64 10.85 2479401.99 IFDA 61.19 30.36 203537.09 21.65 18.23 189960.96 56.24 23.46 2032679.17 Pang’s IFDA 71.64 45.15 204406.03 56.31 49.60 189897.24 64.64 34.80 2036601.02 SoDA 75.27 49.82 12952.07 64.81 59.82 9951.20 66.18 37.11 164475.67 TABLE IX C OMPARISON WITH OFFLINE RE - ID MODELS ON E X M ARKET USING JLH(%). Method CRAFT MLAPG KISSME XQDA SoDA (a) (b) Fig. 10. Effect of the sketch size on rank-1 Matching Rate. (Best viewed in color). rank-1 rank-5 rank-10 rank-20 54.51 69.39 75.56 80.94 50.21 65.29 70.90 77.20 57.42 69.71 74.23 78.83 55.05 68.02 73.10 77.73 66.18 78.36 82.48 86.64 mAP 24.26 25.63 30.03 28.36 37.11 SYSU and spends much less time, where for instance SoDA gains around 60% reduction on the cost of computation time, as compared with Pang’s ILDA. D. SoDA vs. Related Person re-id Models TABLE VII C OMPARISON WITH OFFLINE RE - ID MODELS ON M ARKET-1501 USING JLH (%). Method CRAFT MLAPG KISSME XQDA SoDA rank-1 rank-5 rank-10 rank-20 71.20 87.35 391.69 94.39 69.33 85.63 90.23 93.82 67.99 83.67 88.93 92.79 67.96 83.91 88.95 93.14 75.27 89.28 92.70 95.22 Map 44.24 46.16 39.79 43.89 49.82 TABLE VIII C OMPARISON WITH OFFLINE RE - ID MODELS ON SYSU USING JLH(%). Method CRAFT MLAPG KISSME XQDA SoDA rank-1 rank-5 rank-10 rank-20 24.70 43.03 55.11 67.73 18.46 35.86 47.01 58.83 62.28 79.81 86.06 90.31 64.14 80.88 86.85 91.90 64.81 80.74 87.25 91.77 mAP 23.31 18.03 56.23 59.12 59.82 Comparison with online re-id model. We compared the online re-id method OL-IDM [37] that addresses the same setting as ours in this work. Table III, IV, V and VI tabulate the comparison results. It is noteworthy that our SoDA obtains much more stable results on rank-1 matching rate and mAP performance. Moreover, SoDA is more efficient than OL-IDM, taking 30 times smaller accumulative time. Comparison with related subspace model and classical models. We also compared two related subspace model for person re-identification: 1) CRAFT [5] ; 2) MLAPG [23], and two classical methods: 1) KISSME [16] ; 2) XQDA [22], when the JLH feature was applied on all datasets. All of these methods were learned in an offline way, and the results of these methods on all benchmarks using JLH features are presented in Table VII, VIII and IX. Among all compared methods, the rank-1 matching rate and mAP of SoDA are the highest, and its accumulative time is the lowest. This indicates that SoDA achieves better or comparable performance of the related offline subspace person re-id models. 12 E. Further Evaluation of SoDA We report the performance of SoDA in Figure 10 and Figure 9 when varying two key parameters `. Effect of the sketch size ` using low dimensional feature. On all benchmarks, we conducted experiments using JSTL feature (256−dimensional) for evaluating the effect of the sketch size ` on low dimensional feature. The experimental results in Figure 10(a) indicate that the performance of our proposed SoDA can be improved when ` (i.e. the rank of B) is larger. That is the performance is better when more variations of passed data are remained in the sketch matrix. It is reasonable because when more data variations are reserved, the estimated within-class covariance matrix from the sketch matrix B can approximate the ground-truth one better. However, larger ` indeed increases the accumulative time since the computation complexity and memory depend on ` when the number of samples and the dimensionality of features are determined (Sec. III-D). Fortunately, we empirically find that good performance and low accumulative time can be achieved at the same time when setting the rank of the sketch matrix B to a properly small value, i.e. ` = d = 256. Effect of the sketch size ` using high dimensional features. We also show the effect of ` when using high dimensional features, as some recent proposed state-of-theart person re-id features are of high dimension, such as LOMO (26960−dimensional), HIPHOP (84096−dimensional) and also the JLH (111312−dimensional) formed in this work. High dimensionality will increase the computational and space complexities (e.g., the whole training data matrix of ExMarket is a 112351 × 111312 matrix). Instead of conducting another online learning for dimension reduction, SoDA utilizes a set of orthogonal frequent directions maintained by the sketch matrix B for reducing feature dimension. The experimental results shown in Figure 10(b) and Figure 9 again verify that increasing the sketch size ` can improve the performance of SoDA but also increase the accumulative time due to extra computation for dimension reduction. Also, on high dimensional feature, setting ` to be a properly small value (e.g. ` = 1000) can gain a good balance between good performance and low accumulative computation time. VI. C ONCLUSION We contribute to developing a succinct and effective online person re-identification (re-id) methods namely SoDA. Compared with existing online person re-id models, SoDA performs one-pass online learning without any explicit storage of passed observed data samples, meanwhile preserving a small sketch matrix that describes the main variation of passed observed data samples. And moreover, SoDA is able to be trained on streaming data efficiently with low computational cost, upon on no elaborated human feedback. Compared with the related online FDA models, we take a novel approach by embedding sketch processing into FDA, and we approximately estimate the within-class variation from a sketch matrix and finally derive SoDA for extracting discriminant components. More importantly, we have provided in-depth theoretical analysis on how the sketch information affects the discriminant component analysis. The rigorous upper and lower bounds on how SoDA approaches its offline model (i.e. the classical Fisher Discriminant Analysis) are given and proved. Extensive experimental results have clearly illustrated the effectiveness of our SoDA and verified our theoretical analysis. ACKNOWLEDGEMENT This research was supported by the NSFC (No. 61472456, No. 61573387, No. 61522115). R EFERENCES [1] E. Ahmed, M. Jones, and T. K. Marks. An improved deep learning architecture for person re-identification. In CVPR, 2015. [2] G. Chechik, V. Sharma, U. Shalit, and S. Bengio. Large scale online learning of image similarity through ranking. JMLR, 11(Mar):1109– 1135, 2010. [3] Y.-C. Chen, W.-S. Zheng, and J. Lai. Mirror representation for modeling view-specific transform in person re-identification. In IJCAI, 2015. [4] Y.-C. Chen, W.-S. Zheng, J.-H. Lai, and P. Yuen. 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Human-in-the-loop person re-identification. In ECCV, 2016. [40] M. K. Warmuth and D. Kuzmin. Randomized online pca algorithms with regret bounds that are logarithmic in the dimension. JMLR, 9(Oct):2287– 2320, 2008. [41] A. R. Webb. Statistical pattern recognition. 2003. [42] P. Wu, S. C. Hoi, P. Zhao, C. Miao, and Z.-Y. Liu. Online multi-modal distance metric learning with application to image retrieval. TKDE, 28(2):454–467, 2016. [43] T. Xiao, H. Li, W. Ouyang, and X. Wang. Learning deep feature representations with domain guided dropout for person re-identification. In CVPR, 2016. [44] F. Xiong, M. Gou, O. Camps, and M. Sznaier. Person re-identification using kernel-based metric learning methods. In ECCV, 2014. [45] J. Yan, B. Zhang, S. Yan, Q. Yang, H. Li, Z. Chen, W. Xi, W. Fan, W.-Y. Ma, and Q. Cheng. Immc: incremental maximum margin criterion. In SIGKDD, 2004. [46] J. Yang, A. F. Frangi, J.-y. Yang, D. Zhang, and Z. Jin. Kpca plus lda: a complete kernel fisher discriminant framework for feature extraction and recognition. TPAMI, 27(2):230–244, 2005. [47] H. Yao, S. Zhang, D. Zhang, Y. Zhang, J. Li, Y. Wang, and Q. Tian. Large-scale person re-identification as retrieval. [48] J. Ye, Q. Li, H. Xiong, H. Park, R. Janardan, and V. Kumar. Idr/qr: an incremental dimension reduction algorithm via qr decomposition. TKDE, 17(9):1208–1222, 2005. [49] L. Zhang, T. Xiang, and S. Gong. Learning a discriminative null space for person re-identification. In CVPR, 2016. [50] L. Zheng, Z. Bie, Y. Sun, J. Wang, C. Su, S. Wang, and Q. Tian. Mars: A video benchmark for large-scale person re-identification. In ECCV, pages 868–884. Springer, 2016. [51] L. Zheng, L. Shen, L. Tian, S. Wang, J. Wang, and Q. Tian. Scalable person re-identification: A benchmark. In ICCV, 2015. [52] L. Zheng, S. Wang, L. Tian, F. He, Z. Liu, and Q. Tian. Query-adaptive late fusion for image search and person re-identification. In CVPR, 2015. [53] W.-S. Zheng, S. Gong, and T. Xiang. Person re-identification by probabilistic relative distance comparison. In CVPR, 2011. [54] W.-S. Zheng, X. Li, T. Xiang, S. Liao, J. Lai, and S. Gong. Partial person re-identification. In ICCV, 2015. A PPENDIX Matrix Sketch. The sketch technique we discuss in this work is related to the matrix sketch [24], which is pass-efficient to read streaming data at most a constant number of time. The sketch algorithm learns a set of frequent directions from an N × d matrix X ∈ RN ×d in a stream, where each row of X is a d-dimensional vector. It maintains a sketch matrix B ∈ R`×d containing ` (` << N ) rows and guarantees that: BT B  XT X & ||XT X − BT B|| ≤ 2||X||2f /`. (25) Such a sketch processing is light in both processing time (bounded by O(d`2 ) ) and space (bounded by O(`d)). Wei-Hong Li is currently a postgraduate student majoring in Information and Communication Engineering in School of Electronics and Information Technology at Sun Yat-sen University. He received the bachelor’s degree in intelligence science and technology from Sun Yat-Sen University in 2015. His research interests include person re-identification, object tracking, object detection and image-based modeling. Homepage: https://weihonglee.github.io. Zhuowei Zhong is a student from Sun Yat-sen University under the joint supervision program of the Chinese University of Hong Kong. He is now graduated and received BSc degree in computer science. His research interest is in Artificial Intelligence, especially in machine learning and constraint satisfaction problem. Wei-Shi Zheng is currently a Professor with Sun Yat-sen University. He has joined Microsoft Research Asia Young Faculty Visiting Programme. He has authored over 90 papers, including over 60 publications in main journals (TPAMI, TNN/TNNLS, TIP, TSMC-B, and PR) and top conferences (ICCV, CVPR, IJCAI, and AAAI). His recent research interests include person association and activity understanding in visual surveillance. He was a recipient of Excellent Young Scientists Fund of the National Natural Science Foundation of China, and Royal Society-Newton Advanced Fellowship, U.K. Homepage: http://isee.sysu.edu.cn/∼zhwshi. 

Logical Methods in Computer Science Vol. 5 (3:8) 2009, pp. 1–69 www.lmcs-online.org Submitted Published Jan. 2, 2008 Sep. 11, 2009 FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES NIKOS TZEVELEKOS Oxford University Computing Laboratory e-mail address: nikt@comlab.ox.ac.uk Abstract. Game semantics has been used with considerable success in formulating fully abstract semantics for languages with higher-order procedures and a wide range of computational effects. Recently, nominal games have been proposed for modelling functional languages with names. These are ordinary, stateful games cast in the theory of nominal sets developed by Pitts and Gabbay. Here we take nominal games one step further, by developing a fully abstract semantics for a language with nominal general references. 5.5. Observationality 5.6. Definability and full-abstraction 5.7. An equivalence established semantically 6. Conclusion Appendix A. Deferred proofs References Contents List of Figures 1. Introduction 2. Theory of nominal sets 2.1. Nominal sets 2.2. Strong support 3. The language 3.1. Definitions 3.2. Categorical semantics 4. Nominal games 4.1. The basic category G 4.2. Arena and strategy orders in G 4.3. Innocence: the category V 4.4. Totality: the category Vt 4.5. A monad, and some comonads 4.6. Nominal games à la Laird 5. The nominal games model 5.1. Solving the Store Equation 5.2. Obtaining the νρ-model 5.3. Adequacy 5.4. Tidy strategies 1 2 4 5 7 8 9 12 21 21 29 30 33 38 40 41 42 45 48 50 53 56 61 61 62 67 List of Figures 1 Typing rules. 2 3 Reduction rules. The semantic translation. 10 18 4 The store arena and the type translation. 44 5 6 9 The store monad. Strategies for update, dereferencing and fresh-name creation. 45 47 7 A dialogue in innocent store. 47 8 Store-H’s -Q’s -A’s in arena T 1. 50 1998 ACM Subject Classification: F.3.2. Key words and phrases: game semantics, denotational semantics, monads and comonads, ν-calculus, ML. Research financially supported by the Engineering and Physical Sciences Research Council, the Eugenides Foundation, the A. G. Leventis Foundation and Brasenose College. l LOGICAL METHODS IN COMPUTER SCIENCE c DOI:10.2168/LMCS-5 (3:8) 2009 CC N. Tzevelekos Creative Commons 2 N. TZEVELEKOS 1. Introduction Functional languages constitute a programming paradigm built around the intuitive notion of a computational function, that is, an effectively specified entity assigning values from a codomain to elements of a domain in a pure manner : a pure function is not allowed to carry any notion of state or side-effect. This simple notion reveals great computational power if the domains considered are higher-order, i.e sets of functions: with the addition of recursive constructs, higher-order functional computation becomes Turing complete (PCF [42, 37]). In practice, though, functional programming languages usually contain impure features that make programming simpler (computational effects), like references, exceptions, etc. While not adding necessarily to its computational power, these effects affect the expressivity of a language: two functions which seem to accomplish the same task may have different innerworkings which can be detected by use of effects (e.g. exceptions can distinguish constant functions that do or do not evaluate their inputs). The study of denotational models for effects allows us to better understand their expressive power and to categorise languages with respect to their expressivity. A computational effect present in most functional programming languages is that of general references. General references are references which can store not only values of ground type (integers, booleans, etc.) but also of higher-order type (procedures, higherorder functions) or references themselves. They constitute a very powerful and useful programming construct, allowing us not only the encoding of recursion (see example 3.4) but also the simulation of a wide range of computational effects and programming paradigms (e.g. object-oriented programming [3, section 2.3] or aspect-oriented programming [40]). The denotational modelling of general references is quite demanding since, on top of phenomena of dynamic update and interference, one has to cope with the inherent cyclicity of higher-order storage. In this paper we provide a fully abstract semantics for a language with general references called the νρ-calculus. The νρ-calculus is a functional language with dynamically allocated general references, reference-equality tests and “good variables”, which faithfully reflects the practice of real programming languages such as ML [27]. In particular, it extends the basic nominal language of Pitts and Stark [36], the ν-calculus, by using names for general references. That is, names in νρ are atomic entities which can be (cf. [36]): created with local scope, updated and dereferenced, tested for equality and passed around via function application, but that is all. The fully abstract model of νρ is the first such for a language with general references and good variables.1 Fully abstract models for general references were given via game semantics in [3] and via abstract categorical semantics (and games) in [20]. Neither approach used names. The model of [3] is based on the idea of relaxing strategy conditions in order to model computational effects. In particular, it models references as variables of a read/write product type and it uses strategies which violate visibility in order to use values assigned to references previously in a play. The synchronisation of references is managed by cell strategies which model fresh-reference creation. Because references are modelled by products, and in order to produce a fully abstract semantics, the examined language needs to include bad variables, which in turn yield unwanted behaviours affecting severely the expressivity of the language 1In fact, the νρ-calculus and its fully abstract model were first presented in [46], of which the present paper is an extended and updated version. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 3 and prohibit the use of equality tests for references.2 On the other hand, the approach in [20] bypasses the bad-variables problem by not including types for references (variables and references of the same type coincide). This contributes new intuitions on sequential categorical behaviour (sequoidal category), but we think that it is somehow distanced from the common notion of reference in functional programming. The full-abstraction problem has also been tackled via trace semantics in [23]. The language examined is a version of that in [3] without bad variables. The latter are not needed since the modelling of references is achieved by names pointing to a store (which is analogous to our approach). Of relevance is also the fully abstract trace model for a language with nominal threads and nominal objects presented in [17]. An important difference between trace models and game models is that the former are defined operationally (i.e. traces are computed by using the operational semantics), whereas game models are defined in a purely compositional manner. Nonetheless, trace models and game models have many similarities, deriving mainly from their sequential-interactive representation of computation, and in particular there are connections between [23] and the work herein that should be further examined. The approach. We model nominal computation in nominal games. These were introduced independently in [2, 21] for producing fully abstract models of the ν-calculus and its extension with pointers respectively. Here we follow the formulation of [2] with rectifications pertaining to the issue of unordered state (see remark 4.20).3 Thus, our nominal games constitute a stateful (cf. Ong [34]) version of Honda-Yoshida call-by-value games [15] built inside the universe of nominal sets of Gabbay and Pitts [12, 35]. A particularly elegant approach to the modelling of names is by use of nominal sets [12, 35]. These are sets whose elements involve a finite number of atoms, and which can be acted upon by finite atom-permutations. The expressivity thus obtained is remarkable: in the realm (the category) of nominal sets, notions like atom-permutation, atom-freshness and atom-abstraction are built inside the underlying structure. We therefore use nominal sets, with atoms playing the role of names, as a general foundation for reasoning about names. The essential feature of nominal games is the appearance of names explicitly in plays as constants (i.e. as atoms), which allows us to directly model names and express namerelated notions (name-equality, name-privacy, scope-extrusion, etc.) in the games setting. Thus nominal games can capture the essential features of nominal computation and, in particular, they model the ν-calculus. From that model we can move to a model of νρ by an appropriate effect-encapsulation procedure, that is, by use of a store-monad. A fully abstract model is then achieved by enforcing appropriate store-discipline conditions on the games. 2 By “bad variables” we mean read/write constructs of reference type which are not references. They are necessary for obtaining definability and full-abstraction in [3] since read/write-product semantical objects may not necessarily denote references. 3The nominal games of [2] use moves attached with finite sets of names. It turns out, however, that this yields discrepancies, as unordered name-creation is incompatible with the deterministic behaviour of strategies and, in fact, nominal games in [2] do not form a category. Here (and also in [46]), we recast nominal games using moves attached with name-lists instead of name-sets. This allows us to restrict our attention to strong nominal sets (v. definition 2.6), a restriction necessary for overcoming the complications with determinacy. 4 N. TZEVELEKOS The paper is structured as follows. In section 2 we briefly present nominal sets and some of their basic properties. We finally introduce strong nominal sets, that is, nominal sets with “ordered involvement” of names, and prove the strong support lemma. In section 3 we introduce the νρ-calculus and its operational semantics. We then introduce the notion of a νρ-model, which provides abstract categorical conditions for modelling νρ in a setting involving local-state comonads and a store-monad. We finally show definability and, by use of a quotienting procedure, full-abstraction in a special class of νρ-models. In section 4 we introduce nominal games and show a series of results with the aim of constructing a category Vt of total, innocent nominal strategies. In the end of the section we attempt a comparison with the nominal games presented by Laird in [21, 24]. In section 5 we proceed to construct a specific fully abstract νρ-model in the category Vt . The basic ingredients for such a construction have already been obtained in the previous section, except for the construction of the store-monad, which involves solving a recursive domain equation in Vt . Once this has been achieved and the νρ-model has been obtained, we further restrict legal strategies to tidy ones, i.e. to those that obey a specific store-related discipline; for these strategies we show definability and full-abstraction. We conclude in section 6 with some further directions. The contributions of this paper are: a) the identification of strong nominal sets as the adequate setting for nominal language semantics; b) the abstract categorical presentation in a monadic-comonadic setting of models of a language with nominal general references; c) the rectification of nominal games of [2] and their use in constructing a specific such model; d) the introduction of a game-discipline (tidiness) to capture computation with names-as-references, leading to a definable and hence fully abstract game model. 2. Theory of nominal sets We give a short overview of nominal sets, which form the basis of all constructions presented in this paper; our presentation generally follows [35]. Nominal sets are an inspiring paradigm of the universality (and reusability) of good mathematics: invented in the 1920’s and 1930’s by Fraenkel and Mostowski as a model of set theory with atoms (ZFA) for showing its independence from the Axiom of Choice, they were reused in the late 1990’s by Gabbay and Pitts [12] as the foundation of a general theory of syntax with binding constructs. The central notion of nominal sets is that of atoms, which are to be seen as basic ‘particles’ present in elements of nominal sets, and of atom-permutations which can act upon those elements. Moreover, there is an infinite supply of atoms, yet each element of a nominal set ‘involves’ finitely many of them, that is, it has finite support with regard to atompermutations. We will be expressing the intuitive notion of names by use of atoms, both in the abstract syntax of the language and in its denotational semantics. Perhaps it is not clear to the reader why nominal sets should be used — couldn’t we simply model names by natural numbers? Indeed, numerals could be used for such semantical purposes (see e.g. [24]), but they would constitute an overspecification: numerals carry a linear order and a bottom element, which would need to be carefully nullified in the semantical definitions. Nominal sets factor out this burden by providing the minimal solution to specifying names; in this sense, nominal sets are the intended model for names. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 5 2.1. Nominal sets. Let us fix a countably infinite family (Ai )i∈ω of pairwise disjoint, countably infinite sets of atoms, and let us denote by PERM(Ai ) the group of finite permutations of Ai . Atoms are denoted by a, b, c and variants; permutations are denoted by π and variants; id is the identity permutation and (a b) is the permutation swapping a and b (and fixing all other atoms). We write A for the union of all the Ai ’s. We take M PERM(A) , PERM(Ai ) (2.1) i∈I to be the direct sum of the groups PERM(Ai ), so PERM(A) is a group of finite permutations of A which act separately on each constituent Ai . In particular, each π ∈ PERM(A) is Q an ω-indexed list of permutations, π ∈ i∈ω PERM(Ai ), such that (π)i 6= idAi holds for finitely many indices i. In fact, we will write (non-uniquely) each permutation π as a finite composition π = π1 ◦ · · · ◦ πn such that each πi belongs to some PERM(Aji ) — note that ji ’s need not be distinct. Definition 2.1. A nominal set X is a set |X| (usually denoted X) equipped with an : PERM(A) × X → X such that, for any action of PERM(A), that is, a function ◦ π, π ′ ∈ PERM(A) and x ∈ X, π ◦ (π ′ ◦ x) = (π ◦ π ′ ) ◦ x , id ◦ x = x . Moreover, for any x ∈ X there exists a finite set S such that, for all permutations π, (∀a ∈ S. π(a) = a) =⇒ π ◦ x = x . N For example, A with the action of permutations being simply permutation-application is a nominal set. Moreover, any set can be trivially rendered into a nominal set of elements with empty support. Finite support is closed under intersection and hence there is a least finite support for each element x of a nominal set; this we call the support of x and denote by S(x). Proposition and Definition 2.2 ([12]). Let X be a nominal set and x ∈ X. For any finite S ⊆ A, S supports x iff ∀a, a′ ∈ (A \ S). (a a′ ) ◦ x = x . Moreover, if finite S, S ′ ⊆ A support x then S ∩ S ′ also supports x. Hence, we can define \ S(x) , { S ⊆fin A | S supports x } , which can be expressed also as: S(x) = { a ∈ A | for infinitely many b. (a b) ◦ x 6= x } .  For example, for each a ∈ A, S(a) = {a}. We say that a is fresh for x, written a # x, if a∈ / S(x). x is called equivariant if it has empty support. It follows from the definition that a # x ⇐⇒ for cofinitely many b. (a b) ◦ x = x . (2.2) There are several ways to obtain new nominal sets from given nominal sets X and Y : • The disjoint union X ⊎Y with permutation-action inherited from X and Y is a nominal set. This extends to infinite disjoint unions. • The cartesian product X × Y with permutations acting componentwise is a nominal set; if (x, y) ∈ X ×Y then S(x, y) = S(x) ∪ S(y). 6 N. TZEVELEKOS • The fs-powerset Pfs (X), that is, the set of subsets of X which have finite support, with permutations acting on subsets of X elementwise. In particular, X ′ ⊆ X is a nominal subset of X if it has empty support, i.e. if for all x ∈ X ′ and permutation π, π ◦ x ∈ X ′ . Apart from A, some standard nominal sets are the following. • Using products and infinite unions we obtain the nominal set [ A# , { a1 . . . an | ∀i, j ∈ 1..n. ai ∈ A ∧ (j 6= i =⇒ aj 6= ai ) } , (2.3) n that is, the set of finite lists of distinct atoms. Such lists we denote by ā, b̄, c̄ and variants. • The fs-powerset Pfs (A) is the set of finite and cofinite sets of atoms, and has Pfin (A) as a nominal subset (the set of finite sets of atoms). For X and Y nominal sets, a relation R ⊆ X ×Y is a nominal relation if it is a nominal subset of X×Y . Concretely, R is a nominal relation iff, for any permutation π and (x, y) ∈ X ×Y , xRy ⇐⇒ (π ◦ x)R(π ◦ y) . For example, it is easy to show that # ⊆ A × X is a nominal relation. Extending this reasoning to functions we obtain the notion of nominal functions. Definition 2.3 (The category Nom). We let Nom be the category of nominal sets and nominal functions, where a function f : X → Y between nominal sets is nominal if f (π ◦ x) = π ◦ f (x) for any π ∈ PERM(A) and x ∈ X. N For example, the support function, S( ) : X → Pfin (A) , is a nominal function since S(π ◦ x) = π ◦ S(x) . Nom inherits rich structure from Set and is in particular a topos. More importantly, it contains atom-abstraction mechanisms; we will concentrate on the following. Definition 2.4 (Nominal abstraction). Let X be a nominal set and x ∈ X. For any finite S ⊆ A, we can abstract x to S, by forming [x]S , { y ∈ X | ∃π. (∀a ∈ S ∩ S(x). π(a) = a) ∧ y = π ◦ x } . N The abstraction restricts the support of x to S ∩ S(x) by appropriate orbiting of x (note that [x]S ∈ Pfs (X)). In particular, we can show the following. Lemma 2.5 ([48]). For any x ∈ X, S ⊆fin A and π ∈ PERM(A), π ◦ [x]S = [π ◦ x]π ◦ S ∧ S([x]S ) = S(x) ∩ S .  Two particular subcases of nominal abstraction are of interest. Firstly, in case S ⊆ S(x) the abstraction becomes [x]S = { y ∈ X | ∃π. (∀a ∈ S. π(a) = a) ∧ y = π ◦ x } . (∗) This is the mechanism used in [46]. Note that if S * S(x) ∧ S(x) * S then (∗) does not yield S([x]S ) = S ∩ S(x). The other case is the simplest possible, that is, of S being empty; it turns out that this last constructor is all we need from nominal abstractions in this paper. We define: [x] , { y ∈ X | ∃π. y = π ◦ x } . (2.4) FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 7 2.2. Strong support. Modelling local state in sets of atoms yields a notion of unordered state, which is inadequate for our intended semantics. Nominal game semantics is defined by means of nominal strategies for games that model computation. These strategies, however, are deterministic up to choice of fresh names, a feature which is in direct conflict to unordered state. For example, in unordered state the consecutive creation of two atoms a, b is modelled by adding the set {a, b} to the local state; on the other hand, by allowing strategies to play such moves we lose determinism in strategies.4 Ordered state is therefore more appropriate for our semantical purposes and so we restrict our attention to nominal sets with ordered presence of atoms in their elements. This notion is described as strong support.5 Definition 2.6. For any nominal set X, any x ∈ X and any S ⊆ A, S strongly supports x if, for any permutation π, (∀a ∈ S. π(a) = a) ⇐⇒ π ◦ x = x . We say that X is a strong nominal set if it is a nominal set with all its elements having strong support. N Compare the last assertion above with that of definition 2.1, which employs only the leftto-right implication. In fact, strong support coincides with weak support when the former exists. Proposition 2.7. If X is a nominal set and x ∈ X has strong support S then S = S(x). Proof: By definition, S supports x, so S(x) ⊆ S. Now suppose there exists a ∈ S \ S(x). For any fresh b, (a b) fixes S(x) but not S, so it doesn’t fix x, . Thus, for example, the set {a, b} ⊆ Ai of the previous paragraph does not have strong support, since the permutation (a b) does not fix the atoms in its support (the set {a, b}) but still (a b) ◦ {a, b} = {a, b}. On the other hand, {a, b} strongly supports the list ab. In fact, all lists of (distinct) atoms have strong support and therefore A# is a strong nominal set (but Pfin (A) is not). The main reason for introducing strong nominal sets is the following result, which is a specialised version of the Strong Support Lemma of [48] (with S = ∅). Lemma 2.8 (Strong Support Lemma). Let X be a strong nominal set and let x1 , x2 , y1 , y2 , z1 , z2 ∈ X. Suppose also that S(yi ) ∩ S(zi ) ⊆ S(xi ) , for i = 1, 2, and that there exist πy , πz such that πy ◦ x1 = πz ◦ x1 = x2 , πy ◦ y 1 = y 2 , πz ◦ z1 = z2 . Then, there exists some π such that π ◦ x1 = x2 , π ◦ y1 = y2 and π ◦ z1 = z2 . Proof: Let ∆i , S(zi ) \ S(xi ) , i = 1, 2 , so ∆2 = πz ◦ ∆1 , and let π ′ , πy−1 ◦ πz . By assumption, π ′ ◦ x1 = x1 , and therefore by strong support π ′ (a) = a for all a ∈ S(x1 ). Take any b ∈ ∆1 . Then π ′ (b) # π ′ ◦ x1 = x1 and πz (b) ∈ πz ◦ ∆1 = ∆2 , ∴ πz (b) # y2 , ∴ π ′ (b) # πy−1 ◦ y2 = y1 . Hence, b ∈ ∆1 =⇒ b, π ′ (b) # x1 , y1 . 4The problematic behaviour of nominal games in weak support is discussed again in remark 4.20. 5An even stricter notion of support is linear support, introduced in [31]: a nominal set X is called linear if for each x ∈ X there is a linear order <x of S(x) such that a <x b =⇒ π(a) <π ◦ x π(b). 8 N. TZEVELEKOS Now assume ∆1 = {b1 , ..., bN } and define π1 , ..., πN by recursion: π0 , id , πi+1 , (bi+1 πi ◦ π ′ ◦ bi+1 ) ◦ πi . We claim that, for each 0 ≤ i ≤ N and 1 ≤ j ≤ i, we have π i ◦ π ′ ◦ bj = bj , πi ◦ x1 = x1 , πi ◦ y 1 = y 1 . We do induction on i; the case of i = 0 is trivial. For the inductive step, if πi ◦ π ′ ◦ bi+1 = bi+1 then πi+1 = πi , and πi+1 ◦ π ′ ◦ bi+1 = πi ◦ π ′ ◦ bi+1 = bi+1 . Moreover, by IH, πi+1 ◦ π ′ ◦ bj = bj for all 1 ≤ j ≤ i, and πi+1 ◦ x1 = x1 and πi+1 ◦ y1 = y1 . If πi ◦ π ′ ◦ bi+1 = b′i+1 6= bi+1 then, by construction, πi+1 ◦ π ′ ◦ bi+1 = bi+1 . Moreover, for each 1 ≤ j ≤ i, by IH, πi+1 ◦ π ′ ◦ bj = (bi+1 b′i+1 ) ◦ bj , and the latter equals bj since bi+1 6= bj implies b′i+1 6= πi ◦ π ′ ◦ bj = bj . Finally, for any a ∈ S(x1 ) ∪ S(y1 ), πi+1 ◦ a = (bi+1 b′i+1 ) ◦ πi ◦ a = (bi+1 b′i+1 ) ◦ a, by IH, with a 6= bi+1 . But the latter equals a since π ′ (bi+1 ) 6= a implies that b′i+1 6= πi ◦ a = a, as required. Hence, for each 1 ≤ j ≤ N , π N ◦ π ′ ◦ bj = bj , πN ◦ x1 = x1 , πN ◦ y 1 = y 1 . Moreover, πN ◦ π ′ ◦ z1 = z1 , as we also have b ∈ S(z1 ) ∩ S(x1 ) =⇒ πN ◦ π ′ ◦ b = πN ◦ b = b −1 we have: (again by strong support). Thus, considering π , πy ◦ πN −1 ◦ x1 = πy ◦ x1 = x2 , πy ◦ πN −1 ◦ y 1 = πy ◦ y 1 = y 2 , πy ◦ πN −1 ◦ −1 ◦ πN ◦ π ′ ◦ z1 = πy ◦ π ′ ◦ z1 = πy ◦ πy−1 ◦ πz ◦ z1 = z2 , z1 = πy ◦ πN πy ◦ πN as required. A more enlightening formulation of the lemma can be given in terms of abstractions, as in the following table. In the context of nominal games later on, the strong support lemma will guarantee us that composition of abstractions of plays can be reduced to composition of plays. Strong Support Lemma. Let X be a strong nominal set and x1 , x2 , y1 , y2 , z1 , z2 ∈ X. Suppose also that S(yi ) ∩ S(zi ) ⊆ S(xi ) , for i = 1, 2, and moreover that [x1 , y1 ] = [x2 , y2 ] , [x1 , z1 ] = [x2 , z2 ] . Then, [x1 , y1 , z1 ] = [x2 , y2 , z2 ]. 3. The language The language we examine, the νρ-calculus, is a call-by-value λ-calculus with nominal general references. It constitutes an extension of the ν-calculus [36] and Reduced ML [44, chapter 5] in which names are used for general references. It is essentially the same calculus of [23], that is, the mkvar-free fragment of the language of [3] extended with reference-equality tests and names. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES ā | Γ |− n : N ā | Γ |− skip : 1 ā | Γ, x : A |− x : A ā | Γ |− M : A × B ā | Γ |− M : A × B ā | Γ |− fst M : A ā | Γ |− snd M : B ā | Γ |− M : N ā | Γ |− M : N ā | Γ |− pred M : N ā | Γ |− succ M : N ā | Γ, x : A |− M : B ā | Γ |− M : A ā | Γ |− a : [A] ā | Γ |− M : [A] āa | Γ |− M : B ā | Γ |− N : B ā | Γ |− hM, N i : A × B ā | Γ |− M : N ā | Γ |− Ni : A (i=1,2) ā | Γ |− if0 M then N1 else N2 : A ā | Γ |− M : A → B ā | Γ |− λx.M : A → B a ∈ AA ∧a ∈ ā 9 ā | Γ |− N : A ā | Γ |− M N : B ā | Γ |− M : [A] ā | Γ |− νa.M : B ā | Γ |− N : A ā | Γ |− M := N : 1 ā | Γ |− N : [A] ā | Γ |− [M = N ] : N ā | Γ |− M : [A] ā | Γ |− ! M : A Figure 1: Typing rules. 3.1. Definitions. The syntax of the language is built inside Nom. In particular, we assume there is a set of names (atoms) AA ∈ (Ai )i∈ω for each type A in the language. Types include types for commands, naturals and references, product types and arrow types. Definition 3.1. The νρ-calculus is a typed functional language of nominal references. Its types, terms and values are given as follows. TY ∋ A, B ::= 1 | N | [A] | A → B | A × B TE ∋ M, N ::= x | λx.M | M N hM, N i | fst M | snd N λ-calculus | n | pred M | succ N arithmetic | skip | if0 M then N1 else N2 return / if then else |a reference to type A (a ∈ AA ) | [M = N ] name-equality test | νa.M ν-abstraction | M := N update | !M dereferencing VA ∋ V, W ::= n | skip | a | x | λx.M | hV, W i The typing system involves terms in environments ā | Γ, where ā a list of (distinct) names and Γ a finite set of variable-type pairs. Typing rules are given in figure 1. N The ν-constructor is a name-binder : an occurrence of a name a inside a term M is bound 10 N. TZEVELEKOS if it is in the scope of some νa . We follow the standard convention of equating terms up to α-equivalence, the latter defined with respect to both variable- and name-binding. Note that TE and VA are strong nominal sets: each name a of type A is taken from AA and all terms contain finitely many atoms — be they free or bound — which form their support. Note also the notion of ordered state that is imposed by use of name-lists (instead of name-sets) in type-environments. In fact, we could have used unordered state at the level of syntax (and operational semantics) of νρ, and ordered state at the level of denotational semantics. This already happens with contexts: a context Γ is a set of premises, but JΓK is an (ordered) product of type-translations. Nevertheless, we think that ordered state does not add much complication while it saves us from some informality. The operational semantics of the calculus involves computation in some store environment where created names have their values stored. Formally, we define store environments S to be lists of the form: S ::= ǫ | a, S | a :: V, S . (3.1) Observe that the store may include names that have been created but remain as yet unassigned a value. For each store environment S we define its domain to be the name-list given by: dom(ǫ) , ǫ , dom(a, S) , a, dom(S) , dom(a :: V, S) , a, dom(S) . (3.2) We only consider environments whose domains are lists of distinct names. We write S |=Γ,A M , or simply S |= M , only if dom(S) | Γ |− M : A is valid (i.e., derivable). Definition 3.2. The operational semantics is given in terms of a small-step reduction, the rules of which are given in figure 2. Evaluation contexts E[ ] are of the form: [ = N ] , [a = (λx.N ) , ], ! , N , fst := N , a := , snd , pred , if0 then N1 else N2 , , succ , h , N i , hV, i N We can see that νρ is not strongly normalising with the following example. Recall the NEW EQ IF0 a#S n=0 if a=b PRD S |= [a = b] −→ S |= n n=1 if a6=b S |= if0 n then N1 else N2 −→ S |= Nj UPD SUC S |= νa.M −→ S, a |= M j=1 if n=0 j=2 if n>0 S, a :: V, S ′ |= ! a −→ S, a :: V, S ′ |= V LAM S |= pred (n+1) −→ S |= n PRD FST S, a(:: W ), S ′ |= a := V −→ S, a :: V, S ′ |= skip DRF S |= succ n −→ S |= n+1 SND S |= (λx.M ) V −→ S |= M {V /x} Figure 2: Reduction rules. CTX S |= pred 0 −→ S |= 0 S |= fst hV, W i −→ S |= V S |= snd hV, W i −→ S |= W S |= M −→ S ′ |= M ′ S |= E[M ] −→ S ′ |= E[M ′ ] FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 11 standard CBV encoding of sequencing: M ; N , (λz.N )M (3.3) with z not free in N . Example 3.3. For each type A, take stopA , νb.(b := λx.(! b)skip) ;(! b)skip with b ∈ A1→A . We can see that stopA diverges, since: |= stopA −→ → b :: λx.(! b)skip |= (! b)skip −→ b :: λx.(! b)skip |= (λx.(! b)skip)skip −→ b :: λx.(! b)skip |= (! b)skip .  The great expressive power of general references is seen in the fact that we can encode the Y combinator. The following example is adapted from [3]. Example 3.4. Taking a ∈ AA→A , define: YA , λf.νa.(a := λx.f (! a)x) ; ! a . YA has type ((A → A) → A → A) → A → A and, for any relevant term M and value V , we have |= (YA (λy.M ))V −→ → a :: λx.(λy.M )(! a)x |= (! a)V −→ a :: λx.(λy.M )(! a)x |= (λx.(λy.M )(! a)x)V −→ a :: λx.(λy.M )(! a)x |= (λy.M )(! a)V , and also |= (λy.M )(YA (λy.M ))V −→ → a :: λx.(λy.M )(! a)x |= (λy.M )(! a)V . For example, setting addrecx , λx. if0 snd x then x else xhsucc fst x, pred snd xi , add , Y(λh.addrech ) , S , a :: λx.(λh.addrech )(! a)x , where x is a metavariable of relevant type, we have that, for any n, m ∈ N, |= addhn, mi −→ → S |= (λh.addrech )(! a)hn, mi −→ → S |= addrecS(a) hn, mi −→ → S |= if0 m then hn, mi else S(a)hsucc fst hn, mi, pred snd hn, mii −→ → S |= S(a)hn+1, m−1i −→ S |= (λh.addrech )(! a)hn+1, m−1i · · · −→ → S |= (λh.addrech )(! a)hn+m, 0i −→ → S |= hn+m, 0i .  The notions of observational approximation and observational equivalence are built around the observable type N. Two terms are equivalent if, whenever they are put inside a variable- and name-closing context of resulting type N, called a program context, they reduce to the same natural number. The formal definition follows; note that we usually omit ā and Γ and write simply M / N . Definition 3.5. For typed terms ā | Γ |− M : A and ā | Γ |− N : A , define ā | Γ |− M / N ⇐⇒ ∀C. (∃S ′ . |= C[M ] −→ → S ′ |= 0) =⇒ (∃S ′′ . |= C[N ] −→ → S ′′ |= 0) where C is a program context. Moreover, ≅ , / ∩ ' . N 12 N. TZEVELEKOS 3.2. Categorical semantics. We now examine sufficient conditions for a fully abstract semantics of νρ in an abstract categorical setting. Our aim is to construct fully abstract models in an appropriate categorical setting, pinpointing the parts of structure needed for such a task. In section 5 we will apply this knowledge in constructing a concrete such model in nominal games. Translating each term M into a semantical entity JM K and assuming a preorder “.” in the semantics, full-abstraction amounts to the assertion: M / N ⇐⇒ JM K . JN K (FA) Note that this formulation is weaker than equational full abstraction, which is given by: M ≅ N ⇐⇒ JM K = JN K . (EFA) Nevertheless, once we achieve (FA) we can construct an extensional model, via a quotienting construction, for which EFA holds. Being a quotiented structure, the extensional model does not have an explicit, simple description, and for this reason we prefer working with the intensional model (i.e., the unquotiented one). Of course, an intensional model satisfying (EFA) would be preferred but this cannot be achieved in our nominal games. Therefore, our categorical models will be guided by the (FA) formulation. 3.2.1. Monads and comonads. The abstract categorical semantics we put forward is based on the notions of monads and comonads. These are standard categorical notions (v. [25], and [8, Triples]) which have been used extensively in denotational semantics of programming languages. We present here some basic definitions and properties. Monads. Monads were introduced in denotational semantics through the work of Moggi [29, 30] as a generic tool for encapsulating computational effects. Wadler [49] popularised monads in programming as a means of simulating effects in functional programs, and nowadays monads form part and parcel of the Haskell programming language [18]. Definition 3.6. A strong monad over a category C with finite products is a quadruple (T, η, µ, τ ), where T is an endofunctor in C and η : IdC → T , µ : T 2 → T and τ : × T → T ( × ) are natural transformations such that the following diagrams commute. µT A T 3A / T 2A µA T µA  T 2A  µA (A × B) × T C / TA ηT A / T 2A JJ JJ JJ µA T ηA J  idT A JJ%  / TA T 2A µA A × (B × T C)id 1 × T AO τA,B τ A,T B / T ((A × B) × C) / T (A × T B) A × T 2 BP RRR P P PPP RRR RRRT ∼ idA ×µB PPPP RR=R PPP R( ( / A × T (B × C) / T (A × (B × C)) A × TB A ×τB,C τ1,A / T (1 × A) OOO OOO OO OOO η ∼ idA ×ηB T∼ OOA×B = = OOOO OO' OOO  '  / T (A × B) A × TB TA A × BO τA×B,C ∼ =  TA J τA,B×C T τA,B / T 2 (A × B) µA×B τA,B  / T (A × B) We say that C has T -exponentials if, for every pair B, C of objects, there exists an object T C B such that for any object A there exists a bijection ∼ = ΛTA,B,C : C(A × B, T C) − → C(A, T C B ) natural in A. N FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 13 Given a strong monad (T, η, µ, τ ), we can define the following transformations. ∼ = ∼ = τA,B ′ → T (A × B) , → B × T A −−−→ T (B × A) − τA,B , TA × B − ′ τA,T B T τA,B µA×B τT A,B ′ T τA,B µA×B ψA,B , T A × T B −−−−→ T (A × T B) −−−−→ T 2 (A × B) −−−−→ T (A × B) , (3.4) ′ ψA,B , T A × T B −−−−→ T (T A × B) −−−−→ T 2 (A × B) −−−−→ T (A × B) . Moreover, T -exponentials supply us with T -evaluation arrows, that is, evTB,C : T C B × B → T C , ΛT −1 (idT C B ) (3.5) so that, for each f : A × B → T C, f = ΛT (f ) ×B ; evTB,C . In fact, T -exponentiation upgrades to a functor (T )− : C op × C → C which takes each f : A′ → A and g : B ′ → B to ′ evT id×f Tg −→ T B) . T g f : T B ′ A → T B A , ΛT (T B ′ A × A′ −−−→ T B ′ A × A −−→ T B ′ − (3.6) Naturality of ΛTA,B,C in A implies its naturality in B, C too, by use of the above construct. Comonads. Comonads are the dual notion of monads. They were first used in denotational semantics by Brookes and Geva [9] for modelling programs intensionally, that is, as mechanisms which receive external computation data and decide on an output. Monadiccomonadic approaches were examined by Brookes and van Stone [10]. Definition 3.7. A comonad on a category C is a triple (Q, ε, δ), where Q is an endofunctor in C and ε : Q → IdC , δ : Q → Q2 are natural transformations such that the following diagrams commute. Q3O A o δQA QδA Q2A o Q2O A δA δA QA oeL εQA Q2A O LLL LLL δ L A idQA LLL QA QA QεA / QA r9 r r r rridr r r QA rr Now assume C has binary products. We define a transformation ζ̄ : Q( × ) → hQπ1 ,Qπ2 i × Q( ), εA ×idQB ζ̄ A,B , Q(A × B) −−−−−−→ QA × QB −−−−−−→ A × QB . Q is called a product comonad if ζ̄ is a natural isomorphism, and is written (Q, ε, δ, ζ) where ζ is the inverse of ζ̄. N It is easy to see that the transformation ζ̄ makes the relevant (dualised) diagrams of definition 3.6 commute, even without stipulating the existence of the inverse ζ. Note that we write ζ ′ , ζ̄ ′ for the symmetric counterparts of ζ, ζ̄. Product comonads are a stronger version of “strong comonads” of [10]. A product comonad Q can be written as: Q ∼ = Q1 × 6 hence the name. We say that Q1 is the basis of the comonad . 6Note this is an isomorphism between comonads, not merely between functors. 14 N. TZEVELEKOS Monadic-comonadic setting. In the presence of both a strong monad (T, η, µ, τ ) and a product comonad (Q, ε, δ, ζ) in a cartesian category C, one may want to solely consider arrows from some initial computation data (i.e., some initial state) of type A to some computation of type B, that is, arrows of type: QA → T B T This amounts to applying the biKleisli construction on C, that is, defining the category CQ with the same objects as C, and arrows T CQ (A, B) , C(QA, T B) . For arrow composition to work in the biKleisli category, we need a distributive law between Q and T , that is, a natural transformation ℓ : QT → T Q making the following diagrams commute. QηA εT A / TA / QT A QA M q8 MMM q q MMM qq ℓ qqTq εA ηQA MMMM A q q &  q T QA QT 2 A QµA ℓT A ; T ℓA / QT A δT A / Q2 T A QℓA ; ℓQA ℓA  T 2 QA   µQA / T QA T δA / T Q2 A In this case, composition of f : QA → T B and g : QB → T C is performed as: δ Qf Tg ℓ µC B QA − → Q2A −−→ QT B −→ T QB −− → T 2 C −−→ T C Since we are examining a monadic-comonadic setting for strong monad T and product comonad Q, a distributive law amounts to a natural transformation ℓ : Q1 × T → T (Q1 × ) , which is therefore given for free: take ℓ , τQ1, . The distributivity equations follow straightforwardly from the monadic equations. Exponentials and the intrinsic preorder. The notion of T -exponentials can be generalised to the monadic-comonadic setting as follows. Definition 3.8. Let C be a category with finite products and let (T, η, µ, τ ), (Q, ε, δ) be a strong monad and comonad, respectively, on C. We say that C has (Q, T )-exponentials if, for each pair B, C of in C there exists an object (Q, T )C B such that, for each object A, there exists a bijection ∼ = → C(QA, (Q, T )C B ) φA,B,C : C(Q(A × B), T C) − N natural in A. Assume now we are in a monadic-comonadic setting (C, Q, T ) with T a strong monad with T -exponentials and Q a product comonad. (Q, T )-exponentials then come for free. Proposition 3.9. In the setting of the previous definition, if T is a strong monad with exponentials and Q is a product comonad then C has (Q, T )-exponentials defined by: (Q, T )C B , T C B , ζ′ f φ(f ) , ΛT (QA × B − → Q(A × B) − → T C) . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 15 φ is a bijection with its inverse sending each g : QA → T C B to the arrow: ζ̄ ′ evT g×id Q(A × B) − → QA × B −−−→ T C B × B −−→ T C .  In the same setting, we can define a notion of intrinsic preorder . Assuming an object O of observables and a collection O ⊆ C(1, T O) of observable arrows, we can have the following. Definition 3.10. Let C, Q, T, O, O be as above. We define . to be the union, over all objects A, B, of relations . A,B ⊆ C(QA, T B)2 defined by: f . A,B g ⇐⇒ ∀ρ ∈ C(Q(T B A ), T O). ΛQ,T (f ); ρ ∈ O =⇒ ΛQ,T (g); ρ ∈ O , δ QΛT (ζ ′ ; f ) where ΛQ,T (f ) , Q1 − → Q21 −−−−−−−→ Q(T B A ) . N We have the following enrichment properties. Proposition 3.11. Let C, Q, T, O, O and . be as above. Then, for any f, g : QA → T B and any arrow h, if f . g then: • if h : QB → T B ′ then δ ; Qf ; ℓ ; T h ; µ . δ ; Qg ; ℓ ; T h ; µ , • if h : QA′ → T A then δ ; Qh ; ℓ ; T f ; µ . δ ; Qh ; ℓ ; T g ; µ , • if h : QA → T C then hf, hi ; ψ . hg, hi ; ψ • if A = A1 × A2 then ΛTQA1 ,A2 ,B (ζ ′ ; f ) ; η . and hh, f i ; ψ . hh, gi ; ψ , ΛTQA1 ,A2 ,B (ζ ′ ; g) ; η . 3.2.2. Soundness. We proceed to present categorical models of the νρ-calculus. The approach we take is a monadic and comonadic one, over a computational monad T and a family of local-state comonads Q = (Qā )ā∈A# , so that the morphism related to each ā | Γ |− M : A be of the form JM K : Qā JΓK → T JAK. Computation in νρ is store-update and fresh-name creation, so T is a store monad, while initial state is given by product comonads. Definition 3.12. A νρ-model M is a structure (M, T, Q) such that: I. M is a category with finite products, with 1 being the terminal object and A × B the product of A and B. II. T is a strong monad (T, η, µ, τ ) with exponentials. III. M contains an appropriate natural numbers object N equipped with successor and predecessor arrows and ñ : 1 → N, each n ∈ N. Moreover, for each object A, there is an arrow cndA : N × T A × T A → T A for zero-equality tests. IV. Q is a family of product comonads (Qā , ε, δ, ζ)ā∈A# on M such that: ′ (a) the basis of Qǫ is 1, and Qā = Qā whenever [ā] = [ā′ ] (i.e., whenever π ◦ ā = ā′ ), ′ (b) if S(ā′ ) ⊆ S(ā) then there exists a comonad morphism āā′ : Qā → Qā such that ā ā ′ ′′ ǫ = ε, ā = id and, whenever S(ā ) ⊆ S(ā ) ⊆ S(ā), ā ā′′ ā ; = ′ ā′′ ā′ ā 16 N. TZEVELEKOS (c) for each āa ∈ A# there exists a natural transformation nuāa : Qā → T Qāa such that, for each A, B ∈ Ob(M) and āa, ā′a with S(āa) ⊆ S(ā′a), the following diagrams commute. ā′ Q A ā′ ā hid,nuāa i / Qā A ′ τ nuāa nuāa  ′ T Qā a A ā′a T āa / T Qāa A / Qā (A × B)   A × T Qāa B / T (Qā A × Qāa A) āa T h ā ,idi (N2) nuA×B id×nuB   ζ A × Qā B / Qā A × T Qāa A / T Qāa (A × B) τ ;Tζ V. Setting AA , Qa 1, for each a ∈ AA , there is a name-equality arrow eqA : AA × AA → N such that, for any distinct a, b ∈ AA , the following diagram commutes. ∆ Qa 1 / ab ab ha, b i AA × AA o Qab 1 eqA !  0̃ 1 / (N1) !   1̃ No 1 VI. Setting J1K , 1, JNK , N, J[A]K , AA , JA → BK , T JBK JAK , JA × BK , JAK × JBK, M contains, for each A ∈ TY, arrows drfA : AA → T JAK and updA : AA × JAK → T 1 such that the following diagrams commute, AA × JAK hid,updA i ; τ ; ∼ = / T (AA × JAK) T (π1 ; drfA ) ; µ - 1 T JAK T π2 hid×π1 ;updA ,id×π2 ;updA i AA × JAK × JAK Qab 1 × ψ ;∼ = + / T1 × T1 π2 JAK × JBK ab ab h a ×π1 ;updA , b ×π2 ;updB i / 3 T1 (NR) ψ;∼ = T1 × T1 ψ′ ; ∼ = + 3 T1 and, moreover, āa ′ (nuāa A × updB ) ; ψ = (nuA × updB ) ; ψ , i.e., updates and fresh names are independent effects. (SNR) N The second subcondition of (N2) above essentially states that, for each object A, nuA can be expressed as: ∼ = ∼ = τ′ nu ×id Qā A − → Qā 1 × A −−1−−→ T Qāa 1 × A − → T (Qāa 1 × A) − → T Qāa A It is evident that the role reserved for nu in our semantics is that of fresh name creation. Accordingly, nu gives rise to a categorical name-abstraction operation: for any arrow f : Qāa A → T B in M, we define nu Tf µ A ^ a _ f , Qā A −−→ T Qāa A −−→ T 2 B − → TB . (3.7) FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 17 The (NR) diagrams give the basic equations for dereferencings and updates (cf. [38, definition 1] and [44, section 5.8]). The first diagram stipulates that by dereferencing an updated reference we get the value of the update. The second diagram ensures that the value of a reference is that of the last update: doing two consecutive updates to the same reference is the same as doing only the last one. The last diagram states that updates of distinct references are independent effects. Let us now proceed with the semantics of νρ in νρ-models. Definition 3.13. Let (M, T, Q) be a νρ-model. Recall the type-translation: J 1K , 1 , JNK , N , J[A]K , AA , JA → BK , T JBK JAK , JA × BK , JAK × JBK . A typing judgement ā | Γ |− M : A is translated to an arrow JM Kā|Γ : Qā JΓK → T JAK in M, which we write simply as JM K : Qā Γ → T A, as in figure 3. N We note that the translation of values follows a common pattern: for any ā | Γ |− V : B, we have JV K = |V | ; η , where |x| , Qā π ; āǫ |a| , Qā ! ; āa |ñ| , Qā ! ; āǫ ; ñ |skip| , Qā ! ; āǫ |λx.M | , ΛT (ζ ′ ; JM K) (3.8) |hV, W i| , h|V |, |W |i . We can show the following lemmas, which will be used in the proof of Correctness. ′ Lemma 3.14. For any ā | Γ |− M : A and S(ā) ⊆ S(ā′ ), JM Kā′ |Γ = āā ; JM Kā|Γ . Moreover, if Γ = x1 : B1 , ..., xn : Bn , and ā | Γ |− M : A and ā | Γ |− Vi : Bi are derivable, ζ ′ ; Qā π2 hid,|V1 |,...,|Vn|i JM K ~ /~x}K = Qā Γ −−−−−−−−−→ Qā Γ × Γ −−−−−→ Qā Γ −−→ T A . JM {V  Lemma 3.15. For any relevant f, g, āa   hf, ā ; gi h^ a _ f,gi ψ ψ āa → T (B × C) , ^ a _ Q A −−−−−→ T B × T C − → T (B × C) = Qā A −−−−−→ T B × T C −   ^a_f f Tg µ Tg µ → T B −− → T 2C − → T C = Qā A −−−→ T B −− ^ a _ Qāa A − → T 2C − → TC .  Lemma 3.16. Let ā | Γ |− M : A and ā | Γ |− E[M ] : B be derivable, with E[ ] being an evaluation context. Then JE[M ]K is equal to: hid,JM Ki T ζ′ τ T JE[x]K µ Qā Γ −−−−−→ Qā Γ × T A − → T (Qā Γ × A) −−→ T Qā (Γ × A) −−−−− → T 2B − → TB .  r We write S |= M − → S ′ |= M ′ with r ∈ {NEW,SUC,EQ,...,LAM′ } if the last non-CTX rule in the related derivation is r. Also, to any store S, we relate the term S̄ of type 1 as: ǭ , skip , a, S , S̄ , a :: V, S , (a := V ; S̄) (3.9) Proposition 3.17 (Correctness). For any typed term ā | Γ |− M : A, and S with dom(S) = ā, and r as above, r 1. if r ∈ / {NEW, UPD, DRF} then S |= M − → S |= M ′ =⇒ JM K = JM ′ K , r 2. if r ∈ {UPD, DRF} then S |= M − → S ′ |= M ′ =⇒ JS̄ ; M K = JS̄ ′ ; M ′ K , NEW 3. S |= M −−−→ S, a |= M ′ =⇒ JS̄ ; M K = ^ a _ JS̄ ; M ′ K . Therefore, S |= M − → S ′ |= M ′ =⇒ JS̄ ; M K = ^ ā′ _ JS̄ ′ ; M ′ K , with dom(S ′ ) = āā′ . 18 N. TZEVELEKOS Qā ! ā ǫ ā ā ǫ Qā π η JxK , Qā Γ −−−→ Qā A −→ A − → TA ā Qā ! η a → T AA JaK , Qā Γ −−→ Qā 1 −→ AA − ā Qā ! JM K : Qā Γ − → TN η ñ →N− JñK , Q Γ −−→ Q 1 −→ 1 − → TN ā JMK / TN Qā Γ S S S S S T succ Jsucc MK S S S  ) TN η JM K : Qā Γ − → T AA ǫ 1− → T1 JskipK , Qā Γ −−→ Qā 1 −→ JN K : Qā Γ − → T AA JM K : Qāa Γ − → TA ^ a _ JMK Jνa.M K : Qā Γ −−−−−→ T A Qā Γ K hJMK,JN Ki K K ā K JM K : Q (Γ × A) − → TB / T AA × T AA ψ K K J[M=N ]K K  T (AA × AA ) K ′ T Λ (ζ ; JMK) / TBA Qā Γ T T T T T T η Jλx.MK T T*  T (T B A ) K K T eq K%  TN JM K : Qā Γ − → T AA JN K : Qā Γ − → TA ā A JM K : Q Γ − → T (T B ) JN K : Qā Γ − → TA hJMK,JN Ki ā Q ΓL L LL hJMK,JN Ki K LL L K / T (T B A ) × T A  A L L T (T B × A) LL T evT ; µ L%  TB JM K : Qā Γ − → T (A × B) JMK / T (A × B) Qā Γ U U U U U U T π1 U U Jfst MK U*  TA JM K : Qā Γ − → TA JN K : Q Γ − → TB hJMK,JN Ki / TA × TB Q ΓT T T T T T JhM,N iK T T* ψ  T (A × B) K K  K T (AA × A) K K K T updA ; µ K%  T1 JM K : Qā Γ − → T AA JMK / T AA Qā Γ T T T T T T drfA ; µ J!MK T T T  ) TA JM K : Qā Γ − → TN JNi K : Qā Γ − → TA Qā Γ K ā / T AA × T A ψ K JM:=N K ψ JM N K ā Qā Γ K hJMK,JN1 K,JN2 Ki K K K Jif0 M then N1 else / T N × T A2 τ′  K T ( N × T A2 ) K N2 K K K cndA ; µ K K%  TA K Figure 3: The semantic translation. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 19 Proof: The last assertion follows easily from 1-3. For 1-3 we do induction on the size of the reduction’s derivation. The base case follows from the specifications of definition 3.12 and lemma 3.14. For the inductive step we have that, for any S, M, E, the following diagram commutes. ′ 2 ′ hid,JS̄Ki τ ;Tζ / T QāΓ T hid,JMKi ; T τ / T 2 (Qā Γ × A) T (ζ ; JE[x]K) / T 3 B / Qā Γ × T 1 Qā Γ XXXXX XXXXX XXXXX XXXXX µ µ Tµ XXXXX hid,JS̄ ; MKi XXX+  ′   τ / T (Qā Γ × A) T (ζ ; JE[x]K) / T 2 B Qā Γ × T AT TTTT TTTT µ T (ΛT (ζ ′ ; JE[x]K)×id) ΛT (ζ ′ ; JE[x]K)×id ; τ TTTTT hΛT (ζ ′ ; JE[x]K) ; η,JS̄ ; MKi ; ψ ′ T)   / TB / T ((A −−⊗ T B) × A) T T ev ;µ By the previous lemma, the upper path is equal to hid, JS̄Ki ; τ ; T ζ ′ ; T JE[M ]K ; µ and therefore to JS̄ ; E[M ]K. Hence, we can immediately show the inductive steps of 1-2. For 3, NEW assuming S |= E[M ] −−−→ S, a |= E[M ′ ] and JS̄ ; M K = ^ a _ JS̄ ; M ′ K , we have, using also lemmas 3.14 and 3.15, ^ a _ JS̄ ; E[M ′ ]K = ^ a _ (hΛT (ζ ′ ; JE[x]K) ; η, JS̄ ; M ′ Ki ; ψ′ ; T evT ; µ) = ^ a _ (hΛT (ζ ′ ; JE[x]K) ; η, JS̄ ; M ′ Ki ; ψ ′ ) ; T evT ; µ = hΛT (ζ ′ ; JE[x]K) ; η, ^ a _ JS̄ ; M ′ Ki ; ψ ′ ; T evT ; µ = hΛT (ζ ′ ; JE[x]K) ; η, JS̄ ; M Ki ; ψ ′ ; T evT ; µ = JS̄ ; E[M ]K .  In order for the model to be sound, we need computational adequacy. This is added explicitly as a specification. Definition 3.18. Let M be a νρ-model and J K the respective translation of νρ. M is adequate if → S ′ |= 0̃ , ∃S, b̄. JM K = ^ b̄ _ JS̄ ; 0̃K =⇒ ∃S ′ . ā |= M −→ for any typed term ā | ∅ |− M : N. N Proposition 3.19 (Equational Soundness). If M is an adequate νρ-model, JM K = JN K =⇒ M / N .  3.2.3. Completeness. We equip the semantics with a preorder to match the observational preorder of the syntax as in (FA). The chosen preorder is the intrinsic preorder with regard to a collection of observable arrows in the biKleisli monadic-comonadic setting (cf. definition 3.10). In particular, since we have a collection of monad-comonad pairs, we also need a collection of sets of observable arrows. Definition 3.20. An adequate νρ-model M = (M, T, Q) is observational if, for all ā: • There exists Oā ⊆ M(Qā 1, T N) such that, for all ā | ∅ |− M : N, JM K ∈ Oā ⇐⇒ ∃S, b̄. JM K = ^ b̄ _ JS̄ ; 0̃K . 20 N. TZEVELEKOS • The induced intrinsic preorder on arrows in M(Qā A, T B) defined by f .ā g ⇐⇒ ∀ρ : Qā (T B A ) → T N. (Λā (f ) ; ρ ∈ Oā =⇒ Λā (g) ; ρ ∈ Oā ) with Λā (f ) , ΛQ ā ,T (f ), satisfies, for all relevant a, ā′ , f, f ′ , f .āa f ′ =⇒ ^ a _ f .ā ^ a _ f ′ ∧ f .ā f ′ =⇒ ā′ ā ′ ; f .a ā′ ā ; f′ . N We write M as (M, T, Q, O). Recurring to ΛQ ā ,T of definition 3.10, we have that Λā (f ) is the arrow: δ Qā ΛT (ζ ′ ; f ) Qā 1 − → Qā Qā 1 −−−−−−−→ Qā (T B A ) . (3.10) Hence, Oā contains those arrows that have a specific observable behaviour in the model, and the semantic preorder is built over this notion. In particular, terms that yield a number have observable behaviour. In order to make good use of the semantic preorder we need it to be a congruence with regard to the semantic translation. Congruences for νρ, along with typed contexts, are defined properly in [48]. For now, we state the following. Lemma 3.21. Let (M, T, Q, O) be an observational νρ-model. Then, for any pair ā | Γ |− M, N : A of typed terms and any context C such that ā′ | Γ′ |− C[M ], C[N ] : B are valid, ′ JM K .ā JN K =⇒ JC[M ]K .ā JC[N ]K .  Assuming that we translate νρ into an observational νρ-model, we can now show one direction of (FA). Proposition 3.22 (Inequational Soundness). For typed terms ā | Γ |− M, N : A, JM K . JN K =⇒ M / N . Proof: Assume JM K .ā JN K and |= C[M ] −→ → S ′ |= 0̃ , so JC[M ]K = ^ ā′ _ JS̄ ′ ; 0̃K with ′ ′ ā ā = dom(S ). JM K . JN K implies JC[M ]K . JC[N ]K , and hence JC[N ]K ∈ Oǫ . Thus, by adequacy, there exists S ′′ such that |= C[N ] −→ → S ′′ |= 0̃ . In order to achieve completeness, and hence full-abstraction, we need our semantic translation to satisfy some definability requirement with regard to the intrinsic preorder. Definition 3.23. Let (M, T, Q, O) be an observational νρ-model and let J K be the semantic translation of νρ to M. M satisfies ip-definability if, for any ā, A, B, there exists ā ⊆ M(Qā JAK, T JBK) such that: DA,B ā • For each f ∈ DA,B there exists a term M such that JM K = f . • For each f, g ∈ M(Qā A, T B), ā f .ā g ⇐⇒ ∀ρ ∈ DA→B,N . (Λā (f ) ; ρ ∈ Oā =⇒ Λā (g) ; ρ ∈ Oā ) . We write M as (M, T, Q, O, D). For such a model M we achieve full abstraction. Theorem 3.24 (FA). For typed terms ā | Γ |− M, N : A, JM K . JN K ⇐⇒ M / N . N FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 21 Proof: Soundness is by previous proposition. For completeness (“⇐=”), we do induction on the size of Γ. For the base case suppose ā | ∅ |− M / N and take any ρ ∈ D 1→A,N such that Λā (JM K) ; ρ ∈ Oā . Let ρ = Jā | y : 1 → A |− L : NK , some term L, so Λā (JM K) ; ρ is Λā (JM K) ; JLK = δ ; Qā |λz.M | ; JLK = J(λy.L)(λz.M )K for some z : 1. The latter being in Oā implies that it equals ^ b̄ _ JS̄ ; 0̃K, some S. Now, M / N implies (λy.L)(λz.M ) / (λy.L)(λz.N ) , hence ν b̄.(S̄ ; 0̃) / (λy.L)(λz.N ) , by soundness. But this implies that ā |= (λy.L)(λz.N ) −→ → S ′ |= 0̃ , so J(λy.L)(λz.N )K ∈ Oā , by ā ā ā correctness. Hence, Λ (JN K) ; ρ ∈ O , so JM K . JN K, by ip-definability. For the inductive step, if Γ = x : B, Γ′ then IH ā | Γ |− M / N =⇒ ā | Γ′ |− λx.M / λx.N =⇒ Jλx.M K .ā Jλx.N K =⇒ JM K = J(λx.M )xK .ā J(λx.N )xK = JN K where the last approximation follows from lemma 3.21. 4. Nominal games In this section we introduce nominal games and strategies, and construct the basic structure from which a fully abstract model of νρ will be obtained in the next section. We first introduce nominal arenas and strategies, which form the category G. We afterwards refine G by restricting to innocent, total strategies, obtaining thus the category Vt . Vt is essentially a semantical basis for call-by-value nominal computation in general. In fact, from it we can obtain not only fully abstract models of νρ, but also of the ν-calculus [2], the νερ-calculus [47] (νρ+exceptions), etc. 4.1. The basic category G. The basis for all constructions to follow is the category Nom of nominal sets. We proceed to arenas. Definition 4.1. A nominal arena A , (MA , IA , ⊢A , λA ) is given by: • a strong nominal set MA of moves, • a nominal subset IA ⊆ MA of initial moves, • a nominal justification relation ⊢A ⊆ MA × (MA \ IA ), • a nominal labelling function λA : MA → {O, P } × {A, Q}, which labels moves as Opponent or Player moves, and as Answers or Questions. An arena A is subject to the following conditions. (f) For each m ∈ MA , there exists unique k ≥ 0 such that IA ∋ m1 ⊢A · · · ⊢A mk ⊢A m , for some ml ’s in MA . k is called the level of m, so initial moves have level 0. (l1) Initial moves are P-Answers. (l2) If m1 , m2 ∈ MA are at consecutive levels then they have complementary OP-labels. (l3) Answers may only justify Questions. N We let level-1 moves form the set JA ; since ⊢A is a nominal relation, JA is a nominal subset of MA . Moves in MA are denoted by mA and variants, initial moves by iA and variants, and level-1 moves by jA and variants. By I¯A we denote MA \ IA , and by J¯A the set MA \ JA . 22 N. TZEVELEKOS Note that, although the nominal arenas of [2] are defined by use of a set of weaker conditions than those above, the actual arenas used there fall within the above definition. We move on to prearenas, which are the ‘boards’ on which nominal games are played. Definition 4.2. A prearena is defined exactly as an arena, with the only exception of condition (l1): in a prearena initial moves are O-Questions. Given arenas A and B, construct the prearena A → B as: MA→B , MA + MB IA→B , IA λA→B , [ (iA 7→ OQ , mA 7→ λ̄A (mA )) , λB ] ⊢A→B , {(iA , iB )} ∪ { (m, n) | m ⊢A,B n } N where λ̄A is the OP -complement of λA . It is useful to think of the (pre)arena A as a vertex-labelled directed graph with vertex-set MA and edge-set ⊢A such that the labels on vertices are given by λA (and satisfying (l1-3)). It follows from (f) that the graph so defined is levelled: its vertices can be partitioned into disjoint sets L0, L1, L2,. . . such that the edges may only travel from level i to level i + 1 and only level-0 vertices have no incoming edges (and therefore (pre)arenas are directed acyclic). Accordingly, we will be depicting arenas by levelled graphs or triangles. The simplest arena is 0 , (∅, ∅, ∅, ∅). Other (flat) arenas are 1 (unit arena), N (arena of naturals) and Aā (arena of ā-names), for any ā ∈ A# , which we define as (4.1) MAā = IAā , Aā , where Aā , { π ◦ ā | π ∈ PERM(A) }. Note that for ā empty we get Aǫ = 1, and that we write Ai for Aa with a being of type i. More involved are the following constructions. For arenas A, B, define the arenas A⊗B, A⊥ , A −−⊗ B and A ⇒ B as follows. MA⊗B , IA ×IB + I¯A + I¯B M1 = I1 , {∗} , MN = IN , N , IA⊗B , IA ×IB λA⊗B , [ ((iA , iB ) 7→ P A) , λA ↾ I¯A , λB ↾ I¯B ] 2 2 ⊢A⊗B , { ((iA , iB ), m) | iA ⊢A m ∨ iB ⊢B m } ∪ (⊢A ↾ I¯A ) ∪ (⊢B ↾ I¯B ) B , IB + IA ×JB + I¯A + I¯B ∩ J¯B B , IB B , [ (iB 7→ P A) , ((iA , jB ) 7→ OQ) , λ̄A ↾ I¯A , λB ↾ (I¯B ∩ J¯B ) ] B , { (iB , (iA , jB )) | iB ⊢B jB } ∪ { ((iA , jB ), m) | iA ⊢A m } MA −−⊗ IA −−⊗ λA −−⊗ ⊢A −−⊗ 2 ∪ { ((iA , jB ), m) | jB ⊢B m } ∪ (⊢A ↾ I¯A ) ∪ (⊢B ↾ (I¯B ∩ J¯B )2 ) A B A⊗B A B A −−⊗ B FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES MA⊥ , {∗1 } + {∗2 } + MA IA⊥ , {∗1 } 23 ∗ ∗ ∗ A A⊥ A B A⇒B λA⊥ , [ (∗1 7→ P A) , (∗2 7→ OQ) , λA ] ⊢A⊥ , {(∗1 , ∗2 ), (∗2 , iA )} ∪ (⊢A ↾ MA 2 ) A ⇒ B , A −−⊗ B⊥ In the constructions above it is assumed that all moves which are not hereditarily justified by initial moves are discarded. Hence, for example, for any A, B JB = ∅ =⇒ A −−⊗ B = B Moreover, we usually identify arenas with graph-isomorphic structures; for example, 1 −−⊗ A = A , 0 ⇒ A = A⊥ . Using the latter convention, the construction of A⇒B in the previous definition is equivalent to A ⇒ B of [15, 2] ; concretely, it is given by: (4.2) MA⇒B , {∗} + IA + I¯A + MB IA⇒B , {∗} λA⇒B , [ (∗ 7→ P A) , (iA 7→ OQ) , λ̄A , λB ] ⊢A⇒B , {(∗, iA )} ∪ { (iA , m) | iA ⊢A m ∨ m ∈ IB } ∪ (⊢A ↾ I¯A 2 ) ∪ (⊢B ↾ MB 2 ) Of the previous constructors all look familiar apart from −−⊗ (which in [46] appears as ⇒). ˜ The latter can be seen as a function-space constructor merging the contravariant part of its RHS with its LHS. For example, for any A, B, C, we have A −−⊗ N = N and A −−⊗ (B ⇒ C) = (A⊗B) ⇒ C In the first equality we see that N which appears on the RHS of −−⊗ has no contravariant part, and hence A is redundant. In the second equality B, which is the contravariant part of B ⇒ C, is merged with A. This construction will be of great use when considering a monadic semantics for store. We move on to describe how are nominal games played. Plays of a game consist of sequences of moves from some prearena. These moves are attached with name-lists to the effect of capturing name-environments. Definition 4.3. A move-with-names of a (pre)arena A is a pair, written mā , where m is a move of A and ā is a finite list of distinct names (name-list). N If x is a move-with-names then its name-list is denoted by nlist(x) and its underlying move by x ; therefore, x = xnlist(x) . We introduce some notation for sequences (and lists). Notation 4.4 (Sequences). A sequence s will be usually denoted by xy . . . , where x, y, ... are the elements of s. For sequences s, t, • s ≤ t denotes that s is a prefix of t, and then t = s(t \ s), • s  t denotes that s is a (not necessarily initial or contiguous) subsequence of t, • s− denotes s with its last element removed, • if s = s1 . . . sn then s1 is the first element of s and sn the last. Also, 24 N. TZEVELEKOS ◦ n is the length of s, and is denoted by |s|, ◦ s.i denotes si and s.−i denotes sn+1−i , that is, the i-th element from the end of s (for example, s.−1 is sn ), ◦ s≤si denotes s1 . . . si , and so does s<si+1 , ◦ if s is a sequence of moves-with-names then, by extending our previous notation, we N have s = snlist(s) , where nlist(s) is a list of length |s| of lists of names. A justified sequence over a prearena A is a finite sequence s of OP-alternating moves such that, except for s.1 which is initial, every move s.i has a justification pointer to some s.j such that j < i and s.j ⊢A s.i ; we say that s.j (explicitly) justifies s.i . A move in s is an open question if it is a question and there is no answer inside s justified by it. There are two standard technical conditions that one may want to apply to justified sequences: well-bracketing and visibility . We say that a justified sequence s is wellbracketed if each answer s.i appearing in s is explicitly justified by the last open question in s<i (called the pending question). For visibility, we need to introduce the notions of Player- and Opponent-view . For a justified sequence s, its P-view psq and its O-view xsy are defined as follows. pǫq , ǫ xǫy , ǫ psxq , psq x pxq , x if x a P-move if x is initial psxs′ yq , psq xy if y an O-move expl. justified by x xsxy , xsyx if x an O-move ′ xsxs yy , xsyxy if y a P-move expl. justified by x The visibility condition states that any O-move x in s is justified by a move in xs<xy , and any P-move y in s is justified by a move in ps<yq. We can now define plays. Definition 4.5. Let A be a prearena. A legal sequence on A is sequence of moves-withnames s such that s is a justified sequence satisfying Visibility and Well-Bracketing. A legal sequence s is a play if s.1 has empty name-list and s also satisfies the following Name Change Conditions (cf. [34]): (NC1) The name-list of a P-move x in s contains as a prefix the name-list of the move preceding it. It possibly contains some other names, all of which are fresh for s<x . (NC2) Any name in the support of a P-move x in s that is fresh for s<x is contained in the name-list of x. (NC3) The name-list of a non-initial O-move in s is that of the move justifying it. The set of plays on a prearena A is denoted by PA . N It is important to observe that plays have strong support, due to the tagging of moves with lists of names (instead of sets of names [2]). Note also that plays are the ǫ-plays of [46]. Now, some further notation. Notation 4.6 (Name-introduction). A name a is introduced (by Player) in a play s, written a ∈ L(s), if there exist consecutive moves yx in s such that x is a P-move and a ∈ S(nlist(x) \ nlist(y)). N From plays we move on to strategies. Recall the notion of name-restriction we introduced in definition 2.4; for any nominal set X and any x ∈ X, [x] = { π ◦ x | π ∈ PERM(A) } . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 25 Definition 4.7. Let A be a prearena. A strategy σ on A is a set of equivalence classes [s] of plays in A, satisfying: • Prefix closure: If [su] ∈ σ then [s] ∈ σ. • Contingency completeness: If even-length [s] ∈ σ and sx is a play then [sx] ∈ σ. • Determinacy: If even-length [s1 x1 ], [s2 x2 ] ∈ σ and [s1 ] = [s2 ] then [s1 x1 ] = [s2 x2 ]. We write σ : A whenever σ is a strategy on A. N By convention, the empty sequence ǫ is a play and hence, by prefix closure and contingency completeness, all strategies contain [ǫ] and [iA ]’s. Some basic strategies are the following — note that we give definitions modulo prefix closure. Definition 4.8. For any ā′ , ā ∈ A# with S(ā′ ) ⊆ S(ā), n ∈ N and any arena B, define the following strategies. • ñ : 1 → N , {[∗ n]} • !B : B → 1 , {[iB ∗]} • ā ā′ : Aā → Aā , {[ā ā′ ]} ′ • idB : B → B , { [s] | s ∈ PB(1) →B(2) ∧ ∀t ≤even s. t ↾ B(1) = t ↾ B(2) } N It is easy to see that the aforedefined are indeed strategies. That definitions are given modulo prefix closure means that e.g. ñ is in fact: ñ = { [ǫ], [∗], [∗ n] } . We proceed to composition of plays and strategies. In ordinary games, plays are composed by doing “parallel composition plus hiding” (v. [4]); in nominal games we need also take some extra care for names. Definition 4.9. Let s ∈ PA→B and t ∈ PB→C . We say that: • s and t are almost composable, s ` t, if s ↾ B = t ↾ B. • s and t are composable, s ≍ t, if s ` t and, for any s′ ≤ s, t′ ≤ t with s′ ` t′ : (C1) If s′ ends in a (Player) move in A introducing some name a then a # t′ . Dually, if t′ ends in a move in C introducing some name a then a # s′ . (C2) If both s′ , t′ end in B and s′ ends in a move introducing some name a then a # t′− . Dually, if t′ ends in a move introducing some name a then a # s′− . N The following lemma is taken verbatim from [15], adapted from [7]. Lemma 4.10 (Zipper lemma). If s ∈ PA→B and t ∈ PB→C with s ` t then either s ↾ B = t = ǫ, or s ends in A and t in B, or s ends in B and t in C, or both s and t end in B. Note that in the sequel we will use some standard switching condition results (see e.g. [15, 5]) without further mention. Composable plays are composed as below. Note that we may tag a move m as m(O) (or m(P ) ) to specify it is an O-move (a P-move). Definition 4.11. Let s ∈ PA→B and t ∈ PB→C with s ≍ t . Their parallel interaction s k t and their mix s • t, which returns the final name-list in s k t, are defined by mutual 26 N. TZEVELEKOS recursion as follows. We set ǫ k ǫ , ǫ , ǫ • ǫ , ǫ , and: smb̄A • t smb̄A k t , (s k t)mA smb̄B • tmc̄B smb̄B k tmc̄B , (s k t)mB s • tmc̄C s k tmc̄C , (s k t)mC b̄s b̄ smA(P ) • t , (s • t) b̄ b̄s b̄ c̄ smB(P ) • tmB(O) , (s • t) b̄ c̄t c̄ s • tmC(P ) , (s • t) c̄ smb̄A(O) • t , b̄′ c̄t c̄ smb̄B(O) • tmB(P ) , (s • t) c̄ s • tmc̄C(O) , c̄ ′ , where b̄s is the name-list of the last move in s, and b̄′ is the name-list of mA(O) ’s justifier inside s k t ; similarly for c̄t and c̄ ′ . The composite of s and t is: s ; t , (s k t) ↾ AC . The set of interaction sequences of A, B, C is defined as: ISeq(A, B, C) , { s k t | s ∈ PA→B ∧ t ∈ PB→C ∧ s ≍ t } . N When composing compatible plays s and t, although their parts appearing in the common component (B) are hidden, the names appearing in (the support of) s and t are not lost but rather propagated to the output components (A and C). This is shown in the following lemma (the proof of which is tedious but not difficult, see [48]). Lemma 4.12. Let s ≍ t with s ∈ PA→B and t ∈ PB→C . (a) If s k t ends in a generalised P-move mb̄ then b̄ contains as a prefix the name-list of (s k t).−2 . It possibly contains some other names, all of which are fresh for (s k t)− . (b) If s ; t ends in a P-move mb̄ then b̄ contains as a prefix the name-list of (s ; t).−2 . It possibly contains some other names, all of which are fresh for (s ; t)− . (c) If s k t ends in a move mb̄ then b̄ contains as a prefix the name-list of the move explicitly justifying mb̄ . (d) If s = s′ mb̄ ends in A and t in B then b̄  s • t, if s = s′ mb̄ and t = t′ mc̄ end in B then b̄  s • t and c̄  s • t, if s ends in B and t = t′ mc̄ in C then c̄  s • t. (e) S(s) ∪ S(t) = S(s k t) = S(s ; t) ∪ S(s • t) . Proposition 4.13 (Plays compose). If s ∈ PA→B and t ∈ PB→C with s ≍ t, then s ; t ∈ PA→C . Proof: We skip visibility and well-bracketing, as these follow from ordinary CBV game analysis. It remains to show that the name change conditions hold for s ; t. (NC3) clearly does by definition, while (NC1) is part (b) of previous lemma. For (NC2), let s ; t end in some P-move ms • t and suppose a ∈ S(ms • t ) and a # (s ; t)− . Suppose wlog that s = s′ mb̄ , and so (s ; t)− = s′ ; t. Now, if a # s′ • t then, by part (e) of previous lemma, a # s′ , t and therefore a ∈ b̄ , by (NC2) of s. By part (d) then, a ∈ S(s • t). Otherwise, a ∈ S(s′ • t) and hence, by part (a), a ∈ S(s • t). We now proceed to composition of strategies. Recall that we write σ : A → B if σ is a strategy on the prearena A → B. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 27 Definition 4.14. For strategies σ : A → B and τ : B → C, their composition is defined as σ ; τ , { [s ; t] | [s] ∈ σ ∧ [t] ∈ τ ∧ s ≍ t } , N and is a candidate strategy on A → C. Note that, for any sequence u, if [u] ∈ σ ; τ then u = π ◦ (s ; t) = (π ◦ s) ;(π ◦ t) for some [s] ∈ σ, [t] ∈ τ, s ≍ t and π. Therefore, we can always assume u = s ; t with [s] ∈ σ, [t] ∈ τ and s ≍ t. Our next aim is to show that composites of strategies are indeed strategies. Again, the proofs of the following technical lemmata are omitted for economy (but see [48]). Lemma 4.15. For plays s1 ≍ t1 and s2 ≍ t2 , if s1 k t1 = s2 k t2 then s1 = s2 and t1 = t2 . Hence, if s1 k t1 ≤ s2 k t2 then s1 ≤ s2 and t1 ≤ t2 . Lemma 4.16. Let σ : A → B and τ : B → C be strategies with [s1 ], [s2 ] ∈ σ and [t1 ], [t2 ] ∈ τ . If |s1 k t1 | ≤ |s2 k t2 | and [s1 ; t1 ] = [s2 ; t2 ] then there exists some π such that π ◦ (s1 k t1 ) ≤ s 2 k t2 . Proposition 4.17 (Strategies compose). If σ : A → B and τ : B → C are strategies then so is σ ; τ . Proof: By definition and proposition 4.13, σ ; τ contains equivalence classes of plays. We need also check prefix closure, contingency completeness and determinacy. The former two are rather straightforward, so we concentrate on the latter. Assume even-length [u1 x1 ], [u2 x2 ] ∈ σ ; τ with [u1 ] = [u2 ], say ui xi = si ; ti , [si ] ∈ σ and [ti ] ∈ τ , i = 1, 2 . By prefix-closure of σ, τ we may assume that si , ti don’t both end in B, for i = 1, 2. b̄′ If si end in A then si = s′i nb̄i i and si ; ti = (s′i ; ti )ni i , i = 1, 2 . Now, [s′1 ; t1 ] = [u1 ] = [u2 ] = [s′2 ; t2 ], so, by lemma 4.16 and assuming wlog that |s′1 k t1 | ≤ |s′2 k t2 |, we have ′′ ′ ′′′ π ◦ (s′1 k t1 ) ≤ (s′2 k t2 ), ∴ π ◦ s′1 ≤ s′2 , say s′2 = s′′2 s′′′ 2 with s2 = π ◦ s1 and s2 in B. Then b̄ b̄ ′ ′ ′′′ ′ 1 2 [s′′2 ] = [s′1 ], ∴ [s′′2 (s′′′ 2 n2 ).1] = [s1 n1 ], by determinacy of σ, and hence |s2 | = 0, s2 = π ◦ s1 b̄ b̄ and t2 = π ◦ t1 . Moreover, π ′ ◦ s′1 n11 = s′2 n22 , some permutation π ′ . Now we can apply the Strong Support Lemma, as (C1) implies (S(nb̄i i ) \ S(s′i )) ∩ S(ti ) = ∅. Hence, there exists a permutation π ′′ such that π ′′ ◦ s1 = s2 and π ′′ ◦ t1 = t2 , ∴ [s1 ; t1 ] = [s2 ; t2 ] , as required. If si end in B and ti in C, then work similarly as above. These are, in fact, the only cases we need to check. Because if, say, s2 , t1 end in B, s1 in A and t2 in C then t1 , s2 end in P-moves − − − − − and [s− 1 ; t1 ] = [s2 ; t2 ] implies that s1 , t2 end in O-moves in B. If, say, |s1 k t1 | ≤ |s2 k t2 | − ′ then we have, by lemma 4.16, π ◦ s− 1 ≤ s2 , some permutation π. So if π ◦ s1 = s2 and ′′ ′′ ′ s2 = s2 s2 , determinacy of σ dictates that s2 .1 be in A, to |s1 ; t1 | = |s2 ; t2 | and s2 ; t2 ending in C. In order to obtain a category of nominal games, we still need to show that strategy composition is associative. We omit the (rather long) proof and refer the interested reader to [48]. Proposition 4.18. For any σ : A → B, σ1 : A′ → A and σ3 : B → B ′ , idA ; σ = σ = σ ; idB ∧ (σ1 ; σ) ; σ3 = σ1 ;(σ ; σ3 ) .  Definition 4.19. The category G of nominal games contains nominal arenas as objects and nominal strategies as arrows. N 28 N. TZEVELEKOS In the rest of this section let us examine closer the proof of proposition 4.17 in order identify where exactly is strong support needed, and for which reasons is the nominal games model of [2] flawed. Remark 4.20 (The need for strong support). The nominal games presented here differ from those of [2] crucially in one aspect; namely, the modelling of local state. In [2] local state is modelled by finite sets of names, so a move-with-names is a move attached with a finite set of names, and other definitions differ accordingly. The problem is that thus determinacy is not preserved by strategy composition: information separating freshly created names may be hidden by composition and hence a composite strategy may break determinacy by distinguishing between composite plays that are equivalent. In particular, in the proof of determinacy above we first derived from [s′1 ; t1 ] = [s′2 ; t2 ] that there exists some π so that π ◦ s′1 = s2 and π ◦ t1 = t2 , by appealing to lemma 4.16; in the (omitted) proof of that lemma, the Strong Support Lemma needs to be used several times. In fact, the statement |s′1 k t1 | = |s′2 k t2 | ∧ [s′1 ; t1 ] = [s′2 ; t2 ] =⇒ ∃π. π ◦ s′1 = s′2 ∧ π ◦ t1 = t2 does not hold in a weak support setting such that of [2]. For take some i ∈ ω and consider the following AGMOS-strategies (i.e. strategies of [2]). σ : 1 → Ai , { [∗ a{a,b} ] | a, b ∈ Ai ∧ a 6= b } , (4.20:A) τ : Ai → Ai ⇒ Ai , { [a ∗ c a] | a, c ∈ Ai } . Then, [∗ a{a,b} ; a ∗ b] = [∗ ∗{a,b} b{a,b} ] = [∗ ∗{a,b} a{a,b} ] = [∗ a{a,b} ; a ∗ a] , yet for no π do we have π ◦ (∗ a{a,b} ) = ∗ a{a,b} and π ◦ (a∗b) = a∗a. As a result, determinacy fails for σ ; τ since both [∗ ∗{a,b} b{a,b} a{a,b} ], [∗ ∗{a,b} a{a,b} a{a,b} ] ∈ σ ; τ . Another point where we used the Strong Support Lemma in the proof of determinacy was in showing (the dual of): ∃π, π ′ . π ◦ (s1 , t′1 ) = (s2 , t′2 ) ∧ π ′ ◦ t′1 nb̄11 = t′2 nb̄22 =⇒ ∃π ′′ . π ′′ ◦ (s1 , t′1 nb̄11 ) = (s2 , t′2 nb̄22 ) i.e. [s1 , t′1 ] = [s2 , t′2 ] ∧ [t′1 nb̄11 ] = [t′2 nb̄22 ] =⇒ [s1 , t′1 nb̄11 ] = [s2 , t′2 nb̄22 ] . The above statement does not hold for AGMOS-games. To show this, we need to introduce7 the flat arena Ai ⊙ Ai with MAi ⊙Ai , P2 (Ai ) (the set of 2-element subsets of Ai ). This is not a legal arena in our setting, since its moves are not strongly supported, but it is in the AGMOS setting. Consider the following strategies. σ : Ai ⊗ Ai → Ai ⊙ Ai , { [ (a, b) {a, b}] | a, b ∈ Ai ∧ a 6= b } τ : Ai ⊙ Ai → Ai , { [{a, b} a] | a, b ∈ Ai ∧ a 6= b } (4.20:B) We have that [ (a, b) {a, b}, {a, b}] = [ (a, b) {a, b}, {a, b}] and [{a, b} a] = [{a, b} b] , yet [ (a, b) {a, b}, {a, b} a] 6= [ (a, b) {a, b}, {a, b} b] . N In fact, determinacy is broken since [ (a, b) a], [ (a, b) b] ∈ σ ; τ . 7This is because our presentation of nominal games does not include plays and strategies with non-empty initial local state. In the AGMOS setting we could have used to the same effect the {a, b}-strategies: σ : Ai ⊗ Ai → 1 , { [ (a, b){a,b} ∗{a,b} ]{a,b} } , τ : 1 → Ai , { [∗{a,b} a {a,b} ]{a,b} } . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 29 4.2. Arena and strategy orders in G. G is the raw material from which several subcategories of nominal games will emerge. Still, though, there is structure in G which will be inherited to the refined subcategories we will consider later on. In particular, we consider (subset) orderings for arenas and strategies, the latter enriching G over Cpo.8 These will prove useful for solving domain equations in categories of nominal games. Definition 4.21. For any arenas A, B and each σ, τ ∈ G(A, B) define σ ⊑ τ ⇐⇒ σ ⊆ τ . F S For each ⊑-increasing sequence (σi )i∈ω take i σi , i σi . N F It is straightforward to see that each such i σi is indeed a strategy: prefix closure, contingency completeness and determinacy easily follow from the fact that the sequences we consider are ⊑-increasing. Hence, each G(A, B) is a cpo with least element the empty strategy (i.e. the one containing only [ǫ]). More than that, these cpo’s enrich G. Proposition 4.22. G is Cpo-enriched wrt ⊑. Proof: Enrichment amounts to showing the following straightforward assertions. σ ⊑ σ ′ ∧ τ ⊑ τ ′ =⇒ σ ; τ ⊑ σ ′ ; τ ′ G G (σi ; τ ) (σi )i∈ω an ω-chain =⇒ ( σi ) ; τ ⊑ i∈ω i∈ω (τi )i∈ω an ω-chain =⇒ σ ;( G τi ) ⊑ i∈ω G (σ ; τi ) i∈ω  On the other hand, arenas are structured sets and hence also ordered by a ‘subset relation’. Definition 4.23. For any A, B ∈ Ob(G) define A E B ⇐⇒ MA ⊆ MB ∧ IA ⊆ IB ∧ λA ⊆ λB ∧ ⊢A ⊆ ⊢B , and for any E-increasing sequence (Ai )i∈ω define [ G Ai . Ai , i∈ω i∈ω If A E B then we can define an embedding-projection pair of arrows by setting: inclA,B : A → B , { [s] ∈ [PA→B ] | [s] ∈ idA ∨ (odd(|s|) ∧ [s− ] ∈ idA ) } , projB,A : B → A , { [s] ∈ [PB→A ] | [s] ∈ idA ∨ (odd(|s|) ∧ [s− ] ∈ idA ) } . There is also an indexed version of E, for any k ∈ N, A Ek B ⇐⇒ A E B ∧ { m ∈ MB | level(m) < k } ⊆ MA . N F It is straightforward to see that i∈ω Ai is well-defined, and that E forms a cpo on Ob(G) with least element the empty arena 0. By inclA,B and projB,A being an embeddingprojection pair we mean that: inclA,B ; projB,A = idA ∧ projB,A ; inclA,B ⊑ idB (4.3) 8By cpo we mean a partially ordered set with least element and least upper bounds for increasing ωsequences. Cpo is the category of cpos and continuous functions. 30 N. TZEVELEKOS Note that in essence both inclA,B and projB,A are equal to idA , the latter seen as a partially defined strategy on prearenas A → B and B → A. Finally, it is easy to show the following. A E B E C =⇒ inclA,B ; inclB,C = inclA,C (TRN) 4.3. Innocence: the category V. In game semantics for pure functional languages (e.g. PCF [16]), the absence of computational effects corresponds to innocence of the strategies. Here, although our aim is to model a language with effects, our model will use innocent strategies: the effects will still be achieved, by using monads. Innocence is the condition stipulating that the strategies be completely determined by their behaviour on P-views. In our current setting the manipulation of P-views presents some difficulties, since P-views of plays need not be plays themselves. For example, the P-view of the play on the side (where curved lines represent justification pointers) is ∗ (∗, ∗) ∗ a and violates (NC2). Consequently, we need to explicitly impose innocence on plays. 1 / 1⊥ ⊗ A i ∗ OQ (∗, ∗) PA ∗ OQ ∗a PA ∗ OQ a PA Definition 4.24. A legal sequence s is an innocent play if s.1 has empty name-list and s also satisfies the following Name Change Conditions: (NC1) The name-list of a P-move x in s contains as a prefix the name-list of the move preceding it. It possibly contains some other names, all of which are fresh for s<x . (NC2′ ) Any name in the support of a P-move x in s that is fresh for ps<xq is contained in the name-list of x. (NC3) The name-list of a non-initial O-move in s is that of the P-move justifying it. The set of innocent plays of A is denoted by PAi . N It is not difficult to show now that a play s is innocent iff, for any t ≤ s, ptq is a play. We can obtain the following characterisation of name-introduction in innocent plays. Proposition 4.25 (Name-introduction). Let s be an innocent play. A name a is introduced by Player in s iff there exists a P-move x in s such that a ∈ S(x) and a # ps<xq. Proof: If a is introduced by a P-move x in s then a ∈ nlist(x) and a # nlist(s<x .−1), hence, by (NC1), a # s<x so a # ps<xq. Conversely, if a ∈ S(x) and a # ps<xq then, by (NC2′ ), a ∈ nlist(x), while a # ps<xq implies a # nlist(s<x .−1). Innocent plays are closed under composition (proof omitted, v. [48]). Proposition 4.26. If s ∈ PA→B , t ∈ PB→C are innocent and s ≍ t then s ; t is innocent. We now move on to innocent strategies and show some basic properties. Definition 4.27. A strategy σ is an innocent strategy if [s] ∈ σ implies that s is innocent, and if even-length [s1 x1 ] ∈ σ and odd-length [s2 ] ∈ σ have [ps1q] = [ps2q] then there exists x2 such that [s2 x2 ] ∈ σ and [ps1 x1q] = [ps2 x2q]. N Lemma 4.28. Let σ be an innocent strategy. (1) If [s] ∈ σ then [psq] ∈ σ. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 31 (2) If sy is an even-length innocent play and [s], [psyq] ∈ σ then [sy] ∈ σ. (3) If psyq is even-length with nlist(y) = nlist(s.−1) and [s], [psyq] ∈ σ then [sy] ∈ σ. (4) If s is an even-length innocent play and, for any s′ ≤even s, [ps′q] ∈ σ then [s] ∈ σ. Proof: For (1) we do induction on |s|. The base case is trivial. Now, if s = s′ y with y a P-move then psq = ps′q y and [ps′q] ∈ σ by prefix closure and IH. By innocence, there exists y ′ such that [ps′q y ′ ] ∈ σ and [ps′q y ′ ] = [psyq], so done. If s = s1 ys2 x and x an O-move justified by y then [ps1 yq] ∈ σ by prefix closure and IH, hence [ps1 yq x] ∈ σ by contingency completeness. For (2) note that by innocence we have [sy ′ ] ∈ σ for some y ′ such that [psyq] = [psy ′q]. Then, [psq, y] = [psq, y ′ ] ∧ [psq, s] = [psq, s] ∧ (S(y) \ S(psq)) ∩ S(s) = (S(y ′ ) \ S(psq)) ∩ S(s) = ∅ . Thus we can apply the strong support lemma and get [sy] = [sy ′ ], as required. For (3) it suffices to show that sy is an innocent play. As s, psq y are plays, it suffices to show that sy satisfies the name conditions at y. (NC3) and (NC2′ ) hold because psyq a play. (NC1) also holds, as y is non-introducing. For (4) we do induction on |s|. The base case is encompassed in psq = s, which is trivial. For the inductive step, let s = s− x with psq 6= s. By IH and contingency completeness we have [s− ] ∈ σ, and since [psq] ∈ σ, by (2), [s] ∈ σ. We can now show that innocent strategies are closed under composition (details in [48]). Proposition 4.29. If σ : A → B, τ : B → C are innocent strategies then so is σ ; τ . Definition 4.30. V is the lluf subcategory of G of innocent strategies. N Henceforth, when we consider plays and strategies we presuppose them being innocent. Viewfunctions. We argued previously that innocent strategies are specified by their behaviour on P-views. We formalise this argument by representing innocent strategies by viewfunctions. Definition 4.31. Let A be a prearena. A viewfunction f on A is a set of equivalence classes of innocent plays of A which are even-length P-views, satisfying: • Even-prefix closure: If [s] ∈ f and t is an even-length prefix of s then [t] ∈ f . • Single-valuedness: If [s1 x1 ], [s2 x2 ] ∈ f and [s1 ] = [s2 ] then [s1 x1 ] = [s2 x2 ]. Let σ be an innocent strategy and let f be a viewfunction. Then, we can define a corresponding viewfunction and a strategy by: viewf(σ) , { [s] ∈ σ | |s| even ∧ psq = s } , [ strat(f ) , stratn (f ) , n where strat0 (f ) , {[ǫ]} and: strat2n+1 (f ) , { [sx] | sx ∈ PAi ∧ [s] ∈ strat2n (f ) } , strat2n+2 (f ) , { [sy] | sy ∈ PAi ∧ [s] ∈ strat2n+1 (f ) ∧ [psyq] ∈ f } . N Note in the above definition that, for any even-length s, [s] ∈ strat(f ) implies [psq] ∈ f . We can show that the conversion functions are well-defined inverses. 32 N. TZEVELEKOS Proposition 4.32. For any innocent strategy σ, viewf(σ) is a viewfunction. Conversely, for any viewfunction f , strat(f ) is an innocent strategy. Moreover, f = viewf(strat(f )) ∧ σ = strat(viewf(σ)) .  Recall the subset ordering ⊑ of strategies given in definition 4.21. It is easy to see that the ordering induces a cpo on innocent strategies and that V is Cpo-enriched. We can also show the following. Corollary 4.33. For all viewfunctions f, g and innocent strategies σ, τ , (1) f ⊆ strat(f ) , (2) σ ⊆ τ ⇐⇒ viewf(σ) ⊆ viewf(τ ) , f ⊆ g ⇐⇒ strat(f ) ⊆ strat(g) , (3) viewf(σ) ⊆ τ ∧ viewf(τ ) ⊆ σ =⇒ σ = τ . Moreover, ⊑ yields a cpo on viewfunctions, and viewf and strat are continuous with respect to ⊑. Notation 4.34 (Diagrams of viewfunctions). We saw previously that innocent strategies can be represented by their viewfunctions. A viewfunction is a set of (equivalence classes of) plays, so the formal way to express such a construction is explicitly as a set. For example, we have that viewf(idA ) = { [sm(1) m(2) ] | [s] ∈ viewf(idA ) ∧ (m ∈ IA ∨ (s.−1 ⊢A m(1) ∧ s.−2 ⊢A m(2) )) } . The above behaviour is called copycat (v. [4]) and is perhaps the most focal notion in game semantics. A more convenient way to express viewfunctions is by means of diagrams. For example, for idA we can have the following depiction. idA : A iA /A OQ iA PA The polygonal line in the above depiction stands for a copycat link , meaning that the strategy copycats between the two iA ’s. A more advanced example of this notation is the strategy in the middle below. A⇒B ∗ iA hA,B : (A ⇒ B)⊗A PA (∗, iA ) OQ A− OQ ∗ ∗ iA hA,B : (A ⇒ B)⊗A / B⊥ (∗, iA ) PA OQ ∗ ∗ OQ PQ / B⊥ iA PA OQ PQ B jA OQ jA PQ iB OA iB PA Note first that curved lines (and also the line connecting the two ∗’s) stand for justification pointers. Moreover, recall that the arena A ⇒ B has the form given on the left above, so the leftmost iA (l-iA ) in the diagram of hA,B has two child components, A− and B. Then, FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 33 the copycat links starting from the l-iA have the following meaning. hA,B copycats between the A− -component of l-iA and the other iA , and copycats also between the B-component of l-iA and the lower ∗. That is (modulo prefix-closure), hA,B , strat{ [ (∗, iA ) ∗ ∗ iA s ] | [ iA iA s ] ∈ viewf(idA ) ∨ [s] ∈ viewf(idB ) } . Another way to depict hA,B is by cases with regard to Opponent’s next move after l-iA , as seen on the right diagram above. Finally, we will sometimes label copycat links by strategies (e.g. in the proof of proposition 4.42). Labelling a copycat link by a strategy σ means that the specified strategy plays like σ between the linked moves, instead of doing copycat. In this sense, ordinary copycat links can be seen as links labelled with identities. 4.4. Totality: the category Vt . We introduce the notion of total strategies, specifying those strategies which immediately answer initial questions without introducing fresh names. We extend this type of reasoning level-1 moves, yielding several subclasses of innocent strategies. Note that an arena A is pointed if IA is singleton. Definition 4.35. An innocent strategy σ : A → B is total if for any [iA ] ∈ σ there exists [iA iB ] ∈ σ. A total strategy σ : A → B is: • l4 if whenever [s] ∈ σ and s.−1 ∈ JA then | psq | = 4, b̄ ] ∈ σ, • t4 if for any [iA iB jB ] ∈ σ there exists [iA iB jB jA • tl4 if it is both t4 and l4, • ttotal if it is tl4 and for any [iA iB jB ] ∈ σ there exists [iA iB jB jA ] ∈ σ. A total strategy τ : C ⊗A → B is: • l4* if whenever [s] ∈ τ and s.−1 ∈ JA then | psq | = 4, b̄ ] ∈ τ , • t4* if for any [ (iC , iA )iB jB ] ∈ τ there exists [ (iC , iA )iB jB jA • tl4* if it is both t4* and l4*. We let Vt be the lluf subcategory of V of total strategies, and Vtt its lluf subcategory of ttotal strategies. Vt∗ and Vtt∗ are the full subcategories of Vt and Vtt respectively containing pointed arenas. N The above subclasses of strategies will be demystified in the sequel. For now, we show a technical lemma. Let us define, for each arena A, the diagonal strategy ∆A as follows. ∆A : A → A⊗A , strat{ [ iA (iA , iA ) s ] | [ iA iA s ] ∈ viewf(idA ) } (4.4) Lemma 4.36 (Separation of Head Occurrence). Let A be a pointed arena and let f : A → B be a t4 strategy. There exists a tl4* strategy f˜ : A⊗A → B such that f = ∆ ; f˜. Proof: Let us tag the two copies of A in A ⊗ A as A(1) and A(2) , and take b̄ b̄ ˜ viewf(f ) ∧ ∀i. s.i ∈ / JA(2) } , f˜ , strat{ [ (iA , iA )iB jB jA s ] | [ iA iB jB jA s]∈ (2) (2) ˜ is the composition of de-indexing from MA(1) and MA(2) to MA with ∈. Intuitively, where ∈ f˜ plays the first JA -move of f in A(2) , and then mimics f until the next JA -move of f , which is played in A(1) . All subsequent JA -moves are also played in A(1) . Clearly, f˜ is tl4* and f = ∆ ; f˜. 34 N. TZEVELEKOS We proceed to examine Vt . Eventually, we will see that it contains finite products and that it contains some exponentials, and that lifting promotes to a functor. Lifting and product. We first promote the lifting and tensor arena-constructions to functors. In the following definition recall L from notation 4.6 and note that we write L(m) # m′ for L(m) ∩ S(m′ ) = ∅. Definition 4.37. Let f : A → A′ , g : B → B ′ in Vt . Define the arrows f ⊗g , strat{ [ (iA , iB ) (iA′ , iB ′ ) s ] | ( [ iA iA′ s ] ∈ viewf(f ) ∧ [iB iB ′ ] ∈ g ∧ L(iA iA′ s) # iB ) ∨ ( [ iB iB ′ s ] ∈ viewf(g) ∧ [iA iA′ ] ∈ f ∧ L(iB iB ′ s) # iA ) } , f⊥ , strat{ [∗ ∗′ ∗′ ∗ s] | [s] ∈ viewf(f ) } , of types f ⊗g : A⊗B → A′ ⊗B ′ and f⊥ : A⊥ → A′⊥ . N Let us give an informal description of the above constructions: • f⊥ : A⊥ → A′⊥ initially plays a sequence of asterisks [∗ ∗′ ∗′ ∗] and then continues playing like f . • f ⊗g : A⊗B → A′ ⊗B ′ answers initial moves [ (iA , iB ) ] with f ’s answer to [iA ] and g’s answer to [iB ]. Then, according to whether Opponent plays in JA′ or in JB ′ , Player plays like f or like g respectively. Note that f⊥ is always ttotal. We can show the following. Proposition 4.38. ⊗ : Vt × Vt → Vt and ( )⊥ : Vt → Vtt∗ are functors. Moreover, ⊗ yields products and hence Vt is cartesian. Proposition 4.39. Vt is cartesian: 1 is a terminal object and ⊗ is a product constructor. Proof: Terminality of 1 is clear. Moreover, it is straightforward to see that ⊗ yields a symmetric monoidal structure on Vt , with its unit being 1 and its associativity, leftunit, right-unit and symmetry isomorphisms being the canonical ones. Hence, it suffices to show that there exists a natural coherent diagonal, that is, a natural transformation ∆ : IdVt → ⊗ ◦ hIdVt , IdVt i (where hIdVt , IdVt i is the diagonal functor on Vt ) such that the following diagrams commute for any A, B in Vt . ∆A ⊗∆B / (A⊗A)⊗(B ⊗B) A⊗B TT TTTT TTTT ∼ TTT = ∆A⊗ B TTTT)  (A⊗B)⊗(A⊗B) A qq MMMMM ∼ MM= MMM ∆A MM&  / A⊗1 A⊗A q ∼ =qqqq qq xqqq 1⊗A o !A ⊗idA idA ⊗!A But it is easy to see that the diagonal of (4.4) makes the above diagrams commute. Naturality follows from the single-threaded nature of strategies (v. [14]). FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES π 35 π 1 2 Products are concretely given by triples A ←− A⊗B −→ B, where π1 = strat{ [ (iA , iB ) iA s ] | [iA iA s] ∈ viewf(idA ) } f g and π2 similarly, while for each A ← −C− → B we have hf, gi : C → A⊗B = strat{ [ iC (iA , iB ) s ] | ( [iC iA s] ∈ viewf(f ) ∧ [iC iB ] ∈ viewf(g) ) ∨ ( [iC iA ] ∈ viewf(f ) ∧ [iC iB s] ∈ viewf(g) ) } . Finally, we want to generalise the tensor product to a version applicable to countably many arguments. In arenas, the construction comprises of gluing countably many arenas together at their initial moves. The problem that arises then is that the product of infinitely many (initial) moves need not have finite support, breaking the arena specifications. Nevertheless, in case we are interested only in pointed arenas, this is easily bypassed: a pointed arena has a unique initial move, which is therefore equivariant, and the product of equivariant moves is of course also equivariant. N Proposition and Definition 4.40. For pointed arenas {Ai }i∈ω define i Ai by: ] MNi Ai , {∗} + λNi Ai , [ (∗ 7→ P A), [λAi i∈ω ]] , I¯Ai , i [ ⊢Ni Ai , {(†, ∗)} ∪ { (∗, jAi ) | i ∈ ω } ∪ INi Ai , {∗} , (⊢Ai ↾ I¯Ai 2 ) . i For {fi : Ai → Bi }i∈ω with Ai ’s and Bi ’s pointed define: O fi , strat{ [∗ ∗ s] | ∃k. [iAk iBk s] ∈ viewf(fk ) } . i N Q : Vt∗ → Vt∗ is a functor. Then, In fact, we could proceed and show that the aforedefined tensor yields general products of pointed objects, but this will not be of use here. Partial exponentials. We saw that Vt has products, given by the tensor functor ⊗. We now show that the arrow constructor yields appropriate partial exponentials, which will be sufficient for our modelling tasks. Let us introduce the following transformations on strategies. Definition 4.41. For all arenas A, B, C with C pointed, define a bijection ∼ = → Vt (A, B −−⊗ C) ΛB A,C : Vt (A⊗B, C) − by taking, for each h : A⊗B → C and g : A → B −−⊗ C ,9 −−⊗ C , strat{ [i ΛB A iC (iB , jC ) s] | [ (iA , iB ) iC jC s ] ∈ viewf(h) } , A,C (h) : A → B −1 ΛB A,C (g) : A⊗B → C , strat{ [ (iA , iB ) iC jC s ] | [iA iC (iB , jC ) s] ∈ viewf(g) } . For each (f, g) : (A, B) → (A′ , B ′ ), define the arrows A evA,B : (A −−⊗ B)⊗A → B , ΛA f −−⊗ −1 B,B (idA−−⊗B ) , −−⊗ g : A′ −−⊗ B → A −−⊗ B ′ , ΛA A ′ B,A′ −−⊗B ′ (id⊗f −−⊗ ; ev ; g) . N 9Note the reassignment of pointers that takes place implicitly in the definitions of Λ, Λ−1 , in order e.g. for (iA , iB ) iC jC s to be a play of viewf(h). 36 N. TZEVELEKOS It is not difficult to see that Λ and Λ−1 are well-defined and mutual inverses. What is more, they supply us with exponentials. Proposition 4.42. Vt has partial exponentials wrt to ⊗, in the following sense. For any object B, the functor ⊗B : Vt → Vt has a partial right adjoint B −−⊗ : Vt∗ → Vt , that is, for any object A and any pointed object C the bijection ΛB A,C is natural in A. Proof: It suffices to show that, for any f : A⊗B → C and g : A → B −−⊗ C, Λ(f )⊗id ; ev = f , Λ(f ) ⊗id A⊗B A consequence of partial exponentiation is that : (Vt ) Now, in case g is ttotal, the strategy f strat(φ), where −−⊗ /C (iC , iB ) These equalities are straightforward. For example, the viewfunction of Λ(f )⊗id ; ev is given by the diagram on the side, which also gives the viewfunction of f . −−⊗ ev (iA , iB ) g ⊗id ; ev = Λ−1 (g) . op / (B −−⊗ C)⊗B iC jC (iB , jC ) f −−⊗ naturally upgrades to a functor: × Vt∗ → Vt . g : A′ −−⊗ B → A −−⊗ B ′ is given concretely by φ = { [iB iB ′ (iA , jB ′ ) (iA′ , jB ) s] | ([iA iA′ s] ∈ viewf(f ) ∧ [iB iB ′ jB ′ jB ] ∈ g ∧ L(iA iA′ s)#iB , jB ′ ) ∨ ([iB iB ′ jB ′ jB s] ∈ viewf(g) ∧ [iA iA′ ] ∈ f ∧ L(iB iB ′ jB ′ jB s)#iA ) }. That is, f −−⊗ g answers initial moves [iB ] like g and then responds to [iB iB ′ (iA , jB ′ ) ] with f ’s answer to [iA ] and g’s response to [iB iB ′ jB ′ ] (recall g ttotal). It then plays like f or like g, according to Opponent’s next move. Note that φ is a viewfunction even if B, B ′ are not pointed. A special case of ttotality in the second argument arises in the defined functor: ⇒ : (Vt )op × Vt → Vtt∗ , −−⊗ ( )⊥ . (4.5) Remark 4.43. In the work on CBV games of Honda & Yoshida [15] the following version of partial exponentiation is shown. V(A⊗B, C) ∼ (4.6) = Vt (A, B ⇒ C) Interestingly, that version can be derived from ours (using also another bijection shown in [15]), V(A⊗B, C) ∼ = Vt (A⊗B, C⊥ ) ∼ = Vt (A, B −−⊗ C⊥ ) = Vt (A, B ⇒ C) . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 37 But also vice versa, if C is pointed then C ∼ = C2 ⇒ C1 , for some arenas C1 , C2 ,10 and (4.6) (4.6) Vt (A⊗B, C2 ⇒C1 ) ∼ = Vt (A, (B ⊗ C2 )⇒C1 ) = Vt (A, B −−⊗(C2 ⇒C1 )) . = V(A⊗B ⊗C2 , C1 ) ∼ Strategy and arena orders. Recall the orders defined for strategies (⊑) and arenas (E) in section 4.2. These being subset orderings are automatically inherited by Vt . Moreover, by use of corollary 4.33 we can easily show that the aforedefined functors are continuous. Note that, although the strategy order ⊑ is inherited from V, the least element (the empty strategy) is lost, as it is not total. Proposition 4.44. Vt and Vtt are PreCpo-enriched wrt ⊑.11 Moreover, O Y ( )⊥ : Vt → Vtt∗ , ( ⊗ ) : Vt × Vt → Vt , ( ): Vt∗ → Vt∗ , ( −−⊗ ) : Vtop × Vtt∗ → Vtt∗ , ( ⇒ ) : Vtop × Vt → Vtt∗ are locally continuous functors. The order of arenas in Vt is the same as in G, and therefore Ob(Vt ) is a cpo with least element 0. Note that A E B does not imply that the corresponding projection is a total strategy — but A E1 B does imply it. In fact, A E1 B =⇒ projB,A ∈ Vtt (B, A) ∧ A E2 B =⇒ inclA,B ∈ Vtt (A, B) . Moreover, we have the following. Proposition 4.45. All of the functors of proposition 4.44 are continuous wrt E . Moreover, A E A′ ∧ B E B ′ =⇒ inclA,A′ ⊗inclB,B ′ = inclA⊗B,A′ ⊗B ′ A E1 A′ ∧ B E1 B ′ =⇒ projA′ ,A ⊗projB ′ ,B = projA′ ⊗B ′ ,A⊗B O ∀i ∈ ω. Ai E A′i =⇒ inclAi ,A′i = inclNi Ai ,Ni A′i i O ∀i ∈ ω. Ai E A′i =⇒ projA′ ,Ai = projNi A′ ,Ni Ai i i i A E1 A′ ∧ B E B ′ =⇒ projA′ ,A ⇒ inclB,B ′ = inclA⇒B,A′ ⇒B ′ A E A′ ∧ B E1 B ′ =⇒ inclA,A′ ⇒ projB ′ ,B = projA′ ⇒B ′ ,A⇒B A E1 A′ ∧ B E2 B ′ =⇒ projA′ ,A −−⊗ inclB,B ′ = inclA ′ ′ A E A ∧ B E1 B =⇒ inclA,A′ −−⊗ B,A′ −−⊗B ′ −−⊗ projB ′ ,B = projA′ B ′ ,A−−⊗B −−⊗ . Proof: All the clauses are in effect functoriality statements, since the underlying sets of inclusions and projections correspond to identity strategies. 10 In fact, for C to be expressed as C ⇒ C we need a stronger version of condition (f), namely: 2 1 (f’) For each m ∈ MA , there exists unique k ≥ 0 and a unique sequence x1 . . . xn ∈ {Q, A}∗ such that IA ∋ m1 ⊢A · · · ⊢A mk ⊢A m , for some ml ’s in MA with λQA C (ml ) = xl . A In such a case, C1 and C2 are given by taking KC , { m ∈ MC | ∃jC . jC ⊢C m ∧ λC (m) = P A } and A A A ⊢C1 , ⊢C ↾ (MC1 × I¯C1 ) λC1 , λC ↾ MC1 MC1 , KC + { m ∈ MC | ∃k ∈ KC . k ⊢C · · · ⊢C m } IC1 , KC MC2 , I¯C \ MC1 λC2 , [iC2 7→ P A, m 7→ λ̄C (m) ] IC2 , JC ⊢C2 , ⊢C ↾ (MC2 × I¯C2 ) . 11 By precpo we mean a cpo which may not have a least element. PreCpo is the category of precpos and continuous functions. 38 N. TZEVELEKOS 4.5. A monad, and some comonads. We now proceed to construct a monad and a family of comonads on Vt that will be of use in later sections. Specifically, we will upgrade lifting to a monad and introduce a family of product comonads for initial state. Lifting monad. It is a more-or-less standard result that the lifting functor induces a monad. Definition 4.46. Define the natural transformations up, dn, st as follows. upA : A → A⊥ = strat{ [iA ∗1 ∗2 iA s] | [iA iA s] ∈ viewf(idA ) } dnA : A⊥⊥ → A⊥ , strat{ [∗1 ∗′1 ∗′2 ∗2 ∗3 ∗4 s] | [s] ∈ viewf(idA ) } stA,B : A⊗B⊥ → (A⊗B)⊥ , strat{ [ (iA , ∗1 ) ∗′1 ∗′2 ∗2 iB (iA , iB ) s] | [ (iA , iB ) (iA , iB ) s] ∈ viewf(idA⊗B ) } N (primed asterisks are used for arenas on the RHS, where necessary). Proposition 4.47. The quadruple (( )⊥ , up, dn, st) is a strong monad on Vt . Moreover, it yields monadic exponentials by taking (C⊥ )B to be B ⇒ C, for each B, C. Proof: It is not difficult to see that (( )⊥ , up, dn, st) is a strong monad. Moreover, for each B, C we have that B ⇒ C = B −−⊗ C⊥ is a ( )⊥ -exponential, because of exponentiation properties of −−⊗. Although finding a canonical arrow from A to A⊥ is elementary (upA ), finding a canonical arrow in the inverse direction is not always possible. In some cases, e.g. A = Ai , there is no such arrow at all, let alone canonical. An exception occurs when A is pointed, by setting: puA : A⊥ → A , strat{ [∗ iA jA ∗ iA jA s] | [iA iA jA jA s] ∈ viewf(idA ) } . (4.7) Lemma 4.48. puA yields a natural transformation pu : ( )⊥(Vtt∗ ) → IdVtt∗ . Moreover, for any arenas A, B with B pointed, upA ; puA = idA , puA⊥ = dnA and   puB st′ ev⊥ puA B = Λ (A −−⊗ B)⊥ ⊗A −−→ ((A −−⊗ B)⊗A)⊥ −−→ B⊥ −−→ B .  −−⊗ Initial-state comonads. Our way of modelling terms-in-local-state will be by using initial state comonads, in the spirit of intensional program modelling of Brookes & Geva [9]. In our setting, the initial state can be any list ā of distinct names; we define a comonad for each one of those lists. Definition 4.49 (Initial-state comonads). For each ā ∈ A# define the triple (Qā , ε, δ) and by taking Qā : Vt → Vt , Aā ⊗ π 2 ε : Qā → IdVt , { εA : Aā ⊗A −→ A}, ∆⊗id δ : Qā → (Qā )2 , { δA : Aā ⊗A −−−−→ Aā ⊗ Aā ⊗A } . For each S(ā′ ) ⊆ S(ā) define the natural transformation ā ā′ ′ : Qā → Qā by taking ′ ( āā′ )A : Aā ⊗A → Aā ⊗A , ( āā′ )1 ⊗idA , where ( āā′ )1 is ā ā′ of definition 4.8, that is, ( āā′ )1 , { [ (ā, ∗) (ā′ , ∗) ] } . N FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 39 Note that Qǫ , the comonad for empty initial state, is the identity comonad. Note also that we have suppressed indices ā from transformations ε, δ for notational economy. Clearly, each triple (Qā , ε, δ) forms a product comonad on Vt . Moreover, it is straightforward to show the following. Proposition 4.50 (Chain rule). For each S(ā′ ) ⊆ S(ā) ∈ A# , the transformation āā′ is a comonad morphism. Moreover, āǫ = ε : Qā → IdVt , āā = id : Qā → Qā and, for each ′′ S(ā′ ) ⊆ S(ā′′ ) ⊆ S(ā), āā′′ ; āā′ = āā′ . Finally, for each name-type i, we can define a name-test arrow: eqi : Ai ⊗ Ai → N , { [ (a, a) 0] } ∪ { [ (a, b) 1] | a 6= b } , (4.8) which clearly makes the (N1) diagram (definition 3.12) commute. Fresh-name constructors. Combining the monad and comonads defined previously we can obtain a monadic-comonadic setting (Vt , ( )⊥ , Q), where Q denotes the family (Qā )ā∈A# . This setting, which in fact yields a sound model of the ν-calculus [2, 48], will be used as the basis of our semantics of nominal computation in the sequel. Nominal computation of type A, in name-environment ā and variable-environment Γ, will be translated into the set of strategies { σ : Qā JΓK → JAK ⊥ } . The lifting functor, representing the monadic part of our semantical setting, will therefore incorporate the computational effect of fresh-name creation. We describe in this section the semantical expression of fresh-name creation. Fresh names are created by means of natural transformations which transform a comonad Qā , say, to a monad-comonad composite (Qāa )⊥ . Definition 4.51. Consider the setting (Vt , ( )⊥ , Q). We define natural transformations newāa : Qā → (Qāa )⊥ by newāa ⊗idA st′ 1 ā āa −→ (Aāa ⊗A)⊥ , newāa A , A ⊗A −−−−−−−→ (A )⊥ ⊗A − ā āa a newāa 1 : A ⊗1 → (A ⊗1)⊥ , strat{ [ (ā, ∗) ∗ ∗ (āa, ∗) ] } , for each āa ∈ A# . N That new is a natural transformation is straightforward: for any f : A → B we can form the following commutative diagram. Aā ⊗A new1 ⊗id / (Aāa )⊥ ⊗A st′ / (Aāa ⊗A)⊥ id ⊗f id ⊗f Aā ⊗B (id ⊗f )⊥    new1 ⊗id / (Aāa )⊥ ⊗B st′ / (Aāa ⊗B)⊥ Moreover, we can show the following. Proposition 4.52. In the setting (Vt , ( )⊥ , Q) with new defined as above, the (N2) diagrams (definition 3.12) commute. 40 N. TZEVELEKOS The fresh-name constructor allows us to define name-abstraction on strategies by taking: newāa pu σ C ⊥ ^ a _ σ , Qā B −−−B→ (Qāa B)⊥ −→ C⊥ −−→ C. (4.9) Name-abstraction can be given an explicit description as follows. For any sequence of moves-with-names s and any name a # nlist(a), let sa be s with a in the head of all of its name-lists. Then, for σ as above, we can show that: viewf(^ a _ σ) = { [ (ā, iB ) iC jC mab̄ sa ] | [ (āa, iB ) iC jC mb̄ s] ∈ viewf(σ) ∧ a # iB , jC } (4.10) We end our discussion on fresh-name constructors with a technical lemma stating that name-abstraction and currying commute. Lemma 4.53. Let f : Qāa (A⊗B) → C with C a pointed arena. Then, ^ a _ Λ(ζ ′ ; f ) = Λ(ζ ′ ; ^ a _ f ) : Qā A → B −−⊗ C . Proof: As follows. ′ ^ a _ Λ(ζ ′ ; f ) = newāa A ;(Λ(ζ ; f ))⊥ ; puB C −−⊗ ′ ′ = newāa A ;(Λ(ζ ; f ))⊥ ; Λ(st ; ev ⊥ ; puC ) ′ ′ = Λ(newāa A ⊗idB ;(Λ(ζ ; f ))⊥ ⊗idB ; st ; ev ⊥ ; puC ) ′ ′ = Λ(newāa A ⊗idB ; st ;(Λ(ζ ; f )⊗idB )⊥ ; ev ⊥ ; puC ) (N2) ′ ′ ′ āa = Λ(newāa A ⊗idB ; st ;(ζ ; f )⊥ ; puC ) = Λ(ζ ; newA⊗B ; f⊥ ; puC ) and the latter equals Λ(ζ ′ ; ^ a _ f ). Note that the above result does not imply that ν- and λ-abstractions commute in our semantics of nominal languages, i.e. that we obtain identifications of the form Jνa.λx.M K = Jλx.νa.M K. As we will see in the sequel, λ-abstraction is not simply currying, because of the use of monads. 4.6. Nominal games à la Laird. As aforementioned, there have been two independent original presentations of nominal games, one due to Abramsky, Ghica, Murawski, Ong and Stark (AGMOS) [2] and another one due to Laird [21, 24]. Although Laird’s constructions are are not explicitly based on nominal sets (natural numbers are used instead of atoms), they constitute nominal constructions nonetheless. In this section we highlight the main differences between our nominal games, which follow AGMOS, and those of [21, 24]. Laird’s presentation concerns the ν-calculus with pointers, i.e. with references to names. The main difference in his presentation is in the treatment of name-introduction. In particular, a name does not appear in a play at the point of evaluation of its ν-constructor, but rather at the point of its first use; let us refer to this condition as name-frugality (cf. [31]). An immediate result is that strategies are no longer innocent, as otherwise e.g. νa.λx.a and λx.νa.a would have the same denotation.12 More importantly, name-frugality implies that strategies capture the examined nominal language more accurately: Opponent is not expected to guess names he is not supposed to know and thus, for example, the denotations of νa.skip and skip are identical. In our setting, Player is not frugal with his names 12Non-innocence can be seen as beneficial in terms of simplicity of the model, since strategies then have one condition less. On the other hand, though, innocent strategies are specified by means of their viewfunctions, which makes their presentation simpler. Moreover, non-innocence diminishes the power of definability results, as finitary behaviours are less expressive in absence of innocence. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 41 and therefore the two terms above are identified only at the extensional level (i.e. after quotienting).13 The major difference between [21] and [24] lies in the modelling of (ground-type, namestoring) store. In [21] the store is modelled by attaching to strategies a global, top-level (non-monadic), store arena. Then, a good-store-discipline is imposed on strategies via extra conditions on strategy composition which enforce that hidden store-moves follow the standard read/write pattern. As a result (and in contrast to our model), the model relies heavily on quotienting by the intrinsic preorder in order for the store to work properly. The added accuracy obtained by using frugality conditions is fully exploited in [24], where a carefully formulated setting of moves-with-store14 allows for an explicit characterisation result, that is, a semantic characterisation of operational equality at the intensional level. The contribution of using moves-with-store in that result is that thus the semantics is relieved from the (too revealing) internal workings of store: for example, terms like (a := b) ; λx. ! a ; 0 and (a := b) ; λx.0 are equated semantically at the intensional level, in contrast to what happens in our model.15 Note, though, that in a setting with higher-order store such that of νρ, moves-with-store would not be as simple since stores would need to store higher-order values, that is, strategies. Laird’s approach is therefore advantageous in its use of name-frugality conditions, which allow for more accurate models. At the same time, though, frugality conditions are an extra burden in constructing a model: apart from the fact that they need to be dynamically preserved in play-composition by garbage collection, they presuppose an appropriately defined notion of name-use. In [21, 24], a name is considered as used in a play if it is accessible through the store (in a reflexive transitive manner) from a name that has been explicitly played. This definition, however, does not directly apply to languages with different nominal effects (e.g. higher-order store). Moreover, frugality alone is not enough for languages like Reduced ML or the ν-calculus: a name may have been used in a play but may still be inaccessible to some participant (that is, if it is outside his view [31]). On the other hand, our approach is advantageous in its simplicity and its applicability on a wide rage of nominal effects (see [48]), but suffers from the accuracy issues discussed above. 5. The nominal games model We embark on the adventure of modelling νρ in a category of nominal arenas and strategies. Our starting point is the category Vt of nominal arenas and total strategies. Recall that Vt is constructed within the category Nom of nominal sets so, for each type A, we have an arena AA for references to type A. 13Note here, though, that the semantics being too explicit about the created names can prove beneficial: here we are able to give a particularly concise proof adequacy for νρ (see section 5.3 and compare e.g. with respective proof in [3]) by exploiting precisely this extra information! 14 Inter alia, frugality of names implies that sequences of moves-with-store have strong support even if stores are represented by sets! 15In our model they correspond to the strategies (see also section 5): σ1 , { [ (a, b) ∗ ⊛(∗, ⊛)(n, ⊛) a c 0] } , σ2 , { [ (a, b) ∗ ⊛(∗, ⊛)(n, ⊛) 0] } . Thus, the inner-workings of the store revealled by σ1 (i.e. the moves a c) differentiate it from σ2 . In fact, in our attempts to obtain an explicit characterisation result from our model, we found store-related innaccuracies to be the most stubborn ones. 42 N. TZEVELEKOS The semantics is monadic in a store monad built around a store arena ξ, and comonadic in an initial state comonad. The store monad is defined on top of the lifting monad (see definition 4.46) by use of a side-effect monad constructor, that is, T A , ξ −−⊗ (A ⊗ ξ)⊥ i.e. T A = ξ ⇒ A ⊗ ξ . Now, ξ contains the values assigned to each name (reference), and thus it is of the form O (AA ⇒ JAK) A∈TY where JAK is the translation of each type A. Thus, a recursive (wrt type-structure) definition of the type-translation is not possible because of the following cyclicity. JA → BK = JAK −−⊗ (ξ ⇒ JBK ⊗ ξ) O (SE) ξ= (AA ⇒ JAK) A Rather, both ξ and the type-translation have to be computed as the least solution to the above domain equation. By the way, observe that JA → BK = JAK ⊗ ξ ⇒ JBK ⊗ ξ . 5.1. Solving the Store Equation. The full form of the store equation (SE) is: J1K = 1 , JNK = N , J[A]K = AA , JA → BK = JAK ⊗JBK , N ξ = A(AA ⇒ JAK) . JA → BK = JAK −−⊗ (ξ ⇒ JBK ⊗ ξ) , This can be solved either as a fixpoint equation in the cpo of nominal arenas or as a domain equation in the PreCpo-enriched category Vt . We follow the latter approach, which provides the most general notion of canonical solution (and which incorporates the solution in the cpo of nominal arenas, analogously to [26]). It uses the categorical constructions of [43, 11] for solving recursive domain equations, as adapted to games in [26]. Definition 5.1. Define the category C , Vt × Y Vt A∈TY with objects D of the form (Dξ , DA A∈TY ) and arrows f of the form (fξ , fA A∈TY ). Now take F : (C)op ×C → C to be defined on objects by F (D, E) , (ξD,E , JAK D,E A∈TY ), where: JA × BK D,E , JAK D,E ⊗JBK D,E J[A]K D,E , AA J1K D,E , 1 N JNK D,E , N JA → BK D,E , DA −−⊗ (ξE,D ⇒ EB ⊗ξD,E ) ξD,E , A∈TY(AA ⇒ EA ) and similarly for arrows, with F (f, g) , (ξf,g , JAK f,g A∈TY ) . N Now (SE) has been reduced to: D = F (D, D) (SE∗ ) where F is a locally continuous functor wrt the strategy ordering (proposition 4.44), and continuous wrt the arena ordering (proposition 4.45). The solution to (SE∗ ) is given via a local bilimit construction to the following ω-chain in C.16 16Recall that we call an arrow e : A → B an embedding if there exists eR : B → A such that e ; eR = idA ∧ eR ; e ⊑ idB . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 43 Definition 5.2. In C form the sequence (Di )i∈ω taking D0 as below and Di+1 , F (Di , Di ). D0,N , N D0,1 , 1 D0,A→B , 1 D0,A×B , D0,A ⊗D0,B Moreover, define arrows ei : Di → Di+1 and e0 , inclD0 ,D1 D0,[A] , AA O D0,ξ , (AA ⇒ 0) eR 0 eR i A : Di+1 → Di as: ei+1 , F (eR i , ei ) , projD1 ,D0 R eR i+1 , F (ei , ei ) . N The above inclusion and projection arrows are defined componentwise. In fact, there is a hidden lemma here which allows us to define the projection arrow, namely that D0 E1 D1 (which means D0,ξ E1 D1,ξ and D0,A E1 D1,A for all A). (∆) e0 D0 / D1 e1 / D2 e2 / D3 e3 / ··· Thus, we have formed the ω-chain ∆. We show that ∆ is a E-increasing sequence of objects and embeddings, and proceed to the main result. Lemma 5.3. For (ei , eR i )i∈ω as above and any i ∈ ω, ei = inclDi ,Di+1 ∧ eR i = projDi+1 ,Di . Proof: It is easy to see that Di E1 Di+1 , all i ∈ ω, so the above are well-defined. We now do induction on i; the base case is true by definition. The inductive step follows easily from proposition 4.45. Theorem 5.4. We obtain a local bilimit (D∗ , ηi i∈ω ) for ∆ by taking: G D∗ , Di , ηi , inclDi ,D∗ (each i ∈ ω). i Hence, idD∗ : F (D ∗ , D∗ ) → D ∗ is a minimal invariant for F . Proof: First, note that D0 E1 Di , for all i ∈ ω, implies that all Di ’s share the same initial moves, and hence Di E1 D ∗ . Thus, for each i ∈ ω, we can define ηiR , projD∗ ,Di , and hence each ηi is an embedding. We now need to show the following. (1) (D∗ , ηi i∈ω ) is a cone for ∆, R ;η (2) for all i ∈ ω, ηiR ; ηi ⊑ ηi+1 i+1 , F R ∗ (3) i∈ω (ηi ; ηi ) = idD . ∗ , which follows from For 1, we nts that, for any i, inclD1 ,D∗ = inclDi ,Di+1 ; inclDi+1 ,DS (TRN). For 2 we essentially nts that idDi ⊆ idDi+1 , and for 3 that i idDi = idD∗ ; these are both straightforward. From the local bilimit (D∗ , ηi i∈ω ) we obtain a minimal invariant α : F (D∗ , D∗ ) → D∗ by taking (see e.g. [1]): G prop. 4.45 α , αi , αi , F (ηi , ηiR ) ; ηi+1 = projF (D∗ ,D∗ ),Di+1 ; inclDi+1 ,D∗ . i D∗ R ;η Moreover, = F (D∗ , D∗ ) by the Tarski-Knaster theorem, and therefore αi = ηi+1 i+1 , which implies α = idD∗ . Given an ω-chain ∆ = (Di , ei )i∈ω of objects and embeddings, a cone for ∆ is an object D together with a family (ηi : Di → D)i∈ω of embeddings such that, for all i ∈ ω, ηi = ei ; ηi+1 . Such a cone is a local bilimit for ∆ if, for all i ∈ ω, G R (ηiR ; ηi ) = idD . ηiR ; ηi ⊑ ηi+1 ; ηi+1 ∧ i∈ω 44 N. TZEVELEKOS Thus, D ∗ is the canonical solution to D = F (D, D), and in particular it solves: O DA→B = DA −−⊗ (Dξ ⇒ DB ⊗Dξ ) , Dξ = (AA ⇒ DA ) . A Definition 5.5. Taking D∗ as in the previous theorem define, for each type A, ξ , Dξ∗ , ∗ JAK , DA . N The arena ξ and the translation of compound types N are given explicitly in the following figure. ξ is depicted by means of unfolding it to A(AA ⇒ JAK) : it consists of an initial move ⊛ which justifies each name-question a ∈ AA , all types A, with the answer to the latter being the denotation of A (and modelling the stored value of a). Note that we reserve the symbol “⊛” for the initial move of ξ. ⊛-moves in type-translations can be seen as opening a new store. ξ ⊛ a PA JA × BK (iJAK , iJBK ) PA OQ (a ∈ AA ) JAK JAK − JBK JA → BK ∗ (iJAK , ⊛) (iJBK , ⊛) − JAK − PA OQ PA ξ− JBK − ξ− Figure 4: The store arena and the type translation. The store monad T . There is a standard construction (v. [28]) for defining a monad of Aside-effects (any object A) starting from a given strong monad with exponentials. Here we define a store monad, i.e. a ξ-side-effects monad, from the lifting monad as follows. T : C → C , ξ ⇒ ( ⊗ ξ)   up ηA : A → T A , Λ A ⊗ ξ −→ (A ⊗ ξ)⊥   (5.1) ev⊥ ev dn µA : T 2 A → T A , Λ T 2 A ⊗ ξ −→ (T A ⊗ ξ)⊥ −−→ (A ⊗ ξ)⊥⊥ −→ (A ⊗ ξ)⊥   id⊗ev st τA,B : A ⊗ T B → T (A ⊗ B) , Λ A ⊗ T B ⊗ ξ −−−−→ A ⊗ (B ⊗ ξ)⊥ −→ (A ⊗ B ⊗ ξ)⊥ A concrete description of the store monad is given in figure 5 (the diagrams of strategies depict their viewfunctions, as described in notation 4.34). For the particular case of ⊛moves which appear as second moves in T A’s, let us recall the convention we are following. Looking at the diagram for T A (figure 5), we see that ⊛ justifies a copy of ξ − (left) and a copy of A⊗ξ (right). Thus, a copycat link connecting to the lower-left of a ⊛ expresses a copycat concerning the ξ − justified by ⊛ (e.g. the link between the first two ⊛-moves in the diagram for µA ), and similarly for copycat links connecting to the lower-right of a ⊛. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES ηA : A iA TA ∗ ⊛ ∗ ⊛ OQ (iA , ⊛) PA OQ ∗ PA ⊛ OQ (iA , ⊛) ξ OQ PA / TB Tf : TA ∗ / TA PQ (iA , ⊛) OA (iB , ⊛) f A ξ− µA : T 2 A ∗ τA,B : A⊗T B / TA OQ ∗ (iA , ∗) (∗, ⊛) ⊛ OQ ∗ ⊛ OQ ⊛ PA / T (A⊗B) PA ⊛ PA OQ ⊛ PA − − 45 PA OQ PQ ⊛ PQ OA (iB , ⊛) OA PQ (iA , iB , ⊛) PA Figure 5: The store monad. Thus, for example, µA is given by: µA = strat( { [∗ ∗ ⊛ ⊛ s] | [⊛ ⊛ s] ∈ viewf(idξ ) } ∪ { [∗ ∗ ⊛ ⊛ (∗, ⊛′ ) ⊛′ s] | [⊛′ ⊛′ s] ∈ viewf(idξ ) ∨ [s] ∈ viewf(idA⊗ξ ) } ) . A consequence of lifting being a strong monad with exponentials is that the store monad is also a strong monad with exponentials. T -exponentials are given by: T B A , A −−⊗ T B , ΛT (f : A⊗B → T C) , Λ(f ) . (5.2) Moreover, for each arena A we can define an arrow: (ηA )⊥ pu TA αA , A⊥ −−−−→ (T A)⊥ −−− → TA. (5.3) The transformation pu was introduced in (4.7). Using lemma 4.48 we obtain αA = Λ(st′A,ξ ). Moreover, we can show that α : ( )⊥ → T is a monad morphism. 5.2. Obtaining the νρ-model. Let us recapitulate the structure that we have constructed thus far to the effect of obtaining a νρ-model in Vt . Our numbering below follows that of definition 3.12. I. Vt is a category with finite products (proposition 4.39). II. The store monad T is a strong monad with exponentials. III. Vt contains adequate structure for numerals. IV. There is a family (Qā , ε, δ, ζ)ā∈A# of product comonads, with each Qā having basis Aā (see section 4.5), which fulfils specifications (a,b). There are also fresh-name constructors, newāa : Qā → (Qāa )⊥ , which satisfy (N2). 46 N. TZEVELEKOS V. There are name-equality arrows, eqA for each type A, making the (N1) diagram commute (section 4.5). From new we can obtain a fresh-name transformation for the store monad. Definition 5.6. For each āa ∈ A# , define a natural transformation nuāa : Qā → T Qāa by: new αQāa A A ā nuāa −−→ (Qāa A)⊥ −−−−→ T Qāa A . A , Q A− nu Tf µ B A T Qāa A −−→ T 2 B −−→ TB . N Moreover, for each f : Qāa A → T B, take ^ a _ f , Qā A −−→ Each arrow nuāa A is explicitly given by (note we use the same conventions as in (4.10)): a a nuāa A = strat{ [(ā, iA ) ∗ ⊛ (āa, iA , ⊛) s ] | a # iA ∧ ([iA iA s] ∈ viewf(idA ) ∨ [⊛ ⊛ s] ∈ viewf(idξ )) } and diagrammatically as in figure 6. Moreover, using the fact that α is a monad morphism and lemma 4.48 we can show that, in fact, ^ a _ f is given exactly as in (4.9), that is, ^ a _ f = newA ; f⊥ ; puT B . Finally, α being is a monad morphism implies also the following. Proposition 5.7. The nu transformation satisfies the (N2) diagrams of definition 3.12. What we are only missing for a νρ-model is update and dereferencing maps. Definition 5.8. For any type A we define the following arrows in Vt , drfA , strat{ [ a ∗ ⊛ a iJAK (iJAK , ⊛) s ] | [⊛ ⊛ s] ∈ viewf(idξ ) ∨ [iJAK iJAK s] ∈ viewf(idJAK ) } , updA , strat { [ (a, iJAK ) ∗ ⊛ b b s ] | [⊛ ⊛ b b s] ∈ viewf(idξ ) ∧ b#a } depicted also in figure 6.
∪ { [ (a, iJAK ) ∗ ⊛ a iJAK s ] | [iJAK iJAK s] ∈ viewf(idJAK ) } , N These strategies work as follows. updA responds with the answer (∗, ⊛) to the initial sequence (a, iJAK ) ∗ ⊛ and then: • for any name b # a that is asked by O to (∗, ⊛) (which is a store-opening move), it copies b under the store ⊛ (opened by O) and establishes a copycat link between the two b’s; • if O asks a to (∗, ⊛), it answers iJAK and establishes a copycat link between the two iJAK ’s. On the other hand, drfA does not immediately answer to the initial sequence a ∗ ⊛ but rather asks (the value of) a to ⊛. Upon receiving O’s answer iJAK , it answers (iJAK , ⊛) and establishes two copycat links. We can show by direct computation the following. Proposition 5.9. The (NR) and (SNR) diagrams of definition 3.12 commute. We have therefore established the following. Theorem 5.10. (Vt , T, Q) is a νρ-model. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES drfA : AA updA : AA ⊗JAK / T1 47 / T JAK a (a, iJAK ) OQ ∗ OQ ∗ ⊛ PA ⊛ (∗, ⊛) PA OQ OQ a PQ PA iJAK OA (iJAK , ⊛) b PA OQ b PQ ā nuāa A : Q A / T Qāa A (ā, iA ) a OQ iJAK PA OQ ∗ PA ⊛ OQ (āa, iA , ⊛) a PA Figure 6: Strategies for update, dereferencing and fresh-name creation. We close this section with a discussion on how the store-effect is achieved in our innocent setting, and with some examples of translations of νρ-terms in Vt . Remark 5.11 (Innocent store). The approach to the modelling of store which we have presented differs fundamentally from previous such approaches in game semantics. Those approaches, be they for basic or higher-order store [6, 3], are based on the following methodology. References are modelled by read/write product types, and fresh-reference creation is modelled by a “cell” strategy which creates the fresh cell and imposes a good read/write discipline on it. In order for a cell to be able to return the last stored value, innocence has to be broken since each read-request hides previous write-requests from the P-view. Higher-order cells have to also break visibility in order to establish copycat links between read- and write-requests. Here instead we have only P – What’s the value of a? used innocent strategies and a O – I don’t know, you tell me: what’s the value of a? monad on a store ξ. Because of P – I don’t know, you tell me: what’s the value of a? .. the monad, an arena JAK con. tains several copies of ξ, thereO – I don’t know, you tell me: what’s the value of a? fore several stores are opened P – I know it, it is v. inside a play. The read/ write .. . discipline is then kept in an inO – I know it, it is v. teractive way: when a particiP – I know it, it is v. pant asks (the value of) a name O – I know it, it is v. a at the last (relevant) store,17 Figure 7: A dialogue in innocent store. 17i.e. at the last store-opening move played by the other participant. 48 N. TZEVELEKOS the other participant either answers with a value or asks himself a at the penultimate store, and so on until one of the participants answers or the first store in the play is reached. At each step, a participant answers the question a only if he updated the value of a before opening the current store (of that step, i.e. the last store in the participant’s view) — note that this behaviour does not break innocence. If no such update was made by the participant then he simply passes a to the previous store and establishes a copycat link between the two a’s. These links ensure that when an answer is eventually obtained then it will be copycatted all the way to answer the original question a. Thus, we innocently obtain a read/write discipline: at each question a, the last update of a is returned. Example 5.12. Consider the typed terms: ǫ | ∅ |− νa.a := hfst ! a, snd ! ai , b | ∅ |− b := λx.(! b)skip , b | ∅ |− (! b)skip with a ∈ AN×N and b ∈ A1→B . Their translations in Vt are as follows. A1→B / T1 1 ∗ OQ ∗ b OQ ∗ PA ⊛ OQ aa (n, n ) (∗, ⊛) c PQ PA OQ c a (l, l′ ) (∗, ⊛) b PA OQ ∗ b a (∗, ⊛) OQ ba PQ aa OQ a (n, l′ ) OQ ∗ ⊛ PA OQ b PQ ∗ OA (∗, ⊛) PQ PQ OA a b OQ OA aa / T JBK PA ⊛ PQ ′ a A1→B / T1 b PA (iB , ⊛) (iB , ⊛) OA PA OQ PQ ∗ OA (∗, ⊛) PQ PA In the first example we see that, although the strategy is looking up the fresh (and therefore uninitialised) reference a, the play does not deadlock: if Opponent answered the question aa then the play would proceed as depicted. In practice, however, Opponent will never be able to answer that question and the play will halt indeed (this is because Opponent must play tidily, see section 5.4). Moreover, from the latter two examples we can compute JstopB K : 1 → T JBK = { [∗ ∗ ⊛] } . 5.3. Adequacy. We proceed to show that Vt is adequate (v. definition 3.18). First we characterise non-reducing terms as follows. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 49 Lemma 5.13. Let ā | ∅ |− M : A be a typed term. M is a value iff there exists a store S such that S |= M has no reducts and [(ā, ∗) ∗ ⊛ (iA , ⊛)b̄ ] ∈ JS̄ ; M K , for some iA , b̄. Proof: The “only if”-part is straightforward. For the “if”-part assume that M is a non-value and take any S such that S |= M has no reducts. We show by induction on M that there exist no iA , b̄ such that [(ā, ∗) ∗ ⊛ (iA , ⊛)b̄ ] ∈ JS̄ ; M K. The base case follows trivially from M not being a value. Now, for the inductive step, the specifications of S |= M (and M ) imply that either M ≡ ! a with a not having a value in S, or M ≡ E[K] with E an evaluation context and K a non-value typed as ā | ∅ |− K : B and such that S |= K non-reducing. In case of M ≡ ! a, we have that [(ā, ∗) ∗ ⊛ a] ∈ JS̄ ; M K, which proves the claim because of determinacy. On the other hand, if M ≡ E[K] then, as in proof of proposition 3.17, we have: JS̄ ; M K = hΛ(ζ ′ ; JE[x]K), JS̄ ; KKi ; τ ; T ev ; µ = hid, JS̄ ; KKi ; τ ; T (ζ ′ ; JE[x]K) ; µ By IH, there are no iB , c̄ such that [(ā, ∗) ∗ ⊛ (iB , ⊛)c̄ ] ∈ JS̄ ; KK, which implies that there are no iA , b̄ such that [(ā, ∗) ∗ ⊛ (iA , ⊛)b̄ ] ∈ JS̄ ; M K. Because of the previous result, in order to show adequacy it suffices to show that, whenever JM K = ^ b̄ _ JS̄ ; 0̃K, there is no infinite reduction sequence starting from ā |= M . We will carry out the following reasoning. • Firstly, since the calculus without DRF reductions is strongly normalising — this is inherited from strong normalisation of the ν-calculus — it suffices to show there is no reduction sequence starting from ā |= M and containing infinitely many DRF reduction steps. • In fact, the problem can be further reduced to showing that, whenever [(ā, ∗)∗⊛ (0, ⊛)b̄ ] ∈ JM K, there is no reduction sequence starting from ā |= M and containing infinitely many NEW reduction steps. The latter clearly holds, since M cannot create more than |b̄| fresh names in that case, because of correctness. The reduction to this simpler problem is achieved as follows. For each term M , we construct a term M ′ by adding immediately before each dereferencing in M a freshname construction. The result is that, whenever there is a sequence with infinitely many DRF’s starting from S |= M , there is a sequence with infinitely many NEW’s starting from S |= M ′ . The reduction is completed by finally showing that, whenever we have ′ [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] ∈ JM K, we also have [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] ∈ JM ′ K. The crucial step in the proof is the reduction to “the simpler problem”, and particularly showing the connection between JM K and JM ′ K described above. The latter is carried out by using the observational equivalence relation on strategies, defined later in this section. Note, though, that a direct proof can also be given (see [48]). Proposition 5.14 (Adequacy). (Vt , T, Q) is adequate. Proof: This follows from O-adequacy (lemma 5.28), which is proved independently. Hence, (Vt , T, Q) is a sound model for νρ and thus, for all terms M, N , JM K = JN K =⇒ M / N . 50 N. TZEVELEKOS 5.4. Tidy strategies. Leaving adequacy behind, the route for obtaining a fully abstract model of νρ proceeds to definability. That is, we aim for a model in which elements with finite descriptions correspond to translations of νρ-terms. However, Vt does not satisfy such a requirement: it includes (finitary) store-related behaviours that are disallowed in the operational semantics of νρ. In fact, our strategies treat the store ξ like any other arena, while in νρ the treatment of store follows some basic guidelines. For example, if a store S is updated to S ′ then the original store S is not accessible any more (irreversibility). In strategies we do not have such a condition: in a play there may be several ξ’s opened, yet there is no discipline on which of these are accessible to Player whenever he makes a move. Another condition involves the fact that a store either ‘knows’ the value of a name or it doesn’t know it. Hence, when a name is asked, the store either returns its value or it deadlocks: there is no third option. In a play, however, when Opponent asks the value of some name, Player is free to evade answering and play somewhere else! To disallow such behaviours we will constrain total strategies with further conditions, defining thus what we call tidy strategies. But first, let us specify store-related moves inside type-translating nominal arenas. Definition 5.15. Consider Vνρ , the full subcategory of Vt with objects given by: Ob(Vνρ ) ∋ A, B ::= 1 | N | Aā | A⊗B | A −−⊗ T B For each such arena A we define its set of store-Handles, HA , as follows. H1 = HN = HAā , ∅ , HA⊗B , HA ∪ HB , [ , {(iA , ⊛A ), (iB , ⊛B )} ∪ HA ∪ HB ∪ HξA ∪ HξB with Hξ , HJCK , C N where we write A −−⊗ T B as A −−⊗ (ξA ⇒ B ⊗ξB ), and ξ as C(AC ⇒ JCK). In an arena A ∈ Ob(Vνρ ), a store-Handle justifies (all) questions of the form a, which we call store-Questions. Answers to store-Questions are called store-Answers. N HA TB −−⊗ Note in particular that, for each type A, we have JAK, Qā JAK, T JAK ∈ Ob(Vνρ ), assuming that T JAK is equated with 1 −−⊗ T JAK. Note also there is a circularity in HA T B in the S i and, above definition. In fact, it is a definition by induction: we take HA , i∈ω HA −−⊗ 0 , ∅, H1i = HNi = HAi ā = HA i i i HA⊗B , HA ∪ HB , i+1 i i i+1 i+1 HA T B , {(iA , ⊛A ), (iB , ⊛B )} ∪ HA ∪ HB ∪ HξA ∪ HξB −−⊗ Intuitively, store-H’s are store-opening moves, while store-Q’s and store-A’s are obtained from unfolding the store structure. On the side we give examples of store-related moves in a simple arena. From now on we work in Vνρ , unless stated otherwise. A first property we can show is that a move is exclusively either initial or an element of the aforedefined move-classes. with Hξi+1 , [ C i HJCK . T 1 = ξ ⇒ 1⊗ξ ∗ ⊛ store-Q’s (∗, ⊛) a iA store-A’s store-H’s b iB Figure 8: Store-H’s -Q’s -A’s in arena T 1. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 51 Proposition 5.16. For any A ∈ Ob(Vνρ ), MA = IA ⊎ HA ⊎ { m ∈ MA | m a store-Q} ⊎ {m ∈ MA | m a store-A } . Proof: We show that any m ∈ MA belongs to exactly one of the above sets. We do induction on the level of m, l(m), inside A and on the size of A, |A|, specified by the inductive definition of Ob(Vνρ ). If m is initial then, by definition, it can’t be a store-H. Neither can it be a store-Q or store-A, as these moves presuppose non-initiality. Assume l(m) > 0. If A is base then trivial, while if A = A1 ⊗A2 then use the IH on (l(m), |A|). Now, if A = A1 −−⊗ T A2 then let us write A as A1 −−⊗ (ξ1 ⇒ A2 ⊗ξ2 ); we have the following cases. • If m = (iA1 , ⊛1 ) ∈ HA then m a question and not a store-Q, as store-Q’s are names. • If m = (iA2 , ⊛2 ) ∈ HA then m an answer and not a store-A as its justifier is (iA1 , ⊛1 ). • If m is in A1 or in A2 then use the IH. • If m is in ξ1 then it is either some store-Q a to (iA1 , ⊛1 ) (and hence not a store-H or store-A), or it is in some JCK. In the latter case, if m initial in JCK then a store-A in JAK and therefore not a store-H, as m not a store-H in JCK by IH (on l(m)). If m is non-initial in JCK then use the IH and the fact that store-H’s -Q’s -A’s of JCK are the same in JAK. • Similarly if m is in ξ2 . The notion of store-handles can be straightforwardly extended to prearenas. Definition 5.17. Let A, B ∈ Ob(Vνρ ). The set HA→B of store-handles in prearena A → B is HA ∪ HB . Store-Q’s and store-A’s are defined accordingly. N Using the previous proposition, we can see that, for any A and B, the set MA→B can be decomposed as: IA ⊎ IB ⊎ HA→B ⊎ { m ∈ MA→B | m a store-Q } ⊎ { m ∈ MA→B | m a store-A } (5.4) We proceed to define tidy strategies. We endorse the following notational convention. Since stores ξ may occur in several places inside a (pre)arena we may use parenthesised indices to distinguish identical moves from different stores. For example, the same store-question q may be occasionally denoted q(O) or q(P ) , the particular notation denoting the OP-polarity of the move. Moreover, by O-store-H’s we mean store-H’s played by Opponent, etc. Definition 5.18 (Tidy strategies). A total strategy σ is tidy if whenever odd-length [s] ∈ σ then: (TD1) If s ends in a store-Q q then [sx] ∈ σ , with x being either a store-A to q introducing no new names, or a copy of q. In particular, if q = aā with a # psq− then the latter case holds. (TD2) If [sq(P ) ] ∈ σ with q a store-Q then q(P ) is justified by last O-store-H in psq. (TD3) If psq = s′ q(O) q(P ) t y(O) with q a store-Q then [sy(P ) ] ∈ σ, where y(P ) is justified by psq .−3 . N (TD1) states that, whenever Opponent asks the value of a name, Player either immediately answers with its value or it copycats the question to the previous store-H. The former case corresponds to Player having updated the given name lastly (i.e. between the previous Ostore-H and the last one). The latter case corresponds to Player not having done so and hence asking its value to the previous store configuration, starting thus a copycat between the last and the previous store-H. Hence, the store is, in fact, composed by layers of stores 52 N. TZEVELEKOS — one on top of the other — and only when a name has not been updated in the top layer is Player allowed to search for it in layers underneath. We can say that this is the nominal games equivalent of a memory cell (cf. remark 5.11). (TD3) further guarantees the abovedescribed behaviour. It states that when Player starts a store-copycat then he must copycat the store-A and all following moves he receives, unless Opponent chooses to play elsewhere. (TD2) guarantees the multi-layer discipline in the store: Player can see one store at each time, namely the last played by Opponent in the P-view. The following straightforward result shows that (TD3), as stated, provides the intended copycat behaviour. Proposition 5.19. Let σ be a tidy strategy. If [s′ q(O) q(P ) t] ∈ σ is an even-length P-view and q is a store-Q then q(O) q(P ) t is a copycat. Proof: We do induction on |t|. The base case is straightforward. For the inductive step, let t = t′ xz. Then, by prefix closure, [s′ q(O) q(P ) t′ x] ∈ σ, this latter a P-view. By IH, q(O)q(P ) t′ is a copycat. Moreover, by (TD3), [s′ q(O) q(P ) t′ xx] ∈ σ with last x justified by (q(O) q(P ) t′ x).−3, thus s′ q(O) q(P ) t′ xx a copycat. Now, by determinacy, [s′ q(O) q(P ) t′ xx] = [s′ q(O) q(P ) t′ xz], so there exists π such that π ◦ x = x ∧ π ◦ x = z, ∴ x = z, as required. A good store discipline would guarantee that store-Handles OP-alternate in a play. This indeed happens in P-views played by tidy strategies. In fact, such P-views have canonical decompositions, as we show below. Proposition 5.20 (Tidy Discipline). Let σ : A → B be a tidy strategy and [s] ∈ σ with psq = s. Then, s is decomposed as in the following diagram. @ABC GFED iA HIJK ONML @ABC GFED CC o GFED / @ABC iB S-H HIJK / ONML X e XXXXXXXXXX mm6 OQ m m M XXXXX m m XXXXX mmm XXXXXX m m % mm XXXX, m S-H S-A S-H S-Q HIJK ONML HIJK ONML HIJK ONML HIJK ONML lXRXXXXX PQ hRR hRRR P PA hR l6 P l X RRR XXlXlXlX M RRR l l RRR L RRlRll XXXXXX R ll RRR XXXXlXlXllRlRRRR l RRR l RRR  l RR R vlll lll XXXXXXRXR S-A S-Q S-H HIJK ONML HIJK ONML HIJK ONML O OA O (by CC we mean the state that, when reached by a sequence s = psq, the rest of s is copycat.) Proof: The first two transitions are clear. After them neither P nor O can play initial moves, so all remaining moves in s are store-H -Q -A’s. Assume now O has just played a question x0 which is a store-H and the play continues with moves x1 x2 x3 ... . x1 cannot be a store-A, as this would not be justified by x0 , breaching well-bracketing. If x1 is a store-Q then x2 must be a store-A, by P-view. If x1 is an answer-store-H then x2 is an OQ, while if x1 a question-store-H then x2 is either a store-Q or a store-H. If x2 is a store-Q then, by (TD1), x3 either a store-A or a store-Q, the latter case meaning transition to the CC state. If x2 is not a store-Q then x3 can’t be a store-A: if x3 were a store-A justified by q 6= x2 then, as q wouldn’t have been immediately answered, s≥q would be a copycat and therefore we would be in the CC state right after playing q. Finally, if x3 is a store-A then x4 must be justified by it, so it must be a Q-store-H. Corollary 5.21 (Good Store Discipline). Let [s] ∈ σ with σ tidy and psq = s. Then: • The subsequence of s containing its store-H’s is OP-alternating and O-starting. • If s.−1 = q is a P-store-Q then either q is justified by last store-H in s, or s is in copycat mode at q. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 53 Observe that strategies that mostly do copycats are tidy; in particular, identities are tidy. Moreover, tidy strategies are closed under composition (proof delegated to the appendix). Proposition 5.22. If σ : A → B and τ : B → C are tidy strategies then so is σ ; τ . N Definition 5.23. T is the lluf subcategory of Vνρ of tidy strategies. Finally, we need to check that all structure required for a sound νρ-model pass from Vt to T . It is not difficult to see that all such structure which does not handle the store remains safely within the tidy universe. On the other hand, strategies for update and dereferencing are tidy by construction. (A fully formal proof is given in [48].) Proposition 5.24. T forms an adequate νρ-model by inheriting all the necessary structure from Vt . Henceforth, by strategies we shall mean tidy strategies, unless stated otherwise. 5.5. Observationality. Strategy equality is too fine grained to capture contextual equivalence in a complete manner. For example, even simple contextual equivalences like skip ≅ νa.skip are not preserved by the semantical translation, since strategies include in their name-lists all introduced names, even useless ones. For similar reasons, equivalences like νa.νb.M ≅ νb.νa.M are not valid semantically. In fact, it is not only because of the treatment of name-creation that the semantics is not complete. Terms like a := 1 ; λx. ! a ; 2 ≅ a := 1 ; λx.2 are distinguished because of the ‘explicit’ way in which the store works. So there are many ways in which our semantics is too expressive for our language. We therefore proceed to a quotienting by the intrinsic preorder and prove full-abstraction in the extensional model. Following the steps described in section 3.2, in this section we introduce the intrinsic preorder on T and show that the resulting model is observational. Full-abstraction is then shown in the following section. Definition 5.25. Expand T to (T , T, Q, O) by setting, for each ā ∈ A# , Oā , { f ∈ T (Qā 1, T N) | ∃b̄. [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] ∈ f } . Then, for each f, g ∈ T (Qā A, T B), f .ā g if ∀ρ : Qā (A −−⊗ T B) → T N. (Λā (f ) ; ρ ∈ Oā =⇒ Λā (g) ; ρ ∈ Oā ) . Thus, the observability predicate O is a family (Oā )ā∈A# , and the intrinsic ā family (.ā )ā∈A# . Recall that by Λā (f ) we mean ΛQ ,T (f ), that is, δ N preorder . is a Qā Λ(ζ ′ ; f ) Λā (f ) = Qā 1 − → Qā Qā 1 − −−−−−− → Qā (A −−⊗ T B) . Note in particular that f ⊑ g implies Λā (f ) ; ρ ⊑ Λā (g) ; ρ, for any relevant ρ, and therefore: f ⊑ g =⇒ f .ā g (5.5) The intrinsic preorder is defined by use of test arrows ρ, which stand for possible program contexts. As the following result shows, not all such tests are necessary. 54 N. TZEVELEKOS Lemma 5.26 (tl4 tests suffice). Let f, g ∈ T (Qā 1, B) with B pointed. The following are equivalent (recall definition 4.35). I. ∀ρ : Qā B → T N. δ ; Qā f ; ρ ∈ Oā =⇒ δ ; Qā g ; ρ ∈ Oā II. ∀ρ : Qā B → T N. ρ is tl4 =⇒ (δ ; Qā f ; ρ ∈ Oā =⇒ δ ; Qā g ; ρ ∈ Oā ) Hence, for each ā and f, g ∈ T (Qā A, T B), f .ā g iff ∀ρ : Qā (A −−⊗ T B) → T N. ρ is tl4 =⇒ (Λā (f ) ; ρ ∈ Oā =⇒ Λā (g) ; ρ ∈ Oā ) . Proof: I ⇒ II is trivial. Now assume II holds and let ρ : Qā B → T N be any strategy such that δ ; Qā f ; ρ ∈ Oā . Then, there exist [s] ∈ δ ; Qā f and [t] ∈ ρ such that [s ; t] = [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] ∈ (δ ; Qā f ) ; ρ. We show by induction on the number of JB -moves appearing in s k t that δ ; Qā g ; ρ ∈ Oā . If no such moves appear then t = (ā, iB ) ∗ ⊛ (0, ⊛)b̄ , so done. If n + 1 such moves appear then ρ is necessarily t4, as B is pointed, so by lemma 4.36 there exists tl4* strategy ρ̃ such that ρ = ∆ ; ρ̃. It is not difficult to see that ρ being tidy implies that ρ̃ is tidy. Moreover, δ ; Qā f ; ρ = δ ; Qā f ; ∆ ; ρ̃ = δ ; Qā f ;hid, Qā ! ; δ ; Qā f i ; ρ̃ = δ ; Qā f ; ρ′ , with ρ′ being hid, Qā ! ; δ ; Qā f i ; ρ̃. Now, by definition of ρ̃, [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] = [s′ ; t′ ] ∈ δ ; Qā f ; ρ′ with s′ k t′ containing n JB -moves so, by IH, δ ; Qā g ; ρ′ ∈ Oā . But δ ; Qā g ; ρ′ = δ ; Qā g ;hid, Qā ! ; δ ; Qā f i ; ρ̃ = δ ; Qā f ;hQā ! ; δ ; Qā g, idi ; ρ̃ = δ ; Qā f ; ρ′′ , where ρ′′ is given by hQā ! ; δ ; Qā g, idi ; ρ̃. But ρ′′ is tl4, thus, by hypothesis, Oā ∋ δ ; Qā g ; ρ′′ = δ ; Qā g ; ρ , as required. We can now prove the second half of observationality. Lemma 5.27. For any morphism f : Qāa 1 → B, with B pointed, and any tl4 morphism ρ : Qā B → T N, āa δ ; Qā ^ a _ f ; ρ ∈ Oā ⇐⇒ δ ; Qāa f ; āa ā ; ρ ∈ O Moreover, for each ā and relevant a, ā′ , f, g, f .āa g =⇒ ^ a _ f .ā ^ a _ g , f .ā g =⇒ ā′ ā ′ ; f .ā ā′ ā ;g. Proof: For the first part, ρ being tl4 and B being pointed imply that there exists some b̄ # ā and a ttotal strategy ρ′ such that ρ = ^ b̄ _ ρ′ . Now let δ ; Qā ^ a _ f ; ρ ∈ Oā , so there ¯ exists [s ; t] = [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ac̄ ] ∈ (δ ; Qā ^ a _ f ) ; ρ, and let s = (ā, ∗) (ā, iB ) jB mad s′ and ¯ b̄ t′ . Letting sra be snlist(s)ra , we can see that [(āa, ∗) i j md s′ra ] ∈ f and t = (ā, iB ) ∗ ⊛ jB B B ¯ ′ra āa ′′ ′′ d āa thus [s ] , [(āa, ∗) (ā, iB ) jB m s ] ∈ δ ; Q f ; ā . Hence, [s ; t] = [(āa, ∗) ∗ ⊛ (0, ⊛)b̄c̄ ] ∈ δ ; Qāa f ; āa ā ; ρ, as required. The converse is shown similarly. For the second part, suppose f .āa g : Qāa A → T B and take any tl4 morphism ρ : Qā (A −−⊗ T B) → T N. Then, lem 4.53 Λā (^ a _ f ) ; ρ ∈ Oā ⇐⇒ δ ; Qā Λ(ζ ′ ; ^ a _ f ) ; ρ ∈ Oā ⇐⇒ δ ; Qā ^ a _ (Λ(ζ ′ ; f )) ; ρ ∈ Oā āa ⇐⇒ δ ; Qāa Λ(ζ ′ ; f ) ; āa ā ; ρ ∈ O f .āa g āa ⇐⇒ Λā (^ a _ g) ; ρ ∈ Oā . =⇒ δ ; Qāa Λ(ζ ′ ; g) ; āa ā ; ρ ∈ O For the other claim, let us generalise the fresh-name constructors new to:   ā ′ ′ : Aā → (Aā )⊥ , { [ (ā, ∗) ∗ ∗ (ā′ , ∗)ā r ā ] } ā′ FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 55 ′ for any S(ā) ⊆ S(ā′ ). The above yields a natural transformation of type Qā → Qā⊥ . It is
′ ′ easy to see that, for any h : Qā 1 → T N, h ∈ Oā iff āā′ ; h⊥ ; pu ∈ Oā and, moreover, that the diagram on the right below commutes. Hence, if f .ā g then ′ ′ ′ δ ; Qā Λ(ζ ′ ; āā ; f ) ; ρ ∈ Oā ⇐⇒ ⇐⇒ ⇐⇒ f .ā g ′ ′ ′ ′ δ ; Qā āā ; Qā Λ(ζ ′ ; f ) ; ρ ∈ Oā ā
ā′ ā′ ′ ā′ ā′ ;(δ ; Q ā ; Q Λ(ζ ; f ) ; ρ)⊥ ; pu
δ ; Qā Λ(ζ ′ ; f ) ; āā′ ; ρ⊥ ; pu ∈ Oā ā ′ =⇒ δ ; Q Λ(ζ ; g) ; ā′ ⇐⇒ δ ; Q Λ(ζ ′ ; ā′ ā ā
ā′ ; ρ⊥ ; pu ∈ O ∈ Oā ā Aā ā ā′
ā
h ā′ ,idi / (Aā′ ) ⊗ Aā ⊥ st′  ′ (Aā )⊥ ā′ ; g) ; ρ ∈ O , ā′ hid, ā i⊥  / (Aā′ ⊗ Aā ) ⊥ as required. In order to prove that T is observational, we are only left to show that JM K ∈ Oā ⇐⇒ ∃b̄, S. JM K = ^ b̄ _ JS̄ ; 0K for any ā | ∅ |− M : N. The “⇐=” direction is trivial. For the converse, because of correctness, it suffices to show the following generalisation of adequacy. Lemma 5.28 (O-Adequacy). Let ā | ∅ |− M : N be a typed term. If JM K ∈ Oǫ then there exists some S such that ā |= M −→ → S |= 0. Proof: The idea behind the proof is given above proposition 5.14. It suffices to show that, for any such M , there is a non-reducing sequent S |= N such that ā |= M −→ → S |= N ; therefore, because of Strong Normalisation in the ν-calculus, it suffices to show that there is no infinite reduction sequence starting from ā |= M and containing infinitely many DRF reduction steps. To show the latter we will use an operation on terms adding new-name constructors just before dereferencings. The operation yields, for each term M , a term (M )◦ the semantics of which is equivalent to that of M . On the other hand, ā |= (M )◦ cannot perform infinitely many DRF reduction steps without creating infinitely many new names. For each term M , define (M )◦ by induction as: (a)◦ , a , (! N )◦ (x)◦ , x , ... (λx.M )◦ , λx.(M )◦ , (M N )◦ , (M )◦ (N )◦ , ... νa. !(N )◦ , , some a # N . We show that J(M )◦ K ⋍ JM K, by induction on M ; the base cases are trivial. The induction step follows immediately from the IH and the fact that ⋍ is a congruence, in all ◦ cases except for M being ! N . In the latter case we have that J(M )◦ K = ^ a _ ( āa ā ; J!(N ) K) , ◦ ā while the IH implies that JM K ⋍ J!(N ) K. Hence, it sts that for each f : Q A → T B we have f ⋍ ^ a _ ( āa ā ; f ) . Indeed, for any relevant ρ which is tl4, and lem 5.27 ā āa āa ⇐⇒ δ ; Qāa Λ(ζ ′ ; āa Λā (^ a _ ( āa ā ; f )) ; ρ ∈ O ā ; f ) ; ā ; ρ ∈ O ā ′ āa āa ⇐⇒ δ ; Qāa āa ā ; ā ; Q Λ(ζ ; f ) ; ρ ∈ O ⇐⇒ āa ā ; Λā (f ) ; ρ ∈ Oāa ⇐⇒ Λā (f ) ; ρ ∈ Oā . Now, take any ā | ∅ |− M : N and assume JM K ∈ Oā , and that ā |= M diverges using infinitely many DRF reduction steps. Then, ā |= (M )◦ diverges using infinitely many NEW reduction steps. However, since J(M )◦ K ⋍ JM K, we have J(M )◦ K ∈ Oā and therefore 56 N. TZEVELEKOS [(ā, ∗) ∗ ⊛ (0̃, ⊛)b̄ ] ∈ J(M )◦ K for some b̄. However, ā |= (M )◦ reduces to some S |= M ′ using |b̄|+ 1 NEW reduction steps, so J(M )◦ K = ^ c̄ _ JS̄ ; M ′ K with |c̄| = |b̄|+ 1, to determinacy. We have therefore shown observationality. Proposition 5.29 (Observationality). (T , T, Q, O) is observational. 5.6. Definability and full-abstraction. We now proceed to show definability for T , and through it ip-definability. According to the results of section 3.2.3, this will suffice for full abstraction. We first make precise the notion of finitary strategy, that is, of (tidy) strategy with finite description, by introducing truncation functions that remove inessential branches from a strategy’s description. Definition 5.30. Let σ : A → B in T and let [s] ∈ viewf(σ) be of even length. Define trunc(s) and trunc′ (s) by induction as follows. trunc(ǫ) = trunc′ (ǫ) , ǫ ( ǫ , if x = y are store-Q’s trunc(x(O) y(P ) s′ ) , ′ xy trunc(s ) , o.w.  ǫ , if x = y are store-Q’s    ǫ , if x store-Q , y a store-A and s′ = ǫ trunc′ (x(O) y(P ) s′ ) ,  ǫ , if x ∈ IA , y ∈ IB and s′ = ǫ    xy trunc′ (s′ ) , o.w. Moreover, say σ is finitary if trunc(σ) is finite, where trunc(σ) , { [trunc(s)] | [s] ∈ viewf(σ) ∧ |s| > 3 } . Finally, for any [t] ∈ σ define: σ≤t , strat{ [s] ∈ viewf(σ) | ∃ t′ ≤ t. trunc′ (s) = pt′q } . N Hence, finitary are those strategies whose viewfunctions become finite if we delete all the store-copycats and all default initial answers — the latter dictated by totality. Moreover, the strategy σ≤t is the strategy we are left with if we truncate viewf(σ) by removing all its branches of size greater than 3 that are not contained in t, except for the store-copycats which are left intact and for the store-A’s branches which are truncated to the point of leaving solely the store-A, so that we retain tidiness. Note that, in general, trunc′ (s) ≤ trunc(s) ≤ s. We can then show the following (proof in [48]). Proposition 5.31. If σ is a strategy and [t] ∈ σ is even-length then σ≤t is a finitary strategy with [t] ∈ σ≤t and σ≤t ⊑ σ. FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 57 We proceed to show definability. The proof is facilitated by the following lemma, the proof of which is delegated to the appendix. Note that for economy we define strategies by means of their viewfunctions modulo totality and even-prefix closure. Moreover, we write σ ↾ i for the (total) restriction of a strategy σ to an initial move i, and srb̄ for s with b̄ removed from all of its name-lists. Lemma 5.32 (Decomposition Lemma). Let σ : Qā JAK → T JBK be a strategy. We can decompose σ as follows. 1. If there exists an iA(0) such that ∃x0 . [(ā, iA(0) ) ∗ ⊛x0 ] ∈ σ then Qā JAK MMM ā ′ Mh[x MMM=iA(0) ],hσ0 ,σ ii MMM &  T JBK o N ⊗(T JBK)2 σ cnd where: ā [x = iA(0) ] : Qā JAK → N , { [(ā, iA(0) ) 0 ]} ∪ { [ (ā, iA ) 1 ] | [ (ā, iA ) ] 6= [ (ā, iA(0) ) ] } , ā σ0 : Q JAK → T JBK , strat{ [ (ā, iA(0) ) s ] ∈ viewf(σ) } , σ ′ : Qā JAK → T JBK , strat{ [ (ā, iA ) s ] ∈ viewf(σ) | [ (ā, iA ) ] 6= [ (ā, iA(0) ) ] } . 2. If there exists iA(0) such that ∀iA . (∃x0 . [(ā, iA ) ∗ ⊛ x0 ] ∈ σ) ⇐⇒ [(ā, iA )] = [(ā, iA(0) )] , then σ = ^ b̄ _ σb̄ where: σb̄ : Qāb̄ JAK → T JBK , strat{ [ (āb̄, iA(0) ) ∗ ⊛ m0 srb̄ ] | [ (ā, iA(0) ) ∗ ⊛ mb̄0 s ] ∈ viewf(σ) } . 3. If there exist iA(0) , m0 such that ∀iA , x. [(ā, iA ) ∗ ⊛ x] ∈ σ ⇐⇒ [(ā, iA ) x] = [(ā, iA(0) ) m0 ] , then one of the following is the case. (a) m0 = a, a store-Q of type C under ⊛, in which case σ = σ ′ ↾ (ā, iA(0) ) where σ ′ : Qā JAK → T JBK , hid, φi; τ ; T ζ ′ ; T σa ; µ σa : Qā (JAK ⊗JCK) → T JBK , strat{ [ (ā, iA(0) , iC ) ∗ ⊛ s] | [ (ā, iA(0) ) ∗ ⊛ a iC s ] ∈ viewf(σ) } , ( , if a ∈ S(ā) Qā ! ; aā ; drfC φ : Qā JAK → T JCK , ā ā Q πj ; ǫ ; drfC , if a # ā . (b) m0 = jA ∨ m0 = (iB , ⊛) , a store-H, in which case if [ (ā, iA(0) ) ∗ ⊛ m0 a iC ] ∈ σ, for some store-Q a and store-A iC , then Qā JAK h∆,σa i / Qā JAK ⊗Qā JAK ⊗T JCK τ ;T (id⊗φ;τ );µ σ   T JBK o where: T σ′ ; µ T Qā JAK 58 N. TZEVELEKOS σa : Qā JAK → T JCK , strat{ [ (ā, iA(0) ) ∗ ⊛ (iC , ⊛) s ] | [ (ā, iA(0) ) ∗ ⊛ m0 a iC s ] ∈ viewf(σ) ∨ [ ⊛ ⊛ s ] ∈ viewf(idξ ) } , ′ ā σ : Q JAK → T JBK , strat( { [ (ā, iA(0) ) ∗ ⊛ m0 y s ] ∈ viewf(σ) | y 6= a } ∪ { [ (ā, iA(0) ) ∗ ⊛ m0 a s ] | [ ⊛ ⊛ a s ] ∈ viewf(idξ )} ) , φ : Qā JAK ⊗JCK → T 1 , ( (Qā ! ; aā )⊗idJCK ; updC (Qā πj ; āǫ )⊗idJCK ; updC , if a ∈ S(ā) , if a # ā . In both cases above, we take j = min{ j | (iA(0) )j = a }. The proof of definability is a nominal version of standard definability results in game semantics. In fact, using the Decomposition Lemma we reduce the problem of definability of a finitary strategy σ to that of definability of a finitary strategy σ0 of equal length, with σ0 having no initial effects (i.e. fresh-name creation, name-update or name-dereferencing). On σ0 we then apply almost verbatim the methodology of [15] — itself based on previous proofs of definability. Theorem 5.33 (Definability). Let A, B be types and σ : Qā JAK → T JBK be finitary. Then σ is definable. Proof: We do induction on (|trunc(σ)|, kσk), where we let kσk , max{ |L(s)| | [s] ∈ viewf(σ) }, i.e. the maximum number of names introduced in any play of trunc(σ). If |trunc(σ)| = 0 then σ = JstopB K ; otherwise, there exist x0 , iA(0) such that [(ā, iA(0) ) ∗ ⊛ x0 ] ∈ σ . By Decomposition Lemma, ā σ = h[x = iA(0) ], hσ0 , σ ′ ii; cnd with |trunc(σ ′ )| < |trunc(σ)| and (0, 0) < (|trunc(σ0 )|, kσ0 k) ≤ (|trunc(σ)|, kσk) , so by IH there exists term M ′ such that JM ′ K = σ ′ . Hence, if there exist terms M0 , N0 with ā JM0 K ↾ (ā, iA(0) ) = σ0 and JN0 K = [x = iA(0) ]; η , then we can see that σ = Jif0 N0 then M0 else M ′ K . We first construct N0 . Assume that A = A1 × A2 × · · · × An with Ai ’s non-products, and similarly B = B1 × · · · × Bm . Moreover, assume without loss of generality that A is segmented in four parts: each of A1 , ..., Ak is N; each of Ak+1 , ..., Ak+i , ..., Ak+k′ is [A′′′ i ]; each of Ak+k′ +1 , ..., Ak+k′ +i , ..., Ak+k′ +k′′ is A′i → A′′i ; and the rest are all 1. Take z̄, z̄ ′ , z̄ ′′ , z̄ ′′′ to be variable-lists of respective types. Define φ0 , φ′0 by: φ0 , κ1 , ..., κk , with (κ1 , ..., κk ) being the initial N-segment of iA(0) ,  (iA(0) )k+i , if (iA(0) )k+i ∈ S(ā)    z ′ , if (iA(0) )k+i # ā j φ′0 , κ′1 , ..., κ′k′ , with each κ′i ,  ∧ j = min{ j < i | (iA(0) )k+i = (iA(0) )k+j }    fresh(i) , otherwise . FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 59 fresh(i) is a meta-constant denoting that Opponent has played a fresh name in Ak+i . If the same fresh name is played in several places inside iA(0) then we regard its leftmost occurrence as introducing it — this explains the second item in the cases-definition above. Now, define N0 , [hz̄, z̄ ′ i = hφ0 , φ′0 i] where: [hz̄, z̄ ′ i = h~κ, ~κ′ i] , [z1 = κ1 ] ∧ · · · ∧ [zk = κk ] ∧ [z1′ = κ′1 ] ∧ · · · ∧ [zk′ ′ = κ′k′ ] , ′ [z ′ = fresh(i)] , [z ′ 6= a1 ] ∧ · · · ∧ [z ′ 6= a|ā| ] ∧ [z ′ 6= z1′ ] ∧ · · · ∧ [z ′ 6= zi−1 ], with the logical connectives ∧ and ¬ defined using if0’s, and [zi = κi ] using pred ’s, in the ā standard way. It is not difficult to show that indeed JN0 K = [x = iA(0) ]; η . We proceed to find M0 . By second part of Decomposition Lemma, σ0 = ^ b̄ _ σb̄ with b̄ = nlist(x0 ), |trunc(σb̄ )| = |trunc(σ0 )| and kσb̄ k = kσ0 k − |b̄| . If |b̄| > 0 then, by IH, there exists term Mb̄ such that JMb̄ K = σb̄ , so taking M0 , ν b̄.Mb̄ we have σ0 = JM0 K . Assume now |b̄| = 0, so x0 = m0 . σ0 satisfies the hypotheses of the third part of the Decomposition Lemma. Hence, if m0 = a, a store-Q of type C under ⊛, then σ0 = (hid, φi ; τ ; T ζ ′ ; T σa ; µ) ↾ (ā, iA(0) ) with trunc(σa ) < trunc(σ0 ) . Then, by IH, there exists ā | Γ, y : C |− Ma : B such that σa = JMa K , and taking ( (λy.Ma )(! a) , if a ∈ S(ā) M0 , (λy.Ma )(! zj′ ) , if a # ā ∧ j = min{ j | a = (iA(0) )k+j } we have σ0 = JM0 K ↾ (ā, iA(0) ). Otherwise, m0 = jA ∨ m0 = (iB , ⊛), a store-H. If there exists an a ∈ AC such that σ0 answers to [iA(0) ∗ ⊛ m0 a] then, by Decomposition Lemma, σ0 = h∆, σa i ; τ ; T (id⊗φ ; τ ) ; µ ; T σ ′ ; µ with |trunc(σa )| , |trunc(σ ′ )| < |trunc(σ0 )| . By IH, there exist ā | Γ |− Ma : C and ā | Γ |− M ′ : B such that σa = JMa K and σ ′ = JM ′ K. Taking ( (a := Ma ); M ′ , if a ∈ S(ā) M0 , ′ ′ (zj := Ma ); M , if a # ā ∧ j = min{ j | a = (iA(0) )k+j } we obtain σ0 = JM0 K . Note here that σa blocks initial moves [ā, iA ] 6= [ā, iA(0) ] and hence we do not need the restriction. We are left with the case of m0 being as above and σ0 not answering to any store-Q, which corresponds to the case of Player not updating any names before playing m0 . If m0 = (iB , ⊛) then we need to derive a value term hV1 , ..., Vm i (as B = B1 × · · · × Bm ). For each p, if Bp is a base or reference type then we can choose a Vp canonically so that its denotation be iBp (the only interesting such case is this of iBp being a name a # ā, where we take Vp to be zj′ , for j = min{ j | a = (iA(0) )k+j }). Otherwise, Bp = Bp′ → Bp′′ and from σ0 we obtain the (tidy) viewfunction f : Qā (JAK ⊗JBp′ K) → T JBp′′ K by: f , { [ (ā, iA(0) , iBp′ ) ∗ ⊛ s ] | [ (ā, iA(0) ) ∗ ⊛ (iB , ⊛) (iBp′ , ⊛) s ] ∈ viewf(σ0 ) }. 60 N. TZEVELEKOS Note that, for any [(ā, iA ) ∗ ⊛ (iB , ⊛) (iBp′ , ⊛) s] ∈ viewf(σ0 ), s cannot contain store-Q’s justified by ⊛ , as these would break (TD2). Hence, f fully describes σ0 after (iBp′ , ⊛) . By IH, there exists ā | Γ, y : Bp′ |− N : Bp′′ such that JN K = strat(f ) ; take then Vp , λy.N . Hence, taking M0 , hV1 , ..., Vm i we obtain σ0 = JM0 K ↾ (ā, iA(0) ). If m0 = jA , played in some Ak+k′ +i = A′i → A′′i , then m0 = (iA′i , ⊛) . Assume that A′i = A′i,1 × · · · × A′i,ni with A′i,p ’s being non-products. Now, O can either ask some name a (which would lead to a store-CC), or answer at A′′i , or play at some A′i,p of arrow type, ′ . Hence, say A′i,p = Ci,p → Ci,p [ni viewf(σ0 ) = fA ∪ fp where: p=1 fA , f0 ∪ { [ (ā, iA(0) ) ∗ ⊛ (iA′i , ⊛) (iA′′i , ⊛) s ] ∈ viewf(σ0 ) } fp , f0 ∪ { [ (ā, iA(0) ) ∗ ⊛ (iA′i , ⊛) (iCi,p , ⊛) s ] ∈ viewf(σ0 ) } f0 , { [ (ā, iA(0) ) ∗ ⊛ (iA′i , ⊛) s] | [⊛ ⊛ s ] ∈ viewf(idξ ) } and where we assume fp , f0 if A′i,p is not an arrow type. It is not difficult to see that fA , fp are viewfunctions. Now, from fA we obtain: fA′ : Qā (JAK ⊗JA′′i K) → T JBK , { [ (ā, iA(0) , iA′′i ) ∗ ⊛ s ] | [(ā, iA(0) ) ∗ ⊛ (iA′i , ⊛) (iA′′i , ⊛) s ] ∈ fA } . It is not difficult to see that fA′ is indeed a viewfunction (note that P cannot play a storeQ under ⊛ on the RHS once (iA′′i , ⊛) is played, by tidiness). By IH, there exists some ā | Γ, y : A′′i |− MA : B such that JMA K = strat(fA′ ). ′ K by: From each fp 6= f0 we obtain a viewfunction fp′ : Qā (JAK ⊗JCi,pK) → T JCi,p fp′ , { [ (ā, iA(0) , iCi,p ) ∗ ⊛ s ] | [ (ā, iA(0) ) ∗ ⊛ (iA′i , ⊛) (iCi,p , ⊛) s ] ∈ fp } . ′ such that JM K = strat(f ′ ) , so By IH, there exists some ā | Γ, y ′ : Ci,p |− Mp : Ci,p p p take Vp , λy ′ .Mp . For each A′i,p of non-arrow type, the behaviour of σ0 at A′i,p is fully described by (iA′i )p , so we choose Vp canonically as previously. hV1 , ..., Vni i is now of type A′i and describes σ0 ’s behaviour in A′i . Now, taking M0 , (λy.MA )(zi′′ hV1 , ..., Vni i) we obtain σ0 = JM0 K ↾ (ā, iA(0) ). Finally, using the definability result and proposition 5.31 we can now show the following. Corollary 5.34. T = (T , T, Q, O) satisfies ip-definability. ā Proof: For each ā, A, B, define DA,B , { f : Qā JAK → T JBK | f is finitary } . By definabilā ity, every f ∈ DA,B is definable. We need also show: ā . Λā (f ) ; ρ ∈ Oā =⇒ Λā (g) ; ρ ∈ Oā ) =⇒ f .ā g . (∀ρ ∈ DA→B,N Assume the LHS assertion holds and let Λā (f ) ; ρ ∈ Oā , some ρ : Qā (JAK −−⊗ T JBK) → T N. Then, let [s ; t] = [(ā, ∗) ∗ ⊛ (0, ⊛)b̄ ] ∈ Λā (f ) ; ρ , [s] ∈ Λā (f ) and [t] ∈ ρ. By proposition 5.31, FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 61 ā [t] ∈ ρ≤t , so Λā (f ) ; ρ≤t ∈ Oā . Moreover, ρ≤t ∈ DA→B,N , so Λā (g) ; ρ≤t ∈ Oā , by hypothesis. Finally, ρ≤t ⊑ ρ implies Λā (g) ; ρ≤t ⊑ Λā (g) ; ρ , hence the latter observable, so f .ā g. Hence, we have shown full abstraction. Theorem 5.35. T = (T , T, Q, O) is a fully abstract model of νρ. 5.7. An equivalence established semantically. In this last section we prove that the following terms M and N are equivalent. The particular equivalence exemplifies the fact that exceptional behaviour cannot be simulated in general by use of references, even of higher-order. M , λf. stop : (1 → 1) → 1 , N , λf. f skip ; stop : (1 → 1) → 1 . By full-abstraction, it suffices to show JM K ⋍ JN K, where the latter are given as follows. 1 JM K / T ((1 −−⊗ T 1) −−⊗ T 1) ∗ 1 OQ ∗ OQ (∗, ⊛)(1) ⊥ / T ((1 −−⊗ T 1) −−⊗ T 1) ∗ PA ⊛ JN K OQ ∗ PA ⊛ OQ (∗, ⊛)(1) PA (∗, ⊛)(2) (∗, ⊛)(3) ⊥ PA OQ PQ Bottom links stand for deadlocks: if Opponent plays a move (∗, ⊛)(2) under the last ∗ in JM K (thus providing the function f ) then Player must play JstopK, i.e. remain idle. Similarly for JN K: if Opponent gives an answer to (∗, ⊛)(3) (providing thus the outcome of f skip) then Player deadlocks the play. We have that JM K ⊑ JN K and therefore, by (5.5), JM K . JN K . Conversely, let ρ : T ((1 −−⊗ T 1) −−⊗ T 1) → T N be a tl4 tidy strategy such that [∗ ∗ ⊛ (0, ⊛)ā ] ∈ JN K ; ρ for some ā. Then, because of the form of JN K, ρ can only play initial moves up to (∗, ⊛)(1) , then possibly ask some names to (∗, ⊛)(1) , and finally play (0, ⊛)ā . Crucially, ρ cannot play (∗, ⊛)(2) under ∗: this would introduce a question that could never be answered by JN K, and therefore ρ would not be able to play (0, ⊛)ā without breaking well-bracketing. Hence, JM K and ρ can simulate the whole interaction and therefore [∗ ∗ ⊛ (0, ⊛)ā ] ∈ JM K ; ρ. 6. Conclusion Until recently, names used to be bypassed in Denotational Semantics: most approaches focussed on the effect achieved by use of names rather than names themselves. Characteristic of this attitude was the ‘object-oriented’ modelling of references [6, 3] and exceptions [19] as products of their effect-related methods (in the spirit of [39]). These approaches were unsatisfactory to some extent, due to the need for ‘bad’ syntactic constructors in the examined languages. Moreover, they could not apply to the simplest nominal language, the 62 N. TZEVELEKOS ν-calculus [36], since there the achieved effect could not be given an extensional, name-free description. These issues revealed the need that names be treated as a proper computational effect [44], and led to the advent of nominal games [2, 21]. In this paper we have taken some further steps in the semantics of nominal computation by examining the effect of (nominal) general references. We have shown that nominal games provide a framework expressive enough that, by use of appropriate monadic (and comonadic) constructions, one can model general references without moving too far from the model of the ν-calculus [2]. This approach can be extended to other nominal effects too; e.g. in [47] it is applied to exceptions (with and without references). Moreover, we have examined abstract categorical models for nominal computation, and references in particular (in the spirit of [45, 44]). There are many threads in the semantics of nominal computation which need to be pursued further. Firstly, there are many nominal games models to build yet: research in this direction has already been undertaken in [24, 22, 47, 31]. By constructing models for more nominal languages we better understand the essential features of nominal computation (e.g. name-availability [31]) and build stronger intuitions on nominal games. Another direction for further research is that of characterising the nominal effect — i.e. the computational effect that rises from the use of names — in abstract categorical terms. Here we have pursued this task to some extent by introducing the monadic-comonadic description of nominal computation, but it is evident that the description needs further investigation. We see that there are more monad-comonad connections to be revealed, which will simplify and further substantiate the presentation. The work of Schöpp which examines categories with names [41] seems to be particularly helpful in this direction. A direction which has not been pursued here is that of decidability of observational equivalence in nominal languages. The use of denotational methods, and game semantics in particular, for attacking the problem has been extremely successful in the ‘non-nominal’ case, having characterised decidability of (fragments of) Idealized Algol [13, 34, 32]. It would therefore be useful to ‘nominalise’ that body of work and apply it to nominal calculi. Already from [32] we can deduce that nominal languages with ground store are undecidable, and from [36] we know that equivalence is decidable for programs of first-order type in the ν-calculus, but otherwise the problem remains open. Acknowledgements. I would like to thank Samson Abramsky for his constant encouragement, support and guidance. I would also like to thank Andy Pitts, Andrzej Murawski, Dan Ghica, Ian Stark, Luke Ong, Guy McCusker, Jim Laird, Paul Levy, Sam Sanjabi and the anonymous reviewers for fruitful discussions, suggestions and criticisms. Appendix A. Deferred proofs I. Proof of closure of tidiness under composition. Lemma A.1. Let σ : A → B and τ : B → C be tidy strategies, and let [s ; t] ∈ σ ; τ , [s] ∈ σ and [t] ∈ τ , with ps k tq = s k t ending in a generalised O-move in AB and x, an O-move, being the last store-H in psq. Let x appear in s k t as x̃. Then, x̃ is the last store-H in s k t and if x is in A then all moves after x̃ in s k t are in A. Similarly for BC and t. Proof: We show the (AB, s) case, the other case being entirely dual. Let s = s1 xs2 and let x appear in s k t as some x̃. If x is in A then we claim that s2 is in A. Suppose otherwise, FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 63 so s = s1 xs21 ys22 with s21 in A and y a P-move in B. Since x appears in psq, the whole of s21 y appears in it, as it is in P-view mode already. Since x is last store-H in psq, s21 y is store-H-less. If y a store-Q then it should be justified by last O-store-H in ps<yq, that is x, which is not possible as x is in A. Thus, y must be a store-A, say to some O-store-Q q in B. Now, since q wasn’t immediately answered by P, tidiness dictates that psq be a copycat from move q and on. But then the move following x in s must be a copy of x in B, . Hence, s2 is in A and therefore it appears in psq, which implies that it is store-H-less. Thus, x̃ is last store-H in s k t. If x is in B then we do induction on |s k t|. The base case is encompassed in the case of s2 being empty, which is trivial. So let s2 = s21 ys22 z with y justifying z (since x appears in psq, z has to be justified in s2 ). z is not a store-H and neither is it a store-Q, as then y would be a store-H after x in psq. Thus z a store-A and y a store-Q, the latter justified by last O-store-H in ps<yq = psq<y , that is x, so y, z in B. Now, s = s1 xs21 ys22 z and t = t1 x′ t21 y ′ t22 z ′ ; we claim that s21 and t21 are store-H-less. Indeed, s<y k t<y′ ends in a generalised O-move in AB and x is still the last store-H in ps<yq , from which we have, by IH, that x̃ is the last store-H in s<y k t<y′ . Thus, s k t = (s1 k t1 )x̃v ỹuz̃ with v store-H-less. It suffices to show that u is also storeH-less. In fact, u = ỹ . . . ỹ z̃ . . . z̃ for some n ≥ 0. Indeed, by tidiness of τ , (t22 z ′ ).1 is either | {z } | {z } n n an answer to y ′ , whence t22 = u = ǫ, or a copy of it under the last O-store-H in pt≤y′q. If the latter is in B then σ reacts analogously, and so on, so there is initially a sequence ỹ . . . ỹ in u, played in B. As u finite, at some point σ (or τ ) either answers y (y ′ ) or copycats it in A (in C). In the latter case, O immediately answers, as s (t) is in P-view mode in A (in C). Hence, in either cases there is an answer that is copycatted to all open ỹ in u, yielding thus the required pattern. Therefore, u is store-H-less. Lemma A.2. Let σ : A → B and τ : B → C be tidy strategies, and let [s ; t] ∈ σ ; τ , [s] ∈ σ and [t] ∈ τ , with ps k tq = s k t ending in a generalised O-move. If there exists i ≥ 1 and store-Q’s q̃1 , ..., q̃i with q̃ = q̃j , all 1 ≤ j ≤ i, and q̃1 , ..., q̃i−1 in B and q̃i in AC and [(s k t)q̃1 ...q̃i ] ∈ σ k τ , then q̃i is justified by the last O-store-H in s ; t. Proof: By induction on |s k t|. The base case is encompassed in the case of s ; t containing at most one O-store-H, which is trivial. Now let without loss of generality (s k t)q̃1 ...q̃i = ′ ) with [sq ...q ] ∈ σ and [tq ′ ...q ′ ] ∈ τ , and let each q be justified by (sq1 ...qi ) k(tq1′ ...qi−1 1 i j 1 i−1 xj and each qj′ by x′j . Moreover, by hypothesis, xj = x′j , for 1 ≤ j ≤ i − 1, and therefore each such pair xj , x′j appears in s k t as some x̃j , the latter justifying q̃j in s k t. Now, assume without loss of generality that s k t ends in AB. Then, by tidiness of σ and τ we have that, for each j ≥ 1, q2j+1 = q2j , ′ ′ q2j = q2j−1 , qj = qj′ For each j ≥ 1, q2j+1 is a P-move of σ justified by some store-H, say x2j+1 . By tidiness of σ, x2j+1 is the last O-store-H in ps<q2j+1q = ps≤q2jq, and therefore x2j+1 is the last store-H in ps<x2jq. Then, by previous lemma, x̃2j+1 is the last store-H in s<x2j k t<x′2j = (s k t)<x̃2j . Similarly, x̃2j is the last store-H in (s k t)<x̃2j−1 . Hence, the store-H subsequence of (s k t)≤x̃1 ends in x̃i ...x̃1 . Now, by tidiness of σ, x1 is the last O-store-H in psq. If x1 is also the last store-H in psq then, by previous lemma, x̃1 is the last store-H in s k t, hence x̃i is the last store-H in s ; t. Otherwise, by corollary 5.21, q1 is a copy of s.−1 = q0 . If q0 is in A then its justifier is 64 N. TZEVELEKOS s.−2 = x0 and, because of CC-mode, the store-H subsequence of s k t ends in x̃i ...x̃1 x̃0 , so x̃i is the last O-store-H in s ; t. If q0 is in B then we can use the IH on s− k t− and q̃0 , q̃1 , ..., q̃i , and obtain that x̃i is the last O-store-H in s− ; t− = s ; t. Proposition A.3. If σ : A → B and τ : B → C are tidy strategies then so is σ ; τ . Proof: Take odd-length [s ; t] ∈ σ ; τ with not both s and t ending in B, ps k tq = s k t and |s ; t| odd. We need to show that s ; t satisfies (TD1-3). As (TD2) is a direct consequence of the previous lemma, we need only show the other two conditions. Assume without loss of generality that s ; t ends in A. For (TD1), assume s ; t ends in a store-Q q̃. Then s ends in some q, which is justified by the P-store-H s.−2 = x (also in A). q is either answered or copied by σ ; in particular, if q̃ = aā with a # ps ; tq− = s− ; t then a # s− , t , so σ copies q. If σ answers q with z then z doesn’t introduce new names, so [(s ; t)z̃] ∈ σ ; τ with nlist(z̃) = nlist(q̃) and z̃ = z , as required. Otherwise, let σ copy q as q1 , say, under last O-store-H in psq, say x1 . If x1 is in B then sq1 ≍ tq1′ , with q1 , q1′ in B and q1′ being q1 with name-list that of its justifier, say x′1 , where x1 = x′1 . Now [tq1′ ] ∈ τ and it ends in a store-Q, so τ either answers it or copies it under last O-store-H in ptq1′q. In particular, if q = aā with a # ps ; tq then, as above, a # t and τ copies q1′ . This same reasoning can be applied consecutively, with copycats attaching store-Q’s to store-H’s appearing each time earlier in s and t. As the latter are finite and initial store-H’s are third moves in s and t, at some point either σ plays qi in A or answers it in B, or τ plays qi′ in C or answers it in B. If an answer occurs then it doesn’t introduce new names (by tidiness), so it is copycatted back to q closing all open qj ’s and qj′ ’s. Otherwise, we need only show that, for each j, q̃j = q̃, which we do by induction on j: q̃1 = q s • t,ǫ and (s≤q ) •(t ′ ),ǫ IH ≤q j j q̃j+1 = q = q̃j = q̃. This proves (TD1). For (TD3), assume s ; t = uq̃(O) q̃(P ) v ỹ with q̃(O)q̃(P ) v a copycat. Then, either both q̃(O) , q̃(P ) are in A, or one is in A and the other in C. Let’s assume q̃(O) in A and q̃(P ) in C — the other cases are shown similarly. Then, q̃(O) her(editarily)-justifies ỹ, and let s.−1 = y be justified by some x in s. Now, as above, q̃(O) q̃(P ) is witnessed by some q̃(O)q̃1 . . . q̃i q̃(P ) in s k t, with odd i ≥ 1 and all q̃j ’s in B. We show by induction on 1 ≤ k ≤ i that there exist x1 , ..., xk , x′1 , ..., x′k , y1 , ..., yk , y1′ , ..., yk′ in B such that (sy1 . . . yk k ty1′ . . . yk′ ) ∈ σ k τ and, for each relevant j ≥ 1, yj = yj′ = y , y1 = y , y2j = y2j+1 , ′ ′ y2j−1 = y2j , xj = x′j with qj her-justifying xj in s and xj justifying yj (and qj′ her-justifying x′j in t and x′j justifying yj′ ), and x̃j+1 , x̃j consecutive in s k t, and x̃1 , x̃ also consecutive. For k = 1, let s = s1 q(O) q1 s2 y. Now, q̃(O) her-justifying ỹ implies that q(O) her-justifies y, hence it appears in psq. Thus psq = s′1 q(O) q1 s′2 y, so, by (original definition of) tidiness, [sy1 ] ∈ σ with y1 = y justified by x1 = psq .−3 = s.−3. Then, [ty1′ ] ∈ τ with y1′ = y1 . By proposition 5.19, q(O) q1 s′2 is a copycat, so q1 her-justifies x1 and therefore x1 , y1 in B. Finally, x = psq .−2 = s.−2 is a P-move so x̃1 , x̃ are consecutive in s k t. ′ ) ∈ σ k τ with y ′ For even k > 1 we have, by IH, that (sy1 . . . yk−1 k ty1′ . . . yk−1 k−1 an O′ q, so pty ′ ...y ′ q = ′ ′ move her-justified by qk−1 , an O-move. Then, qk−1 appears in pty1′ ...yk−1 1 k−1 ′ ′ q ′ t y ′ , thus (by tidiness) [ty ′ ...y ′ y ′ ] ∈ τ with y ′ = y ′ t1 qk−1 1 k−1 justified by xk = k k−1 k k 2 k−1 ′ ′ ′ ′ ′ ′ ′ ′ pty1 ...yk−1q .−3. Now, qk−1 qk t2 is a copycat so qk her-justifies xk . Moreover, xk , xk−1 are consecutive in ptq, so, as x′k−1 a P-move, they are consecutive in t, and therefore x̃k , x̃k−1 FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 65 consecutive in s k t. Finally, [sy1 . . . yk−1 yk ] ∈ σ with yk = yk′ . The case of k odd is entirely dual. ′ ]∈ ′ Now, just as above, we can show that there exist x′i+1 , yi+1 in C such that [ty1′ ...yi′ yi+1 ′ ′ ′ τ and yi+1 justified by xi+1 , xi+1 her. justified by q(P ) , etc. Then [(s ; t)ỹi+1 ] ∈ σ ; τ with x̃i+1 , x̃i , ..., x̃1 , x̃ consecutive in s k t, so x̃i+1 = (s ; t).−3. Finally, as above, ỹi+1 = ỹj = ỹ, all j, as required.  II. Proof of Decomposition Lemma 5.32: 1 is straightforward: we just partition σ into σ0 ā and σ ′ and recover it by use of [x = iA(0) ] and cnd. For 2, we just use the definition of name-abstraction for strategies and the condition on σ. For 3, it is clear that m0 is either a store-Q a under ⊛, or a store-H jA , or a store-H (iB , ⊛). In case m0 = a with a ∈ AC , we define σa : Qā (JAK ⊗JCK) → T JBK , strat(fa ) , where fa , { [ (ā, iA(0) , iC ) ∗ ⊛ s ] | [ (ā, iA(0) ) ∗ ⊛ a iC s ] ∈ viewf(σ) } . To see that fa is a viewfunction it suffices to show that its elements are plays, and for that it suffices to show that they are legal. Now, for any [(ā, iA(0) , iC ) ∗ ⊛ s] ∈ fa with [(ā, iA(0) ) ∗ ⊛ a iC s] ∈ viewf(σ), (ā, iA(0) , iC ) ∗ ⊛ s is a justified sequence and satisfies wellbracketing, as its open Q’s outside s are the same as those in (ā, iA(0) ) ∗ ⊛ a iC s , i.e. ⊛. Moreover, visibility is obvious. Hence, fa is a viewfunction, and it inherits tidiness from σ. Moreover, we have the following diagram. Qā JAK hid,φi ; τ ; T ζ ′ / T Qā (JAK ⊗JCK) T σa / T 2 JBK µ / T JBK (ā, iA(0) ) ∗ ∗ ∗ ⊛ ⊛ ⊛ a a a iC iC iC (ā, iA(0) , iC , ⊛) (∗, ⊛) ⊛ 66 N. TZEVELEKOS Because of the copycat links, we see that viewf(hid, φi ; τ ; T ζ ′ ; T σa ; µ) ↾ (ā, iA(0) ) = {[(ā, iA(0) ) ∗ ⊛ a iC s] | [(ā, iA(0) , iC ) ∗ ⊛ s] ∈ viewf(σa )} = viewf(σ) , as required. Note that the restriction to initial moves [ā, iA(0) ] taken above is necessary in case φ contains a projection (in which case it may also answer other initial moves). In case m0 = jA (so m0 a store-H) and [(ā, iA(0) ) ∗ ⊛ m0 a iC ] ∈ σ, we have that σ = strat(fa ∪ (f ′ \ fa′ )) , where fa , f ′ are viewfunctions of type Qā JAK → T JBK, so that fa determines σ’s behaviour if O plays a at the given point, and f ′ \ fa′ determines σ’s behaviour if O plays something else. That is, fa , { [ (ā, iA(0) ) ∗ ⊛ jA a iC s ] ∈ viewf(σ) } fa′ , { [ (ā, iA(0) ) ∗ ⊛ jA a s ] | [⊛ ⊛ a s] ∈ viewf(idξ ) } f ′ , fa′ ∪ { [ (ā, iA(0) ) ∗ ⊛ jA y s ] ∈ viewf(σ) | y 6= a } . f ′ differs from viewf(σ) solely in the fact that it doesn’t answer a but copycats it instead; it is a version of viewf(σ) which has forgotten the name-update of a. On the other hand, fa contains exactly the information for this update. It is not difficult to see that f ′ , fa are indeed viewfunctions. We now define fa′′ : Qā JAK → T JCK , { [ (ā, iA(0) ) ∗ ⊛(iC , ⊛) s ] | [ (ā, iA(0) ) ∗ ⊛jA a iC s ] ∈ fa ∨ [⊛ ⊛ s] ∈ viewf(idξ ) } σa : Qā JAK → T JCK , strat(fa′′ ) σ ′ : Qā JAK → T JBK , strat(f ′ ) σ ′′ : Qā JAK → T JBK , h∆, σa i ; τ ; T (id⊗φ ; τ ) ; µ ; ∼ = ; T σ′ ; µ . We can see that σ ′ is a tidy strategy. For σa , it suffices to show that fa′′ is a viewfunction, since tidiness is straightforward. For that, we note that even-prefix closure and singlevaluedness are clear, so it suffices to show that the elements of fa′′ are plays. So let [(ā, iA(0) ) ∗ ⊛ (iC , ⊛) s] ∈ fa′′ with [(ā, iA(0) ) ∗ ⊛ jA a iC s] ∈ viewf(σ). We have that (ā, iA(0) ) ∗ ⊛ (iC , ⊛) s is a justified sequence, because s does not contain any moves justified by jA or a. In the former case this holds because we have a P-view, and in the latter because a is a closed (answered) Q. Note also that there is no move in s justified by ⊛: such a move (iB , ⊛) would be an A ruining well-bracketing as jA is an open Q, while a store-Q under ⊛ is disallowed by tidiness as s.1 is an O-store-H. Finally, well-bracketing, visibility and NC’s are straightforward. We now proceed to show that σ = σ ′′ . By the previous analysis on fa′′ we have that σa = σa′ ; η (modulo totality) where σa′ is the possibly non-total strategy σa′ : Qā JAK → JCK , strat{ [ (ā, iA(0) ) iC s ] | [ (ā, iA(0) ) ∗ ⊛ jA a iC ] ∈ fa } , ∼ ; T σ ′ ; µ . Analysing the behaviour of the and hence σ ′′ ↾ (ā, iA(0) ) = h∆, σa′ i ; id⊗φ ; τ ; = latter composite strategy and observing that the response of σ ′′ to inputs different than FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES 67 [ā, iA(0) ] is merely the initial answer ∗ imposed by totality, we obtain: viewf(σ ′′ ) = { [ (ā, iA(0) ) ∗ ⊛ jA a s ], [(ā, iA(0) ) ∗ ⊛ jA y s] ∈ viewf(σ ′′ ) | y 6= a } = { [ (ā, iA(0) ) ∗ ⊛ jA a iC s ] | [ (ā, iA(0) ) ∗ ⊛ (iC , ⊛) s] ∈ fa′′ ∧ s.1 ∈ JJCK } ∪ { [ (ā, iA(0) ) ∗ ⊛ jA y s ] ∈ f ′ | y 6= a } = fa ∪ (f ′ \ fa′ ) = viewf(σ) as required. In case x = (iB , ⊛) we work similarly as above. References [1] Abramsky, S. Domain theory. Lecture Notes, Oxford University Computing Laboratory, 2007. [2] Abramsky, S., Ghica, D., Murawski, A., Ong, L., and Stark, I. Nominal games and full abstraction for the nu-calculus. 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Privacy-Enhanced Architecture for Occupancy-based HVAC Control Ruoxi Jia1 , Roy Dong2 , S. Shankar Sastry2 , Costas J. Spanos1 Department of Electrical Engineering and Computer Sciences University of California, Berkeley arXiv:1607.03140v1 [cs.CR] 11 Jul 2016 ruoxijia@berkeley.edu,roydong@eecs.berkeley.edu sastry@eecs.berkeley.edu,spanos@berkeley.edu ABSTRACT CCS Concepts Large-scale sensing and actuation infrastructures have allowed buildings to achieve significant energy savings; at the same time, these technologies introduce significant privacy risks that must be addressed. In this paper, we present a framework for modeling the trade-off between improved control performance and increased privacy risks due to occupancy sensing. More specifically, we consider occupancybased HVAC control as the control objective and the location traces of individual occupants as the private variables. Previous studies have shown that individual location information can be inferred from occupancy measurements. To ensure privacy, we design an architecture that distorts the occupancy data in order to hide individual occupant location information while maintaining HVAC performance. Using mutual information between the individual’s location trace and the reported occupancy measurement as a privacy metric, we are able to optimally design a scheme to minimize privacy risk subject to a control performance guarantee. We evaluate our framework using real-world occupancy data: first, we verify that our privacy metric accurately assesses the adversary’s ability to infer private variables from the distorted sensor measurements; then, we show that control performance is maintained through simulations of building operations using these distorted occupancy readings. •Security and privacy → Privacy protections; •Computing methodologies → Control methods; Modeling methodologies; 1 This research is funded by the Republic of Singapore’s National Research Foundation through a grant to the Berkeley Education Alliance for Research in Singapore (BEARS) for the Singapore-Berkeley Building Efficiency and Sustainability in the Tropics (SinBerBEST) Program. BEARS has been established by the University of California, Berkeley as a center for intellectual excellence in research and education in Singapore. 2 This research is supported in part by FORCES (Foundations Of Resilient CybEr-Physical Systems), which receives support from the National Science Foundation (NSF award numbers CNS-1238959, CNS-1238962, CNS-1239054, CNS1239166). ACM ISBN 978-1-4503-2138-9. . . $15.00 DOI: 10.1145/1235 Keywords Energy; privacy; model predictive control; HVAC; optimization; occupancy 1. INTRODUCTION Large-scale sensing and actuation infrastructures have endowed buildings with the intelligence to perceive the status of their environment, energy usage, and occupancy, and to provide fine-grained and responsive controls over heating, cooling, illumination, and other facilities. However, the information that is collected and harnessed to enable such levels of intelligence may potentially be used for undesirable purposes, thereby raising the question of privacy. To spotlight the value of building sensory data and its potential for exploitation in the inference of private information, we consider as a motivating example the occupancy data, i.e., the number of occupants in a given space over time. Occupancy data is a key component to perform energyefficient and user-friendly building management. Particularly, it offers considerable potential for improving energy efficiency of the heating, ventilation, and air conditioning (HVAC) system, a significant source of energy consumption which contributes to more than 50% of the energy consumed in buildings [12]. Recent papers [4, 24, 13] have demonstrated substantial energy savings of up to 40% by enabling intelligent HVAC control in response to occupancy variations. The value of occupancy data in building management has also inspired extensive research on occupancy sensing [9, 19, 20, 23, 35] as well as a number of commercial products which can provide high accuracy occupancy data. While people have enjoyed the benefits brought by occupancy data, the privacy risks potentially posed by the data are largely overlooked (Figure 1). In effect, location traces of individual occupants can be inferred from the occupancy data with some auxiliary information [34]. Throughout this paper, we refer to the individual location trace as the private information to be protected. The contextual information attached to location traces tells much about the individuals’ habits, interests, activities, and relationships [25]. It can also reveal their personal or corporate secrets, expose them to unwanted advertisement and locationbased spams/scams, cause social reputation or economic HVAC Controller Control signal Auxiliary Knowledge ZN … Office Directory Previous Location Traces Z1 Alice Z2 Chris Z3 Bob Z1 Z1 Z2 Pantry Privacy-Sensitive Information Occupancy Sensor Occupancy 1 1 0 Z2 0 1 2 1 ZN 1 Zone Time 1 0 1 1 2 3 K Z1 Z2 Z1 Z2 Z5 Z3 Z3 Z2 Z2 Z2 Z2 Time 1 2 3 K Alice Adversary Bob … 0 … Building Manager Location Traces Z1 Chris Z1 Figure 1: An overview of the problem of individual occupant location recovery. The building manager collects occupancy data to enable intelligent HVAC controls adapted to occupancy variations. However, an adversary with malicious intent may exploit occupancy data in combination with the auxiliary information to infer privacy details about indoor locations of building users. damage, make them victims of blackmail or even physical violence [31]. At a first glance, it is surprising that occupancy data may incur risks of privacy breach, since it only reports the number of occupants in a given space over time without revealing the identities of the occupants. To illustrate why it is possible to infer location traces from seemingly “anonymized” occupancy data, consider the following scenario. We start by observing two users in one room and then one of them leaves the room and enters another room. We cannot tell which one of the two made this transition by observing the occupancy change. However, if the one who left entered an private office, the user can be identified with high probability based on the ownership of the office. Although a change in occupancy data may correspond to location shifts of many possible potential users, the knowledge of where the individuals mostly spend their time rules out many possibilities and renders the individual who made the transition identifiable. It has been shown in [34] that by simply combining some ancillary information, such as an office directory and user mobility patterns, individual location traces can be inferred from the occupancy data with the accuracy of more than 90%. It is, therefore, the objective of this paper to enable an occupancy-based HVAC control system that provides privacy features for each user on a par with thermal comfort and energy efficiency. A simple yet effective way to preserve privacy is to obfuscate occupancy data by injecting noise to make the data itself less informative. This approach has been widely used in privacy disclosure control of various databases, ranging from healthcare [7], geolocation [2], web-browsing behavior data [14], etc. While reducing the risk of privacy breach, this approach would also deteriorate the utility of the data. There have been attempts to balance learning the statistics of interest reliably with safeguarding the private information [32]. Cryptography [8] and access control [33] are also effective means to ease privacy concerns, but they do not provide protection against all privacy breaches. There may be insiders who can access the private, decrypted data, or the building manager may not want to have access to (and responsibility for) the private data. The objective of this paper cannot be attained by simply extending the techniques developed previously. Our task is more challenging. Firstly, as opposed to learning some fixed statistics from static data in most database applications, the data is used for controlling a highly complex and dynamic system in our case, and the control performance relies on the data fidelity. With highly accurate occupancy data, the infrastructure can correctly sense the environment and enable proper response to occupancy variations; nevertheless, the location privacy is sacrificed. On the other hand, the usage of severely distorted occupancy data reduces the risks of privacy leakage, but may lead to even higher levels of energy consumption and discomfort. Essentially, we need to address the trade-off between the performance of a controller on a dynamical system, and, similarly, privacy of a time-varying signal, i.e. the location traces of individual occupants. Secondly, from the perspective of the building manager, the building performance is paramount: adding the privacy feature into the HVAC control system should not impair the performance of HVAC controller in terms of energy efficiency and thermal comfort. To achieve this, the injected noise should be calculated to minimally affect performance of the controller, while maximizing the amount privacy gained from the distortion. In this paper we develop a method which minimizes the privacy risks incurred by collection of occupancy data while guaranteeing the HVAC system operating in a “nearly” optimal condition. Our solution relies on an occupancy distortion mechanism, which informs the building manager how to distort occupancy data before any form of storage or transport of the data. We draw the inspiration from the information-theoretic approach in [29, 10] for characterizing the privacy-utility trade-off, and choose the mutual information (MI) between reported occupancy measurements and individual location traces as our privacy metric. The design problem of finding the optimal occupancy distortion mechanism is cast as an optimization problem where the privacy risk is minimized for a set of constraints on the controller performance. This allows us to find points on the Pareto frontier in the utility-privacy trade-off, and to further analyze the economic side of privacy concerns [30]. The formulation can be easily generalized to resolve the tension between privacy and data utility in other cases where a control system utilizes some privacy-sensitive information as one of the control inputs, although in this paper we limit our focus to addressing the privacy concern of occupancy-based HVAC controller. In addition, our work here is complementary to the work being done in the cryptography communities: we can use our distortion mechanism to process sensor measurements, and then transmit the processed measurements across secure channels. Our work also serves as a complement for the privacy-preserving access control protocol in [33], as it provides distortion mechanisms against adversaries who might be able to subvert the protocol while still retaining the benefits for the occupancy data. The main contributions of our paper are as follow: • We present a systematic methodology to characterize the privacy loss and control performance loss. • We develop a holistic and tractable framework to balance the privacy pursuit and control performance. • We evaluate the trade-off between privacy and HVAC control performance using the real-world occupancy data and simulated building dynamics. The rest of the paper is organized as follows: Section 2 reviews the existing work on occupancy-based control algorithms and privacy metrics. Section 3 describes the models connecting location and occupancy, and the HVAC system model that will be considered in this paper. In Section 4 we present a framework for quantifying the trade-off between privacy and controller performance. We will evaluate the framework and demonstrate its practical values based on experimental studies in Section 5. Section 6 concludes the paper. 2. 2.1 RELATED WORK Occupancy-based HVAC control Occupancy-based HVAC systems exploit real-time occupancy measurements to condition the space appropriate to usage. The occupancy-based controllers in the existing work can be categorized into two types: rule-based controller and optimization-based controller or model predictive control (MPC). The rule-based controller uses an “if condition then action” logic for decision making in accordance with occupancy variations [13, 4]. MPC is a more advanced control scheme, which employs a model of building thermal dynamics in order to predict the future evolution of the system, and solves an optimization problem in real-time to determine control actions [27]. A number of papers including [16, 17, 3] analyzed in large-scale simulative or experimental studies the energy saving potential in building climate control by using MPC, which was shown to be well-suited for building applications. This leads to our choice of MPC to exemplify the trade-off between controller performance and privacy. Occupancy information can be leveraged in different ways in an MPC-based controller. One approach is to build an occupancy model to predict future occupancy based on which the MPC optimizes control actions [5]. Another method is to use the instantaneous occupancy measurement and consider it to be constant during the control horizon of MPC [15]. This method has been demonstrated to achieve comparable performance with the MPC that exploits occupancy predictions. We will thus without loss of generality follow the latter set-up to avoid explicit modeling of occupancy. 2.2 Privacy Privacy, although not a new topic, has recently developed renewed interest, due in no small part to new technologies and modern infrastructures collecting and storing unprecedented amounts of data. Since privacy is an abstract and subjective concept, it is necessary to develop proper measures for privacy before any privacy protection technique is discussed. Differential privacy [11] is one of the most popular metrics for privacy from the area of statistical databases. It is is typically assured by adding appropriately chosen random noise to the database output. However, calculating optimal noise for differential privacy is very difficult, and research on the applications of differential privacy mostly assumes the injected noise to be an additive zero-mean Gaussian or Laplacian random variable, which offers no guarantee on data utility. As mentioned in the introduction, in our case the performance of HVAC control systems is crucial: as such, our work is an effort to maintain control efficacy by optimally designing noise distribution to maximize privacy subject to a performance guarantee. Recently, MI has become a popular privacy metric [29, 10, 18]. Intuitively, MI reflects the change in the uncertainty of a private variable due to the observation of a public random variable. In fact, it is the only metric of information leakage that satisfies the data processing inequality [18]. Unlike differential privacy, this requires some modeling of the adversary’s available ancillary information; however, in practice, we can suppose an adversary with access to a large amount of ancillary information, which gives a bound on any weaker adversary’s performance. A framework for characterizing privacy-utility trade-off based on MI was proposed in [10], where the MI between a private variable and a distorted measurement is minimized subject to the bound on the value of an exogenous distortion metric that measures the utility loss from replacing a true measurement with a distorted measurement. Our work is an extension of [10] to the situations where dynamics at present. We propose a method to abstract out control performance of a dynamical system into a distortion metric, as well as a set of reasonable assumptions for the probabilistic dependencies between occupancy and location data, which allow us to re-write our privacy metric on time-series data into a static situation akin to that developed in [10]. 3. PRELIMINARIES This section collects the concepts we need before introducing the theoretical framework that characterizes the tradeoff between privacy and control performance in Section 4. Two models are described: the occupancy-location model that formulates the relationship between occupancy observations and individual location traces, and the model for the HVAC system. We will first consider an occupancy detection system that can collect noise-free or true occupancy, which is then processed by a distortion mechanism into the obfuscated data that the controller observes. We will see the distortion can be similarly applied to noisy occupancy, as elaborated in Section 4. 3.1 Occupancy-location model Suppose the building of interest consists of N zones represented by Z = {z0 , z1 , · · · , zN }, where a special zone z0 is added to refer to the outside of the building. Let O = {o1 , · · · , oM } denote the set of occupants. The location of occupant om at time k is a random variable denoted by (m) Xk which takes values in the set Z, for m = 1, · · · , M . The true occupancy of zone zn at time k is denoted by Ykn , n = 0, 1, · · · , N . Ykn takes values from {0, 1, · · · , M }, where M is the total number of occupants in the building. Note that the true occupancy and individual location traces are P (m) connected by Ykn = M = zn ], where 1[·] is the m=1 1[Xk indicator function. Additionally, we suppose that the controller observes a distorted version of the true occupancy, denoted by Vkn which takes values from {0, 1, . . . , M }. P(Vkn |Ykn ) represents the distortion mechanism we wish to design. If no distortion on the occupancy data is applied, then Vkn = Ykn . We fur(1:M ) (1) (M ) ther define some shorthands: Xk := {Xk , · · · , Xk }, 1:N 1 N Vk := {Vk , · · · , Vk }. We make the following assumptions. Assumption 1. The location traces for different occupants Q (1:M ) (m) are mutually independent: P(Xk )= M ). m=1 P(Xk Assumption 2. The location trace for any given occupant om , m ∈ {1, . . . , M }, has the first-order Markov property: (m) (m) (m) (m) P(Xk |Xk−1 , Xk−2 , . . . , X1 ) = (m) (m) P(Xk |Xk−1 ) (1) Assumption 3. The true occupancy Ykn is a sufficient stat(1:M ) istics for Vkn , i.e., P(Vkn |Xk ) = P(Vkn |Ykn ). Assumption 3 is naturally justified since the distribution of Vkn depends only on the value of Ykn in our distortion mechanism. The first two assumptions are necessary to design the optimal distortion method, but we will show that our distortion method will work on the real-world occupancy dataset, which provides a support for Assumption 1 and 2. These assumptions allow us to model occupancy and location traces via the Factorial Hidden Markov model (FHMM), illustrated in Figure 2. The FHMM consists of several independent Markov chains evolving in parallel, representing the location trace of each occupant. Since we only observe the aggregate occupancy information, the location traces are considered to be hidden states. (1) X k−1 … … … Occupant om’s … Location Trace (m ) X k−1 X k(m ) (m ) X k+1 Suppose the thermal comfort of the building space of interest is regulated by the HVAC system shown in Figure 3, which provides a system-wide Air Handling Unit (AHU) and Variable Air Volume (VAV) boxes distributed at the zones. In this type of HVAC system, the outside air is conditioned at the AHU to a setpoint temperature Ta by the cooling coil inside. The conditioned air, which is usually cold, is then supplied to all zones via the VAV box at each zone. The VAV box controls the supply air flow rate to the thermal zone, and heats up the air using the reheat coils at the box, if required. The control inputs are temperature and flow rate of the air supplied to the zone by its VAV box. The AHU outlet air temperature setpoint Ta is assumed to be constant in this paper. The HVAC system models described in the subsequent paragraphs will follow [22, 5, 15] closely1 . Outside Air Conditioned Air AHU VAV VAV Zone 1 Zone N n Vk−1 … … Figure 3: A schematic of a typical multi-zone commercial building with a VAV-based HVAC system. 1 Vk+1 Vk1 … … … … State model. With reference to the notations in Table 1, the continuous time dynamics for the temperature T n of zone zn can be expressed as n Vk+1 Vkn Figure 2: The graphical model representation of the FHMM model. The FHMM model can be specified by the transition probabilities and emission probabilities. The transition probabilities describe the mobility pattern of an occupant, which is denoted as a (N + 1) × (N + 1) transition matrix. We de(m) fine the transition matrix for occupant om as A(m) = [aij ], (m) (m) (m) i, j = 0, 1, · · · , N , where aij = P(Xk+1 = zj |Xk = zi ) for k = 0, 1, · · · , K − 1. The transition parameters can be learned from the occupancy data based on maximum likelihood estimation. If the prior knowledge about the past location traces is also available, it can be encoded as the prior distribution of transition parameters from a Bayesian point of view, and then the transition parameters can be learned via maximum a posteriori (MAP) estimation. We refer the readers to [34] for the details of parameter learning. The emission probabilities characterize the conditional distribution of distorted occupancy given the location of each occupant, defined by )= N Y n=1 (1:M ) P(Vkn |Xk Exhaust Air … Zone zm’s Occupancy (M ) X k+1 X k( M ) … 1 Vk−1 … … … (M ) X k−1 Zone z1’s Occupancy (1) X k+1 X k(1) … … Occupant oM’s … Location Trace (1:M ) HVAC system model Supply Air Occupant o1’s … Location Trace P(Vk1:N |Xk 3.2 )= N Y P(Vkn |Ykn ) (2) n=1 The above equalities result from Assumption 3, which, in other words, indicates that the distorted occupancy depends on individual location traces only via the true occupancy. Cn d n n n T = Rn · T + Qn + ṁn s cp (Ts − T ) dt (3) where the superscript n indicates that the associated quantities are attached to zone zn . T := [T 1 , · · · , T N ] is a vector of temperature of all N zones. Rn indicates the heat transfer among different zones and outside. Qn is the thermal load, which can be obtained by applying a thermal coefficient co to the number of occupants V n , i.e., Qn = co V n . The conn trol inputs U n := [ṁn s , Ts ] are the supply air mass flow rate n n and temperature. Assuming ṁn s , Ts and Q are zero-order held at sample rate ∆t, we can discretize (3) using the trapezoidal method and obtain a discrete-time model, which can be expressed as   n T n −Tkn Tk+1 +Tkn n C n k+1 = Rn·Tk +coVkn + ṁn (4) s,k cp Ts,k− ∆t 2 where k is the discrete time index and Tkn = Ttn |t=k∆t . Qn k, n ṁn and T are similarly defined. s,k s,k Cost function. The control objective is to condition the room while minimizing the energy cost. The power n consumption at time k consists of reheating power Ph,k = cp cp n n n n ṁs,k (Ts,k −Ta ), cooling power Pc,k = ηc ṁs,k (To −Ta ) and ηh 1 Controlling the flow rate is actually more preferable in building codes in consideration of energy efficiency. Herein, we consider both reheat temperature and flow rate are controllable, while the HVAC model with flow rate as the only control input is a simple application of our model. Param. ∆t cp Cn co R ηh ηc β re rh T T Ta ms ms Th Meaning Discretization step Thermal capacity of air Thermal capacity of the env. Thermal load per person Heat transfer vector Heating efficiency Cooling efficiency System parameter Electricity price Heating fuel price Upper bound of comfort zone Lower bound of comfort zone AHU outlet air temperature Minimum air flow rate Maximum air flow rate Heating coil capacity Value & Units 60s 1kJ/(kg · K) 1000kJ/K 0.1kW 0kW/K 0.9 4 0.5kW · s/kg 1.5 · 10−4 $/kJ 5 · 10−6 $/kJ 24◦ C 26◦ C 12.8◦ C 0.0084kg/s 1.5kg/s 40◦ C Table 1: Parameters used in the HVAC controller. 4. PRIVACY-ENHANCED CONTROL With the HVAC model established, we can now develop the mathematical framework to discuss a privacy-enhanced architecture. We will first introduce MI as the metric we use throughout the paper to quantify privacy, and then present a method to optimally design the distortion mechanism which minimizes the privacy loss within a pre-specified constraint on control performance. 4.1 Privacy metric Definition 1. [6] For random variables X and V , the mutual information is given by: I(X; V ) = H(X) − H(X|V ) (5) where H(X) and H(X|V ) represent entropy and conditional entropy, respectively. Let PX (x) = P(X = x), H(X) and H(X|V ) are defined as X H(X) = − PX (x) log(PX (x)) (6) x n Pf,k X  X
H(X|V ) = − (7) PV (v) PX|V (x|v) log PX|V (x|v) β ṁn s,k , fan power = where ηh and ηc capture the efficiencies for heating and cooling side, respectively. β stands for a system dependent constant. We introduce several parameters to reflect utility pricing, re for electricity and rh for heating fuel. These parameters may vary over time. Therefore, the total utility cost of zone zn from time k  =  PK n n n n 1, · · · , K is J = k=1 (re,k Pf,k +rh,k Ph,k +re,k Pc,k )∆t . Constraints. The system states and control inputs are subject to the following constraints: C1: T ≤ Tkn ≤ T , comfort range; C2: ṁs ≤ ṁn s,k ≤ ṁs , minimum ventilation requirement and maximum VAV box capacity; n C3: Ts,k ≥ Ta , heating coils can only increase temperature; n ≤ T h , heating coil capacity. C4: Ts,k These constraints hold at all times k and all zones {zn }N n=1 . MPC controller. Knitting together the models described above, we present an MPC-based control strategy for the HVAC system to efficiently accommodate for occupancy variations. In this control algorithm, we assume that the predicted occupancy during the optimization horizon to be the same as the instanteneous occupancy observed at the beginning of control horizon. It was shown to be in [15] that the control algorithm with this assumption can achieve comparable performance with the MPC that constructs explicit occupancy model to predict occupancy for future time steps. 1:N Let U1:K be the shorthand for {Ukn |k = 1, · · · , K, n = 1, · · · , N }. The optimal control inputs for the next K time P n steps are obtained by solving minU 1:N N n=1 J , subject to 1:K the inequality constraints C1-C4 and the equality constraint n n (4) and T1n = Tinit , ∀n = 1, · · · , N , where Tinit is the initial temperature of zone zn at each MPC iteration. We can see that the optimal control input is a function of the distorted occupancy that the controller sees and the initial temperature. We express this relationship explicitly by denoting the n n n optimal control action at zone zn as UM P C (V , Tinit ) . In addition, the energy cost incurred by applying the optimal n n n n n control action is denoted by JM P C (UM P C (V , Tinit ), Y ), where the second argument stresses that the actual control cost is dependent on the real occupancy. v x Remark. Entropy measures uncertainty about X, and conditional entropy can be interpreted as the uncertainty about X after observing V . By the definition above, MI is a measure of the reduction in uncertainty about X given knowledge of V . We can see that it is a natural measure of privacy since it characterizes how much information one variable tells about another. It is also worth noting that inference technologies evolve and MI as a privacy metric does not depend on any particular adversarial inference algorithm [29] as it models the statistical relationship between two variables. In this paper, we will be using the MI between location (1:M ) traces and occupancy observations, i.e., I(Xk ; Vk1:N ), as a metric of privacy loss. This metric reflects the reduction (1:M ) in uncertainty about location traces Xk due to observa1:N tions of Vk . As a proof of concept, we will verify that this metric serves as an accurate proxy for an adversary’s ability to infer individual location traces in the experiments. We further introduce some assumptions which allow us to simplify the expression of the privacy loss and obtain a form of MI that has direct relationship with the distortion mechanism P (Vkn |Ykn ) we wish to design. Based on results in ergodic theory [21], we know that the probability distribution of individual location traces will converge to a unique stationary distribution under very mild assumptions2 . For more details on stationary distributions, we refer the readesr to [21]. This observation justifies the following: (m) Assumption 4. The Markov chains Xk have a unique stationary distribution for all occupants om and are distributed according to those stationary distributions for all time steps k. Combining this assumption and the occupancy-location model we presented in the preceding section, we present a 2 Since there are only finitely many zones, a sufficient condition is the existence of a path from zi to zj with positive probability for any two zones zi and zj . proposition that allows us to great simplify the form of the privacy loss: Proposition 1. By Assumption 3, we have that: (1:M ) I(Xk ; Vk1:N ) = I(Yk1:N ; Vk1:N ) (8) By Assumption 4, we have that I(Yk1:N ; Vk1:N ) is a constant for all k, so we will drop the subscript: I(Y 1:N ; V 1:N ). Finally, by the various conditional independences introduced in Assumption 3: I(Y 1:N ; V 1:N ) = N X I(Y n ; V n ) (9) n=1 Remark. The result that I(Yk1:N ; Vk1:N ) is a constant value for all k allows us to design a single distortion mechanism P (V n |Y n ) for all time steps (note that we drop the subscript k to indicate the time-homogeneity of the distortion mechanism). By Proposition 1, minimization of privacy loss (1:M ) I(Xk ; Vk1:N ) can be conducted by minimizing a simpler P n n expression N n=1 I(Y ; V ). 4.2 Optimal distortion design We wish to find a distortion mechanism P (Y n |V n ) that can produce some perturbed occupancy data with minimum information leakage, while the performance of the controller using the perturbed occupancy data is on a par with that using true occupancy. To be specific, we will bound the difference of energy costs incurred by the controllers seeing distorted and real occupancy data. Let Tinit1 and Tinit2 be initial temperature of the controller using distorted and real occupancy, respectively. Ren n n n n n n n call that UM P C (V , Tinit ) and JM P C (UM P C (V , Tinit ), Y ) stand for the optimal control actions and the associated cost based on the distorted occupancy; correspondingly, if the controller sees the real occupancy data, the optimal control n n n action and the associated cost will be UM P C (Y , Tinit ) and n n n n n (Y , T ), Y ), respectively. We denote the (U JM init MP C PC resulting temperature after applying optimal control actions n n n n n as TM P C (UM P C (V , Tinit ), Y ), where the second argument emphasizes that the temperature evoluation depends on the true occupancy. We introduce the following constraints: ∀|Tinit1 − Tinit2 | ≤ ∆0T , y = 0, · · · , M , n = 1, · · · , N , C5: Cost difference constraint 
n n n EP(V n |Y n =y) JM P C UM P C (Tinit1 , V ), y − 
n n JM U (T , y), y ≤∆ (10) init2 PC MP C C6: Resulting temperature constraint 
n n n EP(V n |Y n =y) TM P C UM P C (Tinit1 , V ), y − 
n n TM ≤ ∆T P C UM P C (Tinit2 , y), y (11) C5 states that the cost difference between using the distorted occupancy measurements V n and using the ground truth occupancy measurements Y n is bounded by ∆ in expectation, for any possible value of Y n . The cost difference can be regarded as the control performance loss due to the usage of distorted data, and ∆ stands for the tolerance on the control performance loss. C5 alone is a one-step performance guarantee, that is, it only bounds the cost difference associated with a single MPC iteration. In practice, MPC is repeatedly solved from the new initial temperature, yielding new control actions and temperature trajectories. In order to offer a guarantee for future cost difference, we introduce another constraint C6 on the resulting temperature difference of one MPC iteration. The idea is that the resulting temperature will become the new initial temperature of the next MPC iteration. If the resulting temperature difference between using distorted occupancy data and using true occupancy data is bounded within a small interval ∆T , in the next MPC iteration C5 will provide a bound on cost difference for new initial temperatures that do not differ too much, since the cost difference constraint C5 is imposed to hold for all |Tinit1 − Tinit2 | ≤ ∆0T . Typically, ∆0T is set to be similar to ∆T , but a small value of ∆0T is preferred in order to assure the feasibility of the optimization problem (since the number of constraints increases with ∆0T ). Now, we are ready to present the main optimization for privacy-enhanced HVAC controller by combining the privacy metric and performance constraint just presented. Suppose the assumptions of Proposition 1 hold. Given the control performance loss tolerance ∆, the optimal distortion mechanism is given by solving: min n n P(V |Y ) n=1,··· ,N N X I(Y n ; V n ) (12) n=1 subject to the constraint C5-C6. ∆ serves as a knob to adjust the balance between privacy and the controller performance loss. Increasing ∆ leads to larger feasible set for the optimization problem, and thus a smaller value of MI (or privacy loss) is expected. Using the methodology presented in Section 3, we are able to calculate the terms inside the expectation in (10) and (11) for all |Tinit1 − Tinit2 | ≤ ∆0T and y = 0, · · · , M . Treating these as constants, calculating the optimal privacy-aware sensing mechanism is a convex optimization program, and can be efficiently solved. Additionally, since the constraints are enforced for each zone, the optimization (12) can actually be decomposed to N sub-problems and thus we can solve the optimal distortion scheme separately for each zone. Remark on noisy occupancy data. In the preceding privacy-enhanced framework, we consider the occupancy can be accurately detected. In practice, the occupancy data may be noisy itself, and thereby the distortion mechanism will be designed based on noisy occupancy Wkn instead of true occupancy Ykn . In effect, the distortion designed using noisy occupancy provides an upper bound on the privacy loss. That is, in practice we could use noisy occupancy to design the distortion mechanism and the realized privacy loss can only be lower than the minimum privacy loss obtained from the optimization. Note that we have the Markov relationship: Ykn → Wkn → Vkn when the distortion is applied to noisy data. Then the proof follows from the data processing inequality [6]. 5. 5.1 EVALUATION Experiment Setup Occupancy dataset. The occupancy data used in this paper is from the Augsburg Indoor Location Tracking Bench- mark [28], which includes location traces for 4 users in a office building with 15 zones. The location data in the benchmark dataset was recorded every second over a period of 4 to 9 weeks. Since the dataset contains some missing observations due to technical issues or the vacation interruption, we finally use the dataset from November 5th to 24th in our experiment, during which the location traces of all the 4 users are complete, and subsample the dataset with 1-minute resolution. The ground truth occupancy data was synthesized by aggregating the locations trace of each user. Table 2 shows two statistics of the benchmark dataset. Notably, of all transitions per day, 66.7% to 84.6% either start from or end at one’s own office, and office location can divulge one’s identity. This sheds light on why location traces of individual users can be actually inferred from the “anonymized” occupancy data. User 1 2 3 4 avg # of transitions per day 9.3 20.2 9.9 7.6 avg % of transitions from/to office per day 84.6% 75.4% 66.7% 75.5% Table 2: The average number of transitions each user made in each workday, and the average percentage of transitions from or to one’s office. Adversary inference. We consider the adversary to be an insider with authorized building automation system access. One can think of it as the worst case of privacy breach, because insiders not only learn the ancillary information that is public-available, but are familiar with building operation policies. To be specific, the following auxiliary information is assumed to be available to the adversary: (1) Building directory and occupant mobility patterns, encoded by the transition matrix of each occupant3 ; (2) Occupancy distortion mechanism designed by building manager. The adversary attempts to reconstruct the most probable location trace given the occupancy data and the auxiliary information. That is, the attack is to find the MAP of location traces given the other information. The approach to finding MAP is well known as Viterbi algorithm in HMM. However, Viterbi is infeasible in the FHMM case as the location traces to be solved reside in a exponentially large state space (N M × K). We propose a fast inference method based on Mixed Integer Programming, and thus more efficiently evaluate the adversary’s inference attack. The interested readers are referred to the code implementation of this paper for the details of the fast inference algorithm. Controller parameters. Without loss of generality, we consider the zones have the same thermal properties. The comfort range of temperature in the zones is defined to be within 24 − 26◦ C as in [26]. The minimum flow rate is set to be 0.084kg/s to fulfill the minimum ventilation requirement for 25m2 -sized zone as per ASHRAE ventilation standard 62.1-2013 [1]. The optimization horizon of the MPC is 120 min, and the control commands are solved for and updated every 15 min [15]. Other design parameters are shown in Table 1, which bascially follows the choices in [22]. 3 In the experiment, we use 4 days’ occupancy data and 2 days’ location traces to learn these parameters and the rest for evaluating our framework. Platform. The algorithms are implemented in MATLAB; The interior-point algorithm is used to solve the bilinear optimization problem in MPC. To encourage the research on the privacy-preserving controller, the codes involved in this paper will be open-sourced in http://people. eecs.berkeley.edu/˜ruoxijia/code. 5.2 5.2.1 Results MI as proxy for privacy We solve the MI optimization for different tolerance levels of control performance deterioration due to the usage of the distorted data, i.e., ∆, and obtain a set of optimal distortion designs and corresponding optimal values of MI. We then randomly perturb the true occupancy data using the different distortion designs, and infer location traces from the perturbed occupancy data. Monte Carlo (MC) simulations are carried out to assess results under the random distortion design. The inference accuracy is defined to be the ratio between the counts of correct location predictions over the total time steps. Figure 4 demonstrates the monotonically increasing relationship between adversarial location inference accuracy and MI, which justifies the usage of MI as a measure of privacy loss. When the adversary has perfect occupancy data, individual location traces can be inferred with accuracy of 96.81%. On the contrary, when the MI approaches zero, the adversary tends to estimate the location of each user to be constantly outside of the building, which is the best estimate the adversary can generate based on the uninformative occupancy data since people spend most of their time in a day outside. In this case, the inference accuracy is 77% but the adversary actually has no knowledge about users’ movement. This serves as a baseline of the adversarial location inference performance. Figure 4: The adversary location inference accuracy increases as MI increases. The black line and the band around it show the mean and standard deviation of inference accuracy across ten MC simulations, respectively. The black square shows the location inference accuracy if the adversary sees true occupancy data. The black triangle gives the accuracy when the adversary outputs a constant location estimate. 5.2.2 Utility-Privacy Trade-off Figure 5 shows the variation of privacy loss and controller performance loss with respect to different choices of ∆, which is the theoretical guarantee on controller performance loss. It is evident that privacy loss and control performance loss Visualization of Distortion Matrix #10 -3 1.5 3 0 1 0.03 0.00 0.00 0.00 0 0.67 0.33 0.00 0.00 0.00 0 0.07 0.93 0.00 0.00 0.01 1 0.01 0.99 0.00 0.00 0.00 1 0.11 0.88 0.00 0.00 0.01 1 0.07 0.93 0.00 0.00 0.01 2 0.00 0.29 0.44 0.27 0.00 2 0.04 0.69 0.00 0.00 0.27 2 0.07 0.93 0.00 0.00 0.01 3 0.00 0.00 0.01 0.99 0.00 3 0.01 0.42 0.00 0.00 0.58 3 0.00 0.74 0.00 0.00 0.26 0.84 4 0.01 0.06 0.10 0.19 0.64 5.6 5.65 5.7 5.75 Guarantee on Controller Performance Loss 0 5.8 3 #10 -3 Figure 5: The changes of MI and actual control cost difference between using true and perturbed occupancy as the theoretical control cost difference changes. The blue dot line and errorbar demonstrate the mean and standard deviation of actual control cost difference across ten MC simulations, respectively. exhibit opposite trends as ∆ changes. The privacy loss, measured by MI, monotonically decreases as ∆ gets larger. This is the manifestation of the intrinsic utility-privacy trade-off embedded in the main optimization problem (12). As the performance constraint ∆ is more relaxed, a smaller value of MI can be attained and thus privacy can be better preserved. The actual performance loss, measured by the HVAC control cost difference (between using distorted and true data) averaged across different MPC iterations and difference zones, generally increases with ∆ and is upper bounded by ∆. This indicates that the theoretical constraint on controller performance loss in our framework is effective and can actually provide a guarantee on the actual controller performance. We can see that the bound is far from tight, since the framework enforces the constraints on the controller performance for every possible true occupancy value to ensure the robustness while in practice the occupancy distribution is very spiked about the mean occupancy. Figure 6 visualizes the distortion mechanism obtained by solving the MI under different choices of the tolerance on the control performance loss ∆. It can be clearly seen that the mechanism creates a higher level of distortion as ∆ increases. When ∆ is small, the resulting distortion matrix assigns most probability mass on the diagonal, i.e., the occupancy is very likely to keep unperturbed. As ∆ gets larger, the distortion mechanism tends to have the same rows, in which case the distribution of distorted occupancy data is invariant under the change of true occupancy and MI between true occupancy and perturbed occupancy, i.e., the privacy loss, tends to be zero. We also plot the temperature evolution under different distortion levels. Since we enforce a hard constraint on temperature, we can see that the zone temperature stays within the comfort zone for all ∆’s. However, larger ∆ would lead to a larger deviation from the temperature controlled using the true occupancy. Comparison with Other Methods We compare the performance of the HVAC controller using our optimally perturbed data against using unperturbed occupancy data, fixed occupancy schedule as well as randomly perturbed data by other distortion methods. In Figure 7a we plot the privacy loss and control cost for con- 0.00 0.00 0.00 0.00 0.97 0.03 26.5 0.99 0.01 0.00 26 0.29 0.00 0.00 4 0.00 0 0.00 0.02 0.04 0.09 " = 0.0055 2 3 0 1 0.67 0.33 " = 0.0058 2 3 0 1 0.08 0.91 4 gets0.00 larger Δ 0.00 0.00 0.8 0.7 0.6 0.5 0.3 0.00 0.44 0.00 0.27 0.02 0.83 25 0.00 0.00 0.00 0.00 1 0.11 0.88 0.00 0.00 Optimal Controller 0.00 2 0.01 0.75 0.00 0.00 "=0.005402 "=0.0055 3 0.00 0.43 0.00 0.00 0.11 "=0.0058 0.02 0.04 0.09 0.99 4 0.00Zone Comfort 0.1 4 0.9 0 0.00 0.00 0.00 Temperature Evolutions for Different Distortion Levels 0.0025.5 0.05 0.00 4 0.99 " = 0.005402 1 2 3 0 0 4 5.2.3 4 0.2 4 2 5.55 4 0.4 1 5.5 " = 0.0058 2 3 1 0.97 Temperature (°C) bits 0.5 1 dollars per 15 min per zone 1 5.45 " = 0.0055 2 3 0 0 4 0.9 2 0 5.4 " = 0.005402 1 2 3 0 Privacy Loss Actual Performance Loss 0.01 1 0.08 0.91 0.00 0.00 0.00 0.24 2 0.08 0.91 0.00 0.00 0.00 0.57 3 0.00 0.72 0.02 0.03 0.23 0.84 4 0.01 0.06 0.10 0.19 0.64 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 24.5 24 23.5 03:00 06:00 09:00 12:00 15:00 18:00 21:00 24:00 Time of day (hrs) Figure 6: Illustration of distortion matrix P (V |Y ) under different controller performance guarantees. The row index corresponds to the value of Y , while colomn index corresponds to V . The zone temperature traces resulted from the controllers using occupancy data that is randomly distorted by different distortion matrices are also shown. trollers that use the various forms of occupancy data. Fixed occupancy schedule (assuming maximum occupancy during working hours and zero otherwise) exposes zero information about individual location traces, but cannot adapt to occupancy variations and thus incurs considerable control cost. The controller based on clean occupancy data is most costeffective but discloses maximum private information. One of the random distortion method to be compared is uniform distortion scheme in which the true occupancy is perturbed to some value between zero to maximum occupancy with equal probability. We carry out 10 MC simulations to obtain the control cost incurred under this random perturbation scheme. It can be seen that the uniform distortion scheme protects the private information with compromised controller performance. A natural question arising is if the current occupancy sensing systems provide intrinsic privacy-preserving features as there always exists occupancy estimation errors. Can we use a cheaper and inaccurate occupancy sensor to acqiure privacy? As is suggested by the occupancy sensing results in [19], the estimation noise of a real occupancy sensing system can be modeled by a multinomial distribution which has most probability mass at zero. Inspired by this, we use the following multinomial distortion schemes to imitate a real occupancy sensing system with disparate accuracies acc,   acc, V = y n n 1−acc , V = y−1 or y+1 if y 6= 0 (13) P (V |Y = y) = 2  1−acc , V = 1 or 2, if y = 0 2 Again, MC simulations are performed to evaluate the control performance under this random perturbation, and the results are shown in Figure 7a. It can be seen that when the privacy loss is relatively large (or data is slightly distorted), the control cost of our optimal noising scheme and the multinomial noising scheme do not differ too much. This is because at this level of privacy loss the two distortion schemes behave similarly, as shown in Figure 6, where the Utility-Privacy Trade-off for Different Schemes #10 -3 5.8 Optimal Distortion Fixed Schedule Unperturbed Occupancy Data Uniform Distortion Multinomial Distortion 11 6. 0.9 5.75 0.8 5.7 10.5 0.7 5.65 5.6 0.6 9.5 acc 10 " Cumulative Control Cost (dollars/day) 11.5 5.55 0.5 9 5.5 0.4 8.5 8 5.45 0 0.5 1 1.5 2 2.5 3 0.3 3.5 Privacy Loss (bits) (a) Utility-Privacy Trade-off for Different Schemes #10 -3 5.8 Optimal Distortion Fixed Schedule Unperturbed Occupancy Data Uniform Distortion Multinomial Distortion 11.5 11 0.9 5.75 0.8 0.7 10.5 7. 5.6 0.6 acc 5.65 10 9.5 5.55 0.5 9 5.5 0.4 8.5 5.45 8 0 0.5 1 1.5 2 2.5 3 In this paper, we present a tractable framework to model the trade-off between privacy and controller performance in a holistic manner. We take occupancy-based HVAC controller as an example where the objective is to utilize occupancy data to enable smart controls over the HVAC system while protect individual location information from being inferred from the occupancy data. We use MI as the measure of privacy loss, and formulate the privacy-utility trade-off by a convex optimization problem that minimizes the privacy loss subject to a pre-specified controller performance constraint. By solving the optimization problem, we can obtain a mechanism that injects optimal amount of noise to occupancy data to enhance privacy with control performance guarantee. We verify our framework using real-world occupancy data and simulated building dynamics. It is shown that our theoretical framework is able to provide guidelines for practical privacy-enhanced occupancy-based HVAC system design, and reaches a better balance of privacy and control performance compared with other occupancy-based controllers. 5.7 " Cumulative Control Cost (dollars/day) 12 CONCLUSIONS 3.5 0.3 Privacy Loss (bits) (b) Figure 7: Comparison of the privacy-utility trade-off of controllers using different forms of occupancy data, evaluated based on (a) real-world occupancy data and (b) synthesized data. occupancy keeps untainted with high probability. But as the privacy loss decreases, our optimal noising scheme’s intelligent noise placement begins to significantly improve control performance. In addition, our optimal distortion Pareto dominates the other schemes. To investigate the scalability of our proposed scheme, we create synthetic data that simulates location traces for 15 occupants based on the Augsburg dataset. 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1 Fundamental Limits of Covert Communication over MIMO AWGN Channel Amr Abdelaziz and C. Emre Koksal Department of Electrical and Computer Engineering The Ohio State University Columbus, Ohio 43201 arXiv:1705.02303v4 [] 13 Mar 2018 Abstract Fundamental limits of covert communication have been studied for different models of scalar channels. It was shown that, √ over n independent channel uses, O( n) bits can be transmitted reliably over a public channel while achieving an arbitrarily low probability of detection (LPD) by other stations. This result is well known as the square-root law and even to achieve this diminishing rate of covert communication, all existing studies utilized some form of secret shared between the transmitter and the receiver. In this paper, we establish the limits of LPD communication over the MIMO AWGN channel. In particular, using relative entropy as our LPD metric, we study the maximum codebook size for which the transmitter can guarantee reliability and LPD conditions are met. We first show that, the optimal codebook generating input distribution under δ-PD constraint is the zero-mean Gaussian distribution. Then, assuming channel state information (CSI) on only the main channel at the transmitter, we derive the optimal input covariance matrix, hence, establishing scaling laws of the codebook size. We evaluate the codebook scaling rates in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO). We show that, in the asymptotic block-length regime, square-root law still holds for the MIMO AWGN. Meanwhile, in massive √ MIMO limit, the codebook size, while it scales linearly with n, it scales exponentially with the number of transmitting antennas. Further, we derive equivalent results when no shared secret is present. For that scenario, in the massive MIMO limit, higher covert rate up to the non-LPD constrained capacity still can be achieved, yet, with much slower scaling compared to the scenario with shared secret. The practical implication of our result is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay. Index Terms LPD communication, Covert MIMO Communication, MIMO physical layer security, LPD Capacity. I. I NTRODUCTION Conditions for secure communication under a passive eavesdropping attack fall in two broad categories: 1) low probability of intercept (LPI). 2) low probability of detection (LPD). Communication with LPI requires the message exchanged by two legitimate parties to be kept secret from an illegitimate adversary. Meanwhile, LPD constrained communication is more restrictive as it requires the adversary to be unable to decide whether communication between legitimate parties has taken place. Fundamental limits of LPD constrained communication over scalar AWGN has been established in [1] where the square-root law for LPD communication was established. Assuming a shared secret of sufficient length between transmitter √ and receiver, square-root law states that, over n independent channel uses of an AWGN channel, transmitter can send O( n) bits reliably to the receiver while keeping arbitrary low probability of detection at the adversary. In this paper, we study the fundamental limits of communication with LPD over MIMO AWGN channels. Consider the scenario in which a transmitter (Alice) wishes to communicate to a receiver (Bob) while being undetected by a passive adversary (Willie) when all nodes are equipped with multiple antennas. To that end, Alice wish to generate a codebook that satisfies both reliability, in terms of low error probability , over her channel to Bob and, in the same time, ensures, a certain maximum PD, namely δ, at Willie. Denote the maximum possible size of such codebook by Kn (δ, ). In this paper, we are interested in establishing the fundamental limits of Kn (δ, ) in the asymptotic length length regime and in the limit of large number of transmitting antenna. First we show that, the maximum codebook size is attained when the codebook is generated according to zero mean circular symmetric complex Gaussian distribution. We establish this result building upon the the Principle Minimum Relative Entropy [2] and Information Projection [3]. Some p of our findings can be summarized as follows. For an isotropic Willie channel, we show that Alice can transmit O(N n/M ) bits reliably in n independent channel uses, where N and M are the number of active eigenmodes of Bob and Willie channels, respectively. Further, we evaluate δ-PD rates in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO). We show that, while the square-root law still holds for the MIMO AWGN, the number of bits that can be transmitted covertly scales exponentially with the number of transmitting antennas. This work was submitted in part to IEEE CNS-2017. This work was in part supported by the National Science Foundation under Grants NSF NeTs 1618566 and 1514260 and Office of Naval Research under Grant N00014-16-1-2253. 2 TABLE I S UMMARY OF R ESULTS Result Main Channel Adversary Channel Shared Secret Kn (δ, ) Scales Like Theorems 2&3 Deterministic Bounded Spectral norm Yes N Theorem 4 Deterministic and of Unit Rank Deterministic and of Unit Rank Yes Theorems 5&6 Deterministic Bounded Spectral norm No N/M Theorem 7 Deterministic and of Unit Rank Deterministic and of Unit Rank No 1/cos2 (θ) Theorem 8 Deterministic and of Unit Rank Unit Rank chosen uniformly at random Yes Theorem 9 Deterministic and of Unit Rank Unit Rank chosen uniformly at random No √ p n/M n/ cos2 (θ) n c (1 + √ )(Na −2)/2 K 2 Na n where c is constant independent on n and Na . r 1 c √ (1 + )(Na −2)/2 where n K Na c is constant independent on n and Na . r n c (1 + √ )(Na −2)/2 where Na is the K 2 Na n number of transmitting antennas, K is a universal constant and c is constant independent on n and Na . Further, we derive the scaling of Kn (δ, ) with no shared secret between Alice and Bob. In particular, we show that achieving better covert rate is a resource arm race between Alice, Bob and Willie. Alice can transmit O(N/M ) bits reliably in n independent channel uses, i.e., the covert rate is in the order of the ratio between active eigenmodes of both channels. The practical implication of our findings is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay. The results obtained in this paper are summarized in Table I1 . The contributions of this work can be summarized as follows: • Using the Principle Minimum Relative Entropy [2] and Information Projection [3], we show that the Kn (δ, ) is achievable when the codebook is generated according to zero mean complex Gaussian distribution in MIMO AWGN channels. • With the availability of only the main CSI to Alice, we evaluate the optimal input covariance matrix under the assumption that Willie channel satisfies a bounded spectral norm constraint [4], [5]. Singular value decomposition (SVD) precoding is shown to be the optimal signaling strategy and the optimal water-filling strategy is also provided. • We evaluate the block-length and massive MIMO asymptotics for Kn (δ, ). We show that, while the square-root law cannot be avoided, Kn (δ, ) scales exponentially with the number of antennas. Thus, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay. • We evaluate scaling laws of Kn (δ, ) when there is no shared secret between Alice and Bob in both limits of large block length and massive MIMO. Related Work. Fundamental limits of covert communication have been studied in literature for different models of scalar channels. In [6], LPD communication over the binary symmetric channel was considered. It was shown that, square-root law holds for the binary symmetric channel, yet, without requiring a shared secret between Alice and Bob when Willie channel is significantly noisier. Further, it was shown that Alice achieves a non-diminishing LPD rate, exploiting Willie’s uncertainty about his own channel transition probabilities. Recently in [7], LPD communication was studied from a resolvability prespective for the discrete memoryless channel (DMC). Therein, a trade-off between the secret length and asymmetries between Bob and Willie channels has been studied. Later in [8], the exact capacity (using relative entropy instead of total variation distance as LPD measure) of DMC and AWGN have been characterized. For a detailed summary of the recent results for different channel models on the relationship between secret key length, LPD security metric and achievable LPD rate, readers may refer to Table II in [6]. LPD communication over MIMO fading channel was first studied in [9]. Under different assumption of CSI availability, the author derived the average power that satisfies the LPD requirement. However, the authors did not obtain the square-root law, since the focus was not on the achievable rates of reliable LPD communication. Recently in [10], LPD communication with multiple antennas at both Alice and Bob is considered when Willie has only a single antenna over Rayleigh fading channel. An approximation to the LPD constrained rate when Willie employs a radiometer detector and has uncertainty about his noise variance was presented. However, a full characterization of the capacity of MIMO channel with LPD constraint was not established. More precisely, for a unit rank MIMO channel, we show that Kn (δ, ) scales as 1θ is the angle between right singular vectors of main and adversary channels in the unit rank channel model. 3 Despite not explicitly stated, the assumption of keeping the codebook generated by Alice secret from Willie (or at least a secret of sufficient length [1], [7]) is common in all aforementioned studies of covert communication. Without this assumption, LPD condition cannot be met along with arbitrarily low probability of error at Bob. This is because, when Willie is informed about the codebook, he can decode the message using the same decoding strategy as that of Bob [1]. Only in [6], square-root law was obtained over binary symmetric channel without this assumption when Willie channel is significantly noisier than that of Bob, i.e., when there is a positive secrecy rate over the underlying wiretap channel. Despite the availability of the codebook at Willie, [6] uses the total variation distance as the LPD metric. In short, the square root law is shown to be a fundamental upper limitation that cannot be overcome unless the attack model is relaxed to cases such as the lack of CSI or the lack of the knowledge of when the session starts at Willie. Here, we do not make such assumptions on Willie and solely take advantage of increasing spatial dimension via the use of MIMO. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT In the rest of this paper we use boldface uppercase letters for vectors/matrices. Meanwhile, (.)∗ denotes conjugate of complex number, (.)† denotes conjugate transpose, IN denotes identity matrix of size N , tr(.) denotes matrix trace operator, |A| denotes the determinant of matrix A and 1m×n denotes a m × n matrix of all 1’s. We say A  B when the difference A − B is positive semi-definite. The mutual information between two random variables x and y denoted by I(x; y) while lim denotes the limit inferior. We use the standard order notation f (n) = O(g(n)) to denote an upper bound on f (n) that is asymptotically tight, i.e., there exist a constant m and n0 > 0 such that 0 ≤ f (n) ≤ mg(n) for all n > n0 . A. Communication Model We consider the MIMO channel scenario in which a transmitter, Alice, with Na ≥ 1 antennas aims to communicate with a receiver, Bob, having Nb ≥ 1 antennas without being detected by a passive adversary, Willie, equipped with Nw ≥ 1 antennas. The discrete baseband equivalent channel for the signal y and z, received by Bob and Willie, respectively, can be written as: y = Hb x + eb , z = Hw x + ew , (1) where x ∈ CNa ×1 is the transmitted signal vector constrained by an average power constraint E[tr(xx† )] ≤ P . Also, Hb ∈ CNb ×Na and Hw ∈ CNw ×Na are the channel coefficient matrices for Alice-Bob and Alice-Willie channels respectively. Throughout this paper, unless otherwise noted, Hb and Hw are assumed deterministic, also, we assume that Hb is known to all parties, meanwhile, Hw is known only to Willie. We define N , min{Na , Nb } and M , min{Na , Nw }. Finally, eb ∈ CNb ×1 and ew ∈ CNw ×1 are an independent zero mean circular symmetric complex Gaussian random vectors for both destination and adversary channels respectively, where, eb ∼ CN (0, σb2 INb ) and ew ∼ CN (0, σe2 INw ). We further assume Hw lies in the set of matrices with bounded spectral norms: n √ o Sw = Hw : kHw kop ≤ γw n o = Ww , H†w Hw : kWw kop ≤ γw , (2) where kAkop is the operator (spectral) norm of A, i.e., the maximum eigenvalue of A. The set Sw incorporates all possible Ww that is less than or equal to γw Î (in positive semi-definite sense) with no restriction on its eigenvectors, where Î is diagonal matrix with the first M diagonal elements equal to 1 while the rest Na − M elements of the diagonal are zeros. Observe that, kWw kop represents the largest possible power gain of Willie channel. Unless otherwise noted, throughout this paper we will assume that Hw ∈ Sw . B. Problem Statement Our objective is to establish the fundamental limits of reliable transmission over Alice to Bob MIMO channel, constrained by the low detection probability at Willie. Scalar AWGN channel channel have been studied in [8], we use a formulation that follows closely the one used therein while taking into consideration the vector nature of the MIMO channel. Alice employs a stochastic encoder with blocklength2 nNa , where n is the number of channel uses, for message set M consists of: 1) An encoder M 7→ CnNa , m 7→ xn where x ∈ CNa . 2) A decoder CnNb 7→ M, yn 7→ m̂ where y ∈ CNb . Alice chooses a message M from M uniformly at random to transmit to Bob. Let us denote by H0 the null hypothesis under which Alice is not transmitting and denote by P0 the probability distribution of Willie’s observation under the null hypothesis. 2 Note that, when Alice has nN bits to transmit, two alternative options are available for her. Either she splits the incoming stream into N streams of a a n bits each and use each stream to select one from 2n messages for each single antenna, or, use the entire nNa bits to choose from 2nNa message. The latter of these alternatives provides a gain factor of Na in the error exponent, of course, in the expense of much greater complexity [11], [12]. However, in the restrictive LPD scenario, Alice would choose the latter alternative as to achieve the best decoding performance at Bob. 4 Conversely, let H1 be the true hypothesis under which Alice is transmitting her chosen message M and let P1 be the probability distribution of Willie’s observation under the true hypothesis. Further, define type I error α to be the probability of mistakenly accepting H1 and type II error β to be the probability of mistakenly accepting H0 . For the optimal hypothesis test generated by Willie we have [13] α + β = 1 − V(P0 , P1 ), (3) where V(P0 , P1 ) the total variation distance between P0 and P1 and is given by 1 kp0 (x) − p1 (x)k1 , (4) 2 where p0 (x) and p1 (x) are, respectively, the densities of P0 and P1 and k.k1 is the L1 norm. The variation distance between P0 and P1 is related to the Kullback–Leibler Divergence (relative entropy) by the well known Pinsker’s inequality [14]: r 1 V(P0 , P1 ) ≤ D(P0 k P1 ) (5) 2 where V(P0 , P1 ) = D(P0 k P1 ) = EP0 [log P0 − log P1 ] . (6) Note that, since the channel is memoryless, across n independent channel uses, we have D (Pn0 k Pn1 ) = nD (P0 k P1 ) (7) by the chain rule of relative entropy. Accordingly, for Alice to guarantee a low detection probability at Willie’s optimal detector, she needs to bound V(Pn0 , Pn1 ) above by some δ chosen according to the desired probability of detection. Consequently, she ensure that the sum of error probabilities at Willie is bounded as α + β ≥ 1 − δ. Using (5), Alice can achieve her goal by designing her signaling strategy (based on the amount of information available) subject to 2δ 2 . (8) n Throughout this paper, we adopt (8) as our LPD metric. Thus, the input distribution used by Alice to generate the codebook has to satisfy (8). As in [8], our goal is to find the maximum value of log |M| for which a random codebook of length nNa exists and satisfies (8) and whose average probability of error is at most . We denote this maximum by Kn (δ, ) and we define Kn (δ, ) . (9) L , lim lim √ ↓0 n→∞ 2nδ 2 √ Note that L has unit nats. We are interested in the characterization of L under different conditions of Bob and Willie channels in order to derive scaling laws for the number of covert bits over MIMO AWGN channel. We first give the following Proposition which provides a general expression for L by extending Theorem 1 in [8] to the MIMO AWGN channel with infinite input and output alphabet. D(P0 k P1 ) ≤ Proposition 1. For the considered MIMO AWGN channel, r L= max lim {fn (x)} n→∞ tr(En [xx† ])≤P n 2δ 2 I(fn (x), fn (y)) Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0 (10) where {fn (x)} is a sequence of input distributions over CNa and En [·] denotes the expectation with respect to fn (x). Before we give the proof of Proposition 1, we would like to highlight why the second moment constraint on the input signal is meaningful in our formulation. It was explicitly stated in [8] that, an average power constraint on the input signal is to be superseded by the LPD constraint. The reason is that, the LPD constraint requires the average power to tend to zero as the block length tends to ∞. However, unlike the single antenna setting, over a MIMO channel, there exist scenarios in which the LPD constraint can be met without requiring the Alice to reduce her power. In the sequel, we will discuss such scenarios in which the power constraint remains active. Proof. First, using the encoder/decoder structure described above, we see that the converse part of Theorem 1 in [8] can be directly applied here. Meanwhile, the achievability part there was derived based on the finiteness of input and output alphabet. It was not generalized to the continuous alphabet input over scalar AWGN channel. Rather, the achievability over AWGN channel was shown for Gaussian distributed input in Theorem 5. Here, we argue that, showing achievability for Gaussian distributed input is sufficient and, hence, we give achievability proof in Appendix B that follow closely the proof of Theorem 5 5 in [8]. Unlike the non LPD constrained capacity which attains its maximum when the underlying input distribution is zero mean complex Gaussian, it is not straightforward to infer what input distribution is optimal. However, using the Principle Minimum Relative Entropy [2] and Information Projection [3], we verify that, the distribution P1 that minimizes D(P0 k P1 ) is the zero mean circularly symmetric complex Gaussian distribution.  Further, we provide a more convenient expression for L in the following Theorem which provides an extension of Corollary 1 in [8] to the MIMO AWGN channel. Theorem 1. For the considered MIMO AWGN channel, r n L = lim n→∞ 2δ 2 max fn (x) tr(En [xx† ])≤P I(fn (x), fn (y)) Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0 (11) where fn (x) is the input distribution over CNa and En [·] denotes the expectation with respect to fn (x).  Proof. The proof is given in Appendix C. Now, since we now know that zero mean circular symmetric Gaussian input distribution is optimal, the only  complex  remaining task is to characterize the covariance matrix, Q = E xx† , of the optimal input distribution. Accordingly, (11) can be rewritten as: r Wb Q n (12) max log INa + L = lim n→∞ 2δ 2 Q0 σb2 tr(Q)≤P Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0, where Wb , H†b Hb . Further, we can evaluate the relative entropy at Willie as follows (see Appendix A for detailed derivation): 1 Hw QH†w + INw 2 σw ( −1 ) 1 † Hw QHw + INw − Nw . + tr 2 σw D (P0 k P1 ) = log (13) In this paper, we are mainly concerned with characterizing L when Alice knows only Hb . To that end, let us define: Cpd (δ) , max log INa + Q0 tr(Q)≤P Wb Q σb2 (14) Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0. r Clearly, L = limn→∞ n Cpd (δ). In what follows, we characterize Cpd (δ) and, hence, L under different models of Hb and 2δ 2 Hw . Remark 1. Observe that, since Bob and Willie channels are different, Willies does not observe the same channel output as Bob. Hence, there exist situations in which D(Pn0 k Pn1 ) does not increase without bound as n tends to infinity. In the next Section, we provide some examples. III. M OTIVATING E XAMPLES Consider the scenario in which both of Willie’s and Bob’s channels are of unit rank. Accordingly, we can write H◦ = λ◦ v◦ u†◦ , where v◦ ∈ CN◦ and u◦ ∈ CNa are the left and right singular vectors of H◦ where the subscript ◦ ∈ {e, b} used to denote Bob and Willie channels respectively. Under the above settings, consider the scenario in which Alice has a prior (non-causal) knowledge about both channels. Alice task is to find Q∗ that solve (14). Note that, since both channels are of unit rank, so is Q∗ and it can be written as Q∗ = Pth q∗ q∗† where Pth ≤ P is the power threshold above which she will be detected by Willie. Now suppose that Alice choose q∗ to be the solution of the following optimization problem: max < q † , ub > q kqk=1 Subject to < q † , uw >= 0, (15) 6 whose solution is given by   I − uw u†w ub i . q∗ = h I − uw u†w ub (16) The beamforming direction q∗ is known as null steering (NS) beamforming [15], that is, transmission in the direction orthogonal to Willie’s direction while maintaining the maximum possible gain in the direction of Bob. Recall that Willies channel is of unit rank and is in the direction uw , thus, the choice of Q = Q∗ implies that Hw Q∗ H†w = 0. Accordingly, Σ1 = Σ0 , i.e, Willie is kept completely ignorant by observing absolutely no power from Alice’s transmission. More precisely, D (Pn0 k Pn1 ) = 0. However, this doesn’t mean that Alice can communicate at the full rate to Bob as if Willie was not observing, rather, the LPD constraint forced Alice to sacrifice some of its power to keep Willie oblivious of their transmission. More precisely, the effective power seen by Bob scales down with cosine the angle between ub and uw . In the special case when < ub , uw >= 0, Alice communicate at the full rate to Bob without being detected by Willie. In addition, the codebook between Alice and Bob need not to be kept secret from Willie. That is because the power observed at Willie from Alice transmission is, in fact, zero. Fig. 1. Radiation pattern as a function of the number of transmitting antennas. When number of antennas gets large, < ub , uw >→ 0. IV. Cpd (δ) WITH S ECRET C ODEBOOK With uncertainty about Willie’s channel, Hw ∈ Sw , it is intuitive to think that Alice should design her signaling strategy against the worst (stronger) possible Willie channel. We first derive the worst case Willie channel, then, we establish the saddle point property of the considered class of channels in the form of min max = max min, where the maximum is taken over all admissible input covariance matrices and the minimum is over all Hw ∈ Sw . Thus, we show that Cpd (δ) equals to the Cpd (δ) evaluated at the worst possible Hw . A. Worst Willie Channel and Saddle Point Property w To characterize Cpd (δ) when Hw ∈ Sw , we need first to establish the worst case Cpd (δ) denoted by Cpd (δ). Suppose we w have obtained Cpd (δ) for every possible state of Hw , then, Cpd (δ) is the minimum Cpd (δ) over all possible state of Hw . First, let us define Wb Q . (17) R(Ww , Q, δ) = log INa + σb2 w We give Cpd (δ) in the following proposition. Proposition 2. Consider the class of channels in (2), for any Q < 0 satisfies tr{Q} ≤ P and Ww ∈ Sw we have: w Cpd (δ) = min Ww ∈Sw Subject max R(Ww , Q, δ) Q0 tr(Q)≤P to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0 = max R(γw Î, Q, δ) Q0 tr(Q)≤P Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0 (18) 7 i.e., the worst Willie channel is isotropic.  Proof. See Appendix D. w w Proposition 2 establishes Cpd (δ). The following proposition proves that Cpd (δ) = Cpd (δ) by establishing the saddle point property of the considered class of channels. Proposition 3. (Saddle Point Property.) Consider the class of channels in (2), for any Q < 0 satisfies tr{Q} ≤ P and Ww ∈ Sw we have: Cpd (δ) = min Ww ∈Sw Subject = max max R(Ww , Q, δ) Q0 tr(Q)≤P to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0 min R(Ww , Q, δ) Q0 Ww ∈Sw tr(Q)≤P Subject to: D(Pn0 w = Cpd (δ). k Pn1 ) − 2δ 2 ≤ 0 (19) Proof. By realizing that, for any feasible Q, the function D (P0 k P1 (Ww )) is monotonically increasing in Ww , we have that min R(Ww , Q, δ) = R(γw Î, Q, δ) Ww ∈Sw Subject to: D(Pn0 k Pn1 ) − 2δ 2 . (20)  Hence, the required result follows by using proposition 2. B. Evaluation of Cpd (δ) In light of the saddle point property established in the previous Section, in this Section we characterize Cpd (δ) by solving (19) for the optimal signaling strategy, Q∗ . We give the main result of this Section in the following theorem. Theorem 2. The eigenvalue decomposition of the capacity achieving input covariance matrix that solves (14) is given by Q∗ = Ub ΛU†b where Ub ∈ CNa ×Na is the matrix whose columns are the right singular vectors of Hb and Λ is a diagonal matrix whose diagonal entries, Λii , are given by the solution of −1 λ =(σb2 λ−1 i (Wb ) + Λii )  2 −2  2 −1 ! σw σw +η + Λii + Λii − γw γw where λ and η are constants determined from the constraints tr {Q} ≤ P and (8), respectively. Moreover,   N X Λii λi (Wb ) log 1 + Cpd (δ) = σb2 i=1 (21) (22) where λi is the ith non zero eigenvalue of Wb .  Proof. See Appendix E. The result of Theorem 2 provides the full characterization of Cpd (δ) of the considered class of channels. It can be seen that, the singular value decomposition (SVD) precoding [12] is the optimal signaling strategy except for the water filling strategy in (21) which is chosen to satisfy both power and LPD constraints. Unlike both MIMO channel without security constraint and MIMO wiretap channel, transmission with full power is, indeed, not optimal. Let X Pth , Λii , (23) i be the maximum total power that is transmitted by Alice. An equivalent visualization of our problem is that Alice need to choose a certain power threshold, Pth , to satisfy the LPD constraint. However, again, Pth is distributed along the eigenmodes using conventional water filling solution. Although it is not straightforward to obtain a closed form expression3 for Cpd (δ) and, hence, L, we could obtain both upper and lower bounds on Cpd (δ) which leads to upper and lower bounds on L. Based on the obtained bounds, we give the square-root law for MIMO AWGN channel in the following Theorem. 3 Using Mathematica, Λii was found to be an expression of almost 30 lines which does not provide the required insights here. 8 Theorem 3 (Square-root Law of MIMO AWGN channel). For the considered class of channels, the following bounds on Cpd (δ) holds ! √ 2 N X 2σw δλi (Wb ) √ log 1 + ≤ Cpd (δ) σb2 γw nM i=1 ! √ 2 N X 2σw ξδλi (Wb ) √ ≤ (24) log 1 + σb2 γw nM i=1 where ξ ≥ 1 is a function of δ that approaches 1 as δ goes to 0. Moreover, N N 2 2 X X σw λi (Wb ) σw ξλi (Wb ) √ √ ≤ L ≤ . 2γ 2γ σ M σ b w b w M i=1 i=1 Proof. We give both achievability and converse results in Appendix F. (25)  Theorem 3 extends the square-root lawpfor scalar AWGN channel to the MIMO AWGN channel. In particular, it states that Alice can transmit a maximum of O(N n/M ) bits reliably to Bob in n independent channel uses while keeping Willie’s sum of error probabilities lower bounded by 1 − δ. The interesting result here is that, the gain in covert rate scales linearly with number of active eigenmodes of Bob channel. Meanwhile, it scales down with the square-root of the number of active eigenmodes over Willie channel. This fact will be of great importance when we study the case of massive MIMO limit. Further, the bounds on L in (25) can, with small effort, generate the result of Theorem 5 in [8] by setting N = M = 1, λ(Wb ) = γw and σb = σw . It worth mentioning that, in some practical situations, compound MIMO channel can be too conservative for resource allocation. In particular, the bounded spectral norm condition in (2) not only leads us to the worst case Willie channel, but it also does not restrict its eigenvectors leaving the beamforming strategy used by Alice (SVD precoding) to be of insignificant gain in protecting against Willie. Although we believe that the eigenvectors of Willie channel plays an important role in the determination of the achievable covert rate, the ignorance of Alice about Willie channel leaves the compound framework as our best option. V. U NIT R ANK MIMO C HANNEL As pointed out in the previous Section, the distinction between the eigenvectors of Bob and Willie channels would have a considerable effect on the achievable covert rate. However, unavailability of Willie’s CSI left the compound framework as the best model for Hw . In this Section, we consider the case when either Hw or Both Hw and Hb are of unit rank. This scenario not only models the case when both Bob and Willie have a single antenna, but it also covers the case when they have a strong line of sight with Alice. Moreover, this scenario allow us to evaluate the effect of the eigenvectors of Hw and Hb on the achievable covert rate. A. Unit Rank Willie Channel In this Section we analyze the scenario in which only Willie channel is of unit rank. In this case, we can write Hw = 1/2 λw vw u†w , where vw ∈ CNw and uw ∈ CNa are the left and right singular vectors of Hw . Accordingly, Ww = λw uw u†w and the product Ww Q has only one non zero eigenvalue. The nonzero eigenvalue λ(Ww Q) is loosely upper bounded by λw λmax (Q). Accordingly, following the same steps of the proof of Theorem 3 we can get (assuming well conditioned Bob channel, i.e. λi (Wb ) = λb for all i) ! √ 2 2σw δλb N log 1 + 2 √ ≤ Cpd (δ) σb λw n ! √ 2 2σw ξδλb ≤ N log 1 + 2 √ (26) σb λw n consequently, N 2 σw λb σ 2 ξλb ≤ L ≤ N w2 . 2 σb λw σb λw (27) √ Again, Bob gets O(N n) bits in n independent channel uses. We note also that, the achievable covert rate increases linearly with N . 9 B. Both Channels are of Unit Rank Consider the case when both Hb and Hw are of unit rank. In this case, we have N = 1. However, setting N = 1 in the results established so far will yield a loose bounds on the achievable LPD constrained rate. The reason is that, the bound λ(We Q) ≤ λw λmax (Q) is, in fact, too loose especially for large values of Na . Although it is hard to establish a tighter upper bound on λ(We Q) when Q is of high rank, it is straightforward to obtain the exact expression for λ(We Q) when Q is of 1/2 unit rank (which is the case when rank{Hb } = 1). Given that Hb = λb vb u†b , Alice will set Q = Pth ub u†b . Accordingly, we have 2 λ(We Q) = λw Pth |< ub , uw >| = λw Pth cos2 (θ), (28) where θ is the angle between ub and uw . We give the main result of this Section in the following theorem. Theorem 4. If rank{Hb } = rank{He } = 1, then, ( √ ! ) 2 2σw δλb √ min log 1 + 2 , C ≤ Cpd (δ) σb λw cos2 (θ) n ( ! ) √ 2 2σw ξδλb √ ≤ min log 1 + 2 ,C σb λw cos2 (θ) n where C is the non LPD constrained capacity of Alice to Bob channel. Accordingly,  if θ = π/2 L = ∞, 2 2 . σw λb σw ξλb  2 ≤L≤ 2 , otherwise 2 2 σb λw cos (θ) σb λw cos (θ) (29) (30) Proof. Follows directly by substituting (28) into (13) and following the same steps as in the proof of Theorem 3 while realizing that Pth ≤ P .  √ 2 Theorem 4 proves that Alice can transmit a maximum of O( n/ cos (θ)) bits reliably to Bob in n independent channel uses while keeping Willie’s sum of error probabilities lower bounded by 1 − δ. In the statement of Theorem 4, the minimum is taken since the first term diverges as θ → π/2, i.e., when ub and uw are orthogonal. In such case, we will have L = ∞. This fact proves that, over MIMO channel Alice can communicate at full rate to Bob without being detected by Willie. An interesting question is, how rare is the case of having ub and uw to be orthogonal? When the angle of the vectors (i.e., the antenna orientation) are chosen uniformly at random, as the number of antennas at Alice gets large, we will see in Section VII that cos2 (θ) approaches 0 exponentially fast with the number of antennas at Alice. Remark 2. It should not be inferred from the results in this Section that the unit rank Bob channel can offer covert rate better than that of higher rank. In fact, we used a loose upper bound on the eigenvalue of Willie’s channel for higher rank case. That is due to the technical difficulty in setting tight bounds to the power received by Willie when Bob channel has higher rank. Also, we see that unit rank channel offers better covert rate than that shown under the compound settings for Willie channel. That is because, unlike the scenario of this Section, compound settings does not restrict the eigenvectors of Willie’s channel. VI. Cpd (δ) W ITHOUT S HARED S ECRET So far, we have established fundamental limits of covert communication over MIMO AWGN channel under the assumption that the codebook generated by Alice is kept secret from Willie (or at least a secret of sufficiet length). In this Section, we study the LPD communication problem without this assumption. The assumption of keeping the codebook generated by Alice secret from Willie (or at least a secret of sufficient length [1], [7]) is common in all studies of covert communication. Without this assumption, LPD condition cannot be met along with arbitrarily low probability of error at Bob, since, when Willie is informed about the codebook, he can decode the message using the same decoding strategy as that of Bob [1]. Also we note that, acheiving covertness does not require positive secrecy rate over the underlying wiretap channel. Indeed, recalling the expression for relative entropy at Willie 1 D (P0 k P1 ) = log 2 Hw QH†w + INw σw | {z } Willie’s Channel Capacity ( −1 ) 1 † + tr Hw QHw + INw − Nw , 2 σw | {z } ≤0, Willie’s penalty due to codebook ignorance (31) 10 we observe that, the first term in (31) is the channel capacity of Willie channel with the implicit assumption of the knowledge of the codebook generated by Alice. In particular, the first term in (31) equals to I(x; (z, Hw )). Meanwhile, it can be easily verified that the remaining difference term in (31) is always non positive. This term represents Willie’s penalty from his ignorance of the codebook. Analogous result for the scalar AWGN channel can be found in [1] for which the same arguments can be made. This in fact provides an interpretation to the scenario on which the secrecy capacity of the main channel may be zero, meanwhile, Alice still can covertly communicate to Bob. Let us define Kn (δ, ) √ . n 2δ 2 (32) n Cpd (δ). L̂ = lim √ n→∞ 2δ 2 (33) L̂ , lim lim ↓0 n→∞ Following Proposition 1 and theorem 1, we can show that √ Observe that, in the definition of L̂, we used the normalization over n instead of n. Now suppose that Alice chooses Q such that the first term in (31), which is the capacity of Willie’ channel, is upper bounded by 2δ 2 /n. Indeed this signaling strategy satisfies the LPD metric (8) and, thus, achieves covertness. Moreover, we have limn→∞ I(x; (z, Hw )) = 0, thus, strong secrecy condition is also met. In particular, if I(x; (z, Hw )) ≤ 2δ 2 /n, Willie can reliably decode at most 2δ 2 nats of Alice’s message in n independent channel uses. It worth mentioning that, requiring limn→∞ I(x; (z, Hw )) = 0 is more restrictive than the strong secrecy condition. In principle, if Alice has a message m to transmit, strong secrecy condition requires limn→∞ I(m; (z, Hw )) = 0. Meanwhile, since m = f −1 (x) for some encoding function f : m 7→ x, we have I(m; (z, Hw )) ≤ I(x; (z, Hw )). Now, Cpd (δ) without any shared secret can be reformulated as follows: Cpd (δ) = max log INa + Q0 tr(Q)≤P Subject to: log INa + Wb Q σb2 (34) Ww Q − 2δ 2 /n ≤ 0. 2 σw (35) In light of the saddle point property established in Section IV, we characterize Cpd (δ) without shared secret by solving (34) for the optimal signaling strategy, Q∗ . We give the main result of this Section in the following theorem. Theorem 5. The eigenvalue decomposition of the capacity achieving input covariance matrix that solves (34) is given by Q∗ = ub Λu†b where ub ∈ CNa ×Na is the matrix whose columns are the right singular vectors of Hb and Λ is a diagonal matrix whose diagonal entries, Λii , are given by the solution of  2 −1 σw 2 −1 −1 λ =(σb λi (Wb ) + Λii ) − η + Λii (36) γw where λ and η are constants determined from the constraints tr {Q} ≤ P and (8), respectively. Moreover,   N X Λii λi (Wb ) log 1 + Cpd (δ) = σb2 i=1 (37) where λi is the ith non zero eigenvalue of Wb . Proof. See Appendix G.  Theorem 5 provides the full characterization of the Cpd (δ) of the considered class of channels without requiring any shared secret between Alice and Bob. Again, it is not straightforward to obtain a closed form expression for Cpd (δ). Thus, we obtain both upper and lower bounds on Cpd (δ) as we did in Section IV. Based on the obtained bounds, we give the square-root law for MIMO AWGN channel without shared secret in the following theorem. Theorem 6. For the considered class of channels without any shared secret between Alice and Bob, the following bounds on Cpd (δ) holds   N 2 2 X 2σw δ λi (Wb ) log 1 + ≤ Cpd (δ) σb2 γw nM i=1   N X 2σ 2 ξδ 2 λi (Wb ) ≤ log 1 + w 2 (38) σb γw nM i=1 11 where ξ = nM . Accordingly, nM − 2δ 2 N X i=1 √ N 2 X 2σw δλi (Wb ) ≤ L̂ ≤ 2 σb γw M i=1 √ 2 2σw ξδλi (Wb ) 2 σb γw M Proof. We give both achievability and converse results in Appendix H. (39)  Theorem 6 extends the result of Theorem 3 to the scenario when Alice and Bob do not share any form of secret. It proves that Alice can transmit a maximum of O(N/M ) bits reliably to Bob in n independent channel uses while keeping Willie’s sum of error probabilities lower bounded by 1 − δ. Now let us consider the case when both Hb and Hw are of unit rank. Under this assumption, we give converse and achievability results of Cpd (δ) over Alice to Bob channel without a shared secret in the following theorem. Theorem 7. If rank{Hb } = rank{He } = 1, then, δ-PD constrained capacity over Alice to Bob channel without a shared secret between Alice and Bob is bounded as     2σ 2 δ 2 λb , C ≤ Cpd (δ) min log 1 + 2 w 2 σ λw cos (θ)n    b  2 2σw ξδ 2 λb ,C (40) ≤ min log 1 + 2 σb λw cos2 (θ)n where ξ is as defined in Theorem 6 and C is the non LPD constrained capacity of Alice to Bob channel. Accordingly,   = ∞, if θ = π/2 L̂ √ √ 2 2 2σw δλb 2σw ξδλb  ≤ L̂ ≤ 2 , otherwise  2 σb λw cos2 (θ) σb λw cos2 (θ) (41) Proof. Follows directly by substituting (28) into (35) and following the same steps as in the proof of Theorem 6 while realizing that Pth ≤ P .  Again, in (40), the minimum is taken since the first term diverges as θ → π/2, i.e., when ub and uw are orthogonal. The theorem proves that Alice can transmit a maximum of O(1/ cos2 (θ)) bits reliably to Bob in n independent channel uses while keeping Willie’s sum of error probabilities lower bounded by 1 − δ. This fact proves that, over MIMO channel Alice can communicate at full rate to Bob without being detected by Willie without requiring Alice and Bob to have any form of shared secret. VII. C OVERT C OMMUNICATION WITH M ASSIVE MIMO In Theorems 4 and 7, it was shown that Alice can communicate at full rate with Bob without being detected by Willie whenever cos(θ) = 0 regardless of the presence of a shared secret. In this Section, we study the behavior of covert rate as the number of antennas scale, which we call the massive MIMO limit, with and without codebook availability at Willie. In particular, the high beamforming capability of the massive MIMO system can provide substantial gain in the achievable LPD rate. However, a quantitative relation between the achievable LPD rate and the number of transmitting antennas seems to be unavailable. To that end, we address the question: how does the achievable LPD rate scale with the number of transmitting antennas? We also study how does the presence of a shared secret between Alice and Bob affects the scaling of the covert rate in the massive MIMO limit. Before we answer these questions, we state some necessary basic results on the inner product of unit vectors in higher dimensions [16]. A. Basic Foundation In this Section, we reproduce some established results on the inner product of unit vectors in higher dimensions. Lemma 1. [Proposition 1 in [16]] Let a and b any two vectors in the unit sphere in Cp chosen uniformly at random. Let θ = cos−1 (< a, b >) be the angle between them. Then   π √ Pr θ − ≤ ζ ≥ 1 − K p(cos ζ)p−2 (42) 2  π for all p ≥ 2 and ζ ∈ 0, where K is a universal constant. 2 The statement of Lemma 1 states that, the probability that any two vectors chosen uniformly at random being orthogonal increases exponentially fast with the dimension p. Indeed, note that, for any 0 < a < 1, ap has the same decay rate as (2−a)−p . √ Thus, the probability that θ is within ζ from π/2 scales like (2 − cos(ζ))p−2 /K p. 12 Corollary 1. Let a and b any two vectors in the unit sphere in Cp chosen uniformly at random and let θ = cos−1 (< a, b >) be the angle between them. Let A, B ∈ Cp×p be two matrices of unit rank generated as A = λa aa† and B = λb bb† . Then, the probability that the eigenvalue of of the product λ(AB) approaches 0 grows to 1 exponentially fast with the dimension p. Proof. It can be easily verified that λ(AB) = λa λb cos2 (θ). Using Lemma 1, we see that, the probability that θ approaches π/2 increases exponentially with p. Hence, the probability that cos(θ) approaches 0 increases in the same order. Then so is cos2 (θ).  B. Massive MIMO Limit With Shared Secret In the previous Section it was demonstrated that, in higher dimensions every two independent vectors chosen uniformly at random are orthogonal with very high probability. More generally, using spherical invariance [16], given ub , for any uw chosen uniformly at random in CNa , the result of Lemma 1 still holds. This scenario typically models the scenario when Alice knows her channel to Bob, meanwhile, she models uw as a uniform random unit vector. Recall that, when Alice has nNa bits to transmit, two alternative options are available for her. Either she splits the incoming stream into Na streams of n bits each and use each stream to select one from 2n messages for each single antenna, or, use the entire nNa bits to choose from 2nNa message. The latter of these alternatives provides a gain factor of Na in the error exponent, of course, in the expense of much greater complexity [11], [12]. However, in the restrictive LPD scenario, Alice would choose the latter alternative as to achieve the best decoding performance at Bob. Therefore, we need to analyze the LPD rate in the limiting case of the product nNa when both n and Na grow. Of course, the number of antennas at Alice is a physical resource which can not be compared to n that can approach ∞ very fast. The more interesting question is, how fast cos2 (θ) approaches 0 as Na increase. As illustrated in corollary 1, we know that cos(θ) approaches 0 exponentially fast with Na . Consequently, we conclude that cos2 (θ) , also, approaches 0 exponentially fast with Na . For proper handling of the scaling of Kn (δ, ) in massive MIMO limit, let us define S , lim lim ↓0 nNa →∞ Kn (δ, ) √ . Na 2nδ 2 (43) Note that the in the definition of S, both n and Na are allowed to grow without bound compared to L in which only n was allowed to grow while Na was treated as constant. Now observe that, following Proposition 1 and Theorem 1, we can show that r n Cpd (δ). (44) S = lim Na nNa →∞ 2δ 2 We give the result of the massive MIMO limit with a pre-shared secret between Alice and Bob in the following Theorem. Theorem 8. Assume that rank{Hb } = rank{He } = 1. Given ub , for any uw chosen uniformly at random, Cpd (δ) is as given in Theorem 4 and S = ∞. (45) √ ! 2 δ n c (Na −2)/2 2σw √ ) where K is a universal constant and c = . (1 + K 2 Na λw P n r n Proof. Combining the result of Theorem 4 and Corollary 1, multiplying (29) by Na and taking the limit as both of n 2δ 2 and Na tend to infinity we obtain r n lim lim Na Cpd (δ) Na →∞ n→∞ 2δ 2 σ 2 λb = lim Na 2 w 2 Na →∞ σb λw cos (θ) = ∞, (46) r Moreover, Kn grows like where the last equality follow since cos2 (θ) → 0 as Na → ∞. On the other hand, we also can verify that r n lim lim Na Cpd (δ) = ∞. (47) n→∞ Na →∞ 2δ 2 √ To show how Kn scales in this massive MIMO limit, we first note that, for fixed Na , Kn scales like n. Also note that, S = ∞ implies that LPD constraint becomes inactive and full non-LPD capacity is achieved. This happens when the quantity 13 Cpd (δ) = C. The question we adress now is, how does Cpd (δ) behave in between these two extreme regimes. Following the same steps of the proof of Theorem 3, we can obtain the following bound on Pth : ( ) √ 2 2σw δ Pth ≤ min √ ,P . (48) nλw cos2 (θ) Thus, we have Pth = P , and hence Cpd (δ) = C, when √ P ≤√ 2 2σw δ nλw cos2 (θ) (49) equivalently, √ 2 2σ δ cos (θ) ≤ √ w nλw P 2  √ 2 2σ δ √ w . nλw P (50) √ 2 !(Na −2)/2 2σ δ 1− √ w nλw P (51) s ⇒ |θ − π/2| ≤ π/2 − cos−1  This happens with probability no less than: p P r(Cpd (δ) = C) ≥ 1 − K Na where (51) follows by setting ζ in Lemma 1 equal √ to the RHS of (50) and using the following basic trigonometry facts: 2 . It can be seen that, the probability that C (δ) = C scales as cos(π/2 − x) = sin(x) and sin(cos−1 (x)) = 1 − x! pd √ 2 √ 2σw δ (Na −2)/2 .  (1 + g) /K Na up to 1, where g = √ nλw P Theorem 8 states that Alice can communicate at full rate to Bob while satisfying the LPD constraint (8). Note that, the limit in both orders yields S = ∞. As Na → ∞, the radiation pattern of a wireless MIMO transmitter becomes so extremely directive (pencil beam). We call this limit the wired limit of wireless MIMO communication. In the wired limit, Willie cannot detect Alice’s transmission unless he wiretaps this virtual wire. Theorem 8 provides a rigorous characterization of the wired limit of wireless MIMO communication. In principle, it answers the fundamental question: How fast does the LPD constrained rate increase with the√number of antennas at Alice? It can be seen that the probability that Alice fully utilizes the channel scales like 2(Na −2)/2 /K Na n up to 1 using the same justification given after Lemma 1. C. Massive MIMO Limit Without Shared Secret In Section VI it was shown that, only diminishing covert rate, O(N/M ), can be achieved without requiring a shared secret between Alice and Bob. Again, we note that this diminishing rate was shown to be achievable when Willie’s channel is isotropic. Also, we have shown shown that, in the massive MIMO limit, the achievable covert rate grows exponentially with the number of transmitting antennas when there is a shared secret between Alice and Bob. Thus, it is also instructive to consider LPD communication problem without a shared secret in the massive MIMO limit. As illustrated in Section III, if Alice has CSI of both channels, not only can she communicate covertly and reliably at full rate whenever the eigen directions of both channels are orthogonal, but also she does not need a shared secret to achieve this rate. Building on our analysis in Section VII, we give the massive MIMO limit of the δ-PD capacity when there is no shared secret between Alice and Bob. Now, let us consider the scenario in which uw is chosen uniformly at random and fixed once chosen. For proper handling of the scaling of Kn (δ, ) in massive MIMO limit without a shred secret, let us define Kn (δ, ) √ . (52) ↓0 nNa →∞ nNa 2δ 2 √ Observe that, unlike S, Kn (δ, ) is normalized to n instead of n in the expression of Ŝ. Now, following Proposition 1 and Theorem 1, we can show that nNa Ŝ = lim √ Cpd (δ). (53) nNa →∞ 2δ 2 We give the result of this scenario in the following Theorem. Ŝ , lim lim 14 Theorem 9. Assume that rank{Hb } = rank{He } = 1 and suppose that there is no shared secret between Alice and Bob. Given ub , for any uw chosen uniformly at random, Cpd (δ) is as given in Theorem 4 and Ŝ = ∞. r Moreover, Kn grows like 1 c (1 + )(Na −2)/2 where K is a universal constant and c = K 2 Na n Proof. The proof follows exactly the same steps as in the proof of Theorem 8. (54) √ ! 2 2σw δ . λw P  Again, it can be seen that Alice achieve the maximum achievable non-LPD rate even under the LPD constraint.However, the rate at which Cpd (δ) converges to C is much slower, compared to the case with shared secret codebook. Hence, it can be deduced from Theorems 7 and 9 that, in the limit of large Na , Alice can transmit O(n) bits in n independent channel uses while satisfying the LPD constraint without the need for any form of shared secret. Even though, it has to be considered that the number of antennas required at Alice under this scenario is much larger than that when she shares a secret of sufficient length with Bob. The following numerical example demonstrates the covert rates in massive MIMO limit with and without a shared secret between Alice and Bob. Example 1. Assume that Alice intend to use the channel for n = 109 times over a channel of bandwidth of 10M Hz, hence, √ 2 n = 3.1623 × 104 . Suppose that Alice is targeting δ = 10−2 . Let σw = σb2 = 10−2 and λw = λb = 10−3 . Assume that Alice is targeting SN R = 15dB at Bob, hence, P = 316.228. Then, for N a = 100 it can be verified that, Alice can transmit O(n) √ covert bits instead of O( n). Observe that, Alice needed only Na = 100 to communicate covertly at near full rate to Bob. Also note that, at 6GHz, two dimensional array of 100 elements can fit within an area of a single sheet of paper. See Fig. (2) for the relation between the δ-PD capacity and number of transmitting antenna for different values of number of antennas at Willie with δ = 10−2 . Fig. 2. The relation between achievable covert rate with and without a shared secret (plotted in log scale), in bits per second, and Na for different values of Nw with target δ = 10−2 . It shows that Alice can communicate near full rate with Na around 100 when she share a secret with Bob. A large gap can be observed when there is no shared secret. As can be seen from Fig. (2), Alice could achieve a covert rate very close to the non-LPD constrained capacity of her channel to Bob with Na ≥ 100 with a block of length n = 109 . Also we see that there is a significant gap (nearly 4 orders of magnitude) between the achievable covert rate with and without a preshared secret. Although both rates converges to C as Na → ∞, we see that without a shared secret, the number of antennas required to achieve near full rate is significantly greater than that required when Alice and Bob are sharing a secret of sufficient length. For practical consideration, this result leaves the massive MIMO limit of the δ-PD capacity without a shared secret of theoretical interest only. VIII. D ISCUSSION Impact of CSI level. Throughout this paper we have considered perfect CSI at Alice about the her channel to Bob. In practical scenarios, this assumption might not hold true as CSI always suffer from imperfection due to e.g. channel estimation error or non error free CSI feedback link. Despite these potential impairments, we argue that the case of imperfect CSI at Alice does not affect the obtained results. That is because we have made the assumption that Willie has perfect CSI as well about his channel. Further, in case when Alice has absolutely no CSI is of special interest as communication under these critical conditions may not allow Bob to share his √ CSI to Alice, specially when dealing with passive Bob. In this scenario, we can verify that the Alice can transmit O(N/M n) bits in n independent channel uses (details are omitted here). We see that, 15 √ the covert rate scales with N/M compared to N/ M when Alice has CSI. More interestingly, in massive MIMO limit with absolutely no CSI at Alice, we can verify that the covert rate → 0 even with Na → ∞. While it was recognized as the most favorable scenario for Alice when she has CSI, massive MIMO may make matters worse when she has absolutely no CSI. Contrary, when Alice has CSI about both channels, covert rates up to the non LPD constrained rate of the channel may be achieved (under certain conditions, see Section III) without the need for a shared secret. As it is the case for MIMO channel with no secrecy constraint, CSI availability play an important role in achieving higher covert rates. Impact of Willie’s ignorance. All results obtained in this paper assumes that Willie has perfect CSI about his channel and he is aware of his channel noise statistics. Ignorance of Willie about one of these parameters is expected to have positive impact on the achievable covert rate. For example, in [6], it was shown that O(n) bits can be transmitted reliably with low probability of detection over BSC whose error probability is unknown to Willie except that it is drawn from a known interval. This scenario is subject for future research. Length of the Shared Secret. In our analysis, we have considered scenarios in which the entire codebook is either available or unavailable at Willie. Meanwhile, secrets of shorter length were reported to be enough for fulfilling √ LPD requirements. For scalar AWGN channel, it was shown that the required secret secret length is in order of O(log n n) [1]. Similar result was established for DMC in [8]. In this work, while we showed that there is a significant gap between the achievable covert rate with and without a shared secret, the minimum length of the required shared secret has not been addressed. IX. S UMMARY AND C ONCLUSIONS We have established the limits of LPD communication over the MIMO AWGN channel. In particular, using relative entropy as our LPD metric, we studied the maximum codebook size, Kn (δ, ), for which Alice can guarantee reliability and LPD conditions are met. We first showed that, the optimal codebook generating input distribution under δ-PD constraint is the zero-mean Gaussian distribution. We based our argumentspon the the principle of minimum relative entropy. For an isotropic Willie channel, we showed that Alice can transmit O(N n/M ) bits reliably in n independent channel uses, where N and M are the number of active eigenmodes of Bob and Willie channels, respectively. Further, we evaluated the scaling rates of Kn (δ, ) in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO). We showed that, while the square-root law still holds for the MIMO-AWGN channel, the number of bits that can be transmitted covertly scales exponentiallyrwith the number of transmitting antennas. More precisely, for a unit rank MIMO n c channel, we show that Kn (δ, ) scales as (1 + √ )(Na −2)/2 where Na is the number of transmitting antennas, K K 2 Na n is a universal constant and c is constant independent on n and Na . Also, we derived the scaling of Kn (δ, ) with no shared secret between Alice and Bob. In particular, we showed that achieving better covert rate is a resource arm race between Alice, Bob and Willie: Alice can transmit O(N/M ) bits reliably in n independent channel uses, i.e., the covert rate is in the order of the ratio between active eigenmodes of both channels. Despite this diminishing rate, in the massive MIMO limit, Alice can still achieve higher covert rate up to the non LPD constrained capacity of her channel to Bob, yet, with a significantly greater number of antennas. Although the covert rates both with and without a shared secret are shown to converge the non LPD constrained capacity as Na → ∞, numerical evaluations showed that without a shared secret, the number of antennas required to achieve near full rate can be orders of magnitude greater. The practical implication of our result is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay. R EFERENCES [1] B. A. Bash, D. Goeckel, and D. Towsley, “Limits of reliable communication with low probability of detection on awgn channels,” IEEE Journal on Selected Areas in Communications, vol. 31, no. 9, pp. 1921–1930, 2013. [2] A. D. Woodbury, “Minimum relative entropy, bayes and kapur,” Geophysical Journal International, vol. 185, no. 1, pp. 181–189, 2011. [3] I. Csiszár and F. Matus, “Information projections revisited,” IEEE Transactions on Information Theory, vol. 49, no. 6, pp. 1474–1490, 2003. [4] R. F. Schaefer and S. Loyka, “The secrecy capacity of compound gaussian mimo wiretap channels,” IEEE Transactions on Information Theory, vol. 61, no. 10, pp. 5535–5552, 2015. [5] A. Abdelaziz, A. Elbayoumy, C. E. Koksal, and H. El Gamal, “On the compound MIMO wiretap channel with mean feedback,” in 2017 IEEE International Symposium on Information Theory (ISIT) (ISIT’2017), Aachen, Germany, Jun. 2017. [6] P. H. Che, M. Bakshi, and S. Jaggi, “Reliable deniable communication: Hiding messages in noise,” in Information Theory Proceedings (ISIT), 2013 IEEE International Symposium on. IEEE, 2013, pp. 2945–2949. [7] M. R. Bloch, “Covert communication over noisy channels: A resolvability perspective,” IEEE Transactions on Information Theory, vol. 62, no. 5, pp. 2334–2354, 2016. [8] L. Wang, G. W. Wornell, and L. Zheng, “Fundamental limits of communication with low probability of detection,” IEEE Transactions on Information Theory, vol. 62, no. 6, pp. 3493–3503, 2016. [9] A. O. HERo, “Secure space-time communication,” IEEE Transactions on Information Theory, vol. 49, no. 12, pp. 3235–3249, 2003. [10] S. Lee, R. J. Baxley, J. B. McMahon, and R. S. Frazier, “Achieving positive rate with undetectable communication over mimo rayleigh channels,” in Sensor Array and Multichannel Signal Processing Workshop (SAM), 2014 IEEE 8th. IEEE, 2014, pp. 257–260. [11] R. G. Gallager, Information theory and reliable communication. Springer, 1968, vol. 2. [12] E. Telatar, “Capacity of multi-antenna gaussian channels,” European transactions on telecommunications, vol. 10, no. 6, pp. 585–595, 1999. [13] E. L. Lehmann and J. P. Romano, Testing statistical hypotheses. Springer Science & Business Media, 2006. [14] T. M. Cover and J. A. Thomas, Elements of information theory. John Wiley & Sons, 2012. 16 [15] B. Friedlander and B. Porat, “Performance analysis of a null-steering algorithm based on direction-of-arrival estimation,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 4, pp. 461–466, 1989. [16] T. T. Cai, J. Fan, and T. Jiang, “Distributions of angles in random packing on spheres.” Journal of Machine Learning Research, vol. 14, no. 1, pp. 1837–1864, 2013. [17] D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge university press, 2005. A PPENDIX A K ULLBACK –L EIBLER D IVERGENCE AT W ILLIE Assuming that Willie is informed about its own channel to Alice, Willie’s observation when Alice is silent is distributed as P0 , meanwhile, it takes the distribution P1 whenever Alice is active where
−1 P0 = |πΣ0 | exp −z† Σ−1 0 z ,
−1 P1 = |πΣ1 | exp −z† Σ−1 (55) 1 z ,  † 2 † 2 where Σ0 = σw INw and Σ1 = Hw QHw + σw INw . Here, Q = E xx is the covariance matrix of signal transmitted by Alice. Note that, the choice of Q is highly dependent on the amount of CSI available at Alice. Thus, we evaluate the KL divergence at Willie in general, then, in the following Sections we will discuss the effect of CSI availability at Alice on the performance of Willie’s optimal detector. Assuming that Willie channel is known and fixed, the KL divergence between P0 and P1 is given as follows: D = = = = = = = EP0 [log P0 − log P1 ]  EP0 − log |πΣ0 | − z† Σ−1 0 z  † −1 + log |πΣ1 | + z Σ1 z   |Σ1 | † −1 + EP0 z† Σ−1 log 1 z − z Σ0 z |Σ0 | 
 −1 log Σ1 Σ−1 + EP0 z† Σ−1 z 0 1 − Σ0   −1
†  −1 −1 log Σ1 Σ0 + EP0 tr Σ1 − Σ0 zz   −1 log Σ1 Σ−1 + tr Σ−1 0 1 Σ0 − tr Σ0 Σ0  log Σ1 Σ−1 + tr Σ−1 0 1 Σ0 − Nw (56) 1 1 1 Hw QH†w + INw and that 2 Hw QH†w + INw = 2 Ww Q + INa . Define Ww , H†w Hw 2 σw σw σw we note that, the non-zero eigenvalues of Hw QH†w and Ww Q are identical, hence, we ca write: Now observe that, Σ1 Σ−1 0 = D (P0 k P1 ) = 1 log 2 Hw QH†w + INw σw | {z } Willie’s Channel Capacity ( −1 ) 1 † Hw QHw + INw + tr − Nw 2 σw | {z } ≤0, Willie’s penalty due to codebook ignorance Nw  Y  λi (Ww Q) = log 1+ 2 σw i=1 −1 Nw  X λi (Ww Q) + − Nw 1+ 2 σw i=1   Nw X  λi (Ww Q) = log 1 + 2 σw i=1  −1  λi (Ww Q) + 1+ −1 2 σw   Nw X  λi (Ww Q) = log 1 + 2 σw i=1  −1 !−1  λi (Ww Q) − 1+ , 2 σw where λi (Ww Q) is the ith eigenvalue of Ww Q. (57) 17 A PPENDIX B ACHIEVABILITY P ROOF OF P ROPOSITION 1 We show the achievability for Gaussian input. As discussed in [8], the sequence {Kn } is achievable provided that: Kn 1 f ×n (zn |xn ) lim √ ≤ P − lim inf √ log , n→∞ n→∞ f ×n (zn ) n n (58) where P − lim inf denotes the limit inferior in probability, namely, the largest number such that the probability that the random variable in consideration is greater than this number tends to one as n tends to infinity. Meanwhile, f ×n (xn ) denotes the nth extension of the probability density function of a random vector x. Thus, we need to show 1 f ×n (zn |xn ) √ √ log − nI(fn (x), fn (z|x)) → 0 f ×n (zn ) n (59) in probability as n tends to infinity. First observe that, f ×n (zn |xn ) = f ×n (zn ) = n Y i=1 n Y −1
exp −(zi − xi )† Σ−1 0 (zi − xi ) , −1   exp −z†i Σ−1 z , i 1 |πΣ0 | |πΣ1 | (60) i=1   2 2 INw and let Q = E xx† be chosen such that (8) is satisfied. Note that, Q has to INw , Σ1 = Hw QH†w + σw where Σ0 = σw be a decreasing function of n. Then, f ×n (zn |xn ) n = Σ1 Σ− 01 × ×n n f (z ) ! n n X X † † −1 −1 exp tr(Σ1 zi zi ) − tr(Σ0 ei ei ) . i=1 (61) i=1 Accordingly, 1 f ×n (zn |xn ) √ √ log = n log Σ1 Σ− 01 + f ×n (zn ) n ! n n X X 1 † † −1 −1 √ tr(Σ1 zi zi ) − tr(Σ0 ei ei ) , n i=1 i=1 whose expectation can be found as   √ 1 f ×n (zn |xn ) E √ log = n log Σ1 Σ− 01 + f ×n (zn ) n ! n n X X 1 −1 −1 √ tr(Σ1 Σ1 ) − tr(Σ0 Σ0 ) n i=1 i=1 √ = n log Σ1 Σ− 01 √ 1 = n log INw + 2 Hw QH†w σw √ = nI(fn (x), fn (z|x)). It then follows by Chebyshev’s inequality that, for any constant a > 0,   1 f ×n (zn |xn ) √ P r √ log nI(f (x), f (z|x)) − ≥ a n n f ×n (zn ) n   f ×n (zn |xn ) 1 1 ≤ 2 var √ log a f ×n (zn ) n (62) (63) (64) and, it remains to show that  lim var n→∞ 1 f ×n (zn |xn ) √ log f ×n (zn ) n  = 0. (65) 18 Note that,  1 f ×n (zn |xn ) √ log f ×n (zn ) n !! n n X X 1 † −1 † −1 z Σ zi − ei Σ0 ei = var √ n i=1 i 1 i=1 n   1X † −1 = var z†i Σ−1 1 zi − ei Σ0 ei n i=1   † −1 = var z†i Σ−1 1 zi − ei Σ0 ei  h † −1 Σ e z − e = E z†i Σ−1 i i 1 i 0  †  † −1 † −1 zi Σ1 zi − ei Σ0 ei h † −1 = E z†i Σ−1 1 z i z i Σ1 z i  var † −1 − z†i Σ−1 1 zi ei Σ0 ei † −1 − e†i Σ−1 0 ei zi Σ1 zi i † −1 −e†i Σ−1 e e Σ e i i 0 i 0 (66) Now observe that, since Q → 0 as n tends to infinity, we can verify that each term in (66) tends to zero as n tends to infinity.  A PPENDIX C P ROOF OF T HEOREM 1   We show that the limit in (11) always exists. Note that, for every n, fn (x) is zero mean Gaussian. Let Q∗n = En xx† be Q∗n = arg max I(fn (x), fn (y)) (67) Q0 tr(Q)≤P where the maximum is subject to (8). Hence, we have max I(fn (x), fn (y)) = log I + Q0 tr(Q)≤P Hb Q∗n H†b . σb2 (68) Now, we have two cases to consider: 1) D(Pn0 k Pn1 ) = 0. In this case, δ can be made 0 causing the limit to be infinity. 2) D(Pn0 k Pn1 ) > 0. In this case, Q∗n has to be a decreasing function of n, otherwise, the constraint (8) can not be met. In this case, the limit is ≥ 0 and < ∞. In either cases, the limit exist and, hence, limit can be used in place of limit inferior and, also, the order of limit and maximum can be interchanged. A PPENDIX D P ROOF OF P ROPOSITION 2 It is enough to show that ∗ Ww = arg max D (P0 k P1 (Ww )) Ww ∈Sw = γw Î, (69) hence, we need to show that the function D (P0 k P1 (Ww )) is monotonically increasing in Ww , i.e., D1 , D (P0 k P1 (Ww1 )) ≥ D (P0 k P1 (Ww2 )) , D2 whenever Ww1 < Ww2 . Recalling the expression of D (P0 k P1 (Ww )) in (13), we note that the 19 function log |I + WQ| is monotonically increasing in W for any Q. Meanwhile, the second term in (13) is negative and decreases monotonically in W. Even though, we have that D1 − D 2 = EP0 [log P0 − log P1 (Ww1 ) = EP0 [log P1 (Ww2 ) − log P1 (Ww1 )]   P1 (Ww1 ) −EP0 log P1 (Ww2 ) − log P0 + log P1 (Ww2 )] (a) ≥ (b) ≥ 0, (70) where (a) follows from Jensen’s inequality using convexity of log and (b) with some standard matrix algebra  follows because,  P1 (Ww1 ) as in Appendix A, we can show that, for Ww1 < Ww2 , we have EP0 log ≤ 1.  P1 (Ww2 ) A PPENDIX E P ROOF OF T HEOREM 2 ∗ ∗ We solve (14) for Ww = Ww . Without loss of generality, assume that Nw ≥ Na , hence, Ww = γw INw . For Nw < Na , ∗ we can set the γw = 0 for the Na − Nw minimum eigenvalues of Q∗ . Plugging Ww into (13), we get 4 ( ! −1 ) γ γ w w − Nw (71) INa + 2 Q Dn = n log INa + 2 Q + tr σw σw Now observe that, |I + Wb Q| ≤ Na Y (1 + λi (Wb )λi (Wb )) (72) i=1 ∗ is with equality if and only if Wb and Q have the same eigenvectors and note that this choice does not affect (71) since Ww isotropic. Hence, the eigenvectors of Q∗ is the same as the eigenvectors of Wb which is the same as the right singular vectors of Hb . Now, we form the following Lagrange dual problem 1 L = log INa + 2 Wb Q + λ(tr(Q) − P ) − tr(MQ) σb   2δ 2 +η D− , (73) n where λ, η ≥ 0 are the Lagrange multipliers that penalize violating the power and LPD constraints, respectively, and M  0 penalizes the violation of the constraint Q  0. Where the associated KKT conditions can be expressed as: λ, η ≥ 0, λ(tr(Q) − P ) = 0, MQ = 0, Q  0, M  0, tr(Q) ≤ P, (74) where the equality constraints in (74) ar the complementary slackness conditions. Note that, (14) is not a concave problem in general. Thus, KKT conditions are not sufficient for optimality. Yet, since the constraint set is compact and convex and the objective function is continuous, KKT conditions are necessary for optimality. Hence, we proceed by finding the stationary points of the gradient of the dual Lagrange problem in the direction of Q and obtain the stationary points that solve the KKT conditions. By inspecting the objective function at these points, the global optimum can be identified. To identify the stationary points of the Lagrangian (73), we get its gradient with respect to Q as follows:  −1 Wb 1 5Q L = − INa + 2 Wb Q + λINa − M σb σb2  −1  −2 ! ηγw γw γw + 2 INa + 2 Q − INa + 2 Q σb σw σw  −1 1 Wb + λINa − M = − INa + 2 Wb Q σb σb2  2 −1  2 −2 ! σw σw +η IN + Q − IN + Q γw a γw a 4 For n simplicity, we write Dn instead of D Pn 0 k P1
(75) 20 Assume, without loss of generality, that Q  0, then, from MQ = 0 it follows that M = 0. Now, from 5Q L = 0 we obtain  −1 λINa = σb2 Wb−1 + Q −2  2 −1 !  2 σw σw − +η IN + Q IN + Q (76) γw a γw a Since we know that Q∗ and Wb have the same eigenvectors, hence, the eigenvalues of Q∗ , Λii , can be found from −1 λ =(σb2 λ−1 i (Wb ) + Λii )  2 −2  2 −1 ! σw σw +η − + Λii + Λii , γw γw (77) as required.  A PPENDIX F P ROOF OF T HEOREM 3 Achievability. Starting from (71), we obtain (a) D ≤ (b) = = (c) ≤ = (  −1 ) γw γw − Nw Q + tr tr INa + 2 Q 2 σw σw    Nw X  γw Λii   σ2 +  1 − 1  γw Λii w i=1 1+ 2 σw   γw Λii N w X  γw Λii 2  σw    σ2 − γw Λii  w i=1 1+ 2 σw   γw Pth 2   γw Pth N σw  Nw   N σ2 − γw Pth  w 1+ 2 N σw   2 2 γw Pth 4  N 2 σw  , Nw   γw Pth  1+ 2 N σw (78) where (a) follows from the inequality log |A| ≤ tr {A − I}, (b) is straightforward matrix algebra, (c) follows since the RHS is maximized for Λii = Pth /N for all i. For Alice to ensure (8), we need the RHS of (78(e)) to be less than or equal 2δ 2 . After some manipulation, if Alice sets √ 2 2N σw δ √ Pth ≤ , (79) γw nNw we can verify that (8) is satisfied. Now let Alice set Pth for (79) to be met with equality. Given that choice of Pth , the LPD constraint is met and the rest of the problem is that of choosing the input distribution to maximize the achievable rate. The solution of the problem is then the SVD precoding with conventional water filling [12], [17] as follows:  + (µ − σb2 λ−1 for 1 ≤ i ≤ N i (Wb )) Λii = (80) 0 for N < i ≤ Na , where N = min{Na , Nb }, λi is the ith non zero eigenvalue of Wb and x+ = max{0, x}. Further, µ is a constant chosen to satisfy the power constraint tr{Λ} = Pth . Accordingly, the following rate is achievable over Alice to Bob channel:   N + X (µ − σb2 λ−1 i (Wb )) λi (Wb ) Rpd (δ) = log 1 + σb2 i=1  + N  X µλi (Wb ) = log . (81) σb2 i=1 21 However, it is technically difficult to expand (81) to check the applicability of the square-root law. Therefore, we obtain an achievable rate assuming that Alice splits Pth equally across active eigenmodes of her channel to Bob. Note that, this rate is indeed achievable since it is less than or equal to (81). Also note that, when Alice to Bob channel is well conditioned, the power allocation in (80) turns into equal power allocation. Then, the following rate is achievable: ! √ 2 N X 2σw δλi (Wb ) √ R(δ) = log 1 + . (82) σb2 γw nNw i=1 Now suppose that Alice uses a code of rate R̂ ≤ R(δ), then, Bob can obtain ! √ 2 N X 2σw δλi (Wb ) √ nR̂ ≤ n log 1 + σb2 γw nNw i=1 √ 2 N p (a) X 2σw δλi (Wb ) ≤ n/Nw σb2 γw ln 2 i=1 (83) bits in n independent channel uses, where (a) follows from ln(1 + x) ≤ x, and note that the inequality is met with equality for sufficiently large n. Now, assume that λi (Wb ) = λb for all 1 ≤ i ≤ N , i.e., Bob’s channel is well conditioned. Then, Bob can obtain √ 2 p 2σ δλb (84) nR(δ) ≤ N n/Nw 2 w σb γw ln 2 bits in n independent channel uses since the inequality is met with equality for sufficiently large n.  Converse. To show the converse, we assume the most favorable scenario for Alice to Bob channel when Hb is well conditioned. That is because, the rate Alice can achieve over a well conditioned channel to Bob sets an upper bound to that can be achieved over any other channel of the same Frobenius norm [17]. Then, Alice will split her power equally across active eigenmodes of her channel to Bob. Now, Let us choose ξ ≥ 1 such that,   γw γw log INa + 2 Q − Nw ≥ tr Q − ξNw (85) 2 σw σw and note that for small D, ξ is a function of δ that approaches 1 as δ → 0. Hence, combining (71) with (85) we obtain: (  −1 )  (a) γw γw Q + tr INa + 2 Q − ξNw D ≥ tr 2 σw σw   (b) =  γw Pth Nw   N σ2 + w  1 − ξ  γw Pth 1+ 2 N σw (86) following the same steps as in the achievability proof, we can insure that √ 2 2N ξσw δ √ , Pth ≤ γw nNw (87) otherwise, Alice can not ensure that (8) is satisfied. Now let Alice set Pth equals to the RHS of (87). Then, we can verify that: ! √ N 2 X 2ξσw δλb Cpd (δ) ≤ log 1 + 2 √ . (88) σb γw nNw i=1 Now suppose that Alice uses a code of rate R(δ) ≤ Cpd (δ), then, Bob can obtain ! √ 2 2ξσw δλb nR(δ) ≤ nN log 1 + 2 √ σb γw nNw √ p 2ξσ 2 δλb ≤ N n/Nw 2 w σb γw ln 2 bits in n independent channel uses since the inequality is met with equality for sufficiently large n.  (89) 22 A PPENDIX G P ROOF OF T HEOREM 5 ∗ ∗ We solve (34) for Ww = Ww . Without loss of generality, assume that Nw ≥ Na , hence, Ww = γw INw . For Nw < Na , we can set the γw = 0 for the Na − Nw minimum eigenvalues of Q∗ . Now observe that, |I + Wb Q| ≤ Na Y (1 + λi (Wb )λi (Wb )) (90) i=1 ∗ with equality if and only if Wb and Q have the same eigenvectors and note that this choice does not affect (71) since Ww is isotropic. Hence, the eigenvectors of Q∗ is the same as the eigenvectors of Wb which is the same as the right singular vectors of Hb . Now, we form the following Lagrange dual problem 1 L = log INa + 2 Wb Q + λ(tr(Q) − P ) − tr(MQ) σb   1 2 + η log INa + 2 Ww Q − 2δ /n , (91) σw where λ, η ≥ 0 are the Lagrange multipliers that penalize violating the power and LPD constraints, respectively, and M  0 penalizes the violation of the constraint Q  0. Where the associated KKT conditions can be expressed as: λ, η ≥ 0, λ(tr(Q) − P ) = 0, MQ = 0, Q  0, M  0, tr(Q) ≤ P, (92) where the equality constraints in (92) are the complementary slackness conditions. Note that, (34) is not a concave problem in general. Thus, KKT conditions are not sufficient for optimality. Yet, since the constraint set is compact and convex and the objective function is continuous, KKT conditions are necessary for optimality. Hence, we proceed by finding the stationary points of the gradient of the dual Lagrange problem in the direction of Q and obtain the stationary points that solve the KKT conditions. By inspecting the objective function at these points, the global optimum can be identified. To identify the stationary points of the Lagrangian (91), we get its gradient with respect to Q as follows:  −1 1 Wb 5Q L = − IN a + 2 W b Q + λINa − M σb σb2 !  −1 ηγw γw + 2 INa + 2 Q σw σw  −1 1 Wb = − INa + 2 Wb Q + λINa − M σb σb2 −1  2 σ (93) + η w INa + Q γw Assume, without loss of generality, that Q  0, then, from MQ = 0 it follows that M = 0. Now, from 5Q L = 0 we obtain −1  2  −1 σ λINa = σb2 Wb−1 + Q − η w INa + Q (94) γw Since we know that Q∗ and Wb have the same eigenvectors, hence, the eigenvalues of Q∗ , Λii , can be found from  2 −1 σw 2 −1 −1 λ =(σb λi (Wb ) + Λii ) − η + Λii , (95) γw as required.  A PPENDIX H P ROOF OF T HEOREM 6 Achievability. We observe that log INa 1 + 2 Ww Q σw  (a) 1 Ww Q 2 σw ≤ tr = M X γw Λii ( 2 ) σw i=1 (b) ≤ Pth M γw Λii 2 N σw  (96) 23 where (a) follows from the inequality log |A| ≤ tr {A − I}, (b) follows since the RHS is maximized for Λii = Pth /N for all i. For Alice to ensure that the constraint in (34) is satisfied, she needs the RHS of (96(b)) to be less than or equal 2δ 2 /n. Thus, Alice needs Pth ≤ 2 2 2N σw δ . γw nM (97) Now let Alice set Pth for (97) to be met with equality. Given that choice of Pth , the LPD constraint is met and the rest of the problem is that of choosing the input distribution to maximize the achievable rate. The solution of the problem is then the SVD precoding with conventional water filling [12], [17] as follows:  + (µ − σb2 λ−1 for 1 ≤ i ≤ N i (Wb )) Λii = (98) 0 for N < i ≤ Na , where λi is the ith non zero eigenvalue of Wb . Further, µ is a constant chosen to satisfy the power constraint tr{Λ} = Pth . Accordingly, the following rate is achievable over Alice to Bob channel:   N + X (µ − σb2 λ−1 i (Wb )) λi (Wb ) Rpd (δ) = log 1 + σb2 i=1  + N  X µλi (Wb ) = log . (99) σb2 i=1 However, it is technically difficult to expand (99) to check the applicability of the square-root law. Therefore, we obtain an achievable rate assuming that Alice splits Pth equally across active eigenmodes of her channel to Bob. Note that, this rate is indeed achievable since it is less than or equal to (99). Also note that, when Alice to Bob channel is well conditioned, the power allocation in (98) turns into equal power allocation. Then, the following rate is achievable:   N 2 X δλi (Wb ) 2σw R(δ) = log 1 + . (100) σb2 γw nM i=1 Now suppose that Alice uses a code of rate R̂ ≤ R(δ), then, Bob can obtain   N X 2σ 2 δ 2 λi (Wb ) nR̂ ≤ n log 1 + w2 σb γw nM i=1 √ N 2 2 (a) X 2σw δ λi (Wb ) ≤ M σb2 γw ln 2 i=1 (101) bits in n independent channel uses, where (a) follows from ln(1 + x) ≤ x, and note that the inequality is met with equality for sufficiently large n. Now, assume that λi (Wb ) = λb for all 1 ≤ i ≤ N , i.e., Bob’s channel is well conditioned. Then, Bob can obtain √ 2 2 N 2σw δ λb nR(δ) ≤ (102) 2 M σb γw ln 2 bits in n independent channel uses since the inequality is met with equality for sufficiently large n. Converse. To show converse, we assume the most favorable scenario for Alice to Bob channel when Hb is well conditioned. Then, Alice will split her power equally across active eigenmodes of her channel to Bob. Now, (  −1 ) (a) 1 1 log INa + 2 Ww Q ≥ tr INa − INa + 2 Ww Q σw σw   = = Na X  1 −   1  γw Pth  i=1 1+ 2 N σw   M γw Pth 2 +γ P N σw w th (103)  where (a) follows from the inequality log |A| ≥ tr I − A−1 . Following the same steps as in the achievability proof, we can insure that 2 2 2ξN ξσw δ Pth ≤ , (104) γw nM 24 nM where ξ = > 1, otherwise, Alice can not meet the LPD constraint. Now let Alice set Pth equals to the RHS of nM − 2δ 2 (104). Then, we can verify that:   N X 2ξσ 2 δ 2 λb Cpd (δ) ≤ log 1 + 2 w . (105) σb γw nM i=1 Now suppose that Alice uses a code of rate R(δ) ≤ Cpd (δ), then, Bob can obtain   2ξσ 2 δ 2 λb nR(δ) ≤ nN log 1 + 2 w σb γw nM 2 2 2N ξσw δ λb ≤ M σb2 γw ln 2 bits in n independent channel uses since the inequality is met with equality for sufficiently large n.  (106) 

Some HCI Priorities for GDPR-Compliant Machine Learning Michael Veale University College London London, United Kingdom m.veale@ucl.ac.uk Reuben Binns Max Van Kleek University of Oxford Oxford, United Kingdom reuben.binns@cs.ox.ac.uk emax@cs.ox.ac.uk Abstract In this short paper, we consider the roles of HCI in enabling the better governance of consequential machine learning systems using the rights and obligations laid out in the recent 2016 EU General Data Protection Regulation (GDPR)—a law which involves heavy interaction with people and systems. Focussing on those areas that relate to algorithmic systems in society, we propose roles for HCI in legal contexts in relation to fairness, bias and discrimination; data protection by design; data protection impact assessments; transparency and explanations; the mitigation and understanding of automation bias; and the communication of envisaged consequences of processing. Introduction The General Data Protection Regulation: An Opportunity for the CHI Community? (CHI-GDPR 2018), Workshop at ACM CHI’18, 22 April 2018, Montréal, Canada The 2016 EU General Data Protection Regulation (GDPR) is making waves. With all personal data relating to EU residents or processed by EU companies within scope, it seeks to strengthen the rights of data subjects and the obligations of data controllers (see definitions in the box overleaf) in an increasingly data-laden society, newly underpinned with an overarching obligation of data controller accountability as well as hefty maximum fines. Its articles introduce new provisions and formalise existing rights clarified by the European Court of Justice (the Court), such as the “right to be forgotten”, as well as strengthening those already present in the 1995 Data Protection Directive arXiv:1803.06174v1 [cs.HC] 16 Mar 2018 (DPD). Data Subjects & Controllers EU DP law applies whenever personal data is processed either in the Union, or outside the Union relating to an EU resident. Personal data is defined by how much it can render somebody identifiable— going beyond email, phone number, etc to include dynamic IP addresses, browser fingerprints or smart meter readings. The individual data relates to is called the data subject. The organisation(s) who determine ‘the purposes and means of the processing of personal data’ are data controllers. Data subjects have rights over personal data, such as rights of access, erasure, objection to processing, and portability of data elsewhere. Data controllers are subject to a range of obligations, such as ensuring confidentiality, notifying if data is breached, and undertaking risk assessments. Additionally, they must only process data where they have a legal ground—such as consent—to do so, for a specified and limited purpose, and a limited period of storage. The GDPR has been turned to by scholars and activists as a tool for “algorithmic accountability” in a society where machine learning (ML) seems to be increasingly important. Machine learning models—statistical systems which use data to improve their performance on particular tasks—are the approach of choice to generate value from the ‘data exhaust’ of digitised human activities. Critics, however, have framed ML as powerful, opaque, and with potential to endanger privacy [2], equality [10] and autonomy [20]. While the GDPR is intended to govern personal data rather than ML, there are a range of included rights and obligations which might be useful to exert control over algorithmic systems [14]. Given that GDPR rights involve both individual data-subjects and data controllers (see sidebar) interfacing with computers in a wide variety of contexts, it strongly implicates another abbreviation readers will likely find familiar: Human–Computer Interaction (HCI). In this short paper, we outline, non-exhaustively of course, some of the crossovers between the GDPR provisions, HCI and ML that appear most salient and pressing given current legal, social, and technical debates. We group these in two broad categories: those which primarily concern the building and training of models before deployment, and those which primarily concern the post-deployment application of models to data subjects in particular situations. HCI, GDPR and Model Training An increasing proportion of collected personal data1 is used to train machine learning systems, which are in turn used to 1 Note that the GDPR defines personal data broadly—including things like dynamic IP addresses and home energy data—as opposed to the predominantly American notion of personally identifiable information (PII) [25]. make or support decisions in a variety of fields. As model training with personal data is considered data processing (assuming data is not solidly ‘anonymised’), the GDPR does govern it to a varying degree. In this section, we consider to what extent HCI might play a role in promoting the governance of model training under the GDPR. Fairness, discrimination and ‘special category’ data Interest in unfair and/or illegal data-driven discrimination has concerned researchers, journalists, pundits and policy-makers [17, 3], particularly as the ease of transforming seemingly non-sensitive data into potentially damaging, private insights has become clear [9]. Most focus on how to govern data (both in Europe and elsewhere broadly [19]) has been centred on data protection, which is not an anti-discrimination law and does not feature anti-discrimination as a core concept. Yet the GDPR does contain provisions which concern particularly sensitive attributes of data. Several “special” types of data are given higher protection in the GDPR. The 1995 Data Protection Directive (art 8) prohibits processing of data revealing racial or ethnic origin, political opinions, religious or philosophical beliefs and trade-union membership, in addition to data concerning health or sex life. The GDPR (art 9(1)) adds genetic and biometric data (the latter for the purposes of identification), as well as clarifying sex life includes orientation, to create 8 ‘special categories’ of data. This list is similar, but not identical, to the ‘protected characteristics’ in many international anti-discrimination laws. Compared to the UK’s Equality Act 2010, the GDPR omits age, sex and marital status but includes political opinions, trade union membership, and health data more broadly. The collection, inference and processing of special category data triggers both specific provisions (e.g. arts 9, 22) and specific responsibilities (e.g. Data Protection Impact Assessments, art 35 and below), as well as generally heightening the level of risk of processing and therefore the general responsibilities of a controller (art 24). Perhaps the most important difference is that data controllers cannot rely on their own legitimate interests to justify the processing of special category data, which usually will mean they will have to seek explicit, specified consent for the type of processing they intend—which they may not have done for their original data, and may not be built into their legal data collection model. Given that inferred special category data is also characterised as special category data [28], there are important questions around how both controllers and regulators recognise that such inference is or might be happening. Naturally, if a data controller trains a supervised model for the purpose of inferring a special category of data, this is quite a simple task (as long as they are honest about it). Yet when they are using latent characteristics, such as through principal components analysis, or features that are embedded within a machine learning model, this becomes more challenging. In particular it has been repeatedly demonstrated that biases connected to special category data can appear in trained systems even where those special categories are not present in the datasets being used [9]. The difficulty of this task is heightened by how the controller is unlikely to possess ‘ground truth’ special category data in order to assess what it is they are picking up. HCI might play an important role here in establishing what has been described as ‘exploratory fairness analysis’ [27]. The task is to understand potential patterns of discrimination, or to identify certain unexpected but sensitive clusters, with only partial additional information abut the participants. A similar proposal (and prototype of) a visual system, albeit one assuming full information, has been proposed by discrimination-aware data mining researchers concerned that the formal statistical criteria for non-discrimination established by researchers may not connect with ideas of fairness in practice [4, 5]. If we do indeed also know unfairness when we see it, exploratory visual analysis may be a useful tool. A linked set of discussions have been occurring in the information visualisation community around desirable characteristics of feminist data visualisation, which connects feminist principles around marginalisation and dominance in the production of knowledge to information design [11]. Finally, visual tools which help identify misleading patterns in data, such as instances of Simpson’s paradox (e.g. [24]), may prove useful in confirming apparent disparities between groups. Building and testing interfaces which help identify sensitive potential correlations and ask critical questions around bias and discrimination in the data is an important prerequisite to rigorously meeting requirements in the GDPR. Upstream provisions: Data Protection by Design (DPbD) and Data Protection Impact Assessments (DPIAs) The GDPR contains several provisions intended to move considerations of risks to data subjects’ rights and freedoms upstream into the hands of designers. Data Protection by Design (DPbD), a close cousin of privacy by design, is a requirement under the GDPR and means that controllers should use organisational and technical measures to imbue their products and processes with data protection principles [8]. Data Protection Impact Assessments (DPIAs) have a similar motivation [6]. Whenever controllers have reason to believe that a processing activity brings high risks, they must undertake continuous, documented analysis of these, as well as any measures they are taking to mitigate. The holistic nature of both DPbD and DPIAs is emphasised in both the legal text and recent guidance. These are creative processes mixing anticipation and foresight with best practice and documentation. Some HCI research has already addressed this in particular. Luger et al. [23] use ideation cards to engage designers with regulation and co-produce “data protection heuristics”.2 Whether DPIA aides can be built into existing systems and software in a user-centric way is an important area for future exploration. Furthermore, many risks within data, such as bias, poor representation, or the picking up of private features, may be unknown to data controllers. Identifying these is the point of a DPIA, but subtle issues are unlikely to leap out of the page. In times like this, it has been suggested that a shared knowledgebase [27] could be a useful resource, where researchers and data controllers (or their modelling staff) could log risks and issues in certain types of data from their own experiences, creating a resource which might serve useful in new situations. For example, such a system might log found biases in public datasets (particularly when linked to external data) or in whole genres of data, such as Wi-Fi analytics or transport data. Such a system might be a useful starting point for considering issues that may otherwise go undetected, and for supporting low-capacity organisations in their responsible use of analytics. From an HCI perspective though, the design of such a system presents significant challenges. How can often nuanced biases be recorded and communicated both clearly and in such a way that they generalise across applications? How might individuals easily search a system for issues in their own datasets, particularly when they might have a very large number of variables in a combination the system has not seen previously? Making this kind of knowledge accessible to practitioners seems promising, but daunting. 2 The cards are downloadable at https://perma.cc/3VBQ-VVPQ. HCI, GDPR and Model Application The GDPR, and data protection law in general, was not intended to significantly govern decision-making. Already a strange law in the sense that it poses transparency requirement that applies to the public and private sectors alike, it is also a Frankenstein’s monster–style result culminating from the melding of various European law and global principles that preceded it [16]. Modes of Transparency While transparency is generally spoken of as a virtue, the causal link between it and better governance is rarely simple or clear. A great deal of focus has been placed on the so-called “right to an explanation”, where a short paper at a machine learning conference workshop [18] gained sudden notoriety, triggering reactions from lawyers and technologists noting that the existence and applicability of such a right was far from simple [29, 14]. Yet the individualised transparency paradigm has rarely provided much practical use for data subjects in their day-to-day lives (consider the burden of ‘transparent’ privacy policies). Consequently, HCI provides a useful place to start when considering how to make the limited GDPR algorithmic transparency provisions useful governance tools. There are different places in which algorithmic transparency rights can be found in the GDPR [29]. Each bring different important HCI challenges. Meaningful information about the logic of processing Articles 13–14 oblige data controllers to provide information at the time data is collected around the logics of certain automated decision systems that might be applied to this data. Current regulatory guidance [1] states that there is no obligation to tailor this information to the specific situation of a data subject (other than if they might be part of a vulnerable group, like children, which might need further support to make the information meaningful), although as many provisions in data protection law, the Court may interpret this more broadly or narrowly when challenged. This points to an important HCI challenge in making (or visualising) such general information, but with the potential for specific relevance to individuals. Right to be informed In addition, there is a so-called ‘right to be informed’ of automated decision-making [29]: how might an interface seamlessly flag to users when a potentially legally relevant automated decision is being made? This is made more challenging by the potential for adaptive interfaces or targeted advertising to meet the criteria of a ‘decision’. In these cases, it is unclear at what point the ‘decision’ is being made. Decisions might be seen in the design process, or adaptive interfaces may be seen as ‘deciding’ which information to provide or withold [14]. Exercise of data protection rights is different in further ways in ambient environments [13], as smart cities and ambient computing may bring significant challenges, if, for example, they are construed as part of decision-making environments. Existing work in HCI has focussed on the difficulties in identifying “moments of consent” in ubiquitous computing [22, 21]. Not only is this relevant when consent is the legal basis for an automated decision, but additional consideration will be needed in relation to what equivalent “moments” of objection might look like. Given that moments to object likely outnumber moments to consent, this might pose challenges. A right to an explanation? An explicit “right to an explanation” of specific decisions, after they have happened, sits in a non-binding recital in the GDPR [29], and thus its applicability and enforceability depends heavily on regulators and the Court. However, there is support for a parallel right in varying forms in certain other laws, such as French administrative law or the Council of Europe Convention 108 [15], and HCI researchers have already been testing different explanation facilities proposed by machine learning researchers in qualitative and quantitative settings to see how they compare in relation to different notions of procedural justice [7]. Further research on explanation facilities in-the-wild would be strongly welcome, given that most explanation facilities to date have focussed on the user of a decision-support system rather than an individual subject to an automated decision. Mitigating Automation Bias A key trigger condition for the automated decision-making provisions in the GDPR (art 22) [14] centres on the degree of automation of the process. Significant decisions “based solely on automated processing” require at least consent, a contract or a basis in member state law. Recent regulatory guidance indicates that there must be “meaningful” human input undertaken by somebody with “authority and competence” who does not simply “routinely apply” the outputs of the model in order to be able to avoid contestation or challenge [28]. Automation bias has long been of interest to scholars of human factors in computing [26, 12] and the GDPR provides two core questions for HCI in this vein. Firstly, this setup implies that systems that are expected to outperform humans must always be considered “solely” automated [28]. If a decision-making system is expected to legitimately outperform humans it makes meaningful input very difficult. Any routine disagreement would be at best arbitrary and at worst, harmful. This serves as yet another (legal) motivating factor to create systems where human users can augment machine results. Even if this proves difficult, when users contest an automated decision under the GDPR, they have a right to human review. Interfaces need to ensure that even where models may be complex and high-dimensional, decision review systems are rigorous and themselves have “meaningful” human input—or else these reviewed decisions are equally open to contestation. Secondly, how might a data controller or a regulator understand whether systems have “meaningful” human input or not, in order to either obey or enforce the law? How might this input be justified and documented in a useful and user-friendly way which could potentially be provided to the subject of the decision? Recent French law does oblige this in some cases: in the case of algorithmically-derived administrative decisions, information should be provided to decision-subjects on the “the degree and the mode of contribution of the algorithmic processing to the decision-making” [15]. Purpose-built interfaces and increased knowledge from user studies both seem needed for the aim of promoting meaningful, accountable input. Communicating Envisaged Consequences Where significant, automated decision-making using machine learning is expected, the information rights in the GDPR (arts 13–15) provide that a data subject should be provided with the “envisaged consequences” of such decision for her. What this means is far from clear. Recent regulatory guidance provides only the example of giving data subject applying for insurance premiums an app to demonstrate the consequences of dangerous driving [1]. Where users are consenting to complex online personalisation which could potentially bring significant effects to their life, such as content delivery which might lead to echo chambers or “filter bubbles”, it is unclear how complex “envisaged consequences” might be best displayed in order to promote user autonomy and choice. Concluding remarks HCI is well-placed to help enable the regulatory effectiveness of the GDPR in relation to algorithmic fairness and accountability. Here we have touched on different points where governance might come into play—model training and model application—but also different modes of governance. Firstly, HCI might play a role in enabling creative, rigorous, problem solving practices within organisations. Many mechanisms in the GDPR, such as data protection by design and data protection impact assessments, will depend heavily on the communities, practices and technologies that develop around them in different contexts. Secondly, HCI might play a role in enabling controllers do particular tasks better. Here, we discussed the potential for exploratory data analysis tools, such as detecting special category data even when it was not explicitly collected. Finally, it might help data subjects exercise their rights better. It appears especially important to develop new modes and standards for transparency, documentation of human input, and communication of tricky notions such as “envisaged consequences”. As the GDPR often defines data controllers’ obligations as a function of “available technologies” and “technological developments”, it is explicitly enabled and strengthened by computational systems and practices designed with its varied provisions in mind. 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Discrete Mathematics and Theoretical Computer Science DMTCS vol. VOL:ISS, 2015, #NUM arXiv:1603.03019v1 [] 8 Mar 2016 Reducing the generalised Sudoku problem to the Hamiltonian cycle problem Michael Haythorpe1 1 Flinders University, Australia received 2016-03-09, revised xxxx-xx-xx, accepted yyyy-yy-yy. The generalised Sudoku problem with N symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where N is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has O(N 3 ) vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion to undirected Hamiltonian cycle problem is also provided so that it can be submitted to the best heuristics. A simple algorithm for obtaining the valid Sudoku solution from the Hamiltonian cycle is provided. Techniques to reduce the size of the resultant graph are also discussed. Keywords: Sudoku, NP-complete, Reduction, Hamiltonian cycle problem 1 Introduction The generalised Sudoku problem is an NP-complete problem which, effectively, requests a Latin square that satisfies some additional constraints. In addition to the standard requirement that each row and column of the Latin square contains each symbol precisely once, Sudoku also demands block constraints. If there are N symbols, the Latin square size N × N . If N is a perfect square, then the Latin square can be √ is of √ divided into N regions of size N × N , called blocks. Then the block constraints demand that each of these blocks also contain each of the symbols precisely once. Typically, the symbols in a Sudoku puzzle are simply taken as the natural numbers 1 to N . In addition, Sudoku puzzles typically have fixed values in some of the cells, which dramatically limits the number of valid solutions. If the fixed values are such that only a unique solution remains, the Sudoku puzzle is said to be well-formed. The standard version where N = 9 has, in recent years, become a common form of puzzle found in newspapers and magazines the world over. Although variants of the problem have existed for over a century, Sudoku in its current format is a fairly recent problem, first published in 1979 under the name Number Place. The name Sudoku only came into existence in the 1980s. In 2003, the generalised Sudoku problem was shown to be ASP-complete [12], which in turn implies that it is NP-complete. Hence, it is theoretically as difficult as any problems in the set N P of decision problems for which a positive solution can be certified in polynomial time. Note that although there are more general versionsvariants of Sudoku (such as rectangular versions), the square variant described above where N is a perfect square suffices for ISSN subm. to DMTCS c 2015 by the author(s) Distributed under a Creative Commons Attribution 4.0 International License 2 Michael Haythorpe NP-completeness. Hence, for the remainder of this manuscript, it will be assumed that we are restricted to considering the square variant. Since being shown to be NP-complete, Sudoku has subsequently been converted to various NP-complete problems, most notably constraint satisfaction [11], boolean satisfiability [8] and integer programming [2]. Another famous NP-complete problem is the Hamiltonian cycle problem (HCP), which is defined as follows. For a simple graph (that is, one containing no self-loops or multi-edges) containing vertex set V and edge set E : V → V , determine whether any simple cycles containing all vertices in V exist in the graph. Such cycles are called Hamiltonian cycles, and a graph containing at least one Hamiltonian cycle is called Hamiltonian. Although HCP is defined for directed graphs, in practice most heuristics that actually solve HCP are written for undirected graphs. Since both Sudoku and HCP are NP-complete, it should be possible to reduce Sudoku to HCP. In this manuscript, a constructive algorithm that constitutes such a reduction is given. The resultant instance of HCP is a sparse graph or order O(N 3 ). If many values are fixed, it is likely that the resultant graph can be made smaller by clever graph reduction heuristics; to this end, we apply a basic graph reduction heuristic to two example Sudoku instances to investigate the improvement offered. It should be noted that reductions of NP-complete problems to HCP is an interesting but still largely unexplored field of research. Being one of the classical NP-complete problems (indeed, one of the initial 21 NP-complete problems described by Karp [7]), HCP is widely studied and several very efficient algorithms for solving HCP exist. HCP is also an attractive target problem in many cases because the resultant size of the instance is relatively small by comparison to other potential target problems. Indeed, the study of which NP-complete problems provide the best target frameworks for reductions is an ongoing field of research. For more on this topic, as well as examples of other reductions to HCP, the interested reader is referred to [4, 3, 6, 5]. 2 Conversion to HCP At it’s core, a Sudoku problem with N symbols (which we will consider to be the natural numbers from 1 to N ) has three sets of constraints to be simultaneously satisfied. 1. Each of the N blocks must contain each number from 1 to N precisely once. 2. Each of the N rows must contain each number from 1 to N precisely once. 3. Each of the N columns must contain each number from 1 to N precisely once. The variables of the problem are the N 2 cells, which can each be assigned any of the N possible values, although some of the cells may have fixed values depending on the instance. In order to cast an instance of Sudoku as an instance of Hamiltonian cycle problem, we need to first encode every possible variable choice as a subgraph. The idea will be that traversing the various subgraphs in certain ways will correspond to particular choices for each of the variables. Then, we will link the various subgraphs together in such a way that they can only be consecutively traversed if none of the constraints are violated by the variable choices. In the final instance of HCP that is produced, the vertex set V will comprise of the following, where a, i, j and k all take values from 1 to N : • A single starting vertex s and finishing vertex f Reducing the generalised Sudoku problem to the Hamiltonian cycle problem 3 • Block vertices: N 2 vertices bak , corresponding to number k in block a • Row vertices: N 2 vertices rik , corresponding to number k in row i • End Row vertices: N vertices ti corresponding to row i • Column vertices: N 2 vertices cjk corresponding to number k in column j • End Column vertices: N vertices dj corresponding to column j • Puzzle vertices: 3N 3 vertices xijkl corresponding to number k in position (i, j), for l = 1, 2, 3 • End Puzzle vertices: N 2 vertices vij corresponding to position (i, j) • Duplicate Puzzle vertices: 3N 3 vertices yijkl corresponding to number k in position (i, j), for l = 1, 2, 3 • End Duplicate Puzzle vertices: N 2 vertices wij corresponding to position (i, j) The graph will be linked together in such a way that any valid solution to the Sudoku puzzle will correspond to a Hamiltonian cycle in the following manner. 1. The starting vertex s is visited first. 2. For each a and k, suppose number k is placed in position (i, j) in block a. Then, vertex bak is visited, followed by all xijml for m 6= k, followed by all yijml for m 6= k. This process will ensure constraint 1 is satisfied. 3. For each i and k, suppose number k is placed in position (i, j) in row i. Then, vertex rik is visited, followed by xijk3 , xijk2 , xijk1 and then vij . If k = N (ie if i is about to be incremented or we are finished step 3) then this is followed by ti . This process will ensure constraint 2 is satisfied. 4. For each j and k, suppose number k is placed in position (i, j) in column j. Then, vertex cjk is visited, followed by yijk3 , yijk2 , yijk1 and then wij . If k = N (ie if j is about to be incremented or we are finished step 4) then this is followed by dj . This process will ensure constraint 3 is satisfied. 5. The finishing vertex f is visited last and the Hamiltonian cycle returns to s. What follows is a short description of how steps 1–5 are intended to work. A more detailed description follows in the next section. The idea of the above is that we effectively create two identical copies of the Sudoku puzzle. In step 2, we place numbers in the puzzles, which are linked together in such a way to ensure the numbers are placed identically in both copies. Placing a number k into position (i, j), contained in block a, is achieved by first visiting bak , and then proceeding to visit every puzzle vertex xijml except for when m = k, effectively leaving the assigned number “open”, or unvisited. Immediately after visiting the appropriate puzzle vertices, the exact same duplicate puzzle vertices yijml are visited as well, leaving the assigned number unvisited in the second copy as well. Since each block vertex bak is only visited once, each number is placed precisely once in each block, satisfying constraint 1. The hope is, after satisfying constraint 1, 4 Michael Haythorpe that the row and column constraints have also been satisfied. If not, it will prove impossible to complete steps 3 and 4 without needing to revisit a vertex that was visited in step 2. In step 3, we traverse the row vertices one at a time. If number k was placed in position (i, j), then row vertex rik is followed by the unvisited vertices xijk3 , xijk2 , xijk1 , and then by the end puzle vertex vij . Once all rik vertices have been traversed for a given i, we visit the end row vertex ti . Note that the three x vertices visited for each i and k in step 3 are the three that were skipped in step 2. Therefore, every puzzle vertex is visited by the time we finish traversing all the row vertices. However, if row i is missing the number k, then there will be no available unvisited puzzle vertices to visit after rik , so this part of the graph can only be traversed if all the row constraints are satisfied by the choices in step 2. Step 4 evolves analogously to step 3, except for cjk instead of rik , yijkl instead of xijkl , wij instead of vij and dj instead of ti . Hence, this part of the graph can only be traversed if all the column constraints are also satisfied by the choices in step 2. Assuming the graph must be traversed as described above, it is clear that all Hamiltonian cycles in the resultant instance of HCP correspond to valid Sudoku solutions. In order to show this is the case, we first describe the set of directed edges E in the graph. Note that in each of the following, if k + 1 or k + 2 are bigger than N , they should be wrapped back around to a number between 1 and N by subtracting N . For example, if k + 2 = N + 1 then it should be taken as 1 instead. • (s , b11 ), (dN , f ) and (f , s) • (bak , xi,j,(k+1),1 ) for all a, k, and (i, j) contained in block a • (xijk1 , xijk2 ), (xijk2 , xijk1 ), (xijk2 , xijk3 ) and (xijk3 , xijk2 ) for all i, j, k • (xijk3 , xi,j,(k+1),1 ) for all i, j, k • (yijk1 , yijk2 ), (yijk2 , yijk1 ), (yijk2 , yijk3 ) and (yijk3 , yijk2 ) for all i, j, k • (yijk3 , yi,j,(k+1),1 ) for all i, j, k • (xijk3 , yi,j,(k+2),1 ) for all i, j, k • (yijk3 , ba,k+2 ) for all i, j, and for k 6= N − 1, where a is the block containing position (i, j) • (yi,j,N −1,3 , ba+1,1 ) for all i, j except for the case where both i = N and j = N , where a is the block containing position (i, j) • (yN,N,N −1,3 , r11 ) • (rik , xijk3 ) for all i, j, k • (xijk1 , vij ) for all i, j, k • (vij , rik ) for all i, j, k • (vij , ti ) for all i, j • (ti , ri+1,1 ) for all i < N • (tN , c11 ) Reducing the generalised Sudoku problem to the Hamiltonian cycle problem 5 • (cjk , yijk3 ) for all i, j, k • (yijk1 , wij ) for all i, j, k • (wij , cjk for all i, j, k • (wij , dj ) for all i, j • (dj , cj+1,1 ) for all j < N 3 Detailed explanation We need to show that every valid Hamiltonian cycle corresponds to a valid Sudoku solution. Note that at this stage, we have not handled any fixed cells, so any valid Sudoku solution will suffice. Fixed cells will be taken care of in Section 5. Theorem 3.1 Every Hamiltonian cycle in the graph constructed in the previous section corresponds to a valid Sudoku solution, and every valid Sudoku solution has corresponding Hamiltonian cycles. Proof: First of all, note that vertices xijk2 are degree 2 vertices, and so they ensure that if vertex xijk1 is visited before xijk3 , it must be proceeded by xijk2 and then xijk3 . Likewise, if vertex xijk3 is visited before xijk1 , it must be proceeded by xijk2 and xijk1 . The same argument holds for vertices yijk2 . This will ensure that the path any Hamiltonian cycle must take through the x and y vertices is tightly controlled. Each of the block vertices bak links to xi,j,(k+1),1 for all (i, j) contained in block a. One of these edges must be chosen. Suppose number k is to be placed in position (i, j), contained in block a. Then the edge (bak , xi,j,(k+1),1 ) is traversed. From here, the cycle must continue through vertices xi,j,(k+1),2 and xi,j,(k+1),3 . It is then able to either exit to one of the y vertices, or continue visiting x vertices. However, as will be seen later, if it exits to the y vertices at this stage, it will be impossible to complete the Hamiltonian cycle. So instead it continues on to xi,j,(k+2),1 , and so on. Only once all of the xijml vertices for m 6= k have been visited (noting that i and j are fixed here) can it safely exit to the y vertices – refer this as Assumption 1 (we will investigate later what happens if Assumption 1 is violated for any i, j, k). The exit to y vertices will occur immediately after visiting vertex xi,j,(k−1),3 , which is linked to vertex yi,j,(k+1),1 . Note that by Assumption 1, vertices xijkl are unvisited for l = 1, 2, 3. Then, from the y vertices, the same argument as above applies again, and eventually vertex yi,j,(k−1),3 is departed, linking to vertex ba,k+1 if k < N , or to vertex ba+1,1 if k = N . Refer to the equivalent assumption on visiting the y vertices as Assumption 2. This continues until all the block vertices have been traversed, at which time vertex yN,N,N −1,3 links to r11 . Note that, other than by violating Assumptions 1 or 2, it is not possible to have deviated from the above path. By the time we arrive at r11 , all the block vertices bak have been visited. Also, every puzzle vertex xijkl and duplicate puzzle vertex yijkl has been visited other than those corresponding to placing number k in position (i, j). Next, each of the row vertices rik links to xijk3 for all i, j, k. For each i and k, one of these edges must be chosen. However, by Assumption 1, all vertices xijk3 have already been visited except for those corresponding to the number k being placed in position (i, j). If the choices in the previous step violate the row constraints, then there will be a row i that does not contain a number k, and subsequently there will be no valid edge emanating from vertex rik . Hence, if the choices made in step 2 violate the row constraints, and Assumption 1 is correct, it is impossible to complete a Hamiltonian cycle. If the choices 6 Michael Haythorpe in the previous step satisfy the row constraints, then there should always be precisely one valid edge to choose here. Once vertex xijk3 is visited, vertices xijk2 and xijk1 must follow, at which point the only remaining valid choice is to proceed to vertex vij . From here, any row vertex rim that has not yet been visited can be visited. If all, have been visited, then ti can be visited instead. Note that once ti is visited, it is impossible to return to any rik vertices, so they must all be visited before ti is visited. An analogous argument to above can be made for the column vertices cjk . Note that if Assumptions 1 and 2 are correct, then vertex yijkl will be unvisited at the start of step 4 if and only if xijkl was unvisited at the start of step 3. Therefore, we see that if Assumptions 1 and 2 are correct, then it is only possible to complete the Hamiltonian cycle if the choices made in step 2 correspond to a valid Sudoku solution. Now consider the situation where Assumption 1 is violated, that is, after step 2 there exists unvisited vertices xijkl and xijml for some i, j, and k 6= m. Then during step 3, without loss of generality, suppose vertex rik is visited before rim . As argued above, this will be followed by vertices xijk3 , xijk2 , xijk1 , at which point visiting vertex vij is the only available choice. Then later, rim is visited. It must visit xijm3 , xijm2 , xijm1 and is then, again, forced to proceed to vertex vij . However, since vertex vij has already been visited, this is impossible and the Hamiltonian cycle cannot be completed. If Assumption 2 is violated, and it is vertices yijkl and yijml that are unvisited after step 2, an analogous argument can be made involving step 4. Hence, every Hamiltonian cycle in the graph must satisfy Assumptions 1 and 2. This completes the proof. 2 Since any valid Sudoku solution has corresponding Hamiltonian cycles, the resulting instance of HCP is equivalent to a blank Sudoku puzzle. In a later section, the method for removing edges based on fixed numbers for a given Sudoku instance is described. Since the instance of HCP can be constructed, and the relevant edges removed, in polynomial time as a function of N , the algorithm above constitutes a reduction of Sudoku to the Hamiltonian cycle problem. 4 Size of “blank” instance The instance of HCP that emerges from the above conversion consists of 6N 3 + 5N 2 + 2N + 2 vertices, and 19N 3 +2N 2 +2N +2 directed edges. For the standard Sudoku puzzle where N = 9, this corresponds to a directed graph with 4799 vertices and 14033 directed edges. All of the best HCP heuristic currently available assume that the instance is undirected. There is a well-known conversion of directed HCP to undirected HCP which can be performed as follows. First, produce a new graph which has three times as many vertices as the directed graph. Then add edges to this new graph by the following scheme, where n is the number of vertices in the directed graph: 1. Add edges (3i − 1, 3i − 2) and (3i − 1, 3i) for all i = 1, . . . , n. 2. For each directed edge (i, j) in the original graph, add edge (3i, 3j − 2). In the present case, this results in an undirected instance of HCP consisting of 18N 3 + 15N 2 + 6N + 6 vertices and 31N 3 + 12N 2 + 6N + 6 edges. This implies that the average degree in the graph grows monotonically with N , but towards a limit of 31 9 , so the resultant graph instance is sparse. For N = 4, the average degree is just slightly above 3.1, and for N = 9 the average degree is just under 3.3. A trick can be employed to reduce the number of vertices in the undirected graph. Consider the vertices in the undirected graph corresponding to the x and y vertices. In particular, consider the set of 9 Reducing the generalised Sudoku problem to the Hamiltonian cycle problem 7 vertices corresponding to xijk1 , xijk2 and xijk3 . The nine vertices form an induced subgraph such as that displayed at the top of Figure 1. There are incoming edges incident on the first and seventh vertices, and outgoing edges incident on the third and ninth vertices. If the induced subgraph is entered via the first vertex, it must be departed via the ninth vertex, or else a Hamiltonian cycle cannot be completed. Likewise, if the induced subgraph is entered via the seventh vertex, it must be departed via the third vertex. It can be seen by inspecting all cases that if the fifth vertex is removed, and a new edge is introduced between the fourth and sixth vertices, the induced subgraph retains these same properties. This alternative choice is displayed at the bottom of Figure 1. Such a replacement can be made for each triplet xijkl or yijkl . Hence, we can remove 2N 3 vertices and 2N 3 edges from the undirected graph for a final total of 16N 3 + 15N 2 + 6N + 6 vertices and 29N 3 + 12N 2 + 6N + 6, although at the cost of raising the average degree by a small amount (roughly between 0.1 and 0.15, depending on N .) Fig. 1: The induced subgraph created after the conversion to an undirected graph, corresponding to vertices xijk1 , xijk2 and xijk3 , and an alternative subgraph with one vertex removed. 5 Handling fixed numbers In reality, all meaningful instances of Sudoku have fixed values in some of the N 2 cells. Although this could potentially be handled by removing vertices, it would then be necessary to redirect edges appropriately. Instead, it is simpler to remove edges that cannot be used while choosing these fixed values. Once this is performed, a graph simplifying heuristic could then be employed to remove unnecessary vertices if desired. For each fixed value, 12N − 12 edges can be identified as redundant, and be removed. However, when there are multiple fixed values, some edges may be identified as redundant multiple times, so 12N − 12 is only an upper bound on the number of edges that can be removed per fixed value. For example, suppose one cell has a fixed value of 1, and another cell within the same block has a fixed value of 2. From the first fixed value, we know that all other entries in the block must not be 1. From the second fixed value, we know that the second cell must have a value of 2, and hence not 1. Then the edge corresponding to placing a value of 1 in the second cell would be identified as redundant twice. The exact number of redundant edges identified depends on the precise orientation of the fixed values. For each fixed value k in position (i, j), and block a containing position (i, j), the following sets of edges are redundant and may be removed (an explanation for each set follows the list): (1) (bak , xmnk1 ) for all choices of m and n such that block a contains (m, n), and also (m, n) 6= (i, j) 8 Michael Haythorpe (2) (bam , xijm1 ) for m 6= k (3) (xm,n,(k−1),3 , ym,n,(k+1),1 ) for all choices of m and n such that block a contains (m, n), and also (m, n) 6= (i, j) (4) (xi,j,(m−1),3 , yi,j,(m+1),1 ) for m 6= k (5a) If k < N : (ym,n,(k−1),3 , ba,k+1 ) for all choices of m and n such that block a contains (m, n), and also (m, n) 6= (i, j) (5b) If k = N and a < N : (ym,n,(k−1),3 , ba+1,1 ) for all choices of m and n such that block a contains (m, n), and also (m, n) 6= (i, j) (5c) If k = N and a = N : (ym,n,(k−1),3 , r11 ) for all choices of m and n such that block a contains (m, n), and also (m, n) 6= (i, j) (6a) (yi,j,(m−1),3 , ba,m+1 ) for m 6= k and m 6= N (6b) If k < N and a < N : (yi,j,(N −1),3 , ba+1,1 ) (6c) If k < N and a = N : (yi,j,(N −1),3 , r11 ) (7) (rik , ximk3 ) for all m 6= k (8) (ximk1 , vim ) for all m 6= k (9) (rim , xijm3 ) for all m 6= k (10) (cjk , ymjk3 ) for all m 6= k (11) (ymjk1 , wmk ) for all m 6= k (12) (cjm , yijm3 ) for all m 6= k The edges in set (1) correspond to the option of placing a value of k elsewhere in block a. The edges in set (2) correspond to the option of picking a value other than k in position (i, j). Those two sets of incorrect choices would lead to the edges from sets (3) and (4) respectively being used to transfer from the x vertices to the y vertices, and so those edges are also redundant. The edges in (5a)–(5c) correspond to the edges that return from the y vertices to the next block vertex after an incorrect choice is made (corresponding to the set (1)). If k = N then the next block vertex is actually for the following block, rather than for the next number in the same block. If k = N and a = N then all block vertices have been visited and the next vertex is actually the first row vertex. Likewise, the edges in (6a)–(6c) correspond to the edges that return from the y vertices after an incorrect choice is made (corresponding to the set (2)). Note that if k = N , there are N − 1 redundant edges in (6a). If k < N there are N − 2 redundant edges in (6a) and then one additional redundant edge from either (6b) or (6c). The edges in set (7) correspond to the option of finding a value of k in row i at a position other than (i, j), which is impossible. The edges in set (8) correspond to visiting the end puzzle vertex after making an incorrect choice from (7). The edges in set (9) correspond to the option of finding a value other than k Reducing the generalised Sudoku problem to the Hamiltonian cycle problem 9 in row i and position (i, j), which is also impossible. Analogous arguments can be made for the edges in sets (10)–(12), except for columns instead of rows. Each of sets (1)-(4) and (7)-(12) identify N − 1 redundant edges each. As argued above, the relevant sets from (5a)–(5c) will contribute N − 1 more redundant edges, as well the relevant sets from (6a)–(6c). Hence, the maximum number of edges that can be removed per number is 12N − 12 for each fixed value. 6 Recovering the Sudoku solution from a Hamiltonian cycle The constructive algorithm above produces a HCP instance for which each solution corresponds to a valid Sudoku solution Once such a solution is obtained, the following algorithm reconstructs the corresponding Sudoku solution: Denote by h the Hamiltonian cycle obtained. For each i = 1, . . . , N and j = 1, . . . , N , find vertex vij in h. Precisely one of its adjacent vertices in h will be of the form xijk1 for some value of k. Then, number k can be placed in the cell in the ith row and jth column in the Sudoku solution. Suppose that the vertices are labelled in the order given in Section 2. That is, s is labelled as 1, f is labelled as 2, the bak vertices are labelled 3, 4, . . . , N 2 − 2, and so on. Then, for each i and j, vertex vij will be labelled 3N 3 +3N 2 +(i+1)N +(j +2), and vertex xijk1 will be labelled 3iN 2 +(3j −1)N +3k. Of course, if the graph has been converted to an undirected instance, or if it has been reduced in size by a graph reduction heuristic, these labels will need to be adjusted appropriately. 7 Reducing the size of the HCP instances After constructing the HCP instances using the above method, graph reduction techniques can be applied. Most meaningful instances of Sudoku will have many fixed values, which in turn leads to an abundance of degree 2 vertices. In order to test the effectiveness of such techniques, a very simple reduction algorithm was used. Iteratively, the algorithm iteratively checks the following two conditions until there are no applicable reductions remaining: 1. If two adjacent vertices are both degree 2, they can be contracted to a single vertex. 2. If a vertex has two degree 2 neighbours, all of its incident edges going to other vertices can be removed. Note that the second condition above leads to three adjacent degree 2 vertices which will in turn be contracted to a single vertex. The removal of edges when the second condition is satisfied often leads to additional degree 2 vertices being formed which allows the algorithm to continue reducing. Note also that this simple graph reduction heuristic is actually hampered by the graph reduction method described in Section 4, since that method eliminates many degree 2 vertices. It is likely that a more sophisticated graph reduction heuristic could be developed that incorporates both methods. The above heuristic was applied to both a well-formed (that is, uniquely solvable) Sudoku instance with 35 fixed values, as well as one of the Sudoku instances from the repository of roughly 50000 instances maintained by Royle [10]. The instances in that repository all contain precisely 17 fixed numbers, and are all well-formed; it was recently proved via a clever exhaustive computer search that 17 is the minimal 10 Michael Haythorpe Fig. 2: Two well-formed Sudoku instances with 35 fixed values and 17 fixed values respectively. number of fixed values for a well-formed Sudoku problem with 9 symbols [9]. The two instances tested are displayed in Figure 2. After the simple reduction heuristic above was applied to the first Sudoku instance, it had been reduced from an undirected instance with 14397 vertices and 22217 edges, to an equivalent instance with 8901 vertices and 14175 edges. Applying the above reduction algorithm to the second Sudoku instance from Royle’s repository reduced it from an undirected instance with 14397 vertices and 22873 edges, to an equivalent instance with 12036 vertices and 19301 edges. In both cases the reduction is significant, although obviously more there are greater opportunities for reduction when there are more fixed values. Both instances were solved by Concorde [1] which is arguably the best algorithm for solving HCP instances containing large amount of structure, as its branch-and-cut method is very effective at identifying sets of arcs that must be fixed all at once, or not at all, particularly in sparse graphs. Technically, Concorde actually converts the HCP instance to an equivalent TSP instance but does so in an efficient way. The first instance was solved during Concorde’s presolve phase, while the second instance required 20 iterations of Concorde’s branch and cut algorithm(i) to discover a solution. This would seem to indicate that the first Sudoku instance can be solved without requiring any amount of guessing. The two solutions were then interpreted via the algorithm in Section 6 to provide solutions to the initial Sudoku instances; those solutions are displayed in Figure 3. References [1] Applegate, D.L., Bixby, R.B., Chavátal, V., and Cook, W.J.: Concorde TSP Solver: http://www.tsp.gatech.edu/concorde/index.html (2015). Accessed Jan 20, 2016. [2] Bartlett, A., Chartier, T.P., Langville, A.V. and Rankin, T.D.: An integer programming model for the Sudoku problem. Journal of Online Mathematics and its Applications, vol.8, Article ID 1798, 2008. [3] Creignou, N.: The class of problems that are linearly equivalent to Satisfiability or a uniform method for proving NP-completeness, Lect. Notes. Comput. Sc., 145:111-145, 1995. (i) It should be noted that Concorde does use a small amount of randomness in its execution. The random seed used in this experiment was 1453347272. Reducing the generalised Sudoku problem to the Hamiltonian cycle problem 11 Fig. 3: The solutions to the Sudoku instances in Figure 2, as interpreted from the Hamiltonian cycles of the converted HCP instances. [4] Dewdney, A.K.: Linear transformations between combinatorial problems, Int. J. Comput. Math., (11):91–110, 1982. [5] Ejov, V., Haythorpe, M., and Rossomakhine, S.: A Linear-size Conversion of HCP to 3HCP. Australasian Journal of Combinatorics 62(1):45–58, 2015. [6] J. A. Filar and M. Haythorpe, A Linearly-Growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem, in: Optimization and Control methods in Industrial Engineering and Construction, pp. 35–52, 2014. [7] R. M. Karp, Reducibility among combinatorial problems, Springer, New York, 1972. [8] Lynce, I. and Ouaknine, J.: Sudoku as a SAT Problem. In Proceedings of the 9th Symposium on Artificial Intelligence and Mathematics, 2006. [9] McGuire, G., Tugemann, B. and and Civario, G.: There Is No 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem via Hitting Set Enumeration. Exp. Math. 23(2):190–217, 2014. [10] Royle, G.: Minimum Sudoku. http://staffhome.ecm.uwa.edu.au/ 00013890/sudokumin.php (2005). Accessed Jan 20, 2016. [11] Simonis, H.: Sudoku as a constraint problem. In CP Workshop of Modeling and Reformulating Constraint Satisfaction Problems, pages 13–27, 2005. [12] Yato, T. and Seta, T.: Complexity and completeness of finding another solution and its application to puzzles. IEICE T. Fund. Electr., E86-A(5):1052–1060, 2003. 

ON LENGTHS OF HZ-LOCALIZATION TOWERS arXiv:1605.08198v2 [] 27 Jun 2016 SERGEI O. IVANOV AND ROMAN MIKHAILOV Abstract. In this paper, the HZ-length of different groups is studied. By definition, this is the length of HZ-localization tower or the length of transfinite lower central series of HZ-localization. It is proved that, for a free noncyclic group, its HZ-length is ≥ ω + 2. For a large class of Z[C]-modules M, where C is an infinite cyclic group, it is proved that the HZ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its HZ-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M , such that M ⋊ C is finitely presented and H2 (M ⋊ C) is finite. MSC2010: 55P60, 19C09, 20J06 1. Introduction Let R be either a subgring of rationals or a cyclic ring. In his fundamental work [5], A.K. Bousfield introduced the concept of HR-localization. This is a functor in the category of groups, closely related to the functor of homological localization of spaces. In this paper we will study the case R = Z, that is, HZ-localization for different groups. An HZ-map between two groups is a homomorphism which induces an isomorphism on H1 and an epimorphism on H2 . A group Γ is HZ-local if any HZ-map G → H induces a bijection Mor(H, Γ) ≃ Mor(G, Γ). Recall that ([5], Theorem 3.10) the class of HZ-local groups is the smallest class which contains the trivial group and closed under inverse limits and central extensions. Given a group G, the HZ-localization η ∶ G → EG can be uniquely characterized by the following two properties: η is an HZ-map and the group EG is HZ-local. These two properties are given as a definition of HZ-localization in [5]. It is shown in [5] that, for any G, the HZ-localization EG exists, unique and transfiniltely nilpotent. For a group G, denote by {γτ (G)} the transfinite lower central series of G, defined inductively as γτ +1 (G) ∶= [γτ (G), G] and γα = ⋂τ <α γτ (G) for a limit ordinal α. For a G, we will call the length of transfinite lower series of EG, i.e. the least ordinal τ, such that γτ (EG) = 1 by HZ-length of G and denote it as HZ-length(G). Let C be an infinite cyclic group. A Z[C]-module M is tame if and only if M ⋊ C is a finitely presented group [3]. If M is a tame C-module, then dimQ (M ⊗ Q) < ∞ and there exist a generator t ∈ C such that the minimal polynomial of the linear map t ⊗ Q ∶ M ⊗ Q → M ⊗ Q is an integral monic polynomial, which is denoted by µM ∈ Z[x] (see [2, Theorem C] and Lemma 4.7). We prove the following 1 ON LENGTHS OF HZ-LOCALIZATION TOWERS 2 Theorem. Let G be a metabelian group of the form G = M ⋊C, where M is a tame Z[C]module and µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct complex numbers λ1 , . . . , λl and mi ≥ 1. (1) Assume that the equality λi λj = 1 holds only if λi = λj = 1. Then HZ-length(G) ≤ ω. (2) Assume that the equality λi λj = 1 holds only if either mi = mj = 1 or λi = λj = 1. Then HZ-length(G) ≤ ω + 1. As a contrast, we give an example of a finitely presented metabelian group of the form M ⋊C, where M is tame, whose HZ-length is greater than ω +1. In the following example, the Z[C]-module M is tame but it does not satisfy the condition of Theorem 5.6. Let G = ⟨a, b, t ∣ at = a−1 , bt = ab−1 , [a, b] = 1⟩ = Z2 ⋊ C, (1.1) ( −10 −11 ). It is shown in Theorem 5.5 that the HZ-length where C acts on Z2 by the matrix of G is ≥ ω + 2. Let M be a tame Z[C]-module and µM = (x − 1)m f, for some m ≥ 0 and f ∈ Z[x] such that f (1) ≠ 0. Assume that f = f1m1 . . . flml where f1 , . . . , fl ∈ Z[x] are distinct irreducible monic polynomials. If f (1) ∈ {−1, 1}, then HZ-length(M ⋊ C) < ω. (Corollary 4.17). Conjecture. If f (1) ∉ {−1, 1}, then HZ-length(M ⋊ C) ≤ ω + n, where n = max({mi ∣ fi (0) ∈ {−1, 1} ∧ fi (1) ∉ {−1, 1}} ∪ {0}). In particular, for any tame Z[C]-module M, HZ-length(M ⋊ C) < 2ω. It is easy to check that the above theorem together with Corollary 4.17 and Proposition 4.8 imply the conjecture for n = 0, 1. For a group G, denote by Ĝ its pro-nilpotent completion: Ĝ ∶= lim G/γn (G). ←Ð n For a finitely generated group G, there is a natural isomorphism (Prop. 3.14 [5]) EG/γω (EG) = Ĝ. Therefore, for finitely generated groups, HZ-localization gives a natural extension of the pro-nilpotent completion. The pro-nilpotent completion of a finitely generated group G is always HZ-local and the map G → Ĝ induces an isomorphism on H1 . Therefore, for such a group, the following conditions are equivalent: 1) the natural epimorphism EG ↠ Ĝ is an isomorphism; 2) HZ-length of G ≤ ω; 3) The natural map H2 (G) → H2 (Ĝ) is an epimorphism. A simple example of a group with HZ-length ω is the following. Let G = ⟨a, t ∣ at = a3 ⟩ = Z[1/3] ⋊ C. Here C acts on Z[1/3] as the multiplication by 3. Then the pro-nilpotent completion has the structure Ĝ = Z2 ⋊ C, where the cyclic group C = ⟨t⟩ acts on 2-adic integers ON LENGTHS OF HZ-LOCALIZATION TOWERS 3 as the multiplication by 3. Looking at the homology spectral sequence for an extension 1 → Z2 → Ĝ → C → 1, we obtain H2 (Ĝ) = Λ2 (Z2 ) ⊗ Z/9 = 0. Therefore, EG = Ĝ. Since the group G is not pre-nilpotent, HZ-length(G) = ω. The above example is an exception. In most cases, the description of HZ-localization as well as the computation of HZ-length for a given group is a difficult problem. It is shown in [5] that HZ-length of the Klein bottle group GKl ∶= ⟨a, t ∣ at = a−1 ⟩ = Z ⋊ C is greater than ω. As a corollary, it is concluded in [5] that HZ-length of any non-cyclic free group also is greater than ω. Our Theorem 5.6 implies that HZ-length(GKl ) = ω + 1 and that the HZ-localization EGKl lives in the central extension 1 → Λ2 (Z2 ) → EGKl → Z2 ⋊ ⟨t⟩ → 1, where the action of t on 2-adic integers is negation. Moreover, we give a more explicit description of EGKl in Proposition 9.2. The HZ-length of a free non-cyclic group remains a mystery for us, however, we prove the following Theorem. Let F be a free group of rank ≥ 2. Then HZ-length(F ) ≥ ω + 2. Briefly recall the scheme of the proof. Consider an extension of the group (1.1) given by presentation Γ ∶= ⟨a, b, t ∣ [[a, b], a] = [[a, b], b] = 1, at = a−1 , bt = ab−1 ⟩ (1.2) We follow the Bousfield scheme of comparison of pro-nilpotent completions for a free group and the group Γ. Consider a free simplicial resolution of Γ with F0 = F . Group Γ has finite second homology, therefore, lim 1 of its Baer invariants is zero, therefore, π0 of ←Ð the pro-nilpotent completion of the free simplicial resolution equals to the pro-nilpotent completion of Γ. The group Γ has HZ-length greater than ω + 1 and the result about HZ-length of F follows from natural properties of the HZ-localization tower. Observe that the same method does not work for the group (1.1) (as well as for all groups of the type M ⋊ C for abelian M), since lim 1 of Baer invariants of G is huge (this follows from ←Ð proposition 5.8 and an analysis of the tower of Baer invariants for a metabelian group). The paper is organized as follows. In section 2, we present the theory of relative central extensions, which is a generalisation of the standard theory of central extensions. A nonlimit step in the construction of Bousfield’s tower can be viewed as the universal relative extension. Section 2 is technical and introductory, it may be viewed just as a comment to the section 3 of [5]. In section 3 we recall the exact sequences in homology from [9], [10] for central stem-extensions. Observe that the universal relative extensions used for the construction of HZ-tower are stem-extensions. Proposition 3.1 gives the main trick: for a cyclic group C, Z[C]-module M, and a central stem-extension N ↪ G ↠ M ⋊ C, the composite map H3 (M ⋊ C) → (M ⋊ C) ⊗ N ↠ N can be decomposed as H3 (M ⋊ C) → H2 (M)C → H2 (M)C → N. This trick gives a possibility to analyze the homology of the ω + 1-st term of the HZ-localization tower for groups of the type M ⋊ C. Using the properties of tame modules we show in section 4 that the question about the HZ-length of the group M ⋊ C with a tame Z[C]-module M, can be reduced to the same question for the group (M/N) ⋊ C, where N is the largest nilpotent submodule of M. It is shown in [14] that, for a finitely presented metabelian group G, the cokernel (denoted by H2 (ηω )(G)) of the natural map H2 (G) → H2 (Ĝ) is divisible. Using this property we ON LENGTHS OF HZ-LOCALIZATION TOWERS 4 conclude that, HZ-length of M ⋊ C is not greater than ω + 1 if and only if the composite map Λ2 (M̂ )C → Λ2 (M̂ )C → H2 (ηω ) is an epimorphism (see proposition 5.3). Theorem 5.6 is our main result of section 5. There is a simple condition on a tame Z[C]-module M which implies that HZ-length(M ⋊ C) ≤ ω + 1. Theorem 5.6 provides a large class of groups for which one can describe HZlocalization explicitly. In particular, we show that, if the homology H2 (M ⋊ C) is finite, then the module M satisfies the condition (ii) of Theorem 5.6 and therefore, for such M, HZ-length(M ⋊ C) ≤ ω + 1. In section 6 we recall the method of Bousfield from [5], which gives a possibility to study the second homology of the pro-nilpotent completion of a free group. In section 7 we present our root examples (1.1) and (1.2) and prove that they have HZ-length greater than ω + 1. Following the scheme described above, we get the same result for a free non-cyclic group. In the last section of the paper we present an alternative approach for proving that some groups have a long HZ-localization tower. Consider the wreath product i Z ≀ Z = ⟨a, b ∣ [a, ab ] = 1, i ∈ Z⟩ Using functorial technique, we show in theorem 8.2 that HZ-length(Z ≀ Z) ≥ ω + 2. In the last section, as an application of the theory developed in the paper, we give an explicit construction of EGKl . 2. Relative central extensions and HZ-localization Throughout this section G, H denote groups, f ∶ H → G a homomorphism and A an abelian group. 2.1. (Co)homology of a homomorphism. Consider the continuous map between classifying spaces Bf ∶ BH → BG and its mapping cone Cone(Bf ). Following Bousfield [5, 2.14], we define homology and cohomology of f with coefficients in A as follows Hn (f, A) = Hn (Cone(Bf ), A), H n (f, A) = H n (Cone(Bf ), A). Then there are long exact sequences ⋅ ⋅ ⋅ → H2 (H, A) → H2 (G, A) → H2 (f, A) → H1 (H, A) → H1 (G, A) → H1 (f, A) → 0, 0 → H 1 (f, A) → H 1 (G, A) → H 1 (H, A) → H 2 (f, A) → H 2 (G, A) → H 2 (H, A) → . . . . In particular, H1 (f ) = Coker{Hab → Gab } and H1 (f, A) = H1 (f ) ⊗ A. We denote by C̄ ● (G, A) the complex of normalized cochains of G with coefficients in A, [18, 6.5.5] by ∂ n ∶ C̄ n (G, A) → C̄ n+1 (G, A) its differential and by Z̄ n (G, A) and B̄ n (G, A) the groups of normalized cocycles and coboundaries. For a homomorphism f ∶ H → G and an abelian group A we denote by Z̄ n (f, A) and B̄ n (f, A) the following subgroups of Z̄ n (G, A) ⊕ C̄ n−1 (H, A) Z̄ n (f, A) = {(c, α) ∣ f ∗ c = −∂α}, B̄ n (f, A) = {(−∂β, f ∗ β + ∂γ) ∣ β ∈ C̄ n−1 (G, A), γ ∈ C̄ n−2 (H, A)}. Since the map ∂ ∶ C̄ 0 (H, A) → C̄ 1 (H, A) is trivial, we have B̄ 2 (f, A) = {(−∂β, βf ) ∣ β ∈ C̄ 1 (G, A)}. ON LENGTHS OF HZ-LOCALIZATION TOWERS 5 Lemma 2.1. For n ≥ 1 there is an isomorphism H n (f, A) ≅ Z̄ n (f, A)/B̄ n (f, A). Proof. For a space X, we denote by C● (X) the complex of integral chains. Then C● (X, A) = C● (X) ⊗ A and C ● (X, A) = Hom(C● (X), A). For a continuous map F ∶ X → Y we denote by C● (F ) ∶ C● (X) → C● (Y ) the induced morphism of complexes. Then there is a natural homotopy equivalence of complexes Cone(C● (F )) ≃ C● (Cone(F )). It follows that there is a natural homotopy equivalence of complexes C ● (Cone(F ), A) ≃ Cone(C ● (F, A))[−1]. (2.1) Denote by C ● (G, A) the complex of (non-normalised) cochains of the group G. For a homomorphism f ∶ H → G we denote by C ● (f, A) ∶ C ● (G, A) → C ● (H, A) the induced morphism of complexes. There is a natural homotopy equivalence C ● (G, A) ≃ C ● (BG, A). Moreover, there is a natural homotopy equivalence of complexes of normalised and non-normalised cochains C̄ ● (G, A) ≃ C ● (G, A) because that come from two different functorial resolutions. It follows that there is a natural homotopy equivalence Cone(C̄ ● (f, A)) ≃ Cone(C ● (Bf, A)). Combining this with (2.1) we get ∼ C ● (Cone(Bf ), A) Ð → Cone(C̄ ● (f, A))[−1]. The assertion follows.  2.2. Relative central extensions. Definition 2.2. A relative central extension of G by A with respect to f is a couple ι π ι π E = (A ↣ E ↠ G, f˜), where A ↣ E ↠ G is a central extension of G and f˜ ∶ H → E is a homomorphism such that π f˜ = f. H 0 // A ι ❅❅ ❅❅ f ❅❅ f˜ ❅❅  π // // E G // 1 ι1 π1 ι2 π2 Two relative central extensions (A ↣ E1 ↠ G, f˜1 ) and (A ↣ E2 ↠ G, f˜2 ) are said to be ≅ equivalent if there exist an isomorphism θ ∶ E1 → E2 such that θι1 = ι2 , π2 θ = π1 and θf˜1 = f˜2 . Let (c, α) ∈ Z̄ 2 (f, A). Consider the central extension A ↣ Ec ↠ G corresponding to the 2-cocycle c. The underling set of Ec is equal to A × G and the product is given by (a1 , g1 )(a2 , g2 ) = (a1 + a2 + c(g1 , g2 ), g1 g2 ). Denote by f˜α ∶ H → Ec the map given by f˜α (h) = (α(h), f (h)). Note that the equality f ∗ c = −∂α implies the equality c(f (h1 ), f (h2 )) = −α(h1 ) + α(h1 h2 ) − α(h2 ) for all h1 , h2 ∈ H. It follows that f˜α is a homomorphism. Indeed f˜α (h1 )f˜α (h2 ) = (α(h1 ), f (h1 ))(α(h2 ), f (h2 )) = (α(h1 ) + α(h2 ) + c(f (h1 ), f (h2 )), f (h1 )f (h2 )) = (α(h1 h2 ), f (h1 h2 )) = f˜α (h1 h2 ). Then we obtain a relative central extension E(c, α) = (A ↣ Ec ↠ G, f˜α ). ON LENGTHS OF HZ-LOCALIZATION TOWERS 6 Proposition 2.3. The map (c, α) ↦ E(c, α) induces a bijection between elements of H 2 (f, A) and equivalence classes of relative central extensions of G by A with respect to f . Proof. Any central extension is equivalent to the extension A ↣ Ec ↠ G for a normalised 2-cocycle c. Hence, it is sufficient to consider only them. Consider a relative central extension (A ↣ Ec ↠ G, f˜). Define α ∶ H → A so that f˜(h) = (α(h), f (h)). Since f˜(1) = (0, 1), α is a normalised 1-cochain. Since f˜ is a homomorphism, we get (α(h1 ) + α(h2 ) + c(f (h1 ), f (h2 )), f (h1 )f (h2 )) = (α(h1 h2 ), f (h1 h2 )). Thus f ∗ c = −∂α. It follows that any relative central extension is isomorphic to the relative central extension E(c, α) for some (c, α) ∈ Z 2 (f, A). Consider two elements (c, α), (c′ , α′ ) ∈ Z 2 (f, A) such that (c′ , α′ ) − (c, α) = (−∂β, βf ) for some β ∈ C̄ 1 (G, A). It follows that c′ (g1 , g2 ) + β(g1 ) + β(g2 ) = c(g1 , g2 ) + β(g1 g2 ) for any g1 , g2 ∈ G and α′ (h) = α(h) + β(f (h)) for any h ∈ H. Denote by θβ ∶ Ec → Ec′ the map given by θβ (a, g) = (a + β(g), g). Then θβ is a homomorphism. Indeed, θβ (a1 , g1 )θβ (a2 , g2 ) = (a1 + β(g1 ), g1 )(a2 + β(g2 ), g2 ) = (a1 + a2 + β(g1 ) + β(g2 ) + c′ (g1 , g2 ), g1 g2 ) = (a1 + a2 + β(g1 g2 ) + c(g1 , g2 ), g1 g2 ) = θβ ((a1 , g1 )(a2 , g2 )). Moreover, θβ is an isomorphism, because θ−β is its inverse, and it is easy to see that it is an equivalence of the relative central extensions. It follows that the map (c, α) ↦ E(c, α) induces a surjective map from elements of H 2 (f, A) to equivalence classes of relative central extensions. Consider two elements (c, α), (c′ , α′ ) ∈ Z 2 (f, A) such that the relative central extensions E(c, α) and E(c′ , α′ ) are equivalent. Then there is an equivalence θ ∶ Ec → Ec′ . Since θ respects the i njections from A, ϕ(a, 1) = (a, 1) for any a ∈ A. Since θ respects the projections on G, there exist a unique normalised 1-cochain β ∶ G → A such that θ(0, g) = (β(g), g) for any g ∈ G. Using that c is a normalised 2-cocycle, we obtain θ(a, g) = θ((a, 1)(0, g)) = (a, 1)(β(g), g) = (a + β(g), g). Then the fact that θ is a homomorphism implies that c′ (g1 , g2 ) + β(g1 ) + β(g2 ) = c(g1 , g2 ) + β(g1 g2 ), and hence c′ − c = −∂β, and the equality θf˜α = f˜α′ implies α′ − α = βf. The assertion follows.  2.3. Universal relative central extensions. Let A1 and A2 be abelian groups. Recall ι1 π1 ι2 π2 that c morphism from a central extension A1 ↣ E1 ↠ G to a central extension A2 ↣ E2 ↠ G is a couple (ϕ, θ), where ϕ ∶ A1 → A2 and θ ∶ E1 → E2 are homomorphisms such that θι1 = ι2 τ, π2 θ = π1 . Lemma 2.4 (cf. [18, Lemma 6.9.6]). Let (ϕ, θ) and (ϕ′ , θ′ ) be morphisms from a central extension A1 ↣ E1 ↠ G to a central extension A2 ↣ E2 ↠ G. Then the restrictions on the commutator subgroup coincide θ∣[E1 ,E1 ] = θ′ ∣[E1 ,E1 ] . ON LENGTHS OF HZ-LOCALIZATION TOWERS 7 Proof. For the sake of simplicity we identify A1 with the subgroup of E1 and A2 with the subgroup of E2 . Consider the map ρ ∶ E1 → A2 given by ρ(x) = θ(x)θ′ (x)−1 . Since A2 is central, we get ρ(x)ρ(y) = θ(x)θ′ (x)−1 ρ(y) = θ(x)ρ(y)θ′ (x)−1 = θ(x)θ(y)θ′ (y)−1 θ′ (x)−1 = θ(xy)θ′ (xy)−1 = ρ(xy). Hence ρ is a homomorphism to an abelian group. Thus ρ∣[E1 ,E1 ] = 1.  Lemma 2.5. Let (ϕ, θ) be a morphism from a central extension A1 ↣ E1 ↠ G to a central ι2 π2 extension A2 ↣ E2 ↠ G. If ϕ = 0, then A2 ↣ E2 ↠ G splits. ι1 π1 Proof. ϕ = 0 implies θι1 = 0. Since G is a cokernel of ι1 , there exists s ∶ G → E2 such that sπ1 = θ. Then π2 sπ1 = π2 θ = π1 . It follows that π2 s = id.  Definition 2.6. A morphism from a relative central extension (A1 ↣ E1 ↠ G, f˜1 ) to a relative central extension (A2 ↣ E2 ↠ G, f˜2 ) is a morphism (ϕ, θ) from the central extension A1 ↣ E1 ↠ G to the central extension A2 ↣ E2 ↠ G such that θf˜1 = f˜2 . So relative central extensions of G with respect to f form a category. The initial object of this category is called the universal relative central extension of G with respect to f. Example 2.7. For any homomorphism ϕ ∶ A1 → A2 and any (c, α) ∈ Z 2 (f, A1 ) there is a morphism of relative central extensions E(ϕ) = (ϕ, ϕ × id). E(ϕ) ∶ E(c, α) Ð→ E(ϕc, ϕα), (2.2) Definition 2.8. A homomorphism f ∶ H → G is said to be perfect if fab ∶ Hab → Gab is an epimorphism. In other words, f is perfect if and only if H1 (f ) = 0. Lemma 2.9 (cf. [18, Lemma 6.9.6]). Let (ϕ, θ) and (ϕ′ , θ′ ) be morphisms from a relative central extension (A1 ↣ E1 ↠ G, f˜1 ) to a relative central extension (A2 ↣ E2 ↠ G, f˜2 ). If f˜1 is perfect, then (ϕ, θ) = (ϕ′ , θ′ ). Proof. Since f˜1 is perfect, we obtain Im(f˜1 )[E1 , E1 ] = E1 . Since θf˜1 = f˜2 = θ′ f˜1 , we have θ∣Im(f˜1 ) = θ′ ∣Im(f˜1 ) . Lemma 2.4 implies θ∣[E1 ,E1] = θ′ ∣[E1 ,E1 ] . The assertion follows.  Proposition 2.10. The universal relative central extension of G with respect to f exists if and only if f is perfect. Moreover, in this case it is unique (up to isomorphism) and given by a short exact sequence H u 0 // H2 (f )  // U ❄❄ ❄❄ f ❄❄ ❄❄  // G // 1 where u ∶ H → U is a perfect homomorphism. Proof of Proposition 2.10. Assume that there is a universal relative central extension π (A ↣ U ↠ G, u) of G with respect to f and prove that f is perfect. Set B = Coker(fab ∶ Hab → Gab ) and consider the epimorphism τ ∶ U ↠ B given by the composition U ↠ G ↠ Gab ↠ B. Then we have two morphisms (0, (πτ )) and (0, (π0 )) from ON LENGTHS OF HZ-LOCALIZATION TOWERS 8 (A ↣ U ↠ G, u) to the split extension (B ↣ B × G ↠ G, (f0)). The universal property implies that they are equal, and hence B = 0. Thus f is perfect. Assume that f is perfect. Since H1 (Cone(Bf )) = H1 (f ) = 0, the universal coefficient formula for the space Cone(Bf ) implies that there is an isomorphism H 2 (f, A) ≅ Hom(H2 (f ), A) natural by A. Chose an element (cu , αu ) ∈ Z 2 (f, H2 (f )) that represents the element of H 2 (f, H2 (f )) corresponding to the identity map in Hom(H2 (f ), H2 (f )). ιu πu Set U ∶= Ecu , u = f˜αu and Eu = E(cu , αu ). Then Eu = (H2 (f ) ↣ U ↠ G, u). Take a homomorphism ϕ ∶ H2 (f ) → A and consider the commutative diagram Z 2 (f, H2 (f )) // // H 2 (f, H2 (f )) oo ϕ∗ Hom(H2 (f ), H2 (f )) ϕ∗  Z 2 (f, A) ≅ ϕ○−   // // H 2 (f, A) oo ≅ Hom(H2 (f ), A). It shows that the isomorphism Hom(H2 (f ), A) ≅ H 2 (f, A) sends ϕ to the class of (ϕcu , ϕαu ). Combining this with Proposition 2.3 we obtain that any relative central extension is isomorphic to the extension E(ϕcu , ϕαu ) for some ϕ ∶ H2 (f ) → A. For any relative central extension E(ϕcu , ϕαu ) there exists a morphism E(ϕ) ∶ Eu → E(ϕcu , ϕαu ) from Example 2.7. Then we found a morphism from Eu to any other relative central extension. In order to prove that Eu is the universal relative central extension, we have to prove that such a morphism is unique. By Lemma 2.9 it is enough to prove that u ∶ H → U is perfect. Prove that u ∶ H → U is perfect. In other words we prove that Im(u)[U, U] = U. Set ˜ ˜ ˜ E ∶= Im(u)[U, U], A = ι−1 u (E) and f ∶ H → E given by f (h) = u(h). Note that f is perfect. Since f is perfect, πu (E) = Im(f )[G, G] = G. Consider the restriction π = πu ∣E and the π relative central extension E = (A ↣ E ↠ G, f˜) with the obvious embedding E ↪ Eu . Consider the projection ϕ′ ∶ H2 (f ) → H2 (f )/A and take the composition E(ϕ′ ) E ↪ Eu ÐÐÐ→ E(ϕ′ cu , ϕ′ αu ). The composition is equal to (0, (ϕ′ × 1)∣E ). By Lemma 2.5 E(ϕ′ cu , ϕ′ αu ) splits. Thus (ϕ′ cu , ϕ′ αu ) represents 0 in H 2(f, A) ≅ Hom(H2 (f ), A), and hence ϕ′ = 0. It follows that A = H2 (f ). Then the extension A ↣ E ↠ G is embedded into the extension A ↣ U ↠ G. It follows that E = U.  Remark 2.11. If fab ∶ Hab → Gab is an isomorphism, then H2 (f ) = Coker{H2 (H) → H2 (G)}. Remark 2.12. In the proof of Proposition 2.10 we show that, if f is perfect, then the universal relative central extension corresponds to the identity map Hom(H2 (f ), H2 (f )) with respect to the isomorphism H 2 (f, H2 (f )) ≅ Hom(H2 (f ), H2 (f )) that comes from the universal coefficient theorem. 2.4. HZ-localization tower via relative central extensions. Here we give an approach to the HZ-localization tower [5] via relative central extensions. Let G be a group and η ∶ G → EG be its HZ-localization. For an ordinal α we define αth term of the HZ-localization tower by Tα G ∶= EG/γα (EG), where γα (EG) is the αth term of the transfinite lower central series (see [5, Theorem 3.11]). By ηα ∶ G → Tα G we denote the composition of η and the canonical projection, and by tα ∶ Tα+1 G ↠ Tα G we ON LENGTHS OF HZ-LOCALIZATION TOWERS 9 denote the canonical projection. The main point of [5] is that Tα G can be constructed inductively and EG = Tα G for big enough α. We threat the construction of Tα G for a non-limit ordinal α via universal relative central extensions. Proposition 2.13. Let G be a group and α > 1 be an ordinal. Then (ηα )ab ∶ Gab → (Tα G)ab is an isomorphism, the universal central extension of Tα G with respect to ηα is given by G 0 // ●● ●● ● ηα ηα+1 ●●● ●●  ## tα // Tα+1 G // H2 (ηα ) Tα G // 1, and H2 (ηα ) = Coker{H2 (G) → H2 (Tα G)}. Proof. It follows from [5, 3.2], [5, 3.4], Proposition 2.10 and Remarks 2.11, 2.12.  3. Homology of stem-extensions Consider a central extension of groups 1→N →G→Q→1 (3.1) It is shown in [10] that, there is a natural long exact sequence H3 (G) → H3 (Q) → (Gab ⊗ N)/U → H2 (G) → H2 (Q) → N → H1 (G) → H1 (Q) → 0 (3.2) β where U is the image of the natural map H4 K(N, 2) → Gab ⊗ N. Here H4 K(N, 2) is the forth homology of the Eilenberg-MacLane space K(N, 2) which can be described as the Whitehead quadratic functor H4 K(N, 2) = Γ2 N. A central extension (3.1) is called a stem-extension if N ⊆ [G, G]. For a stem extension (3.1), the exact sequence (3.2) has the form (see [9], [10]) H3 (G) → H3 (Q) → Gab ⊗ N → H2 (G) → H2 (Q) → N → 0 δ β (3.3) The map δ is given as follows. We present (3.1) in the form 1 → S/R → F /R → F /S → 1, for a free group F and normal subgroups R, S with R ⊂ S, [F, S] ⊆ R. Then the map δ is induced by the natural epimorphism Sab → S/R: H3 (Q) = H1 (F /S, Sab ) → H1 (F /S, S/R) = Qab ⊗ N = Gab ⊗ N. The isomorphism H1 (F /S, S/R) = Qab ⊗ N follows from the triviality of F /S-action on S/R. In this section we consider the class of metabelian groups of the form Q = M ⋊ C, where C is an infinite cyclic group and M a Z[C]-module. It follows immediately from the homology spectral sequence that, for any n ≥ 2, there is a short exact sequence 0 → H0 (C, Hn (M)) → Hn (Q) → H1 (C, Hn−1 (M)) → 0 ON LENGTHS OF HZ-LOCALIZATION TOWERS 10 which can be presented in terms of (co)invariants as 0 → Hn (M)C → Hn (Q) → Hn (M)C → 0. Composing the last epimorphism with Hn−1 (M)C → Hn−1 (M)C ↪ Hn−1 (Q), we get a natural (in the category of Z[C]-modules) map αn ∶ Hn (Q) → Hn−1 (Q). In the next proposition we will construct a composite map α3′ ∶ H3 (Q) → H1 (C, Λ2 (M)) ↪ Λ2 (M) ↠ Λ2 (M)C ↪ H2 (Q) (3.4) using group-theoretical tools, without spectral sequence. Probably, α3′ coincides with α3 up to isomorphism, but we will not use this comparison later. Proposition 3.1. For a stem extension (3.1) of a group Q = M ⋊ C, there exists a map α3′ ∶ H3 (Q) → H2 (Q), given as a composition (3.4), such that the following diagram is commutative H3 (Q) α′3 H2 (Q) // β δ   Gab ⊗ N // N where the lower horizontal map is the projection Gab ⊗ N → C ⊗ N = N. Proof. We choose a free group F with normal subgroups R ⊂ S ⊂ T such that F /T = C, F /S = Q, F /R = G. In the above notation, we get M = T /S, N = S/R. The proof follows from the direct analysis of the following diagram, which corners are exactly the roots of the diagram given in proposition: α′3 H3 (Q) H1 (F /S, Sab ) H1 (F /T, S∩T S′ ) // ′ H1 (F /T, Sab ) // S∩T H1 (F /T, [S,T ]) ′ (3.5)    H2 (T /S) H2 (Q) // β δ  H1 (F /S, S/R) (F /S)ab ⊗ S/R //  S/R All arrows of this diagram are natural. We will make comments only about two maps from the diagram, other maps are obviously defined. The map S ∩ T′ H1 (F /T, ) → H1 (F /T, Sab ) S′ ON LENGTHS OF HZ-LOCALIZATION TOWERS 11 is an isomorphism since S ) ↪ H1 (F /T, Tab ) = H3 (F /T ) = 0. S ∩ T′ S∩T ′ The vertical map in the diagram H1 (F /T, [S,T ] ) = H1 (F /T, H2 (S/T )) ↪ H2 (S/T ) follows from the identification of H1 of a cyclic group with invariants. The commutativity of (3.5) follows from the commutativity of the natural square H1 (F /T, H1 (F /S, Sab ) // //   H1 (F /S, S/R) H1 (F /T, Sab ) // // H1 (F /T, S/R) and identification of H1 (F /T, −) with invariants of the F /T -action.  4. Tame modules and completions Throughout the section C denotes an infinite cyclic group. If R is a commutative ring, R[C] denotes the group algebra over R. We use only R = Z, Q, C. The augmentation ideal is denoted by I. If t is one of two generators of C, R[C] = R[t, t−1 ] and I = (t − 1). 4.1. Finite dimensional K[C]-modules. Let K be a field (we use only K = Q, C), V be a right K[C]-module such that dimK V < ∞. If we fix a generator t ∈ C we obtain a linear map ⋅t ∶ V → V that defines the module structure. We denote the linear map by aV ∈ GL(V ). The characteristic and minimal polynomials of aV are denoted by χV and µV respectively. These polynomials depend of the choice of t ∈ C. Note that for any such modules V and U we have χV ⊕U = χV χU , µV ⊕U = lcm(µV , µU ). (4.1) Lemma 4.1. Let V be a right K[C]-module such that dimK V < ∞ and t ∈ C be a generator. Then there exist distinct irreducible monic polynomials f1 , . . . , fl ∈ K[x] and an isomorphism V ≅ V1 ⊕ ⋅ ⋅ ⋅ ⊕ Vl , where mi,l m Vi = K[C]/(fi i,1 (t)) ⊕ ⋅ ⋅ ⋅ ⊕ K[C]/(fi i (t)), and mi,1 ≥ mi,2 ≥ ⋅ ⋅ ⋅ ≥ mi,li ≥ 1. Moreover, if we set mi = ∑j mi,j , then χV = f1m1 . . . flml m m and µV = f1 1,1 . . . fl l,1 . Proof. Note that K[C] = K[t, t−1 ] is the polynomial ring K[t] localised at the element t. Then it is a principal ideal domain. Then the isomorphism follows from the structure theorem for finitely generated modules over a principal ideal domain. The statement about χV and µV follows from the fact that both characteristic and minimal polynomials m m  of K[C]/(fi i,j (t)) equal to fi i,j and the formulas (4.1). Let R be a commutative ring and t be a generator of C. For an R[C]-module M and a polynomial f ∈ R[x] we set M f = {m ∈ M ∣ m ⋅ f (t) = 0}. ON LENGTHS OF HZ-LOCALIZATION TOWERS 12 Note that M C = M x−1 . It is easy to see that for any f, g ∈ R[x] we have (M/M f )g = M f g /M f . (4.2) Corollary 4.2. Let V be a right K[C]-module such that dimK V < ∞ and t ∈ C be a generator. Assume that µV = f1m1 . . . flml , where f1 , . . . , fl ∈ K[x] are distinct irreducible monic polynomials. Consider the filtration 0 = F0 V ⊂ F1 V ⊂ ⋅ ⋅ ⋅ ⊂ Fl V = V given by Fi V = V m m f1 1 ...fi i . Then V = ⊕ Fi V /Fi−1 V, i and Fi V /Fj−1 V = (V /Fj−1V )fj mj mi ...fi m µFi V /Fj−1 = fj j . . . fimi . Proof. In the notation of Lemma 4.1 we obtain Fi V = V1 ⊕⋅ ⋅ ⋅⊕Vi . The assertion follows.  4.2. Tame Z[C]-modules. The rank of an abelian group A is dimQ (A ⊗ Q). The torsion subgroup of A is denoted by tor(A). The following statement seems to be well known but we can not find a reference, so we give it with a proof. Lemma 4.3. Let A be a torsion free abelian group of finite rank and B be a finite abelian group. Then A ⊗ B is finite. Proof. It is sufficient to prove that A ⊗ Z/pk is finite for any prime p and k ≥ 1. Consider a p-basic subgroup A′ of A (see [12, VI]). Then A′ ≅ Zr , where r is the rank of A and A/A′ is p-divisible. Thus (A/A′ ) ⊗ Z/pk = 0. Hence the map (Z/pk )r ≅ A′ ⊗ Z/pk ↠ A ⊗ Z/pk is an epimorphism.  A finitely generated Z[C]-module M is said to be tame if the group M ⋊ C is finitely presented. Proposition 4.4 (Theorem C of [2]). Let M be a finitely generated Z[C]-module. Then M is tame if and only if the following properties hold: ● tor(M) is finite; ● the rank of M is finite; ● there is a generator t of C such that χM ⊗Q is integral. Lemma 4.5 (Lemma 3.4 of [2]). Let M and M ′ be tame Z[C]-modules. Then M ⊗ M ′ is a tame Z[C]-module (with the diagonal action). Definition 4.6. Let M be a tame module. The generator t ∈ C such that such that χM ⊗Q is integral is called an integral generator for M. When we consider a tame module, we always denote by t an integral generator for M. We set aM ∶= t ⊗ Q ∶ M ⊗ Q → M ⊗ Q, and denote by χM , µM the characteristic and the minimal polynomial of aM . In other words χM = χM ⊗Q and µM = µM ⊗Q . Lemma 4.7. µM is an integral monic polynomial for any tame Z[C]-module M. Proof. Let χM = (x − λ1 )m1 . . . (x − λl )ml for some distinct λ1 , . . . , λl ∈ C and mi ≥ 1. Then µM = (x − λ1 )k1 . . . (x − λl )kl , where 1 ≤ ki ≤ mi . Since χ is a monic integral polynomial, λ1 , . . . , λl are algebraic integers. It follows that the coefficients of µM are algebraic integers as well. Moreover, they are rational numbers, because aM is defined rationally. Using that a rational number is an algebraic integer iff it is an integer number, we obtain µM ∈ Z[x].  ON LENGTHS OF HZ-LOCALIZATION TOWERS 13 Proposition 4.8. Let M be a torsion free tame Z[C]-module and µM = f1m1 . . . flml where f1 , . . . , fl are distinct irreducible integral monic polynomials and mi ≥ 1 for all i. Consider the filtration 0 = F0 M ⊂ F1 M ⊂ ⋅ ⋅ ⋅ ⊂ Fl M = M m1 mi m given by Fi M = M f1 ...fi . Then Fi M/Fj−1 M is torsion free and µFi M /Fj−1 M = fj j . . . fimi for any i ≥ j. Moreover, the corresponding filtration on M ⊗ Q splits: l M ⊗ Q ≅ ⊕ (Fi M/Fi−1 M) ⊗ Q. i=1 Proof. Prove that Fi M/Fj−1 M is torsion free. Let v + Fj−1 M ∈ Fi M/Fj−1 M and nv + m Fj−1 M = 0. Hence nv ⋅ f1m1 (t) . . . fj j (t) = 0 in M. Using that M is torsion free we get m v ⋅ f1m1 (t) . . . fj j (t) = 0, and hence v + Fj−1 M = 0. Thus Fi M/Fj−1 M is torsion free. Set V = M ⊗Q, K = Q, apply Corollary 4.2 and note that Fi V /Fj−1 V = (Fi M/Fj−1 M)⊗Q. The assertion follows. Here we use Gauss lemma about integral polynomials: an irreducible polynomial in Z[x] is irreducible in Q[x].  Recall that a module N is said to be nilpotent if NI n = 0 for some n, where I is the augmentation ideal. It is easy to see that a Z[C]-module N is nilpotent if and only if n N (x−1) = N for some n. Definition 4.9. A Z[C]-module M is said to be invariant free if M C = 0. Lemma 4.10. Let M be a torsion free tame Z[C]-module. Then the following equivalent. (1) (2) (3) (4) (5) (6) M is invariant free; M does not have non-trivial nilpotent submodules; MC is finite; aM − 1 is an automorphism; χM (1) ≠ 0; µM (1) ≠ 0. Proof. (1) ⇔ (2) and (4) ⇔ (5) ⇔ (6) are obvious. The equality Ker(aM − 1) = M C ⊗ Q implies (1) ⇔ (4). Since M is finitely generated Z[C]-module, MC is a finitely generated abelian group. Then the equality Coker(aM − 1) = MC ⊗ Q implies (3) ⇔ (4).  Corollary 4.11. Let M be a torsion free tame Z[C]-module and µM = (x − 1)m f , where f (1) ≠ 0. Then there exists the largest nilpotent submodule N ≤ M. Moreover, µN = (x − 1)m , µM /N = f, M/N is torsion free and invariant free, and the short exact sequence N ⊗ Q ↣ M ⊗ Q ↠ (M/N) ⊗ Q splits over Q[C]. Proof. If µM (1) ≠ 0, then M is already invariant free, N = 0 and there is nothing to prove. If µM (1) = 0, then we can decompose µM = (x − 1)m1 f2m2 . . . flml into a product of irreducible polynomials such that fi (1) ≠ 0 for i ≥ 2. Consider the filtration from Proposition 4.8. Then N = F1 M.  Corollary 4.12. Let M be a tame Z[C]-module. Then there exists the largest nilpotent submodule N ≤ M. Moreover, M/N is invariant free and (M/N)C is finite. Recall that a module N is said to be prenilpotent if NI n = NI n+1 for n >> 1. ON LENGTHS OF HZ-LOCALIZATION TOWERS 14 Corollary 4.13. Let M be a tame Z[C]-module and µM = (x − 1)m f , where f (1) ≠ 0. Then there exists a prenilpotent submodule N ≤ M such that M/N is torsion free and invariant free. Moreover, tor(N) = tor(M), µN = (x − 1)m , µM /N = f and the sequence N ⊗ Q ↣ M ⊗ Q ↠ (M/N) ⊗ Q splits over Q[C]. Lemma 4.14. Let M be a tame torsion free Z[C]-module. If µM (0) ∈ {−1, 1}, then M is finitely generated as an abelian group. Proof. It follows from the fact that Z[t, t−1 ]/(µM (t)) is a finitely generated abelian group.  4.3. Completion of tame Z[C]-modules. If M is a finitely generated R[C]-module, we set M̂ = lim M/MI i and we denote by ←Ð ϕ = ϕM ∶ M Ð→ M̂ the natural map to the completion. Note that the functor M ↦ M̂ is exact [19, VIII] and M̂ /M̂I i = M/MI i . We set Zn = lim Z/ni ←Ð for any n ∈ Z. In particular, Zn = Z−n , Z0 = Z, Z1 = 0 and, if n ≥ 2, then Zn = ⊕ Zp , where p runs over all prime divisors of n. Lemma 4.15. Let M be a tame torsion free invariant free Z[C]-module and n = χM (1). Then ni ⋅ M ⊆ MI i for any i ≥ 1 and there exists a unique epimorphism of Z[C]-modules ϕ̂ ∶ M ⊗ Zn ↠ M̂ such that the diagrams M ⊗ Zn ϕ̂ // //  M ⊗ Z/ni M̂  // // M/MI i are commutative. Proof. We identify M with the subgroup of M ⊗ Q. Corollary 4.10 implies that n ≠ 0. Set b = aM − 1. Then the characteristic polynomial of b is equal to χb (x) = χM (x + 1) and if χb = ∑di=0 βi xi , then β0 = n. Thus nx = b(∑di=1 βi bi−1 (x)) for any x ∈ M. It follows that nM ⊆ b(M). Hence ni M ⊆ bi (M) = MI i for any i ≥ 1 and we obtain homomorphisms M ⊗ Z/ni → M/MI i . We define ϕ̂ as the composition M ⊗ Zn → lim(M ⊗ Z/ni ) → M̂. ←Ð Since the rank M is finite, the abelian groups M ⊗ Z/ni are finite. Thus we get that the homomorphism lim(M ⊗ Z/ni ) → M̂ is an epimorphism because lim1 of an inverse ←Ð ←Ð sequence of finite groups is trivial. Then it is sufficient to prove that the homomorphism M ⊗Zn → lim(M ⊗ Z/ni ) is an epimorphism. For this it is enough to prove that M ⊗Zp → ←Ð lim(M ⊗ Z/pi ) is an epimorphism for any prime p. Consider a p-basic subgroup B of M ←Ð (see [12, VI]). Since B ≅ Zl , we get B ⊗Zp = lim(B ⊗ Z/pi ). Using that B ⊗Z/pi ↠ M ⊗Z/pi ←Ð are epimorphisms of finite groups, we obtain that lim(B ⊗ Z/pi ) → lim(M ⊗ Z/pi ) is an ←Ð ←Ð ON LENGTHS OF HZ-LOCALIZATION TOWERS 15 epimorphism. Then analysing the diagram B ⊗ Zp // // M ⊗ Zp ≅ lim(B ←Ð   ⊗ Z/pi ) // // lim(M ⊗ Z/pi ) ←Ð we obtain that the right vertical arrow is an epimorphism.  A Z[C]-module is said to be perfect if MI = M. Corollary 4.16. Let M be a torsion free tame Z[C]-module. If µM (1) ∈ {−1, 1}, then M is perfect. Proof. By Lemma 4.10, M is invariant free. Then Lemma 4.15 implies M̂ = 0. Hence M is perfect.  Corollary 4.17. Let M be a tame Z[C]-module. If µM = (x−1)m f, where f is an integral polynomial such that f (1) ∈ {−1, 1}, then M is prenilpotent. Proof. A finite module is always prenilpotent, so we can assume that M has no torsion. Further, by Lemma 4.11, we can consider the largest nilpotent submodule N ≤ M such that µN = (x − 1)m and µM /N = f. Corollary 4.16 implies that M/N is perfect. Then N and M/N are prenilpotent, and hence, M is prenilpotent.  Proposition 4.18. Let M and M ′ be tame Z[C]-modules with the same integral generator t ∈ C, λ1 , . . . , λl ∈ C are eigenvalues of aM and λ′1 , . . . , λ′l′ ∈ C are eigenvalues of aM ′ . Assume that the equality λi λ′j = 1 holds only if λi = λ′j = 1. Then the homomorphism (M ⊗ M ′ )C Ð→ (M̂ ⊗ M̂ ′ )C is an epimorphism. Proof. Note that if M1 ↣ M2 ↠ M3 is a short exact sequence of tame modules and (M1 ⊗ M ′ )C → (M̂1 ⊗ M̂ ′ )C , (M3 ⊗ M ′ )C → (M̂3 ⊗ M̂ ′ )C are epimorphisms, then (M2 ⊗ M ′ )C → (M̂2 ⊗ M̂ ′ )C is an epimorphism. Indeed, since the functor of completion is exact, we have the commutative diagram with exact rows (M1 ⊗ M ′ )C  // (M̂1 ⊗ M̂ ′ )C (M2 ⊗ M ′ )C // //   // (M̂2 ⊗ M̂ ′ )C (M3 ⊗ M ′ )C // // (M̂3 ⊗ M̂ ′ )C that implies this. Then, using Corollary 4.13, we obtain that we can divide our prove into two parts: (1) prove the statement for the case of torsion free invariant free modules M, M ′ ; (2) prove the statement for the case of a prenilpotent module M and arbitrary tame module M ′ . Throughout the proof we use that (M ⊗ M ′ )C ≅ M ⊗Z[C] Mσ′ , where Mσ′ is the module with the same underling abelian group M ′ but with the twisted action of C ∶ m ∗ t = mt−1 . ON LENGTHS OF HZ-LOCALIZATION TOWERS 16 (1) Assume that M, M ′ are torsion free invariant free tame Z[C]-modules. Lemma 4.10 implies that λi ≠ 1 and λ′j ≠ 1 for all i, j. Then we have λi λ′j ≠ 1 for all i, j. Note that the eigenvalues of aM ⊗ aM ′ equal to the products λi λj , and hence 1 is not an eigenvalue of aM ⊗ aM ′ . It follows that det(aM ⊗ aM ′ − 1) ≠ 0. Consider the minimal polynomial µ of the tensor square aM ⊗ aM ′ . Since the aM ⊗ aM ′ is defined over Q, the coefficients of µ are rational (because they are invariant under the action of the absolute Galois group). Moreover, µ = ∏(x − λi λ′j )ki,j for some ki,j , and hence, its coefficients are algebraic integers. It follows that µ is a monic polynomial with integral coefficients. The polynomial µ(x + 1) is the minimal polynomial for aM ⊗ aM ′ − 1. Let µ(x + 1) = ∑ki=0 ni xi . Then n0 = det(aM ⊗aM ′ −1) ≠ 0 and n0 ⋅(M ⊗M ′ ) ⊆ (M ⊗M ′ )(t−1). Since the rank of M ⊗M ′ is finite, (M ⊗M ′ )/n0 (M ⊗M ′ ) is finite, and hence, (M ⊗M ′ )C = (M ⊗M ′ )/(M ⊗M ′ )(t−1) is finite. By Lemma 4.15 we have epimorphisms M ⊗ Zn ↠ M̂ and M ′ ⊗ Zn′ ↠ M̂ ′ , where n = det(aM − 1) and n′ = det(aM ′ − 1). It is easy to see that ((M ⊗ Zn ) ⊗ (M ′ ⊗ Zn′ ))C = (M ⊗ M ′ )C ⊗ (Zn ⊗ Zn′ ). Since (M ⊗M ′ )C is finite, (M ⊗M ′ )C → (M ⊗M ′ )C ⊗(Zn ⊗Zn′ ) is an epimorphism. Then (M ⊗ M ′ )C → (M̂ ⊗ M̂ ′ )C is an epimorphism. (2) Assume that M is a prenilpotent Z[C]-module and M ′ is a tame Z[C]-module. Then there exists i such that M̂ = M/MI i . Since (M̂ ⊗ M̂ ′ )C ≅ M̂ ⊗Z[C] M̂σ′ , we get (M̂ ⊗ M̂ ′ )C ≅ (M/MI i ⊗ M̂ ′ )C ≅ (M/MI i ⊗ M̂ ′ /M̂ ′ I i )C ≅ (M/MI i ⊗ M ′ /M ′ I i )C . It follows that (M ⊗ M ′ )C → (M̂ ⊗ M̂ ′ )C is an epimorphism.  Corollary 4.19. Let M be a tame Z[C]-module and µM = (x − 1)m f1m1 . . . flml for some distinct monic irreducible polynomials f1 , . . . , fl ∈ Z[x] such that fi (1) ≠ 0 and fi (0) ∉ {1, −1} for all 1 ≤ i ≤ l. Then the the homomorphism (M ⊗2 )C Ð→ (M̂ ⊗2 )C is an epimorphism. Proof. Let λ1 , . . . , λk be roots of µM . Assume that λi λj = 1. Then λi is an invertible algebraic integer, and hence, the absolute term of its minimal polynomial equals to ±1. Thus λi can not be a root of fm for 1 ≤ m ≤ l. It follows that it is a root of x − 1. Then λi = λj = 1.  Proposition 4.20. Let M, M ′ be tame Z[C]-modules with the same integral generator t ∈ C, µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct λ1 , . . . , λl ∈ C and µM ′ = (x − ′ ′ λ′1 )m1 . . . (x − λ′l′ )ml′ for some distinct λ′1 , . . . , λ′l′ ∈ C. Assume that the equality λi λ′j = 1 holds only if either mi = m′j = 1 or λi = λ′j = 1. Then the cokernel of the homomorphism (M ⊗ M ′ )C ⊕ (M̂ ⊗ M̂ ′ )C Ð→ (M̂ ⊗ M̂ ′ )C is finite. Proof. Corollary 4.13 implies that the proof can be divided into proofs of the following two statements: (1) the statement for torsion free invariant free modules M, M ′ ; (2) if N ↣ M ↠ M0 is a short exact sequence of tame Z[C]-modules such that N ⊗ Q ↣ M ⊗Q ↠ M0 ⊗Q splits, N is prenilpotent, and the statement holds for the couple M0 , M ′ , then it holds for the couple M, M ′ . ON LENGTHS OF HZ-LOCALIZATION TOWERS 17 (1) Here we prove that the cokernel of (M̂ ⊗ M̂ ′ )C Ð→ (M̂ ⊗ M̂ ′ )C is already finite. Set n = χM (1) and n′ = χM ′ (1). Lemma 4.15 implies that there are epimorphisms M ⊗Zn ↠ M̂ and M ′ ⊗ Zn′ ↠ M̂ ′ . Using that − ⊗ (Zn ⊗ Zn′ ) is an exact functor, we obtain that there is an epimorphism (M ⊗M ′ )C ⊗(Zn ⊗Zn′ ) ↠ (M̂ ⊗ M̂ ′ )C . Moreover, there is an epimorphism Coker((M ⊗ M ′ )C → (M ⊗ M ′ )C ) ⊗ (Zn ⊗ Zn′ ) ↠ Coker((M̂ ⊗ M̂ ′ )C → (M̂ ⊗ M̂ ′ )C ). It follows that it is enough to prove that Coker((M ⊗ M ′ )C → (M ⊗ M ′ )C ) is finite. Lemma 4.5 implies that M ⊗ M ′ is finitely generated, and hence, (M ⊗ M ′ )C is a finitely generated abelian group. It follows that it is enough to prove that (M ⊗ M ′ )C ⊗ C → (M ⊗ M ′ )C ⊗ C is an epimorphism. Eigenvalues of aM ⊗ aM ′ are products λi λ′j . Assume that λi λ′j = 1 for some i, j. Since M and M ′ are invariant free, λi ≠ 1 and λ′j ≠ 1. Then mi = 1 = m′j . It follows that all Jordan blocks of aM ⊗ C corresponding to λi and all Jordan blocks of aM ′ ⊗ C corresponding to λ′j are 1 × 1-matrices. It follows that all Jordan blocks of aM ⊗ aM ′ ⊗ C corresponding to 1 are 1 × 1-matrices. Hence all Jordan blocks of B ∶= aM ⊗ aM ′ ⊗ C − 1 corresponding to 0 are 1 × 1-matrices. It is easy to see that, if all Jordan blocks of a complex linear map B ∶ V → V corresponding to 0 are 1 × 1-matrices, then V = Ker(B)⊕Im(B). It follows that the map Ker(B) → Coker(B) is an isomorphism. Then (M ⊗ M ′ )C ⊗ C → (M ⊗ M ′ )C ⊗ C is an isomorphism. (2) Note that N̂ = N/NI i for some i >> 1. Since, (N̂ ⊗ M̂ ′ )C can be interpret as N̂ ⊗Z[C] M̂σ′ , (tensor product over Z[C]), we obtain (N̂ ⊗ M̂ ′ )C = (N/NI i ⊗ M ′ /M ′ I i )C . It follows that (N̂ ⊗ M̂ ′ )C is a finitely generated abelian group and the map (N ⊗ M ′ )C → (N̂ ⊗ M̂ ′ )C is an epimorphism. Set N ∶= N ⊗ M ′ , M ∶= M ⊗ M ′ , M0 ∶= M0 ⊗ M ′ , Ñ ∶= N̂ ⊗ M̂ ′ , M̃ ∶= M̂ ⊗ M̂ ′ , M̃0 ∶= M̂0 ⊗ M̂ ′ , L̃ ∶= Ker(M̃ ↠ M̃0 ). Then ÑC is a finitely generated abelian group and the map NC → ÑC is an epimorphism. Consider the exact sequence 0 → L̃C → M̃C → M̃C 0 → L̃C → M̃C → (M̃0 )C → 0. Since M̃ ⊗ Q ↠ M̃0 ⊗ Q is a split epimorphism, the image of M̃C 0 → L̃C lies in the torsion subgroup, which is finite because of the epimorphism ÑC ↠ L̃C . Then the cokernel of M̃C → M̃C 0 is finite. Set Q = Coker(MC ⊕ M̃C → M̃C ), Q0 = Coker((M0 )C ⊕ M̃C 0 → (M̃0 )C ). ON LENGTHS OF HZ-LOCALIZATION TOWERS 18 Then we know that Q0 is finite and Coker(M̃C → M̃C 0 ) is finite, and we need to prove that Q is finite. Consider the diagram with exact columns. NC ⊕ Ñ C // MC ⊕ M̃C  ÑC α // (M0 )C ⊕ M̃0   // M̃C C // (M̃0 )C  Q  // Q0 Using the snake lemma, we obtain that Ker(Q → Q0 ) = Coker(α). Since Q0 is finite and Coker(α) = Coker(M̃C → M̃C 0 ) is finite, we get that Q is finite.  Corollary 4.21. Let M be a tame Z[C]-module and µM = (x − 1)m f1m1 . . . flml for some distinct monic irreducible polynomials f1 , . . . , fl ∈ Z[x] such that fi (1) ≠ 0. Assume that for any 1 ≤ i ≤ l either fi (0) ∉ {−1, 1} or mi = 1. Then the cokernel of the homomorphism (M ⊗2 )C ⊕ (M̂ ⊗2 )C Ð→ (M̂ ⊗2 )C is finite. Remark 4.22. We prove Propositions 4.18 and 4.20 for tensor products of some modules and their completions. Further we need the same statements for exterior squares. Of course, the statements for tensor products imply the statements for exterior squares, so it is enough to prove for tensor products. Moreover, it is more convenient to prove such statements for tensor products because they have two advantages. The first obvious advantage is that we can change modules M and M ′ in the tensor product M ⊗ M ′ independently doing some reductions to ‘simpler’ modules. The second less obvious advantage is the following. Let A be an abelian group and M, M ′ are Z[A]-modules. Then we can interpret coinvariants of the tensor product as the tensor product over Z[A] (M ⊗ M ′ )A = M ⊗Z[A] Mσ′ , where Mσ′ is the module M ′ with the twisted module structure m∗a = ma−1 . In particular, there is an additional nontrivial structure of Z[A]-module on (M ⊗ M ′ )A . But there is no such a structure on (Λ2 M)A . More precisely, the kernel of the epimorphism (M ⊗ M)A ↠ (Λ2 M)A is not always a Z[A]-submodule. For example, if A = C = ⟨t⟩, M = Z2 where t acts on M via the matrix ( 10 11 ) , it is easy to check that the kernel is not a submodule. In our article [14] there are two mistakes concerning this that can be fixed easily. (1) On page 562 we define ∧2σ M as a quotient module of M ⊗Λ Mσ by the submodule generated by the elements m ⊗ m. Then we prove Corollaries 3.4 and 3.5 for such a module. In the proof of Proposition 7.2 we assume that ∧2σ M = (∧2 M)A which is the first mistake. In order to fix this mistake we have define ∧2σ M as a quotient ON LENGTHS OF HZ-LOCALIZATION TOWERS 19 of M ⊗Λ M by the abelian group generated by the elements m ⊗ m and prove Corollaries 3.4 and 3.5 using this definition. The prove is the same. We just need to change the meaning of the word ‘generated’ form ‘generated as a module’ to ’generated as an abelian group’. (2) In the proof of Lemma 7.1 we assume that (∧2 M)A is an Z[A]-module. This is the second mistake. In order to fix it, we have replace (∧2 M)A by (M ⊗ M)A in the first sentence of the proof of Lemma 7.1. 5. HZ-localization of M ⋊ C Now consider our group G = M ⋊ C and the maximal nilpotent submodule N ⊆ M, such that (M/N)C is finite. We have a natural commutative diagram N // // G // // G/N ηω ηω   N // // Ĝ // // ̂ G/N Observe that, for any Z[C]-submodule N ′ ⊆ N, ̂′ = Ĝ/N ′ . G/N Lemma 5.1. For any Z[C]-submodule N ′ of N, there is a natural isomorphism H2 (ηω )(G) = H2 (ηω )(G/N ′ ). Proof. We can present the submodule N ′ as a finite tower of central extensions. If we will prove that H2 (ηω )(G) = H2 (ηω )(G/N ′ ) for any N ′ ⊆ N such that N ′ (1−t) = 0, than we will be able to prove the general statement by induction on class of nilpotence of N ′ . The assumption that N ′ (1 − t) = 0 implies that the extensions ̂′ → 1 1 → N ′ → G → G/N ′ → 1, 1 → N ′ → Ĝ → G/N are central. Consider the natural map between sequences (3.2) for these extensions: (Gab ⊗ N ′ )/U // H2 (G) H2 (G/N ′ ) // // N′ H1 (G) //   (Ĝab ⊗ N ′ )/U // H2 (Ĝ)  // H2 (ηω )(G) ̂′ ) H2 (G/N // N′ // H1 (Ĝ) //  H2 (ηω )(G/N ′ ) Elementary diagram chasing implies that the lower horizontal map is an isomorphism and the needed statement follows.  Lemma 5.2. If E(G/N) = Tω+1 (G/N), then EG = Tω+1 (G). ON LENGTHS OF HZ-LOCALIZATION TOWERS 20 Proof. First we observe that, for any N ′ ⊆ N, there is a natural isomorphism Tω+1 (G/N ′ ) = Tω+1 (G)/N ′ . Indeed, lemma 5.1 implies that there is a natural diagram H2 (ηω )(G) // N′ N′     Tω+1 (G) // // //   H2 (ηω )(G/N ′ ) // Tω+1 (G/N ′ ) // Ĝ // // ̂′ G/N Hence, we have a natural diagram N ′ // // G // // G/N ′ ηω+1 ηω+1   N ′ // // Tω+1 (G) // // Tω+1 (G/N ′ ) Again, as in the proof of lemma 5.1, we will assume that N ′ is central and will prove that, in this case, EG = Tω+1 (G) provided E(G/N ′ ) = Tω+1 (G/N ′ ). This follows from comparison of sequences (3.2) applied to the above central extensions: (Gab ⊗ N ′ )/U (Ĝab ⊗ N ′ )/U // H2 (G) // //  H2 (Tω+1 (G)) // N′ //  H2 (Tω+1 (G/N ′ )) //  H2 (ηω+1 )(G) H2 (G/N ′ ) // N′ // H1 (G) H1 (Tω+1 (G)) //  H2 (ηω+1 )(G/N ′ ) Again, elementary diagram chasing shows that the lower horizontal map is an isomorphism and the needed statement follows.  Proposition 5.3. For a tame Z[C]-module M, the following conditions are equivalent: (i) HZ-length(M ⋊ C) ≤ ω + 1; (ii) the composition Λ2 (M̂ )C → Λ2 (M̂ )C → H2 (ηω ) is an epimorphism. ON LENGTHS OF HZ-LOCALIZATION TOWERS 21 Proof. It follows from (3.3) and construction of Tω+1 (G) that we have a natural diagram: H3 (Ĝ) Gab ⊗ H2 (ηω )(G) // // H2 (G) H2 (G)   H2 (Tω+1 (G)) H2 (Ĝ) // // // H2 (ηω )(G)  q qqqq q q qqq qqq H2 (ηω )(G) (5.1) This diagram implies that the condition (i) for G = M ⋊ C is equivalent to the surjectivity of the map δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G). Proposition 3.1 implies the following natural diagram H3 (Ĝ) // Gab ⊗ H2 (ηω )(G)   Λ2 (M̂ )C // H2 (ηω )(G) ♣77 ♣♣♣ ♣ ♣ ♣♣ ♣♣♣ ❏❏ ❏❏ ❏❏ ❏❏ ❏%% %% Λ2 (M̂ )C and the implication (i) ⇒ (ii) follows. Now assume that (ii) holds. Let N be the maximal nilpotent submodule of M such that (M/N)C is finite. Denote H ∶= (M/N) ⋊ C. We have a natural diagram: H3 (Ĝ)  // H3 (Ĥ) // Λ2 (M̂ )C Λ2 (̂ M/N )C // // Λ2 (M̂ )C Λ2 (̂ M/N )C // H2 (ηω )(G) //  H2 (ηω )(H) Lemma 5.1 implies that the right hand vertical map in this diagram is a natural isomorphism. Condition (ii) implies that the composition of three lower arrows in the last diagram must be an epimorphism. Now observe that Hab ⊗ H2 (ηω )(H) = H2 (ηω )(H), since the group H2 (ηω )(H) is divisible and (M/N)C is finite. Therefore, δ ∶ H3 (Ĥ) → Hab ⊗ H2 (ηω )(H) is surjective. The diagram (5.1) with G replaced by H implies that EH = Tω+1 (H). Now the statement (i) follows from lemma 5.2.  Now we will consider the key example of a tame Z[C]-module M, such that HZ-length(M ⋊ C) > ω + 1. For the construction of such an example, recall first certain well-known properties of quadratic functors. Let X1 , . . . , Xn , Y1 , . . . , Ym be abelian groups and X = ⊕ni=1 Xi , Y = ⊕m j=1 Yj . An element of a direct sum will be written as a column (x1 , . . . , xn )T ∈ X and a homomorphism ON LENGTHS OF HZ-LOCALIZATION TOWERS 22 f ∶ X → Y will be written as a matrix f = (fji ), where fji ∶ Xi → Yj and n n f ((x1 , . . . , xn ) ) = (∑ f1i (xi ), . . . , ∑ fmi (xi ))T . T i=1 i=1 For an abelian group X we denote by and X ⊗2 its exterior, symmetric, divided (the same as the Whitehead quadratic functor) and tensor squares respectively. If X is torsion free, then there are short exact sequences Λ2 X, S2 X, ι∧ πS ιΓ π∧ Γ2 X 0 Ð→ Λ2 X ÐÐ→ X ⊗2 ÐÐ→ S2 X Ð→ 0, 0 Ð→ Γ2 X ÐÐ→ X ⊗2 ÐÐ→ Λ2 X Ð→ 0, where π∧ , πS are the canonical projections ι∧ (x1 x2 ) = x1 ⊗ x2 − x1 ⊗ x2 , ιΓ (γ2 (x)) = x ⊗ x (see [13, ch.1 Section 4.3]). We will identify Γ2 X with the kernel of π∧ for torsion free groups. Lemma 5.4. Let M = Z2 be the Z[C]-module with the action of C given by the matrix 1 2n = 4n M for any natural n and M̂ = (Z )2 with the action of c = ( −1 2 0 −1 ) . Then M(t − 1) C given by the same matrix. Moreover, Λ2 M̂ = Λ2 Z2 ⊕ Λ2 Z2 ⊕ Z⊗2 2 , (Λ2 M̂ )C = Λ2 Z2 ⊕ 0 ⊕ Γ2 Z2 , (Λ2 M̂ )C = 0 ⊕ Λ2 Z2 ⊕ S2 Z2 and the cokernel of the natural map (Λ2 M̂)C Ð→ (Λ2 M̂ )C is isomorphic Λ2 Z2 . 1 n n n 2 Z2 /dn (Z2 ). Proof. Set d ∶= c−1 = ( −2 0 −2 ) . Since MI = M(t−1) = d (Z ), we have M̂ = lim ← Ð Computations show that d2 = −4c, and hence d2n = (−4)n cn . Since c induces an automorphism on Z2 , we obtain d2n (Z2 ) = 4n Z2 . Thus the filtration dn (Z2 ) of Z2 is equivalent to the filtration 2n Z2 . It follows that M̂ = (Z2 )2 with the action of C given by c. Then Λ2 M̂ ≅ (Λ2 Z2 )2 ⊕ Z⊗2 2 . We identify the element xe1 ∧ x′ e1 + ye2 ∧ y ′e2 + ze1 ∧ z ′ e2 ∈ ∧2 M̂ with the column (x ∧ x′ , y ∧ y ′ , z ⊗ z ′ )T ∈ (∧2 Z2 )2 ⊕ (Z2 ⊗ Z2 ), where e1 , e2 is the standard basis of (Z2 )2 over Z2 . Let us present the homomorphism ∧2 ĉ ∶ Λ2 M̂ → Λ2 M̂ that defines the action of C as a matrix. Since ∧2 ĉ(xe1 ∧ x′ e1 ) = xe1 ∧ x′ e1 ∧2 ĉ(ye2 ∧ y ′e2 ) = ye1 ∧ y ′e1 + ye2 ∧ y ′ e2 − (ye1 ∧ y ′ e2 − y ′ e1 ∧ ye2 ) ∧2 ĉ(ze1 ∧ z ′ e2 ) = −ze1 ∧ z ′ e1 + ze1 ∧ z ′ e2 , we obtain 1 1 −π∧ 1 0 ), 0 −ι∧ 1 ∧2 ĉ = ( 0 0 1 −π∧ 0 0 ). 0 −ι∧ 0 ∧2 ĉ − 1 = ( 0 ON LENGTHS OF HZ-LOCALIZATION TOWERS 23 Now it is easy to compute that (Λ2 M̂ )C = Ker(∧2 ĉ − 1) = Λ2 Z2 ⊕ 0 ⊕ Γ2 Z2 and Im(∧2 ĉ − 1) = Λ2 Z2 ⊕ 0 ⊕ ι∧ (Λ2 Z2 ). It follows that (Λ2 M̂ )C = Coker(∧2 ĉ − 1) = 0 ⊕ Λ2 Z2 ⊕ S2 Z2 . In order to prove that Coker((Λ2 M̂)C → (Λ2 M̂ )C ) = Λ2 Z2 , 2 we have to prove that Coker(Γ2 Z2 → Z⊗2 2 → S Z2 ) = 0. In other words we have to prove ⊗2 that Z2 is generated by elements x ⊗ x and x ⊗ y − y ⊗ x for x, y ∈ Z2 . Fist note that any 2-divisible element is generated by elements 2x ⊗ y = (x ⊗ y − y ⊗ x) + (x + y) ⊗ (x + y) − x ⊗ x − y ⊗ y. Since any element of Z2 is equal to 2x or 1+2x for some x ∈ Z2 , all other elements of Z2 ⊗Z2 can be presented as sums of elements (1 +2x)⊗(1 +2y) = 1 ⊗1 +2(x⊗1 +1 ⊗y +2x⊗y).  Theorem 5.5. Let M be the Z[C]-module from lemma 5.4 The HZ-length of the group G ∶= M ⋊ C = ⟨a, b, t ∣ [a, b] = 1, at = a−1 , bt = ab−1 ⟩ is greater than ω + 1. Proof. Consider the central sequence 1 → H2 (ηω )(G) → Tω+1 (G) → Ĝ → 1 We will show that the cokernel of the map δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G) from (3.1) contains Λ2 (Z2 ). The theorem will immediately follow. By proposition 3.1, the map δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G) = H2 (ηω )(G) factors as H3 (Ĝ) → (Λ2 M̂ )C → (Λ2 M̂ )C ↠ H2 (ηω )(G) The direct summand Λ2 Z2 from (Λ2 M̂)C maps isomorphically to a direct summand of H2 (ηω )(G) = Λ2 Z2 ⊕ (S2 Z2 )/Z. By lemma 5.4, the summand Λ2 Z2 lies in the cokernel of the map (Λ2 M̂ )C → (Λ2 M̂ )C , therefore, it lies also in the cokernel of the composite map (Λ2 M̂ )C → H2 (ηω )(G) as well as of the map δ.  Theorem 5.6. Let G be a metabelian group of the form G = M ⋊ C, where M is a tame C-module and µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct complex numbers λ1 , . . . , λl and mi ≥ 1. (1) Assume that the equality λi λj = 1 holds only if λi = λj = 1. Then HZ-length(G) ≤ ω. (2) Assume that the equality λi λj = 1 holds only if either mi = mj = 1 or λi = λj = 1. Then HZ-length(G) ≤ ω + 1. Proof. Follows from propositions 5.3, 4.18 and 4.20.  ON LENGTHS OF HZ-LOCALIZATION TOWERS 24 Lemma 5.7. Let M be a tame Z[C]-module such that H2 (M ⋊ C) is finite and µM = (x − λ1 )m1 . . . (x − λl )ml , where λ1 , . . . , λl are distinct complex numbers and mi ≥ 1. Then λi λj = 1 implies mi = mj = 1. Proof. Set V ∶= M ⊗C. Then by Lemma 4.1 we get V = V1 ⊕⋅ ⋅ ⋅⊕Vl such that µVi = (x−λi )mi . The short exact sequence (Λ2 M)C ↣ H2 (M ⋊ C) ↠ M C implies that (∧2 M)C is finite. Then (Λ2C V )C = (Λ2 M)C ⊗ C = 0. Assume the contrary, that there exist i, j such that λi λj = 1 and mi ≥ 2. Consider two cases i = j and i ≠ j. (1) Assume that i = j. Then λi = λj = −1. Since mi ≥ 2, at least one of Jordan blocks of aM ⊗ C corresponding to −1 has size bigger than 1. It follows that there is a epimorphism V ↠ U, where U = C2 and C acts on U by the matrix ( −10 −11 ) . The epimorphism induces a epimorphism 0 = (Λ2C V )C ↠ (∧2C U)C . From the other hand, a simple computation shows that (Λ2C U)C ≅ C. So we get a contradiction. (2) Assume that i ≠ j. Then λi ≠ λj . Because of the isomorphism of C[C]-modules Λ2C V = (⊕ Λ2C Vk ) ⊕ ( ⊕ Vk ⊗C Vk′ ), k k<k ′ we have an epimorphism of Z[C]-modules Λ2C V ↠ Vi ⊗C Vj . Since mi ≥ 2, there is an epimorphism Vi ↠ Ui , where Ui = C2 and C acts on Ui by the matrix ( λ0i λ1i ) . Moreover, there is an epimorphism Vj ↠ Uj , where Uj = C and C acts on Uj by the multiplication on λj . It follows that C acts on Ui ⊗C Uj by the matrix ( 10 λ1j ) . Thus (Ui ⊗C Uj )C ≅ C. From the other hand we have an epimorphism 0 = (Λ2C V )C ↠ (Ui ⊗ Uj )C . So we get a contradiction.  Proposition 5.8. Let G be a metabelian finitely presented group of the form G = M ⋊ C for some Z[C]-module M and H2 (G) is finite. Then HZ-length(G) ≤ ω + 1. Proof. It follows from Lemma 5.7 and Theorem 5.6.  6. Bousfield’s method Let G be a finitely presented group given by presentation ⟨x1 , . . . , xm ∣ r1 , . . . , rk ⟩ Consider the free group F = F (x1 , . . . , xn ) and an epimorphism F → G with the kernel normally generated by k elements ker F → G = ⟨r1 , . . . , rk ⟩F . Here we will study the induced map h ∶ H2 (F̂ ) → H2 (Ĝ). (6.1) ON LENGTHS OF HZ-LOCALIZATION TOWERS 25 We follow the scheme due to Bousfield from [5]. Let Ð→ Ð→ d0 ,d1 Ð→ F∗ ∶ . . . Ð→ F1 Ð→ F0 (= F ) → G ←Ð ←Ð ←Ð be a free simplicial resolution of G, where F1 a free group with m + k generators. The structure of F1 is F (y1 , . . . , yk ) ∗ F , and the maps d0 , d1 are given as id d0 ∶ yi ↦ 1, F ↦ F id d 1 ∶ y i ↦ ri , F ↦ F The following short exact sequence follows from Lemma 5.4 and Proposition 3.13 in [5]: ̂∗ ) → Ĝ → 1 1 → lim 1 π1 (F∗ /γk (F∗ )) → π0 (F ←Ð k (6.2) The first homotopy group π1 (F∗ /γk (F∗ )) is isomorphic to the kth Baer invariant known also as k-nilpotent multiplicator of G (see [7], [11]). If G = F /R for a normal subgroup R ⊲ F , the Baer invariant can be presented as the quotient π1 (F∗ /γ∗(F∗ )) ≃ R ∩ γk (F ) . [R, F, . . . , F ] ´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹¶ k−1 1 Now assume that, for a group G, the lim -term vanishes in (6.2), that is, there is a natural ←Ð isomorphism ̂∗ ) ≃ Ĝ π0 (F There is the first quadrant spectral sequence (see [4], page 108) 1 ̂∗ )) = Hq (F̂p ) ⇒ Hp+q (W (F Ep,q r r → Ep−r,q+r−1 dr ∶ Ep,q As a result of convergence of this spectral sequence, we have the following diagram: 1 E0,1 = H2 (F̂ ) // // ∞ E0,1 (6.3)   ̂∗ )) H2 (W (F // // H2 (Ĝ)  ∞ E1,0 ∞ is a quotient of π ((F ) ) = Since F1 is finitely generated, H1 (F̂1 ) ≃ H1 (F1 ), therefore, E1,0 1 ∗ ab H2 (G). Now the diagram (6.3) implies the following ON LENGTHS OF HZ-LOCALIZATION TOWERS 26 Lemma 6.1. Assuming that lim 1 -term in (6.2) vanishes, the cokernel of the map (6.1) ←Ð is isomorphic to a quotient of the homology group H2 (G). For a group G with H2 (G) = 0, the map h ∶ H2 (F̂ ) → H2 (Ĝ) is an epimorphism. 7. Main construction Lemma 7.1. Let K = ⟨a, b ∣ [[a, b], a] = [[a, b], b] = 1⟩ be the free 2-generated nilpotent group of class 2 and C = ⟨t⟩ acts on K by the following automorphism a ↦ a−1 b ↦ ab−1 . Set K̃ ∶= lim K/γn (K ⋊ C). Then the following holds ←Ð (1) the pronilpotent completion of K ⋊ C is equal to K̃ ⋊ C; (2) Kab is isomorphic to the Z[C]-module from Lemma 5.4; ≅ ̂ (3) the obvious map (K̃)ab Ð → (K ab ) is an isomorphism of Z[C]-modules; (4) either [K̃, K̃] ≅ Z2 or [K̃, K̃] ≅ Z/2m for some m. Proof. For the sake of simplicity we set Kn ∶= γn (K ⋊ C). (1) follows from the equality (K ⋊ C)/Kn = (K/Kn ) ⋊ C for n ≥ 2. (2) is obvious. Prove (3) and (4). It is obvious that γ2 (K) = Z(K) = {[a, b]k ∣ k ∈ Z} ≅ Z and [ai , bj ] = [a, b]ij for all i, j ∈ Z. Moreover, any element of K can be uniquely presented as ai bj [a, b]k for i, j, k ∈ Z. Set M = Kab . We consider M with the additive notation as a module over C. Then γn+1 (M ⋊ C) = M(t − 1)n . Lemma 5.4 implies that γ2n+1 (M ⋊ C) = 4n M. Since the map K2n+1 ↠ γ2n+1 (K ⋊ C) is an epimorphism, there exist k, l ∈ Z (that depend of n n 2n n n n) such that a4 [a, b]k , b4 [a, b]l ∈ K2n+1 . Since, [a, b]4 = [a4 [a, b]k , b4 [a, b]l ], we obtain 2n m(2n+1) [a, b]4 ∈ K2n+1 . Then K2n+1 ∩ γ2 (K) = ⟨[a, b]2 ⟩ for some natural number m(2n + 1) m(n) 2n because 4 is divisible only on powers of 2. Hence Kn ∩ γ2 (K) = ⟨[a, b]2 ⟩ for some m(n). Then the short exact sequence γ2 (K) ↣ K ↠ M induces the short exact sequence ιn 0 Ð→ Z/2m(n) ÐÐÐ→ K/Kn Ð→ M/M(t − 1)n Ð→ 1, where ιn (1) = [a, b]. Since Z/2m(n) and M/M(t − 1)n are finite 2-groups, the order of ′ K/Kn is equal to 2m (n) for some m′ (n). We obtain a short exact sequence 0 Ð→ lim Z/2m(n) ÐÐ→ K̃ Ð→ M̂ Ð→ 1. ←Ð If m(n) → ∞, then lim Z/2m(n) = Z2 , else m(n) stabilizes and lim Z/2m(n) = Z/2m . ←Ð ←Ð Now it is sufficient to prove that [K̃, K̃] = Im(ι̂). Since M̂ is abelian, [K̃, K̃] ⊆ Im(ι̂). Prove that [K̃, K̃] ⊇ Im(ι̂). Any element of K̃ can be presented as a sequence (xn )∞ n=1 , m(n) where xn ∈ K/Kn such that xn ≡ xn+1 mod Kn . Any element of lim Z/2 can be ←Ð ∞ k presented as an image of a 2-adic integer ∑k=0 αk 2 , where αk ∈ {0, 1}. Then an element of Im(ι̂) can be presented as a sequence ι̂ ([a, b]∑k=0 m(n) αk 2k ∞ )n=1 = ([a∑k=0 m(n) αk 2k , b])∞ n=1 . ON LENGTHS OF HZ-LOCALIZATION TOWERS 27 2 Note that the element (a∑k=0 αk 2 )∞ n=1 is a well defined element of K̃ because a It follows that m(n) m(n) k ∑k=0 αk 2k ∞ )n=1 , (b)∞ ([a, b]∑k=0 αk 2 )∞ n=1 ], n=1 = [(a m′ (n) m′ (n) k and hence, [K̃, K̃] ⊇ Im(ι̂). ∈ Kn .  Theorem 7.2. Let F be a free group of rank ≥ 2. Then HZ-length(F ) ≥ ω + 2. Proof. Consider the following group: Γ ∶= ⟨a, b, t ∣ [[a, b], a] = [[b, a], b] = 1, at = a−1 , bt = ab−1 ⟩ Observe that Γ is the semidirect product K ⋊ C from the previous lemma. Consider the natural diagram induced by the projection K → Kab = M, i.e. by Γ = K ⋊ C ↠ G = M ⋊ C: H3 (Γ̂) δ  H3 (Ĝ) δ // H1 (Γ) ⊗ H2 (ηω )(Γ) (7.1) ◗ ◗ ◗ ◗ ◗ ◗ ◗((  // H1 (G) ⊗ H2 (ηω )(G) // // Coker(δ)(G) It is shown in the proof of theorem 5.5 that Coker(δ)(G) contains Λ2 Z2 . This term is an epimorphic image of one of the terms Λ2 Z2 in Λ2 Z2 H2 (M̂ ) = Λ2 M̂ = Λ2 Z2 ⊕ Λ2 Z2 ⊕ Z2 ⊗ Z2 (see lemma 5.4). By lemma 7.1, there is a short exact sequence H2 (K̃) → H2 (M̂ ) ↠ [K̃, K̃] Now, by lemma 7.1, [K̃, K̃] is either Z2 or a finite group, therefore, the image of both terms Λ2 Z2 in [K̃, K̃] are zero (2-adic integers do not contain divisible subgroups). In particular, the term Λ2 Z2 which maps isomorphically onto a subgroup of Coker(δ)(G) lies in the image of H2 (K̃). The natural square H2 (K̃) // //  H2 (M̂ ) H2 (ηω )(Γ)  // // H2 (ηω )(G) is commutative and we conclude that the diagonal arrow in (7.1) maps onto the subgroup Λ2 Z2 of Coker(δ)(G). Hence, Coker(δ)(Γ) maps epimorphically onto Λ2 Z2 . Now lets prove that the second homology H2 (Γ) is finite. Since the group Γ is the semi-direct product K ⋊ C, its homology is given as 0 → H2 (K)C → H2 (Γ) → H1 (C, Kab ) → 0 The right hand term is zero: H1 (C, Kab ) = H1 (C, M) = M C = 0. It follows immediately that H2 (K)C = (γ3 (F (a, b))/γ4 (F (a, b)))C = Z/4. ON LENGTHS OF HZ-LOCALIZATION TOWERS 28 Observe that the group Γ can be defined with two generators only. Let F be a free group of rank ≥ 2. Consider a free simplicial resolution of Γ with F0 = F : F∗ → Γ. Since H2 (Γ) is finite, all Baer invariants of Γ are finite (see, for example, [11]) lim 1 π1 (F∗ /γk (F∗ )) = 0 ←Ð and, by lemma 6.1, the cokernel of the natural map H2 (F̂ ) → H2 (Γ̂) is finite. We have a natural commutative diagram: H3 (F̂ ) δ H1 (F ) ⊗ H2 (ηω )(F ) //  H3 (Γ̂) // // Coker(δ)(F )  δ // H1 (Γ) ⊗ H2 (ηω )(Γ) (7.2)  // // Coker(δ)(Γ) with H2 (ηω )(F ) = H2 (F̂ ). Since the cokernel of H2 (F̂ ) → H2 (Γ̂) is finite, Coker{Coker(δ)(F ) → Coker(δ)(Γ)} also is finite. However, Coker(δ)(Γ) maps onto Λ2 Z2 , as we saw before, hence Coker(δ)(F ) is uncountable. Therefore, by (3.3), H2 (Tω+1 (F )) ≠ 0 and the statement is proved.  8. Alternative approaches In general, given a group, the description of its pro-nilpotent completion is a difficult problem. If a group is not pre-nilpotent, its pro-nilpotent is uncountable and may contain complicated subgroups. In this paper, as well as in [14] we essentially used the explicit structure of pro-nilpotent completion for metabelian groups. Now we observe that, there is a trick which gives a way to show that some groups have HZ-length greater than ω without explicit description of their pro-nilpotent completion. In a sense, the Bousfield scheme described above also gives such a method, however in that way one must compare the considered group with a group with clear completion. The trick given bellow is different. We put Φi H2 (G) = Ker{H2 (G) → H2 (G/γi+1 (G))}. Then Φi H2 (G) is the Dwyer filtration on H2 (G) (see [8]). Proposition 8.1. Let G be a group with the following properties: (i) γω (G) ≠ γω+1 (G) (ii) ∩i Φi H2 (G/γω (G)) = 0. Then the HZ-length of G is greater than ω. ON LENGTHS OF HZ-LOCALIZATION TOWERS 29 Proof. All ingredients of the proof are in the following diagram with exact horizontal and vertical sequences: H2 (G) ▼▼▼ ▼▼▼ ▼▼▼ ▼▼&&  // // H2 (G/γω (G)) ker //   ∩i Φi H2 (G/γω (G)) H2 (Ĝ)  γω (G)/γω+1 (G) The fact that the kernel lies in ∩i Φi H2 (G/γω (G)) follows from the standard property of Dwyer’s filtration: the map G/γω (G) → Ĝ induces isomorphisms on H2 /Φi for all i (see [8]). The vertical exact sequence is the part of 5-term sequence in homology. Now assume that the map H2 (G) → H2 (Ĝ) is an epimorphism. Then H2 (G/γω (G)) → H2 (Ĝ) is an epimorphism as well. However, condition (ii) implies that the last map is a monomorphism as well. Hence, it is an isomorphism and surjectivity of H2 (G) → H2 (Ĝ) contradicts the property (i).  An example of a group with satisfies both conditions (i) and (ii) from proposition, is the group ⟨a, b ∣ [a, b3 ] = [[a, b], a]2 = 1⟩ (see [16], examples 1.70 and 1.85). However, in this example, G/γω (G) is metabelian and one can use explicit description of its pro-nilpotent completion to show the same result. The above proposition can be used for more complicated groups. Now we consider another example of finitely generated metabelian group of the type M ⋊ C with HZ-length greater than ω + 1. In our example, M = Z[C] with a regular action of C as multiplication. This is not a tame Z[C]-module and the group is not finitely presented. The proof of the bellow result uses functorial arguments. Theorem 8.2. Let G = Z[C] ⋊ C = Z ≀ Z = ⟨a, b ∣ [a, ab ] = 1, i ∈ Z⟩, i then HZ-length(G) ≥ ω + 2. Proof. We define a functor from the category fAb of finitely generated free abelian groups to the category of groups as follows: F ∶ A ↦ (A ⊗ M) ⋊ C, A ∈ fAb, where the action of C on A is trivial and M = Z[C]. Now our main example can be written as G = F (Z). Since the action of C on A is trivial, the pro-nilpotent completion of F (A) can be described as follows: ̂ F (A) = (A ⊗ M̂) ⋊ C. One can easily see that H1 (C, Hi (A ⊗ M̂ )) = 0, i ≥ 1. ON LENGTHS OF HZ-LOCALIZATION TOWERS 30 Since the abelian group A ⊗ M̂ is torsion-free, ̂ Hi (F (A)) = Λi (A ⊗ M̂ )C , i ≥ 1. Now we have H2 (ηω )(F (A)) = Coker{Λ2 (A ⊗ M)C → Λ2 (A ⊗ M̂ )C }. Now consider the functor from the category of finitely generated free abelian groups to all abelian groups G ∶ A ↦ H2 (ηω )(F (A)) It follows immediately that G is a quadratic functor. Indeed, it is a proper quotient of the functor A ↦ A ⊗ A ⊗ B for a fixed abelian group B. The exact sequence (3.3) applied to the stem-extension ̂ 1 → H2 (ηω )(F (A)) → Tω+1 (F (A)) → F (A) → 1 can be viewed as an exact sequence of functors in the category sAb. Consider the map δ ̂ as a natural transformation in sAb. The functor H3 (F (A)) = Λ3 (A ⊗ M̂)C is a quotient of the cubic functor A⊗A⊗A⊗D for some fixed abelian group D (which equals to the tensor cube of M̂ ). Recall that any natural transformation of the form A ⊗ A ⊗ A ⊗ D → quadratic functor is zero. This follows from the simple observation that A⊗A⊗A⊗D is a natural epimorphic image of its triple cross-effect. See, for example, [17] for generalizations and detailed discussion of this observation. Therefore, for any non-zero A, the map δ in (3.3) is zero and H2 (Tω+1 (F (A))) contains a subgroup H2 (ηω )(F (A)) which is uncountable. In particular, H2 (Tω+1 (G)) = H2 (Tω+1 (F (Z)) is uncountable and hence H2 (G) → H2 (Tω+1 (G)) is not an epimorphism.  Remark 8.3. If we change the group Z ≀ Z by lamplighter group by adding one more relation, we will obtain another wreath product (for n ≥ 2) Z/n ≀ Z = ⟨a, b ∣ an = 1, [a, ab ] = 1, i ∈ Z⟩ i One can use the scheme of this paper to prove that HZ-length(Z/n≀Z) > ω+1. Essentially it follows from the triviality of Λ2 (M̂ )C as in Theorem 8.2. As it was shown by G. Baumslag [1], there are different ways to embed this group into a finitely-presented metabelian group. For example, Z/n ≀ Z is a subgroup generated by a, b in G[n] ∶= ⟨a, b, c ∣ an = [a, ab ] = [a, ab ] = [b, c] = 1, ac = aab a−b ⟩. 2 2 Now we have a decomposition G[n] = (Z/n)⊕∞ ⋊ (C × C) and it follows immediately from [14] that HZ-length(G[n] ) = ω, since H2 (ηω )(G[n] ) is divisible. ON LENGTHS OF HZ-LOCALIZATION TOWERS 31 9. Concluding remarks Lemma 9.1. For any prime p and n ≥ 2 the embedding Zp ↪ Qp induces an isomorphism Λn Zp ≅ Λn Qp . In particular, Λn Zp is a Q-vector space of countable dimension. ⊗n Proof. Since Zp and Qp are torsion free, the map Z⊗n p → Qp is a monomorphism. Since for torsion free groups the exterior power is embedded into the tensor power, we obtain that the map Λn Zp → Λn Qp is a monomorphism. So we can identify Λn Zp with its image in Λn Qp . Since Qp = ⋃ p1i ⋅ Zp , we obtain Λn Qp = ⋃ p1i ⋅ Λn Zp . Then it is enough to prove that Λn Zp is p-divisible. For any a ∈ Zp we consider the decomposition a = a(0) + a(1) , where a(0) ∈ {0, . . . , p − 1} and a(1) = p ⋅ b for some b ∈ Zp . Then for any a1 , . . . , an ∈ Zp (i ) (i ) we have a1 ∧ ⋅ ⋅ ⋅ ∧ an = ∑ a1 1 ∧ ⋅ ⋅ ⋅ ∧ an n , where (i1 , . . . , in ) runs over {0, 1}n . For any (i ) (i ) sequence (i1 , . . . , in ) except (0, . . . , 0) the element a1 1 ∧ ⋅ ⋅ ⋅ ∧ an n is p divisible because (1) (0) (0) ai is p-divisible. Since Λn Z = 0, we get a1 ∧ ⋅ ⋅ ⋅ ∧ an = 0. It follows that a1 ∧ ⋅ ⋅ ⋅ ∧ an is p-divisible. Thus Λn Zp is p-divisible.  Remind that the Klein bottle group is given by GKl = Z ⋊ C, where C acts on Z by the multiplication on −1. Its pronilpotent completion is equal to Z2 ⋊ C. Consider the map w ∶ Z2 × Z2 → Λ2 Z2 given by w(a, b) = 21 a ∧ b. Here we use that Λ2 Zp is a Q-vector space. It is easy to see that w is a 2-cocycle and we get the corresponding central extension Λ2 Z2 ↣ Nw ↠ Z2 , whose underlying set is (Λ2 Z2 ) × Z2 and the product is given by (α, a)(β, b) = (α + β + 1 ⋅ a ∧ b, a + b) . 2 We define the action of C on Nw by the map (α, a) ↦ (α, −a). Proposition 9.2. There is an isomorphism EGKl = Nw ⋊ C. In other words, EGKl can be described as the set (Λ2 Z2 ) × Z2 × C with the multiplication given by (α, a, ti )(β, b, tj ) = (α + β + (−1)i ⋅ a ∧ b, a + (−1)i b, ti+j ) . 2 Proof. Set G ∶= GKl . Then Ĝ = Z2 ⋊ C. The Lyndon-Hochschild-Serre spectral sequences ≅ imply that H2 (G) = 0 and the map Λ2 Z2 = H2 (Z2 ) Ð → H2 (Ĝ) is an isomorphism. Theorem 5.6 implies that Tω+1 G = EG. Then EG is the universal relative central extension (H2 (ηω ) ↣ EG ↠ Ĝ, ηω+1 ). Since H2 (ηω ) = Coker{H2 (G) → H2 (Ĝ)} and H2 (G) = 0, ≅ ≅ we obtain that H2 (Z2 ) Ð → H2 (Ĝ) Ð → H2 (ηω ) are isomorphisms. The continious maps ON LENGTHS OF HZ-LOCALIZATION TOWERS 32 BZ2 → B Ĝ → Cone(Bηω ) give the commutative diagram for any abelian group A ∶ H 2 (ηω , A) Hom(H2 (ηω ), A) ≅ // (9.1) ≅   H 2 (Ĝ, A) // Hom(H2 (Ĝ), A) ≅  H 2 (Z2 , A) //  Hom(H2 (Z2 ), A) If A is a divisible abelian group, then Ext(H1 (Ĝ), A) = 0 = Ext(Z2 , A), and all morphisms in the diagram (9.1) are isomorphisms. In particular, if A = H2 (ηω ) = H2 (Ĝ) = Λ2 Z2 , then the morphisms induce isomorphisms H 2 (ηω , Λ2 Z2 ) ≅ H 2 (Ĝ, Λ2 Z2 ) ≅ H 2 (Z2 , Λ2 Z2 ) ≅ Hom(Λ2 Z2 , Λ2 Z2 ) and the extension Λ2 Z2 ↣ EG ↠ Ĝ corresponds to the identity map in Hom(Λ2 Z2 , Λ2 Z2 ). Therefore, it is sufficient to prove that the extension Λ2 Z2 ↣ Nw ⋊ C ↠ Ĝ goes to the identity map via the composition H 2 (Ĝ, Λ2 Z2 ) → H 2 (Z2 , Λ2 Z2 ) → Hom(Λ2 Z2 , Λ2 Z2 ). The map H 2 (Ĝ, Λ2 Z2 ) → H 2 (Z2 , Λ2 Z2 ) on the level of extensions is given by the pullback. It follows that the extension Λ2 Z2 ↣ Nw ⋊ C ↠ Ĝ goes to the extension Λ2 Z2 ↣ Nw ↠ Z2 . Then we need to prove that w goes to the identity map under the map H 2 (Z2 , Λ2 Z2 ) → Hom(Λ2 Z2 , Λ2 Z2 ). For any group H and an abelian group A the map H 2 (H, A) → Hom(H2 (H), A) on the level of cocycles is induced by the evaluation map C 2 (H, A) → Hom(C2 (H), A) given by u ↦ ((h, h′ ) ↦ u(h, h′ )). For an abelian group H the map Λ2 H → H2 (H) given by h ∧ h′ ↦ (h, h′ ) − (h′ , h) + B2 (H) is an isomorphism Λ2 H ≅ H2 (H) (see [6, Ch.V §5-6]). Then the map H 2 (H, A) → Hom(Λ2 H, A) is given by u ↦ (h ∧ h′ ↦ u(h, h′ ) − u(h′ , h)). Since w(a, b) − w(b, a) = 21 (a ∧ b − b ∧ a) = a ∧ b, w goes to the identity map.  Remark 9.3. If we identify EGKl with the set (Λ2 Z2 ) × Z2 × C, it is easy to check that γ2 (EGKl ) = (Λ2 Z2 ) × 2Z2 × 1 and [γ2 (EGKl ), γ2 (EGKl )] = (Λ2 Z2 ) × 0 × 1. It follows that EGKl is a solvable group of class 3 but it is not metabelian. In particular, the class of metabelian groups is not closed under the HZ-localization. Finishing this paper, we mention some possibilities of generalization of the obtained results. We conjecture that, the tame Z[C]-module M defined by the k × k-matrix (k ≥ 2) ⎞ ⎛ ⎜−1 1 0 . . . 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 −1 1 0 . . .⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜. . . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ 0 . . . 0 0 −1⎠ ON LENGTHS OF HZ-LOCALIZATION TOWERS 33 defines the group M ⋊C with HZ-length ω +l(k), where l(k) ∈ N and l(k) → ∞ for k → ∞. With the help of this group, one can try to use the same scheme as in this paper to prove that HZ-length of a free non-cyclic group is ≥ 2ω. Acknowledgement The research is supported by the Russian Science Foundation grant N 16-11-10073. References [1] G. Baumslag: Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 16 (1973), 98–110, Collection of articles dedicated to the memory of Hanna Neumann. [2] R. Bieri and K. Strebel: Almost finitely presented soluble groups, Comm. Math. Helv. 53 (1978), 258–278. [3] R. Bieri and R. Strebel: A geometric invariant for modules over an abelian group, J. Reine Angew. Math. 322 (1981), 170–189. [4] A. K. Bousfield and D. Kan: Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, (1972). [5] A. K. Bousfield: Homological localization towers for groups and π-modules, Mem. Amer. Math. Soc, no. 186, 1977. [6] K. Brown, Cohomology of Groups, Springer-Verlag GTM 87, 1982. [7] J. Burns and G. Ellis: On the nilpotent multipliers of a group, Math. Z. 226 (1997), 405–428. [8] W. Dwyer: Homology, Massey products and maps between groups, J. Pure Appl. Algebra, 6 (1975), 177–190. [9] B. Eckmann, P. Hilton and U. Stammbach: On the homology theory of central group extensions: I - the commutator map and stem extensions, Comm. Math. Helv. 47, (1972), 102–122. [10] B. Eckmann, P. Hilton and U. Stammbach: On the homology theory of central group extensions: II - the exact sequence in the general case, Comm. Math. Helv. 47, (1972), 171–178. [11] G. Ellis: A Magnus-Witt type isomorphism for non-free groups, Georgian Math. J. 9 (2002), 703–708. [12] L. Fuchs: Infinite abelian groups, Academic Press, New York and London. [13] L.Illusie: Complexe Cotangent et Déformation I, Lecture Notes in Mathematics, vol. 239, Springer, Berlin, 1971. [14] S. O. Ivanov and R. Mikhailov: On a problem of Bousfield for metabelian groups: Advances in Math. 290 (2016), 552–589. [15] F. Kasch: Modules and Rings, Acd. Press, London New York, 1982. [16] R. Mikhailov and I. B. S. Passi: Lower Central and Dimension Series of Groups, LNM Vol. 1952, Springer 2009. [17] R. Mikhailov: Polynomial functors and homotopy theory, Progress in Math, 311 (2016), arXiv: 1202.0586. [18] C. Weibel: An Introduction to Homological Algebra, Cambridge Univ. Press, 1994. [19] O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, Van Nostrand, Princeton 1960. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia E-mail address: ivanov.s.o.1986@gmail.com Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia and St. Petersburg Department of Steklov Mathematical Institute E-mail address: rmikhailov@mail.ru 

Technical Report: Graph-Structured Sparse Optimization for Connected Subgraph Detection Feng Chen Computer Science Department University at Albany – SUNY Albany, USA fchen5@albany.edu arXiv:1609.09864v1 [] 30 Sep 2016 Baojian Zhou Computer Science Department University at Albany – SUNY Albany, USA bzhou6@albany.edu Abstract—Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured sparsity models have been explored, such as trees, groups, clusters, and paths, connected subgraphs have been rarely explored in the current literature. One of the main technical challenges is that there is no structured sparsity-inducing norm that can directly model the space of connected subgraphs, and there is no exact implementation of a projection oracle for connected subgraphs due to its NP-hardness. In this paper, we explore efficient approximate projection oracles for connected subgraphs, and propose two new efficient algorithms, namely, G RAPH -IHT and G RAPH -GHTP, to optimize a generic nonlinear objective function subject to connectivity constraint on the support of the variables. Our proposed algorithms enjoy strong guarantees analogous to several current methods for sparsity-constrained optimization, such as Projected Gradient Descent (PGD), Approximate Model Iterative Hard Thresholding (AM-IHT), and Gradient Hard Thresholding Pursuit (GHTP) with respect to convergence rate and approximation accuracy. We apply our proposed algorithms to optimize several well-known graph scan statistics in several applications of connected subgraph detection as a case study, and the experimental results demonstrate that our proposed algorithms outperform state-of-the-art methods. I. I NTRODUCTION In recent years, structured sparse methods have attracted much attention in many domains such as bioinformatics, medical imaging, social networks, and astronomy [2], [4], [14], [16], [17]. Structured sparse methods have been shown effective to identify latent patterns in high-dimensional data via the integration of prior knowledge about the structure of the patterns of interest, and at the same time remain a mathematically tractable concept. A number of structured sparsity models have been well explored, such as the sparsity models defined through trees [14], groups [17], clusters [16], and paths [2]. The generic optimization problem based on a structured sparsity model has the form minn f (x) s.t. supp(x) ∈ M x∈R (1) where f : Rn → R is a differentiable cost function, the sparsity model M is defined as a family of structured supports: M = {S1 , S2 , · · · , SL }, where Si ⊆ [n] satisfies a certain structure property (e.g., trees, groups, clusters), [n] = {1, 2, · · · , n}, and the support set supp(x) refers to the set of indexes of non-zero entries in x. For example, the popular k-sparsity model is defined as M = {S ⊆ [n] | |S| ≤ k}. Existing structured sparse methods fall into two main categories: 1) Sparsity-inducing norms based. The methods in this category explore structured sparsity models (e.g., trees, groups, clusters, and paths) [4] that can be encoded as structured sparsity-inducing norms, and reformulate Problem (1) as a convex (or non-convex) optimization problem minx∈Rn f (x) + λ · Ω(x) (2) where Ω(x) is a structured sparsity-inducing norm of M that is typically non-smooth and non-Euclidean and λ is a trade-off parameter. 2) Model-projection based. The methods in this category rely on a projection oracle of M: P(b) = arg minx∈Rn kb − xk22 s.t. supp(x) ∈ M, (3) and decompose the problem into two sub-problems, including unconstrained minimization of f (x) and the projection problem P(b). Most of the methods in this category assume that the projection problem P(b) can be solved exactly, including the forward-backward algorithm [42], the gradient descent algorithm [38], the gradient hard-thresholding algorithms [6], [18], [40], the projected iterative hard thresholding [5], [7], and the Newton greedy pursuit algorithm [41]. However, when an exact solver of P(b) is unavailable and we have to apply approximate projections, the theoretical guarantees of these methods do not hold any more. We note that there is one recent approach named as G RAPH-C OSAMP that admits inexact projections by assuming “head” and “tail” oracles for the projections, but is only applicable to compressive sensing or linear regression problems [15]. We consider an underlying graph G = (V, E) defined on the coefficients of the unknown vector x, where V = [n] and E ⊆ V × V. We focus on the sparsity model of connected subgraphs that is defined as M(G, k) = {S ⊆ V | |S| ≤ k, S is connected}, (4) where k refers to the allowed maximum subgraph size. There are a wide array of applications that involve the search of interesting or anomalous connected subgraphs in networks. The connectivity constraint ensures that subgraphs reflect changes due to localized in-network processes. We describe a few applications below. • • • • • • Detection in sensor networks, e.g., detection of traffic bottlenecks in road networks or airway networks [1]; crime hot spots in geographic networks [22]; and pollutions in water distribution networks [27]. Detection in digital signals and images, e.g., detection of objects in images [15]. Disease outbreak detection, e.g., early detection of disease outbreaks from information networks incorporating data from hospital emergency visits, ambulance dispatch calls and pharmacy sales of over-the-counter drugs [36]. Virus detection in a computer network, e.g., detection of viruses or worms spreading from host to host in a computer network [24]. Detection in genome-scale interaction network, e.g., detection of significantly mutated subnetworks [21]. Detection in social media networks, e.g., detection and forecasting of societal events [8], [9]. To the best of our knowledge, there is no existing approach to Problem (1) for M(G, k) that is computationally tractable and provides performance bound. First, there is no known structured sparsity-inducing norm for M(G, k). P The most relevant norm is fused lasso norm [39]: Ω(x) = (i,j)∈E |xi −xj |, where xi is the i-th entry in x. This norm is able to enforce the smoothness between neighboring entries in x, but has limited capability to recover all the possible connected subsets as described by M(G, k) (See further discussions in Section V Experiments). Second, there is no exact solver for the projection oracle of M(G, k): P(x) = arg minn kb − xk22 s.t. supp(x) ∈ M(G, k), x∈R (5) as this projection problem is NP-hard due to a reduction from classical Steiner tree problem [19]. As most existing modelprojection based methods require an exact solution to the projection oracle P(x), these methods are inapplicable to the problem studied here. To the best of our knowledge, there is only one recent approach named as G RAPH-C OSAMP that admits inexact projections for M(G, k) by assuming “head” and “tail” oracles for the projections, but is only applicable to compression sensing and linear regression problems [15]. The main contributions of our study are summarized as follows: • • Design of efficient approximation algorithms. Two new algorithms, namely, G RAPH -IHT and G RAPH -GHTP, are developed to approximately solve Problem (1) that has a differentiable cost function and a sparsity model of connected subgraphs M(G, k). G RAPH -GHTP is required to minimize f (x) over a projected subspace as an intermediate step, which could be too costly in some applications and G RAPH -IHT could be considered as a fast variant of G RAPH -GHTP. Theoretical guarantees and connections. The convergence rate and accuracy of our proposed algorithms are analyzed under a smoothness condition of f (x) that is more general than popular conditions such as Restricted Strong Convexity/Smoothness (RSC/RSS) and Stable Mode Restricted Hessian (SMRH). We prove that under mild conditions our proposed G RAPH -IHT and G RAPH GHTP enjoy rigorous theoretical guarantees. • Compressive experiments to validate the effectiveness and efficiency of the proposed techniques. Both G RAPH -IHT and G RAPH -GHTP are applied to optimize a variety of graph scan statistic models for the connected subgraph detection task. Extensive experiments on a number of benchmark datasets demonstrate that G RAPH IHT and G RAPH -GHTP perform superior to state-of-theart methods that are designed specifically for this task in terms of subgraph quality and running time. Reproducibility: The implementation of our algorithms and the data sets is open-sourced via the link [11]. The remaining parts of this paper are organized as follows. Sections II introduces the sparsity model of connected subgraphs and statement of the problem. Sections III presents two efficient algorithms and their theoretical analysis. Section IV discusses applications of our proposed algorithms to graph scan statistic models. Experiments on several real world benchmark datasets are presented in Section V. Section VI discusses related work and Section VII concludes the paper and describes future work. II. P ROBLEM F ORMULATION Given an underlying graph G = (V, E) defined on the coefficients of the unknown vector x, where V = [n], E ⊆ V × V, and n is typically large (e.g., n > 10, 000). The sparsity model of connected subgraphs in G is defined in (4), and its projection oracle P(x) is defined in (5). As this projection oracle is NP-hard to solve, we first introduce efficient approximation algorithms for P(x) and then present statement of the problem that will be studied in the paper. A. Approximation algorithms for the projection oracle P(x) There are two nearly-linear time approximation algorithms [15] for P(x) that have the following properties: • Tail approximation (T(x)): Find a S ⊆ V such that • kx − xS k2 ≤ cT · 0 min kx − xS 0 k2 , (6) S ∈M(G,kT ) √ where cT = 7, kT = 5k, and xS is the restriction of x to indices in S: we have (xS )i = xi for i ∈ S and (xS )i = 0 otherwise. Head approximation (H(x)): Find a S ⊆ V such that kxS k2 ≥ cH · 0 max kxS 0 k2 , (7) S ∈M(G,kH ) p where cH = 1/14 and kH = 2k. It can be readily proved that, if cT = cH = 1, then T(x) = H(x) = P(x), which indicates that these two approximations (T(x) and H(x)) stem from the fact that cT > 1 and cH < 1. B. Problem statement Given a predefined cost function f (x) that is differentiable, the input graph G, and the sparsity model of connected Fig. 1. Illustration of G RAPH -IHT and G RAPH -GHTP on the main steps of each iteration. In this example, the gray scale of each node i encodes the weight of this node wi ∈ R, f (x) = −wT x + 12 kxk2 , and the maximum size of subgraphs is set to k = 6, where w = [w1 , · · · , wn ]T . The resulting problem tends to find a connected subgraph with the largest overall weight (See discussion about the effect of 12 kxk2 in Section IV). In each iteration, S i is the connected subset of nodes and its induced subgraph is denoted as GS i . The sequence of intermediate vectors and subgraphs are: (x0 , GS 1 ), · · · , (xi , GS i ). subgraph M(G, k), the problem to be studied is formulated as: min f (x) s.t. supp(x) ∈ M(G, k). x∈Rn (8) Problem (8) is difficult to solve as it involves decision variables from a nonconvex set that is composed of many disjoint subsets. In this paper, we will develop nearly-linear time algorithms to approximately solve Problem (8). The key idea is decompose this problem to sub-problems that are easier to solve. These sub-problems include an optimization sub-problem of f (x) that is independent on M(G, k) and projection approximations for M(G, k), including T(x) and H(x). We will design efficient algorithms to couple these subproblems to obtain global solutions to Problem (8) with good trade-off on running time and accuracy. III. A LGORITHMS This section first presents two efficient algorithms, namely, G RAPH -IHT and G RAPH -GHTP, and then analyzes their time complexities and performance bounds. A. Algorithm G RAPH -IHT The proposed G RAPH -IHT algorithm generalizes the traditional algorithm named as projected gradient descent [5], [7] that requires a exact solver of the projection oracle P(x). The high-level summary of G RAPH -IHT is shown in Algorithm 1 and illustrated in Figure 1 (b). The procedure generates a sequence of intermediate vectors x0 , x1 , · · · from an initial approximation x0 . At the i-th iteration, the first step (Line 5) first calculates the gradient “∇f (xi )”, and then identifies a subset of nodes via head approximation that returns a support set with the head value at least a constant factor of the optimal head value: “Ω ← H(∇f (xi ))”. The support set Ω can be interpreted as the subspace where the nonconvex set “{x | supp(x) ∈ M(G, k)}” is located, and the projected gradient in this subspace is: “∇Ω f (xi )”. The second step (Line 6) calculates the projected gradient descent at the point xi with step-size η: “b ← xi − η · ∇Ω f (xi )”. The third step (Line 7) identifies a subset of nodes via tail approximation that returns a support set with tail value at most a constant times larger than the optimal tail value: “S i+1 ← T(b)”. The last step (Line 8) calculates the intermediate solution xi+1 : xi+1 = bS i+1 . The previous two steps can be interpreted as the projection of b to the nonconvex set “{x | supp(x) ∈ M(G, k)}” using the tail approximation. Algorithm 1 G RAPH -IHT 1: Input: Input graph G, maximum subgraph size k, and step size η (1 by default). 2: Output: The estimated vector x̂ and the corresponding connected subgraph S. 3: i ← 0, xi ← 0; S i ← ∅; 4: repeat 5: Ω ← H(∇f (xi )); 6: b ← xi − η · ∇Ω f (xi ); 7: S i+1 ← T(b); 8: xi+1 ← bS i+1 ; 9: i ← i + 1; 10: until halting condition holds 11: return x̂ = xi and S = GS i ; B. Algorithm G RAPH -GHTP The proposed G RAPH -GHTP algorithm generalizes the traditional algorithm named as Gradient Hard Threshold Pursuit (GHTP) that is designed specifically for the k-sparsity model: M = {S ⊆ [n] | |S| ≤ k} [40]. The high-level summary of G RAPH -GHTP is shown in Algorithm 2 and illustrated in Figure 1 (c). The first two steps (Line 5 and Line 6) in each iteration is the same as the first two steps (Line 5 and Line 6) of G RAPH -IHT, except that we return the support of the projected gradient descent: “Ψ ← supp(xi − η · ∇Ω f (xi ))”, in which pursuing the minimization will be most effective. Over the support set S, the function f is minimized to produce an intermediate estimate at the third step (Line 7): Algorithm 2 G RAPH -GHTP 1: Input: Input graph G, maximum subgraph size k, and step size η (1 by default). 2: Output: The estimated vector x̂ and the corresponding connected subgraph S. 3: i ← 0, xi ← 0; S i ← ∅; 4: repeat 5: Ω ← H(∇f (xi )) 6: Ψ ← supp(xi − η · ∇Ω f (xi )); 7: b ← arg minx∈Rn f (x) s.t. supp(x) ⊆ Ψ; 8: S i+1 ← T(b); 9: xi+1 ← bS i+1 ; 10: i ← i + 1; 11: until halting condition holds 12: return x̂ = xi and S = GS i ; “b ← arg minx∈Rn f (x) s.t. supp(x) ⊆ Ω”. The fourth and fifth steps (Line 8 and Line 9) are the same as the last two steps (Line 7 and Line 8) of G RAPH -IHT in each iteration. C. Relations between G RAPH -IHT and G RAPH -GHTP These two algorithms are both variants of gradient descent. In overall, G RAPH -GHTP converges faster than G RAPH -IHT as it identifies a better intermediate solution in each iteration by minimizing f (x) over a projected subspace {x | supp(x) ⊆ Ω}. If the cost function f (x) is linear or has some special structure, this intermediate step can be conducted in nearlylinear time. However, when this step is too costly in some applications, G RAPH -IHT is preferred. D. Theoretical Analysis of G RAPH -IHT In order to demonstrate the accuracy of estimates using Algorithm 1, we require that the cost function f (x) satisfies the Weak Restricted Strong Convexity (WRSC) condition as follows: Definition III.1 (Weak Restricted Strong Convexity Property (WRSC)). A function f (x) has the (ξ, δ, M)-model-WRSC if ∀x, y ∈ Rn and ∀S ∈ M with supp(x) ∪ supp(y) ⊆ S, the following inequality holds for some ξ > 0 and 0 < δ < 1: kx − y − ξ∇S f (x) + ξ∇S f (y)k2 ≤ δkx − yk2 . (9) The WRSC is weaker than the popular Restricted Strong Convexity/Smoothness (RSC/RSS) conditions that are used in theoretical analysis of convex optimization algorithms [40]. The RSC condition basically characterizes cost functions that have quadratic bounds on the derivative of the objective function when restricted to model-sparse vectors. The RSC/RSS conditions imply condition WRSC, which indicates that WRSC is no stronger than RSC/RSS [40]. In the special case where f (x) = ky−Axk22 and ξ = 1, the condition (ξ, δ, M)-model-WRSC reduces to the well known Restricted Isometry Property (RIP) condition in compressive sensing. Theorem III.1. Consider the sparsity model of connected subgraphs M(G, k) for some k ∈ N and a cost function f : Rn → R that satisfies the (ξ, δ, M(G, 5k))-model-WRSC condition. If η = cH (1 − δ) − δ then for any x ∈ Rn such that supp(x) ∈ M(G, k), with η > 0 the iterates of Algorithm 2 obey kxi+1 − xk2 ≤ αkxi − xk2 + βk∇I f (x)k2 (10) where α0 = cH (1 − δ) − δ, β0 = δ(1 + cH ),  q  2(1 + cT ) η η 2 1 − α0 + (2 − )δ + 1 − , α= 1−δ ξ ξ ! √ √ √ √ 1 + cT 2β0 2α0 β0 β= , (1 + 2 2)ξ + (2 − 2 2)η + +p 1−δ α0 1 − α02 √ and I = arg maxS∈M(G,8k) k∇S f (x)k2 Before we prove this result, we give the following two lemmas III.2 and III.3. Lemma III.2. [40] Assume that f is a differentiable function. If f satisfies condition (ξ, δ, M)-WRSC, then ∀x, y ∈ Rn with supp(x) ∪ supp(y) ⊂ S ∈ M, the following two inequalities hold 1−δ 1+δ kx − yk2 ≤ k∇S f (x) − ∇S f (y)k2 ≤ kx − yk2 ξ ξ 1+δ kx − yk22 f (x) ≤ f (y) + h∇f (y), x − yi + 2ξ Lemma III.3. Let α0 = cH (1 − δ) − δ, β0 = ξ(1 + cH ), ri = xi − x, and Ω = H(∇f (xi )). Then kriΓc k2 ≤ q # β0 α0 β0 k∇I f (x)k2 + +p α0 1 − α02 " 1− α02 kri k2 where I = arg maxS∈M(G,8k) k∇S f (x)k2 . We assume that cH and δ are such that α0 > 0. Proof: Denote Φ = supp(x) ∈ M(G, k), Ω = H(∇f (xi )) ∈ M(G, 2k), ri = xi − x, and Λ = supp(ri ) ∈ M(G, 6k). The component k∇Γ f (xi )k2 can be lower bounded as k∇Γ f (xi )k2 ≥ cH k∇Φ f (xi )k2 ≥ cH (k∇Φ f (xi ) − ∇Φ f (x)k2 − k∇Φ f (x)k2 cH (1 − δ) i kr k2 − cH k∇I f (x)k2 , ξ ≥ where the first inequality follows from the definition of head approximation and the last inequality follows from Lemma III.2 of our paper. The component k∇Γ f (xi )k2 can also be upper bounded as k∇Γ f (xi )k2 ≤ ≤ ≤ ≤ 1 kξ∇Γ f (xi ) − ξ∇Γ f (x)k2 + k∇Γ f (x)k2 ξ 1 kξ∇Γ f (xi ) − ξ∇Γ f (x) − riΓ + riΓ k2 + ξ k∇Γ f (x)k2 1 kξ∇Γ∪Ω f (xi ) − ξ∇Γ∪Ω f (x) − riΓ∪Ω k2 + ξ kriΓ k2 + k∇Γ f (x)k2 δ 1 · kri k2 + kriΓ k2 + k∇I f (x)k2 , ξ ξ where the fourth inequality follows from condition (ξ, δ, M(G, 8k))-WRSC and the fact that riΓ∪Ω = ri . Combining the two bounds and grouping terms, we obtain the inequality: kriΓ k ≥ α0 kri k2 − ξ(1 + cH )k∇I f (x)k2 . have kriΓ k ≥ α0 kri k2 − β0 k∇I f (x)k2 . After a We number of algebraic manipulations, we obtain the inequality kriΓc k2 " # q β0 α0 β0 i 2 k∇I f (x)k2 , +p ≤ 1 − α0 kr k2 + α0 1 − α02 which proves the lemma. We give the formal proof of III.1. Proof: From the traingle inequality, we have kri+1 k2 = kxi+1 − xk2 = kbΨ − xk2 ≤ kb − xk2 + kb − bΨ k2 ≤ (1 + cT )kb − xk2 = = E. Theoretical Analysis of G RAPH -GHTP i i i i i (1 + cT )kx − η∇Ω f (x ) − x k2 (1 + cT )kr − η∇Ω f (x )k2 , i where ∇Ω f (x ) is the projeceted vector of f (xi ) in which the entries outoside Ω are set to zero and the entries in Ω are unchanged. kri − η∇Ω f (xi )k2 has the inequalities kri − η∇Ω f (xi )k2 = kriΩc + riΩ − η∇Ω f (xi )k ≤ kriΩc k2 + kriΩ − η∇Ω f (xi ) + η∇Ω f (x) − η∇Ω f (x)k ≤ kriΩc k2 + kriΩ − η∇Ω f (xi ) + η∇Ω f (x)k + kη∇Ω f (x)k ≤ kriΩc k2 + kriΩ − ξ∇Ω f (xi ) + ξ∇Ω f (x)k+ (ξ − η)k∇Ω f (xi ) − ∇Ω f (x)k2 k + kη∇Ω f (x)k2 ≤ kriΩc k2 + (1 − η/ξ + (2 − η/ξ)δ)kri k2 + ηk∇I f (x)k2 where the last inequality follows from condition (ξ, δ, M)WRSC and Lemma III.2. From Lemma III.3, we have q hβ α0 β0 i 0 +p k∇I f (x)k2 kriΓc k2 ≤ 1 − α02 kri k2 + α0 1 − α02 Combining the above inequalities, we prove the theorem. Our proposed G RAPH -IHT generalizes several existing sparsity-constrained optimization algorithms: 1) Projected Gradient Descent (PGD) [30]. If we redefine H(b) = supp(b) and T(b) = supp(P(b)), where P(b) is the projection oracle defined in Equation (5), then G RAPH -IHT reduces to the PGD method; 2) Approximated Model-IHT(AMIHT) [13]. If the cost function f (x) is defined as the least square cost function f (x) = ky − Axk22 , then ∇f (x) has the specific form −AT (y − A) and G RAPH -IHT reduces to the AM-IHT algorithm, the state-of-the-art variant of IHT for compressive sensing and linear regression problems. In particular, let e = y − Ax. The component k∇f (xi )k2 = kAT ek2 √ is upper bound by bounded by 1 + δkek2 [13], Assume that ξ = 1 and η = 1. Condition (ξ, η, M)-WRSC then reduces to the RIP condition in compressive sensing. The convergence inequality (10) then reduces to kx i+1 0 i 0 − xk2 ≤ α kx − xk2 + β kek2 , (11) h i p 2 where α = (1 + cT ) δ + 1 − α0 and √ h (α + β )√1 + δ α0 β0 ( 1 + δ) i 0 0 0 p β = (1 + cT ) + . α0 1 − α02 0 Surprisingly, the above convergence inequality is identical to the convergence inequality of AM-IHT derived in [13] based on the RIP condition, which indicates that G RAPH -IHT has the same convergence rate and approximation error as AM-IHT, although we did not make any attempt to explore the special properties of the RIP condition. We note that the convergence properties of G RAPH -IHT hold in fairly general setups beyond compressive sensing and linear regression. As we consider G RAPH -IHT as a fast variant of G RAPH -GHTP, due to space limit we ignore the discussions about the convergence condition of G RAPH -IHT. The theoretical analysis of G RAPH -GHTP to be discussed in the next subsection can be readily adapted to the theoretical analysis of G RAPH -IHT. Theorem III.4. Consider the sparsity model of connected subgraphs M(G, k) for some k ∈ N and a cost function f : Rn → R that satisfies the (ξ, δ, M(G, 5k))-model-WRSC condition. If η = cH (1 − δ) − δ then for any x ∈ Rn such that supp(x) ∈ M(G, k), with η > 0 the iterates of Algorithm 2 obey kxi+1 − xk2 ≤ αkxi − xk2 + βk∇I f (x)k2 (12) where α0 = cH (1 − δ) − δ, β0 = δ(1 + cH ),  q  2(1 + cT ) η η 1 − α02 + (2 − )δ + 1 − , α= 1−δ ξ ξ ! √ √ √ √ 1 + cT 2β0 2α0 β0 (1 + 2 2)ξ + (2 − 2 2)η + +p β= , 1−δ α0 1 − α02 √ and I = arg maxS∈M(G,8k) k∇S f (x)k2 Proof: Denote Ω = H(∇f (xi )) and Ψ = supp(xi − η · ∇Ω f (xi )). Let ri+1 = xi+1 − x. kri+1 k2 is bounded as kri+1 k2 = kxi+1 − xk2 ≤ ≤ ≤ kxi+1 − bk2 + kx − bk2 cT kx − bk2 + kx − bk2 (1 + cT )kx − bk2 , (13) where the second inequality follows from the definition of tail approximation. The component k(x − b)Ψ k22 is bounded as k(x − b)Ψ k22 = hb − x, (b − x)Ψ i = hb − x − ξ∇Ψ f (b) + ξ∇Ψ f (x), (b − x)Ψ i − hξ∇Ψ f (x), (b − x)Ψ i ≤ δkb − xk2 k(b − x)Ψ k2 + ξk∇Ψ f (x)k2 k(b − x)Ψ k2 , where the second equality follows from the fact that ∇S f (b) = 0 since b is the solution to the problem in the third Step (Line 7) of G RAPH -GHTP, and the last inequality can be derived from condition (ξ, δ, M(G, 8k))-WRSC. After simplification, we have k(x − b)Ψ k2 ≤ δkb − xk2 + ξk∇Ψ f (x)k2 . It follows that kx − bk2 ≤ k(x − b)Ψ k2 + k(x − b)Ψc k2 ≤ δkb − xk2 + ξk∇Ψ f (x)k2 + k(x − b)Ψc k2 . After rearrangement we obtain kb − xk2 ≤ k(b − x)Ψc k2 ξk∇Ψ f (x)k2 + , 1−δ 1−δ (14) where this equality follows from the fact that supp(b) ⊆ S. Let Φ = supp(x) ∈ M(G, k). k(xi − η∇Ω f (xi ))Φ k2 ≤ k(xi − η∇Ω f (xi ))Ψ k2 , as Ψ = supp(xi −η·∇Ω f (xi )). By eliminating the contribution on Φ ∩ Ψ, we derive k(xi − η∇Ω f (xi ))Φ\Ψ k2 ≤ k(xi − η∇Ω f (xi ))Ψ\Φ k2 For the right-hand side, we have k(xi − η∇Ω f (xi ))Ψ\Φ k2 ≤ k(xi − x − η∇Ω f (xi ) + η∇Ω f (x))Ψ\Φ k2 + ηk∇Ω∪Ψ f (x)k2 , where the inequality falls from the fact that Φ = supp(x). From the left-hand side,i we have i k(x − η∇Ω f (x ))Φ\Ψ k2 ≤ −ηk∇Ω∪Φ f (x)k2 + k(xi − x − η∇Ω f (xi ) + η∇Ω f (x))Φ\Ψ + (x − b)Ψc k2 where the inequality follows from the fact that bΨc = 0, xΦ\Ψ = xΨc , and −xΦ\Ψ + (x − b)Ψc = 0. Let Φ∆Ψ be the symmetric difference of the set Φ and Ψ. It follows that ≤ k(b − x)Ψc k2 √ 2k(xi − x − η∇Ω f (xi ) + η∇Ω f (x))Φ∆Ψ k2 + 2ηk∇I f (x)k2 √ ≤ 2k(xi − x − ξ∇Ω f (xi ) + ξ∇Ω f (x))Φ∆Ψ k2 + √ 2(ξ − η)k(∇Ω f (xi ) + ∇Ω f (x))Φ∆Ψ k + 2ηk∇I f (x)k2 √ ≤ 2k(riΩc + riΩ − ξ∇Ω f (xi ) + ξ∇Ω f (x))Φ∆Ψ k2 + √ 2(ξ − η)k(∇Ω f (xi ) − ∇Ω f (x))Φ∆Ψ k + 2ηk∇I f (x)k2 √ √ ≤ 2kriΩc k + 2k(riΩ − ξ∇Ω f (xi ) + ξ∇Ω f (x))Ψ∆Ψ k2 + √ 2(ξ − η)k(∇Ω f (xi ) − ∇Ω f (x))Ψ∆Ψ k + 2ηk∇I f (x)k2 √ √ i ≤ 2krΩc k + 2kri − ξ∇Ω∪Ψ∪Φ f (xi ) + ξ∇Ω∪Ψ∪Φ f (x)k2 + √ 2(ξ − η)k(∇Ω∪Ψ∪Φ f (xi ) − ∇Ω∪Ψ∪Φ f (x))Ψ∆Ψ k + 2ηk∇I f (x)k2    √ √ η η i c k2 + ≤ 2krΩ δ+1− kri k + 2 2− ξ ξ √ √  2 2ξ + (1 − 2)η k∇I f (x)k2 , where the first inequality follows from the fact that ηk∇Ω∪Φ f (x)k2 + ηk∇Ψ∪Φ∪Ω f (x)k2 ≤ 2ηk∇I f (x)k2 , the third inequality follows as xi − x = ri = riΩc + riΩ , the fourth inequality follows from the fact that k(riΩc )Φ∆Ψ k2 ≤ kriΩc k2 , the fifth inequality follows as ri ⊆ Ω ∪ Ψ ∪ Φ, and the last inequality follows from condition (ξ, δ, M(G, 8k))-WRSC and Lemma III.2. From Lemma III.3, we have " # kriΩc k2 ≤ p ξ(1 + cH ) ξη(1 + cH ) k∇I f (x)k2 1 − η 2 kri k2 + + p η 1 − η2 Combining (14) and above inequalities, we prove the theorem. Theorem III.4 shows the estimator error of G RAPH -GHTP is determined by the multiple of k∇S f (x)k2 , and the convergence rate is geometric. Specifically, if x is an uncontrained minimizer of f (x), then ∇f (x) = 0. It means G RAPH -GHTP is guaranteed to obtain the true x to arbitrary precision. The estimation error is negligible when x is sufficiently close to an unconstrained minimizer of f (x) as k∇S f (x)k2 is a small value. The parameter √  q  2(1 + cT ) η η 1 − α02 + (2 − )δ + 1 − < 1, α= 1−δ ξ ξ controls the convergence rate of G RAPH -GHTP. Our algorithm allows an exact recovery if α < 1. As δ is an arbitrary constant parameter, it can be an arbitrary small positive value. Let η be ξ and δ be an arbitrary small positive value, the parameters cH and cT satisfy the following inequality c2H > 1 − 1/(1 + cT )2 . (15) It is noted that the head and tail approximation algorithms described in [15] do not meet the inequality (15). Nonetheless, the approximation factor cH of any given head approximation algorithm can be boosted to any arbitrary constant c0H < 1, which leads to the satisfaction of the above condition as shown in [15]. Boosting the head-approximation algorithm, though strongly suggested by [13], is not empirically necessary. Our proposed G RAPH -GHTP has strong connections to the recently proposed algorithm named as Gradient Hard Thresholding Pursuit (GHTP) [40] that is designed specifically for the k-sparsity model: M = {S ⊆ [n] | |S| ≤ k}. In particular, if we redefine H(b) = supp(b) and T(b) = supp(P(b)), where P(b) is the projection oracle defined in Equation (5), and assume that there is an algorithm that solves the projection oracle exactly, in which the sparsity model does not require to be the k-sparsity model. It then follows that the upper bound of kriΩc k2 stated in Lemma III.2 in Appendix is updated as kriΩc k2 ≤ 0, since supp(ri ) = Ω and riΩc = 0. In addition, the multiplier (1 + cT ) is replaced as 1 as the first inequality (13) in the proof of Theorem III.4 in Appendix is updated as kri+1 k2 ≤ kx−bk2 , instead of the original version kri+1 k2 ≤ (1 + cT )kx − bk2 . After these two changes, the shrinkage rate α is updated as √   2 η η (2 − )δ + 1 − , (16) α= 1−δ ξ ξ which is the same as the shrinkage rate of G RAPH -GHTP as derived in [40] specifically for the k-sparsity model. The above shrinkage rate α (16) should satisfy the condition α < 1 to ensure the geometric convergence of G RAPH -GHTP, which implies that √ √ √ √ (17) η > ((2 2 + 1)δ + 2 − 1)ξ/( 2 + 2δ). √ It follows that if δ < 1/( 2 + 1), a step-size η < ξ can always be found to satisfy the above inequality. This constant condition of δ is analogous to the constant condition of stateof-the-art compressive sensing methods that consider noisy measurements [23] under the assumption of the RIP condition. We derive the analogous constant using the WRSC condition that weaker than the RIP condition. As discussed above, our proposed G RAPH -GHTP has connections to GHTP on the shrinkage rate of geometric convergence. We note that the shrinkage rate of our proposed G RAPH -GHTP stated in Theorem III.4 is derived based on head and tail approximations of the sparsity model of connected subgraphs M(G, k), instead of the k-sparsity model that has an exact projection oracle solver. Our convergence properties hold in fairly general setups beyond k-sparsity model, as a number of popular structured sparsity models such as the “standard” k-sparsity, block sparsity, cluster sparsity, and tree sparsity can be encoded as special cases of M(G, k). Theorem III.5. Let x ∈ Rn such that supp(x) ∈ M(G, k), and f : Rn → R be cost function that satisfies condition (ξ, δ, M(8k, g))-WRSC. Assuming that α < 1, G RAPH -GHTP (or G RAPH -IHT) returns a x̂ such that, supp(x̂) ∈ M(5k, g) β ) is a fixed and kx − x̂k2 ≤ ck∇I f (x)k2 , where c = (1 + 1−α constant. Moreover, G RAPH -GHTP runs in time
O (T + |E| log3 n) log(kxk2 /k∇I f (x)k2 ) , (18) where T is the time complexity of one execution of the subproblem in Step 6 in G RAPH -GHTP (or Step 5 in G RAPH IHT). In particular, if T scales linearly with n, then G RAPH GHTP (or G RAPH -IHT) scales nearly linearly with n. Proof: The i-th iterate of G RAPH -GHTP (or G RAPH IHT) satisfies β k∇I f (x)k2 . 1 − α m kx − xi k2 ≤ αi kxk2 + (19) l  1 2 iterations, G RAPH After t = log k∇Ikxk f (x)k2 / log α GHTP (or G RAPH -IHT) returns an estimate x̂ satisfying kx− β x̂k2 ≤ (1 + 1−α )k∇I f (x)k2 . The time complexities of both head approximation and tail approximation are O(|E| log3 n). The time complexity of one iteration in G RAPH -GHTP (or 3 G RAPH -IHT)l is  (T + |E| log  n), and m the total number of kxk2 iterations is log k∇I f (x)k2 / log α1 , and the overall time follows. G RAPH -GHTP and G RAPH -IHT are only different in the definition of α and β in this Theorem. As shown in Theorem 18, the time complexity of G RAPH GHTP is dependent on the total number of iterations and the time cost (T ) to solve the subproblem in Step 6. In comparison, the time complexity of G RAPH -IHT is dependent on the total number of iterations and the time cost T to calculate the gradient ∇f (xi ) in Step 5. It implies that, although G RAPH GHTP converges faster than G RAPH -IHT, the time cost to solve the subproblem in Step 6 is often much higher than the time cost to calculate a gradient ∇f (xi ), and hence G RAPH IHT runs faster than G RAPH -GHTP in practice. IV. A PPLICATIONS ON G RAPH S CAN S TATISTICS In this section, we specialize G RAPH-IHT and G RAPHGHTP to optimize a number of well-known graph scan statistics for the task of connected subgraph detection, including elevated mean scan (EMS) statistic [29], Kulldorff’s scan statistic [25], and expectation-based Poisson (EBP) scan statistic [26]. Each graph scan statistic is defined as the generalized likelihood ratio test (GLRT) statistic of a specific hypothesis testing about the distributions of features of normal and abnormal nodes. The EMS statistic corresponds to the following GLRT test: Given a graph G = (V, E), where V = [n] and E ⊆ V × V, each node i is associated with a random variable xi : xi = µ · 1(i ∈ S) + i , i ∈ V, (20) where |µ| represents the signal strength and i ∈ N (0, 1). S is some unknown anomalous cluster that forms as a connected subgraph. The task is to decide between the null hypothesis (H0 ): ci ∈ N (0, 1), ∀i ∈ V and the alternative (H1 (S)): ci ∈ N (µ, 1), ∀i ∈ S and ci ∈ N (0, 1), ∀i ∈ / S. The EMS statistic is defined as the GLRT function under this hypothesis testing: Prob(Data|H1 (S)) 1 X F (S) = ci . (21) =p Prob(Data|H0 ) |S| i∈S The problem of connected subgraph detection based on the EMS statistic is then formulated as 1 X 2 ci ) s.t. S ∈ M(G, k), (22) min − ( S⊆V |S| i∈S where the square of the EMS scan statistic is considered to make the function smooth, and this transformation does not infect the optimum solution. Let the {0, 1}-vectors form of S be x ∈ {0, 1}n , such that supp(x) = S. Problem (22) can be reformulated as min x∈{0,1}n −(cT x)2 /(1T x) s.t. supp(x) ∈ M(G, k), (23) where c = [c1 , · · · , cn ]T . To apply our proposed algorithms, we relax the input domain of x and maximize the strongly convex function [3]: 1 min −(cT x)2 /(1T x) + xT x s.t. supp(x) ∈ M(G, k). (24) x∈Rn 2 The connected subset of nodes can be found as the subset of indexes of positive entries in x̂, where x̂ refers to the solution of the Problem (24). Assume that c is normalized and ci ≤ 1, ∀i. Let ĉ = max{c1 , · · · , cn }. The Hessian matrix of the above objective function satisfies the following conditions cT x cT x (1 − ĉ2 ) · I  I − (c − T 1)(c − T 1)T  1 · I. (25) 1 x 1 x According to Lemma 1 (b) in [40]), the objective function f (x) satisfies condition (ξ, δ, M(G, 8k))-WRSC that p δ = 1 − 2ξ(1 − ĉ2 ) + ξ 2 , for any ξ such that ξ < 2(1 − ĉ2 ). The geometric convergence of G RAPH -GTHP as shown in Theorem III.4 is guaranteed. Different from the EMS statistic that is defined for numerical features based on Gaussian distribution, the Kulldorff’s scan statistic and Expectation Based Poisson statistic (EBP) are defined for count features based on Poisson distribution. In particular, each node i is associated with a feature ci , the count of events (e.g., crimes, flu infections) observed at the current time, and a feature bi , the expected count (or ‘baseline’) of events by using historical data. Let c = [c1 , · · · , cn ]T and b = [b1 , · · · , bn ]T . The Kulldorff’s scan statistic and EBP scan statistics are described Table I. We note that these two scan statistics do not satisfy the WRSC condition, but as demonstrated in our experiments, our proposed algorithms perform empirically well for all the three scan statistics, and in particular, our proposed G RAPH -GHTP converged in less than 10 iterations in all the settings. V. EXPERIMENTS This section evaluates the performance of our proposed methods using four public benchmark data sets for connected subgraph detection. The experimental code and data sets are available from the Link [11] for reproducibility. TABLE I The three typical graph scan statistics that are tested in our experiments (The vectors x, c, and b are defined in Section IV) Score Functions Definition Kulldorff’s Scan Statistic [25] cT x log Expectation-based Poisson Statistic (EBP) [26] Elevated Mean Scan Statsitic (EMS) [29] (1T c − cT x log Applications T cT x c − 1T c log 11T b bT x T T 1 c−c x T c x) log 1T b−bT x cT x bT x + The statistic is used for anomalous pattern detection in graphs with count features, such as detection of traffic bottlenecks in sensor networks [1], [22], detection of anomalous regions in digitals and images [10], detection of attacks in computer networks [24], disease outbreak detection [36], and various others. + bT x − cT x This statistic is used for the same applications as above [1], [12], [25], [36], [37], but has different assumptions on data distribution [25]. cT x/1T x This statistic is used for anomalous pattern detection in graphs with numerical features, such as event detection in social networks, network surveillance, disease outbreak detection, biomedical imaging [29], [35] TABLE II Summary of dataset settings in the experiments. (For the network of each dataset, we only use its maximal connected component if it is not fully connected). Dataset Application Training & Testing Time Periods BWSN Detection of contaminated nodes CitHepPh Detection of emerging research areas Detection of most congested subgraphs Detection of crime hot spots Training: Hours 3 to 5 with 0% noise Testing: Hours 3 to 5 with 2%, 4%, · · · , 10% noises Testing: 1999 to 2002 Traffic ChicagoCrime Observed value at node v: cv Sensor value (0 or 1) Count of citations Testing: Mar.2014, 5AM to 10PM − log(pt (v)/µ) 1 Testing: Year of 2015 Count of burglaries Baseline value at node v: bv # of Nodes # of Edges # of snapshots Average sensor value for EBP Constant ‘1’ for Kulldorff 12,527 14,323 hourly: 3 × 6 Average count of citations for EBP Maximum count of citations for Kulldorff None (EBP and Kulldorff are not applicable) Average count of burglaries for EBP Maximum count of burglaries for Kulldorff 11,895 75,873 yearly: 1 × 11 1,723 5,301 per 15 min: 68 × 304 46,357 168,020 yearly: 1 × 15 pt (v) refers to the statistical p-value of node v at time t that is calculated via empirical calibration based on historical speed values of v from 2013 June. 1st to 2014 Feb. 29; and µ is a significance level threshold and is set to 0.15. The larger this value − log(pt (v)/µ), the more congested in the region near v. 1 A. Experiment Design Datasets: 1) BWSN Dataset. A real-world water network is offered in the Battle of the Water Sensor Networks (BWSN) [28]. That has 12,527 nodes and 14,323 edges. In order to simulate a contaminant sub-area, 4 nodes with chemical contaminant plumes, which were distributed in this sub-area, were generated. We use the water network simulator EPANET [31] that was employed in BWSN for a period of 3 hours to simulate the spreads for contaminant plumes on this graph. If a node is polluted by the chemical, then its sensor reports 1, otherwise, 0, in each hour. To test the tolerance of noise of our methods, K ∈ {2, 4, 6, 8, 10} percent vertices were selected randomly, and their sensor reports were set to 0 if their original reports were 1 and vice versa. Each hour has a graph snapshot. The snapshots corresponding to the 3 hours that have 0% noise are considered for training, and the snapshots that have 2%, · · · , 10% noise reports for testing. The goal is to detect a connected subgraph that is corresponding to the contaminant sub-area. 2) CitHepPh Dataset. We downloaded the high energy physics phenomenology citation data (CitHepPh) from Stanford Network Analysis Project (SNAP) [20]. This citation graph contains 11,897 papers corresponding to graph vertices and 75,873 edges. An undirected edge between two vertices (papers) exists , if one paper is cited by another. The period of these papers published is from January 1992 to April 2002. Each vertex has two attributes for each specific year (t = 1992, · · · , t = 2002). We denote the number of citations of each specific year as the first attribute and the average citations of all papers in that year as the second attribute. The goal is to detect a connected subgraph where the number of citations of vertices (papers) in this subgraph are abnormally high compared with the citations of vertices that are not in this subgraph. This connected subgraph is considered as a potential emerging research area. Since the training data is required for some baseline methods, the data before 1999 is considered as the training data, and the rest from 1999 to 2002 as the testing data. 3) Traffic Dataset. Road traffic speed data from June 1, 2013 to Mar. 31, 2014 in the arterial road network of the Washington D.C. region is collected from the INRIX database (http://inrix.com/publicsector.asp), with 1,723 nodes and 5,301 edges. The database provides traffic speed for each link at a 15-minute rate. For each 15-minute interval, each method identities a connected subgraph as the most congested region. 4) ChicagoCrime Dataset. We collected crime data from City of Chicago [https://data.cityofchicago.org/PublicSafety/Crimes-2001-to-present/ijzp-q8t2] from Jan. 2001 to Dec. 2015. There are 46,357 nodes (census blocks) and 168,020 edges (Two census blocks are connected with each other if they are neighbours). Specifically, we collected all records of burglaries from 2001 to 2015. The data covers burglaries in the period from Jan. 2001 to Dec. 2015. Each vertex has an attribute denoting the number of burglaries in sepcific year and average number of burglaries over 10 years. We aim to detect connected census areas which has anomaly high burglaries accidents. The data before 2010 is considered as training data, and the data from 2011 to 2015 is considered as testing data. Graph Scan Statistics: As shown in Table I, three graph scan statistics were considered as the scoring functions of connected subgraphs, including Kulldorff’s scan statistic [25], expectation-based Poisson (EBP) scan statistic [26], and elevated mean scan (EMS) statistic [29]. The first two statistic functions require that each vertex v has a count cv representing the count of events observed at that vertex, and an expected count (‘baseline‘) bv . For EMS statistic, only cv is used. We need to normalize cv for EMS as it is defined based on the assumptions of standard normal distribution for normal values and shifted-mean normal distribution for abnormal values. Table II provides details about the calculations of cv and bv for each data set. Comparison Methods: We compared our proposed methods with four state-of-the-art baseline methods that are designed specifically for connected subgraph detection, namely, GraphLaplacian [33], EventTree [32], DepthFirstGraphScan [36] and NPHGS [8]. DepthFirstGraphScan is an exact search algorithm based on depth-first search and takes weeks to run on graphs that have more than 1000 nodes. We imposed a maximum limit on the depth of the search to 10 to reduce its time complexity. The basic ideas of these baseline methods are summarized as follows: NPHGS starts from random seeds (nodes) as initial candidate clusters and gradually expends each candidate cluster by including its neighboring nodes that could help improve its BJ statistic score until no new nodes can be added. The candidate cluster with the largest BJ statistic score is returned. DepthFirstGraphScan adopts a similar strategy to NPHGS but expands the initial clusters based on depth-first search. GraphLaplacian uses a graph Laplacian penalty function to replace the connectivity constraint and converts the problem to a convex optimization problem. EventTree reformulates the connected subgraph detection problem as a prize-collecting steiner tree (PCST) problem [19] and apply the Goemans-Williamson (G-W) algorithm for PCST [19] to detect anomalous subgraphs. We also implemented the generalized fused lasso model (GenFusedLasso) for graph scan statistics using the framework of alternating direction method of multipliers (ADMM). GenFusedLasso method solves the following minimization problem X minn −f (x) + λ |xi − xj |, (26) x∈R (i,j)∈E where f (x) is a predefined graph scan statistic and the trade-off parameter λ controls the degree of smoothness of neighboring entries in x. We applied the heuristic rounding step proposed in [29] to x to generate connected subgraphs. Parameter Tunning: We strictly followed strategies recommended by authors in their original papers to tune the related model parameters. Specifically, for EventTree, we tested the set of λ values: {0.02, 0.04, · · · , 2.0, 3.0, · · · , 20}. For Graph-Laplacian, we tested the set of λ values: {0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1} and returned the best result. For GenFusedLasso, we tested the set of λ values: {0.02, 0.04, · · · , 2.0, 3.0, · · · , 20}. For NPHGS, we set the suggested parameters by the authors: αmax = 0.15 and K = 5. Our proposed methods G RAPH -IHT and G RAPH -GHTP have a single parameter k, an upper bound of the subgraph size. We tested the set of k values: {50, 100, · · · , 1000}. As the BWSN dataset has the ground truth about the contaminated nodes, we identified the best parameter for each method that has the largest F-measure in the training data. For the other data sets, as we do not have ground truth about the true subgraphs, for each specific scan statistic, we identified the best parameter for each method that was able to identify the connected subgraph with the largest statistic score. Performance Metrics: 1) Optimization Power. The over- all scores of the three graph scan statistic functions of the connected subgraphs returned by the comparison methods are compared and analyzed. The objective is to identify methods that could find the connected subgraphs with the largest graph scan statistic scores. 2) Precision, Recall, and F-Measure. For the BWSN dataset, as the true anomalous subgraphs are known,we use F-measure that combines precision and recall to evaluate the quality of detected subgraphs by different methods. 3) Run Time. The running times of different methods are compared. B. Evolving Curves of Graph Scan Statistics Figure 3 compares our methods Graph-IHT and G RAPH GHTP with GenFusedLasso on the scores of two graph scan statistics (Kulldorff’s scan statistic and elevated mean scan statistic (EMS)) based on the best connected subgraphs identified by both methods in different iterations. Note that, a heuristic rounding process as proposed in [29] is applied to continuous vector xi estimated by GenFusedLasso at each iteration i in order to identify the best connected subgraph at the current iteration. As the setting of the parameter λ will influence the quality of the detected connected subgraph, the results under different λ values are also shown in Figure 2. The results indicate that our method G RAPH -GHTP converges in less than 10 iterations, and G RAPH -IHT converges in more steps. The qualities (scan statistic scores) of the connected subgraphs identified at different iterations by our two methods are consistently higher than those returned by GenFusedLasso. C. Optimization Power The comparisons between our method and the other baseline methods are shown in Table III and Table IV. The scores of the three graph scan statistics based on the connected subgraphs returned by these methods are reported in these two tables. The results in indicate that our method outperformed all the baseline methods on the three graph scan statistics, except that EventTree achieved the highest Kulldorff score (16738.43) on the CitHepPh dataset, but is only 2.71% larger than the returned score of our method G RAPH -GHTP. We note EventTree is a heuristic algorithm and does not provide theoretical guarantee on the quality of the connected subgraph returned, as measured by the scan statistic scores. D. Water Pollution Detection Figure 4 shows the precision, recall, and F-measure of all the comparison methods on the detection of polluted nodes in the water distribution network in BWSN with respect to different noise ratios. The results indicate that our proposed method G RAPH -GHTP and DepthFirstGraphScan were the best methods on all the three measures for most of the settings. However, DepthFirstGraphScan spent 5929 seconds to finish, and G RAPH -GHTP spent only 166 seconds, 35.8 times faster than DepthFirstGraphScan. EventTree achieved high recalls but low precisions consistently in different settings. In contrast, GraphLaplacian and NPHGS achieved high precisions but low recalls in most settings. Fig. 2. Scability of G RAPH -GHTP with respect to k Fig. 3. Evolving curves of graph scan statistic scores between our methods( G RAPH -IHT and (the upper bound of subgraph size). G RAPH -GHTP) and GenFusedLasso in different iterations. (a) BWSN(precision) Fig. 4. (b) BWSN(recall) (c) BWSN(fmeasure) Precision, Recall, and F-measure curves for water pollution detection in BWSN with respect to different noise ratios. TABLE III Comparison on scores of the three graph scan statistics based on connected subgraphs returned by comparison methods. EMS and EBP are Elevated Mean Scan Statistic and Expectation-Based Poisson Statistic, respectively. G RAPH -GHTP GraphLaplacian EventTree DepthFirstGraphScan NPHGS Kulldorff 1097.15 474.96 834.59 735.85 541.13 BWSN EMS EBP 21.56 79.71 14.89 49.91 20.25 32.13 20.41 79.30 16.90 58.59 E. Scalability Analysis Table III and Table IV also show the comparison between our proposed method G RAPH -GHTP and other baseline methods on the running time. The results indicate that our proposed method G RAPH -GHTP ran faster than all the baseline methods in most of the settings, except for EventTree. EventTree was the fastest method but was unable to detect subgraphs with high qualities. As our method has a parameter on the upper bound (k) of the subgraph returned, we also conducted the scalability of our method with respect to different values of k as shown in Figure 2. The results indicate that the running time of our algorithm is insensitive to the setting of k, which is consistent with the time complexity analysis of G RAPH -GHTP as discussed in Theorem III.5. Run Time 165.86 55315.94 441.74 5929.00 256.91 Kulldorff 16296.40 2585.44 16738.43 9531.19 11965.14 CitHepPh EMS EBP 337.90 9342.94 202.38 2305.05 335.34 9061.56 260.06 5561.66 326.23 9098.22 Run Time 155.74 22424.24 124.28 12183.88 175.08 VI. R ELATED W ORK A. Structured sparse optimization. The methods in this category have been briefly reviewed in the introduction section. The most relevant work is by Hegde et al. [15]. The authors present G RAPH-C OSAMP, a variant of C OSAMP [23], for compressive sensing and linear regression problems based on head and tail approximations of M(G, k). B. Connected subgraph detection. Existing methods in this category fall into three major categories:1) Exact algorithms. The most recent method is a brunch-and-bounding algorithm DepthFirstGraphScan [36] that runs in exponential time in the worst case; 2) Heuristic algorithms. The most recent methods in this category include EventTree [32], NPHGS [8], AdditiveScan [37], TABLE IV Comparison on scores of the three graph scan statistics based on connected subgraphs returned by comparison methods. EMS and EBP are Elevated Mean Scan Statistic and Expectation-Based Poisson Statistic, respectively. GraphLaplacian failed to run on ChicagoCrime due to out-of-memory error. G RAPH -GHTP GraphLaplacian EventTree DepthFirstGraphScan NPHGS EMS 20.45 5.40 12.40 8.13 6.28 Traffic Run Time 22.25 291.75 5.02 47.73 0.22 GraphLaplacian [33], and EdgeLasso [34]; 3) Approximation algorithms that provide performance bounds. The most recent method is presented by Qian et al.. The authors reformulate the connectivity constraint as linear matrix inequalities (LMI) and present a semi-definite programming algorithm based on convex relaxation of the LMI [18, 19] with a performance bound. However, this method is not scalable to large graphs (≥ 1000 nodes). 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A Linear Algorithm for Finding the Sink of Unique Sink Orientations on Grids arXiv:1709.08436v1 [] 25 Sep 2017 Xiaoming Sun, Jialin Zhang, Zhijie Zhang September 26, 2017 Abstract An orientation of a grid is called unique sink orientation (USO) if each of its nonempty subgrids has a unique sink. Particularly, the original grid itself has a unique global sink. In this work we investigate the problem of how to find the global sink using minimum number of queries to an oracle. There are two different oracle models: the vertex query model where the orientation of all edges incident to the queried vertex are provided, and the edge query model where the orientation of the queried edge is provided. In the 2-dimensional case, we design an optimal linear deterministic algorithm for the vertex query model and an almost linear deterministic algorithm for the edge query model, previously the best known algorithms run in O(N log N ) time for the vertex query model and O(N 1.404 ) time for the edge query model. 1 Introduction In this paper we consider a special type of d-dimensional grid, which is the Cartesian product of d complete graphs. Each pair of vertices of the grid has an edge if and only if they are distinct in exactly one coordinate. A subgrid is the Cartesian product of cliques of the original complete graphs. Recall that in an oriented graph a vertex with zero outdegree is called a sink. A unique sink orientation (USO) of a grid is an orientation of its edges such that each of its nonempty subgrids (including the original grid) has a unique sink. Traditionally, an oriented grid with the above property is called a grid USO. The computational problem now is to find the unique global sink of a grid USO. Two different oracle models were introduced in the literature [18, 7], namely the vertex query model and the edge query model. A vertex query 1 Figure 1: (a) a 2-dimensional grid USO. (b) a 3-cube USO with an cycle. reveals the orientation of all incident edges of the queried vertex, whereas an edge query returns the orientation of the queried edge. We count for the time overhead only the number of queries to the oracle. In this paper, we restrict our main attention to the sink-finding problem on a 2-dimensional grid USO, see Figure 1 (a) for an instance. There are two reasons. As is well-known, a planar grid USO must be acyclic [7]. On the contrary, a d-dimensional grid USO with d > 2 may contain cycles. Figure 1 (b) depicts a cyclic cube (a grid with each of its dimensions having size two). The acyclicity of a planar grid USO allows us to design algorithms that enhance the “rank” of queried vertices step by step and finally reach the unique sink. Besides, a fast algorithm running in the lower dimensional case may improve the upper bound in the general case, due to the inherited grid USO introduced by Gärtner et al. [7]. However, the vertex query model seems to be unreasonable when fixing the dimension. Since in practice it takes linear (or polynomial) time to implement a vertex query, while the total number of vertex is polynomial. The time for a vertex query is never negligible compared with the number of queries. There are several reasons to justify the vertex query model. First, the vertex query model is theoretically simpler than the edge query model in that it is easy to formally capture all the information coming from a single vertex query. Second, the number of vertex queries is a good measure of complexity for a general grid USO of d dimension and luckily, due to the inherited grid structure, algorithms running on a fixed-dimensional grid USO may be adapted to a general d-dimensional grid USO. Third, it turns out that our algorithm under the vertex query model serves as a black box when addressing the more natural and practical edge query model. In a word, though being mostly of theoretical interest for the grid USO of fixed 2 dimension, the study of the vertex query model still has potential practical applications. The sink-finding problem on a planar grid USO has an intuitive interpretation. Assume we have a matrix as input. Only numbers in the same row or in the same column are allowed to be compared. Each submatrix has exactly one minimum number. How many comparisons do we need to find the global minimum number? The grid USO of dimension two serves as a simple combinatorial framework to many well-studied optimization problems. The first example is the one line and N points problem, first studied in [9]. Suppose there are N points in general position in the plane and one vertical line to which none of those N points belongs. There is a line segment between a pair of points if and only if this segment intersects with the vertical line. The problem asks to find the unique segment that has the lowest intersection with the vertical line. This problem can be recast to the sink-finding problem of an implicit planar grid USO in the following way [4]. Each line segment defines a grid vertex. Two line segments are adjacent if and only if they share exactly one endpoint. The higher one has an oriented edge to the lower one. The total orientation turns out to be an USO. The lowest segment corresponds to the global sink. Another optimization problem that can be considered as a special case of the sink-finding problem on a planar grid USO is the linear programs on (N − 2)-polytopes with N facets. Felsner et al. [4] showed that the vertexedge graph of such a polytope, oriented by the linear objective function, is isomorphic to a planar grid USO. The global sink, again, corresponds to the optimal solution of this special type of linear programs. It is of interest to take a look at how the algorithms devised for finding the sink of the planar grid USO go on the vertex-edge graph of the above polytope. We provide such an example in Section 3. We have seen some relationship between the grid USO (of dimension two) and linear programs. Indeed, one main motivation of the study of the (general) grid USO is that it is closely related to linear programs. It is generally known that although there are polynomial algorithms for solving linear programs, such as the ellipsoid algorithm by Khachiyan [13] and the interior-point algorithm by Karmarkar [12], both of them are not strongly polynomial. It remains open whether there is such an algorithm. And also it is unknown whether there exists a pivoting rule such that the simplex method runs in polynomial time. Several well-known (randomized) pivoting rules, such as Random-Edge, Random-Facet [11, 15] and Random-Bland [2], have failed to reach such an bound [14, 16, 5, 6]. It turns out that the unique sink orientation may help devise an outperforming algorithm to solve 3 linear programs. Holt and Klee [10] showed that an orientation of a polytope is an acyclic unique sink orientation (AUSO) with the Holt-Klee property if it is induced from an LP instance. The Holt-Klee property states that the number of vertex-disjoint directed paths from the source to the sink equals to the number of the neighbours of the source (or the sink equivalently) in every subgrid. Furthermore, Gärtner and Schurr [8] proved that any LP instance in d nonnegative variables defines a d-dimensional cube USO. The sink of this cube corresponds naturally to an optimal solution to the LP. The vertex query oracle comes down to Gaussian elimination. A polynomial sink-finding algorithm would yield a corresponding algorithm to solve linear programs. Besides linear programs, the underlying combinatorial structures of many other optimization problems are actually a grid USO. An important example is the generalized linear complementarity problem over a P-matrix (PGLCP), first introduced by Cottle and Dantzig [3]. Gärtner et al. [7] showed that this problem can be recast to the sink-finding problem of an implicit grid USO. Previous work. The sink-finding problem of a grid USO was first put forward formally and studied by Szabó and Welzl [18], where they restricted their attention to a d-dimensional cube. They designed the first nontrivial deterministic and randomized algorithms which use O(cd ) vertex queries for some constant c < 2. Later Gärtner et al. [7] extended the two oracle models to a d-dimensional grid USO. In that paper they investigated several properties of a grid USO and introduced randomized algorithms for both oracle models. However, no nontrivial deterministic algorithm was found for a d-dimensional grid USO at that time. Attempting to find such an algorithm, Barba et al. [1] first paid their attention to the planar case. Here we state some known results in the planar case, given a 2-dimensional grid USO with m × n vertices. Assume that N = m + n. In the randomized setting, Gärtner et al. [7] prove an upper bound of O(log m · log n) for the vertex query model, against a lower bound of Ω(log m + log n) claimed by Barba et al. [1]. It is necessary to mention the performance of the most natural Random-Edge algorithm on the planar grid USO, which chooses the next queried vertex randomly from the out-going neighbours of the current one. Gärtner et al. [9] proved it runs in Θ(log2 N ) under the Holt-Klee condition and Milatz [17] extended this result for the general planar grid USO. In the edge query model, the unique sink of a 2-dimensional grid USO can be obtained with Θ(N ) queries in expectation [7]. In the deterministic setting, Barba et al. [1] exhibit an algorithm using O(N log N ) vertex queries to find the sink and another algorithm using O(N log4 7 ) edge queries. In 4 particular, they introduced an O(N ) algorithm for the vertex query model under the Holt-Klee condition [10]. Our contributions. The main contribution of our paper is Lemma 4, which states that we are able to exclude certain row and certain column from being the global sink after querying a linear number of vertices in some way. Based on it, we prove that m + n − 1 vertex queries suffice to determine the global sink in the worst case, which coincide exactly with the lower bound for the vertex query model. Using it as a black box, we are able to exhibit an √ 2 log N ) deterministic algorithm for the edge query model. We note O(N · 2 that for a d-dimensional grid USO our algorithm yields an upper bound of O(N̂ dd/2e ) for the vertex query model, where N̂ denotes the sum of the sizes of every dimension. Paper organization. In Section 2 we establish some notations for a planar grid USO and introduce some known properties. In Section 3 we handle the vertex query model and give an optimal algorithm. We address the edge query model based on the algorithm for the vertex query model in Section 4. And at last we conclude the paper in Section 5 with some open problems. 2 Preliminaries First we provide some definitions and notations for the planar case. Denote by Kn the complete graph with n vertices. An (m, n)-grid is the Cartesian product Km × Kn . Its vertex set is defined to be the Cartesian product [m] × [n], where [n] := {1, 2, . . . , n}. Elements in [n] are called coordinates, and there are N = m + n coordinates. Throughout this paper, we identify [m] × [n] with Km × Kn . A subgrid is then the Cartesian product I × J, where I ⊆ [m] and J ⊆ [n]. For the sake of convenience, we say all vertices with the same first-coordinate form a row. A column is defined analogously. Hence two vertices are adjacent if and only if they are in the same row or in the same column. Denote by uij the vertex at the cross of the i-th row and the j-th column. Let Tv (m, n) be the number of vertex queries needed in the worst case to find the sink of an (m, n)-grid USO deterministically and Tv (n) for short when m = n. Similarly, Te (m, n) and Te (n) are defined for the edge query model. Following the tradition of the previous works [18, 1], in the vertex query model the global sink must be queried even if we have already known its position before it is queried. Thus, for instance, Tv (1) = 1 and Tv (2) = 3, 5 instead of Tv (1) = 0 and Tv (2) = 2. Readers will find the benefit of this tradition shortly. Now we introduce some known properties about the (m, n)-grid USO. Lemma 1 ([7]). Every (m, n)-grid USO is acyclic. Suppose G is an (m, n)-grid USO. This lemma allows us to define a partial order on the vertex set [m] × [n] of G. For arbitrary two vertices u, v ∈ G, define u  v if and only if either u = v or there exists a directed path from u to v. In other word we just say u is larger than v. The unique sink corresponds to the unique minimum vertex. Barba et al. claimed a lower bound of m + n − 1 for the vertex query model without a proof [1]. For the completeness we give a simple adversary argument of this lower bound. Lemma 2. Tv (m, n) ≥ m + n − 1. Proof. Here is the answering strategy of the adversary. Let the first queried vertex be the sink of the first row. Make all vertices in this row point out to their adjacent vertices in other rows. Thus this vertex query eliminates exactly the first row and any query of the other vertices in this row gives no more information. By induction the i-th queried vertex eliminates exactly the i-th row and therefore m − 1 vertex queries are necessary to eliminate all m rows but the last row. At this time, we need n vertex queries instead of n − 1 to find the sink of the last row, recalling the definition of Tv (m, n). Induced grid USO. Barba et al. [1] discovered a simple construction of an induced grid USO from an (m, n)-grid USO, which helped a lot their design of algorithms for both oracle models. It’s worth describing the construction in detail. Assume G is an (m, n)-grid USO. Let P = {P1 , . . . , Pk } be a partition of [m] and Q = {Q1 , . . . , Ql } be a partition of [n]. Each Pi × Qj is a subgrid of G. Let H be a (k, l)-grid with vertex set {Pi × Qj | i = 1, . . . , k, j = 1, . . . , l}. As before, two distinct vertices x = Pi × Qj and y = Pi0 × Qj 0 are adjacent in H if and only if they are in the same row or in the same column, i.e. i = i0 or j = j 0 . Suppose that x and y are adjacent, it remains to determine the orientation of edge xy in H. Recall that x and y are subgrids of G and therefore by the USO property x and y have unique sinks ux and uy in G, respectively. If ux has an outgoing edge to some vertex of y in G, then we make x point to y in H. Otherwise we make v point to u. This orientation is well-defined due to the acyclicity of an (m, n)-grid USO (see figure 2). Such a grid H is called induced grid of G. What’s more, it was proved that the induced grid H also suffices the USO property [1]: 6 Figure 2: (a) let P1 = {1, 2}, P2 = {3}, Q1 = {1} and Q2 = {2, 3}. The (3, 3)-grid USO is divided into four subgrids accordingly. (b) The induced grid USO defined according to the partition. Lemma 3. Let G be a 2-dimensional grid USO and H be an induced grid constructed from G. Then H is also a 2-dimensional grid USO and the sink of H is the subgrid of G which contains the sink of G. 3 Vertex Query Model This section is contributed to the vertex query model. We first explore an important combinatorial property of the 2-dimensional grid USO. Then we exploit this property to obtain an optimal deterministic algorithm using m + n − 1 vertex queries in the worst case. Next we provide an intuitive interpretation of this algorithm in connection with linear programs on N − 2polytope with N facets. To end this section, we adapt our algorithm to higher dimensional case. Let G be an (m, n)-grid USO. For a queried vertex v = uij , let Iv  [m] be the collection of the first-coordinates such that u  v for any u ∈ Iv × {j}. Note that v itself is included in Iv × {j}. Jv  [n] is defined analogously. Clearly v is the unique sink of the subgrid Iv × Jv . Hence if v is not the global sink of G, every vertex in the subgrid Iv × Jv is excluded from being the global sink, since otherwise there would be two sinks in Iv × Jv . Suppose v is not the global sink, the query of v eliminates exactly the corresponding subgrid Iv × Jv . Assume that m = n. First query arbitrary n vertices {v1 , . . . , vn } in distinct rows and distinct columns. Suppose w.l.o.g. that none of them is the sink of G, then the subgrids Ivi × Jvi are eliminated, for i = 1, . . . , n. 7 Figure 3: Induction on a (6, 6)-grid USO. White vertices are eliminated. All vertices in I × {6} and {6} × J are larger than v6 . Now there are a lot of eliminated vertices. The lemma below answers in a way how many such vertices there are. Lemma 4. Let G be an (n, n)-grid USO. After querying arbitrary n vertices of G in distinct rows and distinct columns and eliminating corresponding subgrids, at least one row and one column are eliminated. Proof. We adopt induction on the scale n. There is nothing to prove in the trivial case n = 1. Assume the lemma holds for smaller values of n. Also we assume that none of the n vertices has ever been the global sink, since otherwise the lemma naturally holds. Note that shuffling rows and columns does not change the underlying structure of G. So by rearranging the coordinates, we may assume that all queried vertices {v1 , . . . , vn } lie in the diagonal, i.e. vi = uii , and further assume that for all 1 ≤ i < j ≤ n either vi  vj or vi and vj are incomparable. Consider the (2, 2)-subgrid Hi,j spanned by vi and vj . By the assumption there is no path from vj to vi in Hi,j . Hence by the USO property vj cannot be the source of Hi,j , which implies in Hi,j there is at least one incoming edge of vj . This simple fact was first observed by Barba et al. [1]. In particular, vn has at least one incoming edge in each subgrid Hi,n , which means either i ∈ Ivn or i ∈ Jvn , for 1 ≤ i ≤ n − 1. Let I = Ivn ∩ [n − 1] 8 and J = [n − 1]\I. Clearly J ⊆ Jvn ∩ [n − 1], since J ∩ Ivn = ∅. Accordingly, we divide the (n − 1, n − 1)-grid spanned by {v1 , . . . , vn−1 } into subgrids I × I, J × J, I × J and J × I. The subgrid J × I is of no relevance in our discussion. The subgrid I × J ⊆ Ivn × Jvn and therefore all of its vertices are eliminated. Square subgrids I × I and J × J both contain queried vertices in their diagonals, respectively. So by the inductive assumption both of them contain one eliminated row and one eliminated column, respectively. Let i ∈ I be the coordinate such that {i} × I is the eliminated row in I × I. The subgrid {i} × J ⊆ I × J and therefore is eliminated. The vertex {i} × {n} ∈ Ivn × {n} and is also eliminated. To sum up, the row {i} × [n] = {i} × (I ∪ J ∪ n) is eliminated. Similarly, let j ∈ J be the coordinate such that J × {j} is the eliminated column in J × J. The column [n] × {j} turns out to be an eliminated column of the original grid by the same argument. The proof now is completed. See Figure 3 for an intuitive example. Lemma 4 makes full use of the information from the queried vertices and naturally leads to the following algorithm depicted as Algorithm 1. Algorithm 1: Diagonal Algorithm Input: An (n, n)-grid USO G Output: The unique sink of G 1 query arbitrary n vertices in distinct rows and distinct columns; 2 while the sink has not been queried do 3 eliminate one row and one column, say the i-th row and the j-th column; 4 if two distinct queried vertex uij 0 and ui0 j are eliminated then 5 query vertex ui0 j 0 ; 6 end 7 end We first query arbitrary n vertices in distinct rows and distinct columns (line 1). By Lemma 4, there are one row and one column excluded from being the global sink, say the i-th row and the j-th column (line 3). Now we need to handle a subgrid USO of scale n − 1. Note that at most 2 queried vertices are eliminated, since each row (or column) contains exactly one queried vertex. If two distinct queried vertex uij 0 and ui0 j are eliminated, one more query of vertex ui0 j 0 would make the subgrid contain n − 1 queried vertices in distinct rows and distinct columns (line 4-6). Otherwise, the subgrid has already contained n − 1 such queried vertices. At either case, we can apply Lemma 4 9 again and at most another n − 1 queries suffice to find the global sink. The argument above leads to the theorem below. Theorem 1. There exists a deterministic algorithm using 2n − 1 vertex queries in the worst case to find the unique sink of an (n, n)-grid USO. We can easily extend Algorithm 1 to an arbitrary (m, n)-grid USO. Assume that m < n. First query m vertices in distinct rows and distinct columns. Then the m columns with queried vertices in them form an (m, m)subgrid USO. By Lemma 4, one column is eliminated, and there remains m − 1 queried vertices. Next, one more appropriate vertex query would again exclude one column. Repeat this procedure until there remains exactly m columns, i.e. an (m, m)-subgrid USO, which contains m − 1 queried vertices. Now we can eliminate both one row and one column at the same time after every vertex query and another m queries suffice to determine the global sink. To conclude we have Theorem 2. There exists a deterministic algorithm using m + n − 1 vertex queries in the worst case to find the unique sink of an (m, n)-grid USO. Combined with Lemma 2, this theorem implies that Tv (m, n) = m+n−1, so Algorithm 1 is optimal. As is mentioned before, the vertex-edge graph of a (N − 2)-polytope with N facets is isomorphic to an (m, n)-grid USO G for some m, n with N = m+n. There are totally N coordinates in G. Each coordinate represents a facet of the original polytope. Note that a vertex v = uij of G is the intersection of some (N − 2) facets. The coordinates i and j mean that v does not lie in the corresponding facets. Two vertices are adjacent in the vertex-edge graph if and only if both of them belong to exactly the same N − 3 facets. If in some way (e.g. by Lemma 4), the i-th row (or column) is excluded from being the global sink, we can deduce that the global sink must lie in the corresponding facet. The procedure of Algorithm 1 becomes rather clear. At each step, Algorithm 1 arbitrarily queries a new vertex which is not adjacent to the previously queried vertices (line 1) in order to involve as many facets as possible. Once every vertex is adjacent to at least one queried vertex, it is guaranteed that the global sink must belong to certain facet or (intersection of several facets) (line 3). Later queries are indeed restricted in that facet (line 4-6). From the view of linear programs, such an algorithm is rather interesting. Though being less practical than the edge query model, the vertex query model still has potential applications. One of them is to reach a better upper bound for the general grid USO of d dimension, combined with the inherited 10 grid USO structure. Roughly speaking, fix the coordinates of two dimensions of size n1 and n2 , and vertices share the fixed two coordinates form a subgrid of the original grid. The vertex set of the inherited grid is the collection of the n1 × n2 subgrids. The definition and the orientation of the edges are the same as those in the induced grid USO. Let N̂ be the sum of the sizes of each dimension and T̂v (d) be the time overhead for the grid USO of d dimension in the worst case. Running Algorithm 1 on the inherited grid USO yields the following recurrence T̂v (d) ≤ (n1 + n2 − 1) · T̂v (d − 2). By solving it, we have the following corollary, Corollary 1. There exists a deterministic algorithm using O(N̂ dd/2e ) vertex queries in the worst case to find the unique sink of a d-dimensional grid USO. 4 Edge Query Model Though being optimal, Algorithm 1 may not be a good choice to solve optimization problems like one line and N points, for implementing a vertex query actually takes linear time. However, there are potential applications of this result. An immediate one is a fast algorithm under the edge query model. Throughout this section we assume that m = n, since one can always add rows or columns to make the grid square without changing the structure of the original grid and the position of the global sink. We extend the divideand-conquer strategy in [1] to an almost linear algorithm using Algorithm 1 as a black box. The formal description is depicted as Algorithm 2. Algorithm 2: Divide-and-Conquer Input: An (n, n)-grid USO G Output: The unique sink of G 1 construct an induced (k, k)-grid USO H from G; 2 run Algorithm 1 on H under the vertex query model; Let G be an (n, n)-grid USO, and H be an induced (k, k)-grid USO constructed from G. The construction of H is indeed two respective partitions of [n] and takes constant time (line 1). As is shown in Algorithm 2, the main idea is to run Algorithm 1 on H under the vertex query model (line 2). By Theorem 1, 2k − 1 vertex queries suffice to determine the sink of H. Recall 11 that each vertex in H corresponds to a subgrid in G, and that the sink of H is the subgrid of G which contains the sink of G. Note that a vertex query returns the orientation of all the incident edges. Hence according to the definition of the orientation of edges in H, to implement a vertex query in H, we need to (i) find the sink of the corresponding subgrid in G and (ii) query all edges incident to this local sink in G. In a word, a vertex query in H is equivalent to at most Te ( nk ) + 2n − 2 edge queries in G. The above argument implies the recurrence below, n Te (n) ≤ (2k − 1) Te ( ) + 2n − 2 . k   Note that our careful definition of Tv (m, n), the number of vertex queries in the worst case, pays off here — the sink of H has already been queried, which means that the sink of the corresponding subgrid in G, i.e. the sink of G, has been found. Solve the recurrence will get Te (n) = O(nlogk (2k−1) ) if setting k = O(1), which √ coincides with the result in [1] when k = 4. Furthermore, we set k = 22√ log n , where log means the logarithm to base 2. Assume that Te (n) ≤ cn · 22 log n , where c > 4, for smaller values of n, then we have √ Te (n) ≤ 2kc nk q 2 log n−log k 2 ·2 √ log n−2 + 4kn √ log n 2 log n + 4n · 2 . = 2cn · 2 √ 2 log n To make Te (n) ≤ cn · 2 , we only need to assure q √ √ 2c 2( log n− log n−2 log n) ≥ . 2 c−4 The left side is monotone decreasing and its limit √ is 4, hence setting c ≥ 8 2 log n the inequality holds. √ Therefore Te (n) = O(n · 2 ). Notice that n · 22 log n = o(n1+ ), for any  > 0. Algorithm 2 is mildly superlinear. We conclude this section by the following theorem, √ 2 log N ) deterministic algorithm to find Theorem 3. There exists an O(N · 2 the sink of an (m, n)-grid USO under the edge query model, where N = m+n. 5 Conclusion In this paper, we have discovered a new combinatorial property (Lemma 4) of the 2-dimensional grid USO and developed deterministic algorithms for 12 both oracle models based on it, one optimal, the other nearly optimal. In the randomized setting, however, all the known randomized algorithms only reach an upper bound of O(log2 N ), against the lower bound of Ω(log N ). By further exploiting Lemma 4, one may devise an optimal algorithm to close the gap. In the general d-dimensional grid USO, it is of interest whether there exists a similar combinatorial property. References [1] Luis Barba, Malte Milatz, Jerri Nummenpalo, and Antonis Thomas. Deterministic algorithms for unique sink orientations of grids. Computing and Combinatorics Conference, pages 357–369, 2016. [2] Robert G Bland. New finite pivoting rules for the simplex method. 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arXiv:1702.03150v1 [] 10 Feb 2017 Autocommuting probability of a finite group relative to its subgroups Parama Dutta and Rajat Kanti Nath∗ Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India. Emails: parama@gonitsora.com and rajatkantinath@yahoo.com Abstract Let H ⊆ K be two subgroups of a finite group G and Aut(K) the automorphism group of K. The autocommuting probability of G relative to its subgroups H and K, denoted by Pr(H, Aut(K)), is the probability that the autocommutator of a randomly chosen pair of elements, one from H and the other from Aut(K), is equal to the identity element of G. In this paper, we study Pr(H, Aut(K)) through a generalization. Key words: Automorphism group, Autocommuting probability, Autoisoclinism. 2010 Mathematics Subject Classification: 20D60, 20P05, 20F28. 1 Introduction Let G be a finite group acting on a set Ω. Let Pr(G, Ω) denote the probability that a randomly chosen element of Ω fixes a randomly chosen element of G. In 1975, Sherman [11] initiated the study of Pr(G, Ω) considering G to be an abelian group and Ω = Aut(G), the automorphism group of G. Note that Pr(G, Aut(G)) = |{(x, α) ∈ G × Aut(G) : [x, α] = 1}| |G|| Aut(G)| where [x, α] is the autocommutator of x and α defined as x−1 α(x). The ratio Pr(G, Aut(G)) is called autocommuting probability of G. Let H and K be two subgroups of a finite group G such that H ⊆ K. Motivated by the works in [2, 6], we define Prg (H, Aut(K)) = |{(x, α) ∈ H × Aut(K) : [x, α] = g}| |H|| Aut(K)| (1.1) where g ∈ K. That is, Prg (H, Aut(K)) is the probability that the autocommutator of a randomly chosen pair of elements, one from H and the other from Aut(K), is ∗ Corresponding author 1 equal to a given element g ∈ K. The ratio Prg (H, Aut(K)) is called generalized autocommuting probability of G relative to its subgroups H and K. Clearly, if H = G and g = 1 then Prg (H, Aut(K)) = Pr(G, Aut(G)). We would like to mention here that the case when H = G is considered in [3]. In this paper, we study Prg (H, Aut(K)) extensively. In particular, we obtain some computing formulae, various bounds and a few characterizations of G through a subgroup. We conclude the paper describing an invariance property of Prg (H, Aut(K)). We write S(H, Aut(K)) to denote the set {[x, α] : x ∈ H and α ∈ Aut(K)} and [H, Aut(K)] := hS(H, Aut(K))i. We also write L(H, Aut(K)) := {x ∈ H : [x, α] = 1 for all α ∈ Aut(K)} and L(G) := L(G, Aut(G)), the absolute center of G (see [5]). Note that L(H, Aut(K)) is a normal subgroup of H contained in H ∩ Z(K). Further, L(H, Aut(K)) = ∩ CH (α), where CH (α) = {x ∈ H : [x, α] = 1} is a subgroup α∈Aut(K) of H. Let CAut(K) (x) := {α ∈ Aut(K) : α(x) = x} for x ∈ H and CAut(K) (H) = {α ∈ Aut(K) : α(x) = x for all x ∈ H}. Then CAut(K) (x) is a subgroup of Aut(K) and CAut(K) (H) = ∩ CAut(K) (x). x∈H Clearly, Prg (H, Aut(K)) = 1 if and only if [H, Aut(K)] = {1} and g = 1 if and only if H = L(H, Aut(K)) and g = 1. Also, Prg (H, Aut(K)) = 0 if and only if g ∈ / S(H, Aut(K)). Therefore, we consider H 6= L(H, Aut(K)) and g ∈ S(H, Aut(K)) throughout the paper. 2 Some computing formulae For any x ∈ H, let us define the set Tx,g (H, K) = {α ∈ Aut(K) : [x, α] = g}, where g is a fixed element of K. Note that Tx,1 (H, K) = CAut(K) (x). The following two lemmas play a crucial role in obtaining computing formula for Prg (H, Aut(K)). Lemma 2.1. Let H and K be two subgroups of a finite group G such that H ⊆ K. If Tx,g (H, K) 6= φ then Tx,g (H, K) = σCAut(K) (x) for some σ ∈ Tx,g (H, K) and hence |Tx,g (H, K)| = |CAut(G) (x)|. Proof. Let σ ∈ Tx,g (H, K) and β ∈ σCAut(K) (x). Then β = σα for some α ∈ CAut(K) (x). We have [x, β] = [x, σα] = x−1 σ(α(x)) = [x, σ] = g. Therefore, β ∈ Tx,g (H, K) and so σCAut(K) (x) ⊆ Tx,g (H, K). Again, let γ ∈ Tx,g (H, K) then γ(x) = xg. We have σ −1 γ(x) = σ −1 (xg) = x and so σ −1 γ ∈ CAut(K) (x). Therefore, γ ∈ σCAut(K) (x) which gives Tx,g (H, K) ⊆ σCAut(K) (x). Hence, the result follows. Consider the action of Aut(K) on K given by (α, x) 7→ α(x) where α ∈ Aut(K) and x ∈ K. Let orbK (x) := {α(x) : α ∈ Aut(K)} be the orbit of x ∈ K. Then by orbit-stabilizer theorem, we have | orbK (x)| = | Aut(K)| . |CAut(K) (x)| 2 (2.1) Lemma 2.2. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then Tx,g (H, K) 6= φ if and only if xg ∈ orbK (x). Proof. The result follows from the fact that α ∈ Tx,g (H, K) if and only if xg ∈ orbK (x). The following theorem gives two computing formulae for Prg (H, Aut(K)). Theorem 2.3. Let H and K be two subgroups of a finite group G such that H ⊆ K. If g ∈ K then Prg (H, Aut(K)) = = 1 |H|| Aut(K)| 1 |H| X X |CAut(K) (x)| x∈H xg∈orbK (x) x∈H xg∈orbK (x) 1 . | orbK (x)| Proof. We have {(x, α) ∈ H × Aut(K) : [x, α] = g} = ⊔ ({x} × Tx,g (H, K)), x∈H where ⊔ represents the union of disjoint sets. Therefore, by (1.1), we have X |H|| Aut(K)|Prg (H, Aut(K)) = | ⊔ ({x} × Tx,g (H, K))| = |Tx,g (H, K)|. x∈H x∈H Hence, the result follows from Lemma 2.1, Lemma 2.2 and (2.1). Considering g = 1 in Theorem 2.3, we get the following computing formulae for Pr(H, Aut(K)). Corollary 2.4. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then Pr(H, Aut(K)) = X 1 | orbK (H)| |CAut(K) (x)| = |H|| Aut(K)| |H| x∈H where orbK (H) = {orbK (x) : x ∈ H}. Corollary 2.5. Let H and K be two subgroups of a finite group G such that H ⊆ K. If CAut(K) (x) = {I} for all x ∈ H \ {1}, where I is the identity element of Aut(K), then Pr(H, Aut(K)) = 1 1 1 + − . |H| | Aut(K)| |H|| Aut(K)| Proof. By Corollary 2.4, we have |H|| Aut(K)| Pr(H, Aut(K)) = X |CAut(K) (x)| = | Aut(K)| + |H| − 1. x∈H Hence, the result follows. 3 We also have |{(x, α) ∈ H × Aut(K) : [x, α] = 1}| = P |CH (α)| and hence α∈Aut(K) Pr(H, Aut(K)) = 1 |H|| Aut(K)| X |CH (α)|. (2.2) α∈Aut(K) We conclude this section with the following two results. Proposition 2.6. Let H and K be two subgroups of a finite group G such that H ⊆ K. If g ∈ K then Prg−1 (H, Aut(K)) = Prg (H, Aut(K)). Proof. Let A = {(x, α) ∈ H × Aut(K) : [x, α] = g} and B = {(y, β) ∈ H × Aut(K) : [y, β] = g −1 }. Then (x, α) 7→ (α(x), α−1 ) gives a bijection between A and B. Therefore |A| = |B|. Hence, the result follows from (1.1). Proposition 2.7. Let G1 and G2 be two finite groups. Let H1 , K1 and H2 , K2 be subgroups of G1 and G2 respectively such that H1 ⊆ K1 , H2 ⊆ K2 and gcd(|K1 |, |K2 |) = 1. If (g1 , g2 ) ∈ K1 × K2 then Pr(g1 ,g2 ) (H1 × H2 , Aut(K1 × K2 )) = Prg1 (H1 , Aut(K1 ))Prg2 (H2 , Aut(K2 )). Proof. Let X = {((x, y), αK1 ×K2 ) ∈ (H1 × H2 ) × Aut(K1 × K2 ) : [(x, y), αK1 ×K2 ] = (g1 , g2 )}, Y = {(x, αK1 ) ∈ H1 × Aut(K1 ) : [x, αK1 ] = g1 } and Z = {(y, αK2 ) ∈ H2 × Aut(K2 ) : [y, αK2 ] = g2 }. Since gcd(|K1 |, |K2 |) = 1, by Lemma 2.1 of [1], we have Aut(K1 × K2 ) = Aut(K1 ) × Aut(K2 ). Therefore, for every αK1 ×K2 ∈ Aut(K1 × K2 ) there exist unique αK1 ∈ Aut(K1 ) and αK2 ∈ Aut(K2 ) such that αK1 ×K2 = αK1 × αK2 , where αK1 ×αK2 ((x, y)) = (αK1 (x), αK2 (y)) for all (x, y) ∈ H1 ×H2 . Also, for all (x, y) ∈ H1 × H2 , we have [(x, y), αK1 ×K2 ] = (g1 , g2 ) if and only if [x, αK1 ] = g1 and [y, αK2 ] = g2 . These leads to show that X = Y × Z. Therefore |Y| |Z| |X | = · . |H1 × H2 || Aut(K1 × K2 )| |H1 || Aut(K1 )| |H2 || Aut(K2 )| Hence, the result follows from (1.1). 4 3 Various bounds In this section, we obtain various bounds for Prg (H, Aut(K)). We begin with the following lower bounds. Proposition 3.1. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then, for g ∈ K, we have |CAut(K) (H)|(|H|−|L(H,Aut(K))|) |H|| Aut(K)| (a) Prg (H, Aut(K)) ≥ |L(H,Aut(K))| |H| (b) Prg (H, Aut(K)) ≥ |L(H,Aut(K))||CAut(K) (H)| |H|| Aut(K)| + if g = 1. if g 6= 1. Proof. Let C denote the set {(x, α) ∈ H × Aut(K) : [x, α] = g}. (a) We have (L(H, Aut(K))×Aut(K))∪(H×CAut(K) (H)) is a subset of C and |(L(H, Aut(K)) × Aut(K)) ∪ (H × CAut(K) (H))| = |L(H, Aut(K))|| Aut(K)| + |CAut(K) (H)||H| − |L(H, Aut(K))||CAut(K) (H)|. Hence, the result follows from (1.1). (b) Since g ∈ S(H, Aut(K)) we have C is non-empty. Let (y, β) ∈ C then (y, β) ∈ / L(H, Aut(K)) × CAut(K) (H) otherwise [y, β] = 1. It is easy to see that the coset (y, β)(L(H, Aut(K)) × CAut(K) (H)) is a subset of C having order |L(H, Aut(K))||CAut(K) (H)|. Hence, the result follows from (1.1). Proposition 3.2. Let H and K be two subgroups of a finite group G such that H ⊆ K. If g ∈ K then Prg (H, Aut(K)) ≤ Pr(H, Aut(K)). The equality holds if and only if g = 1. Proof. By Theorem 2.3, we have Prg (H, Aut(K)) = ≤ 1 |H|| Aut(K)| X |CAut(K) (x)| x∈H xg∈orbK (x) X 1 |CAut(K) (x)| = Pr(H, Aut(K)). |H|| Aut(K)| x∈H The equality holds if and only if xg ∈ orbK (x) for all x ∈ H if and only if g = 1. Proposition 3.3. Let H and K be two subgroups of a finite group G such that H ⊆ K. Let g ∈ K and p the smallest prime dividing | Aut(K)|. If g 6= 1 then Prg (H, Aut(K)) ≤ 1 |H| − |L(H, Aut(K))| < . p|H| p 5 Proof. By Theorem 2.3, we have Prg (H, Aut(K)) = 1 |H| X x∈H\L(H,Aut(K)) xg∈orbK (x) 1 | orbK (x)| (3.1) noting that for x ∈ L(H, Aut(K)) we have xg ∈ / orbK (x). Also, for x ∈ H \ L(H, Aut(K)) and xg ∈ orbK (x) we have | orbK (x)| > 1. Since | orbK (x)| is a divisor of | Aut(K)| we have | orbK (x)| ≥ p. Hence, the result follows from (3.1). Proposition 3.4. Let H1 , H2 and K be subgroups of a finite group G such that H1 ⊆ H2 ⊆ K. Then Prg (H1 , Aut(K)) ≤ |H2 : H1 |Prg (H2 , Aut(K)). The equality holds if and only if xg ∈ / orbK (x) for all x ∈ H2 \ H1 . Proof. By Theorem 2.3, we have |H1 || Aut(K)|Prg (H1 , Aut(K)) = X |CAut(K) (x)| X |CAut(K) (x)| x∈H1 xg∈orbK (x) ≤ x∈H2 xg∈orbK (x) = |H2 || Aut(K)|Prg (H2 , Aut(K)). Hence, the result follows. Proposition 3.5. Let H and K be two subgroups of a finite group G such that H ⊆ K. If g ∈ K then Prg (H, Aut(K)) ≤ |K : H| Pr(K, Aut(K)) with equality if and only if g = 1 and H = K. Proof. By Proposition 3.2, we have Prg (H, Aut(K)) ≤ Pr(H, Aut(K)) X 1 = |CAut(K) (x)| |H|| Aut(K)| x∈H X 1 ≤ |CAut(K) (x)|. |H|| Aut(K)| x∈K Hence, the result follows from Corollary 2.4. 6 Note that if we replace Aut(K) by Inn(K), the inner automorphism group of K, in (1.1) then Prg (H, Inn(K)) = Prg (H, K) where Prg (H, K) = |{(x, y) ∈ H × K : x−1 y −1 xy = g}| . |H||K| A detailed study on Prg (H, K) can be found in [2]. The following proposition gives a relation between Prg (H, Aut(K)) and Prg (H, K) for g = 1. Proposition 3.6. Let H and K be two subgroups of a finite group G such that H ⊆ K. If g = 1 then Prg (H, Aut(K)) ≤ Prg (H, K). Proof. If g = 1 then by [2, Theorem 2.3], we have Prg (H, K) = 1 1 X |H| | clK (x)| (3.2) x∈H where clK (x) = {α(x) : α ∈ Inn(K)}. Since clK (x) ⊆ orbK (x) for all x ∈ H, the result follows from (3.2) and Theorem 2.3. Theorem 3.7. Let H and K be two subgroups of a finite group G such that H ⊆ K and p the smallest prime dividing | Aut(K)|. Then Pr(H, Aut(K)) ≥ |L(H, Aut(K))| p(|H| − |XH | − |L(H, Aut(K))|) + |XH | + |H| |H|| Aut(K)| and Pr(H, Aut(K)) ≤ (p − 1)|L(H, Aut(K))| + |H| |XH |(| Aut(K)| − p) − , p|H| p|H|| Aut(K)| where XH = {x ∈ H : CAut(K) (x) = {I}}. Proof. We have XH ∩ L(H, Aut(K)) = φ. Therefore X |CAut(K) (x)| = |XH | + | Aut(K)||L(H, Aut(K))| x∈H X + |CAut(K) (x)|. x∈H\(XH ∪L(H,Aut(K))) For x ∈ H \ (XH ∪ L(H, Aut(K))) we have {I} = 6 CAut(K) (x) 6= Aut(K) | Aut(K)| which implies p ≤ |CAut(K) (x)| ≤ . Therefore p X |CAut(K) (x)| ≥|XH | + | Aut(K)||L(H, Aut(K))| x∈H + p(|H| − |XH | − |L(H, Aut(K))|) 7 (3.3) and X |CAut(K) (x)| ≤|XH | + | Aut(K)||L(H, Aut(K))| x∈H + | Aut(K)|(|H| − |XH | − |L(H, Aut(K))|) . p (3.4) Hence, the result follows from Corollary 2.4, (3.3) and (3.4). We have the following two corollaries. Corollary 3.8. Let H and K be two subgroups of a finite group G such that H ⊆ K. If p and q are the smallest primes dividing | Aut(K)| and |H| respectively then p+q−1 . Pr(H, Aut(K)) ≤ pq In particular, if p = q then Pr(H, Aut(K)) ≤ 2p−1 p2 ≤ 43 . Proof. Since H 6= L(H, Aut(K)) we have |H : L(H, Aut(K))| ≥ q. Therefore, by Theorem 3.7, we have   1 p−1 p+q−1 Pr(H, Aut(K)) ≤ +1 ≤ . p |H : L(H, Aut(K))| pq Corollary 3.9. Let H and K be two subgroups of a finite group G such that H ⊆ K and p, q be the smallest primes dividing | Aut(K)| and |H| respectively. If H is non-abelian then Pr(H, Aut(K)) ≤ q2 + p − 1 . pq 2 In particular, if p = q then Pr(H, Aut(K)) ≤ p2 +p−1 p3 ≤ 58 . Proof. Since H is non-abelian we have |H : L(H, Aut(K))| ≥ q 2 . Therefore, by Theorem 3.7, we have   1 p−1 q2 + p − 1 Pr(H, Aut(K)) ≤ . +1 ≤ p |H : L(H, Aut(K))| pq 2 Now we obtain two lower bounds analogous to the lower bounds obtained in [9, Theorem A] and [8, Theorem 1]. Theorem 3.10. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then   1 |S(H, Aut(K))| − 1 Pr(H, Aut(K)) ≥ 1+ . |S(H, Aut(K))| |H : L(H, Aut(K))| The equality holds if and only if orbK (x) = xS(H, Aut(K)) for all x ∈ H \ L(H, Aut(K)). 8 Proof. For all x ∈ H \ L(H, Aut(K)) we have α(x) = x[x, α] ∈ xS(H, Aut(K)). Therefore orbK (x) ⊆ xS(H, Aut(K)) and so | orbK (x)| ≤ |S(H, Aut(K))| for all x ∈ H \ L(H, Aut(K)). Now, by Corollary 2.4, we have   X X 1  1 1  Pr(H, Aut(K)) = + |H| | orbK (x)| | orbK (x)| x∈L(H,Aut(K)) x∈H\L(H,Aut(K)) 1 |L(H, Aut(K))| + ≥ |H| |H| X x∈H\L(H,Aut(K)) 1 . |S(H, Aut(K))| Hence, the result follows. Lemma 3.11. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then, for any two integers m ≥ n, we have     n−1 1 m−1 1 1+ ≥ 1+ . n |H : L(H, Aut(K))| m |H : L(H, Aut(K))| If L(H, Aut(K)) 6= H then equality holds if and only if m = n. Proof. The proof is an easy exercise. Corollary 3.12. Let H and K be two subgroups of a finite group G such that H ⊆ K. Then   1 |[H, Aut(K)]| − 1 Pr(H, Aut(K)) ≥ 1+ . |[H, Aut(K)]| |H : L(H, Aut(K))| If H 6= L(H, Aut(K)) then the equality holds if and only if [H, Aut(K)] = S(H, Aut(K)) and orbK (x) = x[H, Aut(K)] for all x ∈ H \ L(H, Aut(K)). Proof. Since |[H, Aut(K)]| ≥ |S(H, Aut(K))|, the result follows from Theorem 3.10 and Lemma 3.11. Note that the equality holds if and only if equality holds in Theorem 3.10 and Lemma 3.11. It is worth mentioning that Theorem 3.10 gives better lower bound than the lower bound given by Corollary 3.12. Also   1 |[H, Aut(K)]| − 1 |L(H, Aut(K))| 1+ ≥ |[H, Aut(K)]| |H : L(H, Aut(K))| |H| p(|H| − |L(H, Aut(K))|) . + |H|| Aut(K)| Hence, Theorem 3.10 gives better lower bound than the lower bound given by Theorem 3.7. 9 4 A few Characterizations In this section, we obtain some characterizations of a subgroup H of G if equality holds in Corollary 3.8 and Corollary 3.9. We begin with the following result. Theorem 4.1. Let H and K be two subgroups of a finite group G such that H ⊆ K. If Pr(H, Aut(K)) = p+q−1 for some primes p and q. Then pq divides pq |H|| Aut(K)|. Further, if p and q are the smallest primes dividing | Aut(K)| and |H| respectively, then H ∼ = Zq . L(H, Aut(K)) In particular, if H and Aut(K) are of even order and Pr(H, Aut(K)) = H ∼ L(H,Aut(K)) = Z2 . 3 4 then Proof. By (1.1), we have (p + q − 1)|H|| Aut(K)| = pq|{(x, α) ∈ H × Aut(K) : [x, α] = 1}|. Therefore, pq divides |H|| Aut(K)|. If p and q are the smallest primes dividing | Aut(K)| and |H| respectively then, by Theorem 3.7, we have   p−1 1 p+q−1 ≤ +1 pq p |H : L(H, Aut(K))| which gives |H : L(H, Aut(K))| ≤ q. Hence, H L(H,Aut(K)) ∼ = Zq . Theorem 4.2. Let H ⊆ K be two subgroups of a finite group G such that H 2 is non-abelian and Pr(H, Aut(K)) = q +p−1 for some primes p and q. Then pq2 pq divides |H|| Aut(K)|. Further, if p and q are the smallest primes dividing | Aut(K)| and |H| respectively then H ∼ = Zq × Zq . L(H, Aut(K)) In particular, if H and Aut(K) are of even order and Pr(H, Aut(K)) = H ∼ L(H,Aut(K)) = Z2 × Z2 . 5 8 then Proof. By (1.1), we have (q 2 + p − 1)|H|| Aut(K)| = pq 2 |{(x, α) ∈ H × Aut(K) : [x, α] = 1}|. Therefore, pq divides |H|| Aut(K)|. If p and q are the smallest primes dividing | Aut(K)| and |H| respectively then, by Theorem 3.7, we have   q2 + p − 1 p−1 1 ≤ + 1 pq 2 p |H : L(H, Aut(K))| which gives |H : L(H, Aut(K))| ≤ q 2 . Since H is non-abelian we have |H : H ∼ L(H, Aut(K))| 6= 1, q. Hence, L(H,Aut(K)) = Zq × Zq . 10 The following two results give partial converses of Theorem 4.1 and 4.2 respectively. Proposition 4.3. Let H and K be two subgroups of a finite group G such that H ⊆ K. Let p, q be the smallest prime divisors of | Aut(K)|, |H| respectively and | Aut(K) : CAut(K) (x)| = p for all x ∈ H \ L(H, Aut(K)). (a) If H L(H,Aut(K)) ∼ = Zq then Pr(H, Aut(K)) = (b) If H L(H,Aut(K)) ∼ = Zq × Zq then Pr(H, Aut(K)) = p+q−1 pq . q2 +p−1 pq2 . Proof. Since | Aut(K) : CAut(K) (x)| = p for all x ∈ H \ L(H, Aut(K)) we have |CAut(K) (x)| = | Aut(K)| for all x ∈ H \ L(H, Aut(K)). Therefore, by Corollary p 2.4, we have Pr(H, Aut(K)) = = |L(H, Aut(K))| 1 + |H| |H|| Aut(K)| Pr(H, Aut(K)) = (b) If 5 H L(H,Aut(K)) H L(H,Aut(K)) |CAut(K) (x)| x∈H\L(H,Aut(K)) |L(H, Aut(K))| |H| − |L(H, Aut(K))| + . |H| p|H| Thus (a) If X 1 p   p−1 +1 . |H : L(H, Aut(K))| (4.1) ∼ = Zq then (4.1) gives Pr(H, Aut(K)) = p+q−1 pq . 2 ∼ = Zq × Zq then (4.1) gives Pr(H, Aut(K)) = q +p−1 pq2 . Autoisoclinic pairs In the year 1940, Hall [4] introduced the concept of isoclinism between two groups. Following Hall, Moghaddam et al. [7] have defined autoisoclinism between two groups, in the year 2013. Recall that two groups G1 and G2 G2 G1 → L(G , are said to be autoisoclinic if there exist isomorphisms ψ : L(G 1) 2) β : [G1 , Aut(G1 )] → [G2 , Aut(G2 )] and γ : Aut(G1 ) → Aut(G2 ) such that the following diagram commutes G1 L(G1 ) ψ×γ × Aut(G1 ) −−−−→  a(G ,Aut(G )) 1 y 1 [G1 , Aut(G1 )] β −−−−→ G2 L(G2 ) × Aut(G2 )  a(G ,Aut(G )) 2 y 2 [G2 , Aut(G2 )] Gi × Aut(Gi ) → [Gi , Aut(Gi )], for i = 1, 2, where the maps a(Gi ,Aut(Gi )) : L(G i) are given by a(Gi ,Aut(Gi )) (xi L(Gi ), αi ) = [xi , αi ]. Such a pair (ψ × γ, β) is called an autoisoclinism between the groups G1 and G2 . We generalize the notion of autoisoclinism in the following way: 11 Let H1 , K1 and H2 , K2 be subgroups of the groups G1 and G2 respectively. The pairs of subgroups (H1 , K1 ) and (H2 , K2 ) such that H1 ⊆ K1 and H2 ⊆ K2 H1 are said to be autoisoclinic if there exist isomorphisms ψ : L(H1 ,Aut K1 ) → H2 L(H2 ,Aut(K2 )) , β : [H1 , Aut(K1 )] → [H2 , Aut(K2 )] and γ : Aut(K1 ) → Aut(K2 ) such that the following diagram commutes H1 L(H1 ,Aut(K1 )) ψ×γ × Aut(K1 ) −−−−→  a(H ,Aut(K )) 1 y 1 [H1 , Aut(K1 )] where the maps a(Hi ,Aut(Ki )) : i = 1, 2, are given by H2 L(H2 ,Aut(K2 )) β −−−−→ Hi L(Hi ,Aut(Ki )) × Aut(K2 )  a(H ,Aut(K )) 2 y 2 [H2 , Aut(K2 )] × Aut(Ki ) → (Hi , Aut(Ki )), for a(Hi ,Aut(Ki )) (xi L(Hi , Aut(Ki )), αi ) = [xi , αi ]. Such a pair (ψ × γ, β) is said to be an autoisoclinism between the pairs of groups (H1 , K1 ) and (H2 , K2 ). We conclude this paper with the following generalization of [3, Theorem 5.1] and [10, Lemma 2.5]. Theorem 5.1. Let G1 and G2 be two finite groups with subgroups H1 , K1 and H2 , K2 respectively such that H1 ⊆ K1 and H2 ⊆ K2 . If (ψ × γ, β) is an autoisoclinism between the pairs (H1 , K1 ) and (H2 , K2 ) then, for g ∈ K1 , Prg (H1 , Aut(K1 )) = Prβ(g) (H2 , Aut(K2 )). H1 × Proof. Let us consider the sets Sg = {(x1 L(H1 , Aut(K1 )), α1 ) ∈ L(H1 ,Aut(K 1 )) Aut(K1 ) : [x1 L(H1 , Aut(K1 )), α1 ] = g} and Tβ(g) = {(x2 L(H2 , Aut(K2 )), α2 ) ∈ H2 L(H2 ,Aut(K2 )) × Aut(K2 ) : [x2 L(H2 , Aut(K2 )), α2 ] = β(g)}. Since (H1 , K1 ) is autoisoclinic to (H2 , K2 ) we have |Sg | = |Tβ(g) |. Again, it is clear that |{(x1 , α1 ) ∈ H1 × Aut(K1 ) : [x1 , α1 ] = g}| = |L(H1 , Aut(K1 ))||Sg | (5.1) and |{(x2 , α2 ) ∈ H2 × Aut(K2 ) : [x2 , α2 ] = β(g)}| = |L(H2 , Aut(K2 ))||Tβ(g) |. (5.2) Hence, the result follows from (1.1), (5.1) and (5.2). References [1] C. J. Hillar and D. L. Rhea, Automorphism of finite abelian groups, Amer. Math. Monthly, 114(10), 917–923 (2007). [2] A. K. Das and R. K. Nath, On generalized relative commutativity degree of a finite group, Int. Electron. J. Algebra, 7, 140–151 (2010). 12 [3] P. Dutta and R. K. Nath, Autocommuting probabilty of a finite group, preprint. [4] P. Hall, The classification of prime power groups, J. Reine Angew. Math., 182, 130–141 (1940). [5] P. V. Hegarty, The absolute centre of a group, J. Algebra, 169(3), 929–935 (1994). [6] M. R. R. Moghaddam, F. Saeedi and E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-Eur. J. Math. 4(2), 301308 (2011). [7] M. R. R. Moghaddam, M. J. Sadeghifard and M. Eshrati, Some properties of autoisoclinism of groups, Fifth International group theory conference, Islamic Azad University, Mashhad, Iran, 13-15 March 2013. [8] R. K. Nath and A. K. Das, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo, 59(1), 137–142 (2010). [9] R. K. Nath and M. K. Yadav, Some results on relative commutativity degree, Rend. Circ. Mat. Palermo, 64(2), 229–239 (2015). [10] M. R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacet. J. Math. Stat. 44(4), 893–899 (2015). [11] G. J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly, 82, 261–264 (1975). 13 

arXiv:cs/0601038v1 [cs.LO] 10 Jan 2006 Under consideration for publication in Theory and Practice of Logic Programming 1 Constraint-based Automatic Verification of Abstract Models of Multithreaded Programs GIORGIO DELZANNO Dipartimento di Informatica e Scienze dell’Informazione, Università di Genova via Dodecaneso 35, 16146 Genova - Italy (e-mail: giorgio@disi.unige.it) submitted 17 December 2003; revised 13 April 2005; accepted 15 January 2006 Abstract We present a technique for the automated verification of abstract models of multithreaded programs providing fresh name generation, name mobility, and unbounded control. As high level specification language we adopt here an extension of communication finitestate machines with local variables ranging over an infinite name domain, called TDL programs. Communication machines have been proved very effective for representing communication protocols as well as for representing abstractions of multithreaded software. The verification method that we propose is based on the encoding of TDL programs into a low level language based on multiset rewriting and constraints that can be viewed as an extension of Petri Nets. By means of this encoding, the symbolic verification procedure developed for the low level language in our previous work can now be applied to TDL programs. Furthermore, the encoding allows us to isolate a decidable class of verification problems for TDL programs that still provide fresh name generation, name mobility, and unbounded control. Our syntactic restrictions are in fact defined on the internal structure of threads: In order to obtain a complete and terminating method, threads are only allowed to have at most one local variable (ranging over an infinite domain of names). KEYWORDS: Constraints, Multithreaded Programs, Verification. 1 Introduction Andrew Gordon (Gordon 2001) defines a nominal calculus to be a computational formalism that includes a set of pure names and allows the dynamic generation of fresh, unguessable names. A name is pure whenever it is only useful for comparing for identity with other names. The use of pure names is ubiquitous in programming languages. Some important examples are memory pointers in imperative languages, identifiers in concurrent programming languages, and nonces in security protocols. In addition to pure names, a nominal process calculus should provide mechanisms for concurrency and inter-process communication. A computational model that provides all these features is an adequate abstract formalism for the analysis of multithreaded and distributed software. 2 Giorgio Delzanno The Problem Automated verification of specifications in a nominal process calculus becomes particularly challenging in presence of the following three features: the possibility of generating fresh names (name generation); the possibility of transmitting names (name mobility); the possibility of dynamically adding new threads of control (unbounded control). In fact, a calculus that provides all the previous features can be used to specify systems with a state-space infinite in several dimensions. This feature makes difficult (if not impossible) the application of finite-state verification techniques or techniques based on abstractions of process specifications into Petri Nets or CCS-like models. In recent years there have been several attempts of extending automated verification methods from finite-state to infinite-state systems (Abdulla and Nylén 2000; Kesten et al. 2001). In this paper we are interested in investigating the possible application of the methods we proposed in (Delzanno 2001) to verification problems of interest for nominal process calculi. Constraint-based Symbolic Model Checking In (Delzanno 2001) we introduced a specification language, called MSR(C), for the analysis of communication protocols whose specifications are parametric in several dimensions (e.g. number of servers, clients, and tickets as in the model of the ticket mutual exclusion algorithm shown in (Bozzano and Delzanno 2002)). MSR(C) combines multiset rewriting over first order atomic formulas (Cervesato et al. 1999) with constraints programming. More specifically, multiset rewriting is used to specify the control part of a concurrent system, whereas constraints are used to symbolically specify the relations over local data. The verification method proposed in (Delzanno 2005) allows us to symbolically reason on the behavior of MSR(C) specifications. To this aim, following (Abdulla et al. 1996; Abdulla and Nylén 2000) we introduced a symbolic representation of infinite collections of global configurations based on the combination of multisets of atomic formulas and constraints, called constrained configurations.1 The verification procedure performs a symbolic backward reachability analysis by means of a symbolic pre-image operator that works over constrained configurations (Delzanno 2005). The main feature of this method is the possibility of automatically handling systems with an arbitrary number of components. Furthermore, since we use a symbolic and finite representation of possibly infinite sets of configurations, the analysis is carried out without loss of precision. A natural question for our research is whether and how these techniques can be used for verification of abstract models of multithreaded programs. Our Contribution In this paper we propose a sound, and fully automatic verification method for abstract models of multithreaded programs that provide name generation, name mobility, and unbounded control. As a high level specification language we adopt here an extension with value-passing of the formalism of (Ball et al. 2001) 1 Notice that in (Abdulla et al. 1996; Abdulla and Nylén 2000) a constraint denotes a symbolic state whereas we use the word constraint to denote a symbolic representation of the relation of data variables (e.g. a linear arithmetic formula) used as part of the symbolic representation of sets of states (a constrained configuration). Constraint-based Verification of Abstract Multithreaded Programs 3 based on families of state machines used to specify abstractions of multithreaded software libraries. The resulting language is called Thread Definition Language (TDL). This formalism allows us to keep separate the finite control component of a thread definition from the management of local variables (that in our setting range over a infinite set of names), and to treat in isolation the operations to generate fresh names, to transmit names, and to create new threads. In the present paper we will show that the extension of the model of (Ball et al. 2001) with value-passing makes the model Turing equivalent. The verification methodology is based on the encoding of TDL programs into a specification in the instance MSRN C of the language scheme MSR(C) of(Delzanno 2001). MSRN C is obtained by taking as constraint system a subclass of linear arithmetics with only = and > relations between variables, called name constraints (N C). The low level specification language MSRN C is not just instrumental for the encoding of TDL programs. Indeed, it has been applied to model consistency and mutual exclusion protocols in (Bozzano and Delzanno 2002; Delzanno 2005). Via this encoding, the verification method based on symbolic backward reachability obtained by instantiating the general method for MSR(C) to NC-constraints can now be applied to abstract models of multithreaded programs. Although termination is not guaranteed in general, the resulting verification method can succeed on practical examples as the Challenge-Response TDL program defined over binary predicates we will illustrated in the present paper. Furthermore, by propagating the sufficient conditions for termination defined in (Bozzano and Delzanno 2002; Delzanno 2005) back to TDL programs, we obtain an interesting class of decidable problems for abstract models of multithreaded programs still providing name generation, name mobility, and unbounded control. Plan of the Paper In Section 2 we present the Thread Definition Language (TDL) with examples of multithreaded programs. Furthermore, we discuss the expressiveness of TDL programs showing that they can simulate Two Counter Machines. In Section 3, after introducing the MSRN C formalism, we show that TDL programs can be simulated by MSRN C specifications. In Section 4 we show how to transfer the verification methods developed for MSR(C) to TDL programs. Furthermore, we show that safety properties can be decided for the special class of monadic TDL programs. In Section 5 we address some conclusions and discuss related work. 2 Thread Definition Language (TDL) In this section we will define TDL programs. This formalism is a natural extension with value-passing of the communicating machines used by (Ball et al. 2001) to specify abstractions of multithreaded software libraries. Terminology Let N be a denumerable set of names equipped with the relations = and 6= and a special element ⊥ such that n 6= ⊥ for any n ∈ N . Furthermore, let V be a denumerable set of variables, C = {c1 , . . . , cm } a finite set of constants, and L a finite set of internal action labels. For a fixed V ⊆ V, the set of expressions is 4 Giorgio Delzanno defined as E = V ∪ C ∪ {⊥} (when necessary we will use E(V ) to explicit the set of variables V upon which expressions are defined). The set of channel expressions is defined as Ech = V ∪ C. Channel expressions will be used as synchronization labels so as to establish communication links only at execution time. A guard over V is a conjunction γ1 , . . . , γs , where γi is either true, x = e or x 6= e with x ∈ V and e ∈ E for i : 1, . . . , s. An assignment α from V to W is a conjunction like xi := ei where xi ∈ W , ei ∈ E(V ) for i : 1, . . . k and xr 6= xs for r 6= s. A message template m over V is a tuple m = hx1 , . . . , xu i of variables in V . Definition 1 A TDL program is a set T = {P1 , . . . , Pt } of thread definitions (with distinct names for local variables control locations). A thread definition P is a tuple hQ, s0 , V, Ri, where Q is a finite set of control locations, s0 ∈ Q is the initial location, V ⊆ V is a finite set of local variables, and R is a set of rules. Given s, s′ ∈ Q, and a ∈ L, a rule has one of the following forms2 : a • Internal move: s −−→ s′ [γ, α], where γ is a guard over V , and α is an assignment from V to V ; a • Name generation: s −−→ s′ [x := new], where x ∈ V , and the expression new denotes a fresh name; a • Thread creation: s −−→ s′ [run P ′ with α], where P ′ = hQ′ , t, W, R′ i ∈ T , and α is an assignment from V to W that specifies the initialization of the local variables of the new thread; e!m • Message sending: s −−→ s′ [γ, α], where e is a channel expression, m is a message template over V that specify which names to pass, γ is a guard over V , and α is an assignment from V to V . e?m • Message reception: s −−−→ s′ [γ, α], where e is a channel expression, m is a message template over a new set of variables V ′ (V ′ ∩ V = ∅) that specifies the names to receive, γ is a guard over V ∪ V ′ and α is an assignment from V ∪ V ′ to V . Before giving an example, we will formally introduce the operational semantics of TDL programs. 2.1 Operational Semantics In the following we will use N to indicate the subset of used names of N . Every constant c ∈ C is mapped to a distinct name nc 6= ⊥ ∈ N , and ⊥ is mapped to ⊥. Let P = hQ, s, V, Ri and V = {x1 , . . . , xk }. A local configuration is a tuple p = hs′ , n1 , . . . , nk i where s′ ∈ Q and ni ∈ N is the current value of the variable xi ∈ V for i : 1, . . . , k. A global configuration G = hN, p1 , . . . , pm i is such that N ⊆ N and p1 , . . . , pm are local configurations defined over N and over the thread definitions in T . Note that 2 In this paper we keep assignments, name generation, and thread creation separate in order to simplify the presentation of the encoding into MSR. Constraint-based Verification of Abstract Multithreaded Programs 5 there is no relation between indexes in a global configuration in G and in T ; G is a pool of active threads, and several active threads can be instances of the same thread definition. Given a local configuration p = hs′ , n1 , . . . , nk i, we define the valuation ρp as ρp (xi ) = ni if xi ∈ V , ρp (c) = nc if c ∈ C, and ρp (⊥) = ⊥. Furthermore, we say that ρp satisfies the guard γ if ρp (γ) ≡ true, where ρp is extended to constraints in the natural way (ρp (ϕ1 ∧ ϕ2 ) = ρp (ϕ1 ) ∧ ρp (ϕ2 ), etc.). The execution of x := e has the effect of updating the local variable x of a thread with the current value of e (a name taken from the set of used values N ). On the contrary, the execution of x := new associates a fresh unused name to x. The formula run P with α has the effect of adding a new thread (in its initial control location) to the current global configuration. The initial values of the local variables of the generated thread are determined by the execution of α whose source variables are the local variables of the parent thread. The channel names used in a rendez-vous are determined by evaluating the channel expressions tagging sender and receiver rules. Value passing is achieved by extending the evaluation associated to the current configuration of the receiver so as to associate the output message of the sender to the variables in the input message template. The operational semantics is given via a binary relation ⇒ defined as follows. Definition 2 Let G = hN, . . . , p, . . .i, and p = hs, n1 , . . . , nk i be a local configuration for P = hQ, s, V, Ri, V = {x1 , . . . , xk }, then: a • If there exists a rule s −−→ s′ [γ, α] in R such that ρp satisfies γ, then G ⇒ hN, . . . , p′ , . . .i (meaning that only p changes) where p′ = hs′ , n′1 , . . . , n′k i, n′i = ρp (ei ) if xi := ei is in α, n′i = ni otherwise, for i : 1, . . . , k. a • If there exists a rule s −−→ s′ [xi := new] in R, then G ⇒ hN ′ , . . . , p′ , . . .i ′ ′ ′ ′ where p = hs , n1 , . . . , nk i, ni is an unused name, i.e., n′i ∈ N \ N , n′j = nj for every j 6= i, and N ′ = N ∪ {n′i }; a • If there exists a rule s −−→ s′ [run P ′ with α] in R with P ′ = hQ′ , t0 , W, R′ i, W = {y1 , . . . , yu }, and α is defined as y1 := e1 , . . . , yu := eu then G ⇒ hN, . . . , p′ , . . . , qi (we add a new thread whose initial local configuration is q) where p′ = hs′ , n1 , . . . , nk i, and q = ht0 , ρp (e1 ), . . . , ρp (eu )i. • Let q = ht, m1 , . . . , mr i (distinct from p) be a local configuration in G associated with P ′ = hQ′ , t0 , W, R′ i. e!m e′ ?m′ Let s −−→ s′ [γ, α] in R and t −−−−→ t′ [γ ′ , α′ ] in R′ be two rules such that m = hx1 , . . . , xu i, m′ = hy1 , . . . , yv i and u = v (message templates match). We define σ as the value passing evaluation σ(yi ) = ρp (xi ) for i : 1, . . . , u, and σ(z) = ρq (z) for z ∈ W ′ . Now if ρp (e) = ρp (e′ ) (channel names match), ρp satisfies γ, and that σ satisfies γ ′ , then hN, . . . , p, . . . , q, . . .i ⇒ hN, . . . , p′ , . . . , q′ , . . .i where p′ = hs′ , n′1 , . . . , n′k i, n′i = ρp (v) if xi := v is in α, n′i = ni otherwise for i : 1, . . . , k; q′ = ht′ , m′1 , . . . , m′r i, m′i = σ(v) if ui := v is in α′ , m′i = mi otherwise for i : 1, . . . , r. 6 Giorgio Delzanno Definition 3 An initial global configuration G0 has an arbitrary (but finite) number of threads with local variables all set to ⊥. A run is a sequence G0 G1 . . . such that Gi ⇒ Gi+1 for i ≥ 0. A global configuration G is reachable from G0 if there exists a run from G0 to G. Example 1 Let us consider a challenge and response protocol in which the goal of two agents Alice and Bob is to exchange a pair of new names hnA , nB i, the first one created by Alice and the second one created by Bob, so as to build a composed secret key. We can specify the protocol by using new names to dynamically establish private channel names between instances of the initiator and of the responder. The TDL program in Figure 1 follows this idea. The thread Init specifies the behavior of the initiator. He first creates a new name using the internal action f resh, and stores it in the local variable nA . Then, he sends nA on channel c (a constant), waits for a name y on a channel with the same name as the value of the local variable nA (the channel is specified by variable nA ) and then stores y in the local variable mA . The thread Resp specifies the behavior of the responder. Upon reception of a name x on channel c, he stores it in the local variable nB , then creates a new name stored in local variable mB and finally sends the value in mB on channel with the same name as the value of nB . The thread M ain non-deterministically creates new thread instances of type Init and Resp. The local variable x is used to store new names to be used for the creation of a new thread instance. Initially, all local variables of threads Init/Resp are set to ⊥. In order to allow process instances to participate to several sessions (potentially with different principals), we could also add the following rule restart stopA −−−−−→ initA [nA := ⊥, mA := ⊥] In this rule we require that roles and identities do not change from session to session.3 Starting from G0 = hN0 , hinit, ⊥ii, and running the Main thread we can generate any number of copies of the threads Init and Resp each one with a unique identifier. Thus, we obtain global configurations like hN, hinitM , ⊥i, hinitA , i1 , ⊥, ⊥i, . . . , hinitA , iK , ⊥, ⊥i, hinitB , iK+1 , ⊥, ⊥i, . . . , hinitB , iK+L , ⊥, ⊥i i where N = {⊥, i1, . . . , iK , iK+1 , . . . , iK+L } for K, L ≥ 0. The threads of type Init and Resp can start parallel sessions whenever created. For K = 1 and L = 1 one possible session is as follows. Starting from h{⊥, i1 , i2 }, hinitM , ⊥i, hinitA , i1 , ⊥, ⊥i, hinitB , i2 , ⊥, ⊥ii 3 By means of thread and fresh name creation it is also possible to specify a restart rule in which a given process takes a potential different role or identity. Constraint-based Verification of Abstract Multithreaded Programs 7 T hread Init(local idA , nA , mA ); f resh initA −−−−→ genA c!hnA i genA −−−−→ waitA n ?hyi waitA −−A −−−→ stopA [nA := new] [true] [mA := y] T hread Resp(local id, nB , mB ); c?hxi initB −−−→ genB [nB := x] f resh genB −−−−→ readyB n !hm i [mB := new] B readyB −−B−−−− → stopB [true] T hread M ain(local x); id initM −−→ create newA create −−−−→ initM new → initM create −−−−B [x := new] [run Init with idA := x, nA := ⊥, mA := ⊥, x := ⊥] [run Resp with idB := x, nB := ⊥, mB := ⊥, x := ⊥B ] Fig. 1. Example of thread definitions. if we apply the first rule of thread Init to hinitA , i1 , ⊥, ⊥i we obtain h{⊥, i1 , i2 , a1 }, hinitM , ⊥i, hgenA , i1 , a1 , ⊥i, hinitB , i2 , ⊥, ⊥ii where a1 is the generated name (a1 is distinct from ⊥, i1 , and i2 ). Now if we apply the second rule of thread Init and the first rule of thread Resp (synchronization on channel c) we obtain h{⊥, i1 , i2 , a1 }, hinitM , ⊥i, hwaitA , i1 , a1 , ⊥i, hgenB , i2 , a1 , ⊥ii If we apply the second rule of thread Resp we obtain h{⊥, i1 , i2 , a1 , a2 }, hinitM , ⊥i, hwaitA , i1 , a1 , ⊥i, hreadyB , i2 , a1 , a2 ii Finally, if we apply the last rule of thread Init and Resp (synchronization on channel a1 ) we obtain h{⊥, i1 , i2 , a1 , a2 }, hinitM , ⊥i, hstopA , i1 , a1 , a2 i, hstopB , i2 , a1 , a2 ii Thus, at the end of the session the thread instances i1 and i2 have both a local copy of the fresh names a1 and a2 . Note that a copy of the main thread hinitM , ⊥i is always active in any reachable configuration, and, at any time, it may introduce new threads (either of type Init or Resp) with fresh identifiers. Generation of fresh names is also used by the threads of type Init and Resp to create nonces. Furthermore, threads can restart their life cycle (without changing identifiers). Thus, in this example the set of possible reachable configurations is infinite and contains configurations with arbitrarily many threads and fresh names. Since names are stored in the local variables of active threads, the local data also range over an infinite domain. ✷ 8 Giorgio Delzanno 2.2 Expressive Power of TDL To study the expressive power of the TDL language, we will compare it with the Turing equivalent formalism called Two Counter Machines. A Two Counters Machine configurations is a tuple hℓ, c1 = n1 , c2 = n2 i where ℓ is control location taken from a finite set Q, and n1 and n2 are natural numbers that represent the values of the counters c1 and c2 . Each counter can be incremented or decremented (if greater than zero) by one. Transitions combine operations on individual counters with changes of control locations. Specifically, the instructions for counter ci are as follows Inc: ℓ1 : ci := ci + 1; goto ℓ2 ; Dec: ℓ1 : if ci > 0 then ci := ci − 1; goto ℓ2 ; else goto ℓ3 ; A Two Counter Machine consists then of a list of instructions and of the initial state hℓ0 , c1 = 0, c2 = 0i. The operational semantics is defined according to the intuitive semantics of the instructions. Problems like control state reachability are undecidable for this computational model. The following property then holds. Theorem 1 TDL programs can simulate Two Counter Machines. Proof In order to define a TDL program that simulates a Two Counter Machine we proceed as follows. Every counter is represented via a doubly linked list implemented via a collection of threads of type Cell and with a unique thread of type Last pointing to the head of the list. The i-th counter having value zero is represented as the empty list Cell(i, v, v), Last(i, v, w) for some name v and w (we will explain later the use of w). The i-th counter having value k is represented as Cell(i, v0 , v0 ), Cell(i, v0 , v1 ), . . . , C(i, vk−1 , vk ), Last(i, vk , w) for distinct names v0 , v1 , . . . , vk . The instructions on a counter are simulated by sending messages to the corresponding Last thread. The messages are sent on channel Zero (zero test), Dec (decrement), and Inc (increment). In reply to each of these messages, the thread Last sends an acknowledgment, namely Y es/N o for the zero test, DAck for the decrement, IAck for the increment operation. Last interacts with the Cell threads via the messages tstC, decC, incC acknowledged by messages z/nz, dack. iack. The interactions between a Last thread and the Cell threads is as follows. Zero Test Upon reception of a message hxi on channel Zero, the Last thread with local variables id, last, aux checks that its identifier id matches x - see transition from Idle to Busy - sends a message hid, lasti on channel tstC directed to the cell pointed to by last (transition from Busy to W ait), and then waits for an answer. If the answer is sent on channel nz, standing for non-zero, (resp. z standing for zero) - see transition from W ait to AckN Z (resp. AckZ) - then it sends its identifier on Constraint-based Verification of Abstract Multithreaded Programs 9 Thread Last(local id, last, aux); (Zero test) Zero?hxi Idle −−−−−−→ Busy [id = x] tstC!hid,lasti Busy −−−−−−−−−→ W ait nz?hxi W ait −−−−→ AckN Z [id = x] z?hxi W ait −−−→ AckZ [id = x] Y es!hidi AckZ −−−−−→ Idle No!hidi AckN Z −−−−−→ idle (Decrement) Dec?hxi Idle −−−−−→ Dbusy [id = x] decC!hid,lasti DBusy −−−−−−−−−→ DW ait dack?hx,ui DW ait −−−−−−−→ DAck [id = x, last := u] DAck!hidi DAck −−−−−−−→ Idle (Increment) Inc?hxi Idle −−−−−→ IN ew [id = x] new [aux := new] run [run Cell with idc := id; prev := last; next := aux] IN ew −−−→ IRun IRun −−→ IAck IAck!hidi IAck −−−−−−→ Idle [last := aux] Fig. 2. The process defining the last cell of the linked list associated to a counter channel N o (resp. Y es) as an acknowledgment to the first message - see transition from AckN Z (resp. Z) to Idle. As shown in Fig. 3, the thread Cell with local variables idc, prev, and next that receives the message tstC, i.e., pointed to by a thread Last with the same identifier as idc, sends an acknowledgment on channel z (zero) if prev = next, and on channel nz (non-zero) if prev 6= next. Decrement Upon reception of a message hxi on channel Dec, the Last thread with local variables id, last, aux checks that its identifier id matches x (transition from Idle to Dbusy), sends a message hid, lasti on channel decC directed to the cell pointed to by last (transition from Busy to W ait), and then waits for an answer. 10 Giorgio Delzanno Thread Cell(local idc, prev, next); (Zero test) tstC?hx,ui idle −−−−−−−→ ackZ [x = idc, u = next, prev = next] tstC?hx,ui idle −−−−−−−→ ackN Z [x = idc, u = next, prev 6= next] z!hidci ackZ −−−−→ idle nz!hidci ackN Z −−−−−→ idle (Decrement) dec?hx,ui idle −−−−−−→ dec [x = idc, u = next, prev 6= next] dack!hidc,previ dec −−−−−−−−−−→ idle Fig. 3. The process defining a cell of the linked list associated to a counter If the answer is sent on channel dack (transition from DW ait to DAck) then it updates the local variable last with the pointer u sent by the thread Cell, namely the prev pointer of the cell pointed to by the current value of last, and then sends its identifier on channel DAck to acknowledge the first message (transition from DAck to Idle). As shown in Fig. 3, a thread Cell with local variables idc, prev, and next that receives the message decC and such that next = last sends as an acknowledgment on channel dack the value prev. Increment To simulate the increment operation, Last does not have to interact with existing Cell threads. Indeed, it only has to link a new Cell thread to the head of the list (this is way the Cell thread has no operations to handle the increment operation). As shown in Fig. 2 this can be done by creating a new name stored in the local variable aux (transition from IN ew to IRun) and spawning a new Cell thread (transition from IRun to IAck) with prev pointer equal to last, and next pointer equal to aux. Finally, it acknowledges the increment request by sending its identifier on channel IAck and updates variable last with the current value of aux. Two Counter Machine Instructions We are now ready to use the operations provided by the thread Last to simulate the instructions of a Two Counter Machine. As shown in Fig. 4, we use a thread CM with two local variables id1 , id2 to represent the list of instructions of a 2CM with counters c1 , c2 . Control locations of the Two Counter Machines are used as local states of the thread CM . The initial local state of the CM thread is the initial control location. The increment instruction on counter ci at control location ℓ1 is simulated by an handshaking with the Last thread with identifier idi : we first send the message Inc!hidi i, wait for the acknowledgment on channel IAck and then move to state ℓ2 . Similarly, for the decrement Constraint-based Verification of Abstract Multithreaded Programs 11 Thread CM (local id1 , id2 ); .. . (Instruction : ℓ1 : ci := ci + 1; goto ℓ2 ; ) Inc!hid i ℓ1 −−−−−−i→ waitℓ1 IAck!hxi waitℓ1 −−−−−−→ ℓ2 [x = idi ] .. . (Instruction : ℓ1 : ci > 0 then ci := ci − 1; goto ℓ2 ; else goto ℓ3 ; ) Zero!hid i ℓ1 −−−−−−−i→ waitℓ1 NZAck?hxi waitℓ1 − −−−−−−− → decℓ1 [x = idi ] Dec!hid i decℓ1 −−−−−−i→ wdecℓ1 DAck?hyi wdecℓ1 −−−−−−→ ℓ2 [y = idi ] ZAck?hxi waitℓ1 −−−−−−→ ℓ3 [x = idi ] .. . Fig. 4. The thread associated to a 2CM. Thread Init(local nid1 , p1 , nid2 , p2 ); f reshId init −−−−−→ init1 [nid1 := new] f reshP init1 −−−−−→ init2 [p1 := new] runC init2 − −−−→ init3 [run Cell with idc := nid1 ; prev := p1 ; next := p1 ] runL init3 −−−→ init4 [run Last with idc := nid1 ; last := p1 ; aux := ⊥] f reshId init4 −−−−−→ init5 [nid2 := new] f reshP init5 −−−−−→ init6 [p2 := new] runC init6 − −−−→ init7 [run Cell with idc := nid2 ; prev := p2 ; next := p2 ] runL init7 −−−→ init8 [run Last with idc := nid2 ; last := p2 ; aux := ⊥] runCM init8 −−−−−→ init9 [run 2CM with id1 := nid1 ; id2 := nid2 ] Fig. 5. The initialization thread. instruction on counter ci at control location ℓ1 we first send the message Zero!hidi i. If we receive an acknowledgment on channel N ZAck we send a Dec request, wait for completion and then move to ℓ2 . If we receive an acknowledgment on channel ZAck we directly move to ℓ3 . 12 Giorgio Delzanno Initialization The last step of the encoding is the definition of the initial state of the system. For this purpose, we use the thread Init of Fig. 5. The first four rules of Init initialize the first counter: they create two new names nid1 (an identifier for counter c1 ) and p1 , and then spawn the new threads Cell(nid1 , p1 , p1 ), Last(nid1 , p1 , ⊥). The following four rules spawns the new threads Cell(nid2 , p2 , p2 ), Last(nid2 , p2 , ⊥). After this stage, we create a thread of type 2CM to start the simulation of the instructions of the Two Counter Machines. The initial configuration of the whole system is G0 = hinit, ⊥, ⊥i. By construction we have that an execution step from hℓ1 , c1 = n1 , c2 = n2 i to hℓ2 , c1 = m1 , c2 = m2 i is simulated by an execution run going from a global configuration in which the local state of thread CM is hℓ1 , id1 , id2 i and in which we have ni occurrences of thread Cell with the same identifier idi for i : 1, 2, to a global configuration in which the local state of thread CM is hℓ2 , id1 , id2 i and in which we have mi occurrences of thread Cell with the same identifier idi for i : 1, 2. Thus, every executions of a 2CM M corresponds to an execution of the corresponding TDL program that starts from the initial configuration G0 = hinit, ⊥, ⊥i. As a consequence of the previous theorem, we have the following corollary. Corollary 1 Given a TDL program, a global configurations G, and a control location ℓ, deciding if there exists a run going from G0 to a global configuration that contains ℓ (control state reachability) is an undecidable problem. 3 From TDL to MSRN C As mentioned in the introduction, our verification methodology is based on a translation of TDL programs into low level specifications given in MSRN C . Our goal is to extend the connection between CCS and Petri Nets (German and Sistla 1992) to TDL and MSR so as to be able to apply the verification methods defined in (Delzanno 2005) to multithreaded programs. In the next section we will summarize the main features of the language MSRN C introduced in (Delzanno 2001). 3.1 Preliminaries on MSRN C N C-constraints are linear arithmetic constraints in which conjuncts have one of the following form: true, x = y, x > y, x = c, or x > c, x and y being two variables from a denumerable set V that range over the rationals, and c being an integer. The solutions Sol of a constraint ϕ are defined as all evaluations (from V to Q) that satisfy ϕ. A constraint ϕ is satisfiable whenever Sol(ϕ) 6= ∅. Furthermore, ψ entails ϕ whenever Sol(ψ) ⊆ Sol(ϕ). N C-constraints are closed under elimination of existentially quantified variables. Let P be a set of predicate symbols. An atomic formula p(x1 , . . . , xn ) is such that p ∈ P, and x1 , . . . , xn are distinct variables in V. A multiset of atomic formulas is indicated as A1 | . . . | Ak , where Ai and Aj have distinct variables (we use variable renaming if necessary), and | is the multiset constructor. Constraint-based Verification of Abstract Multithreaded Programs 13 In the rest of the paper we will use M, N , . . . to denote multisets of atomic formulas, ǫ to denote the empty multiset, ⊕ to denote multiset union and ⊖ to denote multiset difference. An MSRN C configuration is a multiset of ground atomic formulas, i.e., atomic formulas like p(d1 , . . . , dn ) where di is a rational for i : 1, . . . , n. An MSRN C rule has the form M −→ M′ : ϕ, where M and M′ are two (possibly empty) multisets of atomic formulas with distinct variables built on predicates in P, and ϕ is an N C-constraint. The ground instances of an MSRN C rule are defined as Inst(M −→ M′ : ϕ) = {σ(M) −→ σ(M′ ) | σ ∈ Sol(ϕ)} where σ is extended in the natural way to multisets, i.e., σ(M) and σ(M′ ) are MSRN C configurations. An MSRN C specification S is a tuple hP, I, Ri, where P is a finite set of predicate symbols, I is finite a set of (initial) MSRN C configurations, and R is a finite set of MSRN C rules over P. The operational semantics describes the update from a configuration M to one of its possible successor configurations M′ . M′ is obtained from M by rewriting (modulo associativity and commutativity) the left-hand side of an instance of a rule into the corresponding right-hand side. In order to be fireable, the left-hand side must be included in M. Since instances and rules are selected in a non deterministic way, in general a configuration can have a (possibly infinite) set of (one-step) successors. Formally, a rule H −→ B : ϕ from R is enabled at M via the ground substitution σ ∈ Sol(ϕ) if and only if σ(H) 4 M. Firing rule R enabled at M via σ yields the new configuration M′ = σ(B) ⊕ (M ⊖ σ(H)) We use M ⇒MSR M′ to denote the firing of a rule at M yielding M′ . A run is a sequence of configurations M0 M1 . . . Mk with M0 ∈ I such that Mi ⇒MSR Mi+1 for i ≥ 0. A configuration M is reachable if there exists M0 ∈ I ∗ ∗ such that M0 ⇒MSR M, where ⇒MSR is the transitive closure of ⇒MSR . Finally, the successor and predecessor operators P ost and P re are defined on a set of configurations S as P ost(S) = {M′ |M ⇒MSR M′ , M ∈ S} and P re(S) = {M|M ⇒MSR M′ , M′ ∈ S}, respectively. P re∗ and P ost∗ denote their transitive closure. As shown in (Delzanno 2001; Bozzano and Delzanno 2002), Petri Nets represent a natural abstractions of MSRN C (and more in general of MSR rule with constraints) specifications. They can be encoded, in fact, in propositional MSR specifications (e.g. abstracting away arguments from atomic formulas). 3.2 Translation from TDL to MSRN C The first thing to do is to find an adequate representation of names. Since all we need is a way to distinguish old and new names, we just need an infinite domain in which the = and 6= relation are supported. Thus, we can interpret names in N 14 Giorgio Delzanno either as integer of as rational numbers. Since operations like variable elimination are computationally less expensive than over integers, we choose to view names as non-negative rationals. Thus, a local (TDL) configuration p = hs, n1 , . . . , nk i is encoded as the atomic formula p• = s(n1 , . . . , nk ), where ni is a non-negative rational. Furthermore, a global (TDL) configuration G = hN, p1 , . . . , pm i is encoded as an MSRN C configuration G• p•1 | . . . | p•m | f resh(n) where the value n in the auxiliary atomic formula f resh(n) is an rational number strictly greater than all values occurring in p•1 , . . . , p•m . The predicate f resh will allow us to generate unused names every time needed. The translation of constants C = {c1 , . . . , cm }, and variables is defined as follows: x• = x for x ∈ V, ⊥• = 0, c•i = i for i : 1, . . . , m. We extend ·• in the natural way on a guard γ, by decomposing every formula x 6= e into x < e• and x > e• . We will call γ • the resulting set of N C-constraints.4 Given V = {x1 , . . . , xk }, we define V ′ as the set of new variables {x′1 , . . . , x′k }. Now, let us consider the assignment α defined as x1 := e1 , . . . , xk := ek (we add assignments like xi := xi if some variable does not occur as target of α). Then, α• is the N C-constraint x′1 = e•1 , . . . , x′k = e•k . The translation of thread definitions is defined below (where we will often refer to Example 1). Initial Global Configuration Given an initial global configuration consisting of the local configurations hsi , ni1 , . . . , niki i with nij = ⊥ for i : 1, . . . , u, we define the following MSRN C rule init → s1 (x11 , . . . , x1k1 ) | . . . | su (xu1 , . . . , xuku ) | f resh(x) : x > C, x11 = 0, . . . , xuku = 0 here C is the largest rational used to interpret the constants in C. For each thread definition P = hQ, s0 , V, Ri in T with V = {x1 , . . . , xk } we translate the rules in R as described below. a Internal Moves For every internal move s −−→ s′ [γ, α], and every ν ∈ γ • we define s(x1 , . . . , xk ) → s′ (x′1 , . . . , x′k ) : ν, α• a Name Generation For every name generation s −−→ s′ [xi := new], we define ^ s(x1 , . . . , xk ) | f resh(x) → s′ (x′1 , . . . , x′k ) | f resh(y) : y > x′i , x′i > x, x′j = xj j6=i f resh For instance, the name generation initA −−−−→ genA [n := new] is mapped into the MSRN C rule initA (id, x, y)| f resh(u) −−→ genA (id′ , x′ , y ′ ) | f resh(u′ ) : ϕ where ϕ 4 As an example, if γ is the constraint x = 1, x 6= z then γ • consists of the two constraints x = 1, x > z and x = 1, z > x. Constraint-based Verification of Abstract Multithreaded Programs 15 is the constraint u′ > x′ , x′ > u, y ′ = y, id′ = id. The constraint x′ > u represents the fact that the new name associated to the local variable n (the second argument of the atoms representing the thread) is fresh, whereas u′ > x′ updates the current value of f resh to ensure that the next generated names will be picked up from unused values. Thread Creation Let P = hQ′ , t0 , V ′ , R′ i and V ′ = {y1 , . . . , yu }. Then, for every a thread creation s −−→ s′ [run P with α], we define s(x1 , . . . , xk ) → s′ (x′1 , . . . , x′k ) | t(y1′ , . . . , yu′ ) : x′1 = x1 , . . . , x′k = xk , α• . new A initM [run Init with id := x, . . .] of Example 1. E.g., consider the rule create −−−−→ Its encoding yields the MSRN C rule create(x) −−→ initM (x′ ) | initA (id′ , n′ , m′ ) : ψ, where ψ represents the initialization of the local variables of the new thread x′ = x, id′ = x, n′ = 0, m′ = 0. Rendez-vous The encoding of rendez-vous communication is based on the use of constraint operations like variable elimination. Let P and P ′ be a pair of thread definitions, with local variables V = {x1 , . . . , xk } and V ′ = {y1 , . . . , yl } with V ∩ e′ ?m′ e!m V ′ = ∅. We first select all rules s −−→ s′ [γ, α] in R and t −−−−→ t′ [γ ′ , α′ ] in R′ , such that m = hw1 , . . . , wu i, m′ = hw1′ , . . . , wv′ i and u = v. Then, we define the new MSRN C rule s(x1 , . . . , xk ) | t(y1 , . . . , yl ) → s′ (x′1 , . . . , x′k ) | t′ (y1′ , . . . , yl′ ) : ϕ for every ν ∈ γ • and ν ′ ∈ γ ′• such that the NC-constraint ϕ obtained by eliminating w1′ , . . . , wv′ from the constraint ν ∧ ν ′ ∧ α• ∧ α′• ∧ w1 = w1′ ∧ . . . ∧ wv = wv′ nA ?hyi is satisfiable. For instance, consider the rules waitA −−−−−→ stopA [mA := y] and nB !hmB i readyB −−−−−−→ stopB [true]. We first build up a new constraint by conjoining the NC-constraints y = mB (matching of message templates), and nA = nB , m′A = y, n′A = nA , m′B = mB , n′B = nB , id′1 = id1 , id′2 = id2 (guards and actions of sender and receiver). After eliminating y we obtain the constraint ϕ defined as nB = nA , m′A = mB , n′A = nA , m′B = mB , n′B = nB , id′1 = id1 , id′2 = id2 defined over the variables of the two considered threads. This step allows us to symbolically represent the passing of names. After this step, we can represent the synchronization of the two threads by using a rule that simultaneously rewrite all instances that satisfy the constraints on the local data expressed by ϕ, i.e., we obtain the rule waitA (id1 , nA , mA )| readyB (id2 , nB , mB ) −→ stopA (id′1 , n′A , m′A ) | stopB (id′2 , n′B , m′B ) : ϕ The complete translation of Example 1 is shown in Fig. 6 (for simplicity we have applied a renaming of variables in the resulting rules). An example of run in the resulting MSRN C specification is shown in Figure 7. Note that, a fresh name is selected between all values strictly greater than the current value of f resh (e.g. in the second step 6 > 4), and then f resh is updated to a value strictly greater than all newly generated names (e.g. 8 > 6 > 4). 16 Giorgio Delzanno init −−→ f resh(x) | initM (y) : x > 0, y = 0. f resh(x) | initM (y) −−→ f resh(x′ ) | create(y ′ ) : x′ > y ′ , y ′ > x. create(x) −−→ initM (x′ ) | initA (id′ , n′ , m′ ) : x′ = x, id′ = x, n′ = 0, m′ = 0. create(x) −−→ initM (x′ ) | initB (id′ , n′ , m′ ) : x′ = x, id′ = x, n′ = 0, m′ = 0. initA (id, n, m)| f resh(u) −−→ genA (id, n′ , m) | f resh(u′ ) : u′ > n′ , n′ > u. genA (id1 , n, m)| initB (id2 , u, v) −−→ waitA (id1 , n, m) | genB (id′2 , u′ , v ′ ) : u′ = n, v ′ = v genB (id, n, m)| f resh(u) −−→ readyB (id, n, m′ ) | f resh(u′ ) : u′ > m′ , m′ > u. waitA (id1 , n, m)| readyB (id2 , u, v) −−→ stopA (id1 , n, m′ ) | stopB (id2 , u, v) : n = u, m′ = v. stopA (id, n, m) −−→ initA (id′ , n′ , m′ ) : n′ = 0, m′ = 0, id′ = id. stopB (id, n, m) −−→ initB (id′ , n′ , m′ ) : n′ = 0, m′ = 0, id′ = id. Fig. 6. Encoding of Example 1: for simplicity we embed constraints like x = x′ into the MSR formulas. init ⇒ . . . ⇒ f resh(4) | initM (0) | initA (2, 0, 0) | initB (3, 0, 0) ⇒ f resh(8) | initM (0) | genA (2, 6, 0) | initB (3, 0, 0) ⇒ f resh(8) | initM (0) | waitA (2, 6, 0) | genB (3, 6, 0) ⇒ . . . ⇒ f resh(16) | initM (0) | waitA (2, 6, 0) | genB (3, 6, 0) | initA (11, 0, 0) Fig. 7. A run in the encoded program. Let T = hP1 , . . . , Pt i be a collection of thread definitions and G0 be an initial global state. Let S be the MSRN C specification that results from the translation described in the previous section. Let G = hN, p1 , . . . , pn i be a global configuration with pi = hsi , vi1 , . . . , viki i, and let h : N ❀ Q+ be an injective mapping. Then, we define G• (h) as the MSRN C configuration s1 (h(v11 ), . . . , h(v1k1 )) | . . . | sn (h(vn1 ), . . . , h(vnkn )) | f resh(v) where v is a the first value strictly greater than all values in the range of h. Given an MSRN C configuration M defined as s1 (v11 , . . . , v1k1 ) | . . . | sn (vn1 , . . . , vnkn ) with sij ∈ Q+ , let V (M) ⊆ Q+ be the set of values occurring in M. Then, given a bijective mapping f : V (M) ❀ N ⊆ N , we define M• (f ) as the global configuration hN, p1 , . . . , pn i where pi = hsi , f (vi1 ), . . . , f (viki )i. Based on the previous definitions, the following property then holds. Theorem 2 For every run G0 G1 . . . in T with corresponding set of names N0 N1 . . ., there exist sets D0 D1 . . . and bijective mappings h0 h1 . . . with hi : Ni ❀ Di ⊆ Q+ for i ≥ 0, such that init G•0 (h0 )G•1 (h1 ) . . . is a run of S. Vice versa, if init M0 M1 . . . is a run of S, then there exist sets N0 N1 . . . in N and bijective mappings f0 f1 . . . with fi : V (Mi ) ❀ Ni for i ≥ 0, such that M•0 (f0 )M•1 (f1 ) . . . is a run in T . Proof We first prove that every run in T is simulated by a run in S. Let G0 . . . Gl be a run in T , i.e., a sequence of global states (with associated set Constraint-based Verification of Abstract Multithreaded Programs 17 of names N0 . . . Nl ) such that Gi ⇒ Gi+1 and Ni ⊆ Ni+1 for i ≥ 0. We prove that it can be simulated in S by induction on its length l. Specifically, suppose that there exist sets of non negative rationals D0 . . . Dl and bijective mappings h0 . . . hl with hi : Ni ❀ Di for 0 ≤ i ≤ l, such that c0 (h0 ) . . . G cl (hl ) init G is a run of S. Furthermore, suppose Gl ⇒ Gl+1 . We prove the thesis by a case-analysis on the type of rule applied in the last step of the run. Let Gl = hNl , p1 , . . . , pr i and pj = hs, n1 , . . . , nk i be a local configuration for the thread definition P = hQ, s, V, Ri with V = {x1 , . . . , xk } and ni ∈ Nl for i : 1, . . . , k. a Assignment Suppose there exists a rule s −−→ s′ [γ, α] in R such that ρpj satisfies γ, Gl = hNl , . . . , pj , . . .i ⇒ hNl+1 , . . . , p′j , . . .i = Gl+1 Nl = Nl+1 , p′j = hs′ , n′1 , . . . , n′k i, and if xi := yi occurs in α, then n′i = ρpj (yi ), otherwise n′i = ni for i : 1, . . . , k. The encoding of the rule returns one M SRN C rule having the form s(x1 , . . . , xk ) → s′ (x′1 , . . . , x′k ) : γ ′ , α b for every γ ′ ∈ γ b. cl (hl ) is a multiset of atomic formulas that contains the By inductive hypothesis, G formula s(hl (n1 ), . . . , hl (nk )). Now let us define hl+1 as the mapping from Nl to Dl such that hl+1 (n′i ) = hl (nj ) if xi := xj is in α and hl+1 (n′i ) = 0 if xi := ⊥ is in α. Furthermore, let us the define the evaluation σ = hx1 7→ hl (n1 ), . . . , xk 7→ hl (nk ), x′1 7→ hl+1 (n′1 ), . . . , x′k 7→ hl+1 (n′k )i Then, by construction of the set of constraints b γ and of the constraint α b, it follows that σ is a solution for γ ′ , α b for some γ ′ ∈ b γ . As a consequence, we have that s(n1 , . . . , nk ) → s′ (n′1 , . . . , n′k ) is a ground instance of one of the considered M SRN C rules. cl (hl ), if we apply a rewriting step Thus, starting from the M SRN C configuration G we obtain a new configuration in which s(n1 , . . . , nk ) is replaced by s′ (n′1 , . . . , n′k ), c [ and all the other atomic formulas in G l+1 (hl+1 ) are the same as in Gl (hl ). The [ resulting M SRN C configuration coincides then with the definition of G l+1 (hl+1 ). Creation of new names Let us now consider the case of fresh name generation. a Suppose there exists a rule s −−→ s′ [xi := new] in R, and let n 6∈ Nl , and suppose hNl , . . . , pj , . . .i ⇒ hNl+1 , . . . , p′j , . . .i where Nl+1 = Nl ∪ {v}, p′j = hs′ , n′1 , . . . , n′k i where n′i = n, and n′j = nj for j 6= i. We note than that the encoding of the previous rule returns the M SRN C rule s(x1 , . . . , xk ) | f resh(x) → s′ (x′1 , . . . , x′k ) | f resh(x′ ) : ϕ where ϕ consists of the constraints y > x′i , x′i > x and x′j = xj for j 6= i. By cl (hl ) is a multiset of atomic formulas that contains the inductive hypothesis, G 18 Giorgio Delzanno formulas s(hl (n1 ), . . . , hl (nk )) and f resh(v) where hl is a mapping into Dl , and v is the first non-negative rational strictly greater than all values occurring in the formulas denoting processes. Let v be a non negative rational strictly greater than all values in Dl . Furthermore, let us define v ′ = v + 1 and Dl+1 = Dl ∪ {v, v ′ }. Furthermore, we define hl+1 as follows hl+1 (n) = hl (n) for n ∈ Nl , and hl+1 (n′i ) = hl+1 (n) = v ′ . Furthermore, we define the following evaluation σ = h x 7→ v, x1 7→ hl (n1 ), . . . , xk 7→ hl (nk ), x′ 7→ v ′ , x′1 7→ hl+1 (n′1 ), . . . , x′k 7→ hl+1 (n′k ) i Then, by construction of σ and α b, it follows that σ is a solution for α b. Thus, s(n1 , . . . , nk ) | f resh(v) → s′ (n′1 , . . . , n′k ) | f resh(v ′ ) is a ground instance of the considered M SRN C rule. cl (hl ), if we apply a rewriting step we Starting from the M SRN C configuration G obtain a new configuration in which s(n1 , . . . , nk ) and f resh(v) are substituted by [ s′ (n′1 , . . . , n′k ) and f resh(v ′ ), and all the other atomic formulas in G l+1 (hl+1 ) are c the same as in Gl (hl ). We conclude by noting that this formula coincides with the [ definition of G l+1 (hl+1 ). For sake of brevity we omit the case of thread creation whose only difference from the previous cases is the creation of several new atoms instead (with values obtained by evaluating the action) of only one. Rendez-vous Let pi = hs, n1 , . . . , nk i and pj = ht, m1 , . . . , mu i two local configurations for threads P 6= P ′ , ni ∈ Nl for i : 1, . . . , k and mi ∈ Nl for i : 1, . . . , u. c!m c?m′ Suppose s −−→ s′ [γ, α] and t −−−→ t′ [γ ′ , α′ ], where m = hxi1 , . . . , xiv i, and m′ = hy1 , . . . , yv i ( all defined over distinct variables) are rules in R. Furthermore, suppose that ρpi satisfies γ, and that ρ′ (see definition of the operational semantics) satisfies γ ′ , and suppose that Gl = hNl , . . . , pi , . . . , pj , . . .i ⇒ hNl+1 , . . . , p′i , . . . , p′j , . . .i = Gl+1 , where Nl+1 = Nl , p′i = hs′ , n′1 , . . . , n′k i, p′j = ht′ , m′1 , . . . , m′u i, and if xi := e occurs in α, then n′i = ρpi (e), otherwise n′i = ni for i : 1, . . . , k; if ui := e occurs in α′ , then m′i = ρ′ (e), otherwise m′i = mi for i : 1, . . . , u. cl (hl ) is a multiset of atomic formulas that contains the By inductive hypothesis, G formulas s(hl (n1 ), . . . , hl (nk )) and t(hl (m1 ), . . . , hl (mu )). Now, let us define hl+1 as the mapping from Nl to Dl such that hl+1 (n′i ) = hl (nj ) if xi := xj is in α, hl+1 (m′i ) = hl (mj ) if ui := uj is in α′ , hl+1 (n′i ) = 0 if xi := ⊥ is in α, hl+1 (m′i ) = 0 if ui := ⊥ is in α′ . Now, let us define σ as the evaluation from Nl to Dl such that σ = σ1 ∪ σ2 σ1 = hx1 7→ hl (n1 ), . . . , xk 7→ hl (nk ), u1 7→ hl (m1 ), . . . , uu 7→ hl (mu )i σ2 = hx′1 7→ hl+1 (n′1 ), . . . , x′k 7→ hl+1 (n′k ), u′1 7→ hl+1 (m′1 ), . . . , u′u 7→ hl+1 (m′u )i. Then, by construction of the sets of constraints γ b, γb′ , α b and αb′ it follows that σ is ′ ′ ′ a solution for the constraint ∃w1 . . . . ∃wp .θ ∧ θ ∧ α b ∧ αb′ ∧ w1 = w1′ ∧ . . . ∧ wp = wp′ for some θ ∈ γ b and θ′ ∈ γb′ . Note in fact that the equalities wi = wi′ express the Constraint-based Verification of Abstract Multithreaded Programs 19 passing of values defined via the evaluation ρ′ in the operational semantics. As a consequence, s(n1 , . . . , nk ) | t(m1 , . . . , mu ) → s′ (n′1 , . . . , n′k ) | t′ (m′1 , . . . , m′u ) is a ground instance of one of the considered M SRN C rules. cl (hl ), if we apply a rewriting Thus, starting from the M SRN C configuration G step we obtain a new configuration in which s(n1 , . . . , nk ) has been replaced by s′ (n′1 , . . . , n′k ), and t′ (m′1 , . . . , m′k ) has been replaced by t(m′1 , . . . , m′u ), and all the cl (hl ). This formula coincides with the definition other atomic formulas are as in G [ of Gl+1 (hl+1 ). The proof of completeness is by induction on the length of an MSR run, and by case-analysis on the application of the rules. The structure of the case analysis is similar to the previous one and it is omitted for brevity. 4 Verification of TDL Programs Safety and invariant properties are probably the most important class of correctness specifications for the validation of a concurrent system. For instance, in Example 1 we could be interested in proving that every time a session terminates, two instances of thread Init and Resp have exchanged the two names generated during the session. To prove the protocol correct independently from the number of names and threads generated during an execution, we have to show that from the initial configuration G0 it is not possible to reach a configuration that violates the aforementioned property. The configurations that violate the property are those in which two instances of Init and Resp conclude the execution of the protocol exchanging only the first nonce. These configurations can be represented by looking at only two threads and at the relationship among their local data. Thus, we can reduce the verification problem of this safety property to the following problem: Given an initial configuration G0 we would like to decide if a global configuration that contains at least two local configurations having the form hstopA , i, n, mi and hstopB , i′ , n′ , m′ i with n′ = n and m 6= m′ for some i, i′ , n, n′ , m, m′ is reachable. This problem can be viewed as an extension of the control state reachability problem defined in (Abdulla and Nylén 2000) in which we consider both control locations and local variables. Although control state reachability is undecidable (see Corollary 1), the encoding of TDL into MSRN C can be used to define a sound and automatic verification methods for TDL programs. For this purpose, we will exploit a verification method introduced for MSR(C) in (Delzanno 2001; Delzanno 2005). In the rest of this section we will briefly summarize how to adapt the main results in (Delzanno 2001; Delzanno 2005) to the specific case of MSRN C . Let us first reformulate the control state reachability problem of Example 1 for the aforementioned safety property on the low level encoding into MSRN C . Given the MSRN C initial configuration init we would like to check that no configuration in P ost∗ ({init}) has the following form {stopA (a1 , v1 , w1 ), stopB (a2 , v2 , w2 )} ⊕ M 20 Giorgio Delzanno for ai , vi , wi ∈ Q i : 1, 2 and an arbitrary multiset of ground atoms M. Let us call U the set of bad MSRN C configurations having the aforementioned shape. Notice that U is upward closed with respect to multiset inclusion, i.e., if M ∈ U and M 4 M′ , then M′ ∈ U . Furthermore, for if U is upward closed, so is P re(U ). On the basis of this property, we can try to apply the methodology proposed in (Abdulla and Nylén 2000) to develop a procedure to compute a finite representation R of P re∗ U ). For this purpose, we need the following ingredients: 1. a symbolic representation of upward closed sets of configurations (e.g. a set of assertions S whose denotation [[S]] is U ); 2. a computable symbolic predecessor operator SP re working on sets of formulas such that [[SP re(S)]] = P re([[S]]); 3. a (decidable) entailment relation Ent to compare the denotations of symbolic representations, i.e., such that Ent(N, M ) implies [[N ]] ⊆ [[M ]]. If such a relation Ent exists, then it can be naturally extended to sets of formulas as follows: EntS (S, S ′ ) if and only if for all N ∈ S there exists M ∈ S ′ such that Ent(N, M ) holds (clearly, if Ent is an entailment, then EntS (S, S ′ ) implies [[S]] ⊆ [[S ′ ]]). The combination of these three ingredients can be used to define a verification methods based on backward reasoning as explained next. Symbolic Backward Reachability Suppose that M1 , . . . , Mn are the formulas of our assertional language representing the infinite set U consisting of all bad configurations. The symbolic backward reachability procedure (SBR) procedure computes a chain {Ii }i≥0 of sets of assertions such that I0 = {M1 , . . . , Mn } Ii+1 = Ii ∪ SP re(Ii ) for i ≥ 0 The procedure SBR stops when SP re produces only redundant information, i.e., EntS (Ii+1 , Ii ). Notice that EntS (Ii , Ii+1 ) always holds since Ii ⊆ Ii+1 . Symbolic Representation In order to find an adequate represention of infinite sets of MSRN C configurations we can resort to the notion of constrained configuration introduced in (Delzanno 2001) for the language scheme MSR(C) defined for a generic constraint system C. We can instantiate this notion with N C constraints as follows. A constrained configuration over P is a formula p1 (x11 , . . . , x1k1 ) | . . . | pn (xn1 , . . . , xnkn ) : ϕ where p1 , . . . , pn ∈ P, xi1 , . . . , xiki ∈ V for any i : 1, . . . n and ϕ is an N C-constraint. . The denotation a constrained configuration M = (M : ϕ) is defined by taking the upward closure with respect to multiset inclusion of the set of ground instances, namely [[M ]] = {M′ | σ(M) 4 M′ , σ ∈ Sol(ϕ)} Constraint-based Verification of Abstract Multithreaded Programs 21 This definition can be extended to sets of MSRN C constrained configurations with disjoint variables (we use variable renaming to avoid variable name clashing) in the natural way. In our example the following set SU of MSRN C constrained configurations (with distinct variables) can be used to finitely represent all possible violations U to the considered safety property SU = { stopA (i1 , n1 , m1 ) | stopB (i2 , n2 , m2 ) : n1 = n2 , m1 > m2 stopA (i1 , n1 , m1 ) | stopB (i2 , n2 , m2 ) : n1 = n2 , m2 > m1 } Notice that we need two formulas to represent m1 6= m2 using a disjunction of > constraints. The MSRN C configurations stopB (1, 2, 6) | stopA (4, 2, 5), and stopB (1, 2, 6) | stopA (4, 2, 5) | waitA (2, 7, 3) are both contained in the denotation of SU . Actually, we have that [[SU ]] = U . This symbolic representation allows us to reason on infinite sets of MSRN C configurations, and thus on global configurations of a TDL program, forgetting the actual number or threads of a given run. To manipulate constrained configurations, we can instantiate to N C-constraints the symbolic predecessor operator SP re defined for a generic constraint system in (Delzanno 2005). Its definition is also given in Section Appendix A in Appendix. From the general properties proved in (Delzanno 2005), we have that when applied to a finite set of MSRN C constrained configurations S, SP reN C returns a finite set of constrained configuration such that [[SP reN C (S)]] = P re([[S]]), i.e., SP reN C (S) is a symbolic representation of the immediate predecessors of the configurations in the denotation (an upward closed set) of S. Similarly we can instantiate the generic entailment operator defined in (Delzanno 2005) to MSRN C constrained configurations so as to obtain an a relation Ent such that EntN C (N, M ) implies [[N ]] ⊆ [[M ]]. Based on these properties, we have the following result. Proposition 1 Let T be a TDL program with initial global configuration G0 , Furthermore, let S be the corresponding MSRN C encoding. and SU be the set of MSRN C constrained configurations denoting a given set of bad TDL configurations. Then, init 6∈ SP re∗N C (SU ) if and only if there is no finite run G0 . . . Gn and mappings h0 , . . . , hn from the names occurring in G to non-negative rationals such that init• G•0 (h0 ) . . . G•n (hn ) is a run in S and G•n (hn ) ∈ [[U ]]. Proof Suppose init 6∈ SP re∗N C (U ). Since [[SP reN C (S)]] = pre([[S]]) for any S, it follows that there cannot exist runs initM0 . . . Mn in S such that Mn ∈ [[U ]]. The thesis then follows from the Theorem 2. As discussed in (Bozzano and Delzanno 2002), we have implemented our verification procedure based on M SR and linear constraints using a CLP system with linear arithmetics. By the translation presented in this paper, we can now reduce the verification of safety properties of multithreaded programs to a fixpoint computation built on constraint operations. As example, we have applied our CLP-prototype to automatically verify the specification of Fig. 6. The unsafe states are those described in Section 4. Symbolic backward reachability terminates after 18 iterations 22 Giorgio Delzanno and returns a symbolic representation of the fixpoint with 2590 constrained configurations. The initial state init is not part of the resulting set. This proves our original thread definitions correct with respect to the considered safety property. 4.1 An Interesting Class of TDL Programs The proof of Theorem 1 shows that verification of safety properties is undecidable for TDL specifications in which threads have several local variables (they are used to create linked lists). As mentioned in the introduction, we can apply the sufficient conditions for the termination of the procedure SBR given in (Bozzano and Delzanno 2002; Delzanno 2005) to identify the following interesting subclass of TDL programs. Definition 4 A monadic TDL thread definition P = hQ, s, V, Ri is such that V is at most a singleton, and every message template in R has at most one variable. A monadic thread definition can be encoded into the monadic fragment of MSRN C studied in (Delzanno 2005). Monadic MSRN C specifications are defined over atomic formulas of the form p or p(x) with p is a predicate symbol and x is a variable, and on atomic constraints of the form x = y, and x > y. To encode a monadic TDL thread definitions into a Monadic MSRN C specification, we first need the following observation. Since in our encoding we only use the constant 0, we first notice that we can restrict our attention to MSRN C specifications in which constraints have no constants at all. Specifically, to encode the generation of fresh names we only have to add an auxiliary atomic formula zero(z), and refer to it every time we need to express the constant 0. As an example, we could write rules like init −−→ f resh(x) | initM (y) | zero(z) : x > z, y = z for initialization, and create(x) | zero(z) −−→ initM (x′ ) | initA (id′ , n′ , m′ ) | zero(z) : x′ = x, id′ = x, n′ = z, m′ = z, z ′ = z for all assignments involving the constant 0. By using this trick an by following the encoding of Section 3, the translation of a collection of monadic thread definitions directly returns a monadic MSRN C specification. By exploiting this property, we obtain the following result. Theorem 3 The verification of safety properties whose violations can be represented via an upward closed set U of global configurations is decidable for a collection T of monadic TDL definitions. Proof Let S be the MSRN C encoding of T and SU be the set of constrained configuration such that SU = U . The proof is based on the following properties. First of all, the MSRN C specification S is monadic. Furthermore, as shown in (Delzanno 2005), Constraint-based Verification of Abstract Multithreaded Programs 23 the class of monadic MSRN C constrained configurations is closed under application of the operator SP reN C . Finally, as shown in (Delzanno 2005), there exists an entailment relation CEnt for monadic constrained configurations that ensures the termination of the SBR procedure applied to a monadic MSRN C specification. Thus, for the monadic MSRN C specification S, the chain defined as I0 = SU , Ii+1 = Ii ∪ SP re(Ii ) always reaches a point k ≥ 1 in which CEntS (Ik+1 , Ik ), i.e. [[Ik ]] is a fixpoint for P re. Finally, we note that we can always check for membership of init in the resulting set Ik . As shown in (Schnoebelen 2002), the complexity of verification methods based on symbolic backward reachability relying on the general results in (Abdulla and Nylén 2000; Finkel and Schnoebelen 2001) is non primitive recursive. 5 Conclusions and Related Work In this paper we have defined the theoretical grounds for the possible application of constraint-based symbolic model checking for the automated analysis of abstract models of multithreaded concurrent systems providing name generation, name mobility, and unbounded control. Our verification approach is based on an encoding into a low level formalism based on the combination of multiset rewriting and constraints that allows us to naturally implement name generation, value passing, and dynamic creation of threads. Our verification method makes use of symbolic representations of infinite set of system states and of symbolic backward reachability. For this reason, it can be viewed as a conservative extension of traditional finite-state model checking methods. The use of symbolic state analysis is strictly related to the analysis methods based on abstract interpretation. A deeper study of the connections with abstract interpretation is an interesting direction for future research. Related Work The high level syntax we used to present the abstract models of multithreaded programs is an extension of the communicating finite state machines used in protocol verification (Bochmann 1978), and used for representing abstraction of multithreaded software programs (Ball et al. 2001). In our setting we enrich the formalism with local variables, name generation and mobility, and unbounded control. Our verification approach is inspired by the recent work of Abdulla and Jonsson. In (Abdulla and Jonsson 2003), Abdulla and Jonsson proposed an assertional language for Timed Networks in which they use dedicated data structures to symbolically represent configurations parametric in the number of tokens and in the age (a real number) associated to tokens. In (Abdulla and Nylén 2000), Abdulla and Nylén formulate a symbolic algorithm using existential zones to represent the state-space of Timed Petri Nets. Our approach generalizes the ideas of (Abdulla and Jonsson 2003; Abdulla and Nylén 2000) to systems specified via multiset rewriting and with more general classes of constraints. In (Abdulla and Jonsson 2001), the authors apply similar ideas to (unbounded) channel systems in which messages can vary over an infinite name domain and can be stored in a finite (and fixed a 24 Giorgio Delzanno priori) number of data variables. However, they do not relate these results to multithreaded programs. Multiset rewriting over first order atomic formulas has been proposed for specifying security protocols by Cervesato et al. in (Cervesato et al. 1999). The relationships between this framework and concurrent languages based on process algebra have been recently studied in (Bistarelli et al. 2005). Apart from approaches based on Petri Net-like models (as in (German and Sistla 1992; Ball et al. 2001)), networks of finite-state processes can also be verified by means of automata theoretic techniques as in (Bouajjani et al. 2000). In this setting the set of possible local states of individual processes are abstracted into a finite alphabet. Sets of global states are represented then as regular languages, and transitions as relations on languages. Differently from the automata theoretic approach, in our setting we handle parameterized systems in which individual components have local variables that range over unbounded values. The use of constraints for the verification of concurrent systems is related to previous works connecting Constraint Logic Programming and verification, see e.g. (Delzanno and Podelski 1999). In this setting transition systems are encoded via CLP programs used to encode the global state of a system and its updates. In the approach proposed in (Delzanno 2001; Bozzano and Delzanno 2002), we refine this idea by using multiset rewriting and constraints to locally specify updates to the global state. In (Delzanno 2001), we defined the general framework of multiset rewriting with constraints and the corresponding symbolic analysis technique. The language proposed in (Delzanno 2001) is given for a generic constraint system C (taking inspiration from CLP the language is called M SR(C)). In (Bozzano and Delzanno 2002), we applied this formalism to verify properties of mutual exclusion protocols (variations of the ticket algorithm) for systems with an arbitrary number of processes. In the same paper we also formulated sufficient conditions for the termination of the backward analysis. The present paper is the first attempt of relating the low level language proposed in (Delzanno 2001) to a high level language with explicit management of names and threads. Acknowledgments The author would like to thank Ahmed Bouajjani, Andrew Gordon, Fabio Martinelli, Catuscia Palamidessi, Luca Paolini, and Sriram Rajamani and the anonymous reviewers for several fruitful comments and suggestions. References Abdulla, P. A., Cerāns, K., Jonsson, B., and Tsay, Y.-K. 1996. General Decidability Theorems for Infinite-State Systems. In Proceedings 11th Annual International Symposium on Logic in Computer Science (LICS’96). IEEE Computer Society Press, New Brunswick, New Jersey, 313–321. Abdulla, P. A. and Jonsson, B. 2001. Ensuring Completeness of Symbolic Verification Methods for Infinite-State Systems. Theoretical Computer Science 256, 1-2, 145–167. Abdulla, P. A. and Jonsson, B. 2003. Model checking of systems with many identical timed processes. Theoretical Computer Science 290, 1, 241–264. Abdulla, P. A. and Nylén, A. 2000. Better is Better than Well: On Efficient Verification of Infinite-State Systems. In Proceedings 15th Annual International Symposium on Constraint-based Verification of Abstract Multithreaded Programs 25 Logic in Computer Science (LICS’00). IEEE Computer Society Press, Santa Barbara, California, 132–140. Ball, T., Chaki, S., and Rajamani, S. K. 2001. Parameterized Verification of Multithreaded Software Libraries. In 7th International Conference on Tools and Algorithms for Construction and Analysis of Systems (TACAS 2001), Genova, Italy, April 2-6,. LNCS, vol. 2031. Springer-Verlag, 158–173. Bistarelli, S., Cervesato, I., Lenzini, G., and Martinelli, F. 2005. Relating multiset rewriting and process algebras for security protocol analysis. Journal of Computer Security 13, 1, 3–47. Bochmann, G. V. 1978. Finite state descriptions of communicating protocols. Computer Networks 2, 46–57. Bouajjani, A., Jonsson, B., Nilsson, M., and Touili, T. 2000. Regular Model Checking. In Proceedings 12th International Conference on Computer Aided Verification (CAV’00), E. A. Emerson and A. P. Sistla, Eds. LNCS, vol. 1855. Springer-Verlag, Chicago, Illinois, 403–418. Bozzano, M. and Delzanno, G. 2002. Algorithmic verification of invalidation-based protocols. In 14th International Conference on Computer Aided Verification, CAV ’02. Lecture Notes in Computer Science, vol. 2404. Springer. Cervesato, I., Durgin, N., Lincoln, P., Mitchell, J., and Scedrov, A. 1999. A Meta-notation for Protocol Analysis. In 12th Computer Security Foundations Workshop (CSFW’99). IEEE Computer Society Press, Mordano, Italy, 55–69. Delzanno, G. 2001. An Assertional Language for Systems Parametric in Several Dimensions. In Verification of Parameterized Systems - VEPAS 2001. ENTCS, vol. 50. Delzanno, G. 2005. Constraint Multiset Rewriting. Tech. Rep. TR-05-08, Dipartimento Informatica e Scienze dell’Informazione, Università di Genova, Italia. Delzanno, G. and Podelski, A. 1999. Model checking in CLP. In Proceedings 5th International Conference on Tools and Algorithms for Construction and Analysis of Systems (TACAS’99). Lecture Notes in Computer Science, vol. 1579. Springer-Verlag, Amsterdam, The Netherlands, 223–239. Finkel, A. and Schnoebelen, P. 2001. Well-Structured Transition Systems Everywhere! Theoretical Computer Science 256, 1-2, 63–92. German, S. M. and Sistla, A. P. 1992. Reasoning about Systems with Many Processes. Journal of the ACM 39, 3, 675–735. Gordon, A. D. 2001. Notes on nominal calculi for security and mobility. In Foundations of Security Analysis and Design, Tutorial Lectures. Lecture Notes in Computer Science, vol. 2171. Springer, 262–330. Kesten, Y., Maler, O., Marcus, M., Pnueli, A., and Shahar, E. 2001. Symbolic model checking with rich assertional languages. Theoretical Computer Science 256, 1, 93–112. Schnoebelen, P. 2002. Verifying Lossy Channel Systems has Nonprimitive Recursive Complexity. Information Processing Letters 83, 5, 251–261. Appendix A Symbolic Predecessor Operator Given a set of MSRN C configurations S, consider the MSRN C predecessor operator P re(S) = {M|M ⇒MSR M′ , M′ ∈ S}. In our assertional language, we can define a symbolic version SP reN C of P re defined on a set S containing MSRN C constrained 26 Giorgio Delzanno multisets (with disjoint variables) as follows: SP reN C (S) = { (A ⊕ N : ξ) | (A −→ B : ψ) ∈ R, (M : ϕ) ∈ S, M′ 4 M, B ′ 4 B, (M′ : ϕ) =θ (B ′ : ψ), N = M ⊖ M′ , ξ ≡ (∃x1 . . . . xk .θ) and x1 , . . . , xk are all variables not in A ⊕ N }. where =θ is a matching relation between constrained configurations that also takes in consideration the constraint satisfaction, namely (A1 | . . . | An : ϕ) =θ (B1 | . . . | Bm : ψ) provided m = n and there exists a permutation j1 , . . . , jn of 1, . . . , n such that Vn the constraint θ = ϕ ∧ ψ ∧ i=1 Ai = Bji is satisfiable; here p(x1 , . . . , xr ) = q(y1 , . . . , ys ) is an abbreviation for the constraints x1 = y1 ∧ . . . ∧ xr = ys if p = q and s = r, f alse otherwise. As proved in (Delzanno 2005), the symbolic operator SP reN C returns a set of MSRN C constrained configurations and it is correct and complete with respect to P re, i.e., [[SP reN C (S)]] = P re([[S]]) for any S. It is important to note the difference between SP reN C and a simple backward rewriting step. For instance, given the constrained configurations M defined as p(x, z) | f (y) : z > y and the rule s(u, m) | r(t, v) → p(u′ , m′ ) | r(t′ , v ′ ) : u = t, m′ = v, v ′ = v, u′ = u, t′ = t (that simulates a rendez-vous (u, t are channels) and value passing (m′ = v)), the application of SP re returns s(u, m) | r(t, v) | f (y) : u = t, v > y as well as s(u, m) | r(t, v) | p(x, z) | f (y) : u = t, x > y (the common multiset here is ǫ). 

R. Isea / Challenges and characterization of a Biological system on Grid Proceedings of the First EELA-2 Conference R. Mayo et al. (Eds.) CIEMAT 2009 © 2009 The authors. All rights reserved Challenges and characterization of a Biological system on Grid by means of the PhyloGrid application R. Isea1, E. Montes2, A.J. Rubio-Montero2 and R. Mayo2 1 Fundación IDEA, Hoyo de la Puerta, Valle de Sartenejal, Baruta 1080 (Venezuela) risea@idea.gob.ve 2 CIEMAT, Avda. Complutense, 22 - 28040 Madrid (Spain) {esther.montes,antonio.rubio,rafael.mayo}@ciemat.es Abstract In this work we present a new application that is being developed. PhyloGrid is able to perform large-scale phylogenetic calculations as those that have been made for estimating the phylogeny of all the sequences already stored in the public NCBI database. The further analysis has been focused on checking the origin of the HIV-1 disease by means of a huge number of sequences that sum up to 2900 taxa. Such a study has been able to be done by the implementation of a workflow in Taverna. 1. Introduction The determination of the evolution history of different species is nowadays one of the more exciting challenges that are currently emerging in the computational Biology [1]. In this framework, Phylogeny is able to determine the relationship among the species and, in this way, to understand the influence between hosts and virus [2]. As an example, we can mention the work published in 2007 related to the origin of the oldest stumps in HIV/AIDS (apart from the African one), which were found in Haiti [3], so it is clear that the test of vaccines for this disease must include both African and Haitian sequences. This work was performed with MrBayes tool, the bayesian inference method used in PhyloGrid. With respect to the computational aspect, the main characteristic of the phylogenetic executions is that they are extremely intensive, even more when the number of sequences increases, so it is crucial to develop efficient tools for obtaining optimized solutions. Thus, 1000 taxa (sequences) generate 2.5·101167 trees, so among all of them the potential consensus tree is found. That is why it is clear to understand the computational challenge that this work represents by studying the origin of the HIV-1 since it is dealing with 2900 sequences. Several techniques for estimating phylogenetic trees have been developed such as the distance based method (which generates a dendrogram, i.e. they do not show the evolution, but the changes among the sequences), the maximum parsimony technique, the maximum likelihood method and the bayesian analysis (mainly Markov Chain Monte Carlo inference). The latest is based on probabilistic techniques for evaluating the topology of the phylogenetic trees; so it is important to point out that the Maximum Parsimony methods are not statistically consistent, this is, the consensus tree cannot be found with the higher probability because of the long branch attraction effect [4]. 1 R. Isea / Challenges and characterization of a Biological system on Grid In this work, we have worked with MrBayes software [5] for obtaining the phylogenetic trees. It is important to indicate that MrBayes is relatively new in the construction of these trees as the reader can check in the pioneering work of Rannala and Yang in 1996 [6]. This methodology works with the Bayesian statistics previously proposed by Felsentein in 1968 as indicated Huelsenbeck [7], a technique for maximizing the subsequent probability. The reason for using this kind of approach is that it deals with higher computational speed methods so the possible values for the generated trees can all be taken into account not being any of them ruling the others. With respect to the consensus tree found in our work, the value of which was obtained with this methodology, known as parametric bootstrap percentages or Bayesian posterior probability values, we can mention that it gives information about the reproducibility of the different part of the trees, but this kind of data does not represent a significant statistical value [8]. Thus, based on MrBayes tool, the PhyloGrid application aims to offer to the scientific community an easy interface for calculating phylogenetic trees by means of a workflow with Taverna [9]. In this way, the user is able to define the parameters for doing the Bayesian calculation, to determine the model of evolution and to check the accuracy of the results in the intermediate stages. In addition to this, no knowledge from his/her side about the computational procedure is required. More details about this workflow can be found in [10]. As a consequence of this development, several biological results have been achieved in the Duffy domain of Malaria [10] or in the HPV classification [11]. In this work, we study the dependence of a successful determination for a biological system with the number of sequences to be aligned. To do so, the HIV case has been selected taking into account over a thousand different sequences. 2. Tools The structure of the implementation of the different tools present in the PhyloGrid application can be seen in Figure 1. We here briefly explain them. 2.1. MrBayes Bayesian inference is a powerful method which is implemented in the program MrBayes [12] for estimating phylogenetic trees that are based on the posterior probability distribution of the trees. Currently is included in different scientific software suites for cluster computing such as Rocks Cluster (Bio Roll) as well as in other Linux distributions where the user doesn’t need to compile it, for example Ubuntu, Gentoo, Mandriva and so on. This easiness allows the program to be ported to the Grid. On the other hand, its main drawback is that its execution for millions of iterations (generations) requires a large amount of computational time and memory. As an example, we can cite that in the work from Cadotte et al. [13] 143 angiosperms were determined by means of MrBayes. In that work, four independent Markov Chains were run, each with 3 heated chains for 100 million generations, the authors sampled the runs every 10,000 generations and used a burning of 70 million steps to generate a majority rule consensus tree that was used to calculate PDC. Such a methodology is basically the same for other calculations and the number of iterations will depend on the convergence from the chains. In fact, this kind of scientific calculation is extremely complex from a mathematical point of view, but also from the computational one where the phylogenetic estimation for a medium size dataset (50 sequences, 300 nucleotides for each sequence) typically requires a simulation for 250,000 generations, which normally runs for 50 hours on a PC with an Intel Pentium4 2.8 GHz processor. 2 R. Isea / Challenges and characterization of a Biological system on Grid Figure 1. EELA Schema of PhyloGrid Even more, the recent publication by Pérez et al. [14] shows that the study of 626 domains of prokaryotic genomes, due to they are a major class of regulatory proteins in eukaryotes, represents by itself a computational challenge that must be limited to the analysis of the results instead of trying to delimit the number of obtained results by the number of sequences that implies the computational time. 2.2. Gridsphere The Gridsphere project aims to develop a standard based portlet framework for building web portals and a set of portlet web applications. It is based on Grid computing and integrates into the GridSphere portal framework a collection of gridportlets provided as an add-on module, i.e. a portfolio of tools that forms a cohesive "grid portal" end-user environment for managing users and provides access to information services. Grid security based on public key infrastructure (PKI) and emerging IETF and OASIS standards are also well-defined characteristics. In this way, it is important to mention that the users are able to access the Grid by means of an implementation of the JSR 168 portlet API standard. As a consequence, they can also interact with other standards such as those of GT4 and all of its capabilities. An important key in this application is that the researcher who use it can do it with his/her personal Grid user certificate (“myproxy” initialization), the execution of which is already integrated in the Gridsphere release. There is also the possibility of running the jobs with a proxy directly managed by the Administrator that would be renewed from time to time in order to allow longer jobs to be ended. Thus, all the technical details are transparent for the user, so all the methodology is automated and the application can either be run directly by a certified user or letting Gridsphere to assign him a provisional proxy, registered in a map log. Within the portal framework and in a future release, the possibility for doing a multiple alignment of the sequences will be available for the user. 3 R. Isea / Challenges and characterization of a Biological system on Grid 2.3. Taverna, a tool for the implementation of Workflows The workflow is fully built in Taverna [9] and structured in different services that are equivalent to the different sections that are run in a common MrBayes job and performs a complete calculation just building the input file by means of the construction of a common Grid jdl file. The front end to the final user is a web portal built upon the Gridsphere framework. This solution makes very easy to the researcher the calculations of molecular phylogenies by no typing at all any kind of command. The main profit that these kind of workflows offer is that they integrate in a common development several modules (tools) connected one to each other. Even more, the deployment of the application with Taverna allows any researcher to download the complete workflow, making easy in this way their spread in the scientific community. Figure 2. The Taverna Workflow used in this work This is so because Taverna allows the user to construct complex analysis workflows from components located on both remote and local machines, run these workflows on their own data and visualise the results. To support this core functionality it also allows various operations on the components themselves such as discovery and description and the selection of personalised libraries of components previously discovered to be useful to a particular application. Finally, we can indicate that Taverna is based on a workbench window where a main section of an Advanced Model Explorer shows a tabular view of the entities within the 4 R. Isea / Challenges and characterization of a Biological system on Grid workflow. This includes all processors, workflow inputs and outputs, data connections and coordination links. For using Taverna, it is necessary to create a web service for the required application that will be integrated into the software. Lately, this web service will call MrBayes inside the workflow. 2.4. The PhyloGrid Workflow: a short description The structure of the Taverna workflow can be seen in Figure 2, the structure of which has been improved since the first works performed with the PhyloGrid application [10]. Once the user has logged into the PhyloGrid portal, registered a valid proxy and introduced all the data needed for his/her job, a background process performs the execution of the PhyloGrid workflow with the input provided by the user. The workflow receives several inputs: the MrBayes parameters (labelled as MrBayes_params in Fig. 2) that define the evolutionary model that will be used in the analysis and the settings for Markov Chain Monte Carlo (MCMC) analysis; the parameters needed to construct the appropriate Grid job file (JobParams); the input file with the aligned sequences (SeqFile) to analyse; and, the format of the file (SeqFileFormat). The first three processors of the workflow (JobParams2jdl, formatFileToNexus and InputParams2runMB) perform some tasks prior to MrBayes execution. Thus, as aforementioned, the processor named JobParams2jdl creates the appropriate file for the job submission to Grid; the formtFileToNexus processor, if necessary, converts the file with the aligned sequences to NEXUS format (not available in the first PhyloGrid release); and the InputParams2runMB constructs an executable file with MrBayes commands from the MrBayes_params. The output of these processors is sent to the core processor of the workflow, i.e. runMrBayes. This processor submits a MrBayes analysis job (see below the Methodology section). A call to a nested workflow (MrBayes_poll_job) is included to check for job status, and wait if job is not ended. When the MrBayes analysis job is finished, the runMrBayes processor receives a notification from the MrByes_poll_job processor. As a final step, an additional processor is included to store the output of the MrBayes analysis in a Storage Element. This step is needed due to the large size that MrBayes output files can reach. Once the output files are stored in a SE, the workflows execution ends and the user can retrieve the results from the PhyloGrid portal. 2.5. GridWay: The next step Even when PhyloGrid is producing reliable scientific results, the improvement of the tool is a key factor for this work team. That is why we are planning to introduce GridWay [15], the standard metascheduler in Globus Toolkit 4, into the whole structure of the application in order to perform the submission of jobs in a more efficient way. Since GridWay enables large-scale, reliable and efficient sharing of heterogeneous computing resources managed by different LRM systems, it is able to determine in real time the nodes that best fit the characteristics of the submitted jobs. In the case of MrBayes and its parallel mode, this functionality is not only limited to match in any resource the adequate tags into the information systems whatever the installed mpich version the node had, but also to select the best resources by means of several statistical methods based on accounting of previous executions. As a consequence, the calculation will be done in the lowest possible time. 5 R. Isea / Challenges and characterization of a Biological system on Grid At the same time, GridWay can recover the information in case of a cancelled job since it creates checkpointers. It also migrates automatically failed jobs to other Grid resources. Thus, it performs a reliable and unattended execution of jobs transparently to Taverna that simplifies the workflow development and increases the global performance of PhyloGrid. The porting process of GridWay into PhyloGrid will be done by means of the plugins already available for Globus in Taverna that allow the use of resources by a standardized way. 3. Methodology Once the user has determined his/her work area and has connected to the Internet Network, new PhyloGrid jobs can be submitted. For doing so, he/she logs in the application and a new window is then open. In this page, the user is able to define the name of the job to be submitted as well as its description, to upload the file with the alignment, to select the model of evolution and the number of iterations with a fixed frequency and, finally, to run the experiment. The Taverna workflow builts the script that will rule the process with the input data once they are set in the MrBayes format. In addition, as parameters the user must configure: the selected model for the evolution of the system (labelled as the lset part/command in the example below); the number of simultaneous, completely independent, analyses (nruns set to 1); the number of generations that will be performed in the calculation (ITER); and, finally, the location of the file where the sequences are present (file-aligned.nex). To the date, this file must be in Nexus format, but in further releases the workflow will be able to translate to a NEXUS format the input alignment if it is written in any other kind of format (Clustal, Phylip, MSA and so on). Thus, our example would be written as: begin mrbayes; set autoclose=yes nowarn=yes; execute /Path-to-file/file-aligned.nex; lset nst=6 rates=gamma; mcmcp nruns=1 ngen=ITER samplefreq=FREQ; mcmc ; mcmc ; mcmc ; end; This is, the workflow must perform its load section by section. Since the first two instructions are always the same for any kind of calculation, the workflow has to begin with the third one (execute...) making a call for bringing the input data, i.e. the aligned sequences to be studied, and following with the rest of commands. PhyloGrid needs to know the evolution model, so the fourth instruction (lset...) is used. Here, two options are possible: to allow the researcher to select a specific one or to allow the workflow to do so. The fifth instruction (mcmcp nruns…) sets the parameters of the MCMC analysis without actually starting the chain. In our example, it sets the number of independent analyses (nruns) and the number of executions to be made (the ITER data, one million of iterations for example) with a concrete frequency (FREQ). This command is identical in all respects to the mcmc one, except that the analysis will not start after this command is issued. The following instructions (mcmc) perform the MCMC analysis (three consecutive ones in our example), which will be able also to be monitored. All these options 6 R. Isea / Challenges and characterization of a Biological system on Grid are able to be changed by the final user, who at the beginning of the process has simply defined the evolutionary model that will be used in the analysis, the settings for MCMC analysis and the name of the input file with the aligned sequences in NEXUS format. By default, if the name of the output files is not provided, MrBayes sets the corresponding extensions for them (files that will be generated adding the .p, .t and .mcmc extensions to the name of the input file). The job must always have the final command end since it is mandatory for MrBayes. Once the workflow has started, MrBayes automatically validates the number of iterations meanwhile it begins to write the output file and sets the burning value for generating the phylogenetic trees. When the whole calculation is ended and the packed output files are stored in an appropriate Storage Element, it can be downloaded by the user for further analysis. 4. Results and conclusions PhyloGrid has been used to calculate the Phylogeny of the HIV-1 sequences stored in the NCBI database, the number of which is up to 2900. All these sequences were previously aligned with MPIClustal. The calculation lasted for 576 hours on 20 cores inside 5 Intel Xeon X5365 (3GHz, 4 MB L2 cache per 2 cores in the quad-core processor) with 2GB of memory per core. The output file had 835 MB, the tree of which can be seen in Figure 3. Figure 3. Phylogenetic tree construction using PhyloGrid of HIV-1 sequences To the knowledge of the authors, these results are the first that are totally performed for the HIV counting on its whole number of sequences. In a future, the results will be analysed separately, but we can point out the periodicity of the branchs in the obtained trees, so a further study on the obtained patterns will be done in order to simplify the analysis of the results. 7 R. Isea / Challenges and characterization of a Biological system on Grid As can be seen for the parameters of the performed calculation (output file size or time consumed), it is clear that Grid is a useful paradigm that allows the Biologists to cope with such a complex research. Acknowledgements This work makes use of results produced by the EELA-2 project (http://www.eu-eela.eu), co-funded by the European Commission within its Seventh Framework Programme and the VII Cuba-Venezuela Scientific Collaboration Framework and the Annual Operative Plan of Fundación IDEA. References [1] C.R. Woese. Procs. Nat. Academy of Sciences 95, 6854-6859 (1998) [2] B. Korber et al. Science 288, 1789-1796 (2000) [3] M. Thomas et al. PNAS 104 (47), 18566-18570 (2007) [4] J. Bergsten. Cladistics 21 (2), 163-193 (2005) [5] The Mrbayes tool, available from http://www.mrbayes.net [6] B. Rannala and Z. Yang. J. Mol. Evolut. 43, 304-311 (1996) [7] J. P. Huelsenbeck et al. Syst. Biol. 51, 673-688 (2002) [8] B.G. Hall and S.J. Salipante. PLoS Comput Biol. March. 3 (3), e51 (2007) [9] T. Oinn et al. Concurrency and Computation: Practice and Experience 18, 1067-1100 (2005) [10] E. Montes, R. Isea and R. Mayo. Iberian Grid Infrastructure Conf. Proc. 2, 378-387 (2008) [11] R. Isea, E. Montes and R. Mayo. UK-AHM 2008, Edinburgh, United Kingdom (2008) [12] F. Ronquist and J. P. Huelsenbeck. Bioinformatics 19, 1572-1574 (2003) [13] M.W. Cadotte, B.J. Cardinale and T.H. Oakley. PNAS 105 (44), 17012-17017 (2008) [14] J. Pérez et al. PNAS 105 (41), 15950-15955 (2008) [15] E. Huedo, R.S. Montero and I.M. Llorente. J. Scalable Comp.–Prac. Exp. 6 (3), 1-8 (2005) 8 

1 Taxi Dispatch with Real-Time Sensing Data in Metropolitan Areas: A Receding Horizon Control Approach arXiv:1603.04418v1 [] 14 Mar 2016 Fei Miao, Student Member, IEEE, Shuo Han, Member, IEEE, Shan Lin, John A. Stankovic, Fellow, IEEE and ACM, Desheng Zhang, Sirajum Munir, Hua Huang, Tian He, and George J. Pappas Fellow, IEEE Abstract—Traditional taxi systems in metropolitan areas often suffer from inefficiencies due to uncoordinated actions as system capacity and customer demand change. With the pervasive deployment of networked sensors in modern vehicles, large amounts of information regarding customer demand and system status can be collected in real time. This information provides opportunities to perform various types of control and coordination for large-scale intelligent transportation systems. In this paper, we present a receding horizon control (RHC) framework to dispatch taxis, which incorporates highly spatiotemporally correlated demand/supply models and real-time GPS location and occupancy information. The objectives include matching spatiotemporal ratio between demand and supply for service quality with minimum current and anticipated future taxi idle driving distance. Extensive trace-driven analysis with a data set containing taxi operational records in San Francisco shows that our solution reduces the average total idle distance by 52%, and reduces the supply demand ratio error across the city during one experimental time slot by 45%. Moreover, our RHC framework is compatible with a wide variety of predictive models and optimization problem formulations. This compatibility property allows us to solve robust optimization problems with corresponding demand uncertainty models that provide disruptive event information. Note to Practitioners—With the development of mobile sensor and data processing technology, the competition between traditional “hailed on street” taxi service and “on demand” taxi service has emerged in the US and elsewhere. In addition, large amounts of data sets for taxi operational records provide potential demand information that is valuable for better taxi dispatch systems. Existing taxi dispatch approaches are usually greedy algorithms focus on reducing customer waiting time instead of total idle driving distance of taxis. Our research is motivated by the increasing need for more efficient, real-time taxi dispatch methods that utilize both historical records and real-time sensing information to match the dynamic customer demand. This paper suggests a new receding horizon control (RHC) framework aiming to utilize the predicted demand information when making taxi dispatch decisions, so that passengers at different areas of a city are fairly served and the total idle distance of vacant taxis are This work was supported by NSF grant numbers CNS-1239483, CNS1239108, CNS-1239226, and CPS-1239152 with project title: CPS: Synergy: Collaborative Research: Multiple-Level Predictive Control of Mobile Cyber Physical Systems with Correlated Context. The preliminary conference version of this work can be found in [19]. F. Miao, S. Han and G. J. Pappas are with Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia PA, 19104, USA 19014. Email: {miaofei, hanshuo, pappasg}@seas.upenn.edu. S. Lin and H. Huang are with Department of Electrical and Computer Engineering, Stony Brook University, Long Island, NY 11794, USA. Email:{shan.x.lin, hua. huang}@stonybrook.edu}. J.A. Stankovic and S. Munir are with Department of Computer Science, University of Virginia, Charlottesville, VA 22904, USA. Email: stankovic@cs.virginia.edu, sm7hr@virginia.edu. D. Zhang and T. He are with Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA. Email:{tianhe, zhang}@cs.umn.edu reduced. We formulate a multi-objective optimization problem based on the dispatch requirements and practical constraints. The dispatch center updates GPS and occupancy status information of each taxi periodically and solves the computationally tractable optimization problem at each iteration step of the RHC framework. Experiments for a data set of taxi operational records in San Francisco show that the RHC framework in our work can redistribute taxi supply across the whole city while reducing total idle driving distance of vacant taxis. In future research, we plan to design control algorithms for various types of demand model and experiment on data sets with a larger scale. Index Terms—Intelligent Transportation System, Real-Time Taxi Dispatch, Receding Horizon Control, Mobility Pattern I. I NTRODUCTION More and more transportation systems are equipped with various sensors and wireless radios to enable novel mobile cyber-physical systems, such as intelligent highways, traffic light control, supply chain management, and autonomous fleets. The embedded sensing and control technologies in these systems significantly improve their safety and efficiency over traditional systems. In this paper, we focus on modern taxi networks, where real-time occupancy status and the Global Positioning System (GPS) location of each taxi are sensed and collected to the dispatch center. Previous research has shown that such data contains rich information about passenger and taxi mobility patterns [31], [24], [23]. Moreover, recent studies have shown that the passenger demand information can be extracted and used to reduce passengers’ waiting time, taxi cruising time, or future supply rebalancing cost to serve requests [16], [25], [32]. Efficient coordination of taxi networks at a large scale is a challenging task. Traditional taxi networks in metropolitan areas largely rely on taxi drivers’ experience to look for passengers on streets to maximize individual profit. However, such self-interested, uncoordinated behaviors of drivers usually result in spatiotemporal mismatch between taxi supply and passenger demand. In large taxi companies that provide dispatch services, greedy algorithms based on locality are widely employed, such as finding the nearest vacant taxi to pick up a passenger [18], or first-come, first-served. Though these algorithms are easy to implement and manage, they prioritize immediate customer satisfaction over global resource utilization and service fairness, as the cost of rebalancing the entire taxi supply network for future demand is not considered. Our goal is to utilize real-time information to optimize taxi network dispatch for anticipated future idle driving cost and global geographical service fairness, while fulfilling current, local passenger demand. To accomplish such a goal, we 2 incorporate both system models learned from historical data and real-time taxi data into a taxi network control framework. To the best of our knowledge, this is the first work to consider this problem. The preliminary version of this work can be found in [19], and more details about problem formulation, algorithm design, and numerical evaluations are included in this manuscript. In this paper, we design a computationally efficient moving time horizon framework for taxi dispatch with large-scale realtime information of the taxi network. Our dispatch solutions in this framework consider future costs of balancing the supply demand ratio under realistic constraints. We take a receding horizon control (RHC) approach to dynamically control taxis in large-scale networks. Future demand is predicted based on either historical taxi data sets [5] or streaming data [31]. The real-time GPS and occupancy information of taxis is also collected to update supply and demand information for future estimation. This design iteratively regulates the mobility of idle taxis for high performance, demonstrating the capacity of large-scale smart transportation management. The contributions of this work are as follows, • To the best of our knowledge, we are the first to design an RHC framework for large-scale taxi dispatching. We consider both current and future demand, saving costs under constraints by involving expected future idle driving distance for re-balancing supply. • The framework incorporates large-scale data in real-time control. Sensing data is used to build predictive passenger demand, taxi mobility models, and serve as real-time feedback for RHC. • Extensive trace driven analysis based on a San Francisco taxi data set shows that our approach reduces average total taxi network idle distance by 52% as in Figure 5, and the error between local and global supply demand ratio by 45% as in Figure 7, compared to the actual historical taxi system performance. • Spatiotemporal context information such as disruptive passenger demand is formulated as uncertainty sets of parameters into a robust dispatch problem. This allows the RHC framework to provide more robust control solutions under uncertain contexts as shown in Figure 8. The error between local and global supply demand ratio is reduced by 25% compared with the error of solutions without considering demand uncertainties. perspective in work [30], but the real-time location information is not utilized in the algorithm. Seow et.al focus on minimizing total customer waiting time by concurrently dispatching multiple taxis and allowing taxis to exchange their booking assignments [27]. A shortest time path taxi dispatch system based on real-time traffic conditions is proposed by Lee et.al [15]. In [26], [13], [25], authors aim to maximize drivers’ profits by providing routing recommendations. These works give valuable results, but they only consider the current passenger requests and available taxis. Our design uses receding horizon control to consider both current and predicted future requests. Various mobility and vehicular network modeling techniques have been proposed for transportation systems [6], [4]. Researchers have developed methods to predict travel time [8], [11] and traveling speed [2], and to characterize taxi performance features [16]. A network model is used to describe the demand and supply equilibrium in a regulated market is investigated [29]. These works provide insights to transportation system properties and suggest potential enhancement on transportation system performance. Our design takes a step further to develop dispatch methods based on available predictive data analysis. There is a large number of works on mobility coordination and control. Different from taxi services, these works usually focus on region partition and coverage control so that coordinated agents can perform tasks in their specified regions [7], [1], [12]. Aircraft dispatch system and air traffic management in the presence of uncertainties have been addressed [3], [28], while the task models and design objectives are different from taxi dispatching problem. Also, receding horizon control (RHC) has been widely applied for process control, task scheduling, and multi-agent transportation networks [20], [14], [17]. These works provide solid results for related mobility scheduling and control problems. However, none of these works incorporates both the real-time sensing data and historical mobility patterns into a receding horizon control design, leveraging the taxi supply based on the spatiotemporal dynamics of passenger demand. The rest of the paper is organized as follows. The background of taxi monitoring system and control problems are introduced in Section II. The taxi dispatch problem is formally formulated in Section III, followed by the RHC framework design in Section IV. A case study with a real taxi data set from San Francisco to evaluation the RHC framework is shown in Section V. Concluding remarks are made in Section VI. A. State-of-the-Art There are three categories of research topics related to our work: taxi dispatch systems, transportation system modeling, and multi-agent coordination and control. A number of recent works study approaches of taxi dispatching services or allocating transportation resources in modern cities. Zhang and Pavone [32] designed an optimal rebalancing method for autonomous vehicles, which considers both global service fairness and future costs, but they didn’t take idle driving distance and real-time GPS information into consideration. Truck schedule methods to reduce costs of idle cruising and missing tasks are designed in the temporal II. TAXI D ISPATCH P ROBLEM : M OTIVATION AND S YSTEM Taxi networks provide a primary transportation service in modern cities. Most street taxis respond to passengers’ requests on their paths when passengers hail taxis on streets. This service model has successfully served up to 25% public passengers in metropolitan areas, such as San Francisco and New York [10], [21]. However, passenger’s waiting time varies at different regions of one city and taxi service is not satisfying. In the recent years, ”on demand” transportation service providers like Uber and Lyft aim to connect a passenger directly with a driver to minimize passenger’s waiting 3 time. This service model is still uncoordinated, since drivers may have to drive idly without receiving any requests, and randomly traverse to some streets in hoping to receive a request nearby based on experience. Our goal in this work is a centralized dispatch framework to coordinate service behavior of large-scale taxi Cyber-Physical system. The development of sensing, data storage and processing technologies provide both opportunities and challenges to improve existing taxi service in metropolitan areas. Figure 1 shows a typical monitoring infrastructure, which consists of a dispatch center and a large number of geographically distributed sensing and communication components in each taxi. The sensing components include a GPS unit and a trip recorder, which provides real-time geographical coordinates and occupancy status of every taxi to the dispatch center via cellular radio. The dispatch center collects and stores data. Then, the monitoring center runs the dispatch algorithm to calculate a dispatch solution and sends decisions to taxi drivers via cellular radio. Drivers are notified over the speaker or on a special display. Given both historical data and real-time taxi monitoring information described above, we are capable to learn spatiotemporal characteristics of passenger demand and taxi mobility patterns. This paper focuses on the dispatch approach with the model learned based on either historical data or streaming data. One design requirement is balancing spatiotemporal taxi supply across the whole city from the perspective of system performance. It is worth noting that heading to the allocated position is part of idle driving distance for a vacant taxi. Hence, there exists trade-off between the objective of matching supply and demand and reducing total idle driving distance. We aim at a scalable control framework that directs vacant taxis towards demand, while balancing between minimum current and anticipated future idle driving distances. III. TAXI D ISPATCH P ROBLEM F ORMULATION Informally, the goal of our taxi dispatch system is to schedule vacant taxis towards predicted passengers both spatially and temporally with minimum total idle mileage. We use supply demand ratio of different regions within a time period as a measure of service quality, since sending more taxis for more requests is a natural system-level requirement to make customers at different locations equally served. Similar Dispatch Center Passenger Distribution Taxi Mobility Real-Time Control Cellular Ratio GPS Pickup & Delivery Figure 1. Occupancy Sensing A prototype of the taxi dispatch system service metric of service node utilization rate has been applied in resource allocation problems, and autonomous driving car mobility control approach [32]. The dispatch center receives real-time sensing streaming data including each taxi’s GPS location and occupancy status with a time stamp periodically. The real-time data stream is then processed at the dispatch center to predict the spatiotemporal patterns of passenger demand. Based on the prediction, the dispatch center calculates a dispatch solution in real-time, and sends decisions to vacant taxis with dispatched regions to go in order to match predicted passenger demands. Besides balancing supply and demand, another consideration in taxi dispatch is minimizing the total idle cruising distance of all taxis. A dispatch algorithm that introduces large idle distance in the future after serving current demands can decrease total profits of the taxi network in the long run. Since it is difficult to perfectly predict the future of a large-scale taxi service system in practice, we use a heuristic estimation of idle driving distance to describe anticipated future cost associated with meeting customer requests. Considering control objectives and computational efficiency, we choose a receding horizon control approach. We assume that the optimization time horizon is T , indexed by k = 1, . . . , T , given demand prediction during time [1, T ]. Notation In this paper, we denote 1N as a length N column vector of all 1s, and 1TN is the transpose of the vector. Superscripts of variables as in X k , X k+1 denote discrete time. We denote the j-th column of matrix X k as X·jk . A. Supply and demand in taxi dispatch We assume that the entire area of a city is divided into n regions such as administrative sub-districts. We also assume that within a time slot k, the total number of passenger requests at the j-th region is denoted by rjk . We also use rk , [r1k , . . . , rnk ] ∈ R1×n to denote the vector of all requests. These are the demands we want to meet during time k = 1, . . . , T with minimal idle driving cost. Then the total number of predicted requests in the entire city is denoted by n P Rk = rjk . j=1 We assume that there are total N vacant taxis in the entire city that can be dispatched according to the real-time occupancy status of all taxis. The initial supply information consists of real-time GPS position of all available taxis, denoted by P 0 ∈ RN ×2 , whose i-th row Pi0 ∈ R1×2 corresponds to the position of the i-th vacant taxi. While the dispatch algorithm does not make decisions for occupied taxis, information of occupied taxis affects the predicted total demand to be served by vacant taxis, and the interaction between the information of occupied taxis and our dispatch framework will be discussed in section IV. The basic idea of the dispatch problem is illustrated in Figure 2. Specifically, each region has a predicted number of requests that need to be served by vacant taxis, as well as locations of all vacant taxis with IDs given by real-time sensing 4 Parameters N n rk ∈ R1×n C k ∈ [0, 1]n×n P 0 ∈ RN ×2 Wi ∈ Rn×2 α ∈ RN β ∈ R+ Rk ∈ R+ Variables X k ∈ {0, 1}N ×n P k ∈ [0, 1]N ×n dki ∈ R+ Description the total number of vacant taxis the number of regions the total number of predicted requests to be served by current vacant taxis at each region matrix that describes taxi mobility patterns during one time slot the initial positions of vacant taxis provided by GPS data preferred positions of the i-th taxi at n regions the upper bound of distance each taxi can drive for balancing the supply the weight factor of the objective function total number of predicted requests in the city Description the dispatch order matrix that represents the region each vacant taxi should go predicted positions of dispatched taxis at the end of time slot k lower bound of idle driving distance of the i-th taxi for reaching the dispatched location Table I PARAMETERS AND VARIABLES OF THE RHC PROBLEM (8). {1,2,3} {4,5} 1 [4] 2 [1] {4,5} 1 [4] 2 [1] adjacent region {4} {2} {1,2,3} {4} {2} dispathing solution 4 [9] 3 [2] 4 [9] {7,8} {6} {7,8} 1TN X·jk 3 [2] {6} demand ratio for time slot k is defined as the total number of vacant taxis decided by the total number of customer requests during time slot k. When the supply demand ratio of every region equals to that of the whole city, we have the following equations for j = 1, . . . , n, k = 1, . . . , T , {6} (a) A dispatch solution – taxi 2 goes (b) A dispatch solution – taxi 2 goes to region 4, and taxi 4 goes to region to region 4, taxi 4 goes to region 3, 4. and taxi 6 goes to region 4. Figure 2. Unbalanced supply and demand at different regions before dispatching and possible dispatch solutions. A circle represents a region, with a number of predicted requests ([·] inside the circle) and vacant taxis ({ taxi IDs } outside the circle) before dispatching. A black dash edge means adjacent regions. A red edge with a taxi ID means sending the corresponding vacant taxi to the pointed region according to the predicted demand. information. We would like to find a dispatch solution that balances the supply demand ratio, while satisfying practical constraints and not introducing large current and anticipated future idle driving distance. Once dispatch decisions are sent to vacant taxis, the dispatch center will wait for future computing a new decision problem until updating sensing information in the next period. B. Optimal dispatch under operational constraints The decision we want to make is the region each vacant taxi should go. With the above initial information about supply and predicted demand, we define a binary matrix X k ∈ {0, 1}N ×n k as the dispatch order matrix, where Xij = 1 if and only if the i-th taxi is sent to the j-th region during time k. Then X k 1n = 1N , k = 1, . . . , T must be satisfied, since every taxi should be dispatched to one region during time k. 1) Two objectives: One design requirement is to fairly serve the customers at different regions of the city — vacant taxis should be allocated to each region according to predicted demand across the entire city during each time slot. To measure how supply matches demand at different regions, we use the metric—supply demand ratio. For region j, its supply rjk = 1TN X·jk rjk N , ⇐⇒ = , Rk N Rk (1) For convenience, we rewrite equation (1) as the following equation about two row vectors 1 T k 1 1 X = k rk , k = 1, · · · , T. (2) N N R However, equation (2) can be too strict if used as a constraint, and there may be no feasible solutions satisfying (2). This is because decision variables X k , k = 1, . . . , T are integer matrices, and taxis’ driving speed is limited that they may not be able to serve the requests from any arbitrary region during time slot k. Instead, we convert the constraint (2) into a soft constraint by introducing a supply-demand mismatch penalty function JE for the requirement that the supply demand ratio should be balanced across the whole city, and one objective of the dispatch problem is to minimize the following function JE = T X k=1 1 T k 1 1N X − k r k N R . (3) 1 The other objective is to reduce total idle driving distance of all taxis. The process of traversing from the initial location to the dispatched region will introduce an idle driving distance for a vacant taxi, and we consider to minimize such idle driving distance associated with meeting the dispatch solutions. We begin with estimate the total idle driving distance associated with meeting the dispatch solutions. For the convenience of routing process, the dispatch center is required to send the GPS location of the destination to vacant taxis. The decision variable X k only provides the region each vacant taxi should go, hence we map the region ID to a specific longitude and latitude position for every taxi. In practice, there are taxi stations on roads in metropolitan areas, and we assume that each taxi has a preferred station or is randomly assigned one at every region by the dispatch system. We denote the preferred geometry location matrix for the i-th taxi by Wi ∈ Rn×2 , and 5 [Wi ]j , where each row of Wi is a two-dimensional geometric position on the map. The j-th row of Wi is the dispatch k position sent to the i-th taxi when Xij = 1. k Once Xi is chosen, then the i-th taxi will go to the location Xik Wi , because equation holds P the following k k Xik Wi = q6=j Xiq [Wi ]q + Xij [Wi ]j = [Wi ]j ∈ R1×2 . k k k With a binary vector Xi that Xij = 1, Xiq = 0 for q 6= j, k we have Xiq Wi = [0 0] for q 6= j. Since Wi does not need to change with time k, the preferred location of each taxi at every region in the city is stored as a matrix W, stored in the dispatch center before the process of calculating dispatch solutions starts. When updating information of vacant taxis, matrix Wi is also updated for every current vacant taxi i. The initial position Pi0 is provided by GPS data. Traversing from position Pi0 to position Xi1 Wi for predicted demand will introduce a cost, since the taxi drives towards the dispatched locations without picking up a passenger. Hence, we consider minimizing the total idle driving distance introduced by dispatching taxis. Driving in a city is approximated as traveling on a grid road. To estimate the distance without knowing the exact path, we use the Manhattan norm or one norm between two geometric positions, which is widely applied as a heuristic distance in path planning algorithms [22]. We define dki ∈ R as the estimated idle driving distance of the i-th taxi for reaching the dispatched location Xik Wi . For k = 1, a lower bound of d1i is given by d1i > kPi0 − Xi1 Wi k1 , i = 1, . . . , N. =f k (Xik−1 Wi ), k f :R 1×2 →R 1×2 , (5) where f k is a convex function, called a mobility pattern function. To reach the dispatched location Xik Wi at the beginning of time k from position Pik−1 , the approximated driving distance is dki > kf k (Xik−1 Wi ) − Xik Wi k1 . (6) The process to calculate a lower bound for dki is illustrated in Figure 3. Within time slot k, the distance that every taxi can drive should be bounded by a constant vector αk ∈ RN : dk 6 α k . Total idle driving distance of all vacant taxis though time k = 1, . . . , T to satisfy service fairness is then denoted by JD = T X N X dki . min. X k ,dk J = JE + βJD (4) For k > 2, to estimate the anticipated future idle driving distance induced by reaching dispatched position Xik Wi at time k, we consider the trip at the beginning of time slot k starts at the end location of time slot k − 1. However, during time k − 1, taxis’ mobility patterns are related to pick-up and drop-off locations of passengers, which are not directly controlled by the dispatch center. So we assume the predicted ending position for a pick-up location Xik−1 Wi during time k − 1 is related to the starting position Xik−1 Wi as follows: Pik−1 distance is nonzero only when a vacant taxi is dispatched to a different region. We also require that the estimated distance is a closed form function of the locations of the original and dispatched regions, without knowledge about accurate traffic conditions or exact time to reach the dispatched region. Hence, in this work we use Manhattan norm to approximate the idle distance—it is a closed form function of the locations of the original and dispatched regions. When accessibility information of the road traffic network is considered in estimating street-level distances, for the case that a taxi may not drive on rectangular grids to pick up a passenger (for instance, when a U-turn is necessary), Lee et.al have proposed a shortest time path approach to pick up passengers in shortest time [15]. 2) An RHC problem formulation: Since there exists a tradeoff between two objectives as discussed in Section II, we define a weight parameter β k when summing up the costs related to both objectives. A list of parameters and variables is shown in Table I. When mixed integer programming is not efficient enough for a large-scale taxi network regarding to the problem size, one standard relaxation method is replacing the k k constraint Xij ∈ {0, 1} by 0 ≤ Xij ≤ 1. To summarize, we formulate the following problem (8) based on the definitions of variables, parameters, constraints and objective function = s.t d1i dki > T X 1 1 T k 1 X − k rk N N R k=1 kPi0 k − Xi1 Wi k1 , (Xik−1 Wi ) − > kf i = 1, . . . , N, k k d 6α , ! dki i=1 i = 1, . . . , N, Xik Wi k1 , k = 1, 2, . . . , T, X 1n = 1N , 06 1 N X k = 2, . . . , T, k k Xij +β k k = 1, 2, . . . , T, 6 1, i ∈ {1, . . . , N }, j ∈ {1, . . . , n}. (8) After getting an optimal solution X 1 of problem (8), for the i-th taxi, we may recover binary solution through rounding by setting the largest value of Xi1 to 1, and the others to 0. This may violate the constraint of d0i , but since we set a conservative upper bound αk , and the rounding process will return a solution that satisfies dki 6 αk +  with bounded , the dispatch solution can still be executed during time slot k. Latitude Estimated distance Possible paths Longitude (7) k=1 i=1 It is worth noting that the idle distance we estimate here is the region-level distance to pick up predicted passengers — the Figure 3. Illustration of the process to estimate idle driving distance to the dispatched location for the i-th taxi at k = 2: predict ending location of k = 1 denoted by EPi1 in (9), get the distance between locations EPi1 and Xi2 Wi denoted by d2i in (10). 6 C. Discussions on the optimal dispatch formulation 1) Why use supply demand ratio as a metric: An intuitive measurement of the difference between the number of vacant taxis and predicted total requests at all regions is: n P e= |skj − rjk |, j=1 where skj is the total number of vacant taxis sent to the j-th region. However, when the total number of vacant taxis and requests are different in the city, this error e can be large even under the case that more vacant taxis are already allocated to busier regions and fewer vacant taxis are left to regions with less predicted demand. We do not have an evidence whether the dispatch center already fairly allocates supply according to varying demand given the value of the above error e. 2) The meaning of αk : For instance, when the length of time slot k is one hour, and αk is the distance one taxi can traverse during 20 minutes of that hour, this constraint means a dispatch solution involves the requirement that a taxi should be able to arrive the dispatched position within 20 minutes in order to fulfill predicted requests. With traffic condition monitoring and traffic speed predicting method [2], αk can be adjusted according to the travel time and travel speed information available for the dispatch system. This constraint also gives the dispatch system the freedom to consider the fact that drivers may be reluctant to drive idly for a long distance to serve potential customers, and a reasonable amount of distance to go according to predicted demand is acceptable. The threshold αk is related to the length of time slot. In general, the longer a time slot is, the larger αk can be, because of constraints like speed limit. 3) One example of mobility pattern function f k : When taxi’s mobility pattern duringPtime slot k is described by a n k matrix C k ∈ Rn×n satisfying j=1 Cij = 1, where Cij is the probability that a vacant taxi starts within region i will end within region j during time k. From the queueing-theoretical perspective such probability transition matrix approximately describes passenger’s mobility [32]. Given Xik−1 and the mobility pattern matrix C k−1 ∈ [0, 1]n×n , the probability of ending at each region for taxi i is p= n X k−1 [C k−1 ]j I(Xij = 1) = Xik−1 C k−1 ∈ R1×n , j=1 k−1 where the indicator function I(Xij = 1) = 1 if and k−1 k−1 only if Xij = 1, and [C ]j is the j-th row of C k−1 . However, introducing a stochastic formula in the objective function will cause high computational complexity for a largescale problem. Hence, instead of involving the probability of taking different paths in the objective function to formulate a stochastic optimization problem, we take the expected value of the position of i-th taxi by the end of time k − 1 Pik−1 = n X pj [Wi ]j = pWi = Xik−1 C k−1 Wi . (9) j=1 Here Pik−1 ∈ R1×2 is a vector representing a predicted ending location of the i-th taxi on the map at each dimension. Then a lower bound of idle driving distance for heading to Xik Wi to meet demand during k is given by the distance between Pik−1 defined as (9) and Xik Wi . dki > k(Xik−1 C k−1 − Xik )Wi k1 . (10) In particular, when the transition probability C k , k = 1, . . . , T is available, we can replace the constraint about dki by dki > k(Xik−1 C k−1 − Xik )Wi k1 . It is worth noting that dki is a function of Xik−1 and k Xi , and the estimation accuracy of idle driving distance to dispatched positions Xik (k = 2, . . . , T ) depends on the predicting accuracy of the mobility pattern during each time slot k, or Pik−1 . The distance d1 is calculated based on realtime GPS location P 0 and dispatch position X 1 , and we use estimations d2 , . . . , dT to measure the anticipated future idle driving distances for meeting requests. The error of estimated C k mainly affects the choice of idle distance dk when the true ending region of a taxi by the end of time slot k is not as predicted based on its starting region at time slot k. This is because C k determines the constraint for dk (k = 2, 3, . . . , T ) as described by inequality (10). However, the system also collects real-time GPS positions to make a new decision based on the current true positions of all taxis, instead of only applying predicted locations provided by the mobility pattern matrix. According to constraint (4) distance d1 is determined by GPS sensing data P 0 and dispatch decision X 1 , and only X 1 will be executed sent to vacant taxis as the dispatch solutions after the system solving problem (8). From this perspective, real-time GPS and occupancy status sensing data is significant to improve the system’s performance when we utilize both historical data and real-time sensing data. We also consider the effect of an inaccurate mobility pattern estimation C k when choosing the prediction time horizon T — large prediction horizon will induce accumulating prediction error in matrix C k and the dispatch performance will even be worse. Evaluation results in Section V show how real-time sensing data helps to reduce total idle driving distance and how the mobility pattern error of different prediction horizon T affects the system’s performance. 4) Information on road congestion and passenger destination: When road congestion information is available to the dispatch system, function in (5) can be generalized to include real-time congestion information. For instance, there is a high probability that a taxi stays within the same region during one time slot under congestions. We do not assume that information of passenger’s destination is available to the system when making dispatch decisions, since many passengers just hail a taxi on the street or at taxi stations instead of reserving one in advance in metropolitan areas. When the destination and travel time of all trips are provided to the dispatch center via additional software or devices as prior knowledge, the trip information is incorporated to the definition of ending position function (5) for problem formulation (8). With more accurate trip information, we get a better estimation of future idle driving distance when making dispatch decisions for k = 1. 5) Customers’ satisfaction under balanced supply demand ratio: The problem we consider in this work is reaching fair service to increase global level of customers’ satisfaction, 7 which is indicated by a balanced supply demand ratio across different regions of one city, instead of minimizing each individual customer’s waiting time when a request arrives at the dispatch system. Similar service fairness metric has been applied in mobility on demand systems [32], and supply demand ratio considered as an indication of utilization ratio of taxis is also one regulating objective in taxi service market [29]. For the situation that taxi i will not pick up passengers in its original region but will be dispatched to another region, the dispatch decision results from the fact that global customers’ satisfaction level will be increased. For instance, when the original region of taxi i has a higher supply demand ratio than the dispatched region, going to the dispatched region will help to increase customer’s satisfaction in that region. By sending taxi i to some other region, customers’ satisfaction in the dispatched region can be increased, and the value of the supply-demand cost-of-mismatch function JE can be reduced without introducing much extra total idle driving distance JD . D. Robust RHC formulations Previous work has developed multiple ways to learn passenger demand and taxi mobility patterns [2], [8], [13], and accuracy of the predicted model will affect the results of dispatch solutions. We do not have perfect knowledge of customer demand and taxi mobility models in practice, and the actual spatial-temporal profile of passenger demands can deviate from the predicted value due to random factors such as disruptive events. Hence, we discuss formulations of robust taxi dispatch problems based on (8). Formulation (8) is one computationally tractable approach to describe the design requirements with a nominal model. One advantage of the formulation (8) is its flexibility to adjust the constraints and objective function according to different conditions. With prior knowledge of scheduled events that disturb the demand or mobility pattern of taxis, we are able to take the effects of the events into consideration by setting uncertainty parameters. For instance, when we have basic knowledge that total demand in the city during time k is about R̃k , but each region rjk belongs to some uncertainty set, denoted by an entry wise inequality R1k  rk  R2k , n k k given R1 ∈ R and R2 ∈ Rn . Then k k rjk ∈ [R1j , R2j ], j = 1, . . . , n Theorem 1. The robust RHC problem (12) is equivalent to the following computationally efficient convex optimization problem   T n N X X X  J0 = min tkj + β k dki  X k ,dk ,tk constraints of problem (8). (12) j=1 k=1 1N X·jk − s.t tkj ≥ N 1N X·jk tkj ≥ − N j = 1, . . . , n, i=1 k R1j R̃k k R2j , tkj ≥ k R1j R̃k k R2j , tkj ≥ R̃k R̃k , k = 1, . . . , T, 1N X·jk , N 1N X·jk − , N − (13) constraints of problem (8). Proof: In the objective function, only the first term is related to rk . To avoid the maximize expression over an uncertain rk , we first optimize the term over rk for any fixed X k . Let X·jk represent the j-th column of X k , then 1 k 1 T k 1N X − r N R̃k 1 n X rjk 1 T k 1 X − = max N N ·j R̃k R1k r k R2k j=1 max R1k r k R2k = n X j=1 max k ,Rk ] rjk ∈[R1j 2j (14) rjk 1 T k 1N X·j − . N R̃k The second equality holds because each rjk can be optimized k k separately in this equation. For R1j ≤ rjk ≤ R2j , we have k R1j ≤ rjk ≤ k R2j . R̃k R̃k R̃k Then the problem is to maximize each absolute value in (14) for j = 1, . . . , n. Consider the following problem for x, a, b ∈ R to examine the character of maximization problem over an absolute value: ( |x − a|, if x > (a + b)/2 max |x − x0 | = x0 ∈[a,b] |x − b|, otherwise (11) is an uncertainty parameter instead of a fixed value as in problem (8). Without additional knowledge about the change of total demand in the whole city, we denote R̃k as the approximated total demand in the city under uncertain rjk for each region. By introducing interval uncertainty (11) to rk and fixing R̃k on the denominator, we have the following robust optimization problem (12) ! N T X X 1 k 1 T k k k 1 X − r +β di min. max N N X k ,dk R1k r k R2k R̃k 1 i=1 k=1 s.t. The robust optimization problem (12) is computationally tractable, and we have the following Theorem 1 to show the equivalent form to provide real-time dispatch decision. = max{|x − a|, |x − b|} = max{x − a, a − x, x − b, b − x}. Similarly, for the problem related to rjk , we have max k ,Rk ] rjk ∈[R1j 2j ( =max rjk 1N X·jk − N R̃k k 1N X·jk R1j − , N R̃k k 1N X·jk R2j − N R̃k (15) ) . Thus, with slack variables tk ∈ Rn , we re-formulate the robust RHC problem as (13). Taxi mobility patterns during disruptive events can not be easily estimated (in general), however, we have knowledge 8 such as a rough number of people are taking part in a conference or competition, or even more customer reservations because of events in the future. The uncertain set of predicted demand rk can be constructed purely from empirical data such as confidence region of the model, or external information about disruptive events. By introducing extra knowledge besides historical data model, the dispatch system responds to such disturbances with better solutions than the those without considering model uncertainties. Comparison of results of (13) and problem (8) is shown in Section V. IV. RHC F RAMEWORK D ESIGN Demand and taxi mobility patterns can be learned from historical data, but they are not sufficient to calculate a dispatch solution with dynamic positions of taxis when the positions of the taxis change in real time. Hence, we design an RHC framework to adjust dispatch solutions according to real-time sensing information in conjunction with the learned historical model. Real-time GPS and occupancy information then act as feedback by providing the latest taxi locations, and demand-predicting information for an online learning method like [31]. Solving problem (8) or (12) is the key iteration step of the RHC framework to provide dispatch solutions. RHC works by solving the cost optimization over the window [1, T ] at time k = 1. Though we get a sequence of optimal solutions in T steps – X 1 , . . . , X T , we only send dispatch decisions to vacant taxis according to X 1 . We summarize the complete process of dispatching taxis with both historical and real-time data as Algorithm 1, followed by a detail computational process of each iteration. The lengths of time slots for learning historical models (t1 ) and updating realtime information (t2 ) do not need to be the same, hence in Algorithm 1 we consider a general case for different t1 , t2 . Algorithm 1: RHC Algorithm for real-time taxi dispatch Inputs: Time slot length t1 minutes, period of sending dispatch solutions t2 minutes (t1 /t2 is an integer); a preferred station location table W for every taxi in the network; estimated request vectors r̂(h1 ), h1 = 1, . . . , 1440/t1 , mobility patterns fˆ(h2 ), h2 = 1, . . . , 1440/t2 ; prediction horizon T ≥ 1. Initialization: The predicted requests vector r = r̂(h1 ) for corresponding algorithm start time h1 . while Time is the beginning of a t2 time slot do (1) Update sensor information for initial position of vacant taxis P 0 and occupied taxis P 00 , total number of vacant taxis N , preferred dispatch location matrices Wi . if time is the beginning of an h1 time slot then Calculate r̂(h1 ) if the system applies an online training method; count total number of occupied taxis no (h1 ); update vector r. end (2) Update the demand vectors rk based on predicted demand r̂(h1 ) and potential service ability of no (h1 ) occupied taxis; update mobility functions f k (·) (for example, C k ), set up values for idle driving distance threshold αk and objective weight β k , k = 1, 2, . . . , T . (3) if there is knowledge of demand rk as an uncertainty set ahead of time then solve problem (13); else solve problem (8) for a certain demand model; end (4) Send dispatch orders to vacant taxis according to the optimal solution of matrix X 1 . Let h2 = h2 + 1. end Return:Stored sensor data and dispatch solutions. A. RHC Algorithm Remark 1. Predicted values of requests r̂(h1 ) depend on the modeling method of the dispatch system. For instance, if the system only applies historical data set to learn each r̂(h1 ), r̂(h1 ) is not updated with real-time sensing data. When the system applies online training method such as [31] to update r̂(h1 ) for each h1 , values of r, rk are calculated based on the real-time value of r̂(h1 ). 1) Update r: We receive sensing data of both occupied and vacant taxis in real-time. Predicted requests that vacant taxis should serve during h1 is re-estimated at the beginning of each h1 time. To approximate the service capability when an occupied taxi turns into vacant during time h1 , we define the total number of drop off events at different regions as a vector dp(h1 ) ∈ Rn×1 . Given dp(h1 ), the probability that a drop off event happens at region j is (16) roj (h1 ) = dpdj (h1 ) × no (h1 )e, where d·e is the ceiling function, no (h1 ) is the total number of current occupied taxis at the beginning of time h1 provided by real-time sensor information of occupied taxis. Let r = r̂(h1 ) − ro (h1 ), then the estimated service capability of occupied taxis is deducted from r for time slot h1 . 2) Update rk for problem (8): We assume that requests are uniformly distributed during h1 . Then for each time k of length t2 , if the corresponding physical time is still in the current h1 time slot, the request is estimated as an average part of r; else, it is estimated as an average part for time slot h1 + 1, h1 + 2, . . . , etc. The rule of choosing rk is (1 if (k + h2 − 1)t2 ≤ h1 t1 H r, k r = 1 l (k+h2 −1)t2 m , otherwise H r̂ t1 where dpj (h1 ) is the number of drop off events at region j during h1 . We assume that an occupied taxi will pick up at least one passenger within the same region after turning vacant, and we approximate future service ability of occupied taxis at region j during time h1 as where H = t1 /t2 . 3) Update rk for robust dispatch (13): When there are disruptive events and the predicted requests number is a range r̂(h1 ) ∈ [R̂1 (h1 ), R̂2 (h1 )], similarly we set the uncertain set of rk as the following interval for the computationally efficient pdj (h1 ) = dpj (h1 )/1n dp(h1 ), 9 form of robust dispatch problem (13) h i  1  R̂ (h ) − r (h ), R̂ (h ) − r (h ) , 1 1 o 1 2 1 o 1   H k r ∈ if (k h +h mi l2 − 1)t2 ≤ mh1 t1 , l    1 R̂ ( (k+h2 −1)t2 ), R̂ ( (k+h2 −1)t2 ) , o.w. 2 1 H t1 t1 4) Spatial and temporal granularity of Algorithm 1: The main computational cost of each iteration is on step (3), and t2 should be no shorter than the computational time of the optimization problem. We regulate parameters according to experimental results based on a given data set, since there are no closed form equations to decide optimal design values of these parameters. For the parameters we estimate from a given GPS dataset, the method we use in the experiments (but not restricted to it) will be discussed in Section V. The length of every time slot depends on the predict precision of prediction, desired control outcome, and the available computational resources. We can set a large time horizon to consider future costs in the long run. However, in practice we do not have perfect predictions, thus a large time horizon may amplify the prediction error over time. Applying real-time information to adjust taxi supply is a remedy to this problem. Modeling techniques are beyond the scope of this work. If we have perfect knowledge of customer demand and taxi mobility models, we can set a large time horizon to consider future costs in the long run. However, in practice we do not have perfect predictions, thus a large time horizon may amplify the prediction error over time. Likewise, if we choose a small look-ahead horizon, then the dispatch solution may not count on idle distance cost of the future. Applying real-time information to adjust taxi supply is a remedy to this problem. With an approximated mobility pattern matrix C k , the dispatch solution with large T is even worse than small T . 5) Selection process of parameters β k , αk , and T : The process of choosing values of parameters for Algorithm 1 is a trial and adjusting process, by increasing/decreasing the parameter value and observing the changing trend of the dispatch cost, till a desired performance is reached or some turning point occurs that the cost is not reduced any more. For instance, objective weight β k is related to the objective of the dispatch system, whether it is more important to reach fair service or reduce total idle distance. Some parameter is related to additional information available to the system besides realtime GPS and occupancy status data; for instance, αk can be adjusted according to the average speed of vehicles or traffic conditions during time k as discussed in subsection III-C2. Adjustments of parameters such as objective weight β k , idle distance threshold αk , prediction horizon T when considering the effects of model accuracy, control objectives are shown in Section V. A formal parameter selection method is a direction for future work. B. Multi-level dispatch framework We do not restrict the data source of customer demand – it can be either predicted results or customer reservation records. Some companies provide taxi service according to the current requests in the queue. For reservations received by the dispatch center ahead of time, the RHC framework in Algorithm 1 is compatible with this type of demand information — we then assign value of the waiting requests vector rk , taxi mobility function f k in (8) according to the reservations, and the solution is subject to customer bookings. For customer requests received in real-time, a multi-level dispatch framework is available based on Algorithm 1. The process is as follows: run Algorithm 1 with predicted demand rk , and send dispatch solutions to vacant taxis. When vacant taxis arrive at dispatched locations, the dispatch center updates real-time demand such as bookings that recently appear in the system, then the dispatch method based on current demand such as the algorithm designed by Lee et al. [15] can be applied. By this multi-level dispatch framework, vacant taxis are pre-dispatched at a regional level according to predicted demand using the RHC framework, and then specific locations to pick up a passenger who just booked a taxi is sent to a vacant taxi according to the shortest time path [15], with the benefit of real-time traffic conditions. V. C ASE S TUDY: M ETHOD E VALUATION We conduct trace-driven simulations based on a San Francisco taxi data set [24] summarized in Table II. In this data set, a record for each individual taxi includes four entries: the geometric position (latitude and longitude), a binary indication of whether the taxi is vacant or with passengers, and the Unix epoch time. With these records, we learn the average requests and mobility patterns of taxis, which serve as the input of Algorithm 1. We note that our learning model is not restricted to the data set used in this simulation, and other models [31] and date sets can also be incorporated. We implement Algorithm 1 in Matlab using the optimization toolbox called CVX [9]. We assume that all vacant taxis follow the dispatch solution and go to suggested regions. Inside a target region, we assume that a vacant taxi automatically picks up the nearest request recorded by the trace data, and we calculate the total idle mileage including distance across regions and inside a region by simulation. The trace data records the change of GPS locations of a taxi in a relatively small time granularity such as every minute. Moreover, there is no additional information about traffic conditions or the exact path between two consecutive data points when they were recorded. Hence, we consider the path of each taxi as connected road segments determined by each two consecutive points of the trace data we use in this section. Assume the latitude and longitude values of two consecutive points in the trace data are [lx1 , ly1 ] and [lx2 , ly2 ], for a short road segment, the mileage distance between the two points (measured in one minute) is approximated as being proportional to the value (|lx1 − lx2 | + |ly1 − ly2 |). The geometric location of a taxi is directly provided by GPS data. Hence, we calculate geographic distance directly from the data first, and then convert the result to mileage. Experimental figures shown in Subsection V-B and V-D are average results of all weekday data from the data set II. Results shown in Subsection V-C are based on weekend data. 10 Taxicab GPS Data set Number of Taxis Data Size 500 90M B Collection Period 05/17/08-06/10/08 Record Number 1, 000, 000 ID Date and Time Format Status Direction Speed GPS Coordinates 600 Average requests number during a weekday Region 3 Region 4 Region 7 Region 10 400 600 Average requests number during weekends Region 3 Region 4 Region 7 Region 10 400 200 0 Average requests number Average requests number Table II S AN F RANCISCO DATA IN THE E VALUATION S ECTION . G IANT BASEBALL GAME IN AT&T PARK ON M AY 31, 2008 IS A DISRUPTIVE EVENT WE USE FOR EVALUATING THE ROBUST OPTIMIZATION FORMULATION . 200 2 4 6 8 10 12 14 16 18 20 22 24 Time: hour 0 2 4 6 Average drop off events number of a weekday 600 Region 3 Region 4 Region 7 400 Region 10 200 0 2 4 6 8 10 12 14 16 18 20 22 24 Time: hour (b) Requests during weekends Average drop off events number Average drop off events number (a) Requests during weekdays 8 10 12 14 16 18 20 22 24 Time: hour Average drop off events during weekends Region 3 Region 4 Region 7 Region 10 600 400 200 0 2 4 6 8 10 12 14 16 18 20 22 24 Time: Hour (c) Drop off during weekdays (d) Drop off during weekends Figure 4. Requests at different hours during weekdays and weekends, for four selected regions. A given historical data set provides basic spatiotemporal information about customer demands, which we utilize with real-time data to dispatch taxis. A. Predicted demand based on historical data Requests during different times of a day in different regions vary a lot, and Figure 4(a) and Figure 4(b) compare bootstrap results of requests r̂(h1 ) on weekdays and weekends for selected regions. This shows a motivation of this work— necessary to dispatch the number of vacant taxis according to the demand from the perspective of system-level optimal performance. The detailed process is described as follows. The original SF data set does not provide the number of pick up events, hence one intuitive way to determine a pick up (drop off) event is as follows. When the occupancy binary turns from 0 to 1 (1 to 0), it means a pick up (drop off) event has happened. Then we use the corresponding geographical position to determine which region this pick up (drop off) belongs to; use the time stamp data to decide during which time slot this pick up (drop off) happened. After counting the total number of pick up and drop off events during each time slot at every region, we obtain a set of vectors rd0 (hk ), dpd0 (hk ), d0 = 1, . . . , d, where d is the number of days for historical data . In the following analysis, each time slot h1 is the time slot of predicting demand model chosen by the RHC framework. The SF data set includes about 24 days of data, so we use d = 18 for weekdays, and d = 6 for weekends. The bootstrap process for a given sample time number B = 1000 is given as follows. (a) Randomly sample a size d dataset with replacement from the data set {r1 (h1 ), . . . , rd (h1 )}, calculate r̂1 (h1 ) = d 1 X rd0 (h1 ), for h1 = 1, . . . , 1440/h1 . d 0 d =1 (b) Repeat step (a) for (B − 1) times, so that we have B estimates for each h1 , r̂b (h1 ), b = 1, . . . , B. The estimated mean value of P r̂(h1 ) based on B samples is B 1 r̂(h1 ) = B l=1 r̂l (h1 ). (c) Calculate the sample variance of the B estimates of r(h1 ) for each h1 , V̂r̂(h1 ) = B B 1 X b 1 X l (r̂ (h1 ) − r̂ (h1 )). B B b=1 (17) l=1 To estimate the demand range for robust dispatch problem (13) according to historical data, we construct the uncertain set of demand rk based on the mean and variance of the bootstrapped demand model. For every region j, the boundary of demand interval is defined as q R̃1,j (h1 ) = r̂j (h1 ) − V̂r̂(h1 ),j , q (18) R̃2,j (h1 ) = r̂j (h1 ) + V̂r̂(h1 ),j , 11 B. RHC with real-time sensor information To estimate a mobility pattern matrix Ĉ(h2 ), we define a matrix T (h2 ), where T (h2 )ij is the total number of passenger trajectories that starting at region i and ending at region j during time slot h2 . We also apply P bootstrap process to get T̂ (h2 ), and Ĉ(h2 )ij = T̂ (h2 )ij /( T̂ (h2 )ij ). Table III shows j Average total idle distance: mile one row of Ĉ(h2 ) for 5:00-6:00 pm during weekdays, the transition probability for different regions. The average cross validation error for estimated mobility matrix Ĉ(h2 ) of time slot h2 , h2 = 1, . . . , 24 during weekdays is 34.8%, which is a reasonable error for estimating total idle distance in the RHC framework when real-time GPS and occupancy status data is available. With only estimated mobility pattern matrix Ĉ(h2 ), the total idle distance is reduced by 17.6% compared with the original record without a dispatch method, as shown in Figure 5. We also tested the case when the dispatch algorithm is provided with the true mobility pattern matrix C k , which is impossible in practice, and the dispatch solution reduces the total idle distance by 68% compared with the original record. When we only have estimated mobility pattern matrices instead of the true value to determine ending locations and potential total idle distance for solving problem (8) or (13), updating real-time sensing data compensates the mobility pattern error and improves the performance of the dispatch framework. Real-time GPS and occupancy data provides latest position information of all vacant and occupied taxis. When dispatching available taxis with true initial positions, the total idle distance 600 480 Idle distance comparison Dispatch with RT information Dispatch without RT information No dispatch 360 240 120 0 2 4 6 8 10 12 14 16 18 20 22 24 Time: Hour Figure 5. Average idle distance comparison for no dispatch, dispatch without real-time data, and dispatch with real-time GPS and occupancy information. Idle distance is reduced by 52% given real-time information, compared with historical data without dispatch solutions. Figure 6. Heat map of passenger picking-up events in San Francisco (SF) with a region partition method. Region 3 covers several busy areas, include Financial District, Chinatown, Fisherman Wharf. Region 7 is mainly Mission District, Mission Bay, the downtown area of SF. is reduced by 52% compared with the result without dispatch methods, as shown in Figure 5, which is compatible with the performance when both true mobility pattern matrix C k and real-time sensing data are available. This is because the optimization problem (8) returns a solution with smaller idle distance cost given the true initial position information P 0 , instead of estimated initial position provided only by mobility pattern of the previous time slot in the RHC framework. Figure 5 also shows that even applying dispatch solution calculated without real-time information is better than non dispatched result. Based on the partition of Figure 6, Figure 7 shows that the supply demand ratio at each region of the dispatch solution with real-time information is closest to the supply demand ratio of the whole city, and the error N1 1TN X k − R1k rk 1 is reduced by 45% compared with no dispatch results. Even the supply demand ratio error of dispatching without realtime information is better than no dispatch solutions. We still allocate vacant taxis to reach a nearly balanced supply demand ratio regardless of their initial positions, but idle distance is increased without real-time data, as shown in Figure 5. Based on the costs of two objectives shown in Figures 5 and 7, the total cost is higher without real-time information, mainly results from a higher idle distance. C. Robust taxi dispatch One disruptive event of the San Francisco data set is Giant baseball at AT&T park, and we choose the historical record on May 31, 2008 as an example to evaluate the robust optimization formulation (12). Customer request number for Supply demand ratio where r̂j (h1 ) is the average value of each step (b) and V̂r̂(h1 ),j is the variance of estimated request number defined in (17). This one standard deviation range is used for evaluating the performance of robust dispatch framework in this work. Estimated drop off events vectors dp(h1 ) are also calculated via a similar process. Figure 4(c) and 4(d) show bootstrap results of passenger drop off events dp(h1 ) on weekdays and weekends for selected regions. For evaluation convenience, we partition the city map to regions with equal area. To get the longitude and latitude position Wi ∈ Rn×2 of vacant taxi i, we randomly pick up a station position in the city drawn from the uniform distribution. 2.0 Supply demand ratio under different conditions Global supply demand ratio 1.7 Dispatch without RT information 1.4 Dispatch with RT information No dispatch 1.1 0.8 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Region ID Figure 7. Supply demand ratio of the whole city and each region for different dispatch solutions. With real-time GPS and occupancy data, the supply demand ratio of each region is closest to the global level. The supply demand ratio mismatch error is reduced by 45% with real-time information, compared with historical data without dispatch solutions. 12 Region ID Transit probability Region ID Transit probability 1 0.0032 9 0.0001 2 0.0337 10 0.0050 3 0.5144 11 0.0340 4 0.0278 12 0.0136 5 0.0132 13 0.0018 6 0.0577 14 0.0082 7 0.1966 15 0.0248 8 0.0263 16 0.0396 Table III A N ESTIMATION OF STATE TRANSITION MATRIX BY BOOTSTRAP : ONE ROW OF MATRIX Ĉ(hk ) 2 1.998 1.718 5.434 10 2.049 1.096 13.009 without dispatch 2.664 4. 519 47.854 Table IV AVERAGE COST COMPARISON FOR DIFFERENT VALUES OF β k . areas near AT&T park is affected, especially Region 7 around the ending time of the game, which increases about 40% than average value. Figure 8 shows that with dispatch solution of the robust optimization formulation (12), the supply demand mismatch error N1 1TN X k − R1k rk 1 is reduced by 25% compared with the solution of (8) and by 46% compared with historical data without dispatch. The performance of robust dispatch solutions does not vary significantly and depends on what type of predicted uncertain demand is available when selecting the formulation of robust dispatch method. Even under solutions of (8), the total supply demand ratio error is reduced 28% compared historical data without dispatch. In general, we consider the factor of disruptive events in a robust RHC iteration, thus the system level supply distribution responses to the demand better under disturbance. D. Design parameters for Algorithm 1 Supply demand ratio Parameters like the length of time slots, the region division function, the objective weight parameter and the prediction horizon T of Algorithm 1 affect the results of dispatching cost in practice. Optimal values of parameters for each individual data set can be different. Given a data set, we change one parameter to a larger/smaller value while keep others the same, and compare results to choose a suboptimal value of the varying parameter. We compare the cost of choosing different parameters for Algorithm 1, and explain how to adjust parameters according to experimental results based on a given historical data set with both GPS and occupancy record. How the objective weight of (8) – β k affects the cost: The cost function includes two parts –the idle geographical distance (mileage) cost and the supply demand ratio mismatch 1.8 1.5 1.3 Supply demand ratio comparison Global supply demand ratio Robust optimization (9) Optimization (8) No dispatch 1.0 0.7 0.4 2 4 6 8 10 12 14 16 Region ID Figure 8. Comparison of supply demand ratio at each region under disruptive events, for solutions of robust optimization problems (12), problem (8) in the RHC framework, and historical data without dispatch. With the roust dispatch solutions of (12), the supply demand ratio mismatch error is reduced by 46%. cost. This trade-off between two parts is addressed by β k , and the weight of idle distance increases with β k . A larger β k returns a solution with smaller total idle geographical distance, while a larger error between supply demand ratio, i.e., a larger 1 k 1 T k value. The two components of the cost N 1N X − R k r 1 with different β k by Algorithm 1, and historical data without Algorithm 1 are shown in Table IV. The supply demand ratio mismatch is shown in the s/d error column. We calculate the total cost as (s/d error +β k × idle distance) (Use β k = 10 for the without dispatch column). Though with β k = 0 we can dispatch vacant taxis to make the supply demand ratio of each region closest to that of the whole city, a larger idle geographical distance cost is introduced compared with β k = 2 and β = 10. Compare the idle distance when β k = 0 with the data without dispatch, we get 23% reduction; compare the supply demand ratio error of β k = 10 with the data without dispatch, we get 32%. Average total idle distance during different hours of one day for a larger β k is smaller, as shown in Figure 10. The supply demand ratio error at different regions of one time slot is increased with larger β k , as shown in Figure 9. How to set idle distance threshold αk : Figure 11 compares the error between local supply demand ratio and global supply demand ratio. Since we directly use geographical distance Supply demand ratio for different Global supply demand ratio k =0 1.5 Supply demand ratio 0 0.645 3.056 0.645 1.3 k 1.1 k =10 k =2 0.9 0.7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Region ID Figure 9. Comparison of supply demand ratios at each region during one time slot for different β k values. When β k is smaller, we put less cost weight on idle distance that taxis are allowed to run longer to some region, and taxi supply matches with the customer requests better. Average total idle distance: mile βk s/d error idle distance total cost Average total idle distance for different =0 =2 180 = 10 240 120 60 0 2 4 6 8 10 12 14 16 18 20 22 24 Time: hour Figure 10. Average total idle distance of taxis at different hours. When β k is larger, the idle distance cost weights more in the total cost, and the dispatch solution causes less total idle distance. 13 Supply demand ratio for different Supply demand ratio 5 k k =0.02 global supply demand ratio No dispatch k =0.002 4 3 k 2 =0.08 1 0 8 10 12 14 16 Region ID Figure 11. Comparison of supply demand ratios at each region during one time slot for different αk . When αk is larger, vacant taxis can traverse longer to dispatched locations and match with customer requests better. 4 6 Total idle distance for different region partitions 4 regions 120 16 regions 64 regions 0 100 regions 256 regions 360 Average total idle distance: mile 90 60 30 0 8 10 12 14 16 18 20 22 24 Time: hour Figure 13. Average total idle distance at different time of one day compared for different prediction horizons.When T = 4, idle distance is decreased at most hours compared with T = 2. For T = 8 the costs are worst. 120 0 240 4 6 8 10 12 14 16 18 20 22 24 Time: hour Figure 12. Average total idle distance of all taxis during one day, for different region partitions. Idle distance decreases with a larger region-division number, till the number increases to a certain level. 6 t2=1 hour t2=30 mins t2=10 mins 180 t2 = 1 min 120 60 2 8 10 12 14 16 18 20 22 24 Time: hour Figure 14. Comparison of average total idle distance for different t2 – the length of time slot for updating sensor information. With a smaller t2 , the cost is smaller. But when t2 = 1 is too small to complete calculating problem (8), the dispatch result is not guaranteed to be better than t2 = 10. 4 6 Supply/demand ratio at each region for different t2 1.7 1.5 1.3 1.1 Global supply/demand ratio t2=10 mins t2=30 mins t2=1 min t2=1h 0.9 0.7 0.5 2 4 Average total idle distance for different t2 240 240 2 distance of vacant taxis at most hours of one day decreases as T increases. For T = 8 the driving distance is the largest. Theoretical reasons are discussed in Section IV. Decide the length of time slot t2 : For simplicity, we choose the time slot t1 as one hour, to estimate requests. A smaller time slot t2 for updating GPS information can reduce the total idle geographical distance with real-time taxi positions. However, one iteration of Algorithm 1 is required to finish in less than t2 time, otherwise the dispatch order will not work for the latest positions of vacant taxis, and the cost will increase. Hence t2 is constrained by the problem size and computation capability. Figure 14 shows that smaller t2 returns a smaller idle distance, but when t2 = 1 Algorithm 1 can not finish one step iteration in one minute, and the idle distance is not reduced. The supply demand ratio at each region does not vary much for t2 = 30, t2 = 10 minutes and t2 = 1 hour, as shown in Figure 15. Comparing two parts of costs, we get that t2 mainly affects the idle driving distance cost in practice. Supply/demand ratio Average total idle distance:mile 2 Average total idle distance for different T T=8 T=2 150 T=4 120 180 Average total idle distance: mile measured by the difference between longitude and latitude values of two points (GPS locations) on the map, the threshold value αk is small — 0.1 difference in GPS data corresponds to almost 7 miles distance on the ground. When αk increase, the error between local supply demand ratio and global supply demand ratio decreases, since vacant taxis are more flexible to traverse further to meet demand. How to choose the number of regions: In general, the dispatch solution of problem (8) for a vacant taxi is more accurate by dividing a city into regions of smaller area, since the dispatch is closer to road-segment level. However, we should consider other factors when deciding the number of regions, like the process of predicting requests vectors and mobility patterns based on historical data. A linear model we assume in this work is not a good prediction for future events when the region area is too small, since pick up and drop off events are more irregular in over partitioned regions. While Increasing n, we also increase the computation complexity. Note that the area of each region does not need to be the same as we divide the city in this experiment. Figure 12 shows that the idle distance will decrease with a larger region division number, but the decreasing rate slows down; while the region number increases to a certain level, the idle distance almost keeps steady. How to decide the prediction Horizon T : In general, when T is larger, the total idle distance to get a good supply demand ratio in future time slots should be smaller. However, when T is large enough, increasing T can not reduce the total idle distance any more, since the model prediction error compensates the advantage of considering future costs. For T = 2 and T = 4, Figure 13 shows that the average total idle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Region ID Figure 15. Comparison of the supply demand ratio at different regions for different lengths of time slot t2 . For t2 = 30, t2 = 10 mins and t2 = 1 hour, results are similar. For t2 = 1 min, the supply demand ratio is even worse at some regions, since the time slot is too short to complete one iteration. 14 VI. C ONCLUSION In this paper, we propose an RHC framework for the taxi dispatch problem. This method utilizes both historical and real-time GPS and occupancy data to build demand models, and applies predicted models and sensing data to decide dispatch locations for vacant taxis considering multiple objectives. From a system-level perspective, we compute suboptimal dispatch solutions for reaching a globally balanced supply demand ratio with least associated cruising distance under practical constraints. 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Control of robotic mobility-ondemand systems: a queueing-theoretical perspective. In Proceedings of Robotics: Science and Systems, July 2014. 15 Fei Miao (S’13) received the B.Sc. degree in Automation from Shanghai Jiao Tong University, Shanghai, China in 2010. Currently, she is working toward the Ph.D. degree in the Department of Electrical and Systems Engineering at University of Pennsylvania. Her research interests include datadriven real-time control frameworks of large-scale interconnected cyber-physical systems under model uncertainties, and resilient control frameworks to address security issues of cyber-physical systems. She was a Best Paper Award Finalist at the 6th ACM/IEEE International Conference on Cyber-Physical Systems in 2015. Shuo Han (S’08-M’14) received the B.E. and M.E. degrees in Electronic Engineering from Tsinghua University, Beijing, China in 2003 and 2006, and the Ph.D. degree in Electrical Engineering from the California Institute of Technology, Pasadena, USA in 2013. He is currently a postdoctoral researcher in the Department of Electrical and Systems Engineering at the University of Pennsylvania. His current research interests include control theory, convex optimization, applied probability, and their applications in largescale interconnected systems. Dr. Shan Lin is an assistant professor of the Electrical and Computer Engineering Department at Stony Brook University. He received his PhD in computer science at the University of Virginia in 2010. His PhD dissertation is on Taming Networking Challenges with Feedback Control. His research is in the area of networked systems, with an emphasis on feedback control based design for cyber-physical systems and sensor systems. He works on wireless network protocols, interoperable medical devices, smart transportation systems, and intelligent sensing systems. Professor John A. Stankovic is the BP America Professor in the Computer Science Department at the University of Virginia. He is a Fellow of both the IEEE and the ACM. He has been awarded an Honorary Doctorate from the University of York. He won the IEEE Real-Time Systems Technical Committee’s Award for Outstanding Technical Contributions and Leadership. He also won the IEEE Technical Committee on Distributed Processing’s Distinguished Achievement Award. He has seven Best Paper awards, including one for ACM SenSys 2006. Stankovic has an h-index of 105 and over 40,000 citations. Prof. Stankovic received his PhD from Brown University. Desheng Zhang is a Research Associate at Department of Computer Science and Engineering of the University of Minnesota. Previously, he was awarded his Ph.D in Computer Science from University of Minnesota. His research includes big data analytics, mobile cyber physical systems, wireless sensor networks, and intelligent transportation systems. His research results are uniquely built upon large-scale urban data from cross-domain urban systems, including cellphone, smartcard, taxi, bus, truck, subway, bike, personal vehicle, electric vehicle, and road networks. Desheng designs and implements large-scale data-driven models and real-world services to address urban sustainability challenges. Sirajum Munir received his PhD in Computer Science from the University of Virginia in 2014. He is currently working at Bosch Research and Technology Center as a Research Scientist. His research interest lies in the areas of cyber-physical systems, wireless sensor and actuator networks, and ubiquitous computing. He has published papers in major conferences in these areas, two of which were nominated for best paper awards at ACM/IEEE ICCPS. Hua Huang received the BE degree from Huazhong University of Science and Technology in 2012, MS degree in Temple University in 2014. He is currently working towards the PhD degree in the Department of Electrical and Computer Engineering in the Stony Brook University. His research interests include activity recognition in wearable devices and smart building, device-free indoor localization, deployment and scheduling in wireless sensor networks. Dr. Tian He is currently an associate professor in the Department of Computer Science and Engineering at the University of Minnesota-Twin City. Dr. He is the author and co-author of over 200 papers in premier network journals and conferences with over 17,000 citations (H-Index 52). Dr. He is the recipient of the NSF CAREER Award, George W. Taylor Distinguished Research Award and McKnight LandGrant Professorship, and many best paper awards in networking. His research includes wireless sensor networks, cyber-physical systems, intelligent transportation systems, real-time embedded systems and distributed systems. George J. Pappas (S’90-M’91-SM’04-F’09) received the Ph.D. degree in electrical engineering and computer sciences from the University of California, Berkeley, CA, USA, in 1998. He is currently the Joseph Moore Professor and Chair of the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA. He also holds a secondary appointment with the Department of Computer and Information Sciences and the Department of Mechanical Engineering and Applied Mechanics. He is a Member of the GRASP Lab and the PRECISE Center. He had previously served as the Deputy Dean for Research with the School of Engineering and Applied Science. His research interests include control theory and, in particular, hybrid systems, embedded systems, cyber-physical systems, and hierarchical and distributed control systems, with applications to unmanned aerial vehicles, distributed robotics, green buildings, and bimolecular networks. Dr. Pappas has received various awards, such as the Antonio Ruberti Young Researcher Prize, the George S. Axelby Award, the Hugo Schuck Best Paper Award, the George H. Heilmeier Award, the National Science Foundation PECASE award and numerous best student papers awards at ACC, CDC, and ICCPS. 

arXiv:1001.0641v1 [cs.LO] 5 Jan 2010 Least and Greatest Fixpoints in Game Semantics Pierre Clairambault pierre.clairambault@pps.jussieu.fr PPS — Université Paris 7 Abstract We show how solutions to many recursive arena equations can be computed in a natural way by allowing loops in arenas. We then equip arenas with winning functions and total winning strategies. We present two natural winning conditions compatible with the loop construction which respectively provide initial algebras and terminal coalgebras for a large class of continuous functors. Finally, we introduce an intuitionistic sequent calculus, extended with syntactic constructions for least and greatest fixed points, and prove it has a sound and (in a certain weak sense) complete interpretation in our game model. 1 Introduction The idea to model logic by game-theoretic tools can be traced back to the work of Lorenzen [21]. The idea is to interpret a formula by a game between two players O and P, O trying to refute the formula and P trying to prove it. The formula A is then valid if P has a winning strategy on the interpretation of A. Later, Joyal remarked [18] that it is possible to compose strategies in Conway games [8] in an associative way, thus giving rise to the first category of games and strategies. This, along with parallel developments in Linear Logic and Geometry of Interaction, led to the more recent construction of compositional game models for a large variety of logics [3, 23, 9] and programming languages [17, 4, 22, 5]. We aim here to use these tools to model an intuitionistic logic with induction and coinduction. Inductive/coinductive definitions in syntax have been defined and studied in a large variety of settings, such as linear logic [6], λ-calculus [1] or Martin-Löf’s type theory [10]. Motivations are multiple, but generally amount to increasing the expressive power of a language without paying the price of exponential modalities (as in [6]) or impredicativity (as in [1] or [10]). However, less work has been carried out when it comes to the semantics of such constructions. Of course we have the famous order-theoretic Knaster-Tarski fixed point theorem [25], the nice categorical theory due to Freyd [12], set-theoretic 1 models [10] (for the strictly positive fragment) or PER-models [20], but it seems they have been ignored by the current trend for intensional models (i.e. games semantics, GoI . . . ). We fix this issue here, showing that (co)induction admits a nice game-theoretic model which arises naturally if one enriches McCusker’s [22] work on recursive types with winning functions inspired by parity games [24]. In Section 2, we first recall the basic definitions of the Hyland-Ong-Nickau setting of game semantics. Then we sketch McCusker’s interpretation of recursive types, and show how most of these recursive types can be modelled by means of loops in the arenas. For this purpose, we define a class of functors called open functors, including in particular all the endofunctors built out of the basic type constructors. We also present a mechanism of winning functions inspired by [16], allowing us to build a category Gam of games and total winning strategies. In section 3, we present µLJ, the intuitionistic sequent calculus with least and greatest fixpoints that we aim to model. We briefly discuss its proof-theoretic properties, then present its semantic counterpart: we show how to build initial algebras and terminal coalgebras to most positive open functors. Finally, we use this semantic account of (co)induction to give a sound and (weakly) complete interpretation of µLJ in Gam. 2 Arena Games 2.1 Arenas and Plays We recall briefly the now usual definitions of arena games, introduced in [17]. More detailed accounts can be found in [22, 14]. We are interested in games with two participants: Opponent (O, the environment ) and Player (P, the program). Possible plays are generated by directed graphs called arenas, which are semantic versions of types or formulas. Hence, a play is a sequence of moves of the ambient arena, each of them being annotated by a pointer to an earlier move — these pointers being required to comply with the structure of the arena. Formally, an arena is a structure A = (MA , λA , ⊢A ) where: • MA is a set of moves, • λA : MA → {O, P } × {Q, A} is a labelling function indicating whether a move is an Opponent or Player move, and whether it is a question (Q) or QA an answer (A). We write λOP A for the projection of λA to {O, P } and λA for its projection on {Q, A}. λA will denote λA where the {O, P } part has been reversed. • ⊢A is a relation between MA + {⋆} to MA , called enabling, satisfying: – ⋆ ⊢ m =⇒ λA (m) = OQ; QA – m ⊢A n ∧ λQA A (n) = A =⇒ λA (m) = Q; OP – m ⊢A n ∧ m 6= ⋆ =⇒ λOP A (m) 6= λA (n). 2 In other terms, an arena is a directed bipartite graph, with a set of distinguished initial moves (m such that ⋆ ⊢A m) and a distinguished set of answers (m such that λQA = A) such that no answer points to another answer. We now A define plays as justified sequences over A: these are sequences s of moves of A, each non-initial move m in s being equipped with a pointer to an earlier move n in s, satisfying n ⊢A m. In other words, a justified sequence s over A is such that each reversed pointer chain sφ(0) ← sφ(1) ← . . . ← sφ(n) is a path on A, viewed as a directed bipartite graph. The role of pointers is to allow reopenings in plays. Indeed, a path on A may be (slightly naively) understood as a linear play on A, and a justified sequence as an interleaving of paths, with possible duplications of some of them. This intuition is made precise in [15]. When writing justified sequences, we will often omit the justification information if this does not cause any ambiguity. ⊑ will denote the prefix ordering on justified sequences. If s is a justified sequence on A, |s| will denote its length. Given a justified sequence s on A, it has two subsequences of particular interest: the P-view and O-view. The view for P (resp. O) may be understood as the subsequence of the play where P (resp. O) only sees his own duplications. In a P-view, O never points more than once to a given P-move, thus he must always point to the previous move. Concretely, P-views correspond to branches of Böhm trees [17]. Practically, the P-view psq of s is computed by forgetting everything under Opponent’s pointers, in the following recursive way: • psmq = psqm if λOP A (m) = P ; • psmq = m if ⋆ ⊢A m and m has no justification pointer; • ps1 ms2 nq = psqmn if λOP A (n) = O and n points to m. The O-view xsy of s is defined dually. Note that in some cases — in fact if s does not satisfies the visibility condition introduced below — psq and xsy may not be correct justified sequences, since some moves may have pointed to erased parts of the play. However, we will restrict to plays where this does not happen. The legal sequences over A, denoted by LA , are the justified sequences s on A satisfying the following conditions: OP • Alternation. If tmn ⊑ s, then λOP A (m) 6= λA (n); • Bracketing. A question q is answered by a if a is an answer and a points to q. A question q is open in s if it has not yet been answered. We require that each answer points to the pending question, i.e. the last open question. • Visibility. If tm ⊑ s and m is not initial, then if λOP A (m) = P the justifier of m appears in ptq, otherwise its justifier appears in xty. 3 2.2 The cartesian closed category of Innocent strategies A strategy σ on A is a prefix-closed set of even-length legal plays on A. A strategy is deterministic if only Opponent branches, i.e. ∀smn, smn′ ∈ σ, n = n′ . Of course, if A represents a type (or formula), there are often many more strategies on A than programs (or proofs) on this type. To address this issue we need innocence. An innocent strategy is a strategy σ such that sab ∈ σ ∧ t ∈ σ ∧ ta ∈ LA ∧ psaq = ptaq =⇒ tab ∈ σ We now recall how arenas and innocent strategies organize themselves into a cartesian closed category. First, we build the product A × B of two arenas A and B: MA×B = MA + MB λA×B ⊢A×B = = [λA , λB ] ⊢A + ⊢B We mention the empty arena I = (∅, ∅, ∅), which will be terminal for the category of arenas and innocent strategies. We mention as well the arena ⊥ = (•, • 7→ OQ, (⋆, •)) with only one initial move, which will be a weak initial object. We define the arrow A ⇒ B as follows: MA⇒B = λA⇒B = m ⊢A⇒B n MA + MB [λA , λB ]  m 6= ⋆ ∧ m ⊢A n    m 6= ⋆ ∧ m ⊢B n ⇔ ⋆ ⊢B m ∧ ⋆ ⊢A n    m = ⋆ ∧ ⋆ ⊢B n We define composition of strategies by the usual parallel interaction plus hiding mechanism. If A, B and C are arenas, we define the set of interactions I(A, B, C) as the set of justified sequences u over A, B and C such that u↾A,B ∈ LA⇒B , u↾B,C ∈ LB⇒C and u↾A,C ∈ LA⇒C . Then, if σ : A ⇒ B and τ : B ⇒ C, we define parallel interaction: σ||τ = {u ∈ I(A, B, C) | u↾A,B ∈ σ ∧ u↾B,C ∈ τ } Composition is then defined as σ; τ = {u↾A,C | u ∈ σ||τ }. It is associative and preserves innocence (a proof of these facts can be found in [17] or [14]). We also define the identity on A as the copycat strategy (see [22] or [14] for a definition) on A ⇒ A. Thus, there is a category Inn which has arenas as objects and innocent strategies on A ⇒ B as morphisms from A to B. In fact, this category is cartesian closed, the cartesian structure given by the arena product above and the exponential closure given by the arrow construction. This category is also equipped with a weak coproduct A + B [22], which is constructed as follows: MA+B = MA + MB + {q, L, R} 4 λA+B m ⊢A+B n 2.3 [λA , λB , q 7→ OQ, L 7→ P A, R 7→ P A]  m, n ∈ MA ∧ m ⊢A n      m, n ∈ MB ∧ m ⊢B n m=⋆∧n=q ⇔   (m = q ∧ n = L) ∨ (m = q ∧ n = R)    (m = L ∧ ⋆ ⊢A n) ∨ (m = R ∧ ⋆ ⊢B n) = Recursive types and Loops Let us recall briefly the interpretation of recursive types in game semantics, due to McCusker [22]. Following [22], we first define an ordering E on arenas as follows. For two arenas A and B, A E B iff MA λA ⊆ = ⊢A = MB λB ↾MA ⊢B ∩ (MA + {⋆} × MA ) This defines a (large) dcpo, with least element I and directed sups given by the componentwise union. If F : Inn → Inn is a functor which is continuous with respect to E, we F∞can find an arena D such that D = F (D) in the usual way by setting D = n=0 F n (I). McCusker showed [22] that when the functors are closed (i.e. their action can be internalized as a morphism (A ⇒ B) → (F A ⇒ F B)), and when they preserve inclusion and projection morphisms (i.e. partial copycat strategies) corresponding to E, this construction defines minimal invariants [12]. Note that the crucial cases of these constructions are the functors built out of the product, sum and function space constructions. We give now a concrete and new (up to the author’s knowledge) description of a large class of continuous functors, that we call open functors. These include all the functors built out of the basic constructions, and allow a rereading of recursive types, leading to the model of (co)induction. 2.3.1 Open arenas. Let T be a countable set of names. An open arena is an arena A with distinguished question moves called holes, each of them labelled by an element of T. We denote by X the holes annotated by X ∈ T. We will sometimes write O P X to denote a hole of Player polarity, or X to denote a hole of Opponent polarity. If A has holes labelled by X1 , . . . , Xn , we denote it by A[X1 , . . . , Xn ]. By abuse of notation, the corresponding open functor we are going to build will be also denoted by A[X1 , . . . , Xn ] : (Inn × Innop )n → Inn. 2.3.2 Image of arenas. If A[X1 , . . . , Xn ] is an open arena and B1 , . . . , Bn , B1′ , . . . , Bn′ are arenas (possibly open as well), we build a new arena A(B1 , B1′ , . . . , Bn , Bn′ ) by replacing each 5 O ′ occurrence of P Xi by Bi and each occurrence of Xi by Bi . More formally: MA(B1 ,B1′ ,...,Bn ,Bn′ ) = (MA \ {X1 , . . . , Xn }) + n X (MBi + MBi′ ) i=1 λA(B1 ,B1′ ,...,Bn ,Bn′ ) = m ⊢A(B1 ,B1′ ,...,Bn ,Bn′ ) p ⇔ [λA , λB1 , λB1′ , . . . , λBn , λBn′ ]  P  Xi ∧ ⋆ ⊢Bi p  m ⊢A  O   m ⊢   A Xi ∧ ⋆ ⊢Bi′ p     ⋆ ⊢Bi m ∧ P Xi ⊢A p ⋆ ⊢Bi′ m ∧ O Xi ⊢A p   m ⊢Bi p     m ⊢Bi′ p    m ⊢A p Note that in this definition, we assimilate all the moves sharing the same hole label Xi and with the same polarity. This helps to clarify notations, and is justified by the fact that we never need to distinguish moves with the same hole label, apart from when they have different polarity. 2.3.3 Image of strategies. If A is an arena, we will, by abuse of notation, denote by IA both the set of initial moves of A and the subarena of A with only these moves. Let A[X1 , . . . , Xn ] be an open arena, B1′ , B1 , . . . , Bn′ , Bn and C1′ , C1 , . . . , Cn′ , Cn be arenas. Consider the application ξ defined on moves as follows:  S Xi if x ∈ i∈{1,...,n} (IB ′ ∪ IBi ∪ IC ′ ∪ ICi ) i i ξ(x) = x otherwise and then extended recursively to an application ξ ∗ on legal plays as follows:  ∗ ξ (s) if a is a non-initial move of Bi , B ′ , Ci or C ′ ∗ i i ξ (sa) = ξ ∗ (s)ξ(a) otherwise ξ ∗ erases moves in the inner parts of Bi′ , Bi , Ci′ , Ci and agglomerates all the initial moves back to the holes. This way we will be able to compare the resulting play with the identity on A[X1 , . . . , Xn ]. Now, if σi : Bi → Ci and τi : Ci′ → Bi′ are strategies, we can now define the action of open functors on them by stating:   ∀i ∈ {1, . . . , n}, s↾Bi ⇒Ci ∈ σi ∀i ∈ {1, . . . , n}, s↾C ′ ⇒B′ ∈ τi s ∈ A(σ1 , τ1 , . . . , σn , τn ) ⇔ i i  ∗ ξ (s) ∈ idA[X1 ,...,Xn ] Proposition 1. For any A[X1 , . . . , Xn ], this defines a functor A[X1 , . . . , Xn ] : (Inn × Innop )n → Inn, which is monotone and continuous with respect to E. 6 Proof sketch. Preservation of identities and composition are rather direct. A little care is needed to show that the resulting strategy is innocent: this relies on two facts: First, for each Player move the three definition cases are mutually exclusive. Second, a P-view of s ∈ A(σ1 , τ1 , . . . , σn , τn ) is (essentially) an initial copycat appended with a P-view of one of σi or τi , hence the P-view of s determines uniquely the P-view presented to one of σi , τi or idA[X1 ,...,Xn ] . Example. Consider the open arena A[X] = X ⇒ X . For any arena B, we have A(B) = B ⇒ B and for any σ : B1 → C1 and τ : C2 → B2 , we have A(σ, τ ) = τ ⇒ σ : (B2 ⇒ B1 ) → (C2 ⇒ C1 ), the strategy which precomposes its argument by τ and postcomposes it by σ. 2.3.4 Loops for recursive types. Since these open functors are monotone and continuous with respect to E, solutions to their corresponding recursive equations can be obtained by computing the infinite expansion of arenas (i.e. infinite iteration of the open functors). However, for a large subclass of the open functors, this solution can be expressed in a simple way by replacing holes with a loop up to the initial moves. Suppose A[X1 , . . . , Xn ] is an open functor, and i is such that Xi appears only in non-initial, positive positions in A. Then we define an arena µXi .A as follows: MµXi .A = λµXi .A = m ⊢µXi .A n (MA \ Xi ) λA↾MµX .A i  m ⊢A n ⇔ m ⊢A Xi ∧ ⋆ ⊢A n A simple argument ensures that the obtained arena is isomorphic to the one obtained by iteration of the functor. For this issue we take inspiration from Laurent [19] and prove a theorem stating that two arenas are isomorphic in the categorical sense if and only if their set of paths are isomorphic. A path in A is a sequence of moves a1 , . . . , an such that for all i ∈ {1, . . . , n − 1} we have ai ⊢A ai+1 . A path isomorphism between A and B is a bijection φ between the set of paths of A and the set of paths on B such that for any non-empty path p on A, φ(ip(p)) = ip(φ(p)) (where ip(p) denotes the immediate prefix of p). We have then the theorem: Theorem 1. Let A and B be two arenas. They are categorically isomorphic if and only if there is a path isomorphism between their respective sets of paths. Now, it is clear by construction that, if A[X] is an open functor such F∞that X appears only in non-initial positive positions in A, the set of paths of n=0 An (I) and of µX.A are isomorphic. Therefore µX.A is solution of the recursive equation X = A(X), and when A[X] is closed and preserves inclusions and projections, µX.A defines as well a minimal invariant for A[X]. But in fact, we have the following fact: 7 Proposition 2. If A[X] is an open functor, then it is closed and preserves inclusions and projections. Hence µX.A is a minimal invariant for A[X]. This interpretation of recursive types as loops preserves finiteness of the arena, and as we shall see, allows to easily express the winning conditions necessary to model induction and coinduction. 2.4 Winning and Totality A total strategy on A is a strategy σ : A such that for all s ∈ σ, if there is a such that sa ∈ LA , then there is b such that sab ∈ σ. In other words, σ has a response to any legal Opponent move. This is crucial to interpret logic because the interpretation of proofs in game semantics always gives total strategies: this is a counterpart in semantics to the cut elimination property in syntax. To model induction and coinduction in logic, we must therefore restrict to total strategies. However, it is well-known that the class of total strategies is not closed under composition, because an infinite chattering can occur in the hidden part of the interaction. This is analogous to the fact that in λ-calculus, the class of strongly normalizing terms is not closed under application: δ = λx.xx is a normal form, however δδ is certainly not normalizable. This problem is discussed in [2, 16] and more recently in [7]. We take here the solution of [16], and equip arenas with winning functions: for every infinite play we choose a loser, hence restricting to winning strategies has the effect of blocking infinite chattering. The definition of legal plays extends smoothly to infinite plays. Let Lω A denote the set of infinite legal plays over A. If s ∈ Lω A , we say that s ∈ σ when for all s ⊏ s, s ∈ σ. We write LA = LA + Lω A . A game will be a pair A = (A, GA ) where A is an arena, and GA is a function from infinite threads on A (i.e. infinite legal plays with exactly one initial move) to {W, L}. The winning function GA extends naturally to potentially finite threads by setting, for each finite s:  W if |s| is even ; GA (s) = L otherwise. Finally, GA extends to legal plays by saying that GA (s) = W iff GA (t) = W for every thread t of s. By abuse of notation, we keep the same notation for this extended function. The constructions on arenas presented in section 2.2 extend to constructions on games as follows: • GA×B (s) = [GA , GB ] (indeed, a thread on A × B is either a thread on A or a thread on B) ; • GA+B (s) = W iff all threads of s↾A are winning for GA and all threads of s↾B are winning for GB . • GA⇒B (s) = W iff if all threads of s↾A are winning for GA , then GB (s↾B ) = W. It is straightforward to check that these constructions commute with the extension of winning functions from infinite threads to potentially infinite legal 8 plays. We now define winning strategies σ : A as innocent strategies σ : A such that for all s ∈ σ, GA (s) = W . Now, the following proposition is satisfied: Proposition 3. Let σ : A ⇒ B and τ : B ⇒ C be two total winning strategies. Then σ; τ is total winning. Proof sketch. If σ; τ is not total, there must be infinite s in their parallel interaction σ||τ , such that s↾A,C is finite. By switching, we have in fact |s↾A | even and |s↾C | odd. Thus GA (s↾A ) = W and GC (s↾C ) = L. We reason then by disjunction of cases. Either GB (s↾B ) = W in which case GB⇒C (s↾B,C ) = L and τ cannot be winning, or GB (s↾B ) = L in which case GA⇒B (s↾A,B ) = L and σ cannot be winning. Therefore σ; τ is total. σ; τ must be winning as well. Suppose there is s ∈ σ; τ such that GA⇒C (s) = L. By definition of GA⇒C , this means that GA (s↾A ) = W and GC (s↾C ) = L. By definition of composition, there is u ∈ σ||τ such that s = u↾A,C . But whatever the value of GB (u↾B ) is, one of σ or τ is losing. Therefore σ; τ is winning. It is clear from the definitions that all plays in the identity are winning. It is also clear that all the structural morphisms of the cartesian closed structure of Inn are winning (they are essentially copycat strategies), thus this defines a cartesian closed category Gam of games and innocent total winning strategies. 3 3.1 3.1.1 Fixpoints µLJ: an intuitionistic sequent calculus with fixpoints Formulas. S ::= S ⇒ T | S ∨ T | S ∧ T | µX.T | νX.T | X | ⊤ | ⊥ A formula F is valid if for any subformula of F of the form µX.F ′ , (1) X appears only positively in F ′ , (2) X does not appear at the root of F ′ (i.e. X appears at least under a ∨ or a ⇒ in the abstract syntax tree of F ′ ). (2) corresponds to the restriction to arenas where loops allow to express recursive types, whereas (1) is the usual positivity condition. We could of course hack the definition to get rid of these restrictions, but we choose not to obfuscate the treatment for an extra generality which is neither often considered in the literature, nor useful in practical examples of (co)induction. 3.1.2 Derivation rules. We present the rules with the usual dichotomy. Identity group A⊢A ax Γ⊢A ∆, A ⊢ B Γ, ∆ ⊢ B 9 Cut Structural group Γ, A, A ⊢ B Γ, A ⊢ B Γ⊢B C Γ, A, B, ∆ ⊢ C W Γ, A ⊢ B Γ, B, A, ∆ ⊢ C γ Logical group Γ, A ⊢ B Γ⊢A⇒B Γ⊢A Γ⊢A ⇒r Γ, ∆, A ⇒ B ⊢ C Γ⊢B Γ ⊢ A∨B ⇒l Γ, A ⊢ C ∧r Γ⊢A∧B Γ⊢A ∆, B ⊢ C Γ, A ∧ B ⊢ C Γ⊢B ← ∨−r Γ⊢A∨B − → ∨ r Γ, ⊥ ⊢ A ⊥l Γ⊢⊤ Γ, B ⊢ C ← ∧−l Γ, A ∧ B ⊢ C Γ, A ⊢ C − → ∧l ∆, B ⊢ C Γ, ∆, A ∨ B ⊢ C ⊤r ∨l Fixpoints Γ ⊢ T [µX.T /X] Γ ⊢ µX.T µr T [A/X] ⊢ A µX.T ⊢ A T [νX.T /X] ⊢ B µl νX.T ⊢ B νl A ⊢ T [A/X] A ⊢ νX.T νr Note that the µl , νl and νr rules are not relative to any context. In fact, the general rules with a context Γ at the left of the sequent are derivable from these ones (even if, for µl and νr , the construction of the derivation requires an induction on T ), and we stick with the present ones to clarify the game model. Cut elimination on the ⇒, ∧, ∨ fragment is the same as usual. For the reduction of µ and ν, we need an additional rule to handle the unfolding of formulas. For this purpose, we add a new rule [T ] for each type T with free variables. This method can already be found in [1] for strictly positive functors: no type variable appears on the left of an implication. From now on, T [A/X] will be abbreviated T (A). This notation implies that, unless otherwise stated, X will be the variable name for which T is viewed as a functor. In the following rules, X appears only positively in T and only negatively in N : Functors A⊢B T (A) ⊢ T (B) A⊢B [T ] N (B) ⊢ N (A) [N ] The dynamic behaviour of this rule is to locally perform the unfolding. We give some of the reduction rules. These are of two kinds: the rules for the elimination of [T ], and the cut elimination rules. Here are the main cases: 10 π π A⊢B T ⊢T T ⊢T [T ](X 6∈ F V (T )) ax A⊢B A⊢B π [X] A⊢B π π π A⊢B A⊢B N (B) ⊢ N (A) N (A) ⇒ T (A) ⊢ N (B) ⇒ T (B) [N ⇒ T ] A⊢B [N ] T (A) ⊢ T (B) N (A) ⇒ T (A), N (B) ⊢ T (B) N (A) ⇒ T (A) ⊢ N (B) ⇒ T (B) [T ] ⇒l ⇒r π π A⊢B A⊢B T (A)[µY.T (B)/Y ] ⊢ T (B)[µY.T (B)/Y ] µY.T (A) ⊢ µY.T (B) [µY.T ] T (A)[µY.T (B)/Y ] ⊢ µY.T (B) µY.T (A) ⊢ µY.T (B) [T [µY.T (B)/Y ]] µr µl We omit the rule for ν, which is dual, and for ∧ and ∨, which are simple pairing and case manipulations. Note also that most of these cases have a counterpart where T is replaced by negative N , which has the sole effect of π being a proof of B ⊢ A instead of A ⊢ B in the expansion rules. With that, we can express the cut elimination rule for fixpoints: π1 π2 Γ ⊢ T [µX.T /X] Γ ⊢ µX.T µr T [A/X] ⊢ A µX.T ⊢ A Γ⊢A µl Cut π2 T [A/X] ⊢ A µl π1 µX.T ⊢ A Γ ⊢ T [µX.T /X] T [µX.T /X] ⊢ T [A/X] Γ ⊢ T [A/X] Γ⊢A [T ] Cut π2 T [A/X] ⊢ A Cut We skip once again the rule for ν, which is dual to µ. We choose consciously not to recall the usual cut elimination rules nor the associated commutation rules, since they are not central to our goals. µLJ, as presented above, does not formally eliminate cuts since there is no rule to reduce the following (and 11 its dual with ν): π1 T (A) ⊢ A µX.T ⊢ A π2 µl Γ, A ⊢ B Γ, µX.T ⊢ B Cut This cannot be reduced without some prior unfolding of the µX.T on the left. This issue is often solved [6] by replacing the rule for µ presented here above by the following: T (A) ⊢ A Γ, A ⊢ B ′ µ Γ, µX.T ⊢ B With the corresponding reduction rule, and analogously for ν. We choose here not to do this, first because our game model will prove consistency without the need to prove cut elimination, and second because we want to preserve the proximity with the categorical structure of initial algebras / terminal coalgebras. 3.2 The games model We present the game model for fixpoints. We wish to model a proof system, therefore we need our strategies to be total. The base arenas of the interpretation of fixpoints will be the arenas with loops presented in section 2.3.4, to which we will adjoin a winning function. While the base arenas will be the same for greatest and least fixpoints, they will be distinguished by the winning function: intuitively, Player loses if a play grows infinite in a least fixpoint (inductive) game, and Opponent loses if this happens in a greatest fixpoint (coinductive) game. The winning functions we are going to present are strongly influenced by Santocanale’s work on games for µ-lattices [24]. A win open functor is a functor T : (Gam × Gamop )n → Gam such that there is an open functor T [X1 , . . . , Xn ] such that for all games A1 , . . . , A2n of base arenas A1 , . . . , A2n , the base arena of T(A1 , . . . , A2n ) is T (A1 , . . . , An ). In other terms, it is the natural lifting of open functors to the category of games. By abuse of notation, we denote this by T[X1 , . . . , Xn ], and T [X1 , . . . , Xn ] will denote its underlying open functor. 3.2.1 Least fixed point. Let T[X1 , . . . , Xn ] be a win open functor such that X1 appears only positively and at depth higher than 0 in T [X1 , . . . , Xn ]. Then we define a new win open functor µX1 .T[X2 , . . . , Xn ] as follows: • Its base arena is µX1 .T [X2 , . . . , Xn ] ; • If A3 , . . . , A2n ∈ Gam, GµX1 .T(A3 ,...,A2n ) (s) = W iff – There is N ∈ N such that no path of s takes the external loop more that N times, and ; 12 – s is winning in the subgame inside the loop, or more formally: GT(I,I,A3 ,...,A2n ) (s↾T(I,I,A3 ,...,A2n ) ) = W . 3.2.2 Greatest fixed point. Dually, if the same conditions are satisfied, we define the win open functor νX1 .T[X1 , . . . , Xn ] as follows: • Its base arena is µX1 .T [X2 , . . . , Xn ] ; • If A3 , . . . , A2n ∈ Gam, GνX1 .T(A3 ,...,A2n ) (s) = W iff – For any N ∈ N, there is a path of s crossing the external loop more than N times, or ; – s is winning in the subgame inside the loop, or more formally: GT(I,I,A3 ,...,A2n ) (s↾T(I,I,A3 ,...,A2n ) ) = W . It is straightforward to check that these are still functors, and in particular win open functors. There is one particular case that is worth noticing: if T[X] has only one hole which appears only in positive position and at depth greater than 0, then µX.T is a constant functor, i.e. a game. Moreover, theorem 1 implies that it is isomorphic in Inn to T(µX.T). It is straightforward to check that this isomorphism iT : T(µX.T) → µX.T is winning (it is nothing but the identity strategy), which shows that they are in fact isomorphic in Gam. Then, one can prove the following theorem: Theorem 2. If T[X] has only one hole which appears only in positive position and at depth greater than 0, then the pair (µX.T, iT ) defines an initial algebra for T[X] and (νX.T, i−1 T ) defines a terminal coalgebra for T[X]. Proof. We give the proof for initial alebras, the second part being dual. Let (A, σ) another algebra of T[X]. We need to show that there is a unique σ † : µX.T ⇒ B such that T(σ† ) T(µX.T) / T(B) σ iT  µX.T σ†  /B commutes. The idea is to iterate σ: ... T3 (σ) / T3 (B) T2 (σ) / T2 (B) T(σ) / T(B) σ /B and somehow to take the limit. In fact we can give a direct definition of σ † : σ σ (1) = σ (n+1) = = Tn (σ); σ (n) {s ∈ LµX.T⇒B | ∃n ∈ N∗ , s ∈ σ (n) } σ † 13 This defines an innocent strategy, since when restricted to plays of µX.T, these strategies agree on their common domain. This strategy is winning. Indeed, take an infinite play s ∈ σ † . Suppose s↾µX.T is winning. By definition of GµX.T , this means that there is N ∈ N such that no path of s↾µX.T takes the external loop more than N times. Thus, s ∈ LTn (I)⇒B . But this implies that s ∈ σ (n) , and σ (n) is a composition of winning strategies thus winning, therefore s is winning. Moreover, σ † is the unique innocent strategy making the diagram commute: suppose there is another f making this square commute. Since T(µX.T) and µX.T have the same set of paths, iT is in fact the identity, thus we have T(f ); σ = f . By applying T and post-composing by σ, we get: T2 (f ); T(σ); σ = T(f ); σ = f And by iterating this process, we get for all n ∈ N: Tn+1 (f ); Tn (σ); . . . ; T(σ); σ = f Thus: Tn+1 (f ); σ (n) = f Now take s ∈ f , and let n be the length of the longest path in s. Since T[X] has no hole at the root, no path of length n can reach B in Tn+1 (B), thus s ∈ σ (n) , therefore s ∈ σ † . The same reasoning also works for the other inclusion. Likewise, if σ : B → T(B), we build a unique σ ‡ : B → νX.T making the coalgebra diagram commute. 3.3 Interpretation of µLJ 3.3.1 Interpretation of Formulas. As expected, we give the interpretation of valid formulas. J⊤K J⊥K JA ∨ BK JA ∧ BK 3.3.2 = = = = JA ⇒ BK JXK JµX.T K JνX.T K I ⊥ JAK + JBK JAK × JBK = = = = JAK ⇒ JBK X µX.JT K νX.JT K Interpretation of Proofs. As usual, the interpretation of a proof π of a sequent A1 , . . . , An ⊢ B will be a morphism JπK : JA1 K × . . . × JAn K −→ JBK. The interpretation is computed by induction on the proof tree. The interpretation of the rules of LJ is standard and its correctness follows from the cartesian closed structure of Gam. Here are the interpretations for the fixpoint and functor rules: u wΓ ⊢ T [µX.T /X] v Γ ⊢ µX.T u } π µr wT [A/X] ⊢ A v  ~ = JπK; iJT K µX.T ⊢ A 14 } π µl  † ~ = JπK u w vT [νX.T /X] ⊢ B νX.T ⊢ B u } π νl w vA ⊢ T [A/X]  −1 ~ = iJT K ; JπK u w v } π A⊢B T (A) ⊢ T (B) } π A ⊢ νX.T νr  ‡ ~ = JπK  ~ = JT K(JπK) [T ] We do not give the details of the proof that this defines an invariant of reduction. The main technical point is the validity of the interpretation of the functor rule; more precisely when the functor is a (least or greatest) fixpoint. Given that, we get the following theorem. Theorem 3. If π π ′ , then JπK = Jπ ′ K. In particular, this proves the following theorem which is certainly worth noticing, because µLJ has large expressive power. In particular, it contains Gödel’s system T [13]. Theorem 4. µLJ is consistent: there is no proof of ⊥. Proof. There is no total strategy on the game ⊥. 3.3.3 Completeness. When it comes to completeness, we run into the issue that the total winning innocent strategies are not necessarily finite, hence the usual definability process does not terminate. Nonetheless, we get a definability theorem in an infinitary version of µLJ. Whether a more precise completeness theorem is possible is a subtle point. First, we would need to restrict to an adequate subclass of the recursive total winning strategies (for example, the Ackermann function is definable in µLJ). Then again, the problem to find a proof whose interpretation is exactly the original strategy would be highly non-trivial: if σ : µX.T ⇒ A, we have to guess an invariant B, a proof π1 of T (B) ⊢ B and a proof π2 of B ⊢ A such that Jπ1 K† ; Jπ2 K = σ. Perhaps it would be more feasible to look for a proof whose interpretation is observationally equivalent to the original strategy, which would be very similar to the universality result in [17]. 4 Conclusion and Future Work We have successfully constructed a games model of a propositional intuitionistic sequent calculus µLJ with inductive and coinductive types. It is striking that the adequate winning conditions on legal plays to model (co)induction are almost identical to those used in parity games to model least and greatest fixpoints, to the extent that the restriction of our winning condition to paths coincides exactly with the winning condition used in [24]. It would be worthwile to investigate this connection further: given a game viewed as a bipartite graph 15 along with winning conditions for infinite plays, under which assumptions can these winning conditions be canonically lifted to the set of legal plays on this graph, viewed as an arena? Results in this direction might prove useful, since they would allow to import many game-theoretic results into game semantics, and thus programming languages. This work is part of a larger project to provide game-theoretic models to total programming languages with dependent types, such as COQ or Agda. In these settings, (co)induction is crucial, since they deliberately lack general recursion. We believe that in the appropriate games setting, we can push the present results further and model Dybjer’s Inductive-Recursive[11] definitions. 4.0.4 Acknowledgements. We would like to thank Russ Harmer, Stephane Gimenez and David Baelde for stimulating discussions, and the anonymous referees for useful comments and suggestions. References [1] A. Abel and T. Altenkirch. A predicative strong normalisation proof for a lambda-calculus with interleaving inductive types. In TYPES, 1991. [2] S. Abramsky. Semantics of interaction: an introduction to game semantics. Semantics and Logics of Computation, pages 1–31, 1996. [3] S. Abramsky and R. Jagadeesan. Games and full completeness for multiplicative linear logic. J. Symb. Log., 59(2):543–574, 1994. [4] S. Abramsky, R. Jagadeesan, and P. Malacaria. Full Abstraction for PCF. Info. & Comp, 2000. [5] S. Abramsky, H. Kohei, and G. McCusker. A fully abstract game semantics for general references. In LICS, pages 334–344, 1998. [6] D. Baelde and D. Miller. Least and greatest fixed points in linear logic. In LPAR, pages 92–106, 2007. [7] P. Clairambault and R. Harmer. Totality in Arena Games. Submitted., 2008. [8] J.H. Conway. On Numbers and Games. AK Peters, Ltd., 2001. [9] J. De Lataillade. Second-order type isomorphisms through game semantics. Ann. Pure Appl. Logic, 151(2-3):115–150, 2008. [10] P. Dybjer. Inductive sets and families in Martin-Löfs Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löfs type theory. Logical Frameworks, 14:59–79, 1991. 16 [11] P. Dybjer. A general formulation of simultaneous inductive-recursive definitions in type theory. J. Symb. Log., 65(2):525–549, 2000. [12] P. Freyd. Algebraically complete categories. In Proc. 1990 Como Category Theory Conference, volume 1488, pages 95–104. Springer, 1990. [13] K. Godel. Über eine bisher noch nicht bentzte Erweiterung des finiten Standpunktes. Dialectica, 1958. [14] R. Harmer. Innocent game semantics. Lecture notes, 2004. [15] R. Harmer, J.M.E. Hyland, and P.-A. Melliès. Categorical combinatorics for innocent strategies. In LICS, pages 379–388, 2007. [16] J.M.E. Hyland. Game semantics. Semantics and Logics of Computation, 1996. [17] J.M.E. Hyland and C.H.L. Ong. On full abstraction for PCF: I, II, and III. Inf. Comput., 163(2):285–408, 2000. [18] A. Joyal. Remarques sur la théorie des jeux à deux personnes. Gaz. Sc. Math. Qu., 1977. [19] O. Laurent. Classical isomorphisms of types. Mathematical Structures in Computer Science, 15(5):969–1004, 2005. [20] R. Loader. Equational theories for inductive types. Ann. Pure Appl. Logic, 84(2):175–217, 1997. [21] P. Lorenzen. Logik und Agon. Atti Congr. Internat. di Filosofia, 1960. [22] G. McCusker. Games and full abstraction for FPC. Inf. Comput., 160(12):1–61, 2000. [23] P.-A. Melliès. Asynchronous games 4: A fully complete model of propositional linear logic. In LICS, pages 386–395, 2005. [24] L. Santocanale. Free µ-lattices. J. Pure Appl. Algebra, 168(2-3):227–264, 2002. [25] A. Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2):285–309, 1955. 17 

arXiv:1512.05105v1 [] 16 Dec 2015 INVARIANTS OF LINKAGE OF MODULES TONY J. PUTHENPURAKAL Abstract. Let (A, m) be a Gorenstein local ring and let M, N be two CohenMacaulay A-modules with M linked to N via a Gorenstein ideal q. Let L be another finitely generated A-module. We show that ExtiA (L, M ) = 0 for all i ≫ 0 if and only if TorA i (L, N ) = 0 for all i ≫ 0. If D is Cohen-Macaulay then we show that ExtiA (M, D) = 0 for all i ≫ 0 if and only if ExtiA (D † , N ) = 0 for all i ≫ 0, where D † = ExtrA (D, A) and r = codim D. As a consequence we get that ExtiA (M, M ) = 0 for all i ≫ 0 if and only if ExtiA (N, N ) = 0 for all i ≫ 0. We also show that EndA (M )/ rad EndA (M ) ∼ = (EndA (N )/ rad EndA (N ))op . We also give a negative answer to a question of Martsinkovsky and Strooker. 1. introduction Let (A, m) be a Gorenstein local ring. Recall an ideal q in A is said to be a Gorenstein ideal if q is perfect and A/q is a Gorenstein local ring. A special class of Gorenstein ideals are CI(complete intersection) ideals, i.e., ideals generated by an A-regular sequence. In this paper, a not necessarily perfect ideal q such that A/q is a Gorenstein ring will be called a quasi-Gorenstein ideal. Two ideals I and J are linked by a Gorenstein ideal q if q ⊆ I ∩ J; J = (q : I) and I = (q : J). We write it as I ∼q J. If q is a complete intersection then we say I is CI-linked to J via q. If q is a quasi-Gorenstein ideal then we say I is quasi-linked to J via q. Note that traditionally only CI-linkage used to be considered. However in recent times more general types of linkage are studied. We say ideals I and J is in the same linkage class if there is a sequence of ideals I0 , . . . , In in A and Gorenstein ideals q0 . . . , qn−1 such that (i) Ij ∼qj Ij+1 , for j = 0, . . . , n − 1. (ii) I0 = I and In = J. If n is even then we say that I and J are evenly linked. We can analogously define CI-linkage class, quasi-linkage class, even CI-linkage class and even quasi-linkage class (of ideals). A natural question is that if I and J are in the same linkage class then what properties of I is shared by J. This was classically done when I and J are in the same CI-linkage class (or even CI-linkage class). In their landmark paper [12], Peskine and Szpiro proved that if I and J are in the same CI-linkage class and I is a Cohen-Macaulay ideal (i.e., A/I is a Cohen-Macaulay ring) then so is J. This can be proved more generally for ideals in a quasi-linkage class, see [10, Corollary 15, p. 616]. In another landmark paper [8, 1.14], Huneke proved that if I is in the CI-linkage class of a complete intersection then the Koszul homology Hi (I) are Date: March 1, 2018. 1991 Mathematics Subject Classification. Primary 13C40; Secondary 13D07. Key words and phrases. liason of modules, Gorenstien ideals, vanishing of Ext, Tor. 1 2 TONY J. PUTHENPURAKAL Cohen-Macaulay for all i ≥ 0. It is known that this result is not true in if I is linked to a complete intersection ( via Gorenstein ideals and not-necessarily CI-ideals). If A = k[[X1 , . . . , Xn ]] (where k is a field or a complete discrete valuation ring) then Huneke defined some invariants of even CI-linkage class of equidimensional unmixed ideals, see [9, 3.2]. Again these invariants are not stable under Gorenstein (even)liason. In a remarkable paper Martsinkovsky and Strooker, [10], introduced liason for modules. See section two for definition. We note here that ideals I and J are linked as ideals if and only if A/I is linked to A/J as modules. One can analogously define linkage class of modules, even linkage of modules etc. We can also define CI linkage of modules, quasi-linkage of modules etc. Thus a natural question arises: If M, N are in the same linkage class of modules (or same even linkage class of modules) then what properties of M are shared by N . The generalization of Peskine and Szpiro’s result holds. If M is Cohen-Macaulay and N is quasi-linked to M then N is also Cohen-Macaulay, see [10, Corollary 15, p. 616]. To state another property which is preserved under linkage first let us recall the definition of Cohen-Macaulay approximation from [1]. A Cohen-Macaulay approximation of a finitely generated A-module M is a exact sequence 0 −→ Y −→ X −→ M −→ 0 where X is a maximal Cohen-Macaulay A-module and Y has finite projective dimension. Such a sequence is not unique but X is known to unique up to a free summand and so is well defined in the stable category CM(A) of maximal CohenMacaulay A-modules. We denote by X(M ) the maximal Cohen-Macaulay approximation of M . In [10, Theorem 13, p. 620], Martsinkovsky and Strooker proved that if M is evenly linked to N then X(M ) ∼ = X(N ) in CM(A). They also asked if this result holds for M and N are in the same even quasi-linkage class, see [10, Question 3, p. 623]. A motivation for this paper was to try and solve this question. We answer this question in the negative. We prove Theorem 1.1. There exists a complete intersection A of dimension one and finite length modules M, N such that M is evenly quasi-linked to N but X(M ) ≇ X(N ). To prove our next result we introduce a construction (essentially due to Ferrand) which is very useful, see section three. Let CMg (A) be the full subcategory of Cohen-Macaulay A-modules of codimension g. We prove: Theorem 1.2. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈ CMg (A). Assume M ∼q N where q is a Gorenstien ideal in A. Let L be a finitely generated A-module and let D ∈ CMr (A). Set D† = ExtrA (D, A). Then (1) ExtiA (L, M ) = 0 for all i ≫ 0 if and only if TorA i (L, N ) = 0 for all i ≫ 0. (2) ExtiA (M, D) = 0 for all i ≫ 0 if and only if ExtiA (D† , N ) = 0 for all i ≫ 0. We should note that this result is new even in the case for cyclic modules. A remarkable consequence of Theorem 1.2 is the following result Corollary 1.3. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈ CMg (A) and C ∈ CMr (A). Assume M ∼q N and C ∼n D where q, n are Gorenstein ideals in A. Then ExtiA (M, C) = 0 for all i ≫ 0 ⇐⇒ ExtiA (D, N ) = 0 for all i ≫ 0. INVARIANTS OF LINKAGE OF MODULES 3 In particular ExtiA (M, M ) = 0 for all i ≫ 0 ⇐⇒ ExtiA (N, N ) = 0 for all i ≫ 0. We also prove the following surprising invariant of quasi-linkage. Theorem 1.4. Let (A, m) be a Gorenstein local ring and let M, N, N ′ ∈ CMg (A). Assume M is quasi-evenly linked to N and that it is quasi-oddly linked to N ′ . Then (1) End(M )/ rad End(M ) ∼ = End(N )/ rad End(N ). op (2) End(M )/ rad End(M ) ∼ = (End(N ′ )/ rad End(N ′ )) . Here if Γ is a ring then Γop is its opposite ring. We now describe in brief the contents of this paper. In section two we recall some preliminaries regarding linkage of modules as given in [10]. In section three we give a construction which is needed to prove our resuts. We prove Theorem 1.2(1) in section four and Theorem 1.2(2) in section five. We recall some facts regarding cohomological operators in section six. This is needed in section seven where we prove Theorem 1.1. Finally in section eight we prove Theorem 1.4. 2. Some preliminaries on Liason of Modules In this section we recall the definition of linkage of modules as given in [10]. Throughout (A, m) is a Gorenstein local ring of dimension d. φ → F0 → M → 0 2.1. Let us recall the definition of transpose of a module. Let F1 − be a minimal presentation of M . Let (−)∗ = Hom(−, A). The transpose Tr(M ) is defined by the exact sequence φ∗ 0 → M ∗ → F0∗ −→ F1∗ → Tr(M ) → 0. Also let Ω(M ) be the first syzygy of M . Definition 2.2. Two A-modules M and N are said to be horizontally linked if M∼ = Ω(Tr(M )). = Ω(Tr(N )) and N ∼ Next we define linkage in general. Definition 2.3. Two A-modules M and N are said to be linked via a Gorenstein ideal q if (1) q ⊆ ann M ∩ ann N , and (2) M and N are horizontally linked as A/q-modules. We write it as M ∼q N . If q is a complete intersection we say M is CI-linked to N via q. If q is a quasi Gorenstein ideal then we say M is quasi-linked to N via q. Remark 2.4. It can be shown that ideals I and J are linked by a quasi-Gorenstein ideal q (definition as in the introduction) if and only if the module A/I is quasilinked to A/J by q, see [10, Proposition 1, p. 592]. 2.5. We say M, N are in same linkage class of modules if there is a sequence of A-modules M0 , . . . , Mn and Gorenstein ideals q0 . . . , qn−1 such that (i) Mj ∼qj Mj+1 , for j = 0, . . . , n − 1. (ii) M0 = M and Mn = N . If n is even then we say that M and N are evenly linked. Analogously we can define the notion of CI-linkage class, quasi-linkage class, even CI-linkage class and even quasi-linkage class (of modules). 4 TONY J. PUTHENPURAKAL 3. A Construction In this section we describe a construction essentially due to Ferrand. Throughout (A, m) is a Gorenstein local ring of dimension d. 3.1. We note the following well-known result, see [4, 3.3.10]. Let D ∈ CMg (A). Then ExtiA (D, A) = 0 for i 6= g. Set D† = ExtgA (D, A). Then D† ∈ CMg (A). Furthermore (D† )† ∼ = D. The following result is well-known. However we give a proof as we do not have a reference. Lemma 3.2. Let (A, m) be a Gorenstein local ring of dimension d and let M ∈ CMg (A). Let q be a quasi-Gorenstein ideal of grade g contained in ann M . Set B = A/q. Then ExtgA (M, A) ∼ = HomB (M, B). Proof. Let y = y1 , . . . , yg ∈ q be a regular sequence. Set C = A/(y). Then we have a natural ring homomorphism C → B. As C, B are Gorenstein rings we have HomC (B, C) ∼ = B, see [4, 3.3.7]. We now note that Extg (M, A) ∼ see [4, 3.1.16] = HomC (M, C), A = HomC (M ⊗B B, C), ∼ = HomB (M, HomC (B, C)), = HomB (M, B).  3.3. Construction: Let M ∈ CMg (A) and let q be a quasi-Gorenstein ideal in A of codimension g contained in ann M . Let M ∼q N . Let P be minimal free resolution of M and let Q be a minimal free resolution of P0 /qP0 . We have a natural map P0 /qP0 → M → 0. We lift this to a chain map φ : Q → P. We then dualize this map to get a chain map φ∗ : P∗ → Q∗ . Let C = cone(φ∗ ). Lemma 3.4. i H (C) = ( N, 0, if i = g, if i = 6 g. Proof. We have an exact sequence of complexes 0 → Q∗ → C → P∗ (−1) → 0. Notice i ∗ H (P ) = ExtiA (M, A) = ( M † = ExtgA (M, A), 0, if i = g, if i = 6 g. We also have H i (Q∗ ) = ExtiA (P0 /qP0 , A) = 0 if i 6= g. It is now immediate that H i (C) = 0 for i 6= g − 1, g. Set B = A/q and P = P0 /qP0 . Then by 3.2 we have Extg (M, A) ∼ = HomB (M, B), and A ExtgA (P , A) ∼ = HomB (P , B). INVARIANTS OF LINKAGE OF MODULES 5 We have an exact sequence of MCM B-modules ǫ → M → 0. 0→K→P − Dualizing with respect to B we get an exact sequence ǫ∗ 0 → HomB (M, B) −→ HomB (P , B) → K ∗ → 0. ∼ K ∗ . We note that the map H g (P∗ ) → H g (Q∗ ) is ǫ∗ . As M ∼q N we get that N = As a consequence we obtain that H g−1 (C) = 0 and H g (C) = N.  Remark 3.5. (1) If A is regular local, M = A/I, q is a complete intersection and I ∼q J, then this construction was used by Ferrand to give a projective resolution of A/J, see [12, Proposition 2.6]. (2) If M is perfect A-module of codimension g and q is a Gorenstein ideal then this construction was used by Martsinkovsky and Strooker to give a projective resolution of N , see [10, Proposition 10, p. 597]. Our interest in this construction is due to the following: 3.6. Observation: Let B g (C) be the module of g-boundaries of C and Z g (C) be the module of g-cocycles of C. Then projdimA B g (C) is finite and Z g (C) is a maximal Cohen-Macaulay A-module. Thus the sequence 0 → B g (C) → Z g (C) → N → 0, is a maximal Cohen-Macaulay approximation of N . Proof This follows from Lemma 3.4. 4. Proof of Theorem 1.2(1) In this section we prove Theorem 1.2(1). It is an easy consequence of the following result: Theorem 4.1. Let (A, m) be a Gorenstein local ring of dimension d. Let M ∈ CMg (A). Assume M ∼q N where q is a Gorenstien ideal in A. Let L be a maximal Cohen-Macaulay A-module. Let P be minimal free resolution of N and let Q be a minimal free resolution of P0 /qP0 . We do the construction as in 3.3 and let C = cone(φ∗ ). Set X = Z g (C) and Y = B g (C). For s ≥ 1, let Xs be the image of the map P∗s−1 → P∗s . Let x = x1 , . . . , xd be a maximal regular A-sequence. Set A = A/(x) and L = L/xL. Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) ExtiA (L, M ) = 0 for all i ≫ 0. ExtiA (L, X) = 0 for all i ≫ 0. For all s ≫ 0 and for all i ≥ 1 we have ExtiA (L, Xs ) = 0. H i (HomA (L, P∗ )) = 0 for all i ≫ 0. H i (HomA (L ⊗A P, A)) = 0 for all i ≫ 0. H i (HomA (L ⊗A P, A)) = 0 for all i ≫ 0. TorA i (L, N ) = 0 for i ≫ 0. TorA i (L, N ) = 0 for i ≫ 0. We need a few preliminary results before we are able to prove Theorem 4.1. 6 TONY J. PUTHENPURAKAL d n Kn+1 → · · · be a co-chain complex. Let s be an integer. 4.2. Let K : · · · → Kn −→ By K≥s we mean the co-chain complex d d n s Kn+1 → · · · Ks+1 → · · · → Kn −→ 0 → Ks −→ Clearly H i (K≥s ) = H i (K) for all i ≥ s + 1. 4.3. Let P, Q, C be as in Theorem 4.1. As q is a Gorenstein ideal, it has in particular finite projective dimension. It follows that for i ≥ d+1, we get Ci = P∗i+1 and the map Ci → Ci+1 is same as the map P∗i+1 → P∗i+2 . In particular if s ≥ d+1 then C≥s = P∗≥s+1 . We need the following: Lemma 4.4. Let (A, m) be a Gorenstein local ring and let D be a chain-complex with Dn = 0 for n ≤ −1. Assume that Dn is a maximal Cohen-Macaulay Amodule for all n ≥ 0. Let x ∈ m be A-regular. Let A = A/(x) and let D be the the ∗ complex D ⊗A A. Let D∗ be the complex HomA (D, A) and let D be the complex HomA (D, A). Then the following are equivalent: (i) H i (D∗ ) = 0 for i ≫ 0. ∗ (ii) H i (D ) = 0 for i ≫ 0. Proof. Let E be a maximal Cohen-Macaulay A-module. Notice x is E-regular. Furthermore HomA (E, A) is a maximal Cohen-Macaulay A-module and ExtiA (M, A) = 0 for i ≥ 1. Set E = E/xE. x The exact sequence 0 → A − → A → A → 0 induces an exact sequence x 0 → HomA (E, A) − → HomA (E, A) → HomA (E, A) → 0. Thus we have an exact sequence of co-chain complexes of A-modules x ∗ 0 → D∗ − → D∗ → D → 0. This in turn induces a long-exact sequence in cohomology (4.4.1) x ∗ · · · → H i (D∗ ) − → H i (D∗ ) → H i (D ) → H i+1 (D∗ ) → · · · We now prove (i) =⇒ (ii). This follows from (4.4.1). x (ii) =⇒ (i). From (4.4.1) we get that for all i ≫ 0 the map H i (D∗ ) − → H i (D∗ ) is surjective. The result now follows from Nakayama’s Lemma.  As an easy consequence of 4.4 we get the following: Corollary 4.5. (with same hypotheses as in Lemma 4.4). Let x = x1 , . . . , xd be a maximal regular sequence in A. Set B = A/(x), K = D ⊗A B, Let K∗ be the complex HomB (K, B). Then the following are equivalent: (i) H i (D∗ ) = 0 for i ≫ 0. (ii) H i (K∗ ) = 0 for i ≫ 0. We now give Proof of Theorem 4.1. (i) ⇐⇒ (ii). By 3.6 we get that X is a maximal CohenMacaulay A-module, projdimA Y is finite and we have an exact sequence 0 → Y → X → M → 0. As A is Gorenstein we have that injdimA Y is finite. It follows that ExtiA (L, X) ∼ = ExtiA (L, M ) for all i ≥ d + 1. The result follows. INVARIANTS OF LINKAGE OF MODULES 7 fs = image(Cs → Cs+1 ). For s ≥ g + 1 by Lemma (ii) ⇐⇒ (iii). For s ≥ 1 let X fs is maximal Cohen-Macaulay. By Lemma 3.4 we also have an 3.4 we get that X exact sequence fs → 0. 0 → X → Cg → Cg+1 → · · · → Cs → X It follows that if s ≫ 0 then ExtiA (L, X) = 0 for i ≫ 0 is equivalent to fs ) = 0 for i ≥ 1. The result now follows as q is a Gorenstein ideal; so we ExtiA (L, X fs = Xs+1 for s ≥ d + 1, see 4.3. have X (iii) =⇒ (iv). Suppose ExtiA (L, Xs ) = 0 for all i ≥ 1 and all s ≥ c. Set a = max{g + 1, c}. As H i (P∗ ) = ExtA i (N, A) = 0 for i ≥ g + 1 we have an exact sequence 0 → Xa → P∗a+1 → P∗a+2 → · · · → P∗n → P∗n+1 → · · · As ExtiA (L, Xs ) = 0 for all i ≥ 1 and all s ≥ a we get that the induced sequence 0 → HomA (L, Xa ) → HomA (L, P∗a+1 ) → · · · is exact. It follows that H i (L, P∗≥a+1 ) = 0 for i ≥ a + 2. Thus by 4.2 we get that H i (L, P∗ ) = 0 for i ≥ a + 2. (iv) =⇒ (iii). Suppose H i (L, P∗ ) = 0 for i ≥ r. If s ≥ r then H i (L, P∗≥s ) = 0 for all i ≥ s + 1. Now let a = max{g + 1, r}. Let s ≥ a. As argued before we have an exact sequence 0 → Xs → P∗s+1 → P∗s+2 → · · · → P∗n → P∗n+1 → · · · As H i (L, P∗≥s ) = 0 for all i ≥ s + 1 we get that the induced sequence 0 → HomA (L, Xs ) → HomA (L, P∗s+1 ) → · · · 2 1 ∼ is exact. In particular ExtA 1 (L, Xs ) = 0. Notice ExtA (L, Xs ) = ExtA (L, Xs+1 ). i The latter module is zero by the same argument. Iterating we get ExtA (L, Xs ) = 0 for all i ≥ 1. (iv) ⇐⇒ (v). We have an isomorphism of complexes HomA (L, P∗ ) ∼ = HomA (L ⊗A P, A). The result follows. (v) ⇐⇒ (vi). D = L⊗A P is a complex of maximal Cohen-Macaulay A-modules since L is maximal Cohen-Macaulay and Pn is a finitely generated free A-module. Also Dn = 0 for n ≤ −1. Set K = D ⊗A A = L ⊗A P. The result now follows from Corollary 4.5. (vi) ⇐⇒ (vii). We note that A is a zero-dimensional Gorenstein local ring and so is injective as an A-module. Furthermore A is the injective hull of its residue field. It follows that H i (Hom (L ⊗A P, A)) ∼ = Hom (Hi (L ⊗A P), A) = Hom (TorA (L, N ), A). A A A i Therefore by [4, 3.2.12] we get the result. (vii) ⇐⇒ (viii). As L is a maximal Cohen-Macaulay A-module we get that x is an L-regular sequence. Set Lj = L/(x1 , . . . , xj ) for j = 1, . . . , d. Note Ld = L. x1 L → L1 → 0. This induces a long exact We have an exact sequence 0 → L −→ sequence x 1 A A A · · · → TorA i (L, N ) −→ Tori (L, N ) → Tori (L1 , N ) → Tori−1 (L, N ) → · · · 8 TONY J. PUTHENPURAKAL A Clearly if TorA i (L, N ) = 0 for i ≫ 0 then Tori (L1 , N ) = 0 for all i ≫ 0. Conversely x1 A A if Tori (L1 , N ) = 0 for all i ≫ 0 then for the map TorA i (L, N ) −→ Tori (L, N ) is A surjective for all i ≫ 0. By Nakayama’s Lemma we get Tori (L, N ) = 0 for i ≫ 0. x2 L1 → L2 → 0. A similar argument We also have an exact sequence 0 → L1 −→ A gives that TorA i (L1 , N ) = 0 for i ≫ 0 if and only if Tori (L2 , N ) = 0 for all i ≫ 0. A Combining with the previous result we get that Tori (L, N ) = 0 for i ≫ 0 if and only if TorA i (L2 , N ) = 0 for all i ≫ 0. Iterating this argument yields the result.  We now give Proof of Theorem 1.2(1). Let 0 → W → E → L → 0 be a maximal CohenMacaulay approximation of L. As projdimA W is finite we get that ExtiA (L, M ) ∼ = A ∼ ExtiA (E, M ) for i ≥ d + 2 and TorA i (L, N ) = Tori (E, N ) for i ≥ d + 2. The result now follows from Theorem 4.1.  5. Proof of Theorem 1.2(2) and Corollary 1.3 In this section we prove Theorem 1.2(2). We need a few preliminary facts to prove this result. 5.1. Let (A, m) be a Noetherian local ring and let E be the injective hull of its residue field. If G is an A-module then set G∨ = HomA (G, E). Let ℓ(G) denote the length of G. The following result is known. We give a proof as we are unable to find a reference. Lemma 5.2. Let (A, m) be a Noetherian local ring and let E be the injective hull of its residue field. Let M be finitely generated A-module. Let L be an A-module with ℓ(L) < ∞. Then for all i ≥ 0 we have an isomorphism Exti (M, L) ∼ = (TorA (M, L∨ ))∨ . A i Proof. We note that (L ) ∼ = L, see [4, 3.2.12]. Let P be a minimal projective resolution of M . We have the following isomorphism of complexes: HomA (P ⊗A L∨ , E) ∼ = HomA (P, HomA (L∨ , E)) ∼ = HomA (P, L). ∨ ∨ We have H i (HomA (P, L)) = ExtA i (M, L). Notice as E is an injective A-module we have H i (HomA (P ⊗A L∨ , E)) ∼ = HomA (Hi (P ⊗A L∨ ), E) ∨ ∼ = HomA (TorA i (M, L ), E) ∨ ∨ = (TorA i (M, L )) .  The following result is also known. We give a proof as we are unable to find a reference. Lemma 5.3. Let (A, m) be a Gorenstein local ring of dimension d and let E be the injective hull of its residue field. Let D ∈ CMr (A) and let x = x1 , . . . , xc ∈ m be a D-regular sequence (note c ≤ d − r). Then (1) x is a D† -regular sequence. (2) (D/xD)† ∼ = D† /xD† . INVARIANTS OF LINKAGE OF MODULES 9 (3) If r = d (and so ℓ(D) < ∞) then D† ∼ = D∨ (= HomA (D, E)). (4) If T is another finitely generated D-module then (a) ExtiA (T, D) = 0 for i ≫ 0 if and only if ExtiA (T, D/xD) = 0 for i ≫ 0. (b) ExtiA (D, T ) = 0 for i ≫ 0 if and only if ExtiA (D/xD, T ) = 0 for i ≫ 0. Proof. (1) and (2) : Let x be D-regular. Then notice D/xD is a Cohen-Macaulay x A-module of codimension r + 1. We have a short-exact sequence 0 → D − →D→ D/xD → 0. Applying the functor HomA (−, A) yields a long exact sequence which after applying 3.1 reduces to a short-exact sequence x 0 → ExtrA (D, A) − → ExtrA (D, A) → Extr+1 A (D/xD, A) → 0. It follows that x is D† -regular and D† /xD† ∼ = (D/xD)† . Iterating this argument yields (1) and (2). (3): Let y = y1 , . . . , yd ⊂ annA D be a maximal A-regular sequence. Set B = A/(y). Then D† = ExtnA (D, A) ∼ = HomB (D, B). We have D∨ = HomA (D, E) = HomA (D ⊗A B, E) ∼ = HomB (M, HomA (B, E)). Now HomA (B, E) is an injective B-module of finite length. As B is an Artin Gorenstein local ring we get that HomA (B, E) is free as a B-module. As ℓ(HomA (B, E)) = ℓ(B) (see [4, 3.2.12]) it follows that HomA (B, E) = B. The result follows. (4): For i = 1, . . . , c set Di = D/(x1 , . . . , xi )D. x1 D → D1 → 0 induces a long exact sequence 4(a): The exact sequence 0 → D −→ (5.3.2) x 1 ExtiA (T, D) → ExtiA (T, D1 ) → Exti+1 · · · ExtiA (T, D) −→ A (T, D) → · · · If Exti (T, D) = 0 for all i ≫ 0 then by (5.3.2) we get ExtiA (T, D1 ) = 0 for all i ≫ 0. Conversely if ExtiA (T, D1 ) = 0 for all i ≫ 0 then by (5.3.2) we get that the map x1 ExtiA (T, D) −→ ExtiA (T, D) is surjective for all i ≫ 0. So by Nakayama’s Lemma i Ext (T, D) = 0 for all i ≫ 0. x2 D1 → D2 → 0. A similar argument We also have an exact sequence 0 → D1 −→ to the above yields that Exti (T, D1 ) = 0 for all i ≫ 0 if and only if Exti (T, D2 ) = 0 for all i ≫ 0. Combining this with the previous result we get Exti (T, D) = 0 for all i ≫ 0 if and only if Exti (T, D2 ) = 0 for all i ≫ 0. Iterating this argument yields the result. 4(b): This is similar to 4(a).  We now give Proof of Theorem 1.2(2). Let x = x1 , . . . , xd−r be a maximal D-regular sequence. Then by 5.3, x is also a D† regular sequence. Also by 5.3, we get D† /xD† = (D/xD)† . Let E be the injective hull of the residue field of A. 10 TONY J. PUTHENPURAKAL We have ExtiA (M, D) = 0 for all i ≫ 0 ⇐⇒ ExtiA (M, D/xD) = 0 for all i ≫ 0; see 5.3, ∨ ⇐⇒ TorA i (M, (D/xD) ) = 0 for all i ≫ 0; see 5.2 and [4, 3.2.12], † † ⇐⇒ TorA i (M, D /xD ) = 0 for all i ≫ 0; see 5.3, † † ⇐⇒ ExtA i (D /xD , N ) = 0 for all i ≫ 0; see Theorem 1.2(1), † ⇐⇒ ExtA i (D , N ) = 0 for all i ≫ 0; see 5.3.  We now give Proof of Corollary 1.3. By Theorem 1.2(2) we get that ExtiA (M, C) = 0 for all i ≫ 0 ⇐⇒ ExtiA (C † , N ) = 0 for all i ≫ 0. We note that C † = ExtrA (C, A) ∼ = HomA/n (C, A/n), see 3.2. As C ∼n D we have † an exact sequence 0 → C → G → D → 0, where G is a finitely generated free A/n-module. As n is a Gorenstein ideal in A we get projdimA G is finite. The result follows.  6. Some preliminaries to prove Theorem 1.1 In this section we discuss a few preliminaries which will enable us to prove Theorem 1.1. More precisely we need the notion of cohomological operators over a complete intersection ring; see [7] and [6]. 6.1. Let f = f1 , . . . , fc be a regular sequence in a local Noetherian ring (Q, n). We assume f ⊆ n2 . Set I = (f ) and A = Q/I. 6.2. The Eisenbud operators, [6] are constructed as follows: ∂ ∂ → Fi → · · · be a complex of free A-modules. → Fi+1 − Let F : · · · → Fi+2 − Step 1: Choose a sequence of free Q-modules Fei and maps ∂e between them: e e ∂ ∂ e : · · · → Fei+2 − → Fei → · · · → Fei+1 − F e so that F = A ⊗ F Pc Step 2: Since ∂e2 ≡ 0 modulo (f ), we may write ∂e2 = j=1 fj e tj where tej : Fei → e Fi−2 are linear maps for every i. Step 3: Define, for j = 1, . . . , c the map tj = tj (Q, f , F) : F → F(−2) by tj = A⊗e tj . 6.3. The operators t1 , . . . , tc are called Eisenbud’s operator’s (associated to f ) . It can be shown that (1) ti are uniquely determined up to homotopy. (2) ti , tj commute up to homotopy. 6.4. Let R = A[t1 , . . . , tc ] be a polynomial ring over A with variables t1 , . . . , tc of degree 2. Let M, N be finitely generated A-modules. By considering a free resolution F of M we get well defined maps tj : ExtnA (M, N ) → Extn+2 R (M, N ) for 1 ≤ j ≤ c and all n, INVARIANTS OF LINKAGE OF MODULES 11 L which turn Ext∗A (M, N ) = i≥0 ExtiA (M, N ) into a module over R. Furthermore these structure depend on f , are natural in both module arguments and commute with the connecting maps induced by short exact sequences. 6.5. Gulliksen, [7, 3.1], proved that if projdimQ M is finite then Ext∗A (M, N ) is a finitely generated R-module. For N = k, the residue field of A, Avramov in [2, 3.10] proved a converse; i.e., if Ext∗A (M, k) is a finitely generated R-module then projdimQ M is finite. 6.6. We need to recall the notion of complexity of a module. This notion was th introduced by Avramov in [2]. Let βiA (M ) = ℓ(TorA Betti number i (M, k)) be the i of M over A. The complexity of M over A is defined by   β A (M ) cxA M = inf b ∈ N | lim n b−1 < ∞ . n→∞ n If A is a local complete intersection of codim c then cxA M ≤ c. Furthermore all values between 0 and c occur. 6.7. Since m ⊆ ann ExtiA (M, k) for all i ≥ 0 we get that Ext∗A (M, k) is a module over S = R/mR = k[t1 , . . . , tc ]. If projdimQ M is finite then Ext∗A (M, k) is a finitely generated S-module of Krull dimension cx M . 6.8. If (Q, n) is regular then by [3, Theorem I(3)] we get that dimS Ext∗A (M, k) = dimS Ext∗A (k, M ). In particular if M is maximal Cohen-Macaulay A-module then cx M = cx M ∗ . Using this fact it is not difficult to show that if M is a CohenMacaulay A-module then cx M = cx M † . 6.9. Let Q is regular local with infinite residue field and let M be a finitely generated A-module with cx(M ) = r. The surjection Q → A factors as Q → R → A, with the kernels of both maps generated by regular sequences, projdimR M < ∞ and cxA M = projdimR A (see [2, 3.9]). We need the following: Proposition 6.10. Let Q = k[x, y, z](x,y,z) where k is an infinite field. Let m be the maximal ideal of Q. Let q be an m-primary Gorenstein ideal such that q is not a complete intersection. Suppose q ⊇ (u, v) where u, v ∈ m2 is an Q-regular sequence. Set A = Q/(u, v) and q = q/(u, v). Then cx A/q = 2. Proof. As codim A = 2 we have that cx A/q = 0, 1 or 2. We prove cx A/q 6= 0, 1. If cx A/q = 0 then A/q has finite projective dimension over A. So by AuslanderBuchsbaum formula projdimA A/q = 1. Therefore q is a principal ideal. It follows that q is a complete intersection, a contradiction. If cx A/q = 1 then by 6.9, the surjection Q → A factors as Q → R → A, with the kernels of both maps generated by regular sequences, projdimR A/q < ∞ and cxA A/q = projdimR A = 1. Thus dim R = 2. So R = Q/(h) for some h. As A/q has finite length, by Auslander-Buchsbaum formula projdimR A/q = 2. Consider the minimal resolution of A/q over R: 0 → Rb → Ra → R → A/q → 0. As A/q is a Gorenstein ring we have b = 1 and so a = 2. Thus there exists α, β ∈ R with A/q = R/(α, β) = Q/(h, α, β). It follows that q is a complete intersection ideal, a contradiction.  12 TONY J. PUTHENPURAKAL 7. Proof of Theorem 1.1 In this section we prove Theorem 1.1. We first make the following: 7.1. Construction: Let Q = Q[x, y, z](x,y,z) and let m be its maximal ideal. We construct a m-primary Gorenstein ideal q in Q such that (1) q is not a complete intersection. (2) There exists a Q-regular sequence f, u, v ∈ m2 such that (f a , v b , u) ⊆ q ⊆ (f, u, v) for some a, b ≥ 2. Set u = x2 + y 2 + z 2 , R = Q/(u), n = m/(u). Then note (x, y) is a reduction of n and n2 = (x, y)n. It follows that x7 , y 7 is a regular sequence in R. So I = (x7 , y 7 ) : (xy + yz + xz) is a Gorenstein ideal. Using Singular, [5], it can be shown that I has 12 minimal generators and I ⊆ n6 . In particular I is not a complete intersection in R. Also note that n6 ⊆ (x, y)5 ⊆ (x2 , y 2 ). Let q be an ideal in Q containing u such that q/(u) = I. Then q has 12 or 13 minimal generators. So q is not a complete intersection. Also clearly q is m-primary. Note ((x2 )4 , (y 2 )4 , u) ⊆ (x7 , y 7 , u) ⊆ q ⊆ (x2 , y 2 , u). Set f = x2 , v = y 2 and a = b = 4. We now give Proof of Theorem 1.1. Let Q = Q[x, y, z](x,y,z) and let m be its maximal ideal. We make the construction as in 7.1. Let E be a non-free stable maximal CohenMacaulay Q/(f )-module. Let 0 → Qr → Qr → E → 0 be a minimal resolution of E as a Q-module. Note u is Q/(f )-regular and so E-regular. Thus we have an exact sequence 0 → (Q/(u))r → (Q/(u))r → E/uE → 0. Set A = Q/(u, f a ) and q = q/(u, f a ). Then note that q is a quasi-Gorenstein ideal in A which is not a complete intersection. Furthermore notice that E/uE is a maximal Cohen-Macaulay Q/(f, u)-module and so a maximal Cohen-Macaulay A-module. Furthermore notice that cx E/uE = 1 as an A-module. We now note that v is A-regular and so E/uE-regular. Set M = E/(u, v)E. Thus M is an A-module of finite length. Also cxA M = 1. Furthermore q ⊆ (f, u, v) ⊆ annQ M. So we have q ⊆ annA M . Thus we have finite length module M of complexity one and a quasi-Gorenstein ideal q of complexity two. Set B = A/q. Clearly M does not have B as a direct summand. Set N = ΩB (TrB (M )). Then M is horizontally linked to N as B-modules, see [10, Proposition 8, p. 596]. So M ∼q N as Amodules. If t ∈ annA M is a regular element then it is not difficult to show that there exists i ≥ 1 such that the C = A/(ti )-module M has no free summands as a C-module. Let M ∼ti L as A-modules. By Lemma 7.2 we have: cxA N = 2 and cxA L = 1. This finishes the proof as for any module P the complexity of a maximal CohenMacaulay approximation of P is equal to complexity of P . We note that N is evenly linked to L and cx X(N ) = cx N = 2 while cx X(L) = cx L = 1. It follows that X(N ) is not stably isomorphic to X(L). INVARIANTS OF LINKAGE OF MODULES 13  We now state and prove the Lemma we need to finish the proof of Theorem 1.1. Lemma 7.2. Let (Q, n) be a regular local ring and let f = f1 , . . . , fc ∈ n2 be a regular sequence. Set A = Q/(f ). Let M ∈ CMg (A). Also let M ∼q N where q is a quasi-Gorenstein ideal in A. Then (1) If q is a Gorenstein ideal then cx M = cx N . (2) If projdim A/q = ∞ and cx A/q > cx M then cx N = cx A/q. Proof. Set B = A/q. Let M † = ExtgA (M, A) ∼ = HomB (M, B), by Lemma 3.2. As M ∼q N we have an exact sequence (7.2.3) 0 → M † → G → N → 0; where G is free B-module. By [3, 3.3] we get cx M † = cx M . The result now follows from the exact sequence 7.2.3 and 6.7.  8. Proof of Theorem 1.4 In this section we prove Theorem 1.4. We need to prove several preliminary results first. 8.1. Let M, N be finitely generated A-modules. By β(M, N ) we mean the subset of HomA (M, N ) which factor through a finitely generated free A-module. We first prove the following Proposition 8.2. Let (A, m) be a Gorenstein local ring. Let M be a maximal Cohen-Macaulay A-module with no free summands. Then β(M, M ) ⊆ rad End(M ). Proof. Let f ∈ β(M, M ). Say f = v ◦ u where u : M → F , v : F → M and F = An . Let u = (u1 , . . . , un ) where ui : M → A. As M does not have a free summand we get that ui (M ) ⊆ m for each i. Thus u(M ) ⊆ mF . It follows that f (M ) ⊆ mM . Thus f ∈ HomA (M, mM ). However it is well-known and easy to prove that HomA (M, mM ) ⊆ rad End(M ).  8.3. It can be easily seen that β(M, M ) is a two-sided ideal in End(M ). Set End(M ) = End(M )/β(M, M ). Assume A is Gorenstein and M is maximal CohenMacaulay with no free summands. Let u π 0→N − →F − → M → 0, be a minimal presentation. Note that N is also maximal Cohen-Macaulay with no free-summnads. We construct a ring homomorphism σ : End(M ) → End(N ) as follows: Let θ ∈ End(M ). Let δ ∈ End(N ) be a lift of θ. We first note that if δ ′ is another lift of θ then it can be easily shown that there exists ξ : F → N such that ξ ◦ u = δ − δ ′ . Thus we have a well defined element σ(θ) ∈ End(N ). Thus we have a map σ e : End(M ) → End(N ) It is easy to see that σ e is a ring homomorphism. We prove Proposition 8.4. (with hypotheses as above) If f ∈ β(M, M ) then σ e (f ) = 0. 14 TONY J. PUTHENPURAKAL Proof. Let F be a minimal resolution of M with F0 = F . Suppose f = ψ ◦ φ where ψ : G → M , φ : M → G and G = Am for some m. We may take G be a minimal resolution of G with G0 = G and Gn = 0 for n > 0. We can construct a lift fe: F → F of f by composing a lift of v with that of u. It follows that fen = 0 for n ≥ 1. An easy computation shows that for this lift fe the corresponding map δ : N → N is infact zero. Thus σ e(f ) = 0.  Thus we have a ring homomorphism σ : End(M ) → End(N ). Our next result is Proposition 8.5. (with hypotheses as above) σ is an isomorphism. Proof. We construct a ring homomorphism τ : End(N ) → End(M ) which we show is the inverse of σ. Let θ : N → N be A-linear. Then θ∗ : N ∗ → N ∗ is also A-linear. We dualize the exact sequence 0 → N → F → M → 0 to get an exact sequence π∗ u∗ 0 → M ∗ −→ F ∗ −→ N ∗ → 0. We can lift θ∗ to an A-linear map δ : M ∗ → M ∗ . Also if δ ′ is another lift then as before it is easy to see δ − δ ′ ∈ β(M ∗ , M ∗ ). We define τe : End(N ) → End(M ) by τe(θ) = δ ∗ . It is clear that τe is a ring homomorphism. Finally as in Proposition 8.4, it can be easily proved that if θ ∈ β(N, N ) then τe(θ) = 0. Thus we have a ring homomorphism τ : End(M ) → End(N ). Finally it is tautalogical that τ ◦ σ = 1End(M) and σ ◦ τ = 1End(N ) .  8.6. If φ : R → S is an isomophism of rings then it is easy to see that φ(rad R) = rad S and thus we have an isomorphism of rings R/ rad R ∼ = S/ rad S. As a corollary we obtain Corollary 8.7. (with hypotheses as above) End(M )/ rad End(M ) ∼ = End(N )/ rad End(N ). Proof. By Proposition 8.5 we have an isomorphism σ : End(M ) → End(N ). By 8.2 we have that β(M, M ) ⊆ rad End(M ). It follows that rad End(M ) = rad End(M )/β(M, M ). Similarly rad End(N ) = rad End(N )/β(N, N ). The result follows from 8.6.  We now give Proof of Theorem 1.4. It suffices to consider to prove that if M0 ∼q M1 ∼ then EndA (M0 )/ rad EndA (M0 ) = (EndA (M0 )/ rad EndA (M0 ))op . By assumption M0 , M1 ∈ CMg (A). It follows that q is a codimension g quasi-Gorenstein ideal [10, Lemma 14, p. 616]. Set B = A/q. Then B is a Gorenstein ring. Notice M0 , M1 are maximal Cohen-Macaulay B-modules. Furthermore they are stable B-modules, see [10, Proposition 3, p. 593]. Notice HomA (Mi , Mi ) = HomB (Mi , Mi ) for i = 0, 1. As M0 is horizontally linked to M1 we get that M0∗ ∼ = Ω(M1 ). By 8.7 we get that End(M1 )/ rad End(M1 ) ∼ = End(M0∗ )/ rad End(M0∗ ). INVARIANTS OF LINKAGE OF MODULES 15 Furthermore it is easy to see that EndB (M0 ) ∼ = EndB (M0∗ )op and this is preserved when we go mod radicals. Thus op End(M1 )/ rad End(M1 ) ∼ = (End(M0 )/ rad End(M0 )) .  References [1] M. Auslander and R-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). Mm. Soc. Math. France (N.S.) No. 38 (1989), 5-37. [2] L. L. Avramov, Modules of finite virtual projective dimension, Invent. math 96 (1989), 71– 101. [3] L. L. Avramov and R-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285-318. [4] W. Bruns and J. Herzog, Cohen-Macaulay Rings, revised edition, Cambridge Stud. Adv. Math., vol. 39, Cambridge University Press, Cambridge, (1998). [5] W. Decker, G. -M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2015). [6] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64. [7] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183. [8] C. Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043-1062. [9] C. Huneke, Numerical invariants of liaison classes, Invent. Math. 75 (1984), no. 2, 301-325. [10] A. Martsinkovsky and J. R. Strooker, Linkage of modules, J. Algebra 271 (2004), no. 2, 587-626. [11] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989. [12] C. Peskine and L. Szpiro, Liaison des varits algbriques. I, Invent. Math. 26 (1974) 271-302. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India E-mail address: tputhen@math.iitb.ac.in 

arXiv:1604.02691v1 [] 10 Apr 2016 On an algorithm for receiving Sudoku matrices Krasimir Yordzhev Faculty of Mathematics and Natural Sciences, South-West University Ivan Mihaylov 66, Blagoevgrad, 2700, Bulgaria E-mail: yordzhev@swu.bg Abstract This work examines the problem to describe an efficient algorithm for obtaining n2 × n2 Sudoku matrices. For this purpose, we define the concepts of n × n Πn -matrix and disjoint Πn -matrices. The article, using the set-theoretical approach, describes an algorithm for obtaining n2 -tuples of n × n mutually disjoint Πn matrices. We show that in input n2 mutually disjoint Πn matrices, it is not difficult to receive a Sudoku matrix. Keywords: Sudoku matrix; S-permutation matrix; Πn -matrix; disjoint matrices; data type set 2010 Mathematics Subject Classification: 05B20, 68Q65 1 Introduction and notation Let n be a positive integer. Throughout [n] denotes the set [n] = {1, 2, . . . , n} , Un = [n] × [n] = {ha, bi | a, b ∈ [n]} and Vn denotes the set of all subsets of Un . Let Pij , 1 ≤ i, j ≤ n, be n2 square n× n matrices, whose entries are elements of the set [n2 ] = {1, 2, . . . , n2 }. The n2 × n2 matrix   P11 P12 · · · P1n  P21 P22 · · · P2n    P = . .. ..  ..  .. . . .  Pn1 Pn2 · · · Pnn is called a Sudoku matrix, if every row, every column and every submatrix Pij , 1 ≤ i, j ≤ n comprise a permutation of the elements of the set [n2 ], i.e., every integer s ∈ {1, 2, . . . , n2 } is found just once in each row, column, and submatrix Pij . Submatrices Pij are called blocks of P . This work is dedicated to the problem of finding an algorithm for getting all n2 × n2 Sudoku matrices for an arbitrary integer n ≥ 2. This task is solved for 1 n = 2 and n = 3 [3]. When n > 3, according to our information, this problem is still open. Finding algorithm to obtain n2 × n2 , n ≥ 4 Sudoku matrices will lead to solving the problem of constructing Sudoku puzzle of higher order, which will increase the interest in this entertaining game. Here we not going to examine and compare different algorithms for solving any Sudoku puzzle. Here we will examine some algebraic properties of n2 × n2 Sudoku matrices, which are the basis for obtaining various Sudoku puzzles. A binary (or boolean, or (0,1)-matrix ) is a matrix all of whose elements belong to the set B = {0, 1}. With Bn we will denote the set of all n × n binary matrices. Two n × n binary matrices A = (aij ) ∈ Bn and B = (bij ) ∈ Bn will be called disjoint if there are not integers i, j ∈ [n] such that aij = bij = 1, i.e. if aij = 1 then bij = 0 and if bij = 1 then aij = 0. A matrix A ∈ Bn2 is called an S-permutation if in each row, in each column, and in each block of A there is exactly one 1. Let the set of all n2 × n2 Spermutation matrices be denoted by Σn2 . A formula for calculating the number of all pairs of disjoint S-permutatiom matrices is given in [11]. S-permutation matrices and their algebraic properties have an important part in the description of the discussed in [6] algorithm. The concept of S-permutation matrix was introduced by Geir Dahl [2] in relation to the popular Sudoku puzzle. It is well known that Sudoku matrices are special cases of Latin squares. It is widespread puzzle nowadays, which presents in the entertaining pages in most of the newspapers and magazines and in entertaining web sites. Sudoku, or Su Doku, is a Japanese word (or phrase) meaning something like Number Place. Obviously a square n2 × n2 matrix M with elements of [n2 ] = {1, 2, . . . , n2 } is a Sudoku matrix if and only if there are matrices A1 , A2 , . . . , An2 ∈ Σn2 , each two of them are disjoint and such that P can be given in the following way: M = 1 · A1 + 2 · A2 + · · · + n2 · An2 (1) Thus, the problem to describe an efficient algorithm for obtaining all n2 tuples of mutually disjoint S-permutation matrices naturally arises. This work is devoted to this task. For this purpose, in the next section using the settheoretical approach, we define the concepts of Πn -matrix and disjoint Πn matrices. We will prove that so defined task can be reduced to the task of receiving all n2 -tuples of mutually disjoint Πn -matrices. In section 3 we will describe an algorithm for obtaining n2 -tuples of n × n mutually disjoint Πn matrices and we will show that in input n2 mutually disjoint Πn matrices, it is not difficult to receive a Sudoku matrix. Described in this article algorithm essentially differs from the algorithm described in [3]. 2 2 A representation of S-permutation matrices Let n be a positive integer. If z1 z2 . . . zn is a permutation of the elements of the set [n] = {1, 2, . . . , n} and let us shortly denote σ this permutation. Then in this case we will denote by σ(i) the i-th element of this permutation, i.e. σ(i) = zi , i = 1, 2, . . . , n. Definition 1 Let Πn denotes the set of all n × n matrices, constructed such that π ∈ Πn if and only if the following three conditions are true: i) the elements of π are ordered pairs of integers hi, ji, where 1 ≤ i, j ≤ n; ii) if [ha1 , b1 i ha2 , b2 i · · · han , bn i] is the i-th row of π for any i ∈ [n] = {1, 2, . . . , n}, then a1 a2 . . . an in this order is a permutation of the elements of the set [n]; iii) if   ha1 , b1 i  ha2 , b2 i      ..   . han , bn i is the j-th column of π for any j ∈ [n], then b1 , b2 , . . . , bn in this order is a permutation of the elements of the set [n]. The matrices of the set Πn we will call Πn -matrices. From Definition 1, it follows that we can represent each row and each column of a matrix M ∈ Πn with the help of a permutation of elements of the set [n]. Conversely for every (2n)-tuple hhρ1 , ρ2 , . . . , ρn i, hσ1 , σ2 , . . . , σn ii, where ρi = ρi (1) ρi (2) . . . ρi (n), 1≤i≤n σj = σj (1) σj (2) . . . σj (n), 1≤j≤n are 2n permutations of elements of [n] (not necessarily different), then the matrix   hρ1 (1), σ1 (1)i hρ1 (2), σ2 (1)i · · · hρ1 (n), σn (1)i  hρ2 (1), σ1 (2)i hρ2 (2), σ2 (2)i · · · hρ2 (n), σn (2)i    π=  .. .. .. ..   . . . . hρn (1), σ1 (n)i hρn (2), σ2 (n)i is matrix of Πn . Hence · · · hρn (n), σn (n)i |Πn | = (n!)2n (2)     Definition 2 We say that matrices π ′ = p′ ij n×n ∈ Πn and π ′′ = p′′ ij n×n ∈ Πn are disjoint, if p′ ij 6= p′′ ij for every i, j ∈ [n]. 3     Definition 3 Let π ′ , π ′′ ∈ Πn , π ′ = p′ ij n×n , π ′′ = p′′ ij n×n and let the integers i, j ∈ [n] are such that p′ ij = p′′ ij . In this case we will say that p′ ij and p′′ ij are component-wise equal elements. Obviously two Πn -matrices are disjoint if and only if they do not have component-wise equal elements. Example 1 We consider the following Π3 -matrices:   h3, 1i h2, 1i h1, 2i π ′ = p′ij =  h2, 3i h3, 2i h1, 1i  h3, 2i h1, 3i h2, 3i   h3, 2i h1, 3i h2, 1i   π ′′ = p′′ij =  h3, 3i h1, 1i h2, 2i  h2, 1i h1, 2i h3, 3i   h3, 1i h1, 3i h2, 2i    h2, 2i h3, 1i h1, 1i  π ′′′ = p′′′ ij = h2, 3i h1, 2i h3, 3i   Matrices π ′ and π ′′ are disjoint, because they do not have component-wise equal elements. Matrices π ′ and π ′′′ are not disjoint, because they have two component-wise ′ ′′′ equal elements: p′11 = p′′′ 11 = h3, 1i and p23 = p23 = h1, 1i. Matrices π ′′ and π ′′′ are not disjoint, because they have three component′′′ ′ ′′′ ′′ wise equal elements: p′′12 = p′′′ 12 = h1, 3i, p32 = p32 = h1, 2i, and p33 = p33 = h3, 3i. The relationship between S-permutation matrices and the matrices from the set Πn are given by the following theorem: Theorem 1 Let n be an integer, n ≥ 2. Then there is one to one correspondence θ : Πn ⇆ Σn2 . Proof. Let π = [pij ]n×n ∈ Πn , where pij = hai , bj i, i, j ∈ [n], ai , bj ∈ [n]. Then for every i, j ∈ [n] we construct a binary n × n matrices Aij with only one 1 with coordinates (ai , bj ). Then we obtain the matrix   A11 A12 · · · A1n  A21 A22 · · · A2n    (3) A= . ..  . .. ..  .. . .  . An1 An2 · · · Ann According to the properties i), ii) and iii), it is obvious that the obtained matrix A is n2 × n2 S-permutation matrix. Conversely, let A ∈ Σn2 . Then A is in the form shown in (3) and for every i, j ∈ [n] in the block Aij there is only one 1 and let this 1 has coordinates (ai , bj ). 4 For every i, j ∈ [n] we obtain ordered pairs of integers hai , bj i corresponding to these coordinates. As in every row and every column of A there is only one 1, then the matrix π = [pij ]n×n , where pij = hai , bj i, 1 ≤ i, j ≤ n, which is obtained by the ordered pairs of integers is matrix of Πn , i.e. matrix for which the conditions i), ii) and iii) are true.  Corollary 1 Let π ′ , π ′′ ∈ Πn and let A′ = θ(π ′ ), A′′ = θ(π ′′ ), where θ is the bijection defined in Theorem 1. Then A′ and A′′ are disjoint if and only if π ′ and π ′′ are disjoint. Proof. It is easy to see that with respect of the described in Theorem 1 one to one correspondence, every pair of disjoint matrices of Πn will correspond to a pair of disjoint matrices of Σn2 and conversely every pair of disjoint matrices of Σn2 will correspond to a pair of disjoint matrices of Πn .  Corollary 2 [2] The number of all n2 × n2 S-permutation matrices is equal to |Σn2 | = (n!) 2n Proof. It follows immediately from Theorem 1 and formula (2). 3 (4)  Description of the algorithm Algorithm 2 Receive n2 mutually disjoint Πn -matrices. Input: Integer n Output: P1 , P2 , . . . , Pn2 ∈ Πn such that Pi and Pj are disjoint when i 6= j 1. Construct n × n arrays P1 , P2 , . . . Pn whose entries assume values of the set Vn ; 2. Initialize all entries of P1 , P2 , . . . , Pn with Un ; 3. For every k = 1, 2, . . . , n2 do loop 4. For every i = 1, 2, . . . , n do loop 5. For every j = 1, 2, . . . , n do loop 6. Choose ha, bi ∈ Pk [i][j]; 7. Pk [i][j] = {ha, bi}; 8. For every t = k + 1, k + 2, . . . n2 from the set Pt [i][j] remove the element ha, bi; 9. For every t = j + 1, j + 2, . . . n from the set Pt [i][j] remove all elements hx, yi such that x = a; 5 10. For every t = i + 1, i + 2, . . . n from the set Pt [i][j] remove all elements hx, yi such that y = b; end loop 5; end loop 4; end loop 3. Algorithm 3 Receive a S-permutation matrix from a Πn -matrix. Input: P = [hak , bl i]n×n ∈ Πn , 1 ≤ k, l ≤ n. Output: S = [sij ]n2 ×n2 ∈ Σn2 , 1 ≤ i, j ≤ n2 . 1. Construct an n2 × n2 integer array S = [sij ], 1 ≤ i, j ≤ n2 and initialize sij = 0 for all i, j ∈ {1, 2, . . . , n2 }; 2. For every k, l ∈ {1, 2, . . . , n} do loop 3. i = (k − 1) ∗ n + ak ; 4. j = (l − 1) ∗ n + bl ; 5. sij = 1 end loop. Algorithm 4 Receive Sudoku matrices. Input: Integer n. Output: Sudoku matrix A. 1. Get n2 mutually disjoint Πn matrices P1 , P2 , . . . Pn2 (Algorithm 2); 2. For every k = 1, 2, . . . , n2 from Pk receive Sk ∈ Σn2 (Algorithm 3); 3. A = 1 ∗ S1 + 2 ∗ S2 + · · · + n2 ∗ Sn2 . 4 Conclusion and remarks • Described in section 3 algorithms will work more efficiently if the programmer uses programming languages and programming environments with integrated tools for working with data structure set [1, 4, 5, 7, 8, 9]. • If in item 6 of Algorithm 2 we choose ordered pair ha, bi ∈ Pk [i][j] randomly, then we will get a random Sudoku matrix [10]. Thus we tested the effectiveness of the algorithm. • If in item 6 of Algorithm 2 we choose all ordered pairs ha, bi ∈ Pk [i][j], then finally we will get all n2 × n2 Sudoku matrices. We do not know a general formula for finding the number θn of all n2 × n2 Sudoku matrices for each integer n ≥ 2. We consider that this is an open mathematical problem. Using a computer program based on described in section 3 algorithms, we calculated that when n = 2, there are θ2 = 288 number of 4 × 4 Sudoku 6 matrices. This number coincides with our results obtained using other methods described in [12]. In [3], it has been shown that there are exactly θ3 = 9! · 722 · 27 · 27 704 267 971 = 6 670 903 752 021 072 936 960 number of 9 × 9 Sudoku matrices. The next step is to calculate the number θ4 of 16 × 16 Sudoku matrices. References [1] Pavel Azalov. Object-oriented programming. Data structures and STL. Ciela, Sofia, 2008. [2] Geir Dahl. Permutation matrices related to sudoku. Linear Algebra and its Applications, 430(8–9):2457–2463, 2009. [3] Bertram Felgenhauer and Frazer Jarvis. Enumerating possible sudoku grids, 2005. [4] Ivor Horton. Beginning STL: Standard Template Library. Apress, 2015. [5] Kathleen Jensen and Niklaus Wirth. PASCAL - User Manual and Report. Lecture Notes in Computer Science. Springer, Berlin Heidelberg, 1985. [6] Pallavi Mishra, D. K. Gupta, and Rakesh P. Badoni. A new algorithm for enumerating all possible sudoku squares. Discrete Mathematics, Algorithms and Applications, 8(2):1650026 (14 pages), 2016. [7] Herbert Sghildt. Java: The Complete Reference, Ninth Edition. McGrawHill Education, 2014. [8] Kiat Shi Tan, Willi-Hans Steeb, and Yorick Hardy. Symbolic C++: An Introduction to Computer Algebra using Object-Oriented Programming. Springer-Verlag, London, 2000. [9] Magdalina Todorova. Data structures and programming in C ++. Ciela, Sofia, 2011. [10] Krasimir Yordzhev. Random permutations, random sudoku matrices and randomized algorithms. International J. of Math. Sci. & Engg. Appls., 6(VI):291 – 302, 2012. [11] Krasimir Yordzhev. Calculation of the number of all pairs of disjoint spermutation matrices. Applied Mathematics and Computation, 268:1 – 11, 2015. [12] Krasimir Yordzhev and Hristina Kostadinova. On some entertaining applications of the concept of set in computer science course. Informational Technologies in Education, (10):24–29, 2011. 7 

Suprisal-Driven Zoneout arXiv:1610.07675v6 [cs.LG] 13 Dec 2016 Kamil Rocki Tomasz Kornuta IBM Research, San Jose, CA 95120, USA KMROCKI @ US . IBM . COM TKORNUT @ US . IBM . COM Tegan Maharaj Ecole Polytechnique de Montreal Abstract We propose a novel method of regularization for recurrent neural networks called suprisal-driven zoneout. In this method, states zoneout (maintain their previous value rather than updating), when the suprisal (discrepancy between the last state’s prediction and target) is small. Thus regularization is adaptive and input-driven on a per-neuron basis. We demonstrate the effectiveness of this idea by achieving state-of-the-art bits per character of 1.31 on the Hutter Prize Wikipedia dataset, significantly reducing the gap to the best known highly-engineered compression methods. 1. Introduction An important part of learning is to go beyond simple memorization, to find as general dependencies in the data as possible. For sequences of information, this means looking for a concise representation of how things change over time. One common way of modeling this is with recurrent neural networks (RNNs), whose parameters can be thought of as the transition operator of a Markov chain. Training an RNN is the process of learning this transition operator. Generally speaking, temporal dynamics can have very different timescales, and intuitively it is a challenge to keep track of long-term dependencies, while accurately modeling more short-term processes as well. The Long-Short Term Memory (LSTM) (Hochreiter and Schmidhuber, 1997) architecture, a type of RNN, has proven to be exceptionally well suited for learning longterm dependencies, and is very widely used to model sequence data. Learned, parameterized gating mechanisms control what is retrieved and what is stored by the LSTM’s state at each timestep via multiplicative interactions with LSTM’s state. There have been many approaches to capturing temporal dynamics at different timescales, e.g. neural networks with kalman filters, clockwork RNNs, narx TEGAN . MAHARAJ @ POLYMTL . CA Memory N or th ern Ireland]] Memory clock = = E xternal links == * (No operation) (No operation) (Predictable sequence) (Predictable sequence) Operation probability Surprisal Input clock Input N o r t h e r n I r e l a n d ] ] = = E x t e r n a l l i n k s = = * t Figure 1.1: Illustration of the adaptive zoneout idea networks, and recently hierarchical multiscale neural networks. It has been proven (Solomonoff, 1964) that the most general solution to a problem is the one with the lowest Kolmogorov complexity, that is its code is as short as possible. In terms of neural networks one could measure the complexity of a solution by counting the number of active neurons. According to Redundancy-Reducing Hypothesis (Barlow, 1961) neurons within the brain can code messages using different number of impulses. This indicates that the most probable events should be assigned codes with fewer impulses in order to minimize energy expenditure, or, in other words, that the more frequently occuring patterns in lower level neurons should trigger sparser activations in higher level ones. Keeping that in mind, we have focused on the problem of adaptive regularization, i.e. minimization of a number of neurons being activated depending on the novelty of the current input. Zoneout is a recently proposed regularizer for recurrent neural networks which has shown success on a variety of benchmark datasets (Krueger et al., 2016; Rocki, 2016a). Zoneout regularizes by zoning out activations, that is: freezing the state for a time step with some fixed probability. This mitigates the unstable behavior of a standard dropout applied to recurrent connections. However, since the zoneout rate is fixed beforehand, one has decide a priori to prefer faster convergence or higher stochasticity, Suprisal-Driven Zoneout The main contribution of this paper is the introduction of Surprisal-Driven Adaptive Zoneout, where each neuron is encouraged to be active as rarely as possible with the most preferred state being no operation. The motivation behind this idea is that low complexity codes will provide better generalization. Normalize every output unit: i eyt pit = P yi t ie (2.13) . z¯t ... ct−1 . . memory ... ct + zt yt−1 ... st ht−1 zt ft internal state control logic whereas we would like to be able to set this per memory cell according to learning phase, i.e. lower initially and higher later to prevent memorization/unnecessary activation. This is why we have decided to add surpisal-driven feedback (Rocki, 2016b), since it gives a measurement of current progress in learning. The provided (negative) feedback loop enables to change the zoneout rate online within the scope of a given cell, allowing the zoneout rate to adapt to current information. As learning progresses, the activations of that cell become less frequent in time and more iterations will just skip memorization, thus the proposed mechanism in fact enables different memory cells to operate on different time scales. The idea is illustrated in Fig. 1.1. . it ot . ut ht ... xt Figure 2.1: Sparse LSTM basic operational unit with suprisal-driven adaptive zoneout. Dashed line denotes the zoneout memory lane. 3. Experiments 3.1. Datasets Hutter Prize Wikipedia Hutter Prize Wikipedia (also known as enwik8) dataset (Hutter, 2012). 2. The model We used the surprisal-feedback LSTM (Rocki, 2016b): st = log pt−1 · xTt (2.1) Linux This dataset comprises approximately 603MB of raw Linux 4.7 kernel source code∗ Next we compute the gate activations: 3.2. Methodology ft = σ(Wf · xt + Uf · ht−1 + Vf · st + bf ) (2.2) it = σ(Wi · xt + Ui · ht−1 + Vi · st + bi ) (2.3) ot = σ(Wo · xt + Uo · ht−1 + Vo · st + bo ) (2.4) ut = tanh(Wu · xt + Uu · ht−1 + Vu · st + bu ) (2.5) The zoneout rate is adaptive; it is a function of st , τ is a threshold parameter added for numerical stability, Wy is a h → y connection matrix: St = pt−1 − xt (2.6) zt = min(τ + |St · WyT |, 1) (2.7) Sample a binary mask Zt according to zoneout probability zt : Zt ∼ zt (2.8) New memory state depends on Zt . Intuitively, Zt = 0 means NOP, that is dashed line in Fig. 2.1. ct = (1 − ft Zt ) ct−1 + Zt ĉt = tanh(ct ) ht = ot ĉt it ut 3.3. Results and discussion (2.11) Remark: Surprisal-Driven Feedback has sometimes been wrongly characterized as a ’dynamic evaluation’ method. This is incorrect for the following reasons : 1. It never actually sees test data during training. 2. It does not adapt weights during testing. 3. The evaluation procedure is exactly the same as using standard RNN - same inputs. Therefore it is fair to compare it to ’static’ methods. (2.12) ∗ http://olab.is.s.u-tokyo.ac.jp/˜kamil. rocki/data/ (2.9) (2.10) Outputs: yt = Wy · ht + by In both cases the first 90% of each corpus was used for training, the next 5% for validation and the last 5% for reporting test accuracy. In each iteration sequences of length 10000 were randomly selected. The learning algorithm used was Adadelta with a learning rate of 0.001. Weights were initialized using the so-called Xavier initialization (Glorot and Bengio, 2010). Sequence length for BPTT was 100 and batch size 128. In all experiments only one layer of 4000 LSTM cells was used. States were carried over for the entire sequence of 10000 emulating full BPTT. Forget bias was set initially to 1. Other parameters were set to zero. The algorithm was written in C++ and CUDA 8 and ran on GTX Titan GPU for up to 2 weeks. Suprisal-Driven Zoneout 2.2 SF-LSTM (Test) LSTM (Test) Sparse Zoneout SF-LSTM (Test) Zoneout SF-LSTM (Test) SF-LSTM (Train) LSTM (Train) Sparse Zoneout SF-LSTM (Train) Zoneout SF-LSTM (Train) 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 (a) (b) 0.9 Colors indicate I/O gate activations: Red – write (i), Green – read (o), Blue - erase (f), White – all, Black – no operation 0.8 4h 12h 24h 32h 48h 72h 96h 144h Figure 3.1: Learning progress on enwik8 dataset Table 3.1: Bits per character on the enwik8 dataset (test) BPC mRNN∗ (Sutskever et al., 2011) GF-RNN (Chung et al., 2015) Grid LSTM (Kalchbrenner et al., 2015) Layer-normalized LSTM (Ba et al., 2016) Standard LSTM‡ MI-LSTM (Wu et al., 2016) Array LSTM (Rocki, 2016a) HM-LSTM (Chung et al., 2016) HyperNetworks (Ha et al., 2016) SF-LSTM (Rocki, 2016b) RHN (Zilly et al., 2016) 1.60 1.58 1.47 1.46 1.45 1.44 1.40 1.40 1.38 1.37 1.32 Surprisal-Driven Zoneout 1.31 † cmix v11 (c) (d) Hidden state 1.245 Table 3.2: Bits per character on the Linux dataset (test) (e) BPC SF-LSTM Surprisal-Driven Zoneout (f) Memory cell state 1.38 1.18 We observed substantial improvements on enwik8 (Table 3.1) and Linux (Table 3.2) datasets. Our hypothesis is that it is due to the presence of memorizable tags and nestedness in this dataset, which are ideal for learning with suprisaldriven zoneout. Patterns such as < timestamp > or long periods of spaces can be represented by a single impulse in this approach, zoning out entirely until the end of the pattern. Without adaptive zoneout, this would have to be controlled entirely by learned gates, while suprisal allows quick adaptation to this pattern. Fig 3.2 shows side by side comparison of a version without and with adaptive zoneout, demonstrating that in fact the dynamic span of memory cells is greater when adaptive zoneout is used. Fur- (g) Mean: 0.27/timestep (h) Mean: 0.092/timestep Memory cell change (L1-norm) Figure 3.2: Visualization of memory and network states in time (left to right, 100 time steps, Y-axis represents cell index); Left: without Adaptive Zoneout, Right: with Adaptive Zoneout Suprisal-Driven Zoneout thermore, we show that the activations using adaptive zoneout are in fact sparser than without it, which supports our intuition about the inner workings of the network. An especially interesting observation is the fact that adaptive zoneout seems to help separate instructions which appear mixed otherwise (see Fig 3.2). A similar approach to the same problem is called Hierarchical Multiscale Recurrent Neural Networks (Chung et al., 2016). The main difference is that we do not design an explicit hierarchy of levels, instead allowing each neuron to operate on arbitrary timescale depending on its zoneout rate. Syntactic patterns in enwik8 and linux datasets are highly nested. For example (< page >, < revision >, < comment >, [[:en, ..., not mentioning parallel semantic context (movie, book, history, language). We believe that in order to learn such complex structure, we need distributed representations with every neuron operating at arbitrary time scale independent of another. Hardcoded hierarchical architecture will have problems solving such a problem. 4. Summary The proposed surprisal-driven zoneout appeared to be a flexible mechanism to control the activation of a given cell. Empirically, this method performs extremely well on the enwik8 and linux datasets. 5. Further work We would like to explore variations of suprisal-driven zoneout on both state and cell. Another interesting direction to pursue is the connection with sparse coding - using suprisal-driven zoneout, the LSTM’s cell contents are more sparsely revealed through time, potentially resulting in information being used more effectively. Acknowledgements This work has been supported in part by the Defense Advanced Research Projects Agency (DARPA). References J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. H. B. Barlow. Possible principles underlying the transformations of sensory messages. 1961. J. Chung, Ç. Gülçehre, K. Cho, and Y. Bengio. Gated feedback recurrent neural networks. CoRR, abs/1502.02367, † ‡ Best known compressor: http://mattmahoney.net/dc/text.html our implementation 2015. URL http://arxiv.org/abs/1502. 02367. J. Chung, S. Ahn, and Y. Bengio. Hierarchical multiscale recurrent neural networks. arXiv preprint arXiv:1609.01704, 2016. X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In In Proceedings of the International Conference on Artificial Intelligence and Statistics (AISTATS10). Society for Artificial Intelligence and Statistics, 2010. D. Ha, A. Dai, and Q. V. Le. 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Event-triggered leader-following tracking control for multivariable multi-agent systems ⋆ arXiv:1603.04125v1 [] 14 Mar 2016 Yi Cheng a and V. Ugrinovskii a a School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia. Abstract The paper considers event-triggered leader-follower tracking control for multi-agent systems with general linear dynamics. For both undirected and directed follower graphs, we propose event triggering rules which guarantee bounded tracking errors. With these rules, we also prove that the systems do not exhibit Zeno behavior, and the bounds on the tracking errors can be tuned to a desired small value. We also show that the combinational state required for the proposed event triggering conditions can be continuously generated from discrete communications between the neighboring agents occurring at event times. The efficacy of the proposed methods is discussed using a simulation example. Key words: Event-triggered control, leader-follower tracking, consensus control, multi-agent systems. 1 Introduction Cooperative control of multi-agent systems has received increasing attention in the past decade, see [1] and references therein. However, many control techniques developed so far rely on continuous communication between agents and their neighbors. This limits practicality of these techniques. To address this concern, several approaches have been proposed in recent years. One approach is to apply sampled control [2]. However in sampled data control schemes control action updates continue periodically with the same frequency even after the system has reached the control goal with sufficient accuracy and no longer requires intervention from the controller. Efforts to overcome this shortcoming have led to the idea of triggered control. Self-triggered control strategies [3,4,9] employ a triggering mechanism to proactively predict the next time for updating the control input ahead of time, using the current measurements. On the other hand, event-triggered controllers [5,6,7,8,9,10] trigger control input updates by reacting to excessive deviations ⋆ This work was supported by the Australian Research Council under the Discovery Projects funding scheme (project DP120102152). Accepted for publication in Automatica on March 14, 2016. Email addresses: y.cheng@adfa.edu.au (Yi Cheng), v.ugrinovskii@gmail.com (V. Ugrinovskii). Preprint submitted to Automatica of the decision variable from an acceptable value, i.e., when a continuously monitored triggering condition is violated. This latter approach is the main focus in this paper. The development of event-triggered controllers remains challenging, because the agents in a multi-agent system do not have access to the complete system state information required to make decisions about control input updates. To prove the concept of eventtriggering, the early work was still assuming continuous communication between the neighboring agents [7,9]. To circumvent this limitation, several approaches have been proposed, e.g., see [11,12,13,14,15,16,17]. For instance, different from [7,9] where state-dependent event triggering conditions were used, [11] proposed an event-triggered control strategy using a time-dependent triggering function which did not require neighbors’ information. In [12], a state-dependent event triggering condition was employed, complemented by an iterative algorithm to estimate the next triggering time, so that continuous communications between neighboring agents were no longer needed. In [13], sampled-data event detection has been used. It must be noted that these results as well as many other results in this area were developed for multi-agent systems with single or double integrator dynamics. Most recently, similar results have been developed for multi-agent systems with general dynamics [14,15] and nonlinear dynamics [16,17]. Accepted for publication in Automatica on March 14, 2016. In comparison with the recent work on event-triggered control for general linear systems [14,30,15,25], the main distinction of our method is computing the combinational state directly using the neighbors’ information. This allowed us to avoid additional sampling when checking event triggering conditions, cf. [14,30]. In contrast in [15], to avoid continuous transmission of information, each agent was equipped with models of itself and its neighbors. In [25], estimators were embedded into each node to enable the agents to estimate their neighbors’ states. Both approaches make the controller rather complex, compared with our controller which does not require additional models or estimators. The leader-follower context and the treatment of both directed and undirected versions of the problem are other distinctions. All the papers mentioned above considered the eventtriggered control problem for leaderless systems. The leader-following control is one of the important problems in cooperative control of multi-agent systems [18,19,20,1], and the interest in event-based solutions to this problem is growing [21,22,23,24]. General multidimensional leader following problems still remain technically challenging, and the development is often restricted to the study of single or double integrator dynamics [21,22,23,24]. Zeno behavior presents another challenge, and is not always excluded [21,22]. Excluding Zeno behavior is an important requirement on control protocols since excessively frequent communications reduce the advantages of using the event-triggered control. In this paper, we also consider the event-triggered leaderfollowing control problem for multi-agent systems. Unlike [21,22,23,24], the class of systems considered allows for general linear dynamics. Also, the leader can be marginally stable or even unstable. For both undirected and directed system interconnections, we propose sufficient conditions for the design of controllers which guarantee that the leader tracking errors are contained within certain bounds; these bounds can be optimized by tuning the parameters of the design procedure. We also show that with the proposed eventtriggered control protocols, the system does not exhibit Zeno behavior. These results are the main contribution of the paper. The paper is organized as follows. Section 2 includes the problem formulation and preliminaries. The main results are given in Sections 3 and 4. In Section 3 we consider the case when the system of followers is connected over a directed graph. Although these results are applicable to systems connected over an undirected graph as well, the symmetry of the graph Laplacian makes it possible to derive an alternative control design scheme in Section 4. In Section 5, the generation of the combinational state is discussed. Section 6 provides an illustrative example. The conclusions are given in Section 7. Throughout the paper, ℜn and ℜn×m are a real Euclidean n-dimensional vector space and a space of real n × m matrices. ⊗ denotes the Kronnecker product of two matrices. λmax (·) and λmin (·) will denote the largest and the smallest eigenvalues of a real symmetric matrix. For q ∈ ℜn , diag{q} denotes the diagonal matrix with the entries of q as its diagonal elements. IN is the N × N identity matrix. When the dimension is clear from the context, the subscript N will be suppressed. Its another contribution is the event-triggered control protocols that do not require the neighboring agents to communicate continuously. Instead, the combinational state to be used in the event triggering condition is generated continuously within the controllers, by integrating the information obtained from the neighbors during their communication events. The idea is inspired by [12], however, the procedure in [12] developed for single integrator systems cannot be applied to multiagent systems with general linear dynamics considered here, since in our case dynamics of the measurement error depend explicitly on the combinational state. Also different from [12], the proposed algorithm involves oneway communications between the neighboring agents. The combinational state is computed continuously by each agent and is broadcast to its neighbors only at the time when the communication event is triggered at this node and only in one direction. The neighbors then use this information for their own computation, and do not send additional requests to measure the combinational state. This is an important advantage of our protocol compared with eventtriggered control strategies proposed in [12,21,23,24,30]. In these references, when an event is triggered at one agent, it must request its neighbors for additional information to update the control signals. Owing to this, our scheme is applicable to systems with a directed graph which only involves one way communications. 2 2.1 Problem formulation and preliminaries Communication graph Consider a communication graph Ḡ = (V̄, Ē, Ā), where V̄ = {0, . . . , N } is a finite nonempty node set, Ē ⊆ V̄ × V̄ is an edge set of pairs of nodes, and Ā is an adjacency matrix. Without loss of generality, node 0 will be assigned to represent the leader, while the nodes from the set V = {1, . . . , N } will represent the followers. The (in general, directed) subgraph G = (V, E, A) obtained from Ḡ by removing the leader node and the corresponding edges describes communications between the followers; the edge set E ⊆ V × V represents the communication links between them, with the ordered pair (j, i) ∈ E indicating that node i obtains information from node j; in this case j is the neighbor of i. The set of neighbors of node i in the graph G is denoted as 2 denoted by ti0 , ti1 , . . ., based on samples zi (tik ) of its combinational state. The value of the combinational state is held constant between updates, thus giving rise to the measurement signal ẑi (t) = zi (tik ), t ∈ [tik , tik+1 ). Based on this model, consider the following control law Ni = {j|(j, i) ∈ E}. Following the standard convention, we assume that G does not have self-loops or repeated edges. The adjacency matrix A = [aij ] ∈ ℜN ×N of G is defined as aij = 1 if (j, i) ∈ E, and aij = 0 otherwise. PN Let di = j=1 aij be the in-degree of node i ∈ V and D = diag{d1 , . . . , dN } ∈ ℜN ×N . Then L = D − A is the Laplacian matrix of the graph G, it is symmetric when G is undirected. ui (t) = −K ẑi (t), t ∈ [tik , tik+1 ), where K ∈ ℜp×n is a feedback gain matrix to be defined later. The problem in this paper is to find a control law (4) and an event triggering strategy which achieve the following leader-following property We assume throughout the paper that the leader is observed by a subset of followers. If the leader is observed by follower i, then the directed edge (0, i) is included in Ē and is assigned with the weighting gi = 1, otherwise we let gi = 0. We refer to node i with gi 6= 0 as a pinned node. Let G = diag{g1 , . . . , gN } ∈ ℜN ×N . The system is assumed to have at least one follower which can observe the leader, hence G 6= 0. lim sup t→∞ i 3 ′ Consider a multi-agent system consisting of a leader agent and N follower agents. Dynamics of the ith follower are described by the equation j∈Ni

xj (t) − xi (t) + gi x0 (t) − xi (t) . (6) and constants ω > 0 and µi > 0 chosen so that ω min (Q) ρi = α1 − µi − α α2 > 0, where α1 = λλmax (Y ) and α2 = (2) λmax (P )λ2max (Y BR−1 B ′ Y ) . ϑλmin (Y ) Let ρmin = min ρi and i select νi > 0, σi ∈ (0, ρmin ) and γ > 0. Introduce the combinational state measurement error for agent i si (t) = ẑi (t) − zi (t). We wish to find a distributed event-triggered control law for each follower to be able to track the leader. For each agent i, introduce a combinational state zi (t), X control Y A + A′ Y − 2ϑmin Y BR−1 B ′ Y + Q ≤ 0, Note that the matrix A is not assumed to be Hurwitz, it can be marginally stable or even unstable. zi (t) = (5) Event-triggered leader-following under a directed graph G (1) where xi ∈ ℜn is the state, ui ∈ ℜp is the control input. Also, the dynamics of the leader agent are given by ẋ0 = Ax0 . − xi (t)k2 ≤ ∆, In this section, we propose an event triggering rule and a leader-following tracking control for multi-agent systems where the followers are connected over a directed graph. Our result will involve certain symmetric positive definite matrices R, Q, and Y related through the following Riccati inequality Problem formulation ẋi = Axi + Bui , i=1 kx0 (t) Definition 1 We say that the leader-follower system (1), (2) with a control law (4) does not exhibit Zeno behavior if over any finite time period there are only a finite number of communication events between the follower systems, i.e., for every agent i the sequence of event times tik has the property inf k (tik+1 − tik ) > 0. and F = (L + G) (L + G). 2.2 PN where ∆ is a given positive constant. Furthermore, the closed loop dynamics of the followers must not exhibit Zeno behavior with the proposed event triggering rule. In addition, we assume the graph G contains a spanning tree rooted at a pinned node ir , i.e., gir > 0. Then, −(L + G) is a Metzler matrix. According to [28], the matrix −(L + G) is Hurwitz stable 1 , which implies that −(L + G) is diagonally stable [29]. That is, there exists a positive definite diagonal matrix Θ = diag{ϑ1 , . . . , ϑN } such that H = Θ−1 (L + G) + (L + G)′ Θ−1 > 0. We will also use the following notation: α = 21 λmin (H), ϑmin = −1 min(ϑi ), ϑ = min(ϑ−1 (L + G)(L + G)′ Θ−1 i ), P = Θ i (4) (7) Theorem 1 Given R = R′ > 0, Q = Q′ > 0, suppose there exists Y = Y ′ > 0 such that (6) holds. Then under the control law (4) with K = − α1 R−1 B ′ Y , the system (1), (2) achieves the leader-follower tracking property Nγ , if the of the form (5) with ∆ = ϑλmin (Y )λ min (F )ρmin communication events are triggered at n tik = inf t > tik−1 : o ′ −σi t ksi k2 ≥ αω(µi ϑ−1 + γ) . (8) i z i Y z i + νi e (3) We seek to develop a control scheme where agent i updates its control input at event times, which are 1 These properties of the matrix L + G can be guaranteed under weaker assumptions on the graph G [28]. 3 on every interval [tik , tik+1 ), then it follows from (13) that In addition, the system does not exhibit Zeno behavior. Remark 1 The Riccati inequality (6) is similar to the Riccati inequality employed in [15]. However, [15] considers an undirected topology, and the design uses the second smallest eigenvalue of L. In contrast, (6) uses ϑmin associated with the directed graph G. Also, the inequality (6) is equivalent to the following LMI in Y −1 , which can be solved using the existing LMI solvers, " AY −1 + Y −1 A′ − 2ϑmin BR−1 B Y −1 Y −1 −Q−1 # N N X X dV (ε) ′ νi e−σi t + N γ ρi ϑ−1 ≤− i zi Y zi + dt i=1 i=1 ≤ −ρmin V (ε) + N X νi e−σi t + N γ. (15) i=1 Thus, we have ≤ 0. (9) N  X V (ε) ≤ e−ρmin t V (ε(0)) − i=1 Remark 2 We note that the event triggering condition (8) involves monitoring of the combinational state zi (t), hence the means for generating zi (t) continuously are needed to implement it. A computational algorithm will be introduced later to generate the combinational state using event-triggered communications. + ≤ V (ε(0)) + kzi (t)k ≤ (17) N X i=1 kzi k2 . (18) p κ(ϑλmin (Y ))−1 = κ̄. (19) Next, we prove that the system does not exhibit Zeno behavior. Suppose t1 , t2 are two adjacent zero points of si (t) on the interval [tik , tik+1 ), tik ≤ t1 < t2 < tik+1 . Then ksi (t)k > 0 for all t ∈ (t1 , t2 ) ⊆ (tik , tik+1 ), and the following inequality holds on the interval (t1 , t2 ) (11) d s′ ṡi ksi kkṡi k d ksi k = (s′i si )1/2 = i ≤ = kṡi k. (20) dt dt ksi k ksi k Furthermore, note that on the interval [t1 , t2 ) ṡi (t) = −żi (t), (12) 2 si (t+ = 0. 1) (21) It follows from (2) that ∀t ∈ [t1 , t2 ) Using (6), it follows from (11) and (12) that kṡi (t)k = (13) X j∈Ni = Azi (t) + BK Since the triggering condition (8) enforces the property ′ −σi t (αω)−1 ksi k2 ≤ µi ϑ−1 +γ i z i Y z i + νi e Nγ νi + = κ, ρmin − σi ρmin This implies that for all i, kzi (t)k is bounded, (10) Since K = − α1 R−1 B ′ Y , the following inequality holds
1 ′ dV (ε) ≤ − z ′ Θ−1 ⊗ Q z + ss dt αω
ω ′ + z P ⊗ (Y BR−1 B ′ Y )2 z. α i=1 (16) κ ≥ V (ε) = z ′ (Θ−1 ⊗ Y )z ≥ ϑλmin (Y ) Let the Lyapunov function candidate for the system comprised of the systems (10) be V (ε) = z ′ (Θ−1 ⊗ Y )z, where z = ((L + G) ⊗ In )ε, and ε = [ε′1 . . . ε′N ]′ . Then 2z ′ ((Θ−1 (L + G)) ⊗ (Y BK))z   1 = −z ′ H ⊗ ( Y BR−1 B ′ Y ) z α   1 ′ ≤ −2αz IN ⊗ ( Y BR−1 B ′ Y ) z. α N X Nγ νi + ρmin − σi ρmin where the constant κ depends on the initial conditions. It then follows from (17) that for all t ≥ 0 Define the tracking error εi (t) = x0 (t) − xi (t) at node i. It follows from (7) and (4) that dV (ε) = 2z ′ (Θ−1 (L + G) ⊗ Y BK)z dt
+ (Θ−1 ⊗ Y A)z + (Θ−1 (L + G) ⊗ Y BK)s . e−σi t i=1 Proof of Theorem 1: We first prove that kzi (t)k are bounded. This fact will then be used to prove that under the proposed control law the system does not exhibit Zeno behavior. Also, the property (5) will be proved after Zeno behavior is excluded. ε̇i (t) =Aεi (t) + BKzi (t) + BKsi (t). N X νi Nγ  − ρmin − σi ρmin ≤ kAkksi (t)k + (14) 2 4

ẋj (t) − ẋi (t) + gi ẋ0 (t) − ẋi (t) X j∈Ni Mki , 
zi (tik ) − ẑj (t) + gi zi (tik ) As usual, s(a+ ) , limt↓a s(t), s(b− ) , limt↑b s(t). (22) where Mki = max t∈[tik ,tik+1 ) kAzi (tik ) + BK  P j∈Ni Proof: According to (26), for q any δ > 0, there exists tδ such that kzi (t)k < (1 + δ) ϑλminN(Yγ )ρmin , ̟1 for all i and t > tδ . Therefore, for
a sufficiently large k, Mki ≤ kAk + (2di + gi )kBKk ̟1 ≤ η̟̄1 . Then (28) follows from (24). ✷ zi (tik ) − 
ẑj (t) + gi zi (tik ) k. Hence, using (20) and (22) we obtain ksi k ≤ Mki kAk(tik+1 −tik ) Mki kAk(t−t1 ) (e − 1) ≤ (e − 1) kAk kAk (23) Remark 3 From (27), the upper bound on the tracking error depends on the parameter γ and the size of the network N . Therefore, the tracking performance can be guaranteed even for larger systems, if γ is sufficiently small. On the other hand, the lower bound on the interevent times in (25) reduces if γ is reduced. This means that a higher tracking precision can be achieved by reducing γ, but the communications may become more frequent. However, Corollary 1 shows that when γ is reduced, the frequency of communication events may increase only on an initial interval [0, tδ ], and after time tδ the minimum inter-event time π is independent of γ. for all t ∈ (t1 , t2 ). Since si (t+ 1 ) = 0, (23) holds for all t ∈ [t1 , t2 ) ⊆ [tik , tik+1 ). The expression on the right hand side of (23) is independent of t; hence the above reasoning applies to all such intervals [t1 , t2 ). Hence, (23) holds for all t ∈ [tik , tik+1 ). Thus, from the definition of the event time tik+1 in (8) and (23) we obtain √ Mki kAk(tik+1 −tik ) αωγ ≤ ksi ((tik+1 )− )k ≤ (e − 1). (24) kAk According
to (19), for any k, Mki ≤ kAk + (2di + gi )kBKk κ̄ = ηi κ̄ ≤ η̄κ̄, where η̄ = max ηi . Hence, it Remark 4 Selecting the parameters for the eventtriggering condition (8) involves the following steps: (a) choose matrices Q > 0 and R > 0 and solve the Riccati inequality (6), or equivalently the LMI (9), to obtain the matrix Y , then compute α1 and α2 ; (b) choose µi > 0 and ω > 0 to compute ρi > 0; (c) choose σi ∈ (0, ρmin ); (d) based on the desired upper bound ∆, select γ, see (27); (e) Lastly, choose νi . Note that the term νi e−σi t in (8) governs the triggering threshold during the initial stage of the tracking process. Thus it determines the frequency of communication events during this stage. The value of νi depends on the selected σi . If σi is large, then typically a relatively large νi must be chosen to ensure the communication events occur less frequently. i follows from (24) that tik+1 − tik √  kAk αωγ  1 . ln 1 + ≥ kAk η̄κ̄ (25) Thus, the inter-event intervals are bounded from below uniformly in k, that is, Zeno behavior does not occur. Since Zeno behavior has been ruled out, it follows from (16) and the rightmost inequality in (18) that for all i, lim sup t→∞ PN i=1 kzi (t)k 2 ≤ N γ(ϑλmin (Y )ρmin )−1 . (26) Since z = ((L + G) ⊗ In )ε, this further implies lim sup t→∞ 4 Nγ . (27) i=1 kεi (t)k ≤ ϑλmin (Y )λmin (F )ρmin PN 2 I.e., (5) holds. This concludes the proof. Theorem 2 Let R = R′ > 0, Q = Q′ > 0 be given matrices. Suppose there exists a matrix Y = Y ′ > 0, Y ∈ ℜn×n , solving the following Riccati inequality Corollary 1 For any δ > 0, there exists a sufficiently large tδ such that with the control law and event triggering condition proposed in Theorem 1, inf Y A + A′ Y − 2λY BR−1 B ′ Y + Q ≤ 0, (29) where λ = λmin (λi ) and λi are the eigenvalues of L + G. Then under the control law (4) with K = −R−1 B ′ Y the system (1), (2) achieves the leaderfollower tracking property of the form (5) with ∆ = N γ(ρλmin (Y )λmin (F ))−1 , if the communication events (tik+1 − tik ) p  kAk αωϑλmin (Y )ρmin  1 √ ≥ = π. ln 1 + kAk η̄(1 + δ) N control Although the problem for an undirected G can be regarded as a special case of the problem in Section 3, an independent derivation is of interest, which uses the symmetry of the matrix L + G. Accordingly, a different event triggering condition is proposed for this case. ✷ According to (25) and (27), the parameter γ not only helps to exclude Zeno behavior, but also determines the upper bound of the tracking errors. We now show that after a sufficiently large time, the lower bound on the inter-event intervals becomes independent of γ. More precisely, the following statement holds. k : tik >tδ Event-triggered leader-following under an undirected graph G (28) 5 coordinate transformation ζ = (T −1 ⊗ In )ε, the identity
′ ′ z = (L + G) ⊗ In ε, z = [z1′ , . . . , zN ] , and condition (30) we can show that on every interval [tik , tik+1 ), are triggered at n µi zi′ Qzi + νi e−σi t + γ o ; tik = inf t > tik−1 : kzi kksi k ≥ 2̟2 (30) N  X V (ζ) ≤e−ρt V (ζ(0)) − here ̟2 = λmax (L + G)λmax (Y BR−1 B ′ Y ), µi , νi , σi and γ are positive constants chosen so that 0 < µi < 1, νi > 0, γ > 0 and σi ∈ (0, ρ), where ρ = (1 − µmax )λmin (Q)/λmax (Y ), µmax = max µi . In addition i=1 + under this control law, Zeno behavior is ruled out: ≤V (ζ(0)) + (31) 5 and s = ... Generation of the combinational state To implement the event triggering conditions (8) and (30) in Theorems 1 and 2, the combinational state zi (t) must be known at all times. We now describe how node i can generate zi (t) continuously using only discrete communications from its neighbors at event times. This eliminates the need for agent i to monitor and communicate with its neighbors continuously. According to (1), for t ∈ [tik , tik+1 ), the dynamics of xi (t) and xj (t), j ∈ Ni , on this interval can be expressed as ε̇ = (IN ⊗ A + (L + G) ⊗ BK)ε + (IN ⊗ BK)s, (32) ... i xi (t) = eA(t−tk ) xi (tik ) − s′N ]′ . i xj (t) = eA(t−tk ) xj (tik ) − It follows from [20] that all the eigenvalues of matrix L + G ar positive. Let T ∈ ℜN ×N be an orthogonal matrix such that T −1 (L+G)T = Λ = diag{λ1 , . . . , λN }. ′ ′ Also, let ζ = (T −1 ⊗ In )ε, ζ = [ζ1′ . . . ζN ] . Using this coordinate transformation, the system (32) can be represented in terms of ζ and s, as
ζ̇ = IN ⊗ A + Λ ⊗ (BK) ζ + (T −1 ⊗ (BK))s. (35) Remark 6 It can be shown that the observation made in Remark 3 applies in this case as well. The parameters in the event triggering conditions (30) can be selected following a process similar to that outlined in Remark 4. Proof of Theorem 2: The proof is similar to the proof of Theorem 1 except for the procedure of obtaining an upper bound of zi (t). Therefore, we only outline the proof of boundedness of zi (t). The closed loop system consisting of error dynamics (10) is represented as where as before ε = (34) PN It then follows from (35) that λmin (Y ) i=1 kzi (t)k2 ≤ V (ζ). The rest of the proof of this theorem is similar to the proof of Theorem 1 and is omitted for brevity. ✷ Y A + A′ Y − 2(λ̂ − ̺1 )Y BR−1 B ′ Y + Q ≤ 0. [s′1 i=1 νi Nγ + = h. ρ − σi ρ V (ζ) = ε′ ((L + G)2 ⊗ Y )ε = z ′ (IN ⊗ Y )z. Remark 5 The Riccati inequality (29) in this theorem is similar to the Riccati inequality employed in [15]. However, our condition (29) depends on the smallest eigenvalue λ of the matrix L + G. In contrast, in [15] the second smallest eigenvalue of the graph Laplacian matrix is required to build the consensus algorithm. When the graph topology is completely known at each node, λ can be readily computed. But even when the graph G is not known at each node, λ can be estimated in a decentralized manner [26]. Errors between the true eigenvalue λ and its estimate λ̂ can be accommodated by replacing (29) with a slightly more conservative condition. Suppose |λ − λ̂| < ̺1 , then the following Riccati inequality can be used in lieu of (29): ε′N ]′ N X V (ζ) can be expressed in terms of ε using the inverse transformation ζ = (T −1 ⊗ In )ε and z = ((L + G) ⊗ In )ε p here ~ = h/λmin (Y ), h is defined in (34) below. [ε′1 Nγ νi + ρ − σi ρ e−σi t i=1 i  1 kAkγ  inf (tik+1 − tik ) ≥ ln 1 + ; k kAk 2̟2 η̄~2 N X νi Nγ  − ρ − σi ρ − t̂j = (33) X m : tik <tjm <t ( Z t Z tik Z eA(t−τ ) BKzi (tik )dτ, t̂j tik min(t,tjm+1 ) tjm (36) eA(t−τ ) BKzj (tjl )dτ eA(t−τ ) BKzj (tjm )dτ, (37) tjl+1 , if j has at least one event on [tik , t), t, otherwise, where tjl+1 = min(tjm : tjm ∈ [tik , t)). Equation (37) accounts for the fact that agent j may experience several Consider the following Lyapunov function candidate for the system (33), V (ζ) = ζ ′ (Λ2 ⊗ Y )ζ. Using (29), the 6 events at times tjm , m = l+1, . . ., within the time interval [tik , tik+1 ). When [tik , t) contains no event triggered by agent j, the last term in (37) vanishes. Similarly, the dynamics of the tracking error εi (t) can be expressed as 1 A2 εi (t) = e A(t−tik ) εi (tik ) + Z 2 t tik eA(t−τ ) BKzi (tik )dτ. (38) t1p A1 t1p+1 z1 (t1p ) z1 (t1p+1) t3q z2 (t2k ) t2k t1p t1p+1 t2k+1 z2 (t2k+1) z3 (t3q ) A3 t3q 3 R t′ ′ Using the notation Φ(t, t′ ) = t eA(t −τ ) BKdτ , it follows from (3), (36), (37) and (38) that (a) (b) Fig. 1. Communication between followers in an undirected network: (a) The graph; (b) Communication events. i zi (t) = eA(t−tk ) zi (tik ) + gi Φ(tik , t)zi (tik ) X X + Φ(tjm , min(t, tjm+1 ))(zi (tik ) − zj (tjm )) j∈Ni m : ti <tjm <t k + X j∈Ni Φ(tik , t̂j )(zi (tik ) − zj (tjl )). 1 (39) A2 2 According to (39), to generate zi (t), agent i must know zi (tik ) and zj (tjm ), tik < tjm < t. It has zi (tik ) in hand and thus it must only receive zj (tjm ) when an event is triggered at node j during [tik , tik+1 ). To ensure this, we propose an algorithm to allow every agent to compute its combinational state and broadcast it to its neighbors at time instants determined by its triggering condition. This algorithm has a noteworthy feature that follows from (39) in that only one-way communications occur between the neighboring agents at the triggering time, even when the graph G is undirected. A3 3 (a) t1p A1 t1p+1 z1 (t1p ) z1 (t1p+1) t2k+1 t2k z2 (t2k ) t1p t1p+1 z2 (t2k+1) z3 (t3q ) t3q (b) Fig. 2. Communication between followers in a directed network: (a) The graph; (b) Communication events. the local event time record tik = 0, and the local measurement error si (t) = 0. (b) Receive xj (0) from all neighbors j ∈ Ni ; (c) Send xi (0) to agents r such that i ∈ Nr ; (d) Compute zi (0) using the received xj (0), j ∈ Ni ; (e) Receive zj (0), j ∈ Ni , and send zi (0) to agents r such that i ∈ Nr . Do While (8) (if G is directed) or (30) (if G is undirected) is not satisfied: (a) Compute zi (t) with the latest received ẑj (t), j ∈ Ni , using (39), then update si (t) using (7); Else (a) Advance the event counter k = k + 1, and set tik = t, si (t) = 0; (b) Set zi (tik ) = zi (t) and send zi (tik ) to agents r such that i ∈ Nr ; (c) Update the control signal ui = −Kzi (tik ). End Before presenting the algorithm formally, let us illustrate using (39) with an example involving three agents, A1 , A2 and A3 ; see Fig. 1 and 2. E.g., consider the timeline in Fig. 1(b). According to the timeline of A2 , an event has been triggered for A2 at time t2k . Until it receives communications from the neighbors, A2 computes z2 (t) using (39) with the information z1 (t1p−1 ) and z3 (t3q−1 ) received from A1 and A3 prior time t2k . This information is used in the third line of (39). Note that t̂1 = t̂3 = t until A2 receives the first message; the terms in the second line are zero until then. At time t1p , an event occurs at node 1, and A2 receives the value of A1 ’s combinational state, z1 (t1p ). From this time on, it starts using Φ(t1p , t)(z2 (t2k ) − z1 (t1p )) in (39), until the next message arrives, this time from A3 . Overall during the interval [t2k , t2k+1 ), A2 receives z1 (t1p ) and z1 (t1p+1 ) from A1 and z3 (t3q ) from A3 , which it uses in (39) to compute z2 (t). When G is directed, A2 computes z2 (t) in the same manner, but it only receives z1 (t1p ) and z1 (t1p+1 ) from A1 , as shown in Fig. 2(b). As one can see, the algorithm uses only one-directional communications between agents at event times: the information is received from j ∈ Ni when an event occurs at node j and is sent to r, i ∈ Nr when an event occurs at node i, e.g., see Figs. 1(b) and 2(b). 6 We conclude this section by summarizing the algorithm for generating the combinational state zi (t) at each node. Example Consider a system consisting of twenty identical pendulums. Each pendulum is subject to an input as shown in Fig. 3. The dynamic of the i-th pendulum is Initialization. (a) Synchronize local clocks to set t = 0 at each node, also set the event counter k = 0, 7 3 [30], also using the directed graph in Fig. 4(b). Out of the results in [30], we chose Theorem 3 for comparison, because it has a way to avoid continuous communications between the followers; this allows for a fair comparison with our methods. l α0 α20 α1 u1 → Leader u20 → Fig. 3. The system consisting of twenty pendulums and the leader pendulum. 1 0 2 8 In the first three simulations, we aimed to restrict the predicted upper bound on the tracking error to ∆ ≤ 0.05. In the design, we chose the same Q matrix and adjusted R to obtain the same control gains in the three simulations. The parameters of the triggering conditions (8) and (30) and the design parameters were set as shown in Table 1. In Simulation 4, using the same matrix Q, we computed the control gain K = [2.63, 7.24] and also chose the parameters required by Theorem 3 of [30] as follows: h = 5, β1 = 0.1, β2 = 0.15, γ = 0.2 and τ = 0 (see [30] for the definition of these parameters). In all simulations we endeavoured to achieve the least number of communications events. 20 12 15 19 (a) Undirected follower graph 1 0 2 8 20 12 15 19 (b) Directed follower graph Fig. 4. Communication graphs for the example. The simulation results achieved in Simulations 1-3 are shown in Table 1. In the table, J[18,20] denotes the PN maximum actual tracking error i=1 kεi (t)k2 observed over the time interval t ∈ [18, 20], t[0,20] , t[0,10] , t[10,20] and E[0,20] , E[0,10] , E[10,20] represent the minimum interevent intervals and the total number of events occurred in the system on the time intervals [0, 20], [0, 10], [10, 20], respectively. The corresponding results of Simulation 4 are J[18,20] = 1.8778 × 10−6 , t[0,20] = 21.1 ms, t[0,10] = 21.1 ms, t[10,20] = 101.5 ms, E[0,20] = 2857, E[0,10] = 1767 and E[10,20] = 1090. The tracking errors are shown in Fig. 5, which illustrates that all four eventtriggered tracking control laws enable all the followers to synchronize to the leader. governed by the following linearized equation ml2 α̈i = − mglαi − ui , i = 1, . . . , 20, (40) where l is the length of the pendulum, g = 9.8 m/s2 is the gravitational acceleration constant, m is the mass of each pendulum and ui is the control torque (realized using a DC motor). In addition, consider the leader pendulum which is identical to those given and whose dynamic is described by the linearized equation ml2 α̈0 = −mglα0 . (41) Choosing the state vectors as xi = (αi , α̇i ), i = 0, . . . , 20, equations (40) and "(41) can#be written in the# form of " (1), (2), where A = 0 1 −g/l 0 ,B= 0 −1/(ml2 ) The first comparison was made between the techniques developed in this paper for systems connected over directed and undirected graphs; these techniques were applied in Simulations 1 and 2, respectively. Although the minimum inter-event intervals in Simulation 2 were observed to be smaller than those in Simulation 1, on average the events were triggered less frequently in Simulation 2. This demonstrates that connecting the followers into an undirected network and using the design scheme based on Theorem 2 may lead to some advantages in terms of usage of communication resources. . In this example, we let m = 1 kg, l = 1 m. Both undirected and directed follower graphs G are considered in the example, shown in Fig. 4(a) and 4(b), respectively. According to Fig. 4, in both cases agents 1, 8, 12 and 15 measure the leader’s state, however in the graph in Fig. 4(b) follower i is restricted to receiving information from follower i−1 only, whereas in Fig. 4(a), it can receive information from both i − 1 and i + 1. Next, we compared Simulations 2 and 3 using the same undirected follower graph in Fig. 4(a) based on Theorems 1 and 2 developed in the paper. Compared with Simulation 2, more communication events and smaller minimum inter-event intervals were observed in Simulation 3. One possible explanation to this is because the method based on Theorem 2 takes an advantage of the symmetry property of the matrix L + G of the undirected follower graph in the derivation. We implemented four simulations to compare the results proposed in this paper and also to compare them with the results in [30]. The directed graph in Fig. 4(b) was employed to illustrate Theorem 1 in Simulation 1. In Simulation 2, we implemented the controller designed using Theorem 2 with the undirected graph in Fig. 4(a). We applied Theorem 1 using the same undirected graph in Fig. 4(a) in Simulation 3. In Simulation 4, we applied the event-based control strategy proposed in Theorem Finally, we compared Simulations 1 and 4 where we 8 Table 1 The design parameters and simulation results. Q 0.8 Simulation 2 Simulation 3 (Theorem 1 (Theorem 2 (Theorem 1 0.7 0.6 Tracking errors Simulation 1 and Fig. 4(b)) and Fig. 4(a)) and Fig. 4(a))       10.59 0.42 10.59 0.42 10.59 0.42       0.42 1.05 0.42 1.05 0.42 1.05 0.5 0.4 0.3 0.2 R 1.1394 0.1 1.5405 K [5.23 13.08] [5.23 13.08] [5.23 13.08] α 0.0877 − 0.0649 ω 0.001 − 0.001 µi 0.1 0.1 0.1 0.8 σi 0.5025 0.7198 0.3007 0.7 νi 2.5 ∆ 2.9769 × 10 0.0462 7.9990 × 10 J[18,20] 3.4524 × 10 3.5259 × 10 5 10 Time (s) 15 20 15 20 15 20 15 20 0.6 −6 3.1507 × 10 0.0462 −6 0 (a) Simulation 1 1.2 −6 0.0462 −7 0 Tracking errors γ 2 −5 0.1 −6 1.5518 × 10 0.5 0.4 0.3 t[0,20] 6.2 ms 5.9 ms 1.9 ms 0.2 t[0,10] 6.2 ms 5.9 ms 1.9 ms 0.1 t[10,20] 30.7 ms 24.9 ms 26.0 ms E[0,20] 2240 2091 2979 E[0,10] 1285 1217 1993 E[10,20] 955 874 986 0 0 5 10 Time (s) (b) Simulation 2 0.8 0.7 used the same directed follower graph in Fig. 4(b) for both designs. Although compared with the method of Theorem 3 of [30], our method produced smaller minimum time intervals between the events, the total number of events occurred during the simulation using our method was also smaller. We remind that we endeavoured to select the simulation parameters and the controller gains for this simulation to reduce the total number of events. We also tried to compare the performance of the two methods by tuning the controller of [30] to almost the same gain as in Simulation 1, but the results in Simulation 4 were even worse, producing a much greater number of communication events (E[0,20] = 5553, E[0,10] = 3100 and E[10,20] = 2453). Tracking errors 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 Time (s) (c) Simulation 3 0.8 0.7 0.6 Conclusions Tracking errors 7 The paper has studied the event-triggered leaderfollower tracking control problem for multi-agent systems. We have presented sufficient conditions to guarantee that the proposed event-triggered control scheme leads to bounded tracking errors. Furthermore, our results show that by adjusting the parameters of the triggering condition, the upper bound on the tracking errors guaranteed by these conditions can be 0.5 0.4 0.3 0.2 0.1 0 0 5 10 Time (s) (d) Simulation 4 Fig. 5. Tracking errors kεi k. 9 tuned to a desired small value, at the expense of more frequent communications during an initial stage of the tracking process. Such conditions have been derived for both undirected and directed follower graphs. Also, we showed that the proposed event triggering conditions do not lead to Zeno behavior even if a tight accuracy requirement on the tracking errors is imposed. In fact, with the proposed triggering rules, such tight accuracy requirements do not impact the inter-event intervals after a sufficiently large time. We also presented a computational algorithm which allows the nodes to continuously generate the combinational state at every node which is needed to implement these event triggering schemes. Thus, continuous monitoring the neighboring states is avoided. The efficacy of the proposed algorithm has been demonstrated using a simulation example. Future work will include the study of robustness of the proposed control scheme. 8 [12] Y. Fan, G. Feng, Y. Wang, and C. Song, “Distributed eventtriggered control for multi-agent systems with combinational measurements,” Automatica, 49, 671–675, 2013. [13] X. Meng and T. Chen, “Event based agreement protocols for multi-agent networks,” Automatica, 49, 2125–2132, 2013. [14] W. Zhu, Z. P. Jiang, and G. Feng, “Event-based consensus of multi-agent systems with general linear models,” Automatica, 50, 552–558, 2014. [15] E. Garcia, Y. Cao, and D. W. 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CROSS-MODAL EMBEDDINGS FOR VIDEO AND AUDIO RETRIEVAL Didac Surı́s1 , Amanda Duarte2 , Amaia Salvador1 , Jordi Torres2 and Xavier Giró-i-Nieto1 1 2 Universitat Politècnica de Catalunya (UPC) Barcelona Supercomputing Center (BSC-CNS) arXiv:1801.02200v1 [cs.IR] 7 Jan 2018 ABSTRACT The increasing amount of online videos brings several opportunities for training self-supervised neural networks. The creation of large scale datasets of videos such as the YouTube8M allows us to deal with this large amount of data in manageable way. In this work, we find new ways of exploiting this dataset by taking advantage of the multi-modal information it provides. By means of a neural network, we are able to create links between audio and visual documents, by projecting them into a common region of the feature space, obtaining joint audio-visual embeddings. These links are used to retrieve audio samples that fit well to a given silent video, and also to retrieve images that match a given a query audio. The results in terms of Recall@K obtained over a subset of YouTube-8M videos show the potential of this unsupervised approach for cross-modal feature learning. We train embeddings for both scales and assess their quality in a retrieval problem, formulated as using the feature extracted from one modality to retrieve the most similar videos based on the features computed in the other modality. Index Terms— Sonorization, embedding, retrieval, cross-modal, YouTube-8M 1. INTRODUCTION Videos have become the next frontier in artificial intelligence. The rich semantics contained in them make them a challenging data type posing several challenges in both perceptual, reasoning or even computational level. Mimicking the learning process and knowledge extraction that humans develop from our visual and audio perception remains an open research question, and video contain all this information in a format manageable for science and research. Videos are used in this work for two main reasons. Firstly, they naturally integrate both visual and audio data, providing a weak labeling of one modality with respect to the other. Secondly, the high volume of both visual and audio data allows training machine learning algorithms whose models are governed by a high amount of parameters. The huge scale video archives available online and the increasing number of video cameras that constantly monitor our world, offer more data than computation power available to process them. The popularization of deep neural networks among the computer vision and audio communities has defined a common framework boosting multimodal research. Tasks like video sonorization, speaker impersonation or self-supervised feature learning have exploited the opportunities offered by artificial neurons to project images, text and audio in a feature space where bridges across modalities can be built. This work exploits the relation between the visual and audio contents in a video clip to learn a joint embedding space with deep neural networks. Two multilayer perceptrons (MLPs), one for visual features and a second one for audio features, are trained to be mapped into the same cross-modal representation. We adopt a self-supervised approach, as we exploit the unsupervised correspondence between the audio and visual tracks in any video clip. We propose a joint audiovisual space to address a retrieval task formulating a query from any of the two modalities. As depicted in Figure 1, whether a video or an audio clip can be used as a query to search its matching pair in a large collection of videos. For example, an animated GIF could be sonorized by finding an adequate audio track, or an audio recording illustrated with a related video. In this paper, we present a simple yet very effective model for retrieving documents with a fast and light search. We do not address an exact alignment between the two modalities that would require a much higher computation effort. The paper is structured as follows. Section 2 introduces the related work on learned audiovisual embeddings with neural networks. Section 3 presents the architecture of our model and Section 4 how it was trained. Experiments are reported in Section 5 and final conclusions drawn in Section 6. The source code and trained model used in this paper is publicly available from https://github.com/surisdi/ youtube-8m. 2. RELATED WORK In the past years, the relationship between the audio and the visual content in videos has been researched in several contexts. Overall, conventional approaches can be divided into four categories according to the task: generation, classification, matching and retrieval. As online music streaming and video sharing websites Fig. 1. A learned cross-modal embedding allows retrieving video from audio, and vice versa. have become increasingly popular, some research has been done on the relationship between music and album covers [1, 2, 3, 4] and also on music and videos (instead of just images) as the visual modality [5, 6, 7, 8] to explore the multimodal information present in both types of data. A recent study [9] also explored the cross-modal relations between the two modalities but using images with people talking and speech. It is done through Canonical Correlation Analysis (CCA) and cross-modal factor analysis. Also applying CCA, [10] uses visual and sound features and common subspace features for aiding clustering in image-audio datasets. In a work presented by [11], the key idea was to use greedy layer-wise training with Restricted Boltzmann Machines (RBMs) between vision and sound. The present work is focused on using the information present in each modality to create a joint embedding space to perform cross-modal retrieval. This idea has been exploited especially using text and image joint embeddings [12, 13, 14], but also between other kinds of data, for example creating a visual-semantic embedding [15] or using synchronous data to learn discriminative representations shared across vision, sound and text [16]. However, joint representations between the images (frames) of a video and its audio have yet to be fully exploited, being [17] the work that most has explored this option up to the knowledge of the authors. In their paper, they seek for a joint embedding space but only using music videos to obtain the closest and farthest video given a query video, only based on either image or audio. The main idea of the current work is borrowed from [14], which is the baseline to understand our approach. There, the authors create a joint embedding space for recipes and their images. They can then use it to retrieve recipes from any food image, looking to the recipe that has the closest embedding. Apart from the retrieval results, they also perform other experiments, such as studying the localized unit activations, or doing arithmetics with the images. images of the video and the vector of features representing the audio. These features are already precomputed and provided in the YouTube-8M dataset [18]. In particular, we use the video-level features, which represent the whole video clip with two vectors: one for the audio and another one for the video. These feature representations are the result of an average pooling of the local audio features computed over windows of one second, and local visual features computed over frames sampled at 1 Hz. The main objective of the system is to transform the two different features (image and audio, separately) to other features laying in a joint space. This means that for the same video, ideally the image features and the audio features will be transformed to the same joint features, in the same space. We will call these new features embeddings, and will represent them with Φi , for the image embeddings, and Φa , for the audio embeddings. The idea of the joint space is to represent the concept of the video, not just the image or the audio, but a generalization of it. As a consequence, videos with similar concepts will have closer embeddings and videos with different concepts will have embeddings further apart in the joint space. For example, the representation of a tennis match video will be close to the one of a football match, but not to the one of a maths lesson. Thus, we use a set of fully connected layers of different sizes, stacked one after the other, going from the original features to the embeddings. The audio and the image network are completely separated. These fully connected layers perform a non-linear transformation on the input features, mapping them to the embeddings, being the parameters of this non-linear mapping learned in the optimization process. After that, a classification from the two embeddings is done, also using a fully connected layer from them to the different classes, using a sigmoid as activation function. We will get more insight on this step in section 4. The number of hidden layers is not necessarily fixed, as well as the number of neurons per layer, since we experimented with different configurations. Each hidden layer uses ReLu as activation function, and all the weights in each layer are regularized using L2 norm. 4. TRAINING In this section we present the used losses as well as their meaning and intuition. 4.1. Similarity Loss 3. ARCHITECTURE In this section we present the architecture for our joint embedding model, which is depicted in the Figure 2. As inputs, we have the vector of features representing the The objective of this work is to get the two embeddings of the same video to be as close as possible (ideally, the same), while keeping embeddings from different videos as far as possible. Formally, we are given a video vk , represented by the audio and visual features vk = {ik , ak } (ik represents the image Fig. 2. Schematic of the used architecture. features and ak the audio features of vk ). The objective is to maximize the similarity between Φik , the embedding obtained by transformations on ik , and Φak , the embedding obtained by transformations on ak . At the same time, however, we have to prevent embeddings from different videos to be “close” in the joint space. In other words, we want them to have low similarity. However, the objective is not to force them to be opposite to each other. Instead of forcing them to have similarity equal to zero, we allow a margin of similarity small enough to force the embeddings to be clearly not in the same place in in the joint space. We call this margin α. During the training, both positive and negative pairs are used, being the positive pairs the ones for which ik and ak correspond to the same video vk , and the negative pairs the ones for which ik1 and ak2 do not correspond to the same video, this is, k1 6= k2. The proportion of negative samples is pnegative . For the negative pairs, we selected random pairs that did not have any common label, in order to help the network to learn how to distinguish different videos in the embedding space. The notion of “similarity” or “closeness” is mathematically translated into a cosine similarity between the embeddings, being the cosine similarity definedNas: P xk zk k=1 s similarity = cos(x, z) = s (1) N N P P 2 2 xk zk k i for any pair of real-valued vectors x and z. From this reasoning we get to the first and most important loss: Lcos ((Φa ,Φi ), y) =  1 − cos(Φa , Φi ), if = max(0, cos(Φa , Φi ) − α), if y=1 y = −1 (2) where y = 1 denotes positive sampling, and y = −1 denotes negative sampling. 4.2. Classification Regularization Inspired by the work presented in [14], we provide additional information to our system by incorporating the video labels (classes) provided by the YouTube-8M dataset. This information is added as a regularization term that seeks to solve the high-level classification problem, both from the audio and from the video embeddings, sharing the weights between the two branches. The key idea here is to have the classification weights from the embeddings to the labels shared by the two modalities. This loss is optimized together with the previously explained similarity loss, serving as a regularization term. Basically, the system learns to classify the audio and the images of a video (separately) into different classes or labels provided by the dataset. We limit its effect by using a regularization parameter λ. To incorporate the previously explained regularization to the joint embedding, we use a single fully connected layer, as shown in Figure 2. Formally, we can obtain the label probabilities as pi = softmax(W Φi ) and pa = softmax(W Φa ), where W represents the learned weights, which are shared between the two branches. The softmax activation is used in order to obtain probabilities at the output. The objective is to make pi as similar as possible to ci , and pa as similar as possible to ca , where ci and ca are the category labels for the video represented by the image features and the audio features, respectively. For positive pairs, ci and ca are the same. The loss function used for the classification is the well known cross entropy loss: L(x, z) = − X k xk log(zk ) (3) Thus, the classification loss is: X Lclass (pi , pa , ci , ca ) = − (pik log(cik )+(pak log(cak )) (4) k Finally, the loss function to be optimized is: L = Lcos + λLclass (5) 4.3. Parameters and Implementation Details For our experiments we used the following parameters: • Batch size of 1024. • We saw that starting with λ different than zero led to a bad embedding similarity because the classification accuracy was preferred. Thus, we began the training with λ = 0 and set it to 0.02 at step number 10,000. • Margin α = 0.2. • Percentage of negative samples pnegative = 0.6. • 4 hidden layers in each network branch, the number of neurons per layer being, from features to embedding, 2000, 2000, 700, 700 in the image branch, and 450, 450, 200, 200 in the audio branch. • Dimensionality of the feature vector = 250. • We trained a single epoch. The simulation was programmed using Tensorflow [19], having as a baseline the code provided by the YouTube-8M challenge authors1 . 5. RESULTS 5.1. Dataset The experiments presented in this section were developed over a subset of 6,000 video clips from the YouTube-8M dataset [18]. This dataset does not contain the raw video files, but their representations as precomputed features, both from audio and video. Audio features were computed using the method explained in [20] over audio windows of 1 second, while visual features were computed over frames sampled at 1 Hz with the Inception model provided in TensorFlow [19]. The dataset provides video-level features, which represent all the video using a single vector (one for audio and another for visual information), and thus does not maintain temporal information; and also provides frame-level features, which consist on a single vector representing each second of audio, and a single vector representing each frame of the video, sampled at 1 frame per second. The main goal of this dataset is to provide enough data to reach state of the art results in video classification. Nevertheless, such a huge dataset also permits approaching other tasks related to videos and cross-modal tasks, such as the one we 1 https://www.kaggle.com/c/youtube8m Table 1. Evaluation of Recall from audio to video Number of elements Recall@1 Recall@5 Recall@10 256 21.5% 52.0% 63.1% 512 15.2% 39.5% 52.0% 1024 9.8% 30.4% 39.6% Table 2. Evaluation of Recall from video to audio Number of elements Recall@1 Recall@5 Recall@10 256 22.3% 51.7% 64.4% 512 14.7% 38.0% 51.5% 1024 10.2% 29.1% 40.3% approach in this paper. For this work, and as a baseline, we only use the video-level features. 5.2. Quantitative Performance Evaluation We divide our results in two different categories: quantitative (numeric) results and qualitative results. To obtain the quantitative results we use the Recall@k metric. We define Recall@k as the recall rate at top K for all the retrieval experiments, this is, the percentage of all the queries where the corresponding video is retrieved in the top K, hence higher is better. The experiments are performed with different dimension of the feature vector. The Table 1 shows the results of recall from audio to video. In other words, from the audio embedding of a video, how many times we retrieve the embedding corresponding to the images of that same video. Table 2 shows the recall from video to audio. To have a reference, the random guess result would be k/Number of elements. The obtained results show a very clear correspondence between the embeddings coming from the audio features and the ones coming from the video features. It is also interesting to notice that the results from audio to video and from video to audio are very similar, because the system has been trained bidirectionally. 5.3. Qualitative Performance Evaluation In addition to the objective results, we performed some insightful qualitative experiments. They consisted on generating the embeddings of both the audio and the video for a list of 6,000 different videos. Then, we randomly chose a video, and from its image embedding, we retrieved the video with the closest audio embedding, and the other way around (from one video’s audio we retrieved the video with the closest image embedding). If the closest embedding corresponded to the same video, we took the second one in the ordered list. The Figure 3 shows some experiments. On the left, we can see the results given a video query and getting the closest audio; and on the right the input query is an audio. Examples depicting the real videos and audio are available online Fig. 3. Qualitative results. On the left we show the results obtained when we gave a video as a query. On the right, the results are based on an audio as a query. 2 . It shows both the results when going from image to audio, and when going from audio to image. Four different random examples are shown in each case. For each result and each query, we also show their YouTube-8M labels, for completeness. The results show that when starting from the image features of a video, the retrieved audio represents a very accurate fit for those images. Subjectively, there are non negligible cases where the retrieved audio actually fits better the video than the original one, for example when the original video has some artificially introduced music, or in cases where there is some background commentator explaining the video in a foreign (unknown) language. This analysis can also be done similarly the other way around, this is, with the audio colorization approach, providing images for a given audio. 6. CONCLUSIONS AND FUTURE WORK We presented an effective method to retrieve audio samples that fit correctly to a given (muted) video. The qualitative results show that the already existing online videos, due to its variety, represent a very good source of audio for new videos, even in the case of only retrieving from a small subset of this large amount of videos. Due to the existing difficulty to create new audio from scratch, we believe that a retrieval approach is the path to follow in order to give audio to videos. The range of possibilities to extend the presented work is excitingly broad. The first idea would be to make use of the YouTube-8M dataset variety and information. The temporal 2 https://goo.gl/NAcJah information provided by the individual image and audio features is not used in the current work. The most promising future work implies using this temporal information to match audio and images, making use of the implicit synchronization the audio and the images of a video have, without needing any supervised control. Thus, the next step in our research is introducing a recurrent neural network, which will allow us to create more accurate representations of the video, and also retrieve different audio samples for each image, creating a fully synchronized system. Also, it would be very interesting to study the behavior of the system depending on the class of the input. Observing the dataset, it is clear that not all the classes have the same degree of correspondence between audio and image, as for example some videos have artificially (posterior) added music, which is not related at all to the images. In short, we believe the YouTube-8M dataset allows for promising research in the future in the field of video sonorization and audio retrieval, for it having a huge amount of samples, and for it capturing multi-modal information in a highly compact way. 7. ACKNOWLEDGEMENTS This work was partially supported by the Spanish Ministry of Economy and Competitivity and the European Regional Development Fund (ERDF) under contract TEC2016-75976R. Amanda Duarte was funded by the mobility grant of the Severo Ochoa Program at Barcelona Supercomputing Center (BSC-CNS). 8. 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arXiv:1304.5833v1 [] 22 Apr 2013 ON THE EQUALITY OF ORDINARY AND SYMBOLIC POWERS OF IDEALS ALINE HOSRY, YOUNGSU KIM AND JAVID VALIDASHTI Abstract. We consider the following question concerning the equality of ordinary and symbolic powers of ideals. In a regular local ring, if the ordinary and symbolic powers of a one-dimensional prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence of a positive answer for classes of prime ideals defining monomial curves or rings of low multiplicities. 1. Introduction Let R be a Noetherian local ring of dimension d and P a prime ideal of R. For a positive integer n, the n-th symbolic power of P , denoted by P (n) , is defined as P (n) := P n RP ∩ R = {x ∈ R | ∃ s ∈ R \ P, sx ∈ P n }. Symbolic powers of ideals are central objects in commutative algebra and algebraic geometry for their tight connection to primary decomposition of ideals and the order of vanishing of polynomials. One readily sees from the definition that P n ⊆ P (n) for all n, but they are not equal in general. Therefore, one would like to compare the ordinary and symbolic powers and provide criteria for equality. This problem has long been a subject of interest, see for instance [2, 8, 11–14, 16, 22]. In this paper, we are interested in criteria for the equality. In particular, we would like to know if P n = P (n) for all n up to some value, then they are equal for all n. The following question was posed by Huneke in this regard. Question 1.1. Let R be a regular local ring of dimension d and P a prime ideal of height d − 1. If P n = P (n) for all n ≤ ht P , then is P n = P (n) for all n? An affirmative answer to Question 1.1 is equivalent to P being generated by a regular sequence [7]. Furthermore, it is equivalent to showing that if P n = P (n) for all n ≤ d − 1, then the analytic spread of P is d − 1. This is not known even when P is the defining ideal of a monomial curve in A4k . Huneke answered Question 1.1 positively in dimension 3, and in dimension 4 if R/P is Gorenstein [15, Corollaries 2.5, 2.6]. One would like to remove the Gorenstein assumption. There are supporting examples showing that the Gorenstein property of R/P might follow from P 2 = P (2) . In fact, this is very close to a conjecture by Vasconcelos which states that if P is syzygetic and R/P and P/P 2 are Cohen-Macaulay, then R/P is Gorenstein [24]. Note that if P has height d − 1, then R/P is Cohen-Macaulay, and P/P 2 being Cohen-Macaulay is equivalent to P 2 = P (2) . Therefore, one is tempted to ask the following question. Question 1.2. Let R be a regular local ring of dimension d and P a prime ideal of height d − 1. If P 2 = P (2) , then is R/P Gorenstein? 1 2 A. HOSRY, Y. KIM AND J. VALIDASHTI By Huneke’s result [15, Corollary 2.6], if Question 1.2 has an affirmative answer, then so does Question 1.1 when dimension of R is 4. The converse of Question 1.2 is true in dimension 4 by Herzog [9]. Also, Question 1.2 has been answered positively for some classes of algebras [19], but it is not true in general (see for instance [19, Example 6.1]). In this paper, we consider the case where P is the defining ideal of a monomial curve k[[ta1 , . . . , tad ]] and we give an affirmative answer to Questions 1.1 and 1.2 when d = 4 and {ai } forms an arithmetic sequence. In higher dimensions, if {ai } contains an arithmetic subsequence of length 5 in which the terms are not necessarily consecutive, we observe that P 2 6= P (2) , hence we have a positive answer to Questions 1.1 and 1.2. We extend these results to certain modifications of arithmetic subsequences. We also consider onedimensional prime ideals P of a regular local ring R in general and we show that if R/P has low multiplicity, then Question 1.1 has a positive answer. We note that if we drop the height d − 1 assumption on P , then this question does not have a positive answer in general, due to a counterexample by Guardo, Harbourne and Van Tuyl [10]. 2. Monomial Curves Let a1 , . . . , ad be an increasing sequence of positive integers with gcd(a1 , . . . , ad ) = 1. Assume that ai ’s generate a numerical semigroup non-redundantly. Consider the monomial curve A = k[[ta1 , . . . , tad ]] ⊂ k[[t]] over a field k, with maximal ideal mA := (ta1 , . . . , tad )A. Let R = k[[x1 , . . . , xd ]] be a formal power series ring with maximal ideal m = (x1 , . . . , xd )R, and let P be the kernel of the homomorphism k[[x1 , . . . , xd ]] −→ k[[ta1 , . . . , tad ]] obtained by mapping xi to tai for all i. Therefore, A is isomorphic to R/P . Note that P ⊂ m2 because of the non-redundancy assumption on ai ’s. We state the following well-known properties about monomial curves. Lemma 2.1. In the above setting, (1) The ideal ta1 A is a minimal reduction of mA . (2) The Hilbert-Samuel multiplicity e(mA , A) of A is a1 . (3) The multiplicity e(mA , A) is at least d, i.e., a1 ≥ d. For the third property in above, we may assume that k is an infinite field. Then by [1, Fact (1)], we have e(mA , A) ≥ λ(mA /m2A ), where λ(−) denotes the length, and observe that λ(mA /m2A ) = d, by the non-redundancy condition on the ai ’s. Note that the third property also follows from Theorem 3.1. We begin with the following result that describes the generators of P when d = 4 and the set of exponents {ai } forms an arithmetic sequence. Proposition 2.2. Let A be the monomial curve k[[ta , ta+r , ta+2r , ta+3r ]], where a and r are positive integers that are relatively prime. Regard A as R/P , where R = k[[x, y, z, w]] ON THE EQUALITY OF ORDINARY AND SYMBOLIC POWERS OF IDEALS 3 and P is the defining ideal of A. Then P is minimally generated by z 2 − yw, yz − xw, y 2 − xz, xk+r − w k , if a = 3k z 2 − yw, yz − xw, y 2 − xz, xk+r z − w k+1, xk+r y − zw k , xk+r+1 − yw k , if a = 3k + 1 z 2 − yw, yz − xw, y 2 − xz, xk+r+1 − zw k , xk+r y − w k+1 , if a = 3k + 2 where k is a positive integer. Proof. Since the numerical semigroup is non-redundantly generated, a is greater than or equal to 4 by Lemma 2.1. Thus k ≥ 2 if a = 3k and k ≥ 1 if a = 3k + 1 or 3k + 2. In each case, let I be the ideal generated by the above-listed elements and m be the maximal ideal (x, y, z, w) of R. One can directly check that I ⊂ P . For all cases, we will use the following method to show I = P . First, we show that (P, x) = (I, x). Then it follows that P = I + x(P : x). But (P : x) = P , since x is not in P . Thus P = I + xP , which implies P = I, by Nakayama’s Lemma. To show (P, x) = (I, x), let I˜ = (I, x). The short exact sequence ·y ˜ y) −→ 0 0 −→ R/(I˜ : y) −−−→ R/I˜ −→ R/(I, ˜ = λR (R/(I˜ : y)) + λR(R/(I, ˜ y)). Since R/P is Cohenyields the length equation λR (R/I) Macaulay and the image of the ideal (x) in R/P is a minimal reduction of m/P by Lemma 2.1, we have ˜ a = e(m, R/P ) = λR (R/(P, x)) ≤ λR (R/I). Thus, it is enough to show ˜ y)) ≤ a. λR (R/(I˜ : y)) + λR (R/(I, ˜ y) = (x, y, z 2 , w k ) and the ideal If a = 3k, then I˜ = (x, z 2 − yw, yz, y 2, w k ). Therefore, (I, I˜ : y contains the ideal (x, y, z, w k ). Thus, ˜ y)) ≤ λR (R/(x, y, z, w k )) + λR (R/(x, y, z 2 , w k )) ≤ k + 2k = a. λR (R/(I˜ : y)) + λR (R/(I, If a = 3k + 1, then I˜ = (x, z 2 − yw, yz, y 2, w k+1, zw k , yw k ). Hence, (x, y, z, w k ) ⊂ I˜ : y ˜ y) = (x, y, z 2 , zw k , w k+1). Note that λR (R/(x, y, z 2 , zw k , w k+1)) = 2k + 1 and and (I, λR (R/(x, y, z, w k )) = k. Thus, ˜ y)) ≤ k + (2k + 1) = a. λR (R/(I˜ : y)) + λR (R/(I, If a = 3k + 2, then I˜ = (x, z 2 − yw, yz, y 2, zw k , w k+1). Therefore, (x, y, z, w k+1) ⊂ I˜ : y ˜ y) = (x, y, z 2 , zw k , w k+1). Similar to the previous case, λR (R/(x, y, z 2 , zw k , w k+1)) and (I, is 2k + 1 and λR (R/(x, y, z, w k+1)) = k + 1. Hence we obtain ˜ y)) ≤ (k + 1) + (2k + 1) = a. λR (R/(I˜ : y)) + λR (R/(I, To show that P is minimally generated by the listed elements in each case, we can compute µ(P ) = λR (P/mP ). In fact, if we let R̄ = R/xR, then µ(P R̄) = λR (P R̄/mP R̄) = λR (P + (x)/mP + (x)) = λR (P/mP + P ∩ (x)). But P ∩ (x) = x(P : x) = xP ⊂ mP . Thus µ(P R̄) = µ(P ). Therefore, to compute the minimal number of generators in each case, we can go modulo (x) first. If a = 3k, we will show in Theorem 2.3 that P 2 6= P (2) , hence P is not a complete intersection ideal. 4 A. HOSRY, Y. KIM AND J. VALIDASHTI Thus it cannot have fewer number of generators than 4. If a = 3k + 2, we will show in Theorem 2.3 that P is generated by the 4 by 4 Pfaffians of a 5 by 5 skew-symmetric matrix. Hence by Buchsbaum-Eisenbud structure theorem for height 3 Gorenstein ideals in [6], P is minimally generated by the listed elements in this case. Thus, we only need to deal with the case a = 3k + 1, where one can check directly that the ideal P R̄ is minimally generated by z 2 − yw, yz, y 2 , w k+1, zw k , yw k .  Theorem 2.3. Let A be the monomial curve A = k[[ta , ta+r , ta+2r , ta+3r ]], where a and r are positive integers that are relatively prime. Regard A as R/P , where R = k[[x, y, z, w]] and P is the defining ideal of A. (1) If a = 3k or 3k + 1, then R/P is not Gorenstein and P 2 6= P (2) . (2) If a = 3k + 2, then R/P is Gorenstein, P 2 = P (2) and P 3 6= P (3) . Proof. If a = 3k, then one can see that P contains the 2 by 2 minors of  x y z . z w M = y k−1 k+r k+r−1 zw x yx  Let D = det(M). Note that D 6∈ P 2, since D is not in P 2 modulo (x, y). We will show that D ∈ P (2) . We have det(adj(M)) · D = D 3 , where adj(M) is the adjoint matrix of M. Note that D 6= 0, for example it is not zero modulo (x, y). Thus D 2 = det(adj(M)). But det(adj(M)) ∈ P 3, since the entries of adj(M) are in P . Hence D 2 ∈ P 3 . Therefore, the image of D 2 in the associated graded ring GP := grP RP (RP ) is zero. Note that GP is a domain as RP is a regular local ring. Hence the image of D is zero in GP , which shows that the image of D in the localization RP is in P 2 RP , i.e., D ∈ P (2) . One could also directly show that w · det(M) ∈ P 2 , hence det(M) ∈ P (2) , as w is not in P . Now by Herzog’s theorem [9, Satz 2.8], we conclude that R/P is not Gorenstein. We note that since in Proposition 2.2 we have shown that P is minimally generated by 4 elements, we could also use Buchsbaum-Eisenbud structure theorem for height 3 Gorenstein ideals in [6], or Bresinsky’s result in [3] which states that if a monomial curve in dimension 4 is Gorenstein, then P is minimally generated by 3 or 5 elements. If a = 3k + 1, then P contains the 2 by 2 minors of  x y z z w . M = y k−1 k k+r zw w x  With a similar argument as in the previous case, one can show that det(M) ∈ P (2) \ P 2. Thus by Herzog’s result, R/P is not Gorenstein. ON THE EQUALITY OF ORDINARY AND SYMBOLIC POWERS OF IDEALS If a = 3k + 2, then by Proposition 2.2, one can see Pfaffians of  0 −w k 0 k k+r  w 0 x  k+r 0 M =  0 −x  −x −y −z −y −z −w 5 that P is generated by the 4 by 4  x y y z   z w  . 0 0  0 0 Thus, by Buchsbaum-Eisenbud structure theorem for height 3 Gorenstein ideals in [6], we obtain that R/P is Gorenstein and P is minimally generated by the 5 listed elements in Proposition 2.2. Hence, P 2 = P (2) by Herzog’s result [9, Satz 2.8], and P 3 6= P (3) by Huneke’s result [15, Corollary 2.6], as P is not a complete intersection ideal.  Corollary 2.4. Question 1.1 and Question 1.2 have affirmative answers for monomial curves as in Theorem 2.3. Now we consider monomial curves in higher dimensions. Theorem 2.5. Let A be the monomial curve k[[ta1 , . . . , tad ]]. Consider A as R/P , where R = k[[x1 , . . . , xd ]] and P is the defining ideal of A. If {ai } has an arithmetic subsequence of length 5, whose terms are not necessarily consecutive, then P 2 6= P (2) . Proof. If {ai } has an arithmetic subsequence {b1 , . . . , b5 } of length 5, without loss of generality we may assume that x1 , . . . , x5 correspond to tb1 , . . . , tb5 . Then, one can see that P contains the 2 by 2 minors of   x1 x2 x3 M =  x2 x3 x4  . x3 x4 x5 We observe that det(M) 6∈ P 2, since det(M) is a polynomial of degree 3 and the generators of P 2 have degree at least 4 as P ⊂ m2 . Also note that det(M) 6= 0, for example it is not zero modulo (x2 , x3 ). Thus, by a similar argument as in the proof of Theorem 2.3, one can show that det(M) ∈ P (2) .  Corollary 2.6. Question 1.1 and Question 1.2 have positive answers for monomial curves as in Theorem 2.5. Using a result of Morales [21, Lemma 3.2], we can extend Theorems 2.3 and 2.5 to a larger class of monomial curves. As before, let A be the monomial curve k[[ta1 , . . . , tad ]]. In the following we will not assume any particular order on the ai ’s. Write A as R/P , where R is k[[x1 , . . . , xd ]] and P is the defining ideal of A. For a positive integer c, relatively prime to a1 , let à be the modified monomial curve k[[ta1 , tca2 , . . . , tcad ]]. Note that a1 , ca2 , . . . , cad non-redundantly generate their numerical semigroup too. Write à as R̃/P̃ , where R̃ denotes k[[x1 , . . . , xd ]] and P̃ is the defining ideal of Ã. Consider R̃ as an R-module via the map φ : R −→ R̃ that sends x1 to xc1 and fixes xi for all i 6= 1. For a polynomial f (x1 , . . . , xd ) ∈ R, let f˜ be the polynomial f (xc1 , . . . , xd ). Lemma 2.7.(Morales) R̃ is a faithfully flat extension of R. Moreover, P R̃ ∩ R = P and P̃ = P R̃. In fact, f ∈ P if and only if f˜ ∈ P̃ , and if {gi } is a minimal generating set for 6 A. HOSRY, Y. KIM AND J. VALIDASHTI P , then {g˜i } is a minimal generating set for P̃ . In addition, for all positive integers k, f ∈ P k if and only if f˜ ∈ P̃ k , and f ∈ P (k) if and only if f˜ ∈ P̃ (k) , i.e., P̃ k ∩ R = P k and P̃ (k) ∩ R = P (k) . Using Lemma 2.7, we obtain the following extension of Theorems 2.3 and 2.5. Corollary 2.8. If Question 1.1 has an affirmative answer for a monomial curve A, then it also has an affirmative answer for the monomial curve Ã. In particular, Question 1.1 has an affirmative answer for successive modifications of the monomial curves as in Theorems 2.3 and 2.5 in the sense of Morales. Proof. If P̃ n = P̃ (n) for all positive integers n ≤ d − 1, then by Lemma 2.7, we obtain that P n = P (n) for all n ≤ d − 1. Thus, by hypothesis, P is a complete intersection and hence, by Lemma 2.7, we obtain that P̃ is a complete intersection.  3. Low Multiplicities Let R be a regular local ring with maximal ideal m and of dimension d. Let P be a prime ideal of height d − 1. We will show that Question 1.1 has an affirmative answer when the Hilbert-Samuel multiplicity e(R/P ) is sufficiently small. Theorem 3.1. Let R be a regular local ring with maximal ideal m and of dimension d. Assume P is a prime ideal of height d − 1 such that P ⊂ m2 . Then P n 6= P (n) for a positive integer n, if d−2 Y 2n + r . e(R/P ) < n+r r=0 Proof. We may assume the residue field of R is infinite, see for instance [17, Lemma 8.4.2]. Thus, as R/P has dimension one, there exists x ∈ R whose image in R/P is a minimal reduction of m/P . Note that x cannot be in m2 by Nakayama’s Lemma, hence R/(x) is regular. Recall that
in a regular local ring S with maximal ideal n and of dimension k, λS (S/nn ) = n+k−1 for all positive integers n. Therefore, since P n ⊂ m2n , we have k
λR (R/(P n , x)) ≥ λR (R/(m2n , x)) = 2n+d−2 . On the other hand, since R/P is a oned−1 dimensional Cohen-Macaulay ring, using the associativity formula for multiplicities, we obtain λR (R/(P (n) , x)) = e((x), R/P (n) ) = λRP (R
P /P n RP ) · e((x), R/P ) = n+d−2 · e(R/P ). d−1

2n+d−2 The multiplicity bound in the statement is equivalent to n+d−2 · e(R/P ) < . d−1 d−1 (n) n n (n) Therefore, λR (R/(P , x)) < λR (R/(P , x)). Thus, P and P cannot be the same.  One can easily observe that the multiplicity bound in Theorem 3.1 is increasing with respect to n. Thus, letting n = d − 1, we obtain the largest bound that guarantees P d−1 6= P (d−1) . Therefore, we have the following corollary. ON THE EQUALITY OF ORDINARY AND SYMBOLIC POWERS OF IDEALS 7 Corollary 3.2. Under the assumptions of Theorem 3.1, Question 1.1 has a positive answer provided d−2 Y 2d + r − 2 e(R/P ) < . d + r − 1 r=0 Note that the multiplicity bound in Corollary 3.2 grows at least exponentially with respect to d, since each term of the product is greater than 32 . The next corollary is an application of Theorem 3.1 in the case of monomial curves in embedding dimension 4. Corollary 3.3. Let A = k[[ta1 , ta2 , ta3 , ta4 ]]. Consider A as R/P , where R = k[[x, y, z, w]] and P is the defining ideal of A. If a1 = 4 or 5, then P 3 6= P (3) . Therefore, Question 1.1 has a positive answer in this case. Proof. Apply Theorem 3.1 for n = 3 and d = 4. On the one hand e(R/P ) = a1 ≤ 5, and on the other hand the multiplicity bound reduces to 5.6. Hence P 3 6= P (3) .  We remark that, by Corollary 2.8, Question 1.1 has an affirmative answer for successive modifications of the monomial curves as in Corollary 3.3 in the sense of Morales. 4. Remarks We end this paper with some remarks and observations on equality of the ordinary and symbolic powers of ideals. Remark 4.1. The multiplicity bound in Theorem 3.1 approaches 2d−1 as n tends to infinity. Thus, if e(R/P ) < 2d−1 , then P n 6= P (n) for n large. Hence, if Question 1.1 has a positive answer and P n = P (n) for all n ≤ d − 1, then e(R/P ) ≥ 2d−1 . This is consistent with the conclusion of Question 1.1, that P is a complete intersection. To see this, suppose P is generated by a regular sequence a1 , . . . , ad−1 and x is a minimal reduction of m/P in R/P . Then, by [20, Theorem 14.9], we have e(m, R/P ) = λR (R/(P, x)) = λR (R/(a1 , . . . , ad )) ≥ d Y ordm (ai ) ≥ 2d−1 , i=1 where ad = x. Note that ordm (x) = 1 and ordm (ai ) ≥ 2 for i = 1, . . . , d − 1, as we are assuming P ⊂ m2 . Remark 4.2. We know that if P n = P (n) for n large, then P is a complete intersection [7]. The conclusion is also true if P n = P (n) for infinitely many n, see for instance Brodmann’s result on stability of associated primes of R/P n in [4]. This can also be obtained using superficial elements, at least when R has infinite residue field and P has positive grade. If P n = P (n) for infinitely many n, then one can show that P n = P (n) for n large. To see this, let x ∈ P be a superficial element, in the sense that P n+1 : x = P n for n large, see [17, Proposition 8.5.7]. Hence, if there exists an element b ∈ P (n) \ P n , then we have xb ∈ P (n+1) \ P n+1 for n large. Remark 4.3. If P n = P (n) for n large, then the analytic spread of P is at most d − 1 [5]. We note that this can also be seen via ε-multiplicity for one-dimensional primes. For 8 A. HOSRY, Y. KIM AND J. VALIDASHTI a prime ideal P of height d − 1, we have H0m (R/P n ) = P (n) /P n , where the left hand side is the zero-th local cohomology of R/P n with support in m. Thus, if P n = P (n) for n large, then ε-multiplicity of P is zero, where ε(P ) = lim sup n d! · λR (H0m (R/P n )). nd Hence, by [18, Theorem 4.7] or [23, Theorem 4.2], the analytic spread of P is at most d−1. Acknowledgements This research was initiated at the MRC workshop on commutative algebra at Snowbird, Utah, in the summer of 2010. We would like to thank AMS, NSF and MRC for providing funding and a stimulating research environment. 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Algebra 140 (1991), no. 1, 12–25. [22] S. Morey, Stability of associated primes and equality of ordinary and symbolic powers of ideals, Comm. Algebra 27 (1999), no. 7, 3221–3231. [23] B. Ulrich and J. Validashti, A criterion for integral dependence of modules, Math. Res. Lett. 15 (2008), no. 1, 149–162. [24] W. V. Vasconcelos, Koszul homology and the structure of low codimension Cohen-Macaulay ideals, Trans. Amer. Math. Soc. 301 (1987), no. 2, 591–613. Department of Mathematics and Statistics, Notre Dame University-Louaize, P.O. Box: 72, Zouk Mikael, Zouk Mosbeh, Lebanon E-mail address: ahosry@ndu.edu.lb Department of Mathematics, Purdue University, West Lafayette, IN 47907 E-mail address: kim455@purdue.edu Department of Mathematics, University of Illinois at Urbana-Champaign, IL 61801 E-mail address: jvalidas@illinois.edu 

A Genetic Algorithm for solving Quadratic Assignment Problem (QAP) H. Azarbonyada, R. Babazadehb a b Department of Electrical and computers engineering, University of Tehran, Tehran, Iran, h.azarbonyad@ece.ut.ac.ir Department of industrial engineering, University of Tehran, Tehran, Iran, r.babazadeh@ut.ac.ir Abstract— The Quadratic Assignment Problem (QAP) is one of the models used for the multi-row layout problem with facilities of equal area. There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the aim of minimizing the sum of the distances multiplied by the corresponding flows. The QAP is among the most difficult NP-hard combinatorial optimization problems. Because of this, this paper presents an efficient Genetic algorithm (GA) to solve this problem in reasonable time. For validation the proposed GA some examples are selected from QAP library. The obtained results in reasonable time show the efficiency of proposed GA. Key words- Genetic Algorithm, QAP, Multi-row layout problems optimization problems. The QAP is NP-hard optimization problem [7], so, to practically solve the QAP one has to apply heuristic algorithms which find very high quality solutions in short computation time. And also there is no known algorithm for solving this problem in polynomial time, and even small instances may require long computation time [2]. The location of facilities with material flow between them was first modeled as a QAP by Koopmans and Beckmann [3]. In a facility layout problem in which there are n facilities to be assigned to n given location. The QAP formulation requires an equal number of facilities and locations. If there are fewer than n, say m<n, facilities to be assigned to n locations, then to use the QAP formulation, n-m dummy facilities should be created and a zero flow between each of these and all others must be 1. INTRODUCTION assigned ( including the other dummy facilities). If there are Some multi-row layout problems are the control layout problem, the machine layout problem in an automated manufacturing system and office layout problem also the Travelling Salesman Problem (TSP) may be seen as a fewer locations than facilities, then the problem is infeasible [4]. Considering previous works in QAP ([2],[8] and [9]) this paper is developed to solve this problem with GA in short computation time. special case of QAP if one assumes that the flows connect all facilities only along a single ring, all flows have the same non- In the next section the QAP is described and formulated. After zero of introducing GA in Section 3, the encoding scheme, solution standard combinatorial optimization problems may be written representation and GA operators are described in this section. in this form (see [1] and [2]). The computational results are reported in Section 4. Finally, The Quadratic Assignment Problem (QAP) is one of the Section 5 concludes this paper and offers some directions for classical combinatorial optimization problems and is known further research. for its diverse applications and is widely regarded as one of 2. Mathematical model (constant) value. Many other problems the most difficult problem in classical combinatorial The following notation is used in formulation of QAP [4]: parameters total number of facilities and locations n flow of material from facility I to facility k fik distance from location j to location l djl Variable 1 If facility I is assigned to location j = xij 0 Otherwise individuals and happens in generations. In each generation, the fitness of every individual in the population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm n n n i j k (1) n ∑∑∑∑ Min f ik d jl x ij x kl terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached l for the population. If the algorithm has terminated due to a n ∑x ij =1 (2) ∀i maximum number of generations, a satisfactory solution may or may not have been reached [5]. j n ∑x ij =1 (3) ∀j i (4) xij is binary 3.1. Encoding scheme In this section we describe the encoding scheme for different component of proposed GA when n=5. For creating initial The objective function (1) minimizes the total distances and flows between facilities. Constraints (2) and (4) ensure that each facility I is assigned to exactly one location. As well as Constraints (3) and (4) assure that each location j has exactly one facility which assigned to it. The term quadratic stems population we consider a chromosome with n gene, as any gene is depictive of assignment each facility to exactly one location. For example the chromosome in Figure 1 show that facilities 2,4,3,1 and 5 is assigned to locations 1,2,3,4 and 5 respectively that it is a feasible solution. from the formulation of the QAP as an integer optimization problem with a quadratic objective function. 3. Genetic Algorithm 2 4 3 1 5 1 2 3 4 5 Fig.1. chromosome representation Genetic Algorithms (Gas) are routinely used to generate useful For making above chromosome (feasible solution) to create solutions to optimization and search problems. Genetic initial population, firstly we assume that all content of algorithms belong to the larger class of evolutionary chromosome are placed equal to 1, then random integer algorithms (EA), which generate solutions to optimization number from 1 to 5 is produced, say that 3, and then it is problems using techniques inspired by natural evolution. In a compared with the content of corresponding square, so if the genetic (called corresponding content is equal to 1, facility 3 is assigned to chromosomes or the genotype of the genome), which encode location 1 and then this content is set equal to zero (see figure candidate solutions (called individuals, creatures) to an 2). Again a random integer number from 1 to 5 is produced, optimization problem, evolves toward better solutions. then If it is equal to 3 because of the corresponding content is Traditionally, solutions are represented in binary as strings of zero (it means that facility 3 is assigned) so, another number 0s and 1s, but other encodings are also possible. The evolution should be produced. This way continues until all facilities are algorithm, a population of strings usually starts from a population of randomly generated assigned to locations and also the determined number of initial 2 5 3 1 4 2 4 3 1 5 population of chromosomes is satisfied. 1 1 1 1 1 1 1 0 1 1 Fig.4. Mutation representation 3.1.3. Selection Fig.2. making chromosome (feasible solution) In selection operator the number of chromosomes required to 3.1.1. Crossover complete next generation from crossover, mutation and We use two point crossovers for implementation of GA. For making child 1 and child 2, two points are randomly selected from 1 to 5 and between these points is hold fixed. Then, child 1 is created according to orders of parent 2 and fixed components of parent 1, as well as child 2 is produced previous generation are determined special mechanism that we use roulette wheel mechanism. In roulette wheel selection, candidate solutions are given a probability of being selected that is directly proportionate to their fitness. The selection probability of a candidate solution I is Fi , where Fi is P ∑ according to order of parent 1 and fixed components of parent Fj j 2 (see Figure 3). After tuning the crossover probability is the fitness of chromosome I (objective function in QAP determined equal to 0.8. This means that 80% of selected formulation) and P is the population size. parents participate in crossover operation. 3 5 2 4 1 5 3 4. Computational Results PARENT 1 2 1 4 PARENT 2 For validation the proposed GA some examples are selected from QAP library. The proposed GA was programmed in Java net beans 6.9.1 on a Pentium dual-core 2.66 GHZ computer with 4 GB RAM. The objective function values of the best 1 5 2 4 3 As it is illustrated in Table 1 the proposed GA results in CHILD 1 3 2 4 known solutions are given by Burkard et al. [6]. 5 1 the short computational time with acceptable GAP. CHILD 2 Table 1: Computational results of proposed GA FIG.3. CROSSOVER REPRESENTATION 3.1.2. Mutation Mutation operator changes the value of each gene in a chromosome with probability 0.2. For mutation two genes are randomly selected from selected chromosome and then their locations are substituted (see Figure 4). Test Problems Nug 12 Nug 17 Nug 20 Nug 24 Nug 28 chr12a.dat chr12b.dat chr15a.dat Global/local optimum global local local local local local local local Gap% 0 .0034 0 .0034 .012 0 0 1 Run time (second) 1 3 3 4 5 1 2 4 5. Conclusion In this paper at first QAP as a special type of multi-row layout problem with facilities of equal area is introduced and its various applications in different fields are described. Then, due to NP-hard nature of QAP an efficient GA algorithm is developed to solve it in reasonable time. Finally, some computational results from data set of QAP library are provided to show the efficiency and capability of proposed GA. Comparing outcome results with those works available in literature justify the proficiency of proposed GA in achieving acceptable solutions in reasonable time. For future research direction, developing other heuristic and/or meta-heuristic such as simulated annealing (SA) algorithm and PSO algorithm and then comparison results of them with results of this paper can be attractive work. References [1] Burkard R.E.(1984). Location with spatial interaction – Quadratic assignment problem in discrete location theory edited by R.L. Francis and P.B. Mirchandani. New York: Academic. [2] Ravindra K. Ahuja, James B. Orlin, Ashish Tiwari (2000). A greedy genetic algorithm for the quadratic assignment Problem. Computers & Operations Research 27. 917-934. [3] Koopmans T.C. and Beckmann M. (1957). Assignment problems and the location of economic activities. Econometrica 25, 1:53-76. [4] Heragu S. (1997). Facilities Design. Boston, PWS Publishing company. [5] Eiben, A. E. et al (1994). Genetic algorithms with multi-parent recombination. PPSN III: Proceedings of the International Conference on Evolutionary Computation. The Third Conference on Parallel Problem Solving from Nature: 78–87. [6] Burkard RE, Karisch SE, Rendl F. (1997). QAPLIB – A quadratic assignment program library. J. Global Optim.10:391- 403. [7] S. Sahni and T. Gonzales, P-Complete Approximation Problems, J. ACM, vol. 23, pp. 555-565, 1976. [8] V. Maniezzo and A. Colorni. The Ant System Applied to the Quadratic Assignment Problem. Accepted for publication in IEEE Transactions on Knowledge and Data Engineering, 1999. [9] P. Merz and B. Freisleben. A Genetic Local Search Approach to the Quadratic Assignment Problem. In T. B¨ack, editor, Proceedings of the Seventh International Conference on Genetic Algorithms (ICGA’97), pages 465–472. Morgan Kaufmann, 1997. 

Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound∗. arXiv:1704.00807v1 [] 3 Apr 2017 Bernhard Haeupler Carnegie Mellon University haeupler@cs.cmu.edu Amirbehshad Shahrasbi Carnegie Mellon University shahrasbi@cs.cmu.edu Abstract We introduce synchronization strings, which provide a novel way of efficiently dealing with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to deal with than more commonly considered half-errors, i.e., symbol corruptions and erasures. For every ε > 0, synchronization strings allow to index a sequence with an ε−O(1) size alphabet such that one can efficiently transform k synchronization errors into (1 + ε)k half-errors. This powerful new technique has many applications. In this paper, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion-deletion channels. While ECCs for both half-errors and synchronization errors have been intensely studied, the later has largely resisted progress. As Mitzenmacher puts it in his 2009 survey [22]: “Channels with synchronization errors . . . are simply not adequately understood by current theory. Given the near-complete knowledge we have for channels with erasures and errors ... our lack of understanding about channels with synchronization errors is truly remarkable.” Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate insdel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regime these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straight forward application of our synchronization strings based indexing method gives a simple black-box construction which transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs into the realm of insdel codes. Most notably, for the complete noise spectrum we obtain efficient “near-MDS” insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0, 1) and ε > 0 we give insdel codes achieving a rate of 1 − δ − ε over a constant size alphabet that efficiently correct a δ fraction of insertions or deletions. ∗ Supported in part by the National Science Foundation through grants CCF-1527110 and CCF-1618280. 1 Introduction Since the fundamental works of Shannon, Hamming, and others the field of coding theory has advanced our understanding of how to efficiently correct symbol corruptions and erasures. The practical and theoretical impact of error correcting codes on technology and engineering as well as mathematics, theoretical computer science, and other fields is hard to overestimate. The problem of coding for timing errors such as closely related insertion and deletion errors, however, while also studied intensely since the 60s, has largely resisted such progress and impact so far. An expert panel [8] in 1963 concluded: “There has been one glaring hole in [Shannon’s] theory; viz., uncertainties in timing, which I will propose to call time noise, have not been encompassed . . . . Our thesis here today is that the synchronization problem is not a mere engineering detail, but a fundamental communication problem as basic as detection itself !” however as noted in a comprehensive survey [21] in 2010: “Unfortunately, although it has early and often been conjectured that error-correcting codes capable of correcting timing errors could improve the overall performance of communication systems, they are quite challenging to design, which partly explains why a large collection of synchronization techniques not based on coding were developed and implemented over the years.” or as Mitzenmacher puts in his survey [22]: “Channels with synchronization errors, including both insertions and deletions as well as more general timing errors, are simply not adequately understood by current theory. Given the near-complete knowledge we have for channels with erasures and errors . . . our lack of understanding about channels with synchronization errors is truly remarkable.” We, too, believe that the current lack of good codes and general understanding of how to handle synchronization errors is the reason why systems today still spend significant resources and efforts on keeping very tight controls on synchronization while other noise is handled more efficiently using coding techniques. We are convinced that a better theoretical understanding together with practical code constructions will eventually lead to systems which naturally and more efficiently use coding techniques to address synchronization and noise issues jointly. In addition, we feel that better understanding the combinatorial structure underlying (codes for) insertions and deletions will have impact on other parts of mathematics and theoretical computer science. In this paper, we introduce synchronization strings, a new combinatorial structure which allows efficient synchronization and indexing of streams under insertions and deletions. Synchronization strings and our indexing abstraction provide a powerful and novel way to deal with synchronization issues. They make progress on the issues raised above and have applications in a large variety of settings and problems. We already found applications to channel simulations, synchronization sequences [21], interactive coding schemes [4–7, 15, 17], edit distance tree codes [2], and error correcting codes for insertion and deletions and suspect there will be many more. In this paper we focus on the last application, namely, designing efficient error correcting block codes over large alphabets for worst-case insertion-deletion channels. The knowledge on efficient error correcting block codes for insertions and deletions, also called insdel codes, severely lacks behind what is known for codes for Hamming errors. While Levenshtein [18] introduced and pushed the study of such codes already in the 60s it took until 1999 for Schulman and Zuckerman [25] to construct the first insdel codes with constant rate, constant distance, and constant alphabet size. Very recent work of Guruswami et al. [10, 13] in 2015 and 2016 gave the first constant rate insdel codes for asymptotically large noise rates, via list decoding. These codes are however still polynomially far from optimal in their rate or decodable distance respectively. In particular, they achieve a rate of Ω(ε5 ) for a relative distance of 1 − ε or a relative distance of O(ε2 ) for a rate of 1 − ε, for asymptotically small ε > 0 (see Section 1.5 for a more detailed discussion of related work). This paper essentially closes this line of work by designing efficient “near-MDS” insdel codes 1 which approach the optimal rate-distance trade-off given by the Singleton bound. We prove that for any 0 ≤ δ < 1 and any constant ε > 0, there is an efficient insdel code over a constant size alphabet with block length n and rate 1 − δ − ε which can be uniquely and efficiently decoded from any δn insertions and deletions. The code construction takes polynomial time; and encoding and decoding can be done in linear and quadratic time, respectively. More formally, let us define the edit distance of two given strings as the minimum number of insertions and deletions required to convert one of them to the other one. Theorem 1.1. For any ε > 0 and δ ∈ (0, 1) there exists an encoding map E : Σk → Σn and a decoding map D : Σ∗ → Σk such that if EditDistance(E(m), x) ≤ δn then D(x) = m. Further k n > 1 − δ − ε, |Σ| = f (ε), and E and D are explicit and can be computed in linear and quadratic time in n. We obtain this code via a black-box construction which transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This transformation, which is a straight forward application of our new synchronization strings based indexing method, is so simple that it can be summarized in one sentence: −1 For any efficient length n ECC with alphabet bit size logεε , attaching to every codeword, symbol by symbol, a random or suitable pseudorandom string over an alphabet of bit size log ε−1 results in an efficient insdel code with a rate and decodable distance that changed by at most ε. Far beyond just implying Theorem 1.1, this allows to instantly transfer much of the highly developed understanding for regular ECCs into the realm of insdel codes. Theorem 1.1 is obtained by using the “near-MDS” expander codes of Guruswami and Indyk [9] as a base ECC. These codes generalize the linear time codes of Spielman [27] and can be encoded and decoded in linear time. Our simple encoding strategy, as outlined above, introduces essentially no additional computational complexity during encoding. Our quadratic time decoding algorithm, however, is slower than the linear time decoding of the base codes from [9] but still pretty fast. In particular, a quadratic time decoding for an insdel code is generally very good given that, in contrast to Hamming codes, even computing the distance between the received and the sent/decoded string is an edit distance computation. Edit distance computations in general do usually not run in sub-quadratic time, which is not surprising given the recent SETH-conditional lower bounds [1]. For the settings of for insertion-only and deletion-only errors we furthermore achieve analogs of Theorem 1.1 with linear decoding complexities. In terms of the dependence of the alphabet bit size on the parameter ε, which characterizes how close a code is to achieving an optimal rate/distance pair summing to one, our transformation seem to inherently produce an alphabet bit size that is near linear in 1ε . However, the same is true for the state of the art linear-time base ECCs [9] which have an alphabet bit size of Θ( ε12 ). Existentially it is known that an alphabet bit size logarithmic in 1ε is necessary and sufficient and ECCs based on algebraic geometry [29] achieving such a bound up to constants are known, but their encoding and decoding complexities are higher. 1.1 High-level Overview, Intuition and Overall Organization While extremely powerful, the concept and idea behind synchronization strings is easily demonstrated. In this section, we explain the high-level approach taken and provide intuition for the formal definitions and proofs to follow. This section also explains the overall organization of the rest of the paper. 2 1.1.1 Synchronization Errors and Half-Errors Consider a stream of symbols over a large but constant size alphabet Σ in which some constant fraction δ of symbols is corrupted. There are two basic types of corruptions we will consider, half-errors and synchronization errors. Half-errors consist of erasures, that is, a symbol being replaced with a special “?” symbol indicating the erasure, and symbol corruptions in which a symbol is replaced with any other symbol in Σ. The wording half-error comes from the realization that when it comes to code distances erasures are half as bad as symbol corruptions. An erasure is thus counted as one half-error while a symbol corruption counts as two half-errors (see Section 2 for more details). Synchronization errors consist of deletions, that is, a symbol being removed without replacement, and insertions, where a new symbol from Σ is added anywhere. It is clear that synchronization errors are strictly more general and harsher than half-errors. In particular, any symbol corruption, worth two half-errors, can also be achieved via a deletion followed by an insertion. Any erasure can furthermore be interpreted as a deletion together with the often very helpful extra information where this deletion took place. This makes synchronization errors at least as hard as half-errors. The real problem that synchronization errors bring with them however is that they cause sending and receiving parties to become “out of synch”. This easily changes how received symbols are interpreted and makes designing codes or other systems tolerant to synchronization errors an inherently difficult and significantly less well understood problem. 1.1.2 Indexing and Synchronization Strings: Reducing Synchronization Errors to Half-Errors There is a simple folklore strategy, which we call indexing, that avoids these synchronization problems: Simply enhance any element with a time stamp or element count. More precisely, consecutively number the elements and attach this position count or index to each stream element. Now, if we deal with only deletions it is clear that the position of any deletion is easily identified via a missing index, thus transforming it into an erasure. Insertions can be handled similarly by treating any stream index which is received more than once as erased. If both insertions and deletions are allowed one might still have elements with a spoofed or incorrectly received index position caused by a deletion of an indexed symbol which is then replaced by a different symbol with the same index. This however requires two insdel errors. Generally this trivial indexing strategy can seen to successfully transform any k synchronization errors into at most k half-errors. In many applications, however, this trivial indexing cannot be used, because having to attach a log n bit1 long index description to each element of an n long stream is prohibitively costly. Consider for example an error correcting code of constant rate R over some potentially large but nonetheless constant size alphabet Σ, which encodes logRn|Σ| bits into n symbols from Σ. Increasing Σ by a factor of n to allow each symbol to carry its log n bit index would destroy the desirable property of having an alphabet which is independent from the block length n and would furthermore reduce the rate of the code from R to Θ( logR n ), which approaches zero for large block lengths. For streams of unknown or infinite length such problems become even more pronounced. This is where synchronization strings come to the rescue. Essentially, synchronization strings allow to index every element in an infinite stream using only a constant size alphabet while achieving an arbitrarily good approximate reduction from synchronization errors to half-errors. In particular, using synchronization strings k synchronization errors can be transformed into 1 Throughout this paper all logarithms are binary. 3 at most (1 + ε)k half-errors using an alphabet of size independent of the stream length and in fact only polynomial in 1ε . Moreover, these synchronization strings have simple constructions and fast and easy decoding procedures. Attaching our synchronization strings to the codewords of any efficient error correcting code, which efficiently tolerates the usual symbol corruptions and erasures, transforms any such code into an efficiently decodable insdel code while only requiring a negligible increasing in the alphabet size. This allows to use the decades of intense research in coding theory for Hamming-type errors to be transferred into the much harder and less well understood insertion-deletion setting. 1.2 Synchronization Strings: Definition, Construction, and Decoding Next, we want to briefly motivate and explain how we arrive at a natural definition of these magical indexing sequences S over a finite alphabet Σ and what intuition lies behind their efficient constructions and decoding procedures. Suppose a sender has attached some indexing sequence S one-by-one to each element in a stream and consider a time t at which a receiver has received a corrupted sequence of the first t index descriptors, i.e., a corrupted version of the length t prefix of S. When the receiver tries to guess or decode the current index it should naturally consider all indexing symbols received so far and find the “best” prefix of S. This suggests that the prefix of length l of a synchronization string S acts as a codeword for the index position l and that one should think of the set of prefixes of S as a code associated with the synchronization string S. Naturally one would want such a code to have good distance properties between any two codewords under some distance measure. While edit distance, i.e., the number of insertions and deletions needed to transform one string into another seems like the right notion of distance for insdel errors in general, the prefix nature of the codes under consideration will guarantee that codewords for indices l and l0 > l will have edit distance exactly l0 − l. This implies that even two very long codewords only have a tiny edit distance. On the one hand, this precludes synchronization codes with a large relative edit distance between its codewords. On the other hand, one should see this phenomenon as simply capturing the fact that at any time a simple insertion of an incorrect symbol carrying the correct next indexing symbol will lead to an unavoidable decoding error. Given this natural and unavoidable sensitivity of synchronization codes to recent corruptions, it makes sense to instead use a distance measure which captures the recent density of errors. In this spirit, we suggest the definition of a, to our knowledge, new string distance measure which we call relative suffix distance, which intuitively measures the worst fraction of insdel errors to transform suffixes, i.e., recently sent parts of two strings, into each other. This natural measure, in contrast to a similar measure defined in [2], turns out to induce a metric space on any set of strings. With this natural definitions for an induced set of codewords and a natural distance metric associated with any such set the next task is to design a string S for which the set of codewords has as large of a minimum pairwise distance as possible. When looking for (infinite) sequences that induce such a set of codewords and thus can be successfully used as synchronization strings it became apparent that one is looking for highly irregular and non-self-similar strings over a fixed alphabet Σ. It turns out that the correct definition to capture these desired properties, which we call ε-synchronization property, states that any two neighboring intervals of S with total length l should require at least (1−ε)l insertions and deletions to transform one into the other, where ε ≥ 0. A one line calculation also shows that this clean property also implies a large minimum relative suffix distance between any two codewords. Not surprisingly, random strings essentially satisfy this ε-synchronization property, except for local imperfections of self-similarity, such as, symbols repeated twice in a row, which would naturally occur in random sequences about every |Σ| positions. 4 This allows us to use the probabilistic method and the general Lovász Local Lemma to prove the existence ε-synchronization strings. This also leads to an efficient randomized construction. Finally, decoding any string to the closest codeword, i.e., the prefix of the synchronization string S with the smallest relative suffix distance, can be easily done in polynomial time because the set of synchronization codewords is linear and not exponential in n and (edit) distance computations (to each codeword individually) can be done via the classical Wagner-Fischer dynamic programming approach. 1.3 More Sophisticated Decoding Procedures All this provides an indexing solution which transforms any k synchronization errors into at most (5 + ε)k half-errors. This already leads to insdel codes which achieve a rate approaching 1 − 5δ for any δ fraction of insdel errors with δ < 51 . While this is already a drastic improvement over the √ previously best 1 − O( δ) rate codes from [10], which worked only for sufficiently small δ, it is a far less strong result than the near-MDS codes we promised in Theorem 1.1 for every δ ∈ (0, 1). We were able to improve upon the above strategy slightly by considering an alternative to the relative suffix distance measure, which we call relative suffix pseudo distance RSPD. RSPD was introduced in [2] and while neither being symmetric nor satisfying the triangle inequality, can act as a pseudo distance in the minimum-distance decoder. For any set of k = ki + kd insdel errors consisting of ki insertions and kd deletions this improved indexing solution leads to at most (1 + ε)(3ki + kd ) half-errors which already implies near-MDS codes for deletion-only channels but still falls short for general insdel errors. We leave open the question whether an improved pseudo distance definition can achieve an indexing solution with negligible number of misdecodings for a minimum-distance decoder. In order to achieve our main theorem we developed an different strategy. Fortunately, it turned out that achieving a better indexing solution and the desired insdel codes does not require any changes to the definition of synchronization codes, the indexing approach itself, or the encoding scheme but solely required a very different decoding strategy. In particular, instead of decoding indices in a streaming manner we consider more global decoding algorithms. We provide several such decoding algorithms in Section 6. In particular, we give a simple global decoding algorithm which for which the number of misdecodings goes to zero as the quality ε of the ε-synchronization string used goes to zero, irrespectively of how many insdel errors are applied. Our global decoding algorithms crucially build on another key-property which we prove holds for any ε-synchronization string S, namely that there is no monotone matching between S and itself which mismatches more than a ε fraction of indices. Besides being used in our proofs, considering this ε-self-matching property has another advantage. We show that this property is achieved easier than the full ε-synchronization property and that indeed a random string satisfies it with good probability. This means that, in the context of error correcting codes, one can even use a simple uniformly random  string as a “synchronization string”. Lastly, we show that even a log n −O(1) n -approximate O log 1 -wise independent random strings satisfy the desired ε-self-matching ε property which, using the celebrated small sample space constructions from [24] also leads to a deterministic polynomial time construction. Lastly, we provide simpler and faster global decoding algorithms for the setting of deletion-only and insertion-only corruptions. These algorithms are essentially greedy algorithms which run in linear time. They furthermore guarantee that their indexing decoding is error-free, i.e., they only output “I don’t know” for some indices but never produce an incorrectly decoded index. Such 5 decoding schemes have the advantage that one can use them in conjunction with error correcting codes that efficiently recover from erasures (and not necessarily also symbol corruptions). 1.4 Organization of this Paper The organization of this paper closely follows the flow of the high-level description above. We start by giving more details on related work in Section 1.5 and introduce notation used in the paper in Section 2 together with a formal introduction of the two different error types as well as (efficient) error correcting codes and insdel codes. In Section 3, we formalize the indexing problem and (approximate) solutions to it. Section 4 shows how any solution to the indexing problem can be used to transform any regular error correcting codes into an insdel code. Section 5 introduces the relative suffix distance and ε-synchronization strings, proves the existence of ε-synchronization strings and provides an efficient construction. Section 5.2 shows that the minimum suffix distance decoder is efficient and leads to a good indexing solution. We elaborate on the connection between ε-synchronization strings and the ε-self-matching property in Section 6.1 and provide our improved decoding algorithms in the remainder of Section 6. 1.5 Related Work Shannon was the first to systematically study reliable communication. He introduced random error channels, defined information quantities, and gave probabilistic existence proofs of good codes. Hamming was the first to look at worst-case errors and code distances as introduced above. Simple counting arguments on the volume of balls around codewords given in the 50’s by Hamming and Gilbert-Varshamov produce simple bounds on the rate of q-ary codes with relative distance δ. In particular, they show the existence of codes with relative distance δ and rate at least 1−Hq (δ) where log(1−x) Hq (x) = x log(q − 1) − x log x−(1−x) is the q-ary entropy function. This means that for any log q δ < 1 and q = ω(1/δ) there exists codes with distance δ and rate approaching 1 − δ. Concatenated codes and the generalized minimum distance decoding procedure introduced by Forney in 1966 led to the first codes which could recover from constant error fractions δ ∈ (0, 1) while having polynomial time encoding and decoding procedures. The rate achieved by √ concatenated codes for large alphabets with sufficiently small distance δ comes out to be 1 − O( δ). On the other hand, for δ sufficiently close to one, one can achieve a constant rate of O(δ 2 ). Algebraic geometry codes suggested by Goppa in 1975 later lead to error correcting codes which for every ε > 0 achieve the optimal rate of 1 − δ − ε with an alphabet size polynomial in ε while being able to efficiently correct for a δ fraction of half-errors [29]. While this answered the most basic questions, research since then has developed a tremendously powerful toolbox and selection of explicit codes. It attests to the importance of error correcting codes that over the last several decades this research direction has developed into the incredibly active field of coding theory with hundreds of researchers studying and developing better codes. A small and highly incomplete subset of important innovations include rateless codes, such as, LT codes [20], which do not require to fix a desired distance at the time of encoding, explicit expander codes [9, 27] which allow linear time encoding and decoding, polar codes [12, 14] which can approach Shannon’s capacity polynomially fast, network codes [19] which allow intermediate nodes in a network to recombine codewords, and efficiently list decodable codes [11] which allow to list-decode codes of relative distance δ up to a fraction of about δ symbol corruptions. While error correcting codes for insertions and deletions have also been intensely studied, our understanding of them is much less well developed. We refer to the 2002 survey by Sloan [26] on single-deletion codes, the 2009 survey by Mitzenmacher [22] on codes for random deletions and 6 the most general 2010 survey by Mercier et al. [21] for the extensive work done around codes for synchronization errors and only mention the results most closely related to Theorem 1.1 here: Insdel codes were first considered by Levenshtein [18] and since then many bounds and constructions for such codes have been given. However, while essentially the same volume and sphere packing arguments as for regular codes show that there exists insdel codes capable of correcting a fraction δ of insdel erros with rate 1−δ, no efficient constructions anywhere close to this rate-distance tradeoff are known. Even the construction of efficient insdel codes over a constant alphabet with any (tiny) constant relative distance and any (tiny) constant rate had to wait until Schulman and Zuckerman gave the first such code in 1999 [25]. Over the last two years Guruswami et al. provided new codes improving over this state of the art the asymptotically small or large noise regime by giving the first codes which achieve a constant rate for noise rates going to one and codes which provide a rate going to one for an asymptotically small noise rate. In particular, [13] gave the first efficient codes codes over fixed alphabets to correct a deletion fraction approaching 1, as well as efficient binary codes to correct a small constant fraction of deletions with rate approaching 1. These codes could, however, only be efficiently decoded for deletions and not insertions. A follow-up work gave new and improved codes with similar rate-distance tradeoffs which can be efficiently decoded √ from insertions and deletions [10]. In particular, these codes achieve a rate of Ω(δ 5 ) and 1 − Õ( δ) while being able to efficiently recover from a δ fraction of insertions and deletions. These works put the current state of the art for error correcting codes for insertions and deletions pretty much equal to what was known for regular error correcting codes 50 years ago, after Forney’s 1965 doctoral thesis. 2 Definitions and Preliminaries In this section, we provide the notation and definitions we will use throughout the rest of the paper. 2.1 String Notation and Edit Distance 0 String Notation. For two strings S ∈ Σn and S 0 ∈ Σn be two strings over alphabet Σ. We define 0 S · S 0 ∈ Σn+n to be their concatenation. For any positive integer k we define S k to equal k copies of S concatenated together. For i, j ∈ {1, . . . , n}, we denote the substring of S from the ith index through and including the j th index as S[i, j]. Such a consecutive substring is also called a factor of S. For i < 1 we define S[i, j] = ⊥−i+1 · S[1, j] where ⊥ is a special symbol not contained in Σ. We refer to the substring from the ith index through, but not including, the j th index as S[i, j). The substrings S(i, j] and S[i, j] are similarly defined. Finally, S[i] denotes the ith symbol of S and |S| = n is the length of S. Occasionally, the alphabets we use are the cross-product of several alphabets, i.e. Σ = Σ1 × · · · × Σn . If T is a string over Σ, then we write T [i] = [a1 , . . . , an ], where ai ∈ Σi . Edit Distance. Throughout this work, we rely on the well-known edit distance metric defined as follows. Definition 2.1 (Edit distance). The edit distance ED(c, c0 ) between two strings c, c0 ∈ Σ∗ is the minimum number of insertions and deletions required to transform c into c0 . It is easy to see that edit distance is a metric on any set of strings and in particular is symmetric and satisfies the triangle inequality property. Furthermore, ED (c, c0 ) = |c| + |c0 | − 2 · LCS (c, c0 ), where LCS (c, c0 ) is the longest common substring of c and c0 . We also use some string matching notation from [2]: 7 Definition 2.2 (String matching). Suppose that c and c0 are two strings in Σ∗ , and suppose that ∗ is a symbol not in Σ. Next, suppose that there exist two strings τ1 and τ2 in (Σ ∪ {∗})∗ such that |τ1 | = |τ2 |, del (τ1 ) = c, del(τ2 ) = c0 , and τ1 [i] ≈ τ2 [i] for all i ∈ {1, . . . , |τ1 |}. Here, del is a function that deletes every ∗ in the input string and a ≈ b if a = b or one of a or b is ∗. Then we say that τ = (τ1 , τ2 ) is a string matching between c and c0 (denoted τ : c → c0 ). We furthermore denote with sc (τi ) the number of ∗’s in τi . Note that the edit distance ED(c, c0 ) between strings c, c, ∈ Σ∗ is exactly equal to minτ :c→c0 {sc (τ1 ) + sc (τ2 )}. 2.2 Error Correcting Codes Next we give a quick summary of the standard definitions and formalism around error correcting codes. This is mainly for completeness and we remark that readers already familiar with basic notions of error correcting codes might want to skip this part. Codes, Distance, Rate, and Half-Errors An error correcting code C is an injective function 0 which takes an input string s ∈ (Σ0 )n over alphabet Σ0 of length n0 and generates a codeword C(s) ∈ Σn of length n over alphabet Σ. The length n of a codeword is also called the block length. The two most important parameters of a code are its distance ∆ and its rate R. The log |Σ| rate R = nn0 log |Σ0 | measures what fraction of bits in the codewords produced by C carries nonredundant information about the input. The code distance ∆(C) = mins,s0 ∆(C(s), C(s0 )) is simply the minimum Hamming distance between any two codewords. The relative distance δ(C) = ∆(C) n measures what fraction of output symbols need to be corrupted to transform one codeword into another. It is easy to see that if a sender sends out a codeword C(s) of code C with relative distance δ a receiver can uniquely recover s if she receives a codeword in which less than a δ fraction of symbols are affected by an erasure, i.e., replaced by a special “?” symbol. Similarly, a receiver can uniquely recover the input s if less than δ/2 symbol corruptions, in which a symbol is replaced by any other symbol from Σ, occurred. More generally it is easy to see that a receiver can recover from any combination of ke erasures and kc corruptions as long as ke + 2kc < δn. This motivates defining half-errors to incorporate both erasures and symbol corruptions where an erasure is counted as a single half-error and a symbol corruption is counted as two half-errors. In summary, any code of distance δ can tolerate any error pattern of less than δn half-errors. We remark that in addition to studying codes with decoding guarantees for worst-case error pattern as above one can also look at more benign error models which assume a distribution over error patterns, such as errors occurring independently at random. In such a setting one looks for codes which allow unique recovery for typical error patterns, i.e., one wants to recover the input with probability tending to 1 rapidly as the block length n grows. While synchronization strings might have applications for such codes as well, this paper focuses exclusively on codes with good distance guarantees which tolerate an arbitrary (worst-case) error pattern. Synchronization Errors In addition to half-errors, we study synchronization errors which consist of deletions, that is, a symbol being removed without replacement, and insertions, where a new symbol from Σ is added anywhere. It is clear that synchronization errors are strictly more general and harsh than half-errors (see Section 1.1.1). The above formalism of codes, rate, and distance works equally well for synchronization errors if one replaces the Hamming distance with edit distance. Instead of measuring the number of symbol corruptions required to transform 8 one string into another, edit distance measures the minimum number of insertions and deletions to do so. An insertion-deletion error correcting code, or insdel code for short, of relative distance δ is a set of codewords for which at least δn insertions and deletions are needed to transformed any codeword into another. Such a code can correct any combination of less than δn/2 insertions and deletions. We remark that it is possible for two codewords of length n to have edit distance up to 2n putting the (minimum) relative edit distance between zero and two and allowing for constant rate codes which can tolerate (1 − ε)n insdel errors. Efficient Codes In addition to codes with a good minimum distance, one furthermore wants efficient algorithms for the encoding and error-correction tasks associated with the code. Throughout this paper we say a code is efficient if it has encoding and decoding algorithms running in time polynomial in the block length. While it is often not hard to show that random codes exhibit a good rate and distance, designing codes which can be decoded efficiently is much harder. We remark that most codes which can efficiently correct for symbol corruptions are also efficient for half-errors. For insdel codes the situation is slightly different. While it remains true that any code that can uniquely be decoded from any δ(C) fraction of deletions can also be decoded from the same fraction of insertions and deletions [18] doing so efficiently is often much easier for the deletion-only setting than the fully general insdel setting. . 3 The Indexing Problem In this section, we formally define the indexing problem. In a nutshell, this problem is that of sending a suitably chosen string S of length n over an insertion-deletion channel such that the receiver will be able to figure out the indices of most of the symbols he receives correctly. This problem can be trivially solved by sending the string S = 1, 2, . . . , n over the alphabet Σ = {1, . . . , n} of size n. Interesting solution to the indexing problem, however, do almost as well while using a finite size alphabet. While very intuitive and simple, the formalization of this problem and its solutions enables an easy use in many applications. To set up an (n, δ)-indexing problem, we fix n, i.e., the number of symbols which are being sent, and the maximum fraction δ of symbols that can be inserted or deleted. We further call the string S the synchronization string. Lastly, we describe the influences of the nδ worst-case insertions and deletions which transform S into the related string Sτ in terms of a string matching τ . In particular, τ = (τ1 , τ2 ) is the string matching from S to Sτ such that del(τ1 ) = S, del(τ2 ) = Sτ , and for every k ( (S[i], ∗) (S[i], Sτ [j]) (τ1 [k], τ2 [k]) = (∗, Sτ [j]) if S[i] is deleted if S[i] is delivered as Sτ [j] if Sτ [j] is inserted where i = |del(τ1 [1, k])| and j = |del(τ2 [1, k])|. Definition 3.1 ((n, δ)-Indexing Algorithm). The pair (S, DS ) consisting of a synchronization string S ∈ Σn and an algorithm DS is called a (n, δ)-indexing algorithm over alphabet Σ if for any set of nδ insertions and deletions represented by τ which alter S to a string Sτ , the algorithm DS (Sτ ) outputs either ⊥ or an index between 1 and n for every symbol in Sτ . The ⊥ symbol here represents an “I don’t know” response of the algorithm while an index j output by DS (Sτ ) for the ith symbol of Sτ should be interpreted as the (n, δ)-indexing algorithm guessing that this was the j th symbol of S. One seeks algorithms that decode as many indices as 9 possible correctly. Naturally, one can only correctly decode indices that were correctly transmitted. Next we give formal definitions of both notions: Definition 3.2 (Correctly Decoded Index). An (n, δ) indexing algorithm (S, DS ) decodes index j correctly under τ if DS (Sτ ) outputs i and there exists a k such that i = |del(τ1 [1, k])|, j = |del(τ2 [1, k])|, τ1 [k] = S[i], τ2 [k] = Sτ [j] We remark that this definition counts any ⊥ response as an incorrect decoding. Definition 3.3 (Successfully Transmitted Symbol). For string Sτ , which was derived from a synchronization string S via τ = (τ1 , τ2 ), we call the j th symbol Sτ [j] successfully transmitted if it stems from a symbol coming from S, i.e., if there exists a k such that |del(τ2 [1, k])| = j and τ1 [k] = τ2 [k]. We now define the quality of an (n, δ)-indexing algorithm by counting the maximum number of misdecoded indices among those that were successfully transmitted. Note that the trivial indexing strategy with S = 1, . . . , n which outputs for each symbol the symbol itself has no misdecodings. One can therefore also interpret our quality definition as capturing how far from this ideal solution an algorithm is (stemming likely due to the smaller alphabet which is used for S). Definition 3.4 (Misdecodings of an (n, δ)-Indexing Algorithm). Let (S, DS ) be an (n, δ)-indexing algorithm. We say this algorithm has at most k misdecodings if for any τ corresponding to at most nδ insertions and deletions the number of correctly transmitted indices that are incorrectly decoded is at most k. Now, we introduce two further useful properties that a (n, δ)-indexing algorithm might have. Definition 3.5 (Error-free Solution). We call (S, DS ) an error-free (n, δ)-indexing algorithm with respect to a set of deletion or insertion patterns if every index output is either ⊥ or correctly decoded. In particular, the algorithm never outputs an incorrect index, even for indices which are not correctly transmitted. It is noteworthy that error-free solutions are essentially only obtainable when dealing with the insertion-only or deletion-only setting. In both cases, the trivial solution with S = 1, · · · , n which decodes any index that was received exactly once is error-free. We later give some algorithms which preserve this nice property, even over a smaller alphabet, and show how error-freeness can be useful in the context of error correcting codes. Lastly, another very useful property of some (n, δ)-indexing algorithms is that their decoding process operates in a streaming manner, i.e, the decoding algorithm decides the index output for Sτ [j] independently of Sτ [j 0 ] where j 0 > j. While this property is not particularly useful for the error correcting block code application put forward in this paper, it is an extremely important and strong property which is crucial in several applications we know of, such as, rateless error correcting codes, channel simulations, interactive coding, edit distance tree codes, and other settings. Definition 3.6 (Streaming Solutions). We call (S, DS ) a streaming solution if the decoded index for the ith element of the received string Sτ only depends on Sτ [1, i]. Again, the trivial solution for (n, δ)-index decoding problem over an alphabet of size n with zero misdecodings can be made streaming by outputting for every received symbols the received symbol itself as an index. This solution is also error-free for the deletion-only setting but not error-free for the insertion-only setting. In fact, it is easy to show that an algorithm cannot be both streaming and error-free in any setting which allows insertions. 10 Overall, the important characteristics of an (n, δ)-indexing algorithm are (a) its alphabet size |Σ|, (b) the bound on the number of misdecodings, (c) the complexity of the decoding algorithm D, (d) the preprocessing complexity of constructing the string S, (e) whether the algorithm works for the insertion-only, the deletion-only or the full insdel setting, and (f) whether the algorithm satisfies the streaming or error-freeness property. Table 1 gives a summary over the different solutions for the (n, δ)-indexing problem we give in this paper. Algorithm Section Section Section Section Section Section 5.2 6.3 6.4 6.5 6.5 6.6 Type ins/del ins/del del ins del ins/del Misdecodings Error-free (2 + ε) · nδ √ ε·n ε · nδ (1 + ε) · nδ ε · nδ (1 + ε) · nδ Streaming Complexity X O(n4 )
√ O n2 / ε O(n) O(n) O(n) O(n4 ) X X X X Table 1: Properties and quality of (n, δ)-indexing algorithms with S being a ε-synchronization string 4 Insdel Codes via Indexing Algorithms Next, we show how a good (n, δ)-indexing algorithms (S, DS ) over alphabet ΣS allows one to transform any regular ECC C with block length n over alphabet ΣC which can efficiently correct half-errors, i.e., symbol corruptions and erasures, into a good insdel code over alphabet Σ = ΣC ×ΣS . To this end, we simply attach S symbol-by-symbol to every codeword of C. On the decoding end, we first decode the indices of the symbols arrived using the indexing part of each received symbol and then interpret the message parts as if they have arrived in the decoded order. Indices where zero or multiple symbols are received get considered as erased. We will refer to this procedure as the indexing procedure. Finally, the decoding algorithm DC for C is used. These two straight forward algorithms are formally described as Algorithm 1 and Algorithm 2. Theorem 4.1. If (S, DS ) guarantees k misdecodings for the (n, δ)-index problem, then the indexing procedure recovers the codeword sent up to nδ + 2k half-errors, i.e., half-error distance of the sent codeword and the one recovered by the indexing procedure is at most nδ +2k. If (S, DS ) is error-free, the indexing procedure recovers the codeword sent up to nδ + k half-errors. Proof. Consider a set insertions and deletions described by τ consisting of Dτ deletions and Iτ insertions. Note that among n encoded symbols, at most Dτ were deleted and less than k of are decoded incorrectly. Therefore, at least n − Dτ − k indices are decoded correctly. On the other hand at most Dτ + k of the symbols sent are not decoded correctly. Therefore, if the output only consisted of correctly decoded indices for successfully transmitted symbols, the output would have contained up to Dτ + k erasures and no symbol corruption, resulting into a total of Dτ + k halferrors. However, any symbol which is being incorrectly decoded or inserted may cause a correctly decoded index to become an erasure by making it appear multiple times or change one of original Iτ + k erasures into a corruption error by making the indexing procedure mistakenly decode an index. Overall, this can increase the number of half-errors by at most Iτ + k for a total of at most Dτ + k + Iτ + k = Dτ + Iτ + 2k = nδ + 2k half-errors. For error-free indexing algorithms, any misdecoding does not result in an incorrect index and the number of incorrect indices is Iτ instead of Iτ + k leading to the reduced number of half-errors in this case. 11 This makes it clear that applying an ECC C which is resilient to nδ + 2k half-errors enables the receiver side to fully recover m. Algorithm 1 Insertion Deletion Encoder Input: n, m = m1 , · · · , mn 1: m̃ = EC (m) 2: for i = 1 to n do 3: Mi = (mi , S[i]) Output: M Algorithm 2 Insertion Deletion Decoder Input: n, M 0 = (m̃0 , S 0 ) 1: Dec ← DS (S 0 ) 2: for i = 1 to n do 3: if there is a unique j for which Dec[j] = i then 4: m0i = m̃0j 5: else 6: m0i = ? 7: m = DC (m0 ) Output: m Next, we formally state how a good (n, δ)-indexing algorithm (S, DS ) over alphabet ΣS allows one to transform any regular ECC C with block length n over alphabet ΣC which can efficiently correct half-errors, i.e., symbol corruptions and erasures, into a good insdel code over alphabet Σ = ΣC × ΣS . The following Theorem is a corollary of Theorem 4.1 and the definition of the indexing procedure: Theorem 4.2. Given an (efficient) (n, δ)-indexing algorithm (S, DS ) over alphabet ΣS with at most k misdecodings, and decoding complexity TDS (n) and an (efficient) ECC C over alphabet ΣC with rate RC , encoding complexity TEC , and decoding complexity TDC that corrects up to nδ + 2k half-errors, one obtains an insdel code that can be (efficiently) decoded from up to nδ insertions and deletions. The rate of this code is   log ΣS RC · 1 − log ΣC The encoding complexity remains TEC , the decoding complexity is TDC +TDS (n) and the preprocessing complexity of constructing the code is the complexity of constructing C and S. Furthermore, if (S, DS ) is error-free, then choosing a C which can recover only from nδ + k erasures is sufficient to produce the same quality code. ΣS Note that if one chooses ΣC such that log log ΣC = o(δ), the rate loss due to the attached symbols will be negligible. With all this in place one can obtain Theorem 1.1 as a consequence of Theorem 4.2. 0 Proof 
ofTheorem 1.1. Given the δ and ε from the statement of Theorem 1.1 we choose ε = 2 O 6ε and use Theorem 6.13 to construct a string S of length n over alphabet ΣS of size ε−O(1) with the ε0 -self-matching property. We then use the (n, δ)-indexing algorithm √ (S, Dε S ) where given in Section 6.3 and line 2 of Table 1 which guarantees that it has at most ε0 = 3 misdecodings. 12 Finally, we choose a near-MDS expander code [9] C which can efficiently correct up to δC = δ + 3ε half-errors and has a rate of RC > 1 − δC − 3ε over an alphabet ΣC = exp(ε−O(1) ) such that ΣS ε log |ΣC | ≥ 3 logε|ΣS | . This ensures that the final rate is indeed at least RC − log log ΣC = 1 − δ − 3 3 ε and the number of insdel errors that can be efficiently corrected is δC − 2 3 ≥ δ. The encoding and decoding complexities are furthermore straight forward and as is the polynomial time preprocessing time given Theorem 6.13 and [9]. 5 Synchronization Strings In this section, we formally define and develop ε-synchronization strings, which can be used as our base synchronization string S in our (n, δ)-indexing algorithms. As explained in Section 1.2 it makes sense to think of the prefixes S[1, l] of a synchronization string S as codewords encoding their length l, as the prefix S[1, l], or a corrupted version of it, will be exactly all the indexing information that has been received by the time the lth symbol is communicated: Definition 5.1 (Codewords Associated with a Synchronization String). Given any synchronization string S we define the set of codewords associated with S to be the set of prefixes of S, i.e., {S[1, l] | 1 ≤ l ≤ |S|}. Next, we define a distance metric on any set of strings, which will be useful in quantifying how good a synchronization string S and its associated set of codewords is: Definition 5.2 (Relative Suffix Distance). For any two strings S, S 0 ∈ Σ∗ we define their relative suffix distance RSD as follows: RSD(S, S 0 ) = max k>0 ED (S(|S| − k, |S|], S 0 (|S 0 | − k, |S 0 |]) 2k Next we show that RSD is indeed a distance which satisfies all properties of a metric for any set of strings. To our knowledge, this metric is new. It is, however, similar in spirit to the suffix “distance” defined in [2], which unfortunately is non-symmetric and does not satisfy the triangle inequality but can otherwise be used in a similar manner as RSD in the specific context here (see also Section 6.6). Lemma 5.3. For any strings S1 , S2 , S3 we have • Symmetry: RSD(S1 , S2 ) = RSD(S2 , S1 ), • Non-Negativity and Normalization: 0 ≤ RSD(S1 , S2 ) ≤ 1, • Identity of Indiscernibles: RSD(S1 , S2 ) = 0 ⇔ S1 = S2 , and • Triangle Inequality: RSD(S1 , S3 ) ≤ RSD(S1 , S2 ) + RSD(S2 , S3 ). In particular, RSD defines a metric on any set of strings. Proof. Symmetry and non-negativity follow directly from the symmetry and non-negativity of edit distance. Normalization follows from the fact that the edit distance between two length k strings can be at most 2k. To see the identity of indiscernibles note that RSD(S1 , S2 ) = 0 if and only if for all k the edit distance of the k prefix of S1 and S2 is zero, i.e., if for every k the k-prefix of S1 and S2 are identical. This is equivalent to S1 and S2 being equal. Lastly, the triangle inequality also essentially follows from the triangle inequality for edit distance. To see this let δ1 = RSD(S1 , S2 ) and δ2 = RSD(S2 , S3 ). By the definition of RSD this implies that for all k the k-prefixes of S1 and 13 S2 have edit distance at most 2δ1 k and the k-prefixes of S2 and S3 have edit distance at most 2δ2 k. By the triangle inequality for edit distance, this implies that for every k the k-prefix of S1 and S3 have edit distance at most (δ1 + δ2 ) · 2k which implies that RSD(S1 , S3 ) ≤ δ1 + δ2 . With these definitions in place, it remains to find synchronization strings whose prefixes induce a set of codewords, i.e., prefixes, with large RSD distance. It is easy to see that the RSD distance for any two strings ending on a different symbol is one. This makes the trivial synchronization string, which uses each symbol in Σ only once, induce an associated set of codewords of optimal minimum-RSD-distance one. Such trivial synchronization strings, however, are not interesting as they require an alphabet size linear in the length n. To find good synchronization strings over constant size alphabets, we give the following important definition of an ε-synchronization string. The parameter 0 < ε < 1 should be thought of measuring how far a string is from the perfect synchronization string, i.e., a string of n distinct symbols. Definition 5.4 (ε-Synchronization String). String S ∈ Σn is an ε-synchronization string if for every 1 ≤ i < j < k ≤ n + 1 we have that ED (S[i, j), S[j, k)) > (1 − ε)(k − i). We call the set of prefixes of such a string an ε-synchronization string. The next lemma shows that the ε-synchronization string property is strong enough to imply a good minimum RSD distance between any two codewords associated with it. Lemma 5.5. If S is an ε-synchronization string, then RSD(S[1, i], S[1, j]) > 1 − ε for any i < j, i.e., any two codewords associated with S have RSD distance of at least 1 − ε. Proof. Let k = j − i. The ε-synchronization string property of S guarantees that ED (S[i − k, i), S[i, j)) > (1 − ε)2k. Note that this holds even if i−k < 1. To finish the proof we note that the maximum in the definition of RSD includes the term ED(S[i−k,i),S[i,j)) > 1 − ε, which implies that RSD(S[1, i], S[1, j]) > 2k 1 − ε. 5.1 Existence and Construction The next important step is to show that the ε-synchronization strings we just defined exist, particularly, over alphabets whose size is independent of the length n. We show the existence of ε-synchronization strings of arbitrary length for any ε > 0 using an alphabet size which is only polynomially large in 1/ε. We remark that ε-synchronization strings can be seen as a strong generalization of square-free sequences in which any two neighboring substrings S[i, j) and S[j, k) only have to be different and not also far from each other in edit distance. Thue [28] famously showed the existence of arbitrarily large square-free strings over a trinary alphabet. Thue’s methods for constructing such strings however turns out to be fundamentally too weak to prove the existence of ε-synchronization strings, for any constant ε < 1. Our existence proof requires the general Lovász local lemma which we recall here first: Lemma 5.6 (General Lovász local lemma). Let A1 , . . . , An be a set of “bad” events. The directed graph G(V, E) is called a dependency graph for this set of events if V = {1, . . . , n} and each event Ai is mutually independent of all the events {Aj : (i, j) 6∈ E}. Now, if there exists x1 , . . . , xn ∈ [0, 1) such that for all i we have Y P [Ai ] ≤ xi (1 − xj ) (i,j)∈E 14 then there exists a way to avoid all events Ai simultaneously and the probability for this to happen is bounded by "n # n Y ^ P Āi ≥ (1 − xi ) > 0. i=1 i=1 Theorem 5.7. For any ε ∈ (0, 1), n ≥ 1, there exists an ε-synchronization string of length n over an alphabet of size Θ(1/ε4 ). Proof. Let S be a string of length n obtained by
concatenating two strings T and R, where T is simply the repetition of 0, . . . , t − 1 for t = Θ ε12 , and R is a uniformly random string of length n over alphabet Σ. In particular, Si = (i mod t, Ri ). We prove that S is an ε-synchronization string by showing that there is a positive probability that S contains no bad triple, where (x, y, z) is a bad triple if ED(S[x, y), S[y, z)) ≤ (1 − ε)(z − x). First, note that a triple (x, y, z) for which z − x < t cannot be a bad triple as it consists of completely distinct symbols by courtesy of T . Therefore, it suffices to show that there is no bad triple (x, y, z) in R for x, y, z such that z − x > t. Let (x, y, z) be a bad triple and let a1 a2 · · · ak be the longest common subsequence of R[x, y) and R[y, z). It is straightforward to see that ED(R[x, y), R[y, z)) = (y − x) + (z − y) − 2k = z − x − 2k. Since (x, y, z) is a bad triple, we have that z −x−2k ≤ (1−ε)(z −x), which means that k ≥ 2ε (z −x). With this observation in mind, we say that R[x, z) is a bad interval if it contains a subsequence a1 a2 · · · ak a1 a2 · · · ak such that k ≥ 2ε (z − x). To prove the theorem, it suffices to show that a randomly generated string does not contain any bad intervals with a non-zero probability. We first upper bound the probability that an interval of length l is bad:   εl l Pr [I is bad] ≤ |Σ|− 2 l εl I∼Σ  εl εl el |Σ|− 2 ≤ εl !εl e p = , ε |Σ| where the first inequality holds because if an interval of length l is bad, then it must contain a repeating subsequence of length lε2 . Any such sequence can be specified via εl positions in the l εl long interval and the probability that a given fixed sequence is valid for a random string is |Σ|− 2 .

k The second inequality comes from the fact that nk < ne . k The resulting inequality shows that the probability of an interval of length l being bad is bounded above by C −εl , where C can be made arbitrarily large by taking a sufficiently large alphabet size |Σ|. To show that there is a non-zero probability that the uniformly random string R contains no bad interval I of size t or larger, we use the general Lovász local lemma stated in Lemma 5.6. Note that the badness of interval I is mutually independent of the badness of all intervals that do not intersect I. We need to find real numbers xp,q ∈ [0, 1) corresponding to intervals R[p, q) for which Y P r [Interval R[p, q) is bad] ≤ xp,q (1 − xp0 ,q0 ). R[p,q)∩R[p0 ,q 0 )6=∅ We have seen that the left-hand side can be upper bounded by C −ε|R[p,q)| = C ε(p−q) . Furthermore, any interval of length l0 intersects at most l + l0 intervals of length l. We propose 15 xp,q = D−ε|R[p,q)| = Dε(p−q) for some constant D > 1. This means that it suffices to find a constant D that for all substrings R[p, q) satisfies C ε(p−q) ≤D ε(p−q) n  Y 1 − D−εl l+(q−p) , l=t or more clearly, for all l0 ∈ {1, · · · , n}, C −l0 ≤D −l0 n  Y 1−D −εl  l+l0 ε , l=t which means that D C≥Q n l=t (1 − D−εl ) 1+l/l0 ε . (1) For D > 1, the right-hand side of Equation (1) is maximized when n = ∞ and l0 = 1, and since we want Equation (1) to hold for all n and all l0 ∈ {1, · · · , n}, it suffices to find a D such that D C≥Q ∞ −εl ) l=t (1 − D l+1 ε . To this end, let ( L = min D>1 ) D Q∞ −εl ) l=t (1 − D . l+1 ε Then, it suffices to have Σ large enough so that p ε |Σ| ≥ L, C= e which means that |Σ| ≥ e2 L2 ε2 suffices to allow us to use the Lovász local lemma. We claim that  1 log ε L = Θ(1), which will complete the proof. Since t = ω , ε ∀l ≥ t D−εl · l+1  1. ε Therefore, we can use the fact that (1 − x)k > 1 − xk to show that: D Q∞ −εl ) l=t (1 − D l+1 ε < Q∞ l=t < = D
−εl 1 − l+1 ε ·D D 1− P∞ 1− 1 ε l=t l+1 ε · D−εl D P∞ l=t (l + 1) · (D−ε )l D = 1− 1 1− 16 2 D− ε ε3 (1−D−ε )2 (3) (4) (5) t 1 2t(D−ε ) ε (1−D−ε )2 D = (2) . (6) Equation (3) is derived using the fact that result of the following equality for x < 1: Q∞ i=1 (1 − xi ) ≥ 1 − P∞ i=1 xi and Equation (5) is a ∞ X xt (1 + t − tx) 2txt (l + 1)xl = < . (1 − x)2 (1 − x)2 l=t  One can see that for D = 7, maxε 1 2 D− ε ε3 (1−D−ε )2  < 0.9, and therefore step (3) is legal and (6) can be upper-bounded by a constant. Hence, L = Θ(1) and the proof is complete. Remarks on the alphabet size: Theorem 5.7 shows that for any ε > 0 there exists an εsynchronization string over alphabets of size O(ε−4 ). A polynomial dependence on ε is also necessary. In particular, there do not exist any ε-synchronization string over alphabets of size smaller than ε−1 . In fact, any consecutive substring of size ε−1 of an ε-synchronization string has to contain completely distinct elements. This can be easily proven as follows: For sake of contradiction let S[i, i + ε−1 ) be a substring of an ε-synchronization string where S[j] = S[j 0 ] for i ≤ j < j 0 < i + ε−1 . Then, ED (S[j], S[j + 1, j 0 + 1))) = j 0 −j −1 = (j 0 +1−j)−2 ≤ (j 0 +1−j)(1−2ε). We believe that using the Lovász Local Lemma together with a more sophisticated non-uniform probability space, which avoids any repeated symbols within a small distance, allows avoiding the use of the string T in our proof and improving the alphabet size to O(ε−2 ). It seems much harder to improved the alphabet size to o(ε−2 ) and we are not convinced that it is possible. This work thus leaves open the interesting question of closing the quadratic gap between O(ε−2 ) and Ω(ε−1 ) from either side. Theorem 5.7 also implies an efficient randomized construction. Lemma 5.8. There exists a randomized algorithm which for any ε > 0 constructs a εsynchronization string of length n over an alphabet of size O(ε−4 ) in expected time O(n5 ). Proof. Using the algorithmic framework for the Lovász local lemma given by Moser and Tardos [23] and the extensions by Haeupler et al. [16] one can get such a randomized algorithm from the proof in Theorem 5.7. The algorithm starts with a random string over any alphabet Σ of size ε−C for some sufficiently large C. It then checks all O(n2 ) intervals for a violation of the ε-synchronization string property. For every interval this is an edit distance computation which can be done in O(n2 ) time using the classical Wagner-Fischer dynamic programming algorithm. If a violating interval is found the symbols in this interval are assigned fresh random values. This is repeated until no more violations are found. [16] shows that this algorithm performs only O(n) expected number of re-samplings. This gives an expected running time of O(n5 ) overall, as claimed. Lastly, since synchronization strings can be encoded and decoded in a streaming fashion they have many important applications in which the length of the required synchronization string is not known in advance. In such a setting it is advantageous to have an infinite synchronization string over a fixed alphabet. In particular, since every consecutive substring of an ε-synchronization string is also an ε-synchronization string by definition, having an infinite ε-synchronization string also implies the existence for every length n, i.e., Theorem 5.7. Interestingly, a simple argument shows that the converse is true as well, i.e., the existence of an ε-synchronization string for every length n implies the existence of an infinite ε-synchronization string over the same alphabet: Lemma 5.9. For any ε ∈ (0, 1) there exists an infinite ε-synchronization string over an alphabet of size Θ(1/ε4 ). 17 Proof of Lemma 5.9. Fix any ε ∈ (0, 1). According to Theorem 5.7 there exist an alphabet Σ of size O(1/ε4 ) such that there exists an at least one ε-synchronization strings over Σ for every length n ∈ N. We will define a synchronization string S = s1 · s2 · s3 . . . with si ∈ Σ for any i ∈ N for which the ε-synchronization property holds for any i, j, k ∈ N. We define this string inductively. In particular, we fix an ordering on Σ and define s1 ∈ Σ to be the first symbol in this ordering such that an infinite number of ε-synchronization strings over Σ starts with s1 . Given that there is an infinite number of ε-synchronization over Σ such an s1 exists. Furthermore, the set of εsynchronization strings over Σ which start with s1 remains infinite by definition, allowing us to define s2 ∈ Σ to be the lexicographically first symbol in Σ such there exists an infinite number of ε-synchronization strings over Σ starting with s1 · s2 . In the same manner, we inductively define si to be the lexicographically first symbol in Σ for which there exists and infinite number of εsynchronization strings over Σ starting with s1 · s2 · . . . · si . To see that the infinite string defined in this manner does indeed satisfy the edit distance requirement of the ε-synchronization property defined in Definition 5.4, we note that for every i < j < k with i, j, k ∈ N there exists, by definition, an ε-synchronization string, and in fact an infinite number of them, which contains S[1, k] and thus also S[i, k] as a consecutive substring implying that indeed ED (S[i, j), S[j, k)) > (1 − ε)(k − i) as required. Our definition thus produces the unique lexicographically first infinite ε-synchronization string over Σ. We remark that any string produced by the randomized construction of Lemma 5.8 is guaranteed to be a correct ε-synchronization string (not just with probability one). This randomized synchronization string construction is furthermore only needed once as a pre-processing step. The encoder or decoder of any resulting error correcting codes do not require any randomization. Furthermore, in Section 6 we will provide a deterministic polynomial time construction of a relaxed version of ε-synchronization strings that can still be used as a basis for good (n, δ)-indexing algorithms thus leading to insdel codes with a deterministic polynomial time code construction as well. It nonetheless remains interesting to obtain fast deterministic constructions of finite and infinite ε-synchronization strings. In a subsequent work we achieve such efficient deterministic constructions for ε-synchronization strings. Our constructions even produce the infinite ε-synchronization string S proven to exist by Lemma 5.9, which is much less explicit: While for any n and any ε an εsynchronization string of length n can in principle be found using an exponential time enumeration there is no straight forward algorithm which follows the proof of Lemma 5.9 and given an i ∈ N produces the ith symbol of such an S in a finite amount of time (bounded by some function in i). Our constructions require significantly more work but in the end lead to an explicit deterministic construction of an infinite ε-synchronization string for any ε > 0 for which the ith symbol can be computed in only O(log i) time – thus satisfying one of the strongest notions of constructiveness that can be achieved. 5.2 Decoding We now provide an algorithm for decoding synchronization strings, i.e., an algorithm that can form a solution to the indexing problem along with ε-synchronization strings. In the beginning of Section 5, we introduced the notion of relative suffix distance between two strings. Theorem 5.5 stated a lower bound of 1 − ε for relative suffix distance between any two distinct codewords associated with an ε-synchronization string, i.e., its prefixes. Hence, a natural decoding scheme for detecting the index of a received symbol would be finding the prefix with the closest relative suffix distance to the string received thus far. We call this algorithm the minimum relative suffix distance 18 decoding algorithm. We define the notion of relative suffix error density at index i which presents the maximized density of errors taken place over suffixes of S[1, i]. We will introduce a very natural decoding approach for synchronization strings that simply works by decoding a received string by finding the codeword of a synchronization string S (prefix of synchronization string) with minimum distance to the received string. We will show that this decoding procedure works correctly as long as the relative suffix error density is not larger than 1−ε 2 . Then, we will show that if adversary is allowed to perform c many insertions or deletions, the relative suffix distance may exceed 1−ε 2 upon arrival 2c of at most 1−ε many successfully transmitted symbols. Finally, we will deduce that this decoding 2c scheme decodes indices of received symbols correctly for all but 1−ε many of successfully transmitted symbols. Formally, we claim that: Theorem 5.10. Any ε-synchronization string of length n along with the minimum relative suffix 2 distance decoding algorithm form a solution to (n, δ)-indexing problem that guarantees 1−ε nδ or less misdecodings. This decoding algorithm is streaming and can be implemented so that it works in O(n4 ) time. Before proceeding to the formal statement and the proofs of the claims above, we first provide the following useful definitions. Definition 5.11 (Error Count Function). Let S be a string sent over an insertion-deletion channel. We denote the error count from index i to index j with E(i, j) and define it to be the number of insdels applied to S from the moment S[i] is sent until the moment S[j] is sent. E(i, j) counts the potential deletion of S[j]. However, it does not count the potential deletion of S[i]. Definition 5.12 (Relative Suffix Error Density). Let string S be sent over an insertion-deletion channel and let E denote the corresponding error count function. We define the relative suffix error density of the communication as: E (|S| − i, |S|) max i≥1 i The following lemma relates the suffix distance of the message being sent by sender and the message being received by the receiver at any point of a communication over an insertion-deletion channel to the relative suffix error density of the communication at that point. Lemma 5.13. Let string S be sent over an insertion-deletion channel and the corrupted message S 0 be received on the other end. The relative suffix distance RSD(S, S 0 ) between the string S that was sent and the string S 0 which was received is at most the relative suffix error density of the communication. Proof. Let τ̃ = (τ̃1 , τ̃2 ) be the string matching from S to S 0 that characterizes insdels that have turned S into S 0 . Then: ED(S(|S| − k, |S|], S 0 (|S 0 | − k, |S 0 |]) k>0 2k minτ :S(|S|−k,|S|]→S 0 (|S 0 |−k,|S 0 |] {sc(τ1 ) + sc(τ2 )} = max k>0 2k 2(sc(τ10 ) + sc(τ20 )) ≤ max ≤ Relative Suffix Error Density k>0 2k RSD(S, S 0 ) = max (7) (8) (9) where τ 0 is τ̃ limited to its suffix corresponding to S(|S|−k, |S|]). Note that Steps (7) and (8) follow from the definitions of edit distance and relative suffix distance. Moreover, to see Step (9), one has 19 to note that one single insertion or deletion on the k-element suffix of a string may result into a string with k-element suffix of edit distance two of the original string’s k-element suffix; one stemming from the inserted/deleted symbol and the other one stemming from a symbol appearing/disappearing at the beginning of the suffix in order to keep the size of suffix k. A key consequence of Lemma 5.13 is that if an ε-synchronization string is being sent over an insertion-deletion channel and at some step the relative suffix error density corresponding to corruptions is smaller than 1−ε 2 , the relative suffix distance of the sent string and the received one at that point is smaller than 1−ε 2 ; therefore, as RSD of all pairs of codewords associated with an ε-synchronization string are greater than 1 − ε, the receiver can correctly decode the index of the corrupted codeword he received by simply finding the codeword with minimum relative suffix distance. The following lemma states that such a guarantee holds most of the time during transmission of a synchronization string: Lemma 5.14. Let ε-synchronization string S be sent over an insertion-channel channel and corrupted string S 0 be received on the other end. If there are ci symbols inserted and cd symbols deleted, then, for any integer t, the relative suffix error density is smaller than 1−ε t upon arrival of all but t(ci +cd ) 1−ε − cd many of the successfully transmitted symbols. Proof. Let E denote the error count function of the communication. We define the potential function Φ over {0, 1, · · · , n} as follows:   t · E(i − s, i) Φ(i) = max −s 1≤s≤i 1−ε Also, set Φ(0) = 0. We prove the theorem by showing the correctness of the following claims: 1. If E(i − 1, i) = 0, i.e., the adversary does not insert or delete any symbols in the interval starting right after the moment S[i − 1] is sent and ending at when S[i] is sent, then the value of Φ drops by 1 or becomes/stays zero, i.e., Φ(i) = max {0, Φ(i − 1) − 1}. 2. If E(i − 1, i) = k, i.e., adversary inserts or deletes k symbols in the interval starting right after the moment S[i − 1] is sent and ending at when S[i] is sent, then the value of Φ increases by tk tk at most 1−ε − 1, i.e., Φ(i) ≤ Φ(i − 1) + 1−ε − 1. 3. If Φ(i) = 0, then the relative suffix error density of the string that is received when S[i] arrives at the receiving side is not larger than 1−ε t . Given the correctness of claims made above, the lemma can be proved as follows. As adversary i +cd ) can apply at most ci + cd insertions or deletions, Φ can gain a total increase of t·(c1−ε . Therefore, the value of Φ can be non-zero for at most t·(ci +cd ) 1−ε many inputs. As value of Φ(i) is non-zero for t·(ci +cd ) − cd indices i where 1−ε t·(ci +cd ) − cd many of correctly 1−ε all i’s where S[i] has been removed by adversary, there are at most Φ(i) is non-zero and i is successfully transmitted. Hence, at most transmitted symbols can possibly be decoded incorrectly. We now proceed to the proof of each of the above claims to finish the proof: 20 1. In this case, E(i − s, i) = E(i − s, i − 1). So,   t · E(i − s, i) Φ(i) = max −s 1≤s≤i 1−ε   t · E(i − s, i − 1) −s = max 1≤s≤i 1−ε    t · E(i − s, i − 1) −s = max 0, max 2≤s≤i 1−ε    t · E(i − 1 − s, i − 1) = max 0, max −s−1 1≤s≤i−1 1−ε = max {0, Φ(i − 1) − 1} 2. In this case, E(i − s, i) = E(i − s, i − 1) + k. So,   t · E(i − s, i) −s Φ(i) = max 1≤s≤i 1−ε    tk t · E(i − s, i − 1) + tk = max − 1, max −s 2≤s≤i 1−ε 1−ε    tk t · E(i − 1 − s, i − 1) tk = max − 1, + max −s−1 1−ε 1 − ε 1≤s≤i−1 1−ε    tk t · E(i − 1 − s, i − 1) = − 1 + max 0, max −s 1≤s≤i−1 1−ε 1−ε tk = − 1 + max {0, Φ(i − 1)} 1−ε tk = Φ(i − 1) + −1 1−ε 3.  ⇒ ⇒ ⇒ ⇒  t · E(i − s, i) Φ(i) = max −s =0 1≤s≤i 1−ε t · E(i − s, i) ∀1 ≤ s ≤ i : −s≤0 1−ε ∀1 ≤ s ≤ i : t · E(i − s, i) ≤ s(1 − ε) ε E(i − s, i) ∀1 ≤ s ≤ i : ≤ s t   E(i − s, i) 1−ε Relative Suffix Error Density = max ≤ 1≤s≤i s t These finish the proof of the lemma. Now, we have all necessary tools to analyze the performance of the minimum relative suffix distance decoding algorithm: Proof of Theorem 5.10. As adversary is allowed to insert or delete up nδ symbols, by Lemma 5.14, 2nδ there are at most 1−ε successfully transmitted symbols during the arrival of which at the receiving 21 side, the relative suffix error density is greater than 1−ε 2 ; Hence, by Lemma 5.13, there are at most 2nδ 1−ε misdecoded successfully transmitted symbols. Further, we remark that this algorithm can be implemented in O(n4 ) as follows: Using dynamic programming, we can pre-process the edit distance of any consecutive substring of S, like S[i, j] to any consecutive substring of S 0 , like S 0 [i0 , j 0 ], in O(n4 ). Then, for each symbol of the received string, like S 0 [l0 ], we can find the codeword with minimum relative suffix distance to S 0 [1, l0 ] by calculating the relative suffix distance of it to all n codewords. Finding suffix distance of S 0 [1, l0 ] 0 (l0 −k,l0 ]) and a codeword like S[1, l] can also be simply done by minimizing ED(S(l−k,l],S over k which k can be done in O(n). With a O(n4 ) pre-process and a O(n3 ) computation as mentioned above, we have shown that the decoding process can be implemented in O(n4 ). We remark that by taking ε = o(1), one can obtain a solution to the (n, δ)-indexing problem with a misdecoding guarantee of 2nδ(1 + o(1)) which, using Theorem 4.1, results into a translation of nδ insertions and deletions into nδ(5 + o(1)) half-errors. As explained in Section 1.3, such a guarantee however falls short of giving Theorem 1.1. In Section 6.6, we show that this guarantee of the min-distance-decoder can be slightly improved to work beyond an RSD distance of 1−ε 2 , at the cost of some simplicity, by considering an alternative distance measure. In particular, the relative suffix pseudo distance RSPD, which was introduced in [2], can act as a metric stand-in for the minimum-distance decoder and lead to slightly improved decoding guarantees, despite neither being symmetric nor satisfying the triangle inequality. For any set of k = ki + kd insdel errors consisting of ki insertions and kd deletions the RSPD based indexing solution leads to at most (1 + ε)(3ki + kd ) half-errors which does imply “near-MDS” codes for deletion-only channels but still falls short for general insdel errors. This leaves open the intriguing question whether a further improved (pseudo) distance definition can achieve an indexing solution with negligible number of misdecodings for the minimum-distance decoder. 6 More Advanced Global Decoding Algorithms Thus far, we have introduced ε-synchronization strings as fitting solutions to the indexing problem. In Section 5.2, we provided an algorithm to solve the indexing problem along with synchronization strings with an asymptotic guarantee of 2nδ misdecodings. As explained in Section 1.3, such a guarantee falls short of giving Theorem 1.1. In this section, we thus provide a variety of more advanced decoding algorithms that provide a better decoding guarantees, in particular achieve a misdecoding fraction which goes to zero as ε goes to zero. We start by pointing out a very useful property of ε-synchronization strings in Section 6.1. We define a monotone matching between two strings as a common subsequence of them. We will next show that in a monotone matching between an ε-synchronization string and itself, the number of matches that both correspond to the same element of the string is fairly large. We will refer to this property as ε-self-matching property. We show that one can very formally think of this ε-self-matching property as a robust global guarantee in contrast to the factor-closed strong local requirements of the ε-synchronization property. One advantage of this relaxed notion of ε-selfmatching is that one can show that a random string over alphabets polynomially large in ε−1 satisfies this property (Section 6.2). This leads to a particularly simple generation process for S. Finally, showing that this property even holds for approximately log n-wise independent strings directly leads to a deterministic polynomial time algorithm generating such strings as well. In Section 6.3, we propose a decoding algorithm for insdel errors that basically works by finding monotone matchings between the received string and the synchronization string. Using the ε-self22 √ matching property we show that this algorithm guarantees O (n ε) misdecodings. This algorithm √ works in time O(n2 / ε) and is exactly what we need to prove our main theorem. Lastly, in Sections 6.4 and 6.5 we provide two simpler linear time algorithms that solve the indexing problem under the assumptions that the adversary can only delete symbols or only insert ε new symbols. These algorithms not only guarantee asymptotically optimal 1−ε nδ misdecodings but are also error-free. Table 1 provides a break down of the decoding schemes presented in this paper, describing the type of error they work under, the number of misdecodings they guarantee, whether they are error-free or streaming, and their decoding complexity. 6.1 Monotone Matchings and the ε-Self Matching Property Before proceeding to the main results of this section, we start by defining monotone matchings which provide a formal way to refer to common substrings of two strings: Definition 6.1 (Monotone Matchings). A monotone matching between S and S 0 is a set of pairs of indices like: M = {(a1 , b1 ), · · · , (am , bm )} where a1 < · · · < am , b1 < · · · < bm , and S[ai ] = S 0 [bi ]. We now point out a key property of synchronization strings that will be broadly used in our decoding algorithms. Basically, Theorem 6.2 states that two similar subsequences of an ε-synchronization string cannot disagree on many positions. More formally, let M = {(a1 , b1 ), · · · , (am , bm )} be a monotone matching between S and itself. We call the pair (ai , bi ) a good pair if ai = bi and a bad pair otherwise. Then: Theorem 6.2. Let S be an ε-synchronization string of size n and M = {(a1 , b1 ), · · · , (am , bm )} be a monotone matching of size m from S to itself containing g good pairs and b bad pairs. Then, b ≤ ε(n − g) Proof. Let (a01 , a02 ), · · · , (a0m0 , b0m0 ) indicate the set of bad pairs in M indexed as a01 < · · · < a0m0 and b01 < · · · < b0m0 . Without loss of generality, assume that a01 < b01 . Let k1 be the largest integer such that a0k1 < b01 . Then, the pairs (a01 , a02 ), · · · , (a0k1 , b0k1 ) form a common substring of size k1 between T1 = S[a01 , b01 ) and T10 = S[b01 , b0k1 ]. Now, the synchronization string guarantee implies that: k1 ≤ LCS(T1 , T10 ) |T1 | + |T10 | − ED (T1 , T10 ) ≤ 2 ε(|T1 | + |T10 |) ≤ 2 Note that the monotonicity of the matching guarantees that there are no good matches occurring on indices covered by T1 and T10 , i.e., a01 , · · · , b0k1 . One can repeat very same argument for the remaining bad matches to rule out bad matches (a0k1 +1 , b0k1 +1 ), · · · , (a0k1 +k2 , b0k1 +k2 ) for some k2 having the following inequality guaranteed: k2 ≤ where ( ε(|T2 | + |T20 |) 2 (10) T2 = [a0k1 +1 , b0k1 +1 ) and T20 = [b0k1 +1 , b0k1 +k2 ] a0k1 +1 < b0k1 +1 T2 = [b0k1 +1 , a0k1 +1 ) and T20 = [a0k1 +1 , a0k1 +k2 ] a0k1 +1 > b0k1 +1 23 𝑆 𝑆 𝑎′1 𝑎′𝑘1 𝑎′𝑘1+𝑘2 𝑎′𝑘1+1 𝑇1 𝑆 𝑇2 𝑇′1 𝑏′1 𝑏′𝑘1 +1 𝑏′𝑘1 +𝑘2 𝑎′𝑘1+1 𝑎′𝑘1+𝑘2 𝑎′𝑘1 𝑇1 𝑆 𝑇′2 𝑏′𝑘1 𝑎′1 𝑇2 𝑇′1 𝑏′1 (a) a0k1 +1 < b0k1 +1 𝑇′2 𝑏′𝑘1 𝑏′𝑘1 +1 𝑏′𝑘1 +𝑘2 (b) a0k1 +1 > b0k1 +1 Figure 1: Pictorial representation of T2 and T20 For a pictorial representation see Figure 1. Continuing the same procedure, one can find k1 , · · · , kl , T1 , · · · , Tl , and T10 , · · · , Tl0 for some l. Summing up all inequalities of form (10), we will have: ! l l l X X X ε 0 |Ti | + ki ≤ · |Ti | (11) 2 i=1 i=1 i=1 Pl Note that i=1 ki = u and Ti s are mutually exclusive and contain no indices where a good pair P P occurs at. Same holds for Ti0 s. Hence, li=1 |Ti | ≤ n − g and li=1 |Ti0 | ≤ n − g. All these along with (11) give that: ε u ≤ · 2 (n − g) = ε(n − g) ⇒ n − g − b ≥ (1 − ε)(n − g) ⇒ b ≤ ε(n − g) 2 We define the ε-self-matching property as follows: Definition 6.3 (ε-self-matching property). String S satisfies ε-self-matching property if any monotone matching between S and itself contains less than ε|S| bad pairs. Note that ε-synchronization property concerns all substrings of a string while the ε-self-matching property only concerns the string itself. Granted that, we now show that ε-synchronization property and satisfying ε-self-matching property on all substrings are equivalent up to a factor of two: Theorem 6.4. ε-synchronization and ε-self matching properties are related as follows: a) If S is an ε-synchronization string, then all substrings of S satisfy ε-self-matching property. b) If all substrings of string S satisfy the 2ε -self-matching property, then S is ε-synchronization string. Proof of Theorem 6.4 (a). This part is a straightforward consequence of Theorem 6.2. Proof of Theorem 6.4 (b). Assume by contradiction that there are i < j < k such that ED(S[i, j), S[j, k)) ≤ (1 − ε)(k − i). Then, LCS(S[i, j), S[j, k)) ≥ k−i−(1−ε)(k−i) = 2ε (k − i). The 2 corresponding pairs of such longest common substring form a monotone matching of size 2ε (k − i) which contradicts 2ε -self-matching property of S. As a matter of fact, the decoding algorithms we will propose for ε-synchronization strings in Sections 6.3, 6.4, and 6.5 only make use of the ε-self-matching property of the ε-synchronization string. We now proceed to the definition of ε-bad-indices which will enable us to show that ε-self matching property, as opposed to the ε-synchronization property, is robust against local changes. 24 Definition 6.5 (ε-bad-index). We call index k of string S an ε-bad-index if there exists a factor S[i, j] of S with i ≤ k ≤ j where S[i, j] does not satisfy the ε-self-matching property. In this case, we also say that index k blames interval [i, j]. Using the notion of ε-bad indices, we now present Lemma 6.6. This lemma suggests that a string containing limited fraction of ε-bad indices would still be an ε0 -self matching string for some ε0 > ε. An important consequence of this result is that if one changes a limited number of elements in a given ε-self matching string, the self matching property will be essentially preserved to a lesser extent. Note that ε-synchronization property do not satisfy any such robustness quality. Lemma 6.6. If the fraction of ε-bad indices in string S is less than γ, then S satisfies (ε + 2γ)-self matching property. Proof. Consider a matching from S to itself. The number of bad matches whose both ends refer to non-ε-bad indices of S is at most |S|(1 − γ)ε by definition. Further, each ε-bad index can appear at most once in each end of bad pairs. Therefore, the number of bad pairs in S can be at most: |S|(1 − γ)ε + 2|S|γ ≤ |S|(ε + 2γ) which, by definition, implies that S satisfies the (ε + 2γ)-self-matching property. On the other hand, in the following theorem, we will show that within a given ε-self matching string, there can be a limited number of ε0 -bad indices for sufficiently large ε0 > ε. Lemma 6.7. Let S be an ε-self matching string of length n. Then, for any 3ε < ε0 < 1, at most 3nε 0 ε0 many indices of S can be ε -bad. Proof. Let s1 , s2 , · · · , sk be bad indices of S and ε0 -bad index si blame substring S[ai , bi ). As intervals S[ai , bi ) are supposed to be bad, there has to be a ε0 -self matching within each S[ai , bi ) like Mi for which |Mi | ≥ ε0 · |[ai , bi )|. We claim that one can choose a subset of [1..k] like I for which • Corresponding intervals to the indices in I are mutually exclusive. In other words, for any i, j ∈ I where i 6= j, [Ai , bi ) ∩ [aj , bj ) = ∅. P k • i∈I |[Ai , bi )| ≥ 3 . S 0 If such I exists, one can take i∈I Mi as a self matching in S whose size is larger than kε3 . As S is an ε-self matching string, kε0 3nε ≤ nε ⇒ k ≤ 0 3 ε which finishes the proof. The only remaining piece is proving the claim. Note that any index in S 0 0 [a i∈I i , bi ) is S a ε -bad index as they by definition belong to an interval with a ε -self matching. Therefore, i∈I [ai , bi ) = k. In order to find set I, we greedily choose the largest substring [ai , bi ), put its corresponding index into I and then remove any interval intersecting [ai , bi ). We continue repeating this procedure until all substrings are removed. The set I obtained by this procedure clearly satisfies the first claimed property. Moreover, note that if li = |[ai , bi )|, any interval intersecting [ai , bi ) falls into [ai − li , bi + li ) which is an interval of length 3li . This certifies the second property and finishes the proof. As the final remark on the ε-self matching property and its relation with the more strict εsynchronization property, we show that using the minimum RSD decoder for indexing together with an ε-self matching string leads to guarantees on the misdecoding performance which are only 25 slightly weaker than the guarantee obtained by ε-synchronization strings. In order to do so, we first show that the (1 − ε) RSD distance property of prefixes holds for any non-ε-bad index in any arbitrary string in Theorem 6.8. Then, using Theorem 6.8 and Lemma 6.7, we upper-bound the number of misdecodings that may happen using a minimum RSD decoder along with an ε-self matching string in Theorem 6.9. Theorem 6.8. Let S be an arbitrary string of length n and 1 ≤ i ≤ n be such that i’th index of S is not an ε-bad index. Then, for any j 6= i, RSD(S[1, i], S[1, j]) > 1 − ε. Proof. Without loss of generality assume that j < i. Consider the interval [2j − i + 1, i]. As i is not an ε-bad index, there is no self matching of size 2ε(i − j) within [2j − i, i]. In particular, the edit distance of S[2j − i + 1, j] and [j + 1, i] has to be larger than (1 − ε) · 2(i − j) which equivalently means RSD(S[1, i], S[1, j]) > 1 − ε. Note that if 2j − i + 1 < 0 the proof goes through by simply replacing 2j − i + 1 with zero. Theorem 6.9. Using any ε-self matching string along with minimum RSD algorithm, one can solve the (n, δ)-indexing problem with a guarantee of n(4δ + 6ε) misdecodings. Proof. Note that applying Lemma 6.7 for ε0 gives that there are at most 3nε ε0 indices in S that are 2nδ 0 ε -bad. Further, using Theorems 5.10 and 6.8, at most 1−ε0 many of the other indices might be decoded incorrectly  arrivals. Therefore, this solution for the (n, δ)-indexing problem can  upon their 2δ 3ε 3ε gives an upper contain at most n ε0 + 1−ε0 many incorrectly decoded indices. Setting ε0 = 3ε+2δ bound of n(4δ + 6ε) on the number of misdecodings. 6.2 Efficient Polynomial Construction of ε-self matching strings In this section, we will use Lemma 6.6 to show that there is a polynomial deterministic construction of a string of length n with the ε-self-matching property, which can then for example be used to obtain a deterministic code construction. We start by showing that even random strings satisfy the ε-selfmatching property for an ε polynomial in the alphabet size: Theorem 6.10. A random string on an alphabet of size O(ε−3 ) satisfies ε-selfmatching property with a constant probability. Proof. Let S be a random string on alphabet Σ of size |Σ| = ε−3 . We are going to find the expected number of ε-bad indices in S. We first count the expected number of ε-bad indices that blame intervals of length 2ε or smaller. If index k blames interval S[i, j] where j − i < 2ε−1 , there has to be two identical symbols appearing in S[i, j] which gives that there are two identical elements in 4ε−1 neighborhood of S. Therefore, the probability of index k being ε-bad blaming S[i, j] for −1
1 j − i < 2ε−1 can be upper-bounded by 4ε2 |Σ| ≤ 8ε. Thus, the expected fraction of ε-bad indices that blame intervals of length 2ε or smaller is less than 8ε. We now proceed to finding the expected fraction of ε-bad indices in S blaming intervals of length 2ε−1 or more. Since every interval of length l which does not satisfy ε-self-matching property causes at most l ε-bad indices, we get that the expected fraction of such indices, i.e., γ 0 , is at most: 26 0 E[γ ] = n n 1 X X l · Pr[S[i, i + l) does not satisfy ε-self-matching property] n −1 l=1 i=2ε = n X l · Pr[S[i, i + l) does not satisfy ε-self-matching property] l=2ε−1 ≤ n X l=2ε−1 l  2 l 1 lε |Σ|lε (12)
2 Last inequality holds because the number of possible matchings is at most lεl . Further, fixing the matching edges, the probability of the elements corresponding to pair (a, b) of the matching being identical is independent from all pairs (a0 , b0 ) where a0 < a and b0 < b. Hence, the probability of the set of pairs being a matching between random string S and itself is |Σ|1lε . Then, E[γ 0 ] ≤ n X  le lε l e p ε |Σ| l=2ε−1 ≤ n X l=2ε−1 ≤ 0 −1 2ε−1 x2ε E[γ ] ≤ 8ε −1 1 |Σ|lε !2εl  ∞ X e p ε |Σ| l l=2ε−1 P l Note that series ∞ l=2ε−1 lx = P∞ l −1 2ε−1 . So, l=l0 lx < 8ε x 2lε l −1 +1 −(2ε−1 −1)x2ε (1−x)2 e p 2ε |Σ| !4εε−1 = !2ε l  for |x| < 1. Therefore, for 0 < x < 1 2, e4 e4 −5 1 ε ≤ ε 2 |Σ|2 2 Using Lemma 6.6, this random structure has to satisfy (ε + 2γ)-self-matching property where E[ε + 2γ] = ε + 16ε + e4 ε = O(ε) Therefore, using Markov inequality, a randomly generated string over alphabet O(ε−3 ) satisfies ε-matching property with constant probability. The constant probability can be as high as one wishes by applying higher constant factor in alphabet size. As the next step,  we prove a similar claim for strings of length n whose symbols are chosen log n from an Θ log(1/ε) -wise independent [24] distribution over a larger, yet still ε−O(1) size, alphabet. This is the key step in allowing for a derandomization using the small sample spaces of Naor and Naor [24]. The proof of Theorem 6.11 follows a similar strategy as was used in [3] to derandomize the constructive Lovász local lemma. In particular the crucial idea, given by Claim 6.12, is to show that for any large obstruction there has to exist a smaller yet not too small obstruction. This allows one to prove that in the absence of any small and medium size obstructions no large obstructions exist either. 27 c log n Theorem 6.11. A log(1/ε) -wise independent random string of size n on an alphabet of size O(ε−6 ) satisfies ε-matching property with a non-zero constant probability. c is a sufficiently large constant. Proof. Let S be a pseudo-random string of length n with c log n log(1/ε) -wise independent symbols. Then, c log n Step (12) is invalid as the proposed upper-bound does not work for l > ε log(1/ε) . To bound the   c log n probability of intervals of size Ω ε log(1/ε) not satisfying ε-self matching property, we claim that: Claim 6.12. Any string of size l > 100m which contains an ε-self-matching contains two subintervals I1 and I2 of size m where there is a matching of size 0.99 mε 2 between I1 and I2 . c log n Using Claim 6.12, one can conclude that any string of size l > 100 ε log(1/ε) which contains an ε-self-matching contains two sub-intervals I1 and I2 of size c log n ε log(1/ε) where there is a matching of c log n 2 log(1/ε) size between I1 and I2 . Then, Step (12) can be revised by upper-bounding the probability of a long interval having an ε-self-matching by a union bound over the probability of pairs of its c log n subintervals having a dense matching. Namely, for l > 100 ε log(1/ε) , let us denote the event of S[i, i + l) containing a ε-self-matching by Ai,l . Then,   c log n c log n and LCS(I1 , I2 ) ≥ 0.99 Pr[Ai,l ] ≤ Pr S contains I1 , I2 : |Ii | = ε log(1/ε) 2 log(1/ε)  −1 2   c log n ε c log n/ log(1/ε) 1 2 log(1/ε) ≤ n2 0.99c log n/2 log(1/ε) |Σ| 2×0.99c log n 6c log n
≤ n2 2.04eε−1 2 log(1/ε) ε 2 log(1/ε) = n2 (2.04e) = n 2+ 0.99c log n log(1/ε) c ln(2.04e) −2.01c log(1/ε) 4.02c log n ε 2 log(1/ε)   0 < n2−c/4 = O n−c where first inequality follows from the fact there can be at most n2 pairs of intervals of size c log n c log n ε log(1/ε) in S and the number of all possible matchings of size log(1/ε) between them is at most ε−1 c log n/ log(1/ε)
2 . Further, for small enough ε, constant c0 can be as large as one desires by c log n/2 log(1/ε) setting constant c large enough. Thus, Step (12) can be revised as:  !2ε l 100c log n/(ε log(1/ε)) n X X e  + E[γ 0 ] ≤ l· p l Pr[Ai,l ] ε |Σ| −1 l=100c log n/(ε log(1/ε)) l=ε l  ! 2ε ∞ X e  + n2 · O(n−c0 ) ≤ O(ε) + O(n2−c0 )  p ≤ l· ε |Σ| l=ε−1 For an appropriately chosen c, 2 − c0 < 0; hence, the later term vanishes as n grows. Therefore, the log n conclusion E[γ] ≤ O(ε) holds for the limited log(1/ε) -wise independent string as well. Proof of Claim 6.12. Let M be a self-matching of size lε or more between S and itself containing only bad edges. We chop S into ml intervals of size m. On the one hand, P the size of M is greater than lε and on the other hand, we know that the size of M is exactly i,j |Ei,j | where Ei,j denotes the number of edges between interval i and j. Thus: P X ε i,j |Ei,j |/m lε ≤ |Ei,j | ⇒ ≤ 2 2l/m i,j 28 Note that |Ei,j | m represents the density of edges between interval i and interval j. Further, Since M |E | 2l is monotone, there are at most m intervals for which |Ei,j | = 6 0 and subsequently mi,j 6= 0. Hence, 2l on the right hand side we have the average of m many non-zero terms which is greater than ε/2. 0 0 So, there has to be some i and j for which: |Ei0 ,j 0 | ε mε ≤ ⇒ ≤ |Ei0 ,j 0 | 2 m 2 To analyze more accurately, if l is not divisible by m, we simply throw out up to m last elements ε of the string. This may decrease ε by ml < 100 . Note that using the polynomial sample spaces of [24] Theorem 6.11 directly leads to a deterministic algorithm for finding a string of size n with ε-self-matching property. For this one simply c log n checks all possible points in the sample space of the log(1/ε) -wise independent strings and finds a string S with γS ≤ E[γ] = O(ε). In  using brute-force, one can find a string satisfying  other words, c log n O(ε)-self-matching property in O |Σ| log(1/ε) = nO(1) . Theorem 6.13. There is a deterministic algorithm running in nO(1) that finds a string of length n satisfying ε-self-matching property over an alphabet of size O(ε−6 ). 6.3 Insdel Errors Now, we provide an alternative indexing algorithm to be used along with ε-synchronization strings. Throughout the following sections, we let ε-synchronization string S be sent as the synchronization string in an instance of (n, δ)-indexing problem and string S 0 be received at the receiving end being affected by up to nδ insertions or deletions. Furthermore, let di symbols be inserted into the communication and dr symbols be deleted from it. The algorithm works as follows. On the first round, the algorithm finds the longest common subsequence between S and S 0 . Note that this common subsequence corresponds to a monotone matching M1 between S and S 0 . On the next round, the algorithm finds the longest common subsequence between S and the subsequence of unmatched elements of S 0 (those that have not appeared in M1 ). This common subsequence corresponds to a monotone matching between S and the elements of S 0 that do not appear in M1 . The algorithm repeats this procedure β1 times to obtain M1 , · · · , M1/β where β is a parameter that we will fix later. In the output of this algorithm, S 0 [ti ] is decoded as S[i] if and only if S[i] is only matched to 0 S [ti ] in all M1 , · · · , M1/β . Note that the longest common subsequence of two strings of length O(n) can
be found in O(n2 ) using dynamic programming. Therefore, the whole algorithm runs in O n2 /β . Now we proceed to analyzing the performance of the algorithm by bounding the number of misdecodings. Theorem 6.14. This decoding algorithm guarantees a maximum misdecoding count of (n + di − √ √ dr )β + βε n. More specifically, for β = ε, the number misdecodings will be O (n ε) and running √
time will be O n2 / ε . Proof. First, we claim that at most (n + di − dr )β many of the symbols that have been successfully transmitted are not matched in any of M1 , · · · , M1/β . Assume by contradiction that more than (n + di − dr )β of the symbols that pass through the channel successfully are not matched in any of M1 , · · · , M1/β . Then, there exists a monotone matching of size greater than (n + di − dr )β 29 between the unmatched elements of S 0 and S after β1 rounds of finding longest common substrings. Hence, size of any of Mi s is at least (n + di − dr )β. So, the summation of their sizes exceeds (n + di − dr )β × β1 = n + di − dr = |S 0 | which brings us to a contradiction. Furthermore, as a result of Theorem 6.4, any of Mi s contain at most εn many incorrectly matched elements. Hence, at least βε n many of the matched symbols are matched to incorrect index. Hence, the total number of misdecodings can be bounded by (n + di − dr )β + βε n. 6.4 Deletion Errors Only We now introduce a very simple linear time streaming algorithm that decodes a received synchronization string of length n which can be affected by up to nδ many deletions. Our scheme is ε guaranteed to have less than 1−ε · nδ misdecodings. Before proceeding to the algorithm description, let dr denote the number of symbols removed by adversary. As adversary is restricted to symbol deletion, each symbol received at the receiver corresponds to a symbol sent by the sender. Hence, there exists a monotone matching of size |S 0 | = n0 = n − dr like M = {(t1 , 1), (t2 , 2), · · · , (tn−dr , n − dr )} between S and S 0 which matches each of the received symbols to their actual indices. Our simple streaming algorithm greedily matches S to the left-most possible subsequence of S. To put it another words, the algorithm matches S 0 [1] to S[t01 ] where S[t01 ] = S 0 [1] and t01 is as small as possible, then matches S 0 [2] to the smallest t02 > t01 where S[t02 ] = S 0 [2] and construct the whole matching M 0 by repeating this procedure. Note that as there is a matching of size |S 0 | between S and S 0 , the size of M 0 will be |S 0 | too. This algorithm clearly works in a streaming manner and runs in linear time. To analyze the performance, we basically make use of the fact that M and M 0 are both monotone matchings of size |S 0 | between S and S 0 . Therefore, M̄ = {(t1 , t01 ), (t2 , t02 ), · · · , (tn−dr , t0n−dr )} is a monotone matching between S and itself. Note that if ti 6= t0i , then the algorithm has decoded the index ti incorrectly. Let p be the number of indices i where ti 6= t0i . Then matching M̄ consists of n − dr − p good pairs and p bad pairs. Therefore, using Theorem 6.2 n − (n − dr − p) − p > (1 − ε)(n − (n − dr − p)) ⇒ dr > (1 − ε)(dr + p) ⇒ p < ε · dr 1−ε This proves the following theorem: Theorem 6.15. Any ε-synchronization string along with the algorithm described in Section 6.4 ε form a linear-time streaming solution for deletion-only (n, δ)-indexing problem guaranteeing 1−ε ·nδ misdecodings. 6.5 Insertion Errors Only We now depart to another simplified case where adversary is restricted to only insert symbols. We propose a decoding algorithm whose output is guaranteed to be error-free and contain less than nδ 1−ε misdecodings. Assume that di symbols are inserted into the string S to turn it in into S 0 of size n + di on the receiving side. Again, it is clear that there exists a monotone matching M of size n like M = {(1, t1 ), (2, t2 ), · · · , (n, tn )} between S and S 0 that matches each symbol in S to its actual index when it arrives at the receiver. The decoding algorithm we present, matches S[i] to S 0 [t0i ] in its output, M 0 , if and only if in all possible monotone matchings between S and S 0 that saturate S (i.e., are of size |S| = n), S[i] 30 is matched to S 0 [t0i ]. Note that any symbol S[i] that is matched to S 0 [t0i ] in M 0 has to be matched to the same element in M ; therefore, the output of this algorithm does not contain any incorrectly decoded indices; therefore, the algorithm is error-free. Now, we are going to first provide a linear time approach to implement this algorithm and then di on the number of misdecodings. To this end, we make use of the prove an upper-bound of 1−ε following lemma: Lemma 6.16. Let ML = {(1, l1 ), (2, l2 ), · · · , (n, ln )} be a monotone matching between S and S 0 such that l1 , · · · , ln has the smallest possible value lexicographically. We call ML the left-most matching between S and S 0 . Similarly, let MR = {(1, r1 ), · · · , (n, rn )} be the monotone matching such that rn , · · · , r1 has the largest possible lexicographical value. Then S[i] is matched to S 0 [t0i ] in all possible monotone matchings of size n between S and S 0 if and only if (i, t0i ) ∈ MR ∩ ML . This lemma can be proved by a simple contradiction argument. Our algorithm starts by computing left-most and right-most monotone matchings between S and S 0 using the trivial greedy algorithm introduced in Section 6.4 on (S, S 0 ) and them reversed. It then outputs the intersection of these two matching as the answer. This algorithm clearly runs in linear time. To analyze this algorithm, we bound the number of successfully transmitted symbols that the algorithm refuses to decode, denoted by p. To bound the number of such indices, we make use of the fact that n − p elements of S 0 are matched to the same element of S in both ML and MR . As there are p elements in S that are matched to different elements in S 0 and there is a total of n + di elements in S 0 , there has to be at least 2p − [(n + di ) − (n − p)] = p − di elements in S 0 who are matched to different elements of S in ML and MR . Consider the following monotone matching from S to itself as follows: M = {(i, i) : If S[i] is matched to the same position of S 0 in both M and M 0 } ∪ {(i, j) : ∃k s.t. (i, k) ∈ ML , (j, k) ∈ MR } Note that monotonicity follows the fact that both ML and MR are both monotone matchings between S and S 0 . We have shown that the size of the second set is at least p − di and the size of the first set is by definition n − p. Also, all pairs in the first set are good pairs and all in the second one are bad pairs. Therefore, by Theorem 6.2: (n − (n − p) − (p − di )) > (1 − ε)(n − (n − p)) ⇒ p < di 1−ε which proves the efficiency claim and gives the following theorem. Theorem 6.17. Any ε-synchronization string along with the algorithm described in Section 6.5 1 form a linear-time error-free solution for insertion-only (n, δ)-indexing problem guaranteeing 1−ε ·nδ misdecodings. Finally, we remark that a similar non-streaming algorithm can be applied to the case of deletiononly errors. Namely, one can compute the left-most and right-most matchings between the received string and string that is supposed to be received and output the common edges. By a similar argument as above, one can prove the following: Theorem 6.18. Any ε-synchronization string along with the algorithm described in Section 6.5 ε form a linear-time error-free solution for deletion-only (n, δ)-indexing problem guaranteeing 1−ε ·nδ misdecodings. 31 In the same manner as Theorem 1.1, we can derive the following theorem concerning deletiononly and insertion-only codes based on Theorems 6.17 and 6.18. Theorem 6.19. For any ε > 0 and δ ∈ (0, 1): • There exists an encoding map E : Σk → Σn and a decoding map D : Σ∗ → Σk such that if x is a subsequence of E(m) where |x| ≥ n − nδ then D(x) = m. Further nk > 1 − δ − ε, |Σ| = f (ε), and E and D are explicit and have linear running times in n. • There exists an encoding map E : Σk → Σn and a decoding map D : Σ∗ → Σk such that if E(m) is a subsequence of x where |x| ≤ n + nδ then D(x) = m. Further nk > 1 − δ − ε, |Σ| = f (ε), and E and D are explicit and have linear running times in n. Finally, we remark that since indexing solutions offered in Theorems 6.17 and 6.18 are error free, it suffices to use good erasure codes along with synchronization strings to obtain Theorem 6.19. 6.6 Decoding Using the Relative Suffix Pseudo-Distance (RSPD) In this section, we show how one can slightly improve the constants in the results obtained in Section 5.2 by replacing RSD with a related notion of “distance” between two strings introduced in [2]. We call this notion relative suffix pseudo-distance or RSPD both to distinguish it from our RSD relative suffix distance and also because RSPD is not a metric distance per se – it is neither symmetric nor satisfies the triangle inequality. Definition 6.20 (Relative Suffix Pseudo-Distance (RSPD)). Given any two strings c, c̃ ∈ Σ∗ , the suffix distance between c and c̃ is    |τ1 | sc (τ1 [i, |τ1 |]) + sc (τ2 [i, |τ2 |]) RSP D (c, c̃) = min max i=1 τ :c→c̃ |τ1 | − i + 1 − sc (τ1 [i, |τ1 |]) We derive our algorithms by proving a useful property of synchronization strings: LemmaS6.21. Let S ∈ Σn be an ε-synchronization string and c̃ ∈ Σm . Then there exists at most one c ∈ ni=1 S[1..i] such that RSP D(c, c̃) ≤ 1 − ε. Before proceeding to the proof of Lemma 6.21, we prove the following lemma: Lemma 6.22. Let RSP D(S, T ) ≤ 1 − ε, then: 1. For every 1 ≤ s ≤ |S|, there exists t such that ED (S[s, |S|], T [t, |T |]) ≤ (1 − ε)(|S| − s + 1). 2. For every 1 ≤ t ≤ |T |, there exists s such that ED (S[s, |S|], T [t, |T |]) ≤ (1 − ε)(|S| − s + 1). Proof. Part 1 Let τ be the string matching chosen in RSP D(S, T ). There exist some r such that del(τ1 [r, |τ1 |]) = S[s, |S|]. Note that del(τ2 [r, |τ2 |]) is a suffix of T . Therefore, there exists some t such that T [t, |T |] = del(τ2 [r, |τ2 |]). Now, ED(S[s, |S|], T [t, |T |]) ≤ sc(del(τ1 [r, |τ1 |])) + sc(del(τ2 [r, |τ1 |])) sc(del(τ1 [r, |τ1 |])) + sc(del(τ2 [r, |τ1 |])) = · (|τ1 | − r + 1 − sc(τ1 [r, |τ1 |])) |τ1 | − r + 1 − sc(τ1 [r, |τ1 |]) ≤ RSP D(S, T ) · (|S| − s + 1) ≤ (1 − ε) · (|S| − s + 1) 32 (13) 𝑠 𝑆 𝑟 ≈ 𝜏1 𝜏2 𝑟 𝑇 𝑡 Figure 2: Pictorial representation of the notation used in Lemma 6.22 Part 2 Similarly, let τ be the string matching chosen in RSP D(S, T ). There exists some r such that del(τ2 [r, |τ2 |]) = T [t, |T |]. Now, del(τ1 [r, |τ1 |]) is a suffix of S. Therefore, there exists some s such that S[s, |S|] = del(τ1 [r, |τ1 |]). Now, all the steps we took to prove equation (13) hold and the proof is complete. Algorithm 3 Synchronization string decode Input: A received message c̃ ∈ Σm and an ε-synchronization string S ∈ Σn 1: ans ← ∅ 2: for Any prefix c of S do |τ1 | sc(τ1 [k..|τ1 |])+sc(τ2 [k..|τ2 |]) 3: d[i][j][l] ← minτ :c(i)→c̃(j) maxk=1 |τ1 |−k+1+sc(τ1 [k..|τ1 |]) sc(τ1 )=l |c̃| RSP D(c, c̃) ← minl0 =0 d[i][|c̃|][l0 ] if RSP D(c, c̃) ≤ 1 − ε then ans ← c Output: ans 4: 5: 6: Proof of Lemma 6.21. For a contradiction, suppose that there exist a c̃, l and l0 such that l < l0 and RSP D(S[1, l], c̃) ≤ 1 − ε and RSP D(S[1, l0 ], c̃) ≤ 1 − ε. Now, using part 1 of Lemma 6.22, there exists k such that ED (S[l + 1, l0 ], c̃[k, |c̃|]) ≤ (1 − ε)(l0 − l). Further, part 2 of Lemma 6.22 gives that there exist l00 such that ED (S[l00 + 1, l], c̃[k, |c̃|]) ≤ (1 − ε)(l − l00 ). Hence, ED(S[l + 1, l0 ], S[l0 + 1, l00 ]) ≤ ED(S[l + 1, l0 ], c̃[k, |c̃|]) + ED(S[l0 + 1, l00 ], c̃[k, |c̃|]) ≤ (1 − ε)(l0 − l00 ) which contradicts the fact that S is an ε-synchronization string. Lemma 6.21 implies a natural algorithm for decoding c̃: simply search over all prefixes of S for the one with small enough suffix distance from c̃. We prove that this is possible in O(n5 ) via dynamic programming. Theorem 6.23. Let S ∈ Σn be an ε-synchronization string, and c̃ ∈ Σm . Then Algorithm 3, given input S and c̃, either returns the unique prefix c of S such that RSP D(c, c̃) ≤ 1 − ε or returns ∅ if 33 no such prefix exists. Moreover, Algorithm 3 runs in time O(n5 ); spending O(n4 ) for each received symbol. Proof. To find c, we calculate the RSPD of c̃ and all prefixes of S one by one. We only need to show that the RSPD of two strings of length at most n can be found in O(n3 ). We do this using dynamic programming. Let us try to find RSP D(s, t). Further, let s(i) represent the suffix of s of length i and t(j) represent the suffix of t of length j. Now, let d[i][j][l] be the minimum string matching (τ1 , τ2 ) from s(i) to t(j) such that sc(τ1 ) = l. In other words, d[i][j][l] = min |τ1 | max τ :s(i)→t(j) k=1 sc(τ1 )=l sc (τ1 [k.. |τ1 |]) + sc (τ2 [k.. |τ2 |]) , |τ1 | − k + 1 + sc (τ1 [k.. |τ1 |]) where τ is a string matching for s(i) and t(j). Note that for any τ : s(i) → t(j), one the following three scenarios might happen: 1. τ1 (1) = τ2 (1) = s (|s| − (i − 1)) = t(|t| − (j − 1)): In this case, removing the first elements of τ1 and τ2 gives a valid string matching from s(i − 1) to t(j − 1). 2. τ1 (1) = ∗ and τ2 (1) = t(|t| − (j − 1)): In this case, removing the first element of τ1 and τ2 gives a valid string matching from s(i) to t(j − 1). 3. τ2 (1) = ∗ and τ1 (1) = s(|s| − (i − 1)): In this case, removing the first element of τ1 and τ2 gives a valid string matching from s(i − 1) to t(j). This implies that ( d[i][j][l] = min d[i − 1][j − 1][l] (Only if s(i) = t(j)),  l + (j − (i − l)) , max d[i][j − 1][l − 1], (i + l) + l )   l + (j − (i − l)) max d[i − 1][j][l], . (i + l) + l  Hence, one can find RSP D(s, t) by minimizing d[|s|][|t|][l] over all possible values of l, as Algorithm 3 does in Step 4 for all prefixes of S. Finally, Algorithm 3 returns the prefix c such that RSP D(s, t) ≤ 1−ε 2 if one exists, and otherwise it returns ∅. We conclude by showing that if an ε-synchronization string of length n is used along with the nδ minimum RSPD algorithm, the number of misdecodings will be at most 1−ε . Theorem 6.24. Suppose that S is an ε-synchronization string of length n over alphabet Σ that is sent over an insertion-deletion channel with ci insertions and cd deletions. By using Algorithm 3 cd ε ci for decoding the indices, the outcome will contain less than 1−ε + 1−ε misdecodings. Proof. The proof of this theorem is similar to the proof of Theorem 5.10. Let prefix S[1, i] be sent through the channel Sτ [1, j] be received on the other end as the result of adversary’s set of actions τ . Further, assume that Sτ [j] is successfully transmitted and is actually S[i] sent be the other end. 34 We first show that RSP D(S[1, i], S 0 [1, j]) is less than the relative suffix error density:    |τ̃1 | sc (τ̃1 [k, |τ̃1 |]) + sc (τ̃2 [k, |τ̃2 |]) 0 RSP D(S[1, i], S [1, j]) = min max τ̃ :c→c̃ k=1 |τ̃1 | − k + 1 − sc (τ̃1 [k, |τ̃1 |])   |τ1 | sc (τ1 [k, |τ1 |]) + sc (τ2 [k, |τ2 |]) ≤ max k=1 |τ1 | − k + 1 − sc (τ1 [k, |τ1 |]) E(j, i) = Relative Suffix Error Density = max j≤i i − j Now, using Theorem 5.14, we know that the relative suffix error density is smaller than 1−ε upon +dd arrival of all but at most ci1−ε − cd of successfully transmitted symbols. Along with Lemma 6.21,   ci 1 this results into the conclusion that the minimum RSPD decoding guarantees 1−ε + cd 1−ε −1 misdecodings. This finishes the proof of the theorem. Acknowledgements The authors thank Ellen Vitercik and Allison Bishop for valuable discussions in the early stages of this work. 35 References [1] Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless seth is false). In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 51–58. ACM, 2015. [2] Mark Braverman, Ran Gelles, Jieming Mao, and Rafail Ostrovsky. Coding for interactive communication correcting insertions and deletions. In Proceedings of the International Conference on Automata, Languages, and Programming (ICALP), 2016. [3] Karthekeyan Chandrasekaran, Navin Goyal, and Bernhard Haeupler. Deterministic algorithms for the lovsz local lemma. SIAM Journal on Computing (SICOMP), pages 2132–2155, 2013. [4] Ran Gelles. Coding for interactive communication: A survey, 2015. [5] Ran Gelles and Bernhard Haeupler. Capacity of interactive communication over erasure channels and channels with feedback. 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arXiv:1709.05061v1 [] 15 Sep 2017 Accelerating Dynamic Graph Analytics on GPUs Technical Report Version 2.0 Mo Sha, Yuchen Li, Bingsheng He and Kian-Lee Tan July 15, 2017 Contents 1 Introduction 1 2 Related Work 2.1 Graph Stream Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Graph Analytics on GPUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Storage Formats on GPUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 3 A dynamic framework on GPUs 5 4 5 GPMA Dynamic Graph Processing 4.1 GPMA Graph Storage on GPUs . . . . 4.2 Adapting Graph Algorithms to GPMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 9 GPMA+: GPMA Optimization 5.1 5.2 11 Bottleneck Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lock-Free Segment-Oriented Updates . . . . . . . . . . . . . . . . . . . . . . 12 6 Experimental Evaluation 6.1 Experimental Setup . . . . . . . . . . 6.2 The Performance of Handling Updates 6.3 Application Performance . . . . . . . . 6.4 Scalability . . . . . . . . . . . . . . . . 6.5 Overall Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 18 19 21 22 7 Conclusion & Future Work 22 Appendices A TryInsert+ Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . B Additional Experimental Results For Data Transfer . . . . . . . . . . . . . . C The Performance of Handling Updates on Sorted Graphs . . . . . . . . . . D Additional Experimental Results For Graph Streams with Explicit Deletions 23 23 27 27 28 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Abstract As graph analytics often involves compute-intensive operations, GPUs have been extensively used to accelerate the processing. However, in many applications such as social networks, cyber security, and fraud detection, their representative graphs evolve frequently and one has to perform a rebuild of the graph structure on GPUs to incorporate the updates. Hence, rebuilding the graphs becomes the bottleneck of processing high-speed graph streams. In this paper, we propose a GPU-based dynamic graph storage scheme to support existing graph algorithms easily. Furthermore, we propose parallel update algorithms to support efficient stream updates so that the maintained graph is immediately available for high-speed analytic processing on GPUs. Our extensive experiments with three streaming applications on large-scale real and synthetic datasets demonstrate the superior performance of our proposed approach. 1 Introduction Due to the rising complexity of data generated in the big data era, graph representations are used ubiquitously. Massive graph processing has emerged as the de facto standard of analytics on web graphs, social networks (e.g., Facebook and Twitter), sensor networks (e.g., Internet of Things) and many other application domains which involve high-dimen-sional data (e.g., recommendation systems). These graphs are often highly dynamic: network traffic data averages 109 packets/hour/router for large ISPs [23]; Twitter has 500 million tweets per day [40]. Since real-time analytics is fast becoming the norm [26, 12, 35, 42], it is thus critical for operations on dynamic massive graphs to be processed efficiently. Dynamic graph analytics has a wide range of applications. Twitter can recommend information based on the up-to-date TunkRank (similar to PageRank) computed based on a dynamic attention graph [14] and cellular network operators can fix traffic hotspots in their networks as they are detected [27]. To achieve real-time performance, there is a growing interest to offload the graph analytics to GPUs due to its much stronger arithmetical power and higher memory bandwidth compared with CPUs [43]. Although existing solutions, e.g. Medusa [57] and Gunrock [48], have explored GPU graph processing, we are aware that the only one work [29] has considered a dynamic graph scenario which is a major gap for running analytics on GPUs. In fact, a delay in updating a dynamic graph may lead to undesirable consequences. For instance, consider an online travel insurance system that detects potential frauds by running ring analysis on profile graphs built from active insurance contracts [5]. Analytics on an outdated profile graph may fail to detect frauds which can cost millions of dollars. However, updating the graph will be too slow for issuing contracts and processing claims in real time, which will severely influence legitimate customers’ user experience. This motivates us to develop an update-efficient graph structure on GPUs to support dynamic graph analytics. There are two major concerns when designing a GPU-based dynamic graph storage scheme. First, the proposed storage scheme should handle both insertion and deletion operations efficiently. Though processing updates against insertion-only graph stream could be handled by reserving extra spaces to accommodate the updates, this naı̈ve approach fails to preserve the locality of the graph entries and cannot support deletions efficiently. Considering a common sliding window model on a graph edge stream, each element in the stream is an edge in a graph and analytic tasks are performed on the graph induced by all edges in the up-to-date window [49, 15, 17]. A naı̈ve approach needs to access the entire graph in 1 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. the sliding window to process deletions. This is obviously undesirable against high-speed streams. Second, the proposed storage scheme should be general enough for supporting existing graph formats on GPUs so that we can easily reuse existing static GPU graph processing solutions for graph analytics. Most large graphs are inherently sparse. To maximize the efficiency, existing works [6, 32, 31, 29, 51] on GPU sparse graph processing rely on optimized data formats and arrange the graph entries in certain sorted order, e.g. CSR [32, 6] sorts the entries by their row-column ids. However, to the best of our knowledge, no schemes on GPUs can support efficient updates and maintain a sorted graph format at the same time, other than a rebuild. This motivates us to design an update-efficient sparse graph storage scheme on GPUs while keeping the locality of the graph entries for processing massive analytics instantly. In this paper, we introduce a GPU-based dynamic graph analytic framework followed by proposing the dynamic graph storage scheme on GPUs. Our preliminary study shows that a cache-oblivious data structure, i.e., Packed Memory Array (PMA [10, 11]), can potentially be employed for maintaining dynamic graphs on GPUs. PMA, originally designed for CPUs [10, 11], maintains sorted elements in a partially contiguous fashion by leaving gaps to accommodate fast updates with a constant bounded gap ratio. The simultaneously sorted and contiguous characteristic of PMA nicely fits the scenario of GPU streaming graph maintenance. However, the performance of PMA degrades when updates occur in locations which are close to each other, due to the unbalanced utilization of reserved spaces. Furthermore, as streaming updates often come in batches rather than one single update at a time, PMA does not support parallel insertions and it is non-trivial to apply PMA to GPUs due to its intricate update patterns which may cause serious thread divergence and uncoalesced memory access issues on GPUs. We thus propose two GPU-oriented algorithms, i.e. GPMA and GPMA+, to support efficient parallel batch updates. GPMA explores a lock-based approach which becomes increasingly popular due to the recent GPU architectural evolution for supporting atomic operations [18, 28]. While GPMA works efficiently for the case where few concurrent updates conflict, e.g., small-size update batches with random updating edges in each batch, there are scenarios where massive conflicts occur and hence, we propose a lock-free approach, i.e. GPMA+. Intuitively, GPMA+ is a bottom-up approach by prioritizing updates that occur in similar positions. The update optimizations of our proposed GPMA+ are able to maximize coalesced memory access and achieve linear performance scaling w.r.t the number of computation units on GPUs, regardless of the update patterns. In summary, the key contributions of this paper are the following: • We introduce a framework for GPU dynamic graph analytics and propose, the first of its kind, a GPU dynamic graph storage scheme to pave the way for real-time dynamic graph analytics on GPUs. • We devise two GPU-oriented parallel algorithms: GPMA and GPMA+, to support efficient updates against high-speed graph streams. • We conduct extensive experiments to show the performance superiority of GPMA and GPMA+. In particular, we design different update patterns on real and synthetic graph streams to validate the update efficiency of our proposed algorithms against their CPU counterparts as well as the GPU rebuild baseline. In addition, we implement three real world graph analytic applications on the graph streams to demonstrate the efficiency and broad applicability of our proposed solutions. In order to support larger graphs, we extend our proposed formats to multiple GPUs and demonstrate the scalability of our 2 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. approach with multi-GPU systems. The remainder of this paper is organized as follows. The related work is discussed in Section 2. Section 3 presents a general workflow of dynamic graph processing on GPUs. Subsequently, we describe GPMA and GPMA+ in Sections 4-5 respectively. Section 6 reports results of a comprehensive experimental evaluation. We conclude the paper and discuss some future works in Section 7. 2 Related Work In this section, we review related works in three different categories as follows. 2.1 Graph Stream Processing Over the last decade, there has been an immense interest in designing efficient algorithms for processing massive graphs in the data stream model (see [35] for a detailed survey). This includes the problems of PageRank-styled scores [38], connectivity [21], spanners [20], counting subgraphs e.g. triangles [46] and summarization [44]. However, these works mainly focus on the theoretical study to achieve the best approximation solution with linear bounded space. Our proposed methods can incorporate existing graph stream algorithms with ease as our storage scheme can support most graph representations used in existing algorithms. Many systems have been proposed for streaming data processing, e.g. Storm [45], Spark Streaming [54], Flink [1]. Attracted by its massively parallel performance, several attempts have successfully demonstrated the advantages of using GPUs to accelerate data stream processing [47, 56]. However, the aforementioned systems focus on general stream processing and lack support for graph stream processing. Stinger [19] is a parallel solution to support dynamic graph analytics on a single machine. More recently, Kineograph [14], CellIQ [27] and GraphTau [26] are proposed to address the need for general time-evolving graph processing under the distributed settings. However, to our best knowledge, existing works focusing on CPU-based time-evolving graph processing will be inefficient on GPUs, because CPU and GPU are two architectures with different design principles and performance concerns in the parallel execution. We are aware that only one work [29] explores the direction of using GPUs to process real-time analytics on dynamic graphs. However, it only supports insertions and lacks an efficient indexing mechanism. 2.2 Graph Analytics on GPUs Graph analytic processing is inherently data- and compute-intensive. Massively parallel GPU accelerators are powerful to achieve supreme performance of many applications. Compared with CPU, which is a general-purpose processor featuring large cache size and high single core processing capability, GPU devotes most of its die area to a large number of simple Arithmetic Logic Units (ALUs), and executes code in a SIMT (Single Instruction Multiple Threads) fashion. With the massive amount of ALUs, GPU offers orders of magnitude higher computational throughput than CPU in applications with ample parallelism. This leads to a spectrum of works which explore the usage of GPUs to accelerate graph analytics and demonstrate immense potentials. Examples include breath-first search 3 Accelerating Dynamic Graph Analytics on GPUs Graph Stream M. Sha et al. Streaming Applications Graph Stream Buffer Dynamic Query Buffer Graph Update CPU Continuous Monitoring Graph Analytics GPU Active Graph Structure Figure 1: The dynamic graph analytic framework (BFS) [32], subgraph query [31], PageRank [6] and many others. The success of deploying specific graph algorithms on GPUs motivates the design of general GPU graph processing systems like Medusa [57] and Gunrock [48]. However, the aforementioned GPU-oriented graph algorithms and systems assume static graphs. To handle dynamic graph scenario, existing works have to perform a rebuild on GPUs against each single update. DCSR [29] is the only solution, to the best of our knowledge, which is designed for insertion-only scenarios as it is based on linked edge block and rear appending technique. However, it does not support deletions or efficient searches. We propose GPMA to enable efficient dynamic graph updates (i.e. insertions and deletions) on GPUs in a fine-grained manner. In addition, existing GPU-optimized graph analytics and systems can replace their storage layers directly with ease since the fundamental graph storage schemes used in existing works can be directly implemented on top of our proposed storage scheme. 2.3 Storage Formats on GPUs Sparse matrix representation is a popular choice for storing large graphs on GPUs [3, 2, 57, 48] The Coordinate Format [16] (COO) is the simplest format which only stores non-zero matrix entries by their coordinates with values. COO sorts all the non-zero entries by the entries’ row-column key for fast entry accesses. CSR [32, 6] compresses COO’s row indices into an offset array to reduce the memory bandwidth when accessing the sparse matrix. To optimize matrices with different non-zero distribution patterns, many customized storage formats were proposed, e.g., Block COO [50] (BCCOO), Blocked Row-Column [7] (BRC) and Tiled COO [52] (TCOO). Existing formats require to maintain a certain sorted order of their storage base units according to the unit’s position in the matrix, e.g. entries for COO and blocks for BCCOO, and still ensure the locality of the units. As mentioned previously, few prior schemes can handle efficient sparse matrix updates on GPUs. To the best of our knowledge, PMA [10, 11] is a common structure which maintains a sorted array in a contiguous manner and supports efficient insertions/deletions. However, PMA is designed for CPU and no concurrent updating algorithm is ever proposed. Thus, we are motivated to propose GPMA and GPMA+ for supporting efficient concurrent updates on all existing storage formats. 4 Accelerating Dynamic Graph Analytics on GPUs Graph Stream Graph Stream Transfer (host to device) Active Graph Update (Summarization & Update) Query Transfer (host to device) Query Stream Results Transfer (device to host) Step 1 Step 2 M. Sha et al. Graph Stream Transfer (host to device) Active Graph Update (Summarization & Update) Query Transfer (host to device) Graph Analytic Processing Results Transfer (device to host) Step 3 Data transfer on PCIe Repeat… GPU computation Figure 2: Asynchronous streams 3 A dynamic framework on GPUs To address the need for real-time dynamic graph analytics, we offload the tasks of concurrent dynamic graph maintenance and its corresponding analytic processing to GPUs. In this section, we introduce a general GPU dynamic graph analytic framework. The design of the framework takes into account two major concerns: the framework should not only handle graph updates efficiently but also support existing GPU-oriented graph analytic algorithms without forfeiting their performance. Model: We adopt a common sliding window graph stream model [35, 27, 44]. The sliding window model consists of an unbounded sequence of elements (u, v)t 1 which indicates the edge (u, v) arrives at time t, and a sliding window which keeps track of the most recent edges. As the sliding window moves with time, new edges in the stream keep being inserted into the window and expiring edges are deleted. In real world applications, the sliding window of a graph stream can be used to monitor and analyze fresh social actions that appearing on Twitter [49] or the call graph formed by the most recent CDR data [27]. In this paper, we focus on presenting how to handle edge streams but our proposed scheme can also handle the dynamic hyper graph scenario with hyper edge streams. Apart from the sliding window model, the graph stream model which involves explicit insertions and deletions (e.g., a user requests to add or delete a friend in the social network) is also supported by our scheme as the proposed dynamic graph storage structure is designed to handle random update operations. That is, our system supports two kinds of updates, implicit ones generated from the sliding window mechanism and explicit ones generated from upper level applications or users. The overview of the dynamic graph analytic framework is presented in Figure 1. Given a graph stream, there are two types of streaming tasks supported by our framework. The first type is the ad-hoc queries such as neighborhood and reachability queries on the graph which is constantly changing. The second type is the monitoring tasks like tracking PageRank scores. We present the framework by illustrating how to handle the graph streams and the corresponding queries while hiding data transfer between CPU and GPU, as follows: Graph Streams: The graph stream buffer module batches the incoming graph streams on the CPU side (host) and periodically sends the updating batches to the graph update module located on GPU (device). The graph update module updates the “active” graph stored on the device by using the batch received. The “active” graph is stored in the format of our proposed GPU dynamic graph storage structure. The details of the graph storage 1 Our framework handles both directed and undirected edges. 5 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. non-zero entry Level 3 balanced [0,31] Level 2 rebalanced [0,15] [0,7] Level 1 [0,3] Leaf [8,15] [4,7] [8,11] [16,23] [12,15] [16,19] [20,23] segment size density lower bound ρ density upper bound τ min # of entries max # of entries unbalanced [16,31] [24,31] [24,27] [28,31] Original 2 5 8 13 16 17 23 27 28 31 34 37 42 46 51 62 Inserted 2 5 8 13 16 17 23 27 28 31 34 37 42 46 51 62 48 Final 2 5 8 13 16 17 23 27 28 31 34 37 42 46 48 51 62 Leaf 4 0.08 0.92 1 3 Level 1 8 0.19 0.88 2 6 Level 2 16 0.29 0.84 4 12 Level 3 32 0.40 0.80 8 24 Figure 3: PMA insertion example (Left: PMA for insertion; Right: predefined thresholds) structure and how to update the graph efficiently on GPUs will be discussed extensively in later sections. Queries: Like the graph stream buffer, the dynamic query buffer module batches adhoc queries submitted against the stored active graph, e.g., queries to check the dynamic reachability between pairs of vertices. The tracking tasks will also be registered in the continuous monitoring module, e.g., tracking up-to-date PageRank. All ad-hoc queries and monitoring tasks will be transferred to the graph analytic module for GPU accelerated processing. The analytic module interacts with the active graph to process the queries and the tracking tasks. Subsequently, the query results will be transferred back to the host. As most existing GPU graph algorithms use optimized array formats like CSR to accelerate the performance [18, 28, 34, 52], our proposed storage scheme provides an interface for storing the array formats. In this way, existing algorithms can be integrated into the analytic module with ease. We describe the details of the integration in Section 4.2. Hiding Costly PCIe Transfer: Another critical issue on designing GPU-oriented systems is to minimize the data transfer between the host and the device through PCIe. Our proposed batching approach allows overlapping data transfer by concurrently running analytic tasks on the device. Figure 2 shows a simplified schedule with two asynchronous streams: graph streams and query streams respectively. The system is initialized at Step 1 where the batch containing incoming graph stream elements is sent to the device. At Step 2, while PCIe handles bidirectional data transfer for previous query results (device to host) and freshly submitted query batch (host to device), the graph update module updates the active graph stored on the device. At Step 3, the analytic module processes the received query batch on the device and a new graph stream batch is concurrently transferred from the host to the device. It is clear to see that, by repeating the aforementioned process, all data transfers are overlapped with concurrent device computations. Stroa 4 GPMA Dynamic Graph Processing To support dynamic graph analytics on GPUs, there are two major challenges discussed in the introduction. The first challenge is to maintain the dynamic graph storage in the device memory of GPUs for efficient update as well as compute. The second challenge is that the storage strategy should show its good compatibility with existing graph analytic algorithms on GPUs. In this section, we discuss how to address the challenges with our proposed scheme. First, we introduce GPMA for GPU resident graph storage to simultaneously achieve update and 6 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. compute efficiency (Section 4.1). Subsequently, we illustrate GPMA’s generality in terms of deploying existing GPU based graph analytic algorithms (Section 4.2). 4.1 GPMA Graph Storage on GPUs In this subsection, we first discuss the design principles our proposed dynamic graph storage should follow. Then we introduce how to implement our proposal. Design Principles. The proposed graph storage on GPUs should take into account the following principles: • The proposed dynamic graph storage should efficiently support a broad range of updating operations, including insertions, deletions and modifications. Furthermore, it should have a good locality to accommodate the highly parallel memory access characteristic of GPUs, in order to achieve high memory efficiency. • The physical storage strategy should support common logical storage formats and the existing graph analytic solutions on GPUs based on such formats can be adapted easily. Background of PMA. GPMA is primarily motivated by a novel structure, Packed Memory Array (PMA [10, 11]), which is proposed to maintain sorted elements in a partially continuous fashion by leaving gaps to accommodate fast updates with a bounded gap ratio. PMA is a self-balancing binary tree structure. Given an array of N entries, PMA separates the whole memory space into leaf segments with O(log N ) length and defines non-leaf segments as the space occupied by their descendant segments. For any segment located at height i (leaf height is 0), PMA designs a way to assign the lower and upper bound density thresholds for the segment as ρi and τi respectively to achieve O(log2 N ) amortized update complexity. Once an insertion/deletion causes the density of a segment to fall out of the range defined by (ρi , τi ), PMA tries to adjust the density by re-allocating all elements stored in the segment’s parent. The adjustment process is invoked recursively and will only be terminated if all segments’ densities fall back into the range defined by PMA’s density thresholds. For an ordered array, modifications are trivial. Therefore, we mainly discuss insertions because deletions are the dual operation of insertions in PMA. Example 1. Figures 3 presents an example for PMA insertion. Each segment is uniquely identified by an interval (starting and ending position of the array) displayed in the corresponding tree node, e.g., the root segment is segment-[0,31] as it covers all 32 spaces. All values stored in PMA are displayed in the array. The table in the figure shows predefined parameters including the segment size, the assignment of density thresholds (ρi , τi ) and the corresponding minimum and maximum entry sizes at different heights of the tree. We use these setups as a running example throughout the paper. To insert an entry. i.e. 48, into PMA, the corresponding leaf segment is firstly identified by a binary search, and the new entry is placed at the rear of leaf segment. The insertion causes the density of the leaf segment 4 to exceed the threshold 3. Thus, we need to identify the nearest ancestor segment which can accommodate the insertion without violating the thresholds, i.e., the segment-[16,31]. Finally, the insertion is completed by re-distpatching all entries evenly in segment-[16,31]. Lemma 1 ([10, 11]). The amortized update complexity of PMA is proved to be O(log2 N ) in the worst case and O(log N ) in the average case. It is evident that PMA could be employed for dynamic graph maintenance as it maintains sorted elements efficiently with high locality on CPU. However, the update procedure described in [11] is inherently sequential and no concurrent algorithms have been proposed. 7 Accelerating Dynamic Graph Analytics on GPUs Level 3 [0,31] Level 2 [0,15] Level 1 Leaf [16,31] [0,7] [0,3] [8,15] [4,7] Original 2 5 8 13 non-zero entry balanced rebalanced Round1 1 M. Sha et al. 2 5 8 [8,11] 16 17 [12,15] 23 [16,19] 27 28 [20,23] 31 [24,31] [24,27] 34 37 42 16 17 …… 1 4 23 9 35 48 27 28 [28,31] 46 51 62 balanced unbalanced rebalanced trylock failed Thread Pool Insertion Buffer 9 13 [16,23] …… 31 34 37 42 46 48 51 62 Figure 4: GPMA concurrent insertions To support batch updates of edge insertions and deletions for efficient graph stream analytic processing, we devise GPMA to support concurrent PMA updates on GPUs. Note that we focus on the insertion process for a concise presentation because the deletion process is a dual process w.r.t. the insertion process in PMA. Concurrent Insertions in GPMA. Motivated by PMA on CPUs, we propose GPMA to handle a batch of insertions concurrently on GPUs. Intuitively, GPMA assigns an insertion to a thread and concurrently executes PMA algorithm for each thread with a lock-based approach to ensure consistency. More specifically, all leaf segments of insertions are identified in advance, and then each thread checks whether the inserted segments still satisfy their thresholds from bottom to top. For each particular segment, it is accessed in a mutually exclusive fashion. Moreover, all threads are synchronized after updating all segments located at the same tree height to avoid possible conflicts as segments at a lower height are fully contained in the segments at a higher level. Algorithm 1 presents the pseudocode for GPMA concurrent insertions. We highlight the lines added to the original PMA update algorithm in order to achieve concurrent update of GPMA. As shown in line 2, all entries in the insertion set are iteratively tried until all of them take effect. For each iteration shown in line 9, all threads start at leaf segments and attempt the insertions in a bottom-up fashion. If a particular thread fails the mutex competition in line 11, it aborts immediately and waits for the next attempt. Otherwise, it inspects the density of the current segment. If the current segment does not satisfy the density requirement, it will try the parent segment in the next loop iteration (lines 13-14). Once an ancestor segment is able to accommodate the insertion, it merges the new entry in line 16 and the entry is removed from the insertion set. Subsequently, the updated segment will re-dispatch all its entries evenly and the process is terminated. Example 2. Figure 4 illustrates an example with five insertions, i.e. {1, 4, 9, 35, 48}, for concurrent GPMA insertion. The initial structure is the same as in Example 1. After identifying the leaf segment for insertion, threads responsible for Insertion-1 and Insertion-4 compete for the same leaf segment. Assuming Insertion-1 succeeds in getting the mutex, Insertion-4 is aborted. Due to enough free space of the segment, Insertion-1 is successfully inserted. Even though there is no leaf segment competition for Insertions9,35,48, they should continue to inspect the corresponding parent segments because all the leaf segments do not satisfy the density requirement after the insertions. Insertions35,48 still compete for the same level-1 segment and Insertion-48 wins. For this example, three of the insertions are successful and the results are shown in the bottom of Figure 4. Insertions-4,35 are aborted in this iteration and will wait for the next attempt. 8 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Algorithm 1 GPMA Concurrent Insertion procedure GPMAInsert(Insertions I) 2: while I is not empty do 3: parallel for i in I 1: Seg s ← BinarySearchLeafSegment(i) TryInsert(s, i, I) synchronize release locks on all segments 4: 5: 6: 7: function TryInsert(Seg s, Insertion i, Insertions I) while s 6= root do 10: synchronize 11: if fails to lock s then 12: return 8: 9: if (|s| + 1)/capacity(s) < τ then s ← parent segment of s else Merge(s, i) re-dispatch entries in s evenly remove i from I return double the space of the root segment 13: 14: 15: 16: 17: 18: 19: 20: 4.2 . insertion aborts . insertion succeeds Adapting Graph Algorithms to GPMA Existing graph algorithms often use sparse matrix format to store the graph entries since most large graphs are naturally sparse[5]. Although many different sparse storage formats have been proposed, most of the formats assume a specific order to organize the nonzero entries. These formats enforce the order of the graph entries to optimize their specific access patterns, e.g., row-oriented (COO2 ), diagonal-oriented (JAD), and block-/tile-based (BCCOO, BRC and TCOO). It is natural that the ordered graph entries can be projected into an array and these similar formats can be supported by GPMA easily. Among all formats, we choose CSR as an example to illustrate how to adapt the format to GPMA. CSR as a case study. CSR is most widely used by existing algorithms on sparse matrices or graphs. CSR compresses COO’s row indices into an offset array, which contributes to reducing the memory bandwidth when accessing the sparse matrix, and achieves a better workload estimation for skewed graph distribution (e.g., power-law distribution). The following example demonstrates how to implement CSR on GPMA. Example 3. In Figure 5, we have a graph of three vertices and six edges. The number on each edge denotes the weight of the corresponding edge. The graph is represented as a sparse matrix and is further transformed to the CSR format shown in the upper right. CSR sorts all non-zero entries in the row-orient order, and compresses row indices into intervals as a row offset array. The lower part denotes the GPMA representation of this 2 Generally, COO means ordered COO and it can also be column-oriented. 9 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Algorithm 2 Breadth-First Search 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: procedure BFS(Graph G, Vertex s) for each vertex u ∈ G.V − {s} do u.visited = false Q←φ s.visited ← true ENQUEUE(Q, s) while Q 6= φ do u ← DEQUEUE(Q) for each v ∈ G.Adj[u] do if IsEntryExist(v) then if v.visited = false then v.visited ← true ENQUEUE(v) Algorithm 3 GPU-based BFS Neighbour Gathering procedure Gather(Vertex frontier, Int csrOffset) {r, rEnd} ← csrOffset[frontier, frontier + 1] 3: for (i ← r+threadId; i<rEnd; i+=threadNum) do 4: if IsEntryExist(i) then ParallelGather(i) 1: 2: graph. In order to maintain the row offset array without synchronization among threads, we add a guard entry whose column index is ∞ during concurrent insertions. That is to say, when the guard is moved, the corresponding element in row offset array will change. Given a graph stored on GPMA, the next step is to adapt existing graph algorithms to GPMA. In particular, how existing algorithms access the graph entries stored on GPMA is of vital importance. As for the CSR example, most algorithms access the entries by navigating through CSR’s ordered array[18, 28, 34, 52]. We note that a CSR stored on GPMA is also an array which has bounded gaps interleaved with the graph entries. Thus, we are able to efficiently replace the operations of arrays with the operations of GPMA. We will demonstrate how we can do this replacement as follows. Algorithm 2 illustrates the pseudocode of the classic BFS algorithm. We should pay attention to line 10, which is highlighted. Compared with the raw adjacency list, the applications based on GPMA need to guarantee the current vertex being traversed is a valid neighbour instead of an invalid space in GPMA’s gap. Algorithm 2 provides a high-level view for GPMA adaption. Furthermore, we present how it adapts GPMA in the parallel GPU environment with some low-level details. Algorithm 3 is the pseudocode of the Neighbour Gathering parallel procedure, which is a general primitive for most GPU-based vertex-centric graph processing models [36, 18, 22]. This primitive plays a role similar to line 10 of Algorithm 2 but in a parallel fashion in accessing the neighbors of a particular vertex. When traversing all neighbours of frontiers, Neighbour Gathering follows the SIMT manner, which means that there are threadNum threads as a group assigned to one of the vertex frontier and the procedure in Algorithm 3 is executed 10 Accelerating Dynamic Graph Analytics on GPUs 2 1 0 6 5 1 2 4 Row Offset Column Index Value M. Sha et al. Row Offset [0 2 3 6] [0 2 2 0 1 2] [1 2 3 4 5 6] 4 8 14 [0,15] [0,7] 3 [8,15] [0,3] Example Graph CSR Format (0,0) (0,2) 1 2 [4,7] (0,∞) (1,2) 3 [8,11] (1,∞) (2,0) 4 [12,15] (2,1) (2,2) (2,∞) 5 6 Figure 5: GPMA based on CSR in parallel. For the index range (in the CSR on GPMA) of the current frontier given by csrOffset (shown in line 2), each thread will handle the corresponding tasks according to its threadId. For GPU-based BFS, the visited labels of neighbours for all frontiers will not be judged immediately after neighbours are accessed. Instead, they will be compacted to contiguous memory in advance for higher memory efficiency. Similarly, we can also carry out the entry existing checking for other graph applications to adapt them to GPMA. To summarize, GPMA can be adapted to common graph analytic applications which are implemented in different representation and execution models, including matrix-based (e.g., PageRank), vertex-centric (e.g., BFS) and edge-centric (e.g., Connected Component). 5 GPMA+: GPMA Optimization Although GPMA can support concurrent graph updates on GPUs, the update algorithm is basically a lock-based approach and can suffer from serious performance issue when different threads compete for the same lock. In this section, we propose a lock-free approach, i.e. GPMA+, which makes full utilization of GPU’s massive multiprocessors. We carefully examine the performance bottleneck of GPMA in Section 5.1. Based on the issues identified, we propose GPMA+ for optimizing concurrent GPU updates with a lock-free approach in Section 5.2. 5.1 Bottleneck Analysis The following four critical performance issues are identified for GPMA: • Uncoalesced Memory Accesses: Each thread has to traverse the tree from the root segment to identify the corresponding leaf segment to be updated. For a group of GPU threads which share the same memory controller (including access pipelines and caches), memory accesses are uncoalesced and thus, cause additional IO overheads. • Atomic Operations for Acquiring Lock: Each thread needs to acquire the lock before it can perform the update. Frequently invoking atomic operations for acquiring locks will bring huge overheads, especially for GPUs. • Possible Thread Conflicts: When two threads conflict on a segment, one of them has to abort and wait for the next attempt. In the case where the updates occur on segments which are located proximately, GPMA will end up with low parallelism. As most real world large graphs have the power law property, the effect of thread conflicts can be exacerbated. • Unpredictable Thread Workload: Workload balancing is another major concern for optimizing concurrent algorithms [43]. The workload for each thread in GPMA is unpredictable because: (1) It is impossible to obtain the last non-leaf segment traversed 11 Accelerating Dynamic Graph Analytics on GPUs Level 3 non-zero entry [0,31] Level 2 balanced [0,15] Level 1 [16,31] [0,7] M. Sha et al. [8,15] [16,23] rebalanced [24,31] Update Segments: 0 Leaf [0,3] Round 2 1 2 Round 1 1 2 Original 2 4 [4,7] [8,11] [12,15] [16,19] [20,23] [24,27] [28,31] 4 24 28 Successful Flag: Y N N N 4 Update Segments: 0 24 4 5 8 9 13 16 17 23 27 28 31 34 35 37 42 46 48 51 62 Update Offsets: 0 2 3 5 8 13 16 17 23 27 28 31 34 37 42 46 51 62 Update Keys: 1 4 9 35 48 5 8 13 16 17 23 27 28 31 34 37 42 46 51 62 Original Update Segments: 0 16 Successful Flag: N N 5 Update Offsets: 0 1 Successful Flag: Y Y 3 Update Offsets: 0 1 Update Keys: 9 35 48 Round 1 3 Update Keys: 9 35 48 Round 2 Figure 6: GPMA+ concurrent insertions (best viewed in color) by each thread in advance; (2) The result of lock competition is random. The unpredictable nature triggers the imbalanced workload issue for GPMA. In addition, threads are grouped as warps on GPUs. If a thread has a heavy workload, the remaining threads of the same warp are idle and cannot be re-scheduled. 5.2 Lock-Free Segment-Oriented Updates Based on the discussion above, we propose GPMA+ to lift all bottlenecks identified. The proposed GPMA+ does not rely on lock mechanism and achieves high thread utilization simultaneously. Existing graph algorithms can be adapted to GPMA+ in the same manner as GPMA. Compared with GPMA, which handles each update separately, GPMA+ concurrently processes updates based on the segments involved. It breaks the complex update pattern into existing concurrent GPU primitives to achieve maximum parallelism. There are three major components in the GPMA+ update algorithm: (1) The updates are first sorted by their keys and then dispatched to GPU threads for locating their corresponding leaf segments according to the sorted order. (2) The updates belonging to the same leaf segment are grouped for processing and GPMA+ processes the updates level by level in a bottom-up manner. (3) In any particular level, we leverage GPU primitives to invoke all computing resources for segment updates. We note that, the issue of uncoalesced memory access in GPMA is resolved by component (1) as the updating threads are sorted in advance to achieve similar traversal paths. Component (2) completely avoids the use of locks, which solves the problem of atomic operations and thread conflicts. Finally, component (3) makes use of GPU primitives to achieve workload balancing among all GPU threads. We present the pseudocode for GPMA+’s segment-oriented insertion in the procedure GpmaPlusInsertion of Algorithm 4. Note that, similar to Section 4 (GPMA), we focus on presenting the insertions for GPMA+ and the deletions could be naturally inferred. The inserting entries are first sorted by their keys in line 2 and the corresponding segments are then identified in line 3. Given the update set U , GPMA+ processes updating segments level by level in lines 4-15 until all updates are executed successfully (line 11). In each iteration, UniqueInsertion in line 7 groups update entries belonging to the same segments into unique segments, i.e., S ∗ , and produces the corresponding index set I for quick accesses of updates entries located in a segment from S ∗ . As shown in lines 19-20, UniqueSegments only utilizes standard GPU primitives, i.e. RunLenghtEncoding and ExclusiveScan. RunLenghtEncoding compresses an input array by merging runs of an element into a 12 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Algorithm 4 GPMA+ Segment-Oriented Insertion 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: procedure GpmaPlusInsertion(Updates U ) Sort(U ) Segs S ← BinarySearchLeafSegments(U ) while root segment is not reached do Indices I ← ∅ Segs S ∗ ← ∅ (S ∗ , I) ← UniqueSegments(S) parallel for s ∈ S ∗ TryInsert+(s, I, U ) if U = ∅ then return parallel for s ∈ S if s does not contain any update then remove s from S s ← parent segment of s r ← double the space of the old root segment TryInsert+(r, ∅, U ) function UniqueSegments(Segs S) 19: (S ∗ , Counts) ← RunLengthEncoding(S) 20: Indices I ← ExclusiveScan(Counts) 21: return (S ∗ , I) 18: 22: 23: 24: 25: 26: 27: 28: function TryInsert+(Seg s, Indices I, Updates U ) ns ← CountSegment(s) Us ← CountUpdatesInSegment(s,I,U ) if (ns + |Us |)/capacity(s) < τ then Merge(s, Us ) re-dispatch entries in s evenly remove Us from U single element. It also outputs a count array denoting the length of each run. ExclusiveScan calculates, for each entry e in an array, the sum of all entries before e. Both primitives have very efficient parallelized GPU-based implementation which makes full utilization of the massive GPU cores. In our implementation, we use the NVIDIA CUB library [4] for these primitives. Given a set of unique updating segments, TryInsert+ first checks if a segment s has enough space for accommodating the updates by summing the valid entries in s (CountSegment) and the number of updates in s (CountUpdatesInSegment). If the density threshold is satisfied, the updates will be materialized by merging the inserting entries with existing entries in the segment (as shown in line 26). Subsequently, all entries in the segment will be re-dispatched to balance the densities. After TryInsert+, the algorithm will terminate if there are no entries to be updated. Otherwise, GPMA+ will advance to higher levels by setting all remaining segments to their parent segments (lines 12-15). The following example illustrates GPMA+’s segment-oriented updates. Example 4. Figure 6 illustrates an example for GPMA+ insertions with the same setup as 13 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. in example 2. The left part is GPMA+’s snapshots in different rounds during this batch of insertions. The right part denotes the corresponding array information after the execution of each round. Five insertions are grouped into four corresponding leaf segments (denoted in different colours). For the first iteration at the leaf level, Insertions-1,4 of the first segment (denoted as red) is merged into the corresponding leaf segment, then its success flag is marked and will not be considered in the next round. The remaining intervals fail in this iteration and their corresponding segments will upgrade to their parent segments. It should be noted that the blue and the green grids belong to the same parent segment and therefore, will be merged and then dispatched to their shared parent segment (as shown in Round1). In this round, both segments (denoted as yellow and blue) cannot satisfy the density threshold, and their successful flags are not checked. In Round2, both update segments can be merged by the corresponding insertions and no update segments will be considered in the next round since all of them are flagged. In Algorithm 4, TryInsert+ is the most important function as it handles all the corresponding insertions with no conflicts. Moreover, it achieves a balanced workload for each concurrent task. This is because GPMA+ handles the updates level by level and each segment to be updated in a particular level has exactly the same capacity. However, segments in different levels have different capacities. Intuitively, the probability of updating a segment with a larger size (a segment closer to the root) is much lower than that of a segment with a smaller size (a segment closer to the leaf). To optimize towards the GPU architecture, we propose the following optimization strategies for TryInsert+ for segments with different sizes. • Warp-Based: For a segment with entries not larger than the warp size, the segment will be handled by a warp. Since all threads in the same warp are tied together and warp-based data is held by registers, updating a segment by a warp does not require explicit synchronization and will obtain superior efficiency. • Block-Based: For a segment of which the data can be loaded in GPU’s shared memory, block-based approach is chosen. Block-based approach executes all updates in the shared memory. As shared memory has much larger size than warp registers, block-based approach can handle large segments efficiently. • Device-Based: For a segment with the size larger than the size of the shared memory, we handle them via global memory and rely on kernel synchronization. Device-based approach is slower than the two approaches above, but it has much less restriction on memory size (less than device memory amount) and is not invoked frequently. We refer interested readers to Appendix A for the detailed algorithm of the optimizations above. Theorem 1. Given there are K computation units in the GPU, the amortized update 2 performance of GPMA+ is O(1 + logK N ), where N is the maximum number of edges in the dynamic graph. Proof. Let X denote the set of updating entries contained in a batch. We consider the case where |X| ≥ K as it is rare to see |X| < K in real world scenarios. In fact, our analysis works for cases where |X| = O(K). The total update complexity consists of three parts: (1) sorting the updating entries; (2) searching the position of the entries in GPMA; (3) inserting the entries. We study these three parts separately. For part (1), the sorting complexity of |X| entries on the GPU is O( |X| K ) since parallel radix 14 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Table 1: Experimented Graph Algorithms and the Compared Approaches Compared Approaches CPU Approaches GPU Approaches Graph Container AdjLists PMA [10, 11] Stinger [19] cuSparseCSR [3] GPMA/GPMA+ BFS ConnectedComponent PageRank Standard Single Thread Algorithms Stinger built-in Parallel Algorithms D. Merrill et al.[36] J. Soman et al.[41] CUSP SpMV [2] sort is used (keys in GPMA are integers for storing edges). Then, the amortized sorting complexity is O( |X| K )/|X| = O(1). ) For part (2), the complexity of concurrently searching |X| entries on GPMA is O( |X|·logN K since each entry is assigned to one thread and the depth of traversal is the same for one thread (GPMA is a balanced tree). Thus, the amortized searching complexity is O( |X|·logN )/|X| = O( logN K K ). For part (3), we need to conduct a slightly complicated analysis. We denote the total insertion complexity of X with GPMA+ as cX GPMA+ . As GPMA+ is updated level by level, X X cGPMA+ can be decomposed into: cGPMA+ = c0 + c1 + ... + ch where h is the height of the PMA tree. Given any level i, let zi denote the number of segments to be updated by GPMA+. Since all segments at level i have the same size, we denote pi as the sequential complexity to update any segment si,j at level i (TryInsert+ in Algorithm 4). GPMA+ evenly distributes the computing resources to each segment. As processing each segment only requires a constant number of scans on the segment by GPU primitives, the complexity for GPMA+ to process level i is ci = piK·zi . Thus we have: cX GPMA+ = X pi · zi 1 X x ≤ cPMA K K x∈X i=0,..,h where cxPMA is the sequential complexity for PMA to process the update of a particular entry x ∈ X. The inequality holds because for each segment updated by GPMA+, it must be updated at least once by a sequential PMA process. With Lemma 1, we have |X|·log 2 N cxPMA = O(log 2 N ) and thus cX ). Then the amortized complexity to GPMA+ = O( K 2 update one single entry under the GPMA scheme naturally follows as O(1 + logK N ). Finally, we conclude the proof by combining the complexities from all three parts. Theorem 1 proves that the speedups of GPMA+ over sequential PMA is linear to the number of processing units available on GPUs, which showcases the theoretical scalability of GPMA+. 6 Experimental Evaluation In this section, we present the experimental evaluation of our proposed methods. First, we present the setup of the experiments. Second, we examine the update costs of different schemes for maintaining dynamic graphs. Finally, we implement three different applications to show the performance and the scalability of the proposed solutions. 15 Accelerating Dynamic Graph Analytics on GPUs 102 101 100 0 102 101 100 0 101 103 Sliding Size 105 Reddit 104 103 Time (ms) Time (ms) Time (ms) Graph500 104 103 102 101 100 0 101 AdjLists 103 Sliding Size PMA 105 103 102 101 100 0 101 Stinger Pokec 104 103 Time (ms) Uniform Random 104 M. Sha et al. cuSparseCSR 103 Sliding Size 105 GPMA 101 103 Sliding Size 105 GPMA+ Figure 7: Performance comparison for updates with different batch sizes. The dashed lines represent CPU-based solutions whereas the solid lines represent GPU-based solutions. Table 2: Statistics of Datasets Datasets Reddit Pokec Graph500 Random 6.1 |V | 2.61M 1.60M 1.00M 1.00M |E| 34.4M 30.6M 200M 200M |E|/|V | 13.2 19.1 200 200 |Es | 17.2M 15.3M 100M 100M |Es |/|V | 6.6 9.6 100 100 Experimental Setup Datasets. We collect two real world graphs (Reddit and Pokec) and synthesize two random graphs (Random and Graph500) to test the proposed methods. The datasets are described as follows and their statistics are summarized in Table 2. • Reddit is an online forum where user actions include post and comment. We collect all comment actions from a public resource3 . Each comment of a user b to a post from another user a is associated with an edge from a to b, and the edge indicates an action of a has triggered an action of b. As each comment is labeled with a timestamp, it naturally forms a dynamic influence graph. • Pokec is the most popular online social network in Slovakia. We retrieve the dataset from SNAP [30]. Unlike other online datasets, Pokec contains the whole network over a span of more than 10 years. Each edge corresponds to a friendship between two users. • Graph500 is a synthetic dataset obtained by using the Graph500 RMAT generator [37] to synthesize a large power law graph. • Random is a random graph generated by the Erdős-Renyi model. Specifically, given a graph with n vertices, the random graph is generated by including each edge with probability p. In our experiments, we generate a Erdős-Renyi random graph with 0.02% of non-zero entries against a full clique. Stream Setup. In our datasets, Reddit has a timestamp on every edge whereas the other datasets do not possess timestamps. As commonly used in existing graph stream algorithms [55, 53, 38], we randomly set the timestamps of all edges in the Pokec, Graph500 and Random datasets. Then, the graph stream of each dataset receives the edges with increasing timestamps. For each dataset, a dynamic graph stream is initialized with a subgraph consisting of the dataset’s first half of its total edges according to the timestamps, i.e., Es in Table 2 denotes the initial edge set of a dynamic graph before the stream starts. To demonstrate the update performance of both insertions and deletions, we adopt a sliding window setup where the window contains a fixed number of edges. Whenever the window slides, we need 3 https://www.kaggle.com/reddit/reddit-comments-may-2015 16 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. to update the graph by deleting expired edges and inserting arrived edges until there are no new edges left in the stream. Applications. We conduct experiments on three most widely used graph applications to showcase the applicability and the efficiency of GPMA+. • BFS is a key graph operation which is extensively studied in previous works on GPU graph processing [24, 33, 13]. It begins with a given vertex (or root) of an unweighted graph and iteratively explores all connected vertices. The algorithm will assign a minimum distance away from the root vertex to every visited vertex after it terminates. In the streaming scenario, after each graph update, we select a random root vertex and perform BFS from the root to explore the entire graph. • Connected Component is another fundamental algorithm which has been extensively studied under both CPU [25] and GPU [41] µ environment. It partitions the graph in the way that all vertices in a partition can reach the others in the same partition and cannot reach vertices from other partitions. In the streaming context, after each graph update, we run the ConnectedComponent algorithm to maintain the up-to-date partitions. • PageRank is another popular benchmarking application for large scale graph processing. Power iteration method is a standard method to evaluate the PageRank where the Sparse Matrix Vector Multiplication (SpMV) kernel is recursively executed between the graph’s adjacency matrix and the PageRank vector. In the streaming scenario, whenever the graph is updated, the power iteration is invoked and it obtains the up-to-date PageRank vector by operating on the updated graph adjacency matrix and the PageRank vector obtained in the previous iteration. In our experiments, we follow the standard setup by setting the damping factor to 0.85 and we terminate the power iteration once the 1-norm error is less than 10−3 . These three applications have different memory and computation requirements. BFS requires little computation but performs frequent random memory accesses, and PageRank using SpMV accesses the memory sequentially and it is the most compute-intensive task among all three applications. Maintaining Dynamic Graph. We adopt the CSR [32, 6] format to represent the dynamic graph maintained. Note that all approaches proposed in the paper are not restricted to CSR but general enough to incorporate any popular representation formats like COO [16], JAD [39], HYB [9, 34] and many others. To evaluate the update performance of our proposed methods, we compare different graph data structures and respective approaches on both CPUs and GPUs. • AdjLists (CPU). AdjLists is a basic approach for CSR graph representation. As the CSR format sorts all entries according to their row-column indices, we implement AdjLists with a vector of |V | entries for |V | vertices and each entry is a RB-Tree to denote all (out)neighbors of each vertex. The insertions/deletions are operated by TreeSet insertions/deletions. • PMA (CPU). We implement the original CPU-based PMA and adopt it for the CSR format. The insertions/deletions are operated by PMA insertions/deletions. • Stinger (CPU). We compare the graph container structure used in the state-of-the-art CPU-based parallel dynamic graph analytic system, Stinger [19]. The updates are handled by the internal logic of Stinger. • cuSparseCSR (GPU). We also compare with the GPU-based CSR format used in the NVIDIA cuSparse library [3]. The updates are executed by calling the rebuild function 17 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. in the cuSparse library. • GPMA/GPMA+. They are our proposed approaches. Although insertions and deletions could be handled similarly, in the sliding window models where the numbers of insertions and deletions are often equal, the lazy deletions can be performed via marking the location as deleted without triggering the density maintenance and recycling for new insertions. Note that we do not compare with DCSR [29] because, as discussed in Section 2.2, the scheme can neither handle deletions nor support efficient searches, which makes it incomparable to all schemes proposed in this paper. To validate if using the dynamic graph format proposed in this paper affects the performance of graph algorithms, we implement the state-of-the-art GPU-based algorithms on the CSR format maintained by GPMA/GPMA+ as well as cuSparseCSR. Meanwhile, we invoke Stinger’s built-in APIs to handle the same workloads of the graph algorithms, which are considered as the counterpart of GPU-based approaches in highly parallel CPU environment. Finally, we implement the standard single-threaded algorithms for each application in AdjLists and PMA as baselines for thorough evaluation. The details of all compared solutions for each application is summarized in Table 1. Experimental Environment. All algorithms mentioned in the remaining part of this section are implemented with CUDA 7.5 and GCC 4.8.4 with -O3 optimization. All experiments except Stinger run on a CentOS server which has Intel(R) Core i7-5820k (6-cores, 3.30GHz) with 64GB main memory and three GeForce TITAN X GPUs (each has 12GB device memory), connected with PCIe v3.0. Stinger baselines run on a multi-core server which is deployed 4-way Intel(R) Xeon(R) CPU E7-4820 v3 (40-cores, 1.90GHz) with 128GB main memory. 6.2 The Performance of Handling Updates In this subsection, we compare the update costs for different update approaches. As previously mentioned, we start with the initial subgraph consisting of each dataset’s first half of total edges. We measure the average update time where the sliding window iteratively shifts for a batch of edges. To evaluate the impact of update batch sizes, the batch size is set to range from one edge and exponentially grow to one million edges with base two. Figure 7 shows the average latency for all approaches with different sliding batch sizes. Note that the x-axis and y-axis are plotted in log scales. We have also tested sorted graph streams to evaluate extreme cases. We omit the detailed results and we refer interested readers to Appendix C. We observe that, PMA-based approaches are very efficient in handling updates when the batch size is small. As batch size becomes larger, the performance of PMA and GPMA quickly degrades to the performance of simple rebuild. Although GPMA achieves better performance than GPMA+ for small batches since the concurrent updating entries are unlikely to conflict, thread conflicts become serious for larger batches. Due to its lock-free characterstic, GPMA+ shows superior performance over PMA and GPMA. In particular, GPMA+ has speedups of up to 20.42x and 18.30x against PMA and GPMA respectively. Stinger shows impressive update performance in most cases as Stinger efficiently updates its dynamic graph structure in a parallel fashion and the code runs on a powerful multi-core CPU system. For now, multi-core CPU system is considered more powerful than GPUs for pure random data structure maintainance but cost more (in our experimental setup, 18 Accelerating Dynamic Graph Analytics on GPUs Slide Size Uniform Random M. Sha et al. Graph500 Reddit Pokec 1% 15 4.4 0.7 1% 14 4.8 23 1% 1.6 1.2 0.4 1% 1.6 0.8 0.4 0.1% 14 2.1 0.3 0.1% 18 1.7 4.4 0.1% 1.7 0.7 0.4 0.1% 1.9 0.6 0.5 0.01% 19 1.4 2.1 0.01% 1.7 0.6 0.4 0.01% 1.4 0.4 0.6 12 2.4 0.01% 0.0 0.05 0.1 0.15 0.2 0.25 0.3 Time (second) AdjLists PMA 0.0 0.05 0.1 0.15 0.2 0.25 0.3 Time (second) Stinger cuSparseCSR 0.0 0.01 0.02 0.03 0.04 0.05 Time (second) BFS (patterned) 0.0 GPMA+ 0.01 0.02 0.03 0.04 0.05 Time (second) Update (unpatterned) Figure 8: Streaming BFS Slide Size Uniform Random Graph500 1% 15 4.4 1.5 0.1% 0.01% 0.0 0.1 0.2 0.3 0.4 Time (second) AdjLists Reddit 1% 16 5.7 23 16 1.9 1.3 0.1% 11 1.9 1.0 0.01% 0.5 0.0 PMA 0.1 0.2 0.3 0.4 Time (second) cuSparseCSR Stinger Pokec 1% 1.8 0.9 1.0 1% 1.5 0.7 0.5 19 2.5 4.9 0.1% 2.0 0.6 0.9 0.1% 1.6 0.4 0.5 14 1.9 1.9 0.01% 2.3 0.5 0.6 0.01% 1.7 0.4 0.5 0.5 0.0 GPMA+ 0.02 0.04 0.06 0.08 0.1 Time (second) ConnectedComponent (patterned) 0.0 0.02 0.04 0.06 0.08 Time (second) Update (unpatterned) 0.1 Figure 9: Streaming Connected Component Uniform Random Slide Size 1% 42 4.5 0.1% 35 4.0 0.01% 0.0 Graph500 44 9.2 3.6 0.5 1.0 1.5 2.0 2.5 3.0 Time (second) AdjLists Reddit 1% 70 21 25 1% 0.1% 37 5.1 4.8 0.1% 30 4.9 0.01% 0.0 PMA 0.5 1.0 Stinger 8.1 2.2 6.0 1.6 0.01% 1.5 2.0 2.5 3.0 Time (second) cuSparseCSR 0.0 GPMA+ Pokec 12 4.9 1.0 0.2 0.4 0.6 0.8 1.0 Time (second) PageRank (patterned) 14 3.0 1% 9.7 1.8 0.1% 6.9 1.2 0.01% 0.0 0.2 0.4 0.6 0.8 Time (second) Update (unpatterned) 1.0 Figure 10: Streaming PageRank our CPU server costs more than 5 times that of the GPU server). Moreover, we also note that, Stinger shows extremely poor performance in the Graph500 dataset. According to the previous study [8], the phenomenon is due to the fact that Stinger holds a fixed size of each edge block. Since Graph500 is a heavily skewed graph as the graph follows the power law model, the skewness causes severe performance deficiency on the utilization of memory for Stinger. We observe the sharp increase for GPMA+ performance curves occur when the batch size is 512. This is because the multi-level strategy is used in GPMA+ (which is mentioned in Section 5.2) and shared-memory constraint cannot support batch size which is more than 512 on our hardware. Finally, the experiments show that, GPMA is faster than GPMA+ when the update batch is smaller and leads to few thread conflicts, because the GPMA+ logic is more complicated and includes overheads by a number of kernel calls. However, using GPMA only benefits when the update batch is extremely small and the performance gain in such extreme case is also negligible compared with GPMA+. Hence, we can conclude that GPMA+ shows its stability and efficiency across different update patterns compared with GPMA, and we will only show the results of GPMA+ in the remaining experiments. 6.3 Application Performance As previously mentioned, all compared application-specific approaches are summarized in Table 1. We find that integrating GPMA+ into an existing GPU-based implementation requires little modification. The main one is in transforming the array operations in 19 Accelerating Dynamic Graph Analytics on GPUs Slide Size Uniform Random M. Sha et al. Graph500 Reddit Pokec 1% 1% 1% 1% 0.1% 0.1% 0.1% 0.1% 0.01% 0.01% 0 20 40 60 Time (ms) 80 100 0.01% 0 20 40 60 80 Time (ms) GPMA+ Update BFS 100 0.01% 0 5 10 15 20 25 Time (ms) Fetch BFS Distance Vector Send Updates 0 5 10 15 Time (ms) 20 25 Figure 11: Concurrent data transfer and BFS computation with asynchronous stream Number of Edges PMA Update PageRank BFS Connected Component 600M 600M 600M 600M 1.2B 1.2B 1.2B 1.2B 1.8B 1.8B 1.8B 1.8B 0 20 40 60 80 Throughput (million edges / second) 0 2 4 6 8 0 Throughput (billion edges / second) 1 GPU 2 GPUs 1 2 3 4 5 Throughput (billion edges / second) 3 GPUs 0 1 2 3 Throughput (billion edges / second) Figure 12: Multi-GPU performance on different sizes of Graph500 datasets the original implementation to the operations on GPMA+, as presented in Section 4.2. The intentions of this subsection are two-fold. First, we test if using the PMA-like data structure to represent the graph brings significant overheads for the graph algorithms. Second, we demonstrate how the update performance affects the overall efficiency of dynamic graph processing. In the remaining part of this section, we present the performance of different approaches by showing their average elapsed time to process a shift of the sliding window with three different batch sizes, i.e., the batches contain 0.01%, 0.1% and 1% edges of the respective dataset. We have also tested the graph stream with explicit random insertions and deletions for all applications as an extended experiment. We omit the detailed results here since they are similar to the results of the sliding window model and we refer interested readers to Appendix D. We distinguish the time spent on updates and analytics with different patterns among all figures. BFS Results: Figure 8 presents the results for BFS. Although processing BFS only accesses each edge in the graph once, it is still an expensive operation because BFS can potentially scan the entire graph. This has led to the observation that CPU-based approach takes significant amount of time for BFS computation whereas the update time is comparatively negligible. Thanks to the massive parallelism and high memory bandwidth of GPUs, GPU-based approaches are much more efficient than CPU-based approaches for BFS computation as well as the overall performance. For the cuSparseCSR approach, the rebuild process is the bottleneck as the update needs to scan the entire group multiple times. In contrast, GPMA+ takes much shorter time for the update and has nearly identical BFS performance compared with cuSparseCSR. Thus, GPMA+ dominates the comparisons in terms of the overall processing efficiency. We have also tested our framework in terms of hiding data transfer over PCIe by using asynchronous streams to concurrently perform GPU computation and PCIe transfer. In Figure 11, we show the results when running concurrent execution by using the GPMA+ approach. The data transfer consists of two parts: sending graph updates and fetching updated distance vector (from the query vertex to all other vertices). It is clear from the figure that, under any circumstances, sending graph updates is overlapped by GPMA+ update processing and fetching the distance vector is overlapped by BFS computation. 20 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Thus, the data transfer is completely hidden in the concurrent streaming scenario. As the observations remain similar in other applications, we omit their results and explanations, and the details can be found in Appendix B. Connected Component Results: Figure 9 presents the results for running Connected Component on the dynamic graphs. The results show different performance patterns compared with BFS as ConnectedComponent takes more time in processing which is caused by a number of graph traversal passes to extract the partitions. Meanwhile, the update cost remains the same. Thus, GPU-based solutions enhance their performance superiority over CPU-based solutions. Nevertheless, the update process of cuSparseCSR is still expensive compared with the time spent on Connected- Component. GPMA+ is very efficient in processing the updates. Although we have observed that, in the Reddit and the Pokec datasets, GPMA+ shows some discrepancies for running the graph algorithm against cuSparseCSR due to the “holes” introduced in the graph structure, the discrepancies are insignificant considering the huge performance boosts for updates. Thus, GPMA+ still dominates the rebuild approach for overall performance. PageRank Results: Figure 10 presents the results for Page- Rank. PageRank is a compute-intensive task where the SpMV kernel is iteratively invoked on the entire graph until the PageRank vector converges. The pattern follows from previous results: CPUbased solutions are dominated by GPU-based approaches because iterative SpMV is a more expensive process than BFS and ConnectedComponent, and GPU is designed to handle massively parallel computation like SpMV. Although cuSparseCSR shows inferior performance compared with GPMA+, the improvement brought by GPMA+’s efficient update is not as significant as that in previous applications since the update costs are small compared with the cost of iterative SpMV kernel calls. Nevertheless, the dynamic structure of GPMA+ does not affect the efficiency of the SpMV kernel and GPMA+ outperforms other approaches in all experiments. 6.4 Scalability GPMA and GPMA+ can also be extended to multiple GPUs to support graphs with size larger than the device memory of one GPU. To showcase the scalability of our proposed framework, we implement the multi-GPU version of GPMA+ and carry out experiments of the aforementioned graph applications. We generate three large datasets using Graph500 with increasing numbers of edges (600 Million, 1.2 Billion and 1.8 Billion) and conduct the same performance experiments in section 6.3 with 1% slide size, on 1, 2 and 3 GPUs respectively. We evenly partition graphs according to the vertex index and synchronize all devices after each iteration. For fair comparison among different datasets, we use throughput as our performance metric.The experimental results of GPMA+ updates and application performance are illustrated in Figure 12. We do not compare with Stinger because in this subsection, we focus on the evaluation on the scalability of GPMA+. The memory consumption of Stinger exceeds our machine’s 128GB main memory based on its default configuration in the standalone mode. Multiple GPUs can extend the memory capacity so that analytics on larger graphs can be executed. According to Figure 12, the improvement in terms of throughput for multiple 21 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. GPUs behaves differently in various applications. For GPMA+ update and PageRank, we achieve a significant improvement with more GPUs, because their workloads between communications are relatively compute-intensive. For BFS and ConnectedComponent, the experimental results demonstrate a tradeoff between overall computing power and communication cost with increasing number of GPUs, as these two applications incur larger communication cost. Nevertheless, multi-GPU graph processing is an emerging research area and more effectiveness optimizations are left as future work. Overall, this set of preliminary experiments shows that our proposed scheme is capable of supporting large scale dynamic graph analytics. 6.5 Overall Findings We summarize our findings in the subsection. First, GPU-based approaches (cuSparseCSR and GPMA+) outperform CPU-based approaches thanks to our optimizations in taking advantage of the superior hardware of the GPUs, even compared with Stinger running on a 40-core CPU server. One of the key reasons is that GPMA+ and graph analytics can exploit the superb high memory bandwidth and massive parallelism of the GPU, as many graph applications are data- and compute-intensive. Second, GPMA+ is much more efficient than cuSparseCSR as maintaining the dynamic updates is often the bottleneck of continuous graph analytic processing and GPMA+ avoids the costly process of rebuilding the graph via incremental updates while bringing minimal overheads for existing graph algorithms running its graph structure. 7 Conclusion & Future Work In this paper, we address how to dynamically update the graph structure on GPUs in an efficient manner. First, we introduce a GPU dynamic graph analytic framework, which enables existing static GPU-oriented graph algorithms to support high-performance evolving graph analytics. Second, to avoid the rebuild of the graph structure which is a bottleneck for processing dynamic graphs on GPUs, we propose GPMA and GPMA+ to support incremental dynamic graph maintenance in parallel. We prove the scalability and complexity of GPMA+ theoretically and evaluate the efficiency through extensive experiments. As the future work, we would like to explore the hybrid CPU-GPU supports for dynamic graph processing and more effectiveness optimizations for involved applications. 22 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. Appendices A TryInsert+ Optimizations Based on different segment sizes, we propose three optimizations of Function TryInsert+ in Algorithm 4. The motivation is to obtain better memory access efficiency and lower cost of synchronization by balancing between problem scale and hardware hierarchy on GPU. The key computation logic of TryInsert+ is to merge two sorted arrays, i.e., existing segment entries and entries to be inserted. Standard approach for parallel merging needs to identify the position in merged array by binary search and then to execute parallel map, which requires heavy and uncoalesced memory accesses. Thus, depending on the size of the merge, we wish to employ different hardware hierarchies on GPU (i.e. warp, block and device) to minimize the cost of memory accesses. Before presenting the details of our optimizations, Algorithm 5 illustrates how to group threads according to their positions in different hierarchies of GPU architecture and how to target the groups to their assigned segments. In particular, each thread is assigned with a lane id, a block id and a global thread id to indicate the position of the thread in the corresponding warp, block and device work group. Each thread is assigned for one GPMA+ segment and the thread will ask other threads in the same work group to cooperate for its task. This means that each thread tries to drive a group of threads to deal with the assigned segment. Such a strategy lifts thread divergences caused by different execution branches. Note that this assignment policy will be used in our warp and block based optimizations as an initialization function. Algorithm 6 shows the Warp-Based optimization for any segments with entries no larger than the warp size. This implementation has high efficiency because explicit synchronization is not needed and all data is stored in registers. For each iteration, all threads of a particular warp will compete for the control of the warp as shown in line 11. The winner will drive the other threads in this warp to handle its required computation steps of the corresponding segment. As an example, line 27 counts valid entries in the segment concurrently. Lines 32-34 omit the remaining computation steps in TryInsert+, such as merging insertions and redistributing entries of segments, as their computation paradigm is similar to counting entries. Algorithm 7 shows the Block-Based optimization. It utilizes the shared memory, which has a higher volume than registers, to store data. Even though explicit synchronization is needed in line 12 and line 32 to guarantee consistency, synchronization in a block is highly optimized in GPU hardware and thus it does little effect to the overall performance. Both Warp-Based and Block-Based optimizations explicitly accommodate GPU features. As discussed in Section 5.2, although these two methods have limited memory for efficient access, they can handle most of the update requests. Algorithm 8 shows the Device-Based implementation. The implementation is different from the ones in Warp and Block based approaches, because it is designed for segments having a size larger than the shared memory size. Under this scenario, we have to handle them in the GPU’s global memory. One possible approach is to invoke an independent kernel for each large segment, but it will take considerable costs to initialize and schedule for multiple kernels. Hence, we propose an approach to handle a large number segments by only invoking a few unique kernel calls. 23 Accelerating Dynamic Graph Analytics on GPUs M. Sha et al. We illustrate the idea by showing how to perform counting segments which are valid for insertions as an example. As shown in lines 5 and 12, all valid entries stored in GPMA+ segments are first marked, and then all valid entry counts are calculated by SumReduce in one kernel call. Line 16 generates valid indexes for segments which have enough free space to receive their corresponding insertions, which is used by the rest computation steps. Simply speaking, our approach executes in horizontal steps of the execution logic, in order to avoid load imbalance caused by branch divergences. Finally, merging and segment entries redistribution use the same mechanism and thus are omitted. 24 Accelerating Dynamic Graph Analytics on GPUs Algorithm 5 TryInsert+ Initialization 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 inline function ConstInit( void ) { // cuda protocol variables WARPS = blockDim / 32; warp_id = threadIdx / 32; lane_id = threadIdx % 32; thread_id = threadIdx; block_width = gridDim; grid_width = gridDim * blockDim; global_id = block_width * blockIdx + threadIdx; // infos for assigned segment seg_beg = segments[global_id]; seg_end = seg_beg + segment_width; // infos for insertions belong current segment ins_beg = offset[global_id]; ins_end = offset[global_id + 1]; insert_size = ins_end - ins_beg; // the upper number that current segment can hold upper_size = tau * segment_size; } Algorithm 6 TryInsert+ Warp-Based Optimization 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 kernel TryInsert+(int segments[m], int offsets[m], int insertions[n], int segment_width) { ConstInit(); volatile shared comm[WARPS][5]; warp_shared_register pma_buf[32]; while (WaryAny(seg_end - seg_beg)) { // compete for warp comm[warp_id][0] = lane_id; // winner controls warp in this iteration if (comm[warp_id][0] == lane_id) { seg_beg = seg_end; comm[warp_id][1] = seg_beg; comm[warp_id][2] = seg_end; comm[warp_id][3] = ins_beg; comm[warp_id][4] = ins_end; } memcpy(pma_buf, pma[seg_beg], segment_width); // count valid entries in this segment entry_num = 0; if (lane_id < segment_width) { valid = pma_buf[lane_id] == NULL ? 0 : 1; entry_num = WarpReduce(valid); } // check upper density if insert if (entry_num + insert_size) < upper_size) { // merge insertions with pma_buf // evenly redistribute pma_buf // mark all insertions successful memcpy(pma[seg_beg], pma_buf, segment_width); } } } 25 M. Sha et al. Accelerating Dynamic Graph Analytics on GPUs Algorithm 7 TryInsert+ Block-Based Optimization 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 kernel TryInsert+(int segments[m], int offsets[m], int insertions[n], int segment_width) { ConstInit(); volatile shared comm[5]; volatile shared pma_buf[segment_width]; while (BlockAny(seg_end - seg_beg)) { // compete for block comm[0] = thread_id; BlockSynchronize(); // winner controls block in this iteration if (comm[0] == lane_id) { seg_beg = seg_end; comm[1] = seg_beg; comm[2] = seg_end; comm[3] = ins_beg; comm[4] = ins_end; } memcpy(pma_buf, pma[seg_beg], segment_width); // count valid entries in this segment entry_num = 0; ptr = thread_id; while (ptr < segment_width) { valid = pma_buf[ptr] == NULL ? 0 : 1; entry_num += BlockReduce(valid); thread_id += block_width; } BlockSynchronize(); // same as lines 30-37 in Algorithm 5 } } Algorithm 8 TryInsert+ Device-Based Optimization 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 function TryInsert+(int segments[m], int offsets[m], int insertions[n], int segment_width) { int valid_flags[m * segment_width]; parallel foreach i in range(m): parallel foreach j in range(segment_width): if (pma[segments[i] + j] != NULL) { valid_flags[i * segment_width + j] = 1; } DeviceSynchronize(); int entry_nums[m]; DeviceSegmentedReduce(valid_flags, m, segment_size, entry_nums); DeviceSynchronize(); int valid_indexes[m]; parallel foreach i in range(m): if (entry_nums[i] + insert_size < upper_size) { valid_indexes[i] = i; } DeviceSynchronize(); RemoveIfTrue(valid_indexes); DeviceSynchronize(); // according to valid_indexes, segmentedly to: // merge insertions into segments // evenly redistribute segments // mark all insertions successful } 26 M. Sha et al. Accelerating Dynamic Graph Analytics on GPUs B M. Sha et al. Additional Experimental Results For Data Transfer We show the experimental results for using asynchronous streams for concurrently transmitting data on PCIe and running computations on the GPU. We only show the results for GPMA+. In ConnectedComponent, the data transferred on PCIe consists of two parts: the graph updates and the component label vector to all vertices computed by ConnectedComponent. In PageRank, the result vector to be fetched indicates PageRank scores, which has the same size as ConnectedComponent’s. The results in Figures 13 and 14 have shown that the data transfer is completely hidden by analytic processing on GPU and GPMA+ update. Slide Size Uniform Random Graph500 Reddit Pokec 1% 1% 1% 1% 0.1% 0.1% 0.1% 0.1% 0.01% 0.01% 0.01% 0.01% 0 25 50 75 100 125 0 Time (ms) GPMA+ Update 25 50 75 100 125 Time (ms) ConnectedComponent 0 10 20 30 Time (ms) Fetch Component Label Vector 40 0 10 20 Time (ms) 30 Send Updates Figure 13: Concurrent data transfer and Connected Component computation with asynchronous stream Slide Size Uniform Random Graph500 Reddit Pokec 1% 1% 1% 1% 0.1% 0.1% 0.1% 0.1% 0.01% 0.01% 0.01% 0.01% 0 50 100 150 200 250 Time (ms) 0 100 GPMA+ Update 200 300 400 Time (ms) PageRank 0 25 50 75 100 125 Time (ms) Fetch PageRank Vector Send Updates 0 25 50 75 100 Time (ms) 125 Figure 14: Concurrent data transfer and PageRank computation with asynchronous stream C The Performance of Handling Updates on Sorted Graphs Uniform Random Graph500 106 Reddit 106 105 104 104 104 104 102 101 1000 103 102 101 1000 10 1 3 10 Sliding Size 10 5 103 102 101 1000 10 AdjLists Time (ms) 105 Time (ms) 105 103 1 3 10 Sliding Size PMA 10 5 103 102 101 1000 10 Stinger Pokec 106 105 Time (ms) Time (ms) 106 cuSparseCSR 1 3 10 Sliding Size GPMA 10 5 101 103 Sliding Size 105 GPMA+ Figure 15: Performance comparison for updates with different batch sizes. The dashed lines represent CPU-based solutions whereas the solid lines represent GPU-based solutions. For the update results with sorted streaming orders, AdjLists performs the best among all approaches due to its efficient balanced binary tree update mechanism. Meanwhile, a batch of sorted updates makes GPMA very inefficient as all updating threads within the batch conflict. Thanks to the non-locking optimization introduced, the update performance of GPMA+ is still significantly faster than that of the rebuild approach (cuSparseCSR) with orders of magnitude speedups for small batch sizes. 27 Accelerating Dynamic Graph Analytics on GPUs D M. Sha et al. Additional Experimental Results For Graph Streams with Explicit Deletions We present the experimental results for graph streams which involve explicit deletions. In this section, we use the same stream setup which is mentioned in Section 6.1. However, for each iteration of sliding, we will randomly pick a set of edges belonging to current sliding window insteading of the head part as edges to be deleted. Slide Size Uniform Random Graph500 1% 15 4.4 0.7 0.1% 14 2.1 0.3 12 2.4 0.01% 0.0 0.05 0.1 0.15 0.2 0.25 0.3 Time (second) AdjLists Reddit 1% 14 4.8 23 0.1% 0.01% PMA Pokec 1% 1.6 1.2 0.4 1% 1.6 0.8 0.4 18 1.7 4.4 0.1% 1.7 0.7 0.4 0.1% 1.9 0.6 0.5 19 1.4 2.1 0.01% 1.7 0.6 0.4 0.01% 1.4 0.4 0.6 0.0 0.05 0.1 0.15 0.2 0.25 0.3 Time (second) Stinger cuSparseCSR 0.0 0.01 0.02 0.03 0.04 0.05 Time (second) BFS (patterned) 0.0 GPMA+ 0.01 0.02 0.03 0.04 0.05 Time (second) Update (unpatterned) Figure 16: Streaming BFS with explicit deletions Slide Size Uniform Random Graph500 Reddit Pokec 1% 15 4.4 1.5 1% 16 5.7 23 1% 1.8 0.9 1.0 1% 1.5 0.7 0.5 0.1% 16 1.9 1.3 0.1% 19 2.5 4.9 0.1% 2.0 0.6 0.9 0.1% 1.6 0.4 0.5 0.01% 11 1.9 1.0 0.01% 14 1.9 1.9 0.01% 2.3 0.5 0.6 0.01% 1.7 0.4 0.5 0.0 0.1 0.2 0.3 0.4 Time (second) AdjLists 0.5 0.0 PMA 0.1 0.2 0.3 0.4 Time (second) cuSparseCSR Stinger 0.5 0.0 GPMA+ 0.02 0.04 0.06 0.08 0.1 Time (second) ConnectedComponent (patterned) 0.0 0.02 0.04 0.06 0.08 Time (second) Update (unpatterned) 0.1 Figure 17: Streaming Connected Component with explicit deletions Uniform Random Slide Size 1% 42 4.5 0.1% 35 4.0 0.01% 0.0 Graph500 44 9.2 3.6 0.5 1.0 1.5 2.0 2.5 3.0 Time (second) AdjLists Reddit 1% 70 21 25 1% 0.1% 37 5.1 4.8 0.1% 30 4.9 0.01% 0.0 PMA 0.5 1.0 Stinger 8.1 2.2 6.0 1.6 0.01% 1.5 2.0 2.5 3.0 Time (second) cuSparseCSR 0.0 GPMA+ Pokec 12 4.9 1.0 0.2 0.4 0.6 0.8 1.0 Time (second) PageRank (patterned) 14 3.0 1% 9.7 1.8 0.1% 6.9 1.2 0.01% 0.0 0.2 0.4 0.6 0.8 Time (second) Update (unpatterned) 1.0 Figure 18: Streaming PageRank with explicit deletions Figures 16, 17 and 18 illustrate the results of three streaming applications respectively. 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Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 Vers 02 Nov 2016 A State Space Approach for Piecewise‐Linear Recurrent Neural Networks for Reconstructing Nonlinear Dynamics from Neural Measurements Short title: Nonlinear State Space Model for Reconstructing Computational Dynamics Daniel Durstewitz Dept. of Theoretical Neuroscience, Bernstein Center for Computational Neuroscience Heidelberg‐ Mannheim, Central Institute of Mental Health, Medical Faculty Mannheim/ Heidelberg University daniel.durstewitz@zi‐mannheim.de 1 Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 2 Abstract The computational and cognitive properties of neural systems are often thought to be implemented in terms of their network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single‐unit recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a (lower‐ dimensional) state space representation of the dynamics, but would wish to have access to its (computational) governing equations for in‐depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi‐analytical maximum‐likelihood estimation scheme for piecewise‐linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation‐Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, and using inference through particle filters for comparison, the approach is applied to multiple single‐unit recordings from the rodent anterior cingulate cortex (ACC) obtained during performance of a classical working memory task, delayed alternation. A model with 5 states turned out to be sufficient to capture the essential computational dynamics underlying task performance, including stimulus‐selective delay activity. The estimated models were rarely multi‐stable, however, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi‐analytical (thus reasonably fast) maximum‐likelihood estimation framework for PLRNNs that may enable to recover the computationally relevant dynamics underlying observed neuronal time series, and directly link them to computational properties. Author Summary Neuronal dynamics mediate between the physiological and anatomical properties of a neural system and the computations it performs, in fact may be seen as the ‘computational language’ of the brain. It is therefore of great interest to recover from experimentally recorded time series, like multiple single‐unit or neuroimaging data, the underlying network dynamics and, ideally, even its governing equations. This is not at all a trivial enterprise, however, since neural systems are very high‐ dimensional, come with considerable levels of intrinsic (process) noise, are usually only partially observable, and these observations may be further corrupted by noise from measurement and preprocessing steps. The present article embeds piecewise‐linear recurrent neural networks (PLRNNs) within a state space approach, a statistical estimation framework that deals with both process and observation noise. PLRNNs are computationally and dynamically powerful model systems. Their statistically principled estimation from multivariate neuronal time series thus may provide access to some essential features of the neuronal dynamics, like attractor states, their governing equations, and their computational implications. The approach is exemplified on multiple single‐unit recordings from the rat prefrontal cortex during working memory. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 3 Introduction Neural dynamics mediate between the underlying biophysical and physiological properties of a neural system and its computational and cognitive properties (e.g. [1‐4]). Hence, from a computational perspective, we are often interested in recovering the neural network dynamics of a given brain region or neural system from experimental measurements. Yet, experimentally, we commonly have access only to noisy recordings from a relatively small proportion of neurons (compared to the size of the brain area of interest), or to lumped surface signals like local field potentials or the EEG. Inferring from these the computationally relevant underlying dynamics is therefore not trivial, especially since both the neural system itself (e.g., stochastic synaptic release; [5]) as well as the recorded signals (e.g., spike sorting errors; [6]) come with a good deal of noise. Speaking in statistical terms, 'model‐free' techniques which combine state space reconstruction methods (delay embeddings) from nonlinear dynamics with nonlinear basis expansions and kernel techniques have been one approach to the problem [7, 8]. These techniques provide informative lower‐dimensional visualizations of population trajectories and approximations to the neural flow field, but they highlight only certain, salient aspects of the dynamics and do not return its governing equations (e.g. [9]) or underlying computations. Alternatively, state space models, a statistical framework particularly popular in engineering and ecology (e.g. [10]), have been adapted to extract lower‐dimensional neural trajectory flows from higher‐dimensional recordings [11‐21]. State space models link a process model of the unobserved (latent) underlying dynamics to the experimentally observed time series via observation equations, and differentiate between process noise and observation noise (e.g. [22]). So far, with few exceptions (e.g. [19, 23]), these models assumed linear latent dynamics, however. Although this may be sufficient to yield smoothed trajectories and reduced state space representations, it implies that the recovered dynamical model by itself is not powerful enough to reproduce a range of important dynamical and computational phenomena in the nervous system, among them multi‐stability which has been proposed to underlie neural activity during working memory [24‐28]. Here we derive a new state space algorithm based on piecewise‐linear (PL) recurrent neural networks (RNN). It has been shown that RNNs with nonlinear activation functions can, in principle, approximate any dynamical system's trajectory or, in fact, dynamical system itself (given some general conditions; [29‐31]). Thus, in theory, they are powerful enough to recover whatever dynamical system is underlying the experimentally observed time series. Piecewise linear activation functions, in particular, are by now the most popular choice in deep learning algorithms [32, 33], and considerably simplify some of the derivations within the state space framework (as shown later). They may also be more apt for producing working memory‐type activity with longer delays if for some units the transfer function happens to coincide with the bisectrix (cf. [34]), and ease the analysis of fixed points and stability. We then apply this newly derived algorithm to multiple single‐ unit recordings from the rat prefrontal cortex obtained during a classical delayed alternation working memory task [35]. Results State space model Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 4 This article considers simple discrete‐time piecewise‐linear (PL) recurrent neural networks (RNN) of the form (1) z t  Az t 1  W max{0, z t 1  θ}  s t  ε t , ε t ~ N (0, Σ) , where z t  ( z1t ...z Mt )T is the (M1)‐dimensional (latent) neural state vector at time t=1…T, A  diag ([a11...aMM ]) is an MM diagonal matrix of auto‐regression weights, W  (0 w12 ...w1M , w21 0 w23 ...w2 M , w31 w32 0 w34 ...w3 M ,...) is an MM off‐diagonal matrix of connection weights, θ  (1... M )T is a set of (constant) activation thresholds, s t is a sequence of (known) external inputs, and ε t denotes a Gaussian white noise process with diagonal covariance 2 ]) . The max‐operator is assumed to work element‐wise. matrix Σ  diag ([ 112 ... MM Before proceeding further, two things are worth pointing out: First, more complicated PL functions may, in principle, be constructed from (1) by properly connecting simple PL units, combined with an appropriate choice of activation thresholds  (and acknowledging the activation lags among units). Second, all fixed points (in the absence of external input) of the PLRNN (1) could be obtained by solving the 2M linear equations (2) z *  ( A  W  I ) 1 Wθ , where  is to denote the set of indices of units for which we assume z m   m , and W the respective connectivity matrix in which all columns from W corresponding to units   are set to 0. Obviously, to make z * a true fixed point of (1), the solution to (2) has to be consistent with the defined set , that is z*m   m has to hold for all m   and z*m   m for all m   . For networks of moderate size (say M<30) it is thus computationally feasible to explicitly check for all fixed points and their stability. Here, latent state model (1) is then connected to some N‐dimensional observed vector time series X={xt} via a simple linear‐Gaussian model, (3) x t  B (z t )  ηt , ηt ~ N (0, Γ) , where  (z t ) : max{0, z t  θ} , {ηt } is the (white Gaussian) observation noise series with diagonal 2 covariance matrix Γ  diag ([ 112 ... NN ]) , and B an NM matrix of regression weights. Thus, the idea is that only the PL‐transformed activation  (z t ) reaches the ‘observation surface’ (as, e.g., with spiking activity when the underlying membrane dynamics itself is not visible). We further assume for the initial state, (4) z1 ~ N (μ 0  s1 , Σ) , with, for simplicity, the same covariance matrix as for the process noise in general (reducing the number of to be estimated parameters). In the case of multiple, temporally separated trials, we allow each one to have its own individual initial condition μ k , k  1...K . Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 5 The general goal here is to determine both the model’s unknown parameters Ξ  {μ 0 , A, W, Σ, B, Γ} (assuming fixed thresholds  for now) as well as the unobserved, latent state path Z : {z t } (and its second‐order moments) from the experimentally observed time series {xt}. These could be, for instance, properly transformed multivariate spike time series or neuroimaging data. This is accomplished here by the Expectation‐Maximization (EM) algorithm which iterates state (E) and parameter (M) estimation steps and is developed in detail for model (1), (3), in the Methods. In the following I will first discuss state and parameter estimation separately for the PLRNN, before describing results from the full EM algorithm in subsequent sections. This will be done along two toy problems, an higher‐order nonlinear oscillation (stable limit cycle), and a simple 'working memory' paradigm in which one of two discrete stimuli had to be retained across a temporal interval. Finally, the application of the validated PLRNN EM algorithm will be demonstrated on multiple single‐unit recordings obtained from rats on a standard working memory task (delayed alternation; data from [35], kindly provided by Dr. James Hyman, University of Nevada, Las Vegas). State estimation The latent state distribution, as explained in Methods, is a high‐dimensional (piecewise) Gaussian mixture with the number of components growing as 2TM with sequence length T and number of latent states M. Here a semi‐analytical, approximate approach was developed that treats state estimation as a combinatorial problem by first searching for the mode of the full distribution (cf. [36, 37]; in contrast, e.g., to a recursive filtering‐smoothing scheme that makes local [linear‐Gaussian] approximations, e.g. [11], cf. [22]). This approach amounts to solving a high (2MT)‐dimensional piecewise linear problem (due to the piecewise quadratic, in the states Z, log‐likelihood eq. 6, 7). Here this was accomplished by alternating between (1) solving the linear set of equations implied by a given set of linear constraints Ω : {(m, t ) | z mt   m } (cf. eq. 7 in Methods) and (2) flipping the sign of the constraints violated by the current solution z * () to the linear equations, thus following a path through the (MT)‐dimensional binary space of linear constraints using Newton‐type iterations (similar as in [38], see Methods). Given the mode and state covariance matrix (evaluated at the mode from the negative inverse Hessian), all other expectations needed for the EM algorithm were then derived analytically, with one exception that was approximated (see Methods for full details). The toy problems introduced above were used to assess the quality of these approximations. For the first toy problem, an order‐15 limit cycle was produced with a PLRNN consisting of three recurrently coupled units, inputs to units #1 and #2, and parameter settings as indicated in Fig. 1 and provided Matlab file ‘PLRNNoscParam.mat’. The limit cycle was repeated for 50 full cycles (giving 750 data points) and corrupted by process noise (cf. Fig. 1). These noisy states (arranged in a (3 x 750) matrix Z) were then transformed into a (3 x 750) output matrix X, to which observation noise was added, through a randomly drawn (3 x 3) regression weight matrix B. State estimation was started from a random initial condition. True (but noise‐corrupted) and estimated states for this particular problem are illustrated in Fig. 1A, indicating a tight fit (although some fraction of the linear constraints were still violated, 0.27% in the present example and <2.3% in the working memory example below; see Methods on this issue). To examine more systematically the quality of the approximate‐analytical estimates of the first and second order moments of the joint distribution across states z and their piecewise linear Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 6 transformations (z), samples from p(Z|X) were simulated using bootstrap particle filtering (see Methods). Although these simulated samples are based only on the filtering (not the smoothing) steps (and (re‐)sampling schemes may have issues of their own; e.g. [22]), analytical and sampling estimates were in tight agreement, correlating almost to 1 for this example, as shown in Fig. 2. Fig. 3A illustrates the setup of the ‘two‐cue working memory task’, chosen for later comparability with the experimental setup. A 5‐unit PLRNN was first trained by conventional gradient descent (‘real‐time recurrent learning’ (RTRL), see [39, 40]) to produce a series of six 1’s on unit #3 and six 0’s on unit #4 five time steps after an input (of 1) occurred on unit #1, and the reverse pattern (six 0’s on unit #3 and six 1’s on unit #4) five time steps after an input occurred on unit #2. A stable PLRNN with a reasonable solution to this problem was then chosen for further testing the present algorithm (cf. Fig. 3C). (While the RTRL approach was chosen to derive a working memory circuit with reasonably ‘realistic’ characteristics like a wider distribution of weights, it is noted that a multi‐stable network is relatively straightforward to construct explicitly given the analytical accessibility of fixed points [see Methods]; for instance, choosing θ  (0.5 0.5 0.5 0.5 2) , A  (0.9 0.9 0.9 0.9 0.5) , and W  (0        ,  0       ,     0    ,      0   , 11 11 0) with   0.2 , yields a tri‐stable system.) Like for the limit cycle problem before, the number of observations was taken to be equal to the number of latent states, and process and observation noise were added (see Fig. 4 and Matlab file ‘PLRNNwmParam.mat’ for specification of parameters). The system was simulated for 20 repetitions of each trial type (i.e., cue‐1 or cue‐2 presentations) with different noise realizations and each ‘trial’ started from its own initial condition μ k (see Methods), resulting in a total series length of T=20220=800 (although, importantly, in this case the time series consisted of distinct, temporally segregated trials, instead of one continuous series, and was treated as such an ensemble of series by the algorithm). As before, state estimation started from random initial conditions and was provided with the correct parameters, as well as with the observation matrix X. While Fig. 3B illustrates the correlation between true (i.e., simulated) and estimated states across all trials and units, Fig. 3C shows true and estimated states for a representative cue‐1 (left) and cue‐2 (right) trial, respectively. Again, our procedure for obtaining the maximum a‐posteriori (MAP) estimate of the state distribution appears to work quite well (in general, only locally optimal solutions can be guaranteed, however, and the algorithm may have to be repeated with different state initializations; see Methods). Parameter estimation Given the true states, how well would the algorithm retrieve the parameters of the PLRNN? To assess this, the actual model states (which generated the observations X) from simulation runs of the oscillation and the working memory task described above were provided as initialization for the E‐ step. Based on these, the algorithm first estimated the state covariances for z and (z) (see above), and then the parameters in a second step (i.e., the M‐step). Note that the parameters can all be computed analytically given the state distribution (see Methods), and, provided the state covariance matrices (summed across time) as required in eqn. 17a,d,f are non‐singular, have a unique solution. Hence, in this case, any misalignment with the true model parameters can only come from one of two sources: i) estimation was based on one finite‐length noisy realization of the PLRNN process, ii) all second order moments of the state distribution were still estimated based on the true state vectors. However, as can be appreciated from Fig. 1B (oscillation) and Fig. 4 (working memory), for Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 7 the two example scenarios studied here, all parameter estimates still agreed tightly with those describing the true underlying model. In the more general case where both the states and the parameters are unknown and only the observations are given, note that the model as stated in eqns. 1 & 3 is over‐specified as, for instance, at the level of the observations, additional variance placed into  can be compensated for by adjusting  accordingly (cf. [41, 42]). In the following we therefore always arbitrarily fixed Σ  I 10 2 , as common in many latent variable models (like factor analysis), including state space models (e.g. [23, 43]). Joint estimation of states and parameters by EM The observations above confirm that our algorithm finds satisfactory approximations to the underlying state path and state covariances when started with the right parameters, and ‐ vice versa ‐ identifies the correct parameters when provided with the true states. Indeed, the M‐step, since it is exact, can only increase the expected log‐likelihood eq. 5 with the present state expectancies fixed. However, due to the system's piecewise‐defined discrete nature, modifying the parameters may lead to a new set of constraint violations, that is may throw the system into a completely different linear subspace which may imply a decrease in the likelihood in the E‐step. It is thus not guaranteed that a straightforward EM algorithm converges (cf. [44, 45]), or that the likelihood would even monotonically increase with each EM iteration. To examine this issue, full EM estimation of the WM model (as specified in Fig. 4, using N=20 outputs in this case) was run 240 times, starting from different random, uniformly distributed initializations for the parameters. Fig. 5B gives, for the maximum likelihood solution across all 240 runs (Fig. 5A), the correlations between true and estimated states for all five state variables of the WM model. Note that estimated and true model states may not be in the same order, as any permutation of the latent state indices together with the respective columns of observation matrix B will be equally consistent with the data X (see also [23]). For the WM model examined here, however, partial order information is implicitly provided to the EM algorithm through the definition of unit‐specific inputs sit . For the present example, true and estimated states were nicely linearly correlated for all 5 latent variables (Fig. 5B), but some of the regression slopes significantly differed from 1, indicating a degree of freedom in the scaling of the states. More generally, there may not even be a clear linear relationship with a single latent state, although, if the estimation was successful, a linear ˆ may usually map the estimated onto the true states. This is because, if the transformation VZ observation eq. were strictly linear, any linear transformation of the latent states by some matrix V could essentially be reversed at the level of the outputs by back‐multiplying B with V‐1 (cf. [23]; note that here the piecewise linearity through  (z ) in eq. 1, 3, ignored in the argument above, complicates matters, however). This implies that the model is (at most) identifiable only up to this linear transformation, which might not be a serious issue, however, if one is interested primarily in the latent dynamics (rather than in the exact parameters). Fig. 6 illustrates the distribution of initial and final parameter estimates around their true values across all 240 runs (before and after reordering the estimated latent states based on the rotation that would be required for achieving the optimal mapping onto the true states, as determined through Procrustes analysis). Fig. 6 reveals that a) the EM algorithm does clearly improve the estimates and b) these final estimates seemed to be largely unbiased (deviations centered around 0). Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 8 Application to experimental recordings I next was interested in what kind of structure the present PLRNN approach would retrieve from experimental multiple (N=19) single‐unit recordings obtained while rats were performing a simple and well‐examined working memory task, namely spatial delayed alternation [35] (see Methods). (Note that in the present context this analysis is mainly meant as an exemplification of the current model approach, not as a detailed examination of the working memory issue itself.) The delay was always initiated by a nose poke of the animal into a port located on the side opposite from the response levers, and had a minimum length of 10 s. Spike trains were first transformed into kernel density estimates by convolution with a Gaussian kernel (see Methods), as done previously (e.g. [8, 46, 47]), and binned with 500 ms resolution. This also renders the observed data more suitable to the Gaussian noise assumptions of the present observation model, eq. 3. Models with 5 and 8 latent states were estimated, but only results from the former will be reported here (as 8 states did not appear to yield any additional insight). Periods of cue presentation were indicated to the model by setting external inputs sit  1 to units i=1 (left lever) or i=2 (right lever) for three 500 ms time bins surrounding the event (and sit  0 otherwise). The EM algorithm was started from 36 different initializations of the parameters (including thresholds ), and the 5 highest likelihood solutions were considered further. Fig. 7A gives the model log‐likelihoods across EM iterations for these 5 highest‐likelihood solutions. Interestingly, there were single neurons whose responses were predicted very well by the estimated model despite large trial‐to‐trial fluctuations (Fig. 7B, top row), while there were others with similar trial‐to‐trial fluctuations for which the model only captured the general trend (Fig. 7B, bottom row). This could potentially suggest that trial‐to‐trial fluctuations in single neurons could be for very different reasons: In those cases where strongly varying single unit responses are nevertheless highly predictable, at least a considerable proportion of their trial‐to‐trial fluctuations must have been captured by the deterministic part of the model’s latent state dynamics, hence may be due to different (trial‐unique) initializations of the states (recall that the states are not free to vary in accounting for the observations, but are tightly constrained by the model’s temporal consistency requirements). In contrast, when only the average trend is captured, the neuron’s trial‐to‐trial fluctuations likely represent true intrinsic (or measurement) noise sources that the model’s deterministic part cannot account for. This observation highlights that (nonlinear) state space models could potentially also provide new insights into other long‐standing questions in neurophysiology. Fig. 8 shows the five trial‐averaged latent states for both left‐ and right‐lever trials for one of the highest likelihood solutions. Not surprisingly, the first two state variables (receiving external input) exhibit a strong cue response for the left vs. right lever, respectively. The third latent variable appears to reflect the (end‐of‐trial) motor response, while the fourth and fifth state variable clearly distinguish between the left and right lever options throughout the delay period of the task, in this sense carrying a memory of the cue (previous response) within the delay. Hence, for this particular data set, the extracted latent states appear to summarize quite well the most salient computational features of this simple working memory task. Further insight might be gained by examining the system’s fixed points and their eigenvalue spectrum. For this purpose, the EM algorithm was started from 200 different initial conditions (that is, initial parameter estimates and threshold settings ) with maximum absolute eigenvalues (of the corresponding fixed points) drawn from a relatively uniform distribution within the interval [0 3]. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 9 Although the estimation process rarely returned truly multi‐stable solutions (just 2% of all cases), there was a clear trend for the final maximum eigenvalues to aggregate around 1 (Fig. 9), that is to produce models with very slow dynamics. Indeed, effectively slow dynamics is all that is needed to bridge the delays (see also [1]), while true multi‐stability may perhaps even be the physiologically less likely scenario (e.g. [48, 49]). (Reducing the bin width from 500 ms to 100 ms appeared to produce solutions with eigenvalues even closer to 1 while retaining stimulus selectivity across the delay, but this observation was not followed up more systematically here). Discussion Reconstructing neuronal dynamics parametrically and non‐parametrically In the present work, a semi‐analytical, maximum‐likelihood (ML) approach for estimating piecewise‐ linear recurrent neural networks (PLRNN) from brain recordings was developed. The idea is that such models would provide 1) a representation of neural trajectories and computationally relevant dynamical features underlying high‐dimensional experimental time series in a much lower‐ dimensional latent variable space (cf. [16, 19]), and 2) more direct access to the neural system’s computational properties. Specifically, once estimated to reproduce the data (in the ML sense), such models may allow for more detailed analysis and in depth insight into the system’s computational dynamics, e.g. through an analysis of fixed points and their linear stability (e.g. [24, 26, 28, 50‐56]), which is not directly accessible from the experimental time series. Model‐free (non‐parametric) techniques, usually based on Takens’ delay embedding theorem [57] and extensions thereof [58, 59], have also frequently been applied to gain insight into neuronal dynamics and its essential features, like attracting states associated with different task phases from in‐vivo multiple single‐unit recordings [7, 8] or unstable periodic orbits extracted from relatively low‐ noise slice recordings [60]. In neuroscience, however, one commonly deals with high‐dimensional observations, as provided by current multiple single‐unit or neuroimaging techniques (which still usually constitute just a minor subset of all the system’s dynamical variables). In addition, there is a large variety of both process and measurement noise sources. The former include potential thermal noise sources and the probabilistic behavior of single ion channel gating [61], probabilistic synaptic release [5], fluctuations in neuromodulatory background and hormone levels, and a large variety of uncontrollable external noise sources via the sensory surfaces, including somatosensory and visceral feedback from within the body. Measurement noise may come from direct physical sources like, for instance, instabilities and movement in the tissue surrounding the recording electrodes, noise properties of the recording devices themselves, the mere fact that only a fraction of all system variables is experimentally accessed (‘sampling noise’), or may result from preprocessing steps like spike sorting (e.g. [62, 63]). This is therefore a quite different scenario from the comparatively low‐ dimensional and low‐noise situations in, e.g., laser physics [64], and delay‐embedding‐based approaches to the reconstruction of neural dynamics may have to be augmented by machine learning techniques to retrieve at least some of its most salient features [7, 8]. Of course, model‐based approaches like the one developed here are also plagued by the high dimensionality and high noise levels inherent in neural data, but perhaps to a lesser extent than approaches like delay embeddings that aim to directly construct the state space from the observations (see also [65]). This is because models as pursued in the statistical state space Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 10 framework explicitly incorporate both process and measurement noise into the system’s description. Also, as long as the latent variable space itself is relatively small and related to the observations by simple linear equations, as here, the high dimensionality of the observations themselves does not constitute a too serious issue for estimation. More importantly, however, it is of clear advantage to have access to the governing equations themselves, as only this allows for an in depth analysis of the system’s dynamics and its relation to neural computation (e.g. [2, 24, 26, 53, 54, 56, 66]). For instance, recurrent network models have been trained in the past to perform behavioral tasks or reproduce behavioral data to infer the dynamical mechanisms potentially underlying working memory [67] or context‐dependent decision making [54], although commonly, as in the cited cases, not within a statistical framework (or not even by direct estimation from experimental data). There are also approaches which are somewhat in between, attempting to account for the observations directly, without reference to an underlying latent variable model, through differential equations expressed in terms of nonlinear basis expansions in the observations, estimated through strongly regularized (penalized) regression methods ([7], see also [9]). It remains to be investigated how well such methods, which go without a noise model and face high data dimensionality directly, transfer to neuroscience problems. Estimation of neural state space models State space models are a popular statistical tool in many fields of science (e.g. [10, 68]), although their applications in neuroscience are of more recent origin [11, 12, 14, 17‐19, 21]. The Dynamic Causal Modeling (DCM) framework advanced in the human fMRI literature to infer the functional connectivity of brain networks and their dependence on task conditions [68, 69] may be seen as a state space approach, although these models usually do not contain process noise (except for the recently proposed ‘stochastic DCM’ [69]) and are commonly estimated through Bayesian inference, which imposes more constraints (via computational burden) on the complexity of the models that can reasonably be dealt with in this framework. In neurophysiology, Smith & Brown [11] were among the first to suggest a state space model for multivariate spike count data by coupling a linear‐ Gaussian transition model with Poisson observations, with state estimation achieved by making locally Gaussian approximations to eq. 18. Similar models have variously been used subsequently to infer local circuit coding properties [14] or, e.g., biophysical parameters of neurons or synaptic inputs from postsynaptic voltage recordings [70, 12]. Yu et al. [21] proposed Gaussian Process Factor Analysis (GPFA) for retrieving lower‐dimensional, smooth latent neural trajectories from multiple spike train recordings. In GPFA, the correlation structure among the latent variables is specified (parameterized) explicitly rather than being given through a transition model. Buesing et al. [16] discuss regularized forms of neural state space models to enforce their stability, while Macke et al. [18] review different estimation methods for such models like the Laplace approximation or variational inference methods. By far most of the models discussed above are linear in their latent dynamics, however (although observations may be non‐Gaussian). Although this may often be sufficient to uncover important properties of underlying latent processes or structures, like connectivity or synaptic/neuronal parameters, or to obtain lower‐dimensional representations of the observed process, it is not suitable for retrieving the system dynamics or computations, as linear systems are strongly limited in the repertoire of dynamics (and computations) they can produce (e.g. [50, 71]). There are a few exceptions, however, the current work builds on: Yu et al. [19] suggested a RNN with sigmoid‐type activation function (using the error function), coupled to Poisson spike count outputs, Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 11 and used it to reconstruct the latent neural dynamics underlying motor preparation and planning in non‐human primates. In their work, they combined the Gaussian approximation suggested by Smith & Brown [11] with the Extended Kalman Filter (EKF) for estimation within the EM framework. These various approximations in conjunction with the iterative EKF estimation scheme may be quite prone to numerical instabilities and accumulating errors, however (cf. [22]). Earlier work by Roweis & Ghahramani [23] used radial basis function (RBF) networks as a partly analytically tracktable approach. Nonlinear extensions to DCM, incorporating quadratic terms, have been proposed as well recently [72]. State and parameter estimation has also been attempted in (noisy) nonlinear biophysical models [73,74], but these approaches are usually computationally expensive, especially when based on numerical sampling [74], while at the same time pursuing objectives somewhat different from those targeted here (i.e., less focused on computational properties). Nevertheless, nonlinear neural state space models remain an under‐researched topic in theoretical neuroscience. In the present work, PLRNNs were therefore chosen as a mathematically comparatively tracktable, yet computationally powerful nonlinear recurrent network approach that can reproduce a wide range of nonlinear dynamical phenomena [75‐78]. Given its semi‐analytical nature, the present algorithm runs reasonably fast (and starting it from a number of different initializations to approach a globally optimal solution is computationally very feasible). However, its mathematical properties (among them, issues of convergence/monotonicity, local maxima/uniqueness and existence of solutions, and identifiability [of dynamics]) certainly require further illumination and may lead to further algorithmic improvements. Mechanisms of working memory Although the primary focus of this work was to develop and evaluate a state space framework for PLRNNs, some discussion of the applicational example chosen here, working memory, is in order. Working memory is generally defined as the ability to actively hold an item in memory, in the absence of guiding external input, for short‐term reference in subsequent choice situations [79]. Various neural mechanisms have been proposed to underlie this cognitive capacity, most prominently multi‐stable neural networks which retain short‐term memory items by switching into one of several stimulus‐selective attractor states (e.g. [24, 25, 28]). These attractors usually represent fixed points in the firing rates, with assemblies of recurrently coupled stimulus‐selective cells exhibiting high rates while those cells not coding for the present stimulus in short‐term memory remaining at a spontaneous low‐rate base level. These models were inspired by the physiological observation of ‘delay‐active’ cells [80‐82], that is cells that switch into a high‐rate state during the delay periods of working memory tasks, and back to a low‐rate state after completion of a trial, similar to the ‘delay‐active’ latent states observed in Fig. 8. Nakahara & Doya [83] were among the first to point out, however, that, for working memory, it may be completely sufficient (or even advantageous) to tune the system close to a bifurcation point where the dynamics becomes very slow (see also [1]), and true multi‐stability may not be required. This is supported by the present observation that most of the estimated PLRNN models had fixed points with eigenvalues close to 1 but were not truly bi‐ or multi‐stable (cf. Fig. 9), yet this was sufficient to account for maintenance of stimulus‐selectivity throughout the 10 s delay of the present task (cf. Fig. 8) and for experimental observations (cf. Fig. 7). Recently, a number of other mechanisms for supporting working memory, however, including sequential activation of cell populations [84] or synaptic mechanisms [85] have been discussed. Thus, the neural mechanisms of working memory remain an active research area to Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 12 which statistical model estimation approaches as developed here may significantly contribute, but too broad a topic in its own right to be covered in more depth by this mainly methodological work. Models and Methods Expectation‐Maximization Algorithm: State estimation As with most previous work on estimation in (neural) state space models [16, 18, 19, 22], we use the Expectation‐Maximization (EM) framework for obtaining estimates of both the model parameters and the underlying latent state path. Due to the piecewise‐linear nature of model (1), however, the conditional latent state path density p(Z|X) is a high‐dimensional ‘mixture’ of partial Gaussians, with the number of integrations to perform to obtain moments of p(Z|X) scaling as 2TM. Although analytically accessible, this will be computationally prohibitive for almost all cases of interest. Our approach therefore focuses on a computationally reasonably efficient way of searching for the mode (maximum a‐posteriori, MAP estimate) of p(Z|X) which was found to be in good agreement with E(Z|X) in most cases. Covariances were then approximated locally around the MAP estimate. More specifically, the EM algorithm maximizes the expected log‐likelihood of the joint distribution p(X,Z) as a lower bound on log p ( X | Ξ) [23], where Ξ  {μ 0 , A, W, Σ, B, Γ} denotes the set of to‐be‐optimized‐for parameters (note that we dropped the thresholds  from this for now): Q(Ξ, Z) : E[log p (Z, X | Ξ)]  1   E  (z1  μ 0  s1 )T Σ 1 (z1  μ 0  s1 )  2   1 T   E   (z t  Az t 1  W (z t 1 )  s t )T Σ 1 (z t  Az t 1  W (z t 1 )  s t )  2 t 2  (5)  1 T  T  E   (x t  B (z t ))T Γ 1 (x t  B (z t ))  (log | Σ |  log | Γ |) .  2 t 1  2 For state estimation (E‐step), if  were a linear function, obtaining E (Z | X, Ξ) would be equivalent to maximizing the argument of the expectancy in (5) w.r.t. Z, i.e., E[ Z | X, Ξ]  arg max Z log p (Z, X | Ξ) (see [12]; see also [86]). This is because for a Gaussian mean and mode coincide. In our case, p(X,Z) is piecewise Gaussian, and we still take the approach (suggested in [12]) of maximizing log p(Z, X | Ξ) directly w.r.t. Z (essentially a Laplace approximation of p ( X | Ξ) where we neglect the Hessian which is constant around the maximizer; cf. [12,37]). Let (t )  {1...M } be the set of all indices of the units for which we have z mt   m at time t, and W ( t ) and B  ( t ) the matrices W and B , respectively, with all columns with indices  (t ) set to 0. The state estimation problem can then be formulated as 13 Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 (6) maximize 1 QΩ* (Z) :  (z1  μ 0  s1 )T Σ 1 (z1  μ 0  s1 ) 2 1 T   [z t  ( A  W ( t 1) )z t 1  W ( t 1)θ  s t ]T Σ 1[z t  ( A  W ( t 1) )z t 1  W (t 1)θ  s t ] 2 t 2 1 T (x t  B  ( t ) z t  B  ( t )θ)T Γ 1 (x t  B  (t ) z t  B  ( t )θ)  2 t 1 w.r.t. (Ω, Z) , subject to ztm   m t , m  (t ) AND ztm   m t , m  (t )  Let us concatenate all state variables into one long column vector, z  (z1 ,..., z T )  ( z11...z mt ...z MT )T , and unwrap the sums across time into large, block‐banded MTMT matrices (see [12, 71]) in which we combine all terms quadratic or linear in z, or  (z ) , respectively. Further, define d  as the binary (MT1) indicator vector which has 1s everywhere except for the entries with indices  Ω  {1...MT } which are set to 0, and let D  : diag (d  ) the MTMT diagonal matrix formed from d  . Let Θ : (θ, θ,..., θ) ( MT 1) , and Θ  M the same vector shifted downward by M positions, with the first M entries set to 0. One may then rewrite QΩ* (Z ) in the form QΩ* (Z)   (7)  1 T z (U 0  D U1  U1T D  D U 2 D  )z 2  z T ( v 0  D  v1  V2 diag[d  M ]Θ  M  V3 DΘ  D V4 D Θ)   ( v 0  D v1  V2 diag[d  M ]Θ  M  V3 DΘ  D V4 D Θ)T z  const. The MTMT matrices U{0...2} separate product terms that do not involve  (z ) ( U 0 ), involve multiplication by  (z ) only from the left‐hand or right‐hand side ( U1 ), or from both sides ( U 2 ). Likewise, for the terms linear in z, vector and matrix terms were separated that involved z mt or  m conditional on z mt   m (please see the provided MatLab code for the exact composition of these matrices). For now, the important point is that we have 2M T different quadratic equations, depending on the bits on and off in the binary vector d  . Consequently, to obtain the MAP estimator for z, in theory, one may consider all 2MT different settings for d  , for each solve the linear equations implied by QΩ* (Z) / Z  0 , and select among those for which the solution z * is consistent with the considered set  the one which produces the largest value QΩ* (z * ) . In practice, this is generally not feasible. Various solution methods for piecewise linear equations have been suggested in the mathematical programming literature in the past [87, 88]. For instance, some piecewise linear problems may be recast as a linear complementarity problem [89], but the pivoting methods often used to solve it work (numerically) well only for smaller scale settings [38]. Here we therefore settled on a similar, simple Newton‐type iteration scheme as proposed in Brugnano & Casulli [38]. Specifically, if we denote by z * () the solution to eq. 7 obtained with the set of constraints  active, the present scheme initializes with a random drawing of the {z mt } , sets the components of d  for which z mt   m to 1 and all others to 0, and then keeps on alternating Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 14 between (1) solving QΩ* (Z) / Z  0 for z * () and (2) flipping the bits in d  for which sgn[2d ( k )  1]  sgn[ z*k ()   k ] , that is, for which the components of the vector (8) c : (2d   1)T  (θ  z * ()) are positive, until the solution to QΩ* (Z) / Z  0 is consistent with set  (i.e., c  0 ). For the problem as formulated in Brugnano & Casulli [38], these authors proved that such a solution always exists, and that the algorithm will always terminate after a finite (usually low) number of steps, given certain assumptions and provided the matrix that multiplies with the states z in QΩ* (Z) / Z  0 (i.e., the Hessian of QΩ* (z * ) ), fulfills certain conditions (Stieltjes‐type; see [38] for details). This will usually not be the case for the present system; although the Hessian of QΩ* (z * ) will be symmetric and positive‐definite (with proper parameter settings), its off‐diagonal elements may be either larger or smaller than 0. Moreover, for the problem considered here, all elements of the Hessian in (7) depend on  , while in [38] this is only the case for the on‐diagonal elements (i.e., in [38] D  enters the Hessian only in additive, not multiplicative form as here). For these reasons, the Newton‐type algorithm outlined above may not always converge to an exact solution (if one exists in this case) but may eventually cycle among non‐solution configurations, or may not even always increase Q(Z) (i.e., eq. 5!). To bypass this, the algorithm was always terminated if one of the following three conditions was met: (i) A solution to QΩ* (Z) / Z  0 consistent with  was encountered; (ii) a previously probed set  was revisited; (iii) the constraint violation error defined by c  1 , the l1 norm of the positive part of c defined in eq. 8, went up beyond a pre‐specified tolerance level. With these modifications, we found that the algorithm would usually terminate after only a few iterations (<10 for the examined toy examples) and yield approximate solutions with only a few constraints still violated (<3% for the toy examples). For these elements k of z for which the constraints are still violated, that is for which ck  0 in eq. 8, one may explicitly enforce the constraints by setting the violating states z {k }  θ{k } , but either way it was found that even these approximate (and potentially only locally optimal) solutions were generally (for the problems studied) in sufficiently good agreement with E(Z|X). In the case of full EM iterations (with the parameters unknown as well), it appeared that flipping violated constraints in d  one by one may often (for the scenarios studied here) improve overall performance, in the sense of yielding higher‐likelihood solutions and less numerical problems (although it may leave more constraints violated in the end). Hence, this scheme was adopted here for the full EM, that is only the single bit k* corresponding to the maximum element of vector c in eq. 8 was inverted on each iteration (the one with the largest wrong‐side deviation from θ ). In general, however, the resultant slow‐down in the algorithm may not always be worth the performance gains; or a mixture of methods, with d kl *1  1  d kl * with k * : arg max k {ck  0} early on, and d{l k1}  1  d{l k }k : ck  0 during later iterations, may be considered. Once a (local) maximum z max has been obtained, the (local) covariances may be read off from the inverse negative Hessian at z max , i.e. the elements of 15 Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 (9) V : ( U 0  D  U1  U1T D   D  U 2 D  ) 1 . We then use these covariance estimates to obtain (estimates of) E[ ( z )] , E[z (z T )] , and  E[ (z ) (z T )] , as required for the maximization step. Denoting by F ( ;  ,  2 ) :  N ( x;  ,  2 )dx  the complementary cumulative Gaussian, to ease subsequent derivations, let us introduce the following notation: (10) N k : N ( k ; z kmax ,  k2 ) , Fk : F ( k ; z kmax ,  k2 ) ,  kl2 : cov( z kmax , zlmax )  vkl . The elements of the expectancy vectors and matrices above are computed as (11) E[ ( z k )]   k2 N k  ( z kmax   k ) Fk , E[ ( z k ) 2 ]  ([ z kmax ]2   k2   k2  2 k z kmax ) Fk  ( z kmax   k ) k2 N k , E[ z k  ( zl )]  ( kl2   l z kmax  z kmax zlmax ) Fl  z kmax l2 N l . The terms E[ ( z k ) ( zl )] , for k  l , are more tedious, and cannot be (to my knowledge and insight) computed exactly (analytically), so we develop them in a bit more detail here: (12) E[ ( z k ) ( zl )]      p (z k , zl )( z k   k )( zl   l ) dz k dzl k l      k l p (z k , zl ) z k zl dz k dzl   k      k l   k l      k l p (z k , zl ) zl dz k dzl   l      k l p (z k , zl ) z k dz k dzl p (z k , zl ) dz k dz l The last term is just a (complementary) cumulative bivariate Gaussian evaluated with parameters specified through the MAP solution ( z max , V ) (and multiplied by the thresholds). The first term we may rewrite as follows:   k l     k l p (z k , zl ) z k zl dz k dzl   p ( z k ) z k  p (zl | z k ) zl dz k dzl l   p ( z k ) z k  N ( l ;  l |k , l1 )   l |k (1   N ( zl ;  l |k , lk1 ) dzl dz k    k 1 max  l lk ( z k  z k )  (13) where l |k : zlmax l :  l2 /( k2 l2   kl4 ) lk :  kl2 /( k2 l2   kl4 ) These are just standard results one can derive by the reverse chain rule for integration, with the ’s the elements of the inverse bivariate (k,l)‐covariance matrix. Note that if the variable z k were removed from the first integrand in eq. 13, i.e. as in the second term in eq. 12, all terms in eq. 13 would just come down to uni‐ or bivariate Gaussians (times some factor) or a univariate Gaussian expectancy value, respectively. Noting this, one obtains for the second (and correspondingly for the third) term in eq. 12: 16 Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 k     p (z k , zl ) zl dz k dzl   k k N l F ( k ; lk , l1 )   k ( zlmax Fk   kl2 N k ) F ( l ; zlmax , k1 )  k l (14) with  kl : zlmax   kl2 /  k2 ( k  z kmax ) The problematic bit is the product term   k l p ( z k ) z k  l|k  N ( zl ;  l|k , lk1 ) dz l dz k in eq. 13, which we  resolve by making the approximation  l |k  l  z   k l   (15) max l . This way we have for the first term in eq. 12:  p (z k , zl ) z k zl dz k dzl  k N l l1 N ( k ;  lk , l1 )   lk F ( k ;  lk , l1 )     ( k2 zlmax   kl2 z kmax ) N k  ( z kmax zlmax   kl2 ) Fk F ( l ;  lk , k1 ) Putting (13)‐(15) together with the bivariate cumulative Gaussian yields an analytical approximation to eq. 12 that can be computed based on the quantities obtained from the MAP estimate ( z max , V ) . Expectation‐Maximization Algorithm: Parameter estimation Once we have estimates for E[z ] , E[ zz T ] , E[ ( z )] , E[z (z T )] , and E[ (z ) (z T )] , the maximization step is standard and straightforward, so for convenience we just state the results here, using the notation T  T E1, :  E[ (z t ) (z )] , t 1 T 1 E 4 :  E[ (z t )z Tt ] , t 1 (16) E 2 :  E[z z ] , T t t 2 E3, : T t t 1 T 1   E[z z t 1  t T t ], T E5 :  E[z t (z Tt1 )] , t 2 T T t 1 t 1 F1 :  x t E[ (z Tt )] , F2 :  x t xTt T T t 2 t 1 T , F3 :  s t E[z Tt1 ] , F4 :  s t E[ (z Tt1 )] , F5 :  E[z t ]sTt t 2 T , F6 :  s t sTt t 1 With these expectancy sums defined, one has (17a) B  F1E1,10 (17b) Γ (17c) μ 0  E[z1 ]  s1 (17d) A  [(E 2  WE4  F3 )  I ][E3, 0  I]1 1 (F2  F1BT  BF1T  BE1T,0 BT )  I T (17e) Σ  1 var(z1 )  μ 0s1T  s1μT0  ET3,1  F5  F5T  F6  (F3  E 2 ) AT  A(F3T  ET2 )  AET3,0 AT T  (F4  E5 ) W T  W (F4T  ET5 ) WE1T W T  AET4 W T  WE 4 A T  I  Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 17 Note that to avoid redundancy in the parameters, here we usually fixed Σ  I 10 2 . For W, since we assumed this matrix to have an off‐diagonal structure (i.e., with zeros on the diagonal), we solve for each row of W separately: P ( 0) : (E3,0  I) 1 ET4 (17f) P (1) : E5  (E 2  F3 )  I P ( 0 )  F4 m  {1...M } : Wm ,{1:M }\ m  Pm(1,{) 1:M }\ m ([E1  E 4, m Pm( 0) ]{1:M }\ m,{1:M }\ m ) 1 where the subscripts indicate the matrix elements to be pulled out, with the subscript dot denoting all elements of the corresponding column or row (e.g., ‘m’ takes the mth column of that matrix). Starting from a number of different random parameter initializations, the E‐ and M‐steps are alternated until the log‐likelihood ratio falls below a predefined tolerance level (while still increasing) or a preset maximum number of allowed iterations are exceeded. For reasons mentioned in the Results, sometimes it can actually happen that the log‐likelihood ratio temporarily decreases, in which case the iterations are continued. If ( N  M ) 2  N  M , factor analysis may be used to derive initial estimates for the latent states and observation parameters in (3) [23], although this was not attempted here. For further implementational details see the MatLab code provided at www.zi‐ mannheim.de/en/research/departments‐research‐groups‐institutes/theor‐neuroscience‐e.html [upon publication]. Particle filter To validate the approximations from our semi‐analytical procedure developed above, a bootstrap particle filter as given in Durbin & Koopman [22] was implemented. In bootstrap particle filtering, the state posterior distribution at time t, pΞ ( z t | x1 ,..., x t )  (18)  pΞ (x t | z t ) pΞ (z t | x1 ,..., x t 1 ) pΞ (x t | x1 ,..., x t 1 ) pΞ (x t | z t )  pΞ (z t | z t 1 ) pΞ (z t 1 | x1 ,..., x t 1 )d z t 1 z t 1 pΞ ( x t | x1 ,..., x t 1 ) is numerically approximated through a set of ‘particles’ (samples) {z t(1) ,..., z t( K ) } , drawn from pΞ (z t | x1 ,..., x t 1 ) , together with a set of normalized weights {wt(1) ,..., wt( K ) } ,  wt( r ) : pΞ (x t | z t( r ) ) k 1 pΞ (x t | z t( k ) ) K  1 . Based on this representation, moments of pΞ ( z t | x1:t ) and pΞ ( (z t ) | x1:t ) can be easily obtained by evaluating  (or any other function of z) on the set of samples {z t(r ) } and summing the outcomes weighted with their respective normalized observation likelihoods {wt(r ) } . A new set of samples {z t( r 1) } for t+1 is then generated by first drawing K times from {z t(k ) } with replacement according to the weights {wt(k ) } , and then drawing K new samples according to the transition probabilities pΞ (z t( k1) | z t( k ) ) (thus approximating the integral in eq. 18). Here we used K=104 samples. Note that this numerical sampling scheme, like a Kalman filter, but Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 18 unlike the procedure outlined above, only implements the filtering step (i.e., yields pΞ (z t | x1:t ) , not pΞ (z t | x1T: ) ). On the other hand, it gives (weakly) consistent (asymptotically unbiased; [90, 91]) estimates of all expectancies across this distribution, that is, it does not rely on the type of approximations and locally optimal solutions of our semi‐analytical approach that almost inevitably will come with some bias (since, among other factors, the mode would usually deviate from the mean by some amount for the present model). Experimental data sets Details of the experimental task and electrophysiological data sets used here could be found in Hyman et al. [35]. Briefly, rats had to alternate between left and right lever presses in a Skinner box to obtain a food reward dispensed on correct choices, with a  10 s delay enforced between consecutive lever presses. While the levers were located on one side of the Skinner box, animals had to perform a nosepoke on the opposite side of the box in between lever presses for initiating the delay period, to discourage them from developing an external coding strategy (e.g., through maintenance of body posture during the delay). While animals were performing the task, multiple single units were recorded with a set of 16 tetrodes implanted bilaterally into the anterior cingulate cortex (ACC, a subdivision of rat prefrontal cortex). For the present analyses, a data set from only one of the four rats recorded on this task was selected for the present exemplary purposes, namely the one where the clearest single unit traces of delay activity were observed in the first place. This data set consisted of 30 simultaneously recorded units, of which the 19 units with spiking rates >1 Hz were retained, on 14 correct trials (only correct response trials were analyzed). The trials had variable length, but were all cut down to the same length of 14 s, including 2 s of pre‐nosepoke, 5 s extending into the delay from the nosepoke, 5 s preceding the next lever press, and 2 s of post‐response phase (note that this may imply temporal gaps in the middle of the delay on some trials, which were ignored here for convenience). All spike trains were convolved with Gaussian kernels (see, e.g., [8, 46, 92]), with the kernel standard deviation set individually for each unit to one half of its mean interspike‐interval. Note that this also brings the observed series into tighter agreement with the Gaussian assumptions of the observation model, eq. 3. Finally, the spike time series were binned into 500 ms bins (corresponding roughly to the inverse of the overall [across all 30 recorded cells] average neural firing rates of 2.2 Hz), which resulted in 14 trials of 28 time bins each submitted to the estimation process. As indicated in the section ‘State space model’, a trial‐unique initial state mean μ k , k  1...14 , was assumed for each of the 14 temporally segregated trials. Acknowledgements I thank Dr. Georgia Koppe for her feedback on this manuscript, and Drs. James Hyman and Jeremy Seamans for lending me their in‐vivo electrophysiological recordings from rat ACC as an analysis testbed. Funding statement Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 19 This work was funded through two grants from the German Research Foundation (DFG, Du 354/8‐1, and within the Collaborative Research Center 1134) to the author, and by the German Ministry for Education and Research (BMBF, 01ZX1314G) within the e:Med program. References 1. Durstewitz D. Self‐organizing neural integrator predicts interval times through climbing activity. J Neurosci. 2003;23: 5342‐5353. 2. Wang XJ. Probabilistic decision making by slow reverberation in cortical circuits. 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Contextual encoding by ensembles of medial prefrontal cortex neurons. Proc Natl Acad Sci USA. 2013;109: 5086-5091. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 25 Figure Legends Fig. 1. State and parameter estimates for nonlinear cycle example. (A) True (solid‐circle lines) and estimated (dashed‐star lines) states over some periods of the simulated limit cycle generated by a 3‐ state PLRNN when true parameters were provided (for this example, θ  (0.86,0.09,0.85) ; all other parameters as in B, see also provided Matlab file ‘PLRNNoscParam.mat’). ‘True states’ refers to the actual states from which the observations X were generated. Inputs of sit  1 were provided to units i=1 and i=2 on time steps 1 and 10 of each cycle, respectively. (B) True and estimated model parameters for (from top‐left to bottom‐right) μ 0 , A, W, Σ, B, Γ , when true states (but not their higher‐order moments) were provided. Bisectrix lines (black) indicate identity. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 26 Fig. 2. Agreement between simulated (x‐axes) and semi‐analytical (y‐axes) solutions for state expectancies for the model from Fig. 1 across all three state variables and T=750 time steps. Here,  ( zi ) : max{0, zi  θi } is the PL activation function. Simulated state paths and their moments were generated using a bootstrap particle filter with 104 particles. Bisectrix lines in gray indicate identity. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 27 Fig. 3. State estimation for ‘working memory’ example when true parameters were provided. (A) Setup of the simulated working memory task: Stimulus inputs (green bars, sit  1 , and 0 otherwise) and requested outputs (black = 1, light‐gray = 0, dark‐grey = no output required) across the 20 time points of a working memory trial (with two different trial types) for the 5 PLRNN units. (B) Correlation between estimated and true states (i.e., those from which the observations X were generated) across all five state variables and T=800 time steps. Bisectrix in black. (C) True (circle‐solid lines) and estimated (star‐dashed lines) states for output units #3 (blue) and #4 (red) when s15  1 (left) or s25  1 (right) for single example trials. Note that although working memory PLRNNs may, in principle, be explicitly designed, here a 5‐state PLRNN was first trained by conventional gradient descent (real‐time recurrent‐learning; [39]) to perform the task in A, to yield more ‘natural’ and less uniform ground truth states and parameters. Here, all θi  0 (implying that there can only be one stable fixed point). See Matlab file ‘PLRNNwmParam.mat’ and Fig. 4 for details on parameters. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 28 Fig. 4. True and estimated parameters for the working memory PLRNN (cf. Fig. 3) when true states were provided. From top‐left to bottom‐right, estimates for: μ 0 , A, W, Σ, B, Γ . Note that most parameter estimates were highly accurate, although all state covariance matrices still had to be estimated as well (i.e., with the true states provided as initialization for the E‐step). Bisectrix lines in black indicate identity. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 29 Fig. 5. Full EM algorithm on working memory model: State estimates for ML solution. (A) Log‐ likelihood as a function of EM iteration for the highest‐likelihood run out of all 240 initializations. As in this example, the log‐likelihood, although generally increasing, was not always monotonic (note the little ripples; see discussion in Results). (B) In this example, true and estimated states were nicely linearly related, although not with a regression slope of 1 (in general, as discussed in the text, the sets of true and estimated states may be related by some linear transformation). State estimation in this case was performed by inverting only the single constraint corresponding to the largest deviation on each iteration (see Methods). Bisectrix lines in black indicate identity. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 30 Fig. 6. Full EM algorithm on working memory model. (A) Parameter estimates for ML solution from Fig. 5. True parameters (on x‐axes or as blue bars, respectively), initial (gray circles or green bars) and final (black circles or yellow bars) parameter estimates for (from left to right) μ 0 , A, W, B, Γ . Bisectrix lines in blue. (B) Distributions of initial (gray curves), final (black‐solid curves), and final after reordering of states (black‐dashed curves), deviations between estimated and true parameters across all 240 EM runs from different initial conditions. All final distributions were centered around 0, indicating that final parameter estimates were largely unbiased. Note that partial information about state assignments was implicitly provided to the network through the unit‐specific inputs (and, more generally, may also come from the unit‐specific thresholds θi , although these were all set to 0 for the present example), and hence state reordering only produced slight improvements in the parameter estimates. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 31 Fig. 7. Prediction of single unit responses. (A) Examples of log‐likelihood curves across EM iterations from the 5/36 highest‐likelihood runs for a 5‐state PLRNN estimated from 19 simultaneously recorded prefrontal neurons on a working memory task. (B) Example of an ACC unit predicted extremely well by the estimated PLRNN despite considerable trial to trial fluctuations (3 consecutive trials shown). (C) Example of another ACC unit on the same three trials where only the average trend was captured by the PLRNN. Gray vertical bars in B and C indicate times of cue/ response. State estimation in this case was performed by inverting only the single constraint corresponding to the largest deviation on each iteration (see Methods). Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 32 Fig. 8. Example for latent states of PLRNN estimated from ACC multiple single‐unit recordings during working memory (cf. Fig. 7). Shown are trial averages for left‐lever (black) and right‐lever (gray) trials with SEMs computed across trials. Dashed vertical lines flank the 10 s period of the delay phase used for model estimation. Note that latent variables z4 and z5, in particular, differentiate between left and right lever responses throughout most of the delay period. Durstewitz, Nonlinear State Space Model for Reconstructing Computational Dynamics, 11/02/16 33 Fig. 9. Initial (gray) and final (black) distributions of maximum (absolute) eigenvalues associated with all fixed points of 200 PLRNNs estimated from the experimental data (cf. Figs. 7 & 8) with different initializations of parameters, including the (fixed) threshold parameters θi . Initial parameter configurations were deliberately chosen to yield a rather uniform distribution of absolute eigenvalues  3. 

Published as a conference paper at ICLR 2018 arXiv:1705.07904v3 [cs.LG] 22 Feb 2018 S EMANTICALLY D ECOMPOSING THE L ATENT S PACES OF G ENERATIVE A DVERSARIAL N ETWORKS Chris Donahue Department of Music University of California, San Diego cdonahue@ucsd.edu Zachary C. Lipton Carnegie Mellon University Amazon AI zlipton@cmu.edu Akshay Balsubramani Department of Genetics Stanford University abalsubr@stanford.edu Julian McAuley Department of Computer Science University of California, San Diego jmcauley@eng.ucsd.edu A BSTRACT We propose a new algorithm for training generative adversarial networks that jointly learns latent codes for both identities (e.g. individual humans) and observations (e.g. specific photographs). By fixing the identity portion of the latent codes, we can generate diverse images of the same subject, and by fixing the observation portion, we can traverse the manifold of subjects while maintaining contingent aspects such as lighting and pose. Our algorithm features a pairwise training scheme in which each sample from the generator consists of two images with a common identity code. Corresponding samples from the real dataset consist of two distinct photographs of the same subject. In order to fool the discriminator, the generator must produce pairs that are photorealistic, distinct, and appear to depict the same individual. We augment both the DCGAN and BEGAN approaches with Siamese discriminators to facilitate pairwise training. Experiments with human judges and an off-the-shelf face verification system demonstrate our algorithm’s ability to generate convincing, identity-matched photographs. 1 I NTRODUCTION In many domains, a suitable generative process might consist of several stages. To generate a photograph of a product, we might wish to first sample from the space of products, and then from the space of photographs of that product. Given such disentangled representations in a multistage generative process, an online retailer might diversify its catalog, depicting products in a wider variety of settings. A retailer could also flip the process, imagining new products in a fixed setting. Datasets for such domains often contain many labeled identities with fewer observations of each (e.g. a collection of face portraits with thousands of people and ten photos of each). While we may know the identity of the subject in each photograph, we may not know the contingent aspects of the observation (such as lighting, pose and background). This kind of data is ubiquitous; given a set of commonalities, we might want to incorporate this structure into our latent representations. Generative adversarial networks (GANs) learn mappings from latent codes z in some low-dimensional space Z to points in the space of natural data X (Goodfellow et al., 2014). They achieve this power through an adversarial training scheme pitting a generative model G : Z 7→ X against a discriminative model D : X 7→ [0, 1] in a minimax game. While GANs are popular, owing to their ability to generate high-fidelity images, they do not, in their original form, explicitly disentangle the latent factors according to known commonalities. In this paper, we propose Semantically Decomposed GANs (SD-GANs), which encourage a specified portion of the latent space to correspond to a known source of variation.1,2 The technique 1 2 Web demo: https://chrisdonahue.github.io/sdgan Source code: https://github.com/chrisdonahue/sdgan 1 Published as a conference paper at ICLR 2018 Figure 1: Generated samples from SD-BEGAN. Each of the four rows has the same identity code zI and each of the fourteen columns has the same observation code zO . decomposes the latent code Z into one portion ZI corresponding to identity, and the remaining portion ZO corresponding to the other contingent aspects of observations. SD-GANs learn through a pairwise training scheme in which each sample from the real dataset consists of two distinct images with a common identity. Each sample from the generator consists of a pair of images with common zI ∈ ZI but differing zO ∈ ZO . In order to fool the discriminator, the generator must not only produce diverse and photorealistic images, but also images that depict the same identity when zI is fixed. For SD-GANs, we modify the discriminator so that it can determine whether a pair of samples constitutes a match. As a case study, we experiment with a dataset of face photographs, demonstrating that SD-GANs can generate contrasting images of the same subject (Figure 1; interactive web demo in footnote on previous page). The generator learns that certain properties are free to vary across observations but not identity. For example, SD-GANs learn that pose, facial expression, hirsuteness, grayscale vs. color, and lighting can all vary across different photographs of the same individual. On the other hand, the aspects that are more salient for facial verification remain consistent as we vary the observation code zO . We also train SD-GANs on a dataset of product images, containing multiple photographs of each product from various perspectives (Figure 4). We demonstrate that SD-GANs trained on faces generate stylistically-contrasting, identity-matched image pairs that human annotators and a state-of-the-art face verification algorithm recognize as depicting the same subject. On measures of identity coherence and image diversity, SD-GANs perform comparably to a recent conditional GAN method (Odena et al., 2017); SD-GANs can also imagine new identities, while conditional GANs are limited to generating existing identities from the training data. 2 S EMANTICALLY D ECOMPOSED G ENERATIVE A DVERSARIAL N ETWORKS Before introducing our algorithm, we briefly review the prerequisite concepts. 2.1 GAN PRELIMINARIES GANs leverage the discriminative power of neural networks to learn generative models. The generative model G ingests latent codes z, sampled from some known prior PZ , and produces G(z), a sample of an implicit distribution PG . The learning process consists of a minimax game between G, parameterized by θG , and a discriminative model D, parameterized by θD . In the original formulation, the discriminative model tries to maximize log likelihood, yielding min max V (G, D) = Ex∼PR [log D(x)] + Ez∼PZ [log(1 − D(G(z)))]. G D (1) Training proceeds as follows: For k iterations, sample one minibatch from the real distribution PR and one from the distribution of generated images PG , updating discriminator weights θD to increase V (G, D) by stochastic gradient ascent. Then sample a minibatch from PZ , updating θG to decrease V (G, D) by stochastic gradient descent. 2 Published as a conference paper at ICLR 2018 Algorithm 1 Semantically Decomposed GAN Training 1: for n in 1:NumberOfIterations do 2: for m in 1:MinibatchSize do 3: Sample one identity vector zI ∼ Uniform([−1, 1]dI ). 4: Sample two observation vectors z1O , z2O ∼ Uniform([−1, 1]dO ). 5: z1 ← [zI ; z1O ], z2 ← [zI ; z2O ]. 6: Generate pair of images G(z1 ), G(z2 ), adding them to the minibatch with label 0 (fake). 7: 8: 9: 10: 11: 12: 13: for m in 1:MinibatchSize do Sample one identity i ∈ I uniformly at random from the real data set. Sample two images of i without replacement x1 , x2 ∼ PR (x|I = i). Add the pair to the minibatch, assigning label 1 (real). Update discriminator weights by θD ← θD + ∇θD V (G, D) using its stochastic gradient. Sample another minibatch of identity-matched latent vectors z1 , z2 . Update generator weights by stochastic gradient descent θG ← θG − ∇θG V (G, D). Zhao et al. (2017b) propose energy-based GANs (EBGANs), in which the discriminator can be viewed as an energy function. Specifically, they devise a discriminator consisting of an autoencoder: D(x) = Dd (De (x)). In the minimax game, the discriminator’s weights are updated to minimize the reconstruction error L(x) = ||x − D(x)|| for real data, while maximizing the error L(G(z)) for the generator. More recently, Berthelot et al. (2017) extend this work, introducing Boundary Equilibrium GANs (BEGANs), which optimize the Wasserstein distance (reminiscent of Wasserstein GANs (Arjovsky et al., 2017)) between autoencoder loss distributions, yielding the formulation: VBEGAN (G, D) = L(x) − L(G(z)). (2) Additionally, they introduce a method for stabilizing training. Positing that training becomes unstable when the discriminator cannot distinguish between real and generated images, they introduce a new hyperparameter γ, updating the value function on each iteration to maintain a desired ratio between the two reconstruction errors: E[L(G(z))] = γ E[L(x)]. The BEGAN model produces what appear to us, subjectively, to be the sharpest images of faces yet generated by a GAN. In this work, we adapt both the DCGAN (Radford et al., 2016) and BEGAN algorithms to the SD-GAN training scheme. 2.2 SD-GAN FORMULATION Consider the data’s identity as a random variable I in a discrete index set I. We seek to learn a latent representation that conveniently decomposes the variation in the real data into two parts: 1) due to I, and 2) due to the other factors of variation in the data, packaged as a random variable O. Ideally, the decomposition of the variation in the data into I and O should correspond exactly to a decomposition of the latent space Z = ZI × ZO . This would permit convenient interpolation and other operations on the inferred subspaces ZI and ZO . A conventional GAN samples I, O from their joint distribution. Such a GAN’s generative model samples directly from an unstructured prior over the latent space. It does not disentangle the variation in O and I, for instance by modeling conditional distributions PG (O | I = i), but only models their average with respect to the prior on I. Our SD-GAN method learns such a latent space decomposition, partitioning the coordinates of Z into two parts representing the subspaces, so that any z ∈ Z can be written as the concatenation [zI ; zO ] of its identity representation zI ∈ RdI = ZI and its contingent aspect representation zO ∈ RdO = ZO . SD-GANs achieve this through a pairwise training scheme in which each sample from the real data consists of x1 , x2 ∼ PR (x | I = i), a pair of images with a common identity i ∈ I. Each sample from the generator consists of G(z1 ), G(z2 ) ∼ PG (z | ZI = zI ), a pair of images generated from a common identity vector zI ∈ ZI but i.i.d. observation vectors z1O , z2O ∈ ZO . We assign identity-matched pairs from PR the label 1 and zI -matched pairs from PG the label 0. The discriminator can thus learn to reject pairs for either of two primary reasons: 1) not photorealistic or 2) not plausibly depicting the same subject. See Algorithm 1 for SD-GAN training pseudocode. 3 Published as a conference paper at ICLR 2018 (a) DCGAN (b) SD-DCGAN (c) BEGAN (d) SD-BEGAN Figure 2: SD-GAN architectures and vanilla counterparts. Our SD-GAN models incorporate a decomposed latent space and Siamese discriminators. Dashed lines indicate shared weights. Discriminators also observe real samples in addition to those from the generator (not pictured for simplicity). 2.3 SD-GAN DISCRIMINATOR ARCHITECTURE With SD-GANs, there is no need to alter the architecture of the generator. However, the discriminator must now act upon two images, producing a single output. Moreover, the effects of the two input images x1 , x2 on the output score are not independent. Two images might be otherwise photorealistic but deserve rejection because they clearly depict different identities. To this end, we devise two novel discriminator architectures to adapt DCGAN and BEGAN respectively. In both cases, we first separately encode each image using the same convolutional neural network De (Figure 2). We choose this Siamese setup (Bromley, 1994; Chopra et al., 2005) as our problem is symmetrical in the images, and thus it’s sensible to share weights between the encoders. To adapt DCGAN, we stack the feature maps De (x1 ) and De (x2 ) along the channel axis, applying one additional strided convolution. This allows the network to further aggregate information from the two images before flattening and fully connecting to a sigmoid output. For BEGAN, because the discriminator is an autoencoder, our architecture is more complicated. After encoding each image, we concatenate the representations [De (x1 ); De (x2 )] ∈ R2(dI +dO ) and apply one fully connected bottleneck layer R2(dI +dO ) ⇒ RdI +2dO with linear activation. In alignment with BEGAN, the SD-BEGAN bottleneck has the same dimensionality as the tuple of latent codes (zI , z1O , z2O ) that generated the pair of images. Following the bottleneck, we apply a second FC layer RdI +2dO ⇒ R2(dI +dO ) , taking the first dI + dO components of its output to be the input to the first decoder and the second dI + dO components to be the input to the second decoder. The shared intermediate layer gives SD-BEGAN a mechanism to push apart matched and unmatched pairs. We specify our exact architectures in full detail in Appendix E. 3 E XPERIMENTS We experimentally validate SD-GANs using two datasets: 1) the MS-Celeb-1M dataset of celebrity face images (Guo et al., 2016) and 2) a dataset of shoe images collected from Amazon (McAuley et al., 2015). Both datasets contain a large number of identities (people and shoes, respectively) with multiple observations of each. The “in-the-wild” nature of the celebrity face images offers a richer test bed for our method as both identities and contingent factors are significant sources of variation. In contrast, Amazon’s shoe images tend to vary only with camera perspective for a given product, making this data useful for sanity-checking our approach. Faces From the aligned face images in the MS-Celeb-1M dataset, we select 12,500 celebrities at random and 8 associated images of each, resizing them to 64x64 pixels. We split the celebrities into subsets of 10,000 (training), 1,250 (validation) and 1,250 (test). The dataset has a small number of duplicate images and some label noise (images matched to the wrong celebrity). We detect and 4 Published as a conference paper at ICLR 2018 Figure 3: Generated samples SD-DCGAN model trained on faces. from Figure 4: Generated samples from SD-DCGAN model trained on shoes. remove duplicates by hashing the images, but we do not rid the data of label noise. We scale the pixel values to [−1, 1], performing no additional preprocessing or data augmentation. Shoes Synthesizing novel product images is another promising domain for our method. In our shoes dataset, product photographs are captured against white backgrounds and primarily differ in orientation and distance. Accordingly, we expect that SD-GAN training will allocate the observation latent space to capture these aspects. We choose to study shoes as a prototypical example of a category of product images. The Amazon dataset contains around 3,000 unique products with the category “Shoe” and multiple product images. We use the same 80%, 10%, 10% split and again hash the images to ensure that the splits are disjoint. There are 6.2 photos of each product on average. 3.1 T RAINING DETAILS We train SD-DCGANs on both of our datasets for 500,000 iterations using batches of 16 identitymatched pairs. To optimize SD-DCGAN, we use the Adam optimizer (Kingma & Ba, 2015) with hyperparameters α = 2e−4, β1 = 0.5, β2 = 0.999 as recommended by Radford et al. (2016). We also consider a non-Siamese discriminator that simply stacks the channels of the pair of real or fake images before encoding (SD-DCGAN-SC). As in (Radford et al., 2016), we sample latent vectors z ∼ Uniform([−1, 1]100 ). For SD-GANs, we partition the latent codes according to zI ∈ RdI , zO ∈ R100−dI using values of dI = [25, 50, 75]. Our algorithm can be trivially applied with k-wise training (vs. pairwise). To explore the effects of using k > 2, we also experiment with an SD-DCGAN where we sample k = 4 instances each from PG (z | ZI = zI ) for some zI ∈ ZI and from PR (x | I = i) for some i ∈ I. For all experiments, unless otherwise stated, we use dI = 50 and k = 2. We also train an SD-BEGAN on both of our datasets. The increased complexity of the SD-BEGAN model significantly increases training time, limiting our ability to perform more-exhaustive hyperparameter validation (as we do for SD-DCGAN). We use the Adam optimizer with the default hyperparameters from (Kingma & Ba, 2015) for our SD-BEGAN experiments. While results from our SD-DCGAN k = 4 model are compelling, an experiment with a k = 4 variant of SD-BEGAN resulted in early mode collapse (Appendix F); hence, we excluded SD-BEGAN k = 4 from our evaluation. We also compare to a DCGAN architecture trained using the auxiliary classifier GAN (AC-GAN) method (Odena et al., 2017). AC-GAN differs from SD-GAN in two key ways: 1) random identity codes zI are replaced by a one-hot embedding over all the identities in the training set (matrix of size 10000x50); 2) the AC-GAN method encourages that generated photos depict the proper identity by tasking its discriminator with predicting the identity of the generated or real image. Unlike SD-GANs, the AC-DCGAN model cannot imagine new identities; when generating from AC-DCGAN (for our quantitative comparisons to SD-GANs), we must sample a random identity from those existing in the training data. 5 Published as a conference paper at ICLR 2018 Table 1: Evaluation of 10k pairs from MS-Celeb-1M (real data) and generative models; half have matched identities, half do not. The identity verification metrics demonstrate that FaceNet (FN) and human annotators on Mechanical Turk (MT) verify generated data similarly to real data. The sample diversity metrics ensure that generated samples are statistically distinct in pixel space. Data generated by our best model (SD-BEGAN) performs comparably to real data. * 1k pairs, † 200 pairs. Identity Verification Dataset 3.2 Mem Sample Diversity Judge AUC Acc. FAR ID-Div All-Div MS-Celeb-1M AC-DCGAN SD-DCGAN SD-DCGAN-SC SD-DCGAN k=4 SD-DCGAN dI =25 SD-DCGAN dI =75 SD-BEGAN 131 MB 57 MB 47 MB 75 MB 57 MB 57 MB 68 MB FN FN FN FN FN FN FN FN .913 .927 .823 .831 .852 .835 .816 .928 .867 .851 .749 .757 .776 .764 .743 .857 .045 .083 .201 .180 .227 .222 .268 .110 .621 .497 .521 .560 .523 .526 .517 .588 .699 .666 .609 .637 .614 .615 .601 .673 †MS-Celeb-1M *MS-Celeb-1M *AC-DCGAN *SD-DCGAN k=4 *SD-BEGAN 131 MB 75 MB 68 MB Us MT MT MT MT - .850 .759 .765 .688 .723 .110 .035 .090 .147 .096 .621 .621 .497 .523 .588 .699 .699 .666 .614 .673 E VALUATION The evaluation of generative models is a fraught topic. Quantitative measures of sample quality can be poorly correlated with each other (Theis et al., 2016). Accordingly, we design an evaluation to match conceivable uses of our algorithm. Because we hope to produce diverse samples that humans deem to depict the same person, we evaluate the identity coherence of SD-GANs and baselines using both a pretrained face verification model and crowd-sourced human judgments obtained through Amazon’s Mechanical Turk platform. 3.2.1 Q UANTITATIVE Recent advancements in face verification using deep convolutional neural networks (Schroff et al., 2015; Parkhi et al., 2015; Wen et al., 2016) have yielded accuracy rivaling humans. For our evaluation, we procure FaceNet, a publicly-available face verifier based on the Inception-ResNet architecture (Szegedy et al., 2017). The FaceNet model was pretrained on the CASIA-WebFace dataset (Yi et al., 2014) and achieves 98.6% accuracy on the LFW benchmark (Huang et al., 2012).3 FaceNet ingests normalized, 160x160 color images and produces an embedding f (x) ∈ R128 . The training objective for FaceNet is to learn embeddings that minimize the L2 distance between matched pairs of faces and maximize the distance for mismatched pairs. Accordingly, the embedding space yields a function for measuring the similarity between two faces x1 and x2 : D(x1 , x2 ) = ||f (x1 ) − f (x2 )||22 . Given two images, x1 and x2 , we label them as a match if D(x1 , x2 ) ≤ τv where τv is the accuracy-maximizing threshold on a class-balanced set of pairs from MS-Celeb-1M validation data. We use the same threshold for evaluating both real and synthetic data with FaceNet. We compare the performance of FaceNet on pairs of images from the MS-Celeb-1M test set against generated samples from our trained SD-GAN models and AC-DCGAN baseline. To match FaceNet’s training data, we preprocess all images by resizing from 64x64 to 160x160, normalizing each image individually. We prepare 10,000 pairs from MS-Celeb-1M, half identity-matched and half unmatched. From each generative model, we generate 5,000 pairs each with z1I = z2I and 5,000 pairs with z1I 6= z2I . For each sample, we draw observation vectors zO randomly. We also want to ensure that identity-matched images produced by the generative models are diverse. To this end, we propose an intra-identity sample diversity (ID-Div) metric. The multi-scale structural similarity (MS-SSIM) (Wang et al., 2004) metric reports the similarity of two images on a scale from 0 (no resemblance) to 1 (identical images). We report 1 minus the mean MS-SSIM for all pairs 3 “20170214-092102” pretrained model from https://github.com/davidsandberg/facenet 6 Published as a conference paper at ICLR 2018 of identity-matched images as ID-Div. To measure the overall sample diversity (All-Div), we also compute 1 minus the mean similarity of 10k pairs with random identities. In Table 1, we report the area under the receiver operating characteristic curve (AUC), accuracy, and false accept rate (FAR) of FaceNet (at threshold τv ) on the real and generated data. We also report our proposed diversity statistics. FaceNet verifies pairs from the real data with 87% accuracy compared to 86% on pairs from our SD-BEGAN model. Though this is comparable to the accuracy achieved on pairs from the AC-DCGAN baseline, our model produces samples that are more diverse in pixel space (as measured by ID-Div and All-Div). FaceNet has a higher but comparable FAR for pairs from SD-GANs than those from AC-DCGAN; this indicates that SD-GANs may produce images that are less semantically diverse on average than AC-DCGAN. We also report the combined memory footprint of G and D for all methods in Table 1. For conditional GAN approaches, the number of parameters grows linearly with the number of identities in the training data. Especially in the case of the AC-GAN, where the discriminator computes a softmax over all identities, linear scaling may be prohibitive. While our 10k-identity subset of MS-Celeb-1M requires a 131MB AC-DCGAN model, an AC-DCGAN for all 1M identities would be over 8GB, with more than 97% of the parameters devoted to the weights in the discriminator’s softmax layer. In contrast, the complexity of SD-GAN is constant in the number of identities. 3.2.2 Q UALITATIVE In addition to validating that identity-matched SD-GAN samples are verified by FaceNet, we also demonstrate that humans are similarly convinced through experiments using Mechanical Turk. For these experiments, we use balanced subsets of 1,000 pairs from MS-Celeb-1M and the most promising generative methods from our FaceNet evaluation. We ask human annotators to determine if each pair depicts the “same person” or “different people”. Annotators are presented with batches of ten pairs at a time. Each pair is presented to three distinct annotators and predictions are determined by majority vote. Additionally, to provide a benchmark for assessing the quality of the Mechanical Turk ensembles, we (the authors) manually judged 200 pairs from MS-Celeb-1M. Results are in Table 1. For all datasets, human annotators on Mechanical Turk answered “same person” less frequently than FaceNet when the latter uses the accuracy-maximizing threshold τv . Even on real data, balanced so that 50% of pairs are identity-matched, annotators report “same person” only 28% of the time (compared to the 41% of FaceNet). While annotators achieve higher accuracy on pairs from ACDCGAN than pairs from SD-BEGAN, they also answer “same person” 16% more often for ACDCGAN pairs than real data. In contrast, annotators answer “same person” at the same rate for SD-BEGAN pairs as real data. This may be attributable to the lower sample diversity produced by AC-DCGAN. Samples from SD-DCGAN and SD-BEGAN are shown in Figures 3 and 1 respectively. 4 R ELATED WORK Style transfer and novel view synthesis are active research areas. Early attempts to disentangle style and content manifolds used factored tensor representations (Tenenbaum & Freeman, 1997; Vasilescu & Terzopoulos, 2002; Elgammal & Lee, 2004; Tang et al., 2013), applying their results to face image synthesis. More recent work focuses on learning hierarchical feature representations using deep convolutional neural networks to separate identity and pose manifolds for faces (Zhu et al., 2013; Reed et al., 2014; Zhu et al., 2014; Yang et al., 2015; Kulkarni et al., 2015; Oord et al., 2016; Yan et al., 2016) and products (Dosovitskiy et al., 2015). Gatys et al. (2016) use features of a convolutional network, pretrained for image recognition, as a means for discovering content and style vectors. Since their introduction (Goodfellow et al., 2014), GANs have been used to generate increasingly highquality images (Radford et al., 2016; Zhao et al., 2017b; Berthelot et al., 2017). Conditional GANs (cGANs), introduced by Mirza & Osindero (2014), extend GANs to generate class-conditional data. Odena et al. (2017) propose auxiliary classifier GANs, combining cGANs with a semi-supervised discriminator (Springenberg, 2015). Recently, cGANs have been used to ingest text (Reed et al., 2016) and full-resolution images (Isola et al., 2017; Liu et al., 2017; Zhu et al., 2017) as conditioning information, addressing a variety of image-to-image translation and style transfer tasks. Chen et al. (2016) devise an information-theoretic extension to GANs in which they maximize the mutual information between a subset of latent variables and the generated data. Their unsupervised method 7 Published as a conference paper at ICLR 2018 Figure 5: Linear interpolation of zI (identity) and zO (observation) for three pairs using SD-BEGAN generator. In each matrix, rows share zI and columns share zO . appears to disentangle some intuitive factors of variation, but these factors may not correspond to those explicitly disentangled by SD-GANs. Several related papers use GANs for novel view synthesis of faces. Tran et al. (2017); Huang et al. (2017); Yin et al. (2017a;b); Zhao et al. (2017a) all address synthesis of different body/facial poses conditioned on an input image (representing identity) and a fixed number of pose labels. Antipov et al. (2017) propose conditional GANs for synthesizing artificially-aged faces conditioned on both a face image and an age vector. These approaches all require explicit conditioning on the relevant factor (such as rotation, lighting and age) in addition to an identity image. In contrast, SD-GANs can model these contingent factors implicitly (without supervision). Mathieu et al. (2016) combine GANs with a traditional reconstruction loss to disentangle identity. While their approach trains with an encoder-decoder generator, they enforce a variational bound on the encoder embedding, enabling them to sample from the decoder without an input image. Experiments with their method only address small (28x28) grayscale face images, and their training procedure is complex to reproduce. In contrast, our work offers a simpler approach and can synthesize higher-resolution, color photographs. One might think of our work as offering the generative view of the Siamese networks often favored for learning similarity metrics (Bromley, 1994; Chopra et al., 2005). Such approaches are used for discriminative tasks like face or signature verification that share the many classes with few examples structure that we study here. In our work, we adopt a Siamese architecture in order to enable the discriminator to differentiate between matched and unmatched pairs. Recent work by Liu & Tuzel (2016) propose a GAN architecture with weight sharing across multiple generators and discriminators, but with a different problem formulation and objective from ours. 5 D ISCUSSION Our evaluation demonstrates that SD-GANs can disentangle those factors of variation corresponding to identity from the rest. Moreover, with SD-GANs we can sample never-before-seen identities, a benefit not shared by conditional GANs. In Figure 3, we demonstrate that by varying the observation vector zO , SD-GANs can change the color of clothing, add or remove sunnies, or change facial pose. They can also perturb the lighting, color saturation, and contrast of an image, all while keeping the apparent identity fixed. We note, subjectively, that samples from SD-DCGAN tend to appear less photorealistic than those from SD-BEGAN. Given a generator trained with SD-GAN, we can independently interpolate along the identity and observation manifolds (Figure 5). On the shoe dataset, we find that the SD-DCGAN model produces convincing results. As desired, manipulating zI while keeping zO fixed yields distinct shoes in consistent poses (Figure 4). The identity code zI appears to capture the broad categories of shoes (sneakers, flip-flops, boots, etc.). Surprisingly, neither original BEGAN nor SD-BEGAN can produce diverse shoe images (Appendix G). In this paper, we presented SD-GANs, a new algorithm capable of disentangling factors of variation according to known commonalities. We see several promising directions for future work. One logical extension is to disentangle latent factors corresponding to more than one known commonality. We also plan to apply our approach in other domains such as identity-conditioned speech synthesis. 8 Published as a conference paper at ICLR 2018 ACKNOWLEDGEMENTS The authors would like to thank Anima Anandkumar, John Berkowitz and Miller Puckette for their helpful feedback on this work. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI1053575 (Towns et al., 2014). GPUs used in this research were donated by the NVIDIA Corporation. R EFERENCES Grigory Antipov, Moez Baccouche, and Jean-Luc Dugelay. Face aging with conditional generative adversarial networks. In ICIP, 2017. Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein GAN. In ICML, 2017. David Berthelot, Tom Schumm, and Luke Metz. BEGAN: Boundary equilibrium generative adversarial networks. arXiv:1703.10717, 2017. Jane Bromley. Signature verification using a "Siamese" time delay neural network. In NIPS, 1994. 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In CVPR, 2015. Jost Tobias Springenberg. Unsupervised and semi-supervised learning with categorical generative adversarial networks. In ICLR, 2015. Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alex Alemi. Inception-v4, Inception-ResNet and the impact of residual connections on learning. In AAAI, 2017. Yichuan Tang, Ruslan Salakhutdinov, and Geoffrey Hinton. Tensor analyzers. In ICML, 2013. Joshua B Tenenbaum and William T Freeman. Separating style and content. In NIPS, 1997. Lucas Theis, Aäron van den Oord, and Matthias Bethge. A note on the evaluation of generative models. In ICLR, 2016. John Towns, Timothy Cockerill, Maytal Dahan, Ian Foster, Kelly Gaither, Andrew Grimshaw, Victor Hazlewood, Scott Lathrop, Dave Lifka, Gregory D Peterson, et al. XSEDE: Accelerating scientific discovery. Computing in Science & Engineering, 2014. Luan Tran, Xi Yin, and Xiaoming Liu. Disentangled representation learning GAN for pose-invariant face recognition. In CVPR, 2017. M Vasilescu and Demetri Terzopoulos. Multilinear analysis of image ensembles: Tensorfaces. In ECCV, 2002. Zhou Wang, Eero P Simoncelli, and Alan C Bovik. Multiscale structural similarity for image quality assessment. In Asilomar Conference on Signals, Systems and Computers, 2004. Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao. A discriminative feature learning approach for deep face recognition. In ECCV, 2016. Xinchen Yan, Jimei Yang, Kihyuk Sohn, and Honglak Lee. Attribute2Image: Conditional image generation from visual attributes. In ECCV, 2016. Jimei Yang, Scott E Reed, Ming-Hsuan Yang, and Honglak Lee. Weakly-supervised disentangling with recurrent transformations for 3D view synthesis. In NIPS, 2015. 10 Published as a conference paper at ICLR 2018 Dong Yi, Zhen Lei, Shengcai Liao, and Stan Z Li. Learning face representation from scratch. arXiv:1411.7923, 2014. Weidong Yin, Yanwei Fu, Leonid Sigal, and Xiangyang Xue. Semi-latent GAN: Learning to generate and modify facial images from attributes. arXiv:1704.02166, 2017a. Xi Yin, Xiang Yu, Kihyuk Sohn, Xiaoming Liu, and Manmohan Chandraker. Towards large-pose face frontalization in the wild. In ICCV, 2017b. Bo Zhao, Xiao Wu, Zhi-Qi Cheng, Hao Liu, and Jiashi Feng. Multi-view image generation from a single-view. arXiv:1704.04886, 2017a. Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. In ICLR, 2017b. Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In ICCV, 2017. Zhenyao Zhu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning identity-preserving face space. In ICCV, 2013. Zhenyao Zhu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Multi-view perceptron: a deep model for learning face identity and view representations. In NIPS, 2014. 11 Published as a conference paper at ICLR 2018 A E STIMATING LATENT CODES Figure 6: Linear interpolation of both identity (vertical) and observation (horizontal) on latent codes recovered for unseen images. All rows have the same identity vector (zI ) and all columns have the same observation vector (zO ). We estimate latent vectors for unseen images and demonstrate that the disentangled representations of SD-GANs can be used to depict the estimated identity with different contingent factors. In order to find a latent vector ẑ such that G(ẑ) (pretrained G) is similar to an unseen image x, we can minimize the distance between x and G(ẑ): minẑ ||G(ẑ) − x||22 (Lipton & Tripathi, 2017). In Figure 6, we depict estimation and linear interpolation across both subspaces for two pairs of images using SD-BEGAN. We also display the corresponding source images being estimated. For both pairs, ẑI (identity) is consistent in each row and ẑO (observation) is consistent in each column. B PAIRWISE DISCRIMINATION OF EMBEDDINGS AND ENCODINGS In Section 3.1, we describe an AC-GAN (Odena et al., 2017) baseline which uses an embedding matrix over real identities as latent identity codes (G : i, zO 7→ x̂). In place of random identity vectors, we tried combining this identity representation with pairwise discrimination (in the style of SD-GAN). In this experiment, the discriminator receives either either two real images with the same identity (x1i , x2i ), or a real image with label i and synthetic image with label i (x1i , G(i, zO )). All other hyperparameters are the same as in our SD-DCGAN experiment (Section 3.1). We show results in Figure 7. Figure 7: Generator with a one-hot identity embedding trained against a pairwise discriminator. Each row shares an identity vector and each column shares an observation vector. Random sample of 4 real images of the corresponding identity on the right. In Appendix C, we detail a modification of the DR-GAN (Tran et al., 2017) method which uses an encoding network Ge to transform images to identity representations (Gd : Ge (x), zO 7→ x̂). We also tried combining this encoder-decoder approach with pairwise discrimination. The discriminator 12 Published as a conference paper at ICLR 2018 receives either two real images with the same identity (x1i , x2i ), or (x1i , Gd (Ge (x1i ), zO ). We show results in Figure 8. Figure 8: Generator with an encoder-decoder architecture trained against a pairwise discriminator. Each row shares an identity vector and each column shares an observation vector. Input image on the right. While these experiments are exploratory and not part of our principle investigation, we find the results to be qualitatively promising. We are not the first to propose pairwise discrimination with pairs of (real, real) or (real, fake) images in GANs (Pathak et al., 2016; Isola et al., 2017). C E XPLORATORY EXPERIMENT WITH DR-GAN S Tran et al. (2017) propose Disentangled Representation learning-GAN (DR-GAN), an approach to face frontalization with similar setup to our SD-GAN algorithm. The (single-image) DR-GAN generator G (composition of Ge and Gd ) accepts an input image x, a pose code c, and a noise vector z. The DR-GAN discriminator receives either x or x̂ = Gd (Ge (x), c, z). In the style of (Springenberg, 2015), the discriminator is tasked with determining not only if the image is real or fake, but also classifying the pose c, suggesting a disentangled representation to the generator. Through their experiments, they demonstrate that DR-GAN can explicitly disentangle pose and illumination (c) from the rest of the latent space (Ge (x); z). Figure 9: Generated samples from cGAN trained only to disentangle identity. Each row shares an identity vector and each column shares an observation vector; input image on the right. In addition to our AC-DCGAN baseline (Odena et al., 2017), we tried modifying DR-GAN to only disentangle identity (rather than both identity and pose in the original paper). We used the DCGAN (Radford et al., 2016) discriminator architecture (Table 4) as Ge , linearly projecting the final convolutional layer to Ge (x) ∈ R50 (in alignment with our SD-GAN experiments). We altered the discriminator to predict the identity of x or x̂, rather than pose information (which is unknown in our experimental setup). With these modifications, Ge (x) is analogous to zI in the SD-GAN generator, and z is analogous to zO . Furthermore, this setup is identical to the AC-DCGAN baseline 13 Published as a conference paper at ICLR 2018 except that the embedding matrix is replaced by an encoding network Ge . Unfortunately, we found that the generator quickly learned to produce a single output image x̂ for each input x regardless of observation code z (Figure 9). Accordingly, we excluded this experiment from our evaluation (Section 3.2). D I MAGINING IDENTITIES WITH AC-GAN Figure 10: AC-DCGAN generation with random identity vectors that sum to one. Each row shares an identity vector and each column shares an observation vector. Figure 11: AC-DCGAN generation with one-hot identity vectors. Each row shares an identity vector and each column shares an observation vector. As stated in Section 3.1, AC-GANs Odena et al. (2017) provide no obvious way to imagine new identities. For our evaluation (Section 3.2), the AC-GAN generator receives identity input zI ∈ [0, 1]10000 : a one-hot over all identities. One possible approach to imagining new identities would be to query a P10000 trained AC-GAN generator with a random vector zI such that i=1 zI [i] = 1. We found that this strategy produced little identity variety (Figure 10) compared to the normal one-hot strategy (Figure 11) and excluded it from our evaluation. E A RCHITECTURE D ESCRIPTIONS We list here the full architectural details for our SD-DCGAN and SD-BEGAN models. In these descriptions, k is the number of images that the generator produces and discriminator observes per identity (usually 2 for pairwise training), and dI is the number of dimensions in the latent space ZI (identity). In our experiments, dimensionality of ZO is always 100 − dI . As a concrete example, the bottleneck layer of the SD-BEGAN discriminator autoencoder (“fc2” in Table 6) with k = 2, dI = 50 has output dimensionality 150. We emphasize that generators are parameterized by k in the tables only for clarity and symmetry with the discriminators. Implementations need not modify the generator; instead, k can be collapsed into the batch size. For the stacked-channels versions of these discriminators, we simply change the number of input image channels from 3 to 3k and set k = 1 wherever k appears in the table. 14 Published as a conference paper at ICLR 2018 Table 2: Input abstraction for both SD-DCGAN and SD-BEGAN generators during training (where zO is always different for every pair or set of k) Operation [zi ; zo ] dup zi concat Input Shape [(dI ,);(k,100-dI )] [(dI ,);(k,100-dI )] [(k,dI );(k,100-dI )] Kernel Size Output Shape [(dI ,);(k,100-dI )] [(k,dI );(k,100-dI )] (k,100) Table 3: SD-DCGAN generator architecture Operation z fc1 reshape bnorm relu upconv1 bnorm relu upconv2 bnorm relu upconv3 bnorm relu upconv4 tanh Input Shape (k,100) (k,8192) (k,8192) (k,4,4,512) (k,4,4,512) (k,4,4,512) (k,8,8,256) (k,8,8,256) (k,8,8,256) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,32,32,64) (k,32,32,64) (k,32,32,64) (k,64,64,3) Kernel Size (100,8192) (5,5,512,256) (5,5,256,128) (5,5,128,64) (5,5,64,3) Output Shape (k,100) (k,8192) (k,4,4,512) (k,4,4,512) (k,4,4,512) (k,8,8,256) (k,8,8,256) (k,8,8,256) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,32,32,64) (k,32,32,64) (k,32,32,64) (k,64,64,3) (k,64,64,3) Table 4: SD-DCGAN discriminator architecture Operation x or G(z) downconv1 lrelu(a=0.2) downconv2 bnorm lrelu(a=0.2) downconv3 bnorm lrelu(a=0.2) downconv4 stackchannels downconv5 flatten fc1 sigmoid Input Shape (k,64,64,3) (k,64,64,3) (k,32,32,64) (k,32,32,64) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,8,8,256) (k,8,8,256) (k,8,8,256) (k,4,4,512) (4,4,512k) (2,2,512) (2048,) (1,) Kernel Size (5,5,3,64) (5,5,64,128) (5,5,128,256) (5,5,256,512) (3,3,512k,512) (2048,1) 15 Output Shape (k,64,64,3) (k,32,32,64) (k,32,32,64) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,8,8,256) (k,8,8,256) (k,8,8,256) (k,4,4,512) (4,4,512k) (2,2,512) (2048,) (1,) (1,) Published as a conference paper at ICLR 2018 Table 5: SD-BEGAN generator architecture Operation z fc1 reshape conv2d elu conv2d elu upsample2 conv2d elu conv2d elu upsample2 conv2d elu conv2d elu upsample2 conv2d elu conv2d elu conv2d Input Shape (k,100) (k,100,) (k,100,8192) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) Kernel Size (100,8192) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,128) (3,3,128,3) Output Shape (k,100) (k,100,8192) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,8,8,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,16,16,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,32,32,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,3) Table 6: SD-BEGAN discriminator autoencoder architecture. The decoder portion is equivalent to, but does not share weights with, the SD-BEGAN generator architecture (Table 5). Operation x or G(z) conv2d elu conv2d elu conv2d elu downconv2d elu conv2d elu conv2d elu downconv2d elu conv2d elu conv2d elu downconv2d elu conv2d elu conv2d elu flatten fc1 concat fc2 fc3 split G (Table 5) Input Shape (k,64,64,3) (k,64,64,3) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,32768) (k,100) (100k,) (dI +(100-dI )k,) (100k,) (k,100) Kernel Size (3,3,3,128) (3,3,128,128) (3,3,128,128) (3,3,128,256) (3,3,256,256) (3,3,256,256) (3,3,256,384) (3,3,384,384) (3,3,384,384) (3,3,384,512) (3,3,512,512) (3,3,512,512) (32768,100) (100k,dI +(100-dI )k,) (dI +(100-dI )k,100k,) 16 Output Shape (k,64,64,3) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,64,64,128) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,32,32,256) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,16,16,384) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,8,8,512) (k,32768) (k,100) (100k,) (dI +(100-dI )k,) (100k,) (k,100) (k,64,64,3) Published as a conference paper at ICLR 2018 F FACE S AMPLES We present samples from each model reported in Table 1 for qualitative comparison. In each matrix, zI is the same across all images in a row and zO is the same across all images in a column. We draw identity and observation vectors randomly for these samples. Figure 12: Generated samples from AC-DCGAN (four sample of real photos of ID on right) Figure 13: Generated samples from SD-DCGAN Figure 14: Generated samples from SD-DCGAN with stacked-channel discriminator 17 Published as a conference paper at ICLR 2018 Figure 15: Generated samples from SD-DCGAN with k = 4 Figure 16: Generated samples from SD-DCGAN with dI = 25 Figure 17: Generated samples from SD-DCGAN with dI = 75 Figure 18: Generated samples from SD-DCGAN trained with the Wasserstein GAN loss (Arjovsky et al., 2017). This model was optimized using RMS-prop (Hinton et al.) with α = 5e−5. In our evaluation (Section 3.2), FaceNet had an AUC of .770 and an accuracy of 68.5% (at τv ) on data generated by this model. We excluded it from Table 1 for brevity. 18 Published as a conference paper at ICLR 2018 Figure 19: Generated samples from SD-BEGAN Figure 20: Generated samples from SD-BEGAN with k = 4, demonstrating mode collapse G S HOE S AMPLES We present samples from an SD-DCGAN and SD-BEGAN trained on our shoes dataset. Figure 21: Generated samples from SD-DCGAN Figure 22: Generated samples from SD-BEGAN 19 

arXiv:1705.07962v2 [cs.LG] 19 Sep 2017 pix2code: Generating Code from a Graphical User Interface Screenshot Tony Beltramelli UIzard Technologies Copenhagen, Denmark tony@uizard.io Abstract Transforming a graphical user interface screenshot created by a designer into computer code is a typical task conducted by a developer in order to build customized software, websites, and mobile applications. In this paper, we show that deep learning methods can be leveraged to train a model end-to-end to automatically generate code from a single input image with over 77% of accuracy for three different platforms (i.e. iOS, Android and web-based technologies). 1 Introduction The process of implementing client-side software based on a Graphical User Interface (GUI) mockup created by a designer is the responsibility of developers. Implementing GUI code is, however, time-consuming and prevent developers from dedicating the majority of their time implementing the actual functionality and logic of the software they are building. Moreover, the computer languages used to implement such GUIs are specific to each target runtime system; thus resulting in tedious and repetitive work when the software being built is expected to run on multiple platforms using native technologies. In this paper, we describe a model trained end-to-end with stochastic gradient descent to simultaneously learns to model sequences and spatio-temporal visual features to generate variable-length strings of tokens from a single GUI image as input. Our first contribution is pix2code, a novel approach based on Convolutional and Recurrent Neural Networks allowing the generation of computer tokens from a single GUI screenshot as input. That is, no engineered feature extraction pipeline nor expert heuristics was designed to process the input data; our model learns from the pixel values of the input image alone. Our experiments demonstrate the effectiveness of our method for generating computer code for various platforms (i.e. iOS and Android native mobile interfaces, and multi-platform web-based HTML/CSS interfaces) without the need for any change or specific tuning to the model. In fact, pix2code can be used as such to support different target languages simply by being trained on a different dataset. A video demonstrating our system is available online1 . Our second contribution is the release of our synthesized datasets consisting of both GUI screenshots and associated source code for three different platforms. Our datasets and our pix2code implemention are publicly available2 to foster future research. 2 Related Work The automatic generation of programs using machine learning techniques is a relatively new field of research and program synthesis in a human-readable format have only been addressed very recently. 1 2 https://uizard.io/research#pix2code https://github.com/tonybeltramelli/pix2code A recent example is DeepCoder [2], a system able to generate computer programs by leveraging statistical predictions to augment traditional search techniques. In another work by Gaunt et al. [5], the generation of source code is enabled by learning the relationships between input-output examples via differentiable interpreters. Furthermore, Ling et al. [12] recently demonstrated program synthesis from a mixed natural language and structured program specification as input. It is important to note that most of these methods rely on Domain Specific Languages (DSLs); computer languages (e.g. markup languages, programming languages, modeling languages) that are designed for a specialized domain but are typically more restrictive than full-featured computer languages. Using DSLs thus limit the complexity of the programming language that needs to be modeled and reduce the size of the search space. Although the generation of computer programs is an active research field as suggested by these breakthroughs, program generation from visual inputs is still a nearly unexplored research area. The closest related work is a method developed by Nguyen et al. [14] to reverse-engineer Android user interfaces from screenshots. However, their method relies entirely on engineered heuristics requiring expert knowledge of the domain to be implemented successfully. Our paper is, to the best of our knowledge, the first work attempting to address the problem of user interface code generation from visual inputs by leveraging machine learning to learn latent variables instead of engineering complex heuristics. In order to exploit the graphical nature of our input, we can borrow methods from the computer vision literature. In fact, an important number of research [21, 4, 10, 22] have addressed the problem of image captioning with impressive results; showing that deep neural networks are able to learn latent variables describing objects in an image and their relationships with corresponding variable-length textual descriptions. All these methods rely on two main components. First, a Convolutional Neural Network (CNN) performing unsupervised feature learning mapping the raw input image to a learned representation. Second, a Recurrent Neural Network (RNN) performing language modeling on the textual description associated with the input picture. These approaches have the advantage of being differentiable end-to-end, thus allowing the use of gradient descent for optimization. (b) Sampling (a) Training Figure 1: Overview of the pix2code model architecture. During training, the GUI image is encoded by a CNN-based vision model; the context (i.e. a sequence of one-hot encoded tokens corresponding to DSL code) is encoded by a language model consisting of a stack of LSTM layers. The two resulting feature vectors are then concatenated and fed into a second stack of LSTM layers acting as a decoder. Finally, a softmax layer is used to sample one token at a time; the output size of the softmax layer corresponding to the DSL vocabulary size. Given an image and a sequence of tokens, the model (i.e. contained in the gray box) is differentiable and can thus be optimized end-to-end through gradient descent to predict the next token in the sequence. During sampling, the input context is updated for each prediction to contain the last predicted token. The resulting sequence of DSL tokens is compiled to the desired target language using traditional compiler design techniques. 3 pix2code The task of generating computer code written in a given programming language from a GUI screenshot can be compared to the task of generating English textual descriptions from a scene photography. In both scenarios, we want to produce a variable-length strings of tokens from pixel values. We can thus divide our problem into three sub-problems. First, a computer vision problem of understanding the given scene (i.e. in this case, the GUI image) and inferring the objects present, their identities, 2 positions, and poses (i.e. buttons, labels, element containers). Second, a language modeling problem of understanding text (i.e. in this case, computer code) and generating syntactically and semantically correct samples. Finally, the last challenge is to use the solutions to both previous sub-problems by exploiting the latent variables inferred from scene understanding to generate corresponding textual descriptions (i.e. computer code rather than English) of the objects represented by these variables. 3.1 Vision Model CNNs are currently the method of choice to solve a wide range of vision problems thanks to their topology allowing them to learn rich latent representations from the images they are trained on [16, 11]. We used a CNN to perform unsupervised feature learning by mapping an input image to a learned fixed-length vector; thus acting as an encoder as shown in Figure 1. The input images are initially re-sized to 256 × 256 pixels (the aspect ratio is not preserved) and the pixel values are normalized before to be fed in the CNN. No further pre-processing is performed. To encode each input image to a fixed-size output vector, we exclusively used small 3 × 3 receptive fields which are convolved with stride 1 as used by Simonyan and Zisserman for VGGNet [18]. These operations are applied twice before to down-sample with max-pooling. The width of the first convolutional layer is 32, followed by a layer of width 64, and finally width 128. Two fully connected layers of size 1024 applying the rectified linear unit activation complete the vision model. (b) Code describing the GUI written in our DSL (a) iOS GUI screenshot Figure 2: An example of a native iOS GUI written in our markup-like DSL. 3.2 Language Model We designed a simple lightweight DSL to describe GUIs as illustrated in Figure 2. In this work we are only interested in the GUI layout, the different graphical components, and their relationships; thus the actual textual value of the labels is ignored. Additionally to reducing the size of the search space, the DSL simplicity also reduces the size of the vocabulary (i.e. the total number of tokens supported by the DSL). As a result, our language model can perform token-level language modeling with a discrete input by using one-hot encoded vectors; eliminating the need for word embedding techniques such as word2vec [13] that can result in costly computations. In most programming languages and markup languages, an element is declared with an opening token; if children elements or instructions are contained within a block, a closing token is usually needed for the interpreter or the compiler. In such a scenario where the number of children elements contained in a parent element is variable, it is important to model long-term dependencies to be able to close a block that has been opened. Traditional RNN architectures suffer from vanishing and exploding gradients preventing them from being able to model such relationships between data points spread out in time series (i.e. in this case tokens spread out in a sequence). Hochreiter and Schmidhuber proposed the Long Short-Term Memory (LSTM) neural architecture in order to address this very problem [9]. The different LSTM gate outputs can be computed as follows: 3 it ft ot ct ht = φ(Wix xt + Wiy ht−1 + bi ) = φ(Wf x xt + Wf y ht−1 + bf ) = φ(Wox xt + Woy ht−1 + bo ) = ft • ct−1 + it • σ(Wcx xt + Wcy ht−1 + bc ) = ot • σ(ct ) (1) (2) (3) (4) (5) With W the matrices of weights, xt the new input vector at time t, ht−1 the previously produced output vector, ct−1 the previously produced cell state’s output, b the biases, and φ and σ the activation functions sigmoid and hyperbolic tangent, respectively. The cell state c learns to memorize information by using a recursive connection as done in traditional RNN cells. The input gate i is used to control the error flow on the inputs of cell state c to avoid input weight conflicts that occur in traditional RNN because the same weight has to be used for both storing certain inputs and ignoring others. The output gate o controls the error flow from the outputs of the cell state c to prevent output weight conflicts that happen in standard RNN because the same weight has to be used for both retrieving information and not retrieving others. The LSTM memory block can thus use i to decide when to write information in c and use o to decide when to read information from c. We used the LSTM variant proposed by Gers and Schmidhuber [6] with a forget gate f to reset memory and help the network model continuous sequences. 3.3 Decoder Our model is trained in a supervised learning manner by feeding an image I and a contextual sequence X of T tokens xt , t ∈ {0 . . . T − 1} as inputs; and the token xT as the target label. As shown on Figure 1, a CNN-based vision model encodes the input image I into a vectorial representation p. The input token xt is encoded by an LSTM-based language model into an intermediary representation qt allowing the model to focus more on certain tokens and less on others [8]. This first language model is implemented as a stack of two LSTM layers with 128 cells each. The vision-encoded vector p and the language-encoded vector qt are concatenated into a single feature vector rt which is then fed into a second LSTM-based model decoding the representations learned by both the vision model and the language model. The decoder thus learns to model the relationship between objects present in the input GUI image and the associated tokens present in the DSL code. Our decoder is implemented as a stack of two LSTM layers with 512 cells each. This architecture can be expressed mathematically as follows: p = CN N (I) qt = LST M (xt ) rt = (q, pt ) (6) (7) (8) yt = sof tmax(LST M 0 (rt )) xt+1 = yt (9) (10) This architecture allows the whole pix2code model to be optimized end-to-end with gradient descent to predict a token at a time after it has seen both the image as well as the preceding tokens in the sequence. The discrete nature of the output (i.e. fixed-sized vocabulary of tokens in the DSL) allows us to reduce the task to a classification problem. That is, the output layer of our model has the same number of cells as the vocabulary size; thus generating a probability distribution of the candidate tokens at each time step allowing the use of a softmax layer to perform multi-class classification. 3.4 Training The length T of the sequences used for training is important to model long-term dependencies; for example to be able to close a block of code that has been opened. After conducting empirical experiments, the DSL input files used for training were segmented with a sliding window of size 48; in other words, we unroll the recurrent neural network for 48 steps. This was found to be a satisfactory trade-off between long-term dependencies learning and computational cost. For every 4 token in the input DSL file, the model is therefore fed with both an input image and a contextual sequence of T = 48 tokens. While the context (i.e. sequence of tokens) used for training is updated at each time step (i.e. each token) by sliding the window, the very same input image I is reused for samples associated with the same GUI. The special tokens < ST ART > and < EN D > are used to respectively prefix and suffix the DSL files similarly to the method used by Karpathy and Fei-Fei [10]. Training is performed by computing the partial derivatives of the loss with respect to the network weights calculated with backpropagation to minimize the multiclass log loss: L(I, X) = − T X xt+1 log(yt ) (11) t=1 With xt+1 the expected token, and yt the predicted token. The model is optimized end-to-end hence the loss L is minimized with regard to all the parameters including all layers in the CNN-based vision model and all layers in both LSTM-based models. Training with the RMSProp algorithm [20] gave the best results with a learning rate set to 1e − 4 and by clipping the output gradient to the range [−1.0, 1.0] to cope with numerical instability [8]. To prevent overfitting, a dropout regularization [19] set to 25% is applied to the vision model after each max-pooling operation and at 30% after each fully-connected layer. In the LSTM-based models, dropout is set to 10% and only applied to the non-recurrent connections [24]. Our model was trained with mini-batches of 64 image-sequence pairs. Dataset type iOS UI (Storyboard) Android UI (XML) web-based UI (HTML/CSS) 3.5 Table 1: Dataset statistics. Training set Synthesizable Instances Samples 26 × 105 1500 93672 21 × 106 1500 85756 31 × 104 1500 143850 Test set Instances Samples 250 15984 250 14265 250 24108 Sampling To generate DSL code, we feed the GUI image I and a contextual sequence X of T = 48 tokens where tokens xt . . . xT −1 are initially set empty and the last token of the sequence xT is set to the special < ST ART > token. The predicted token yt is then used to update the next sequence of contextual tokens. That is, xt . . . xT −1 are set to xt+1 . . . xT (xt is thus discarded), with xT set to yt . The process is repeated until the token < EN D > is generated by the model. The generated DSL token sequence can then be compiled with traditional compilation methods to the desired target language. (b) Micro-average ROC curves (a) pix2code training loss Figure 3: Training loss on different datasets and ROC curves calculated during sampling with the model trained for 10 epochs. 5 4 Experiments Access to consequent datasets is a typical bottleneck when training deep neural networks. To the best of our knowledge, no dataset consisting of both GUI screenshots and source code was available at the time this paper was written. As a consequence, we synthesized our own data resulting in the three datasets described in Table 1. The column Synthesizable refers to the maximum number of unique GUI configuration that can be synthesized using our stochastic user interface generator. The columns Instances refers to the number of synthesized (GUI screenshot, GUI code) file pairs. The columns Samples refers to the number of distinct image-sequence pairs. In fact, training and sampling are done one token at a time by feeding the model with an image and a sequence of tokens obtained with a sliding window of fixed size T . The total number of training samples thus depends on the total number of tokens written in the DSL files and the size of the sliding window. Our stochastic user interface generator is designed to synthesize GUIs written in our DSL which is then compiled to the desired target language to be rendered. Using data synthesis also allows us to demonstrate the capability of our model to generate computer code for three different platforms. Table 2: Experiments results reported for the test sets described in Table 1. Error (%) Dataset type greedy search beam search 3 beam search 5 iOS UI (Storyboard) 22.73 25.22 23.94 Android UI (XML) 22.34 23.58 40.93 web-based UI (HTML/CSS) 12.14 11.01 22.35 Our model has around 109 × 106 parameters to optimize and all experiments are performed with the same model with no specific tuning; only the training datasets differ as shown on Figure 3. Code generation is performed with both greedy search and beam search to find the tokens that maximize the classification probability. To evaluate the quality of the generated output, the classification error is computed for each sampled DSL token and averaged over the whole test dataset. The length difference between the generated and the expected token sequences is also counted as error. The results can be seen on Table 2. Figures 4, 5, and 6 show samples consisting of input GUIs (i.e. ground truth), and output GUIs generated by a trained pix2code model. It is important to remember that the actual textual value of the labels is ignored and that both our data synthesis algorithm and our DSL compiler assign randomly generated text to the labels. Despite occasional problems to select the right color or the right style for specific GUI elements and some difficulties modelling GUIs consisting of long lists of graphical components, our model is generally able to learn the GUI layout in a satisfying manner and can preserve the hierarchical structure of the graphical elements. (a) Groundtruth GUI 1 (b) Generated GUI 1 (c) Groundtruth GUI 2 Figure 4: Experiment samples for the iOS GUI dataset. 6 (d) Generated GUI 2 5 Conclusion and Discussions In this paper, we presented pix2code, a novel method to generate computer code given a single GUI image as input. While our work demonstrates the potential of such a system to automate the process of implementing GUIs, we only scratched the surface of what is possible. Our model consists of relatively few parameters and was trained on a relatively small dataset. The quality of the generated code could be drastically improved by training a bigger model on significantly more data for an extended number of epochs. Implementing a now-standard attention mechanism [1, 22] could further improve the quality of the generated code. Using one-hot encoding does not provide any useful information about the relationships between the tokens since the method simply assigns an arbitrary vectorial representation to each token. Therefore, pre-training the language model to learn vectorial representations would allow the relationships between tokens in the DSL to be inferred (i.e. learning word embeddings such as word2vec [13]) and as a result alleviate semantical error in the generated code. Furthermore, one-hot encoding does not scale to very big vocabulary and thus restrict the number of symbols that the DSL can support. (a) Groundtruth GUI 5 (b) Generated GUI 5 (c) Groundtruth GUI 6 (d) Generated GUI 6 Figure 5: Experiment samples from the web-based GUI dataset. Generative Adversarial Networks GANs [7] have shown to be extremely powerful at generating images and sequences [23, 15, 25, 17, 3]. Applying such techniques to the problem of generating computer code from an input image is so far an unexplored research area. GANs could potentially be used as a standalone method to generate code or could be used in combination with our pix2code model to fine-tune results. A major drawback of deep neural networks is the need for a lot of training data for the resulting model to generalize well on new unseen examples. One of the significant advantages of the method we described in this paper is that there is no need for human-labelled data. In fact, the network can model the relationships between graphical components and associated tokens by simply being trained on image-sequence pairs. Although we used data synthesis in our paper partly to demonstrate the capability of our method to generate GUI code for various platforms; data synthesis might not be needed at all if one wants to focus only on web-based GUIs. In fact, one could imagine crawling the World Wide Web to collect a dataset of HTML/CSS code associated with screenshots of rendered websites. Considering the large number of web pages already available online and the fact that new websites are created every day, the web could theoretically supply a virtually unlimited amount of 7 training data; potentially allowing deep learning methods to fully automate the implementation of web-based GUIs. (a) Groundtruth GUI 3 (b) Generated GUI 3 (c) Groundtruth GUI 4 (d) Generated GUI 4 Figure 6: Experiment samples from the Android GUI dataset. References [1] D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. [2] M. Balog, A. L. 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A Statistical Perspective on Inverse and Inverse Regression Problems Debashis Chatterjeea , Sourabh Bhattacharyaa,∗ arXiv:1707.06852v1 [stat.ME] 21 Jul 2017 a Interdisciplinary Statistical Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata - 700108, India Abstract Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. However, in mainstream statistics such inverse problem paradigm does not seem to be as popular. In this article we provide a brief overview of such problems from a statistical, particularly Bayesian, perspective. We also compare and contrast the above class of problems with the perhaps more statistically familiar inverse regression problems, arguing that this class of problems contains the traditional class of inverse problems. In course of our review we point out that the statistical literature is very scarce with respect to both the inverse paradigms, and substantial research work is still necessary to develop the fields. Keywords: Bayesian analysis, Inverse problems, Inverse regression problems, Regularization, Reproducing Kernel Hilbert Space (RKHS), Palaeoclimate reconstruction 1. Introduction The similarities and dissimilarities between inverse problems and the more traditional forward problems are usually not clearly explained in the literature, and often “ill-posed” is the term used to loosely characterize inverse problems. We point out that these two problems may have the same goal or different goal, while both consider the same model given the data. We first elucidate using the traditional case of deterministic differential equations, that the goals of the two problems may be the same. Consider a dynamical system dxt = G(t, xt , θ), (1.1) dt where G is a known function and θ is a parameter. In the forward problem the goal is to obtain the solution xt ≡ xt (θ), given θ and the initial conditions, whereas, in ∗ Corresponding author. Email addresses: debashis1chatterjee@gmail.com (Debashis Chatterjee), bhsourabh@gmail.com, sourabh@isical.ac.in (Sourabh Bhattacharya) Preprint submitted to XYZ July 24, 2017 the inverse problem, the aim is to obtain θ given the solution process xt . Realistically, the differential equation would be perturbed by noise, and so, one observes the data y = (y1 , . . . , yT )T , where yt = xt (θ) + t , (1.2) for noise variables t having some suitable independent and identical (iid) error distribution q, which we assume to be known for simplicity of illustration. A typical method of estimating θ, employed by the scientific community, is the method of calibration, where the solution of (1.1) would be obtained for each θ-value on a proposed grid of plausible values, and a set ỹ(θ) = (ỹ1 (θ), . . . , ỹT (θ))T is generated from the iid model (1.2) for every such θ after simulating, for i = 1, . . . , T , ˜t ∼ q; then forming ỹt (θ) = xt (θ) + ˜t , and finally reporting that value θ in the grid as an estimate of the true values for which ky − ỹ(θ)k is minimized, given some distance measure k·k; maximization of the correlation between y and ỹ(θ) is also considered. In other words, the calibration method makes use of the forward technique to estimate the desired quantities of the model. On the other hand, the inverse problem paradigm attempts to directly estimate θ from the observed data y usually by minimizing some discrepancy measure between y and x(θ), where x(θ) = (x1 (θ), . . . , xT (θ))T . Hence, from this perspective the goals of both forward and inverse approaches are the same, that is, estimation of θ. However, the forward approach is well-posed, whereas, the inverse approach is often ill-posed. To clarify, note that within a grid, there always exists some θ̂ that minimizes ky − ỹ(θ)k among all the grid-values. In this sense the forward problem may be thought of as well-posed. However, direct minimization of the discrepancy between y and x(θ) with respect to θ is usually difficult and for high-dimensional θ, the solution to the minimization problem is usually not unique, and small perturbations of the data causes large changes in the possible set of solutions, so that the inverse approach is usually ill-posed. Of course, if the minimization is sought over a set of grid values of θ only, then the inverse problem becomes well-posed. From the statistical perspective, the unknown parameter θ of the model needs to be learned, in either classical or Bayesian way, and hence, in this sense there is no real distinction between forward and inverse problems. Indeed, statistically, since the data are modeled conditionally on the parameters, all problems where learning the model parameter given the data is the goal, are inverse problems. We remark that the literature usually considers learning unknown functions from the data in the realm of inverse problems, but a function is nothing but an infinite-dimensional parameter, which is a very common learning problem in statistics. We now explain when forward and inverse problems can differ in their aims, and are significantly different even from the statistical perspective. To give an example, consider the palaeoclimate reconstruction problem discussed in Haslett et al. [18] where the reconstruction of prehistoric climate at Glendalough in Ireland from fossil pollen is of interest. The model is built on the realistic assumption that pollen abundance depends upon climate, not the other way around. The compositional pollen data with the modern climates are available at many modern sites but the climate values associated with the fossil pollen data are missing. The inverse nature of the problem is associated with the fact that it is of interest to predict the fossil climate values, given the pollen assemblages. The forward problem would result, if given the fossil climate values (if 2 known), the fossil pollen abundances (if unknown), were to be predicted. Technically, given a data set y that depends upon covariates x, with a probability distribution f (y|x, θ) where θ is the model parameter, we call the problem ‘inverse’ if it is of interest to predict the corresponding unknown x̃ given a new observed ỹ (see Bhattacharya and Haslett [9]), after eliminating θ. On the other hand, the more conventional forward problem considers the prediction of ỹ for given x̃ with the same probability distribution, again, after eliminating the unknown parameter θ. This perspective clearly distinguishes the forward and inverse problems, as opposed to the other parameter-learning perspective discussed above, which is much more widely considered in the literature. In fact, with respect to predicting unknown covariates from the responses, mostly inverse linear regression, particularly in the classical set-up, has been considered in the literature. To distinguish the traditional inverse problems from the covariate-prediction perspective, we use the phrase ‘inverse regression’ to refer to the latter. Other examples of inverse regression are given in Section 7. Our discussion shows that statistically, there is nothing special about the existing literature on inverse problems that considers estimation of unknown (perhaps, infinitedimensional) parameters, and the only class of problems that can be truly regarded as inverse problems as distinguished from forward problems are those which consider prediction of unknown covariates from the dependent response data. However, for the sake of completeness, the traditional inverse problems related to learning of unknown functions shall occupy a significant portion of our review. The rest of the paper is structured as follows. In Section 2 we discuss the general inverse model, providing several examples. In Section 3 we focus on linear inverse problems, which constitute the most popular class of inverse problems, and review the links between the Bayesian approach based on simple finite difference priors and the deterministic Tikhonov regularization. Connections between Gaussian process based Bayesian inverse problems and deterministic regularizations are reviewed in Section 4. In Section 5 we provide an overview of the connections between the Gaussian process based Bayesian approach and regularization using differential operators, which generalizes the discussion of Section 3 on the connection between finite difference priors and the Tikhonov regularization. The Bayesian approach to inverse problems in Hilbert spaces is discussed in Section 6. We then turn attention to inverse regression problems, providing an overview of such problems and discussing the links with traditional inverse problems in Section 7. Finally, we make concluding remarks in Section 8. 2. Traditional inverse problem Suppose that one is interested in learning about the function θ given the noisy observed responses y n = (y1 , . . . , yn )T , where the relationship between θ and y n is governed by following equation (2.1) : yi = G(xi , θ) + i , (2.1) for i = 1, . . . , n, where xi are known covariates or design points, i are errors associated with the i-th observation and G is a forward operator defined appropriately, which is usually allowed to be non-injective. 3 Note that since n = (1 , . . . , n )T is unknown, the noisy observation vector y n itself may not be in the image set of G. If θ is a p-dimensional parameter, then there will often be situations when the number of equations is smaller than the number of unknowns, in the sense that p > n (see, for example, Dashti and Stuart [14]). Modern statistical research is increasingly coming across such inverse problems termed as “ill-posed” which are not in the exact domain of statistical estimation procedures (O’Sullivan [27]) where the maximum likelihood solution or classical least squares may not be uniquely defined and with very bad perturbation sensitivity of the classical solution. However, although such problematic issues are said to characterize inverse problems, the problems in fact fall in the so-called “large p small n” paradigm and has received wide attention in statistics; see, for example, Bühlmann and van de Geer [10], Giraud [17]. A key concept involved in handling such problems is inclusion of some appropriate penalty term in the discrepancy to be minimized with respect to θ. Such regularization methods are initiated by Tikhonov [34] and Tikhonov and Arsenin [35]. Under this method, usually a criterion of the following form is chosen for the minimization purpose: n 1X 2 [yi − G(xi , θ)] + λJ(θ), λ > 0. n i=1 (2.2) The functional J is chosen such that highly implausible or irregular values of θ has large values (O’Sullivan [27]). Thus, depending on the problem at hand, J(θ) can be used to induce “sparsity” in an appropriate sense so that the minimization problem may be well-defined. We next present several examples of classical inverse problems based on Aster et al. [3]. 2.1. Examples of inverse problems 2.1.1. Vertical seismic profiling In this scientific field, one wishes to learn about the vertical seismic velocity of the material surrounding a borehole. A source generates downward-propagating seismic wavefront at the surface, and in the borehole, a string of seismometers sense these seismic waves. The arrival times of the seismic wavefront at each instrument are measured from the recorded seismograms. These times provide information on the seismic velocity for vertically traveling waves as a function of depth. The problem is nonlinear if it is expressed in terms of seismic velocities. However, we can linearize this problem via a simple change of variables, as follows. Letting z denote the depth, it is possible to parameterize the seismic structure in terms of slowness, s(z), which is the reciprocal of the velocity v(z). The observed travel time at depth z can then be expressed as: Z z Z ∞ t(z) = s(u)du = s(u)H(z − u)du, (2.3) 0 0 where H is the Heaviside step function. The interest is to learn about s(z) given observed t(z). Theoretically, s(z) = dt(z) dz , but in practice, simply differentiating the observations need not lead to useful solutions because noise is generally present in the observed times t(z), and naive differentiation may lead to unrealistic features of the solution. 4 2.1.2. Estimation of buried line mass density from vertical gravity anomaly Here the problem is to estimate an unknown buried line mass density m(x) from data on vertical gravity anomaly, d(x), observed at some height, h. The mathematical relationship between d(x) and m(x) is given by Z ∞ h d(x) = 3 m(u)du. −∞ [(u − x)2 + h2 ] 2 As before, noise in the data renders the above linear inverse problem difficult. Variations of the above example has been considered in Aster et al. [3]. 2.1.3. Estimation of incident light intensity from diffracted light intensity Consider an experiment in which an angular distribution of illumination passes through a thin slit and produces a diffraction pattern, for which the intensity is observed. The data, d(s), are measurements of diffracted light intensity as a function of the outgoing angle −π/2 ≤ s ≤ π/2. The goal here is to obtain the intensity of incident light on the slit, m(θ), as a function of the incoming angle −π/2 ≤ θ ≤ π/2, using the following mathematical relationship: Z π/2 d(s) = (cos(s) + cos(θ)) 2 −π/2  sin(π (sin(s) + sin(θ))) π (sin(s) + sin(θ)) 2 m(θ)dθ. 2.1.4. Groundwater pollution source history reconstruction problem Consider the problem of recovering the history of groundwater pollution at a source site from later measurements of the contamination at downstream wells to which the contaminant plume has been transported by advection and diffusion. The mathematical model for contamination transport is given by the following advection-diffusion equation with respect to t and transported site x: ∂C ∂2C ∂C =D 2 −ν ∂t ∂x ∂x C(0, t) = Cin (t) C(x, t) → 0 as x → ∞. In the above, D is the diffusion coefficient, ν is the velocity of the groundwater flow, and Cin (t) is the time history of contaminant injection at x = 0. The solution to the above advection-diffusion equation is given by Z T Cin (t)f (x, T − t)dt, C(x, T ) = 0 where " (x − ν(T − t)) f (x, T − t) = p exp 3 4D(T − t) 2 πD(T − t) x 2 It is of interest to learn about Cin (t) from data observed on C(x, T ). 5 # . 2.1.5. Transmission tomography The most basic physical model for tomography assumes that wave energy traveling between a source and receiver can be considered to be propagating along infinitesimally narrow ray paths. In seismic tomography, if the slowness at a point x is s(x), and the ray path is known, then the travel time for seismic energy transiting along that ray path is given by the line integral along `: Z t = s(x(l))dl. (2.4) ` Learning of s(x) from t is required. Note that (2.4) is a high-dimensional generalization of (2.3). In reality, seismic ray paths will be bent due to refraction and/or reflection, resulting in nonlinear inverse problem. The above examples demonstrate the ubiquity of linear inverse problems. As a result, in the next section we take up the case of linear inverse problems and illustrate the Bayesian approach in details, also investigating connections with the deterministic approach employed by the general scientific community. 3. Linear inverse problem The motivating examples and discussions in this section are based on Bui-Thanh [11]. Let us consider the following one-dimensional integral equation on a finite interval as in equation (3.1): Z G(x, θ) = K(x, t) θ(t) dt, (3.1) where K(x, ·) is some appropriate, known, real-valued function given x Now, let the dataset be y n = (y1 , y2 , . . . , yn )T . Then for a known system response K(xi , t) for the dataset, the equation can be written as follows: Z yi = G(xi , θ) + i ; i ∈ {1, 2, . . . , n} (3.2) R1 As a particular example, let G(x, θ) = 0 K(x, t) θ(t) dt, where K(x, t) =  √ 1 2 exp −(x − t)2 /2ψ 2 is the Gaussian kernel and θ : [0, 1] 7→ R is to be 2πψ learned given the data y n and xn = (x1 , . . . , xn )T . We first illustrate the Bayesian approach and draw connections with the traditional approach of Tikhonov’s regularization when the integral in G is discretized. In this regard, let xi = (i − 1)/n, for i = 1, . . . , n. Letting θ = (θ(x1 ), . . . , θ(xn ))T and K be the n × n matrix with the (i, j)-th element K(xi , xj )/n, and n = (1 , . . . , n )T , the discretized version of (3.2) can be represented as y n = Kθ + n . (3.3)
We assume that n ∼ Nn 0n , σ 2 I n , that is, an n-variate normal with mean 0n , an n-dimensional vector with all components zero, and covariance σ 2 I n , where I n is the n-th order identity matrix. 6 3.1. Smooth prior on θ To reflect the belief that the function θ is smooth, one may presume that θ(xi−1 ) + θ(xi+1 ) + ˜i , (3.4) 2
iid where, for i = 1, . . . , n, ˜i ∼ N 0, σ̃ 2 . Thus, a priori, θ(xi ) is assumed to be an average of its nearest neighbors to quantify smoothness, with an additive random perturbation term. Letting   −1 2 −1 0 · · · · · ·  0 −1 2 −1 0 · · ·    .. .. .. .. .. ..  1  . . . . .  (3.5) L=  . , 2 .  . . . . . .. .. .. .. ..   .. 0 0 · · · −1 2 −1 θ(xi ) = and ˜ = (˜ 1 , . . . , ˜n )T , it follows from (3.4) that Lθ = ˜, (3.6) Now, noting that the Laplacian of a twice-differentiable real-valued function f with Pk 2 independent arguments z1 , . . . , zk is given by ∆f = i=1 ∂∂zf2 , we have i ∆θ(xj ) ≈ n2 (Lθ)j , (3.7) where (Lθ)j is the j-th element of Lθ. However, the rank of L is n−1, and boundary conditions on the Laplacian operator is necessary to ensure positive definiteness of the operator. In our case, we assume that θ ≡ 0 outside [0, 1], so that we now assume θ(0) = θ(x2 1 ) + ˜0 and θ(xn ) =
θ(xn−1 ) + ˜n , where ˜0 and ˜n are iid N 0, σ̃ 2 . With this modification, the prior on 2 θ is given by   1 2 π(θ) ∝ exp − 2 kL̃θk , (3.8) 2σ̃ where k · k is the Euclidean norm and  2 −1  −1 2   0 −1  .. .. 1 . . L̃ =   2 . . ..  ..   0 0 0 0 0 −1 2 .. . .. . ··· ··· 7 0 0 −1 .. . .. . −1 0 ··· ··· 0 .. . .. . 2 −1 ··· ··· ··· .. . .. . −1 2       .      (3.9) Rather than assuming zero boundary conditions,   moregenerally  one may assume σ̃ 2 σ̃ 2 that θ(0) and θ(xn ) are distributed as N 0, δ2 and N 0, δ2 , respectively. The n 0 resulting modified matrix is then given by   2δ0 0 0 0 ··· ···  −1 2 −1 0 · · · · · ·     0 −1 2 −1 0 · · ·    .. .. .. .. ..  1  .. . . . . . . . (3.10) L̂ =   2   . . . . . . .. .. .. .. ..   ..    0 0 · · · −1 2 −1  0 0 ··· 0 0 2δn To choose δ0 and δn , one may assume that V ar [θ(0)] =  T −1   σ̃ 2 σ̃ 2 2 T [n/2] , = V ar [θ(x )] = = V ar θ(x ) = σ̃  L̂ L̂ n [n/2] [n/2] δ02 δn2 where [n/2] is the largest integer not exceeding n/2, and [n/2] is the [n/2]-th canonical basis vector in Rn+1 . It follows that δ02 = δn2 = 1 . −1 [n/2] T[n/2] L̂ L̂  T Since this requires solving a non-linear equation (since L̂ contains δ0 and δn ), for avoiding computational complexity one may simply employ the approximation δ02 = δn2 = 1 , −1 T[n/2] L̃ L̃ [n/2]  T where L̃ is given by (3.9). 3.2. Non-smooth prior on θ To begin with, let us assume that θ has several points of discontinuities on the grid of points {x0 , . . . , xn }. To reflect this information in the prior, one may assume that θ(0) = 0
and for i = 1, . . . , n, θ(xi ) = θ(xi−1 ) + ˜i , where, as before, ˜i are iid N 0, σ̃ 2 . Then, with   1 0 0 0 ··· ···  −1 1 0 0 ··· ···     0 −1 1 0 0 ···    .. .. .. .. ..  1  .. , . . . . . . L∗ =  (3.11)  2  .  . . . . . .. .. .. .. ..   ..    0 0 · · · −1 1 0  0 0 ··· 0 − 1 8 the prior is given by   1 ∗ 2 π(θ) ∝ exp − 2 kL θk . 2σ̃ (3.12) One may also flexibly account for any particular big jump. For instance, if for some ` ∈ {0, . . . , n}, the jump θ(x` ) − θ(x`−1 ) is particularly large compared  to the  other 2 jumps, then it can be assumed that θ(x` ) = θ(x`−1 ) + ∗` , with ∗` ∼ N 0, σ̃ξ2 , where ξ < 1. Letting D ` be the diagonal matrix with ξ 2 being the `-th diagonal element and 1 being the other diagonal elements, the prior is then given by   1 π(θ) ∝ exp − 2 kD ` L∗ θk2 . (3.13) 2σ̃ A more general prior can be envisaged where the number and location of the jump discontinuities are unknown. Then we may consider a diagonal matrix D = diag{ξ1 , . . . , ξn }, so that conditionally on the hyperparameters ξ1 , . . . , ξn , the prior on θ is given by   1 ∗ 2 (3.14) π(θ|ξ1 , . . . , ξn ) ∝ exp − 2 kDL θk . 2σ̃ Prior on ξ1 , . . . , ξn may be considered to complete the specification. These may also be estimated by maximizing the marginal likelihood obtained by integrating out θ, which is known as the ML-II method; see Berger [6]. Calvetti and Somersalo [12] also advocate likelihood based methods. 3.3. Posterior distribution For convenience, let us generically denote the matrices L, L̃, L̂, L∗ , D ` L∗ , DL∗ , 1 by Γ− 2 . Then it can be easily verified that the posterior of θ admits the following generic form:    1 1 − 21 2 2 π (θ|y n , xn ) ∝ exp − ky − Kθk + kΓ θk . (3.15) 2σ 2 n 2σ̃ 2 Note that the exponent of the posterior is of the form of the Tikhonov functional, which we denote by T (θ). The maximizer of the posterior, commonly known as the maximum a posteriori (MAP) estimator, is given by θ̂ M AP = arg max π (θ|y n , xn ) = arg min T (θ). θ (3.16) θ In other words, the deterministic solution to the inverse problem obtained by Tikhonov’s regularization is nothing but the Bayesian MAP estimator in our context. Writing H = σ12 K T K + σ̃12 Γ−1 , which is the Hessian of the Tikhonov functional 1 (regularized misfit), and writing k·kH = kH 2 ·k, it is clear that (3.15) can be simplified to the Gaussian form, given by ( ) 2 1 −1 −1 π (θ|y n , xn ) ∝ exp − θ − 2 H K y n . (3.17) σ H 9 It follows from (3.17) that the inverse of the Hessian of the regularized misfit is the posterior covariance itself. From the above posterior it also trivially follows that θ̂ M AP 1 1 = 2 H −1 K −1 y n = 2 σ σ  1 T 1 K K + 2 Γ1 σ2 σ̃ −1 K T Y n, (3.18) which coincides with the Tikhonov solution for linear inverse problems. The connection between the traditional deterministic Tikhonov regularization approach with Bayesian analysis continues to hold even if the likelihood is non-Gaussian. 3.4. Exploration of the smoothness conditions For deeper investigation of the smoothness conditions, let us write   1 1 1 2 2 2 2 θ̂ M AP = arg min T (θ) = σ ky − ỹ n k + %kΓ̃ θk , 2 n 2 θ 1 (3.19) 1 where ỹ n = Kθ, % = σ 2 /σ̃ 2 and Γ̃ 2 = Γ− 2 . Now, from (3.7) it follows that for the smooth priors with the zero boundary conditions, our Tikhonov functional discretizes T∞ (θ) = 1 1 ky − ỹ n k2 + %k∆θk2L2 (0,1) , 2 n 2 (3.20) R1 where k · k2L2 (0,1) = 0 (·)2 dt. On the other hand, for the non-smooth prior (3.12), rather than discretizing ∆θ, ∇θ, that is, the gradient of θ, is discretized. In other words, for non-smooth priors, our Tikhonov functional discretizes T∞ (θ) = 1 1 ky − ỹ n k2 + %k∇θk2L2 (0,1) . 2 n 2 (3.21) Hence, realizations of prior (3.12) is less smooth compared to those of our smooth priors. However, the realizations (3.12) must be continuous. The priors given by (3.13) and (3.14) also support continuous functions as long as the hyperparameters are bounded away from zero. These facts, although clear, can be rigorously justified by functional analysis arguments, in particular, using the Sobolev imbedding theorem (see, for example, Arbogast and Bona [1]). 4. Links between Bayesian inverse problems based on Gaussian process prior and deterministic regularizations In this section, based on Rasmussen and Williams [31], we illustrate the connections between deterministic regularizations such as those obtained from differential operators as above, and Bayesian inverse problems based on the very popular Gaussian process prior on the unknown function. A key tool for investigating such relationship is the reproducing kernel Hilbert space (RKHS). 10 4.1. RKHS We adopt the following definition of RKHS provided in Rasmussen and Williams [31]: Definition 4.1 (RKHS). Let H be a Hilbert space of real functions θ defined on an index set X . Then H is called an RKHS endowed with an inner product h·, ·iH (and norm kθkH = hθ, θiH ) if there exists a function K : X × X 7→ R with the following properties: (a) for every x, K(·, x) ∈ H, and (b) K has the reproducing property hθ(·), K(·, x)iH = θ(x). Observe that since K(·, x), K(·, x0 ) ∈ H, it follows that hK(·, x), K(·, x0 )iH = K(x, x0 ). The Moore-Aronszajn theorem asserts that the RKHS uniquely determines K, and vice versa. Formally, Theorem 1 (Aronszajn [2]). . Let X be an index set. Then for every positive definite function K(·, ·) on X × X there exists a unique RKHS, and vice versa. R Here, by positive definite function K(·, ·) on X×X, we mean K(x, x0 )g(x)g(x0 )dν(x)dν(x0 ) > 0 for all non-zero functions g ∈ L2 (X, ν), where L2 (X, ν) denotes the space of functions square-integrable on X with respect to the measure ν. Indeed, the subspace H0 of H spanned by the functions {K(·, xi ); i = 1, 2, . . .} is dense in H in the sense that every function in H is a pointwise limit of a Cauchy sequence from H0 . To proceed, we require the concepts of eigenvalues and eigenfunctions associated with kernels. In the following section we provide a briefing on these. 4.2. Eigenvalues and eigenfunctions of kernels We borrow the statements of the following definition of eigenvalue and eigenfunction, and the subsequent statement of Mercer’s theorem from Rasmussen and Williams [31]. Definition 4.2. A function ψ(·) that obeys the integral equation Z C(x, x0 )ψ(x)dν(x) = λψ(x0 ), (4.1) X is called an eigenfunction of the kernel C with eigenvalue λ with respect to the measure ν. We assume that the ordering is chosen such that λ1 ≥ λ2 ≥ · · · . The eigenfunctions are orthogonal with respect to ν and can be chosen to be normalized so that R ψ (x)ψ i j (x)dν(x) = δij , where δij = 1 if i = j and 0 otherwise. X The following well-known theorem (see, for example, König [21]) expresses the positive definite kernel C in terms of its eigenvalues and eigenfunctions. 11 Theorem 2 (Mercer’s theorem). Let (X, ν) be a
finite measure space and C ∈ L∞ X2 , ν 2 be a positive definite kernel. By L∞ X2 , ν 2 we mean the set of all measurable functions C : X2 7→ R which are essentially bounded, that is, bounded up to a set of ν 2 -measure zero. For any function C in this set, its essential supremum, given by inf {C ≥ 0 : |C(x1 , x2 )| < C, for almost all (x1 , x2 ) ∈ X × X} serves as the norm kCk. Let ψj ∈ L2 (X, ν) be the normalized eigenfunctions of C associated with the eigenvalues λj (C) > 0. Then ∞ (a) the eigenvalues {λj (C)}j=1 are absolutely summable. P∞ (b) C(x, x0 ) = j=1 λj (C)ψj (x)ψ¯j (x0 ) holds ν 2 -almost everywhere, where the series converges absolutely and uniformly ν 2 -almost everywhere. In the above, ψ¯j denotes the complex conjugate of ψj . It is important to note the difference between the eigenvalue λj (C) associated with the kernel C and λj (Σn ) where Σn denotes the n × n Gram matrix with (i, j)-th element C(xi , xj ). Observe that (see Rasmussen and Williams [31]): 0 Z λj (C)ψj (x ) = n C(x, x0 )ψj (x)dν(x) ≈ X 1X C(xi , x0 )ψj (xi , x0 ), n i=1 (4.2) where, for i = 1, . . . , n, xi ∼ ν, assuming that ν is a probability measure. Now substituting x0 = xi ; i = 1, . . . , n in (4.2) yields the following approximate eigen system for the matrix Σn : Σn uj ≈ nλj (C)uj , (4.3) where the i-th component of uj is given by uij = ψj (xi ) √ . n (4.4) Since ψj are normalized to have unit norm, it holds that n uTj uj = 1X 2 ψ (xi ) ≈ n i=1 j Z ψ 2 (x)dν(x) = 1. (4.5) X From (4.5) it follows that λj (Σn ) ≈ nλj (C). −1 (4.6) Indeed, Theorem 3.4 of Baker [5] shows that n λj (Σn ) → λj (C), as n → ∞. For our purposes the main usefulness of the RKHS framework is that kθk2H can be perceived as a generalization of θ T K −1 θ, where θ = (θ(x1 ), . . . , θ(xn ))T and K = (K(xi , xj ))i,j=1,...,n , is the n × n matrix with (i, j)-th element K(xi , xj ). 12
4.3. Inner product Consider a real positive semidefinite kernel K(x, x0 ) with an eigenfunction exPN pansion K(x, x0 ) = i=1 λi φi (x)φi (x0 ) relative to a measure µ. Mercer’s theorem ensures that the eigenfunctions are orthonormal with respect to µ, that is, we have R φi (x)φj (x)dµ(x) = δij . Consider a Hilbert space of linear combinations of the PN PN θi2 eigenfunctions, that is, θ(x) = i=1 θi φi (x) with i=1 λi < ∞. Then the inner PN PN product hθ1 , θ2 iH between θ1 = i=1 θ1i φi (x), and θ2 = i=1 θ2i φi (x) is of the form N X θ1i θ2i hθ1 , θ2 iH = . (4.7) λi i=1 PN θ2 This induces the norm k · kH , where kθk2H = i=1 λii . A smoothness condition on the space is immediately imposed by requiring the norm to be finite – the eigenvalues must decay sufficiently fast. The Hilbert space defined above is a unique RKHS with respect to K, in that it satisfies the following reproducing property: hθ, K(·, x)i = N X θi λi φi (x) i=1 λi = θ(x). (4.8) = K(x, x0 ). (4.9) Further, the kernel satisfies the following: hK(x, ·), K(x0 , ·)i = N X λ2 φi (x) i i=1 λi PN Now, with reference to (4.6), observe that the square norm kθk2H = i=1 θi2 /λi and the quadratic form θ T Kθ have the same form if the latter is expressed in terms of the eigenvectors of K, albeit the latter has n terms, while the square norm has N terms. 4.4. Regularization The ill-posed-ness of inverse problems can be understood from the fact that for any given data set y n , all functions that pass through the data set minimize any given measure of discrepancy D(y n , θ) between the data y n and θ. To combat this, one considers minimization of the following regularized functional: R(θ) = D(y n , θ) + τ kθk2H , 2 (4.10) where the second term, which is the regularizer, controls smoothness of the function and τ is the appropriate Lagrange multiplier. The well-known representer theorem (see, for example, Kimeldorf and Wahba [20], O’Sullivan et al. [28], Wahba [38], Schölkopf and Pn Smola [32]) guarantees that each minimizer θ ∈ H can be represented as θ(x) = i=1 ci K (x, xi ), where K is the corresponding reproducing kernel. If D (y n , θ) is convex, then there is a unique minimizer θ̂. 13 4.5. Gaussian process modeling of the unknown function θ For simplicity, let us consider the model yi = θ(xi ) + i , (4.11) iid for i = 1, . . . , n, where i ∼ N (0, σ 2 ), where we assume σ to be known for simplicity of illustration. Let θ(x) be modeled by a Gaussian process with mean function µ(x) and covariance kernel K associated with the RKHS. In other words, for any x ∈ X, E [θ(x)] = µ(x) and for any x1 , x2 ∈ X, Cov (θ(x1 ), θ(x2 )) = K(x1 , x2 ). Assuming for convenience that µ(x) = 0 for all x ∈ X, it follows that the posterior distribution of θ(x∗ ) for any x∗ ∈ X is given by
π(θ(x∗ )|y n , xn ) ≡ N µ̂(x∗ ), σ̂ 2 (x∗ ) , (4.12) where, for any x∗ ∈ X, µ̂(x∗ ) = sT (x∗ ) K + σ 2 In
−1 yn ; σ̂ 2 (x∗ ) = K(x∗ , x∗ ) − sT (x∗ ) bK + σ 2 In (4.13)
−1 s(x∗ ), (4.14) T with s(x∗ ) = (K(x∗ , x1 ), . . . , K(x∗ , xn )) . Observe that the posterior mean admits the following representation: µ̂(x∗ ) = n X c̃i K(x∗ , xi ), (4.15) i=1
−1 where c̃i is the i-th element of K + σ 2 In yn . In other words, the posterior mean of the Gaussian process based model is consistent with the representer theorem. 5. Regularization using differential operators and connection with Gaussian process For x = (x1 , . . . , xd )T ∈ Rd , let m 2 Z kL θk = X ∂ m θ(x) j1 +···+jd =m ∂xj11 · · · ∂xjdd and kPθk2 = M X bm kLm θk2 , !2 , (5.1) (5.2) m=0 for some M > 0, where the co-efficients bm ≥ 0. In particular, we assume for our purpose that b0 > 0. It is clear that kPθk2 is translation and rotation invariant. This norm penalizes θ in terms of its derivatives up to order M . 14 5.1. Relation to RKHS It can be shown, using the fact that the complex exponentials exp(2πisT x) are eigen functions of the differential operator, that 2 kPθk = Z X M 2
m bm 4π 2 sT s θ̃(s) ds, (5.3) m=0 where θ̃(s) is the Fourier transform of θ(s). Comparison of (5.3) with (4.7) yields the hP
m i−1 M 2 T power spectrum of the form b 4π s s which yields the following m=0 m kernel by Fourier inversion: Z exp(2πisT (x − x0 )) 0 0 ds. (5.4) K(x, x ) = K(x − x ) = PM 2 T m m=0 bm (4π s s) Calculus of variations can also be used to minimize R(θ) with respect to θ, which yields (using the Euler-Lagrange equation) θ(x) = n X bi G(x − xi ), (5.5) i=1 with m X (−1)m bm ∇m G = δx−x0 , (5.6) i=1 where G is known as the Green’s function. Using Fourier transform on (5.6) it can be shown that the Green’s function Pm is nothing but the kernel K given by (5.4). Moreover, it follows from (5.6) that i=1 (−1)m bm ∇m and K are inverses of each other. Examples of kernels derived from differential operators are as follows. For d = 1, setting b0 = b2 , b1 = 1 and bm = 0 for m ≥ 2, one obtains K(x, x0 ) = K(x − x0 ) = 1 0 2b exp (−b|x − x |), which is the covariance of the Ornstein-Uhlenbeck process. For m 0 0 general d dimension, bm = b2m  1 setting  /(m!2 ), yields K(x, x ) = K(x − x ) = 1 0 T 0 exp − 2b2 (x − x ) (x − x ) . (2πb2 )d/2 Considering a grid xn , note that ! M M X X T T kPθk2 ≈ bm (Dm θ) (Dm θ) = θ T Dm Dm θ, (5.7) m=0 m=0 where Dm is a suitable finite-difference approximation of the differential operator. Note that such finite-difference approximation has been explored in Section 3, which we now investigate in a rigorous setting. Also, since (5.7) is quadratic in θ, assuming a prior for θ, the logarithm of which has this form, and further assuming that log [D(y n , θ)] is a log-likelihood quadratic in θ, a Gaussian posterior results. 15 5.2. Spline models and connection with Gaussian process Let us consider the penalty function to be kLm θk2 . Then polynomials up to degree m − 1 are not penalized and so, are in the null space of the regularization operator. In this case, it can be shown that a minimizer of R(θ) is of the form θ(x) = k X dj ψj (x) + j=1 n X ci G(x, xi ), (5.8) i=1 where {ψ1 , . . . , ψk } are polynomials that span the null space and the Green’s function G is given by (see Duchon [15], Meinguet [23])  cm,d |x − x0 |2m−d log |x − x0 | if 2m > d and d even G(x, x0 ) = G(x − x0 ) = , cm,d |x − x0 |2m−d otherwise. (5.9) where cm,D are constants (see Wahba [38] for the explicit form). We now specialize the above arguments to the spline set-up. As before, let us con
iid sider the model yi = θ(xi ) + i , where, for i = 1, . . . , n, i ∼ N 0, σ 2 . For simplicity, we consider the one-dimensional set-up, and consider the cubic spline smoothing problem that minimizes Z n X 2 R(θ) = (yi − θ(xi )) + τ 1 2 [θ00 (x)] dx, (5.10) 0 i=1 where 0 < x1 < · · · < xn < 1. The solution to this minimization problem is given by θ(x) = 1 X dj xj + n X ci (x − xi )3+ , (5.11) i=1 j=0 where, for any x, (x)+ = x if x > 0 and zero otherwise. Following Wahba [37], let us consider f (x) = 1 X βj xj + θ(x), (5.12) j=0   where β = (β0 , β1 )T ∼ N 0, σβ2 I2 , and θ is a zero mean Gaussian process with covariance   Z 1 v3 |x − x0 |v 2 + , (5.13) σθ2 K(x, x0 ) = (x − u)+ (x0 − u)+ du = σθ2 2 3 0 where v = min{x, x0 }. Taking σβ2 → ∞ makes the prior of β vague, so that penalty on the polynomial terms in the null space is effectively washed out. It follows that   −1 E [θ(x∗ )|y n , xn ] = h(x∗ )T β̂ + s(x∗ )T K̂ y n − H T β̂ , (5.14) 16 where, for any x, h(x) = (1, x)T , H = (h(x1 ), . . . , h(xn )), K̂ is the covariance  −1 −1 −1 matrix corresponding to σθ2 K(xi , xj ) + σ 2 δij , and β̂ = H K̂ H H K̂ y n . Since the elements of s(x∗ ) are piecewise cubic polynomials, it is easy to see that the posterior mean (5.14) is also a piecewise cubic polynomial. It is also clear that (5.14) is a first order polynomial on [0, x1 ] and [xn , 1]. 5.2.1. Connection with the `-fold integrated Wiener process Shepp [33] considered the `-fold integrated Wiener process, for ` = 0, 1, 2 . . ., as follows: Z 1 (x − u)`+ Z(u)du, (5.15) W` (x) = `! 0 where Z is a Gaussian white noise process with covariance δ(u − u0 ). As a special case, note that W0 is the standard Wiener process. In our case, note that K(x, x0 ) = Cov (W1 (x), W1 (x0 )) . (5.16) 2 R The above ideas can be easily extended to the case of the regularizer f (m) (x) dx, for m ≥ 1 by replacing (x − u)+ with (x − u)m−1 /(m − 1)! and letting h(x) = +
m−1 T 1, x, . . . , x . 6. The Bayesian approach to inverse problems in Hilbert spaces We assume the following model y = G(θ) + , (6.1) where y, θ and  are in Banach or Hilbert spaces. 6.1. Bayes theorem for general inverse problems We will consider the model stated by equation (6.1). Let Y and Θ denote the sample spaces for y and θ, respectively. Let us first assume that both are separable Banach spaces. Assume µ0 to be the prior measure for θ. Assuming well-defined joint distribution for (y, θ), let us denote the posterior of θ given y as µy . Let  ∼ Q0 where Q0 such that  and θ are independent. Let Q0 be the distribution of . Let us denote the conditional distribution of y given θ by Qθ , obtained from a translation of Q0 by G(θ). Assume that Qθ  Q0 . Thus, for some potential Φ : Θ × Y 7→ R, dQθ = exp (−Φ(θ, y)) . dQ0 (6.2) Thus, for fixed θ, Φ(θ, ·) : Y 7→ R is measurable and EQ0 [exp (−Φ(θ, y))] = 1. Note that −Φ(·, y) is nothing but the log-likelihood. Let ν0 denote the product measure ν0 (dθ, dy) = µ0 (dθ)Q0 (dy), 17 (6.3) and let us assume that Φ is ν0 -measurable. Then (θ, y) ∈ Θ×Y is distributed according to the measure ν(dθ, dy) = µ0 (dθ)Qθ (dy). It then also follows that ν  ν0 , with dνθ (θ, y) = exp (−Φ(θ, y)) . dν0 (6.4) Then we have the following statement of Bayes’ theorem for general inverse problems: Theorem 3 (Bayes theorem for general inverse problems). Assume that Φ : Θ × Y 7→ R is ν0 -measurable and Z C= exp (−Φ(θ, y)) µ0 (dy) > 0, (6.5) Θ for Q0 -almost surely all y. Then the posterior of θ given y, which we denote by µy , exists under ν. Also, µy  µ0 and for all y ν0 -almost surely, dµyθ 1 (θ) = exp (−Φ(θ, y)) . dµ0 C (6.6) Now assume that Θ and Y are Hilbert spaces. Suppose  ∼ N(0, Γ). Then the following theorem holds: Theorem 4 (Vollmer [36]).   dµy 1 ∝ exp − kG(θ)k2Γ + hy, G(θ)iΓ , dµ0 2 (6.7) where h·, ·iΓ = hΓ−1 ·, ·i, and k · kΓ is the norm induced by h·, ·iΓ .
iid For the model yi = θ(xi ) + i for i = 1, . . . , n, with i ∼ N 0, σ 2 , the posterior is of the form ! n 2 X dµy (yi − θ(xi )) . (6.8) ∝ exp − dµ0 2σ 2 i=1 6.2. Connection with regularization methods It is not immediately clear if the Bayesian approach in the Hilbert space setting has connection with the deterministic regularization methods, but Vollmer [36] prove consistency of the posterior assuming certain stability results which are used to prove convergence of regularization methods; see Engl et al. [16]. We next turn to inverse regression. 7. Inverse regression We first provide some examples of inverse regression, mostly based on Avenhaus et al. [4]. 18 7.1. Examples of inverse regression 7.1.1. Example 1: Measurement of nuclear materials Measurement of the amount of nuclear materials such as plutonium by direct chemical means is an extremely difficult exercise. This motivates model-based methods. For instance, there are physical laws relating heat production or the number of neutrons emitted (the dependent response variable y) to the amount of material present, the latter being the independent variable x. But any measurement instrument based on the physical laws first needs to be calibrated. In other words, the unknown parameters of the model needs to be learned, using known inputs and outputs. However, the independent variables are usually subject to measurement errors, motivating a statistical model. Thus, conditionally on x and parameter(s) θ, y ∼ P (·|x, θ), where P (·|x, θ) denotes some appropriate probability model. Given y n and xn , and some specific ỹ, the corresponding x̃ needs to be predicted. 7.1.2. Example 2: Estimation of family incomes Suppose that it is of interest to estimate the family incomes in a certain city through public opinion poll. Most of the population, however, will be unwilling to provide reliable answers to the questionnaires. One way to extract relatively reliable figures is to consider some dependent variable, say, housing expenses (y), which is supposed to strongly depend on family income (x); see Muth [26], and such that the population is less reluctant to divulge the correct figures related to y. From past survey data on xn and y n , and using current data from families who may provide reliable answers related to both x and y, a statistical model may be built, using which the unknown family incomes may be predicted, given their household incomes. 7.1.3. Example 3: Missing variables In regression problems where some of the covariate values xi are missing, they may be estimated from the remaining data and the model. In this context, Press and Scott [29] considered a simple linear regression problem in a Bayesian framework. Under special assumptions about the error and prior distributions, they showed that an optimal procedure for estimating the linear parameters is to first estimate the missing xi from an inverse regression based only on the complete data pairs. 7.1.4. Example 4: Bioassay It is usual to investigate the effects of substances (y) given in several dosages on organisms (x) using bioassay methods. In this context it may be of interest to determine the dosage necessary to obtain some interesting effect, making inverse regression relevant (see, for example, Rasch et al. [30]). 7.1.5. Example 5: Learning the Milky Way The modelling of the Milky Way galaxy is an integral step in the study of galactic dynamics; this is because knowledge of model parameters that define the Milky Way directly influences our understanding of the evolution of our galaxy. Since the nature of the Galaxy’s phase space, in the neighbourhood of the Sun, is affected by distinct Milky Way features, measurements of phase space coordinates of individual stars that 19 live in this neighbourhood of the Sun, will bear information about the influence of such features. Then, inversion of such measurements can help us learn the parameters that describe such Milky Way features. In this regard, learning about the location of the Sun with respect to the center of the galaxy, given the two-component velocities of the stars in the vicinity of the Sun, is an important problem. For k such stars, Chakrabarty et al. [13] model the k × 2-dimensional velocity matrix V as a function of the galactocentric location (S) of the Sun, denoted by V = ξ(S). For a given observed value V ∗ of V , it is then of interest to obtain the corresponding S ∗ . Since ξ is unknown, Chakrabarty et al. [13] model ξ as a matrix-variate Gaussian process, and consider the Bayesian approach to learning about S ∗ , given data {(S i , V i ) : i = 1, . . . , n} simulated from established astrophysical models, and the observed velocity matrix V ∗ . We now provide a brief overview of of the methods of inverse linear regression, which is the most popular among inverse regression problems. Our discussion is generally based on Hoadley [19] and Avenhaus et al. [4]. 7.2. Inverse linear regression Let us consider the following simple linear regression model: for i = 1, . . . , n, yi = α + βxi + σi , (7.1) iid where i ∼ N (0, 1). For simplicity, let us consider a single unknown x̃, associated with a further set of m responses {ỹ1 , . . . , ỹm }, related by ỹi = α + β x̃ + τ ˜i , (7.2) iid for i = 1, . . . , m, where ˜i ∼ N (0, 1) and are independent of the i ’s associated with (7.1). The interest in the above problem is inference regarding the unknown x. Based on (7.1), first least squares estimates of α and β are obtained as Pn (y − ȳ)(xi − x̄) Pn i ; (7.3) β̂ = i=1 2 i=1 (xi − x̄) α̂ = ȳ − β̂ x̄, (7.4) Pn Pn where ȳ = i=1 yi /n and ȳ = i=1 xi /n. Then, letting ỹ¯ = i=1 ỹi /n, a ‘classical’ estimator of x is given by ỹ¯ − α̂ , (7.5) x̂C = β̂ which is also the maximum likelihood estimator for the likelihood associated with (7.1) and (7.2), assuming known σ and τ . However, h i 2 E (x̂C − x) |α, β, σ, τ, x = ∞, (7.6) Pn which prompted Krutchkoff [22] to propose the following ‘inverse’ estimator: ¯ x̂I = γ̂ + δ̂ ỹ, 20 (7.7) where Pn (y − ȳ)(xi − x̄) Pn i ; δ̂ = i=1 2 i=1 (yi − ȳ) (7.8) γ̂ = x̄ − δ̂ ȳ, (7.9) are the least squares estimators of the slope and intercept when the xi are regressed on the yi . It can be shown that the mean square error of this inverse estimator is finite. However, Williams [39] showed that if σ 2 = τ 2 and if the sign of β is known, then the unique unbiased estimator of x has infinite variance. Williams advocated the use of confidence limits instead of point estimators. Hoadley [19] derive Pn confidence limits setting σ = τ and assuming without loss of generality that i=1 xi = 0. Under these assumptions, the maximum likelihood estimators of σ 2 with xn and y n only, ỹ n = (ỹ1 , . . . , ỹn )T only, and with the entire available data set are, respectively, n σ̂12 = 2 1 X yi − α̂ − β̂xi ; n − 2 i=1 σ̂22 = 1 X ¯ 2; (ỹi − ỹ) m − 1 i=1 (7.11) σ̂ 2 =   1 (n − 2)σ12 + (m − 1)σ22 . n−2+m−1 (7.12) (7.10) n 2 Now consider the F -statistic F = nσ̂β̂2 for testing the hypothesis β = 0. Note that under the null hypothesis this statistic has the F distribution with 1 and n + m degrees of freedom. For m = 1, r n β̂ (x̂C − x) σ 2 (n + 1 + x2 ) has a t distribution with n − 2 degrees of freedom. Letting Fα;1,ν denote the upper α point of the F distribution with 1 and ν degrees of freedom, a confidence set S can be derived as follows:  {x : xL ≤ x ≤ xU } if F > Fα;1,n−2 ;   n+1 {x : x ≤ x } ∪ {x ≥ x } if F ≤ F < Fα;1,n−2 ; 2 L U α;1,n−2 S= n+1+x̂C  n+1  (−∞, ∞) if F < n+1+x̂2 Fα;1,n−2 , C (7.13) where xL and xU are given by    1 Fα;1,n−2 (n + 1) (F − Fα;1,n−2 ) + F x̂2C 2 F x̂C ± . F − Fα;1,n−1 F − Fα;1,n−2 n+1 Hence, if F < n+1+x̂ 2 Fα;1,n−2 , then the associated confidence interval is S = C (−∞, ∞), which is of course useless. Hoadley [19] present a Bayesian analysis of this problem, presented below in the form of the following two theorems. 21 Theorem 5 (Hoadley [19]). Assume that σ = τ , and let x be independent of (α, β, σ 2 ) a priori. With any prior π(x) on x and the prior π(α, β, σ 2 ) ∝ 1 σ2 on (α, β, σ 2 ), the posterior density of x given by π(x|y n , xn , ỹ n ) ∝ π(x)L(x), where L(x) = h 1+ n m
m+n−3 n 2 + x2 1+ m ,   i m+n−2 2 2 F 2 + Rx̂C + m+n−3 + 1 (x − Rx̂C ) where R= F . F +m+n−3 For m = 1, Hoadley [19] present the following result characterizing the inverse estimator x̂I : Theorem 6 (Hoadley [19]). Consider the following informative prior on x: x = tn−3 n+1 , n−3 where tν denotes the t distribution with ν degrees of freedom. Then the posterior distribution of x given y n , xn and ỹ n has the same distribution as s x̂2 n + 1 + RI . x̂I + tn−2 F +n−2 In particular, it follows from Theorem 6 that the posterior mean of x is x̂I when m = 1. In other words, the inverse estimator x̂I is Bayes with respect to the squared error loss and a particular informative prior distribution for x. Since the goal of Hoadley [19] was to provide a theoretical justification of the inverse estimator, he had to choose a somewhat unusual prior so that it leads to x̂I as the posterior mean. In general it is not necessary to confine ourselves to any specific prior for Bayesian analysis of inverse regression. It is also clear that the Bayesian framework is appropriate for any inverse regression problem, not just linear inverse regression; indeed, the palaeoclimate reconstruction problem (Haslett et al. [18]) and the Milky Way problem (Chakrabarty et al. [13]) are examples of very highly non-linear inverse regression problems. 7.3. Connection between inverse regression problems and traditional inverse problems Note that the class of inverse regression problems includes the class of traditional inverse problems. The Milky Way problem is an example where learning the unknown, 22 matrix-variate function ξ (inverse problem) was required, even though learning about S, the galactocentric location of the sun (inverse regression problem) was the primary goal. The Bayesian approach allowed learning both S and ξ simultaneously and coherently. In the palaeoclimate models proposed in Haslett et al. [18], Bhattacharya [7] and Mukhopadhyay and Bhattacharya [25], although species assemblages are modeled conditionally on climate variables, the functional relationship between species and climate are not even approximately known. In all these works, it is of interest to learn about the functional relationship as well as to predict the unobserved climate values, the latter being the main aim. Again, the Bayesian approach facilitated appropriate learning of both the unknown quantities. 7.4. Consistency of inverse regression problems In the above linear inverse regression, notice that if τ > 0, then the variance of the estimator of x can not tend to zero, even as the data size tends to infinity. This shows that no estimator of x can be consistent. The same argument applies even to Bayesian approaches; for any sensible prior on x that does not give point mass to the true value of x, the posterior of x will not converge to the point mass at the true value of x as the data size increases indefinitely. The arguments remain valid for any inverse regression problem where the response variable y probabilistically depends upon the independent variable x. Not only in inverse regression problems, even in forward regression problems where the interest is in prediction of y given x, any estimate of y or any posterior predictive distribution y will be inconsistent. To give an example of inconsistency in non-linear and non-normal inverse problem, iid consider the following set-up: yi ∼ Poisson (θxi ), for i = 1, . . . , n, where θ > 0 and xi > 0 for each i. Let us consider the prior π(θ) ≡ 1 for all θ > 0. For some i∗ ∈ {1, . . . , n} let us assume the leave-one-out cross-validation set-up in that we wish to learn x = xi∗ assuming it is unknown, from the rest of the data. Putting the prior π(x) ≡ 1 for x > 0, the posterior of x is given by (see Bhattacharya and Haslett [9], Bhattacharya [8]) π(x|xn \xi , y n ) ∝ xyi (x + P j6=i xj )( Pn j=1 yj +1) . (7.14) Figure 7.1 displays the posterior of x when i∗ = 10, for increasing sample size. Observe that the variance of the posterior does not decrease even with sample size as large as 100, 000, clearly demonstrating inconsistency. Hence, special, innovative priors are necessary for consistency in such cases. 8. Conclusion In this review article, we have clarified the similarities and dissimilarities between the traditional inverse problems and the inverse regression problems. In particular, we have argued that only the latter class of problems qualify as authentic inverse problems in they have significantly different goals compared to the corresponding forward problems. Moreover, they include the traditional inverse problems on learning unknown 23 0.30 Posterior of x n=10 0.25 n=100 n=10000 0.15 n=100000 0.00 0.05 0.10 posterior density 0.20 n=1000 0 5 10 15 20 x Figure 7.1: Demonstration of posterior inconsistency in inverse regression problems. The vertical line denotes the true value. functions as a special case, as exemplified by our palaeoclimate and Milky Way examples. We advocate the Bayesian paradigm for both classes of problems, not only because of its inherent flexibility, coherency and posterior uncertainty quantification, but also because the prior acts as a natural penalty which is very important to regularize the so-called ill-posed inverse problems. The well-known Tikhonov regularizer is just a special case from this perspective. It is important to remark that the literature on inverse function learning problems and inverse regression problems is still very young and a lot of research is necessary to develop the fields. Specifically, there is hardly any well-developed, consistent model adequacy test or model comparison methodology in either of the two fields, although Mohammad-Djafari [24] deal with some specific inverse problems in this context, and Bhattacharya [8] propose a test for model adequacy in the case of inverse regression problems. Moreover, as we have demonstrated, inverse regression problems are inconsistent in general. The general development in these respects will be provided in the PhD thesis of the first author. 24 References References [1] Arbogast, T., and Bona, J. L. [2008], “Methods of Applied Mathematics,”. University of Texas at Austin. [2] Aronszajn, N. [1950], “Theory of Reproducing Kernels,” Transactions of the American Mathematical Society, 68, 337–404. [3] Aster, R. 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Overcommitment in Cloud Services – Bin packing with Chance Constraints Maxime C. Cohen Google Research, New York, NY 10011, maxccohen@google.com Philipp W. Keller arXiv:1705.09335v1 [] 25 May 2017 Google, Mountain View, CA 94043, pkeller@google.com Vahab Mirrokni Google Research, New York, NY 10011, mirrokni@google.com Morteza Zadimoghaddam Google Research, New York, NY 10011, zadim@google.com This paper considers a traditional problem of resource allocation, scheduling jobs on machines. One such recent application is cloud computing, where jobs arrive in an online fashion with capacity requirements and need to be immediately scheduled on physical machines in data centers. It is often observed that the requested capacities are not fully utilized, hence offering an opportunity to employ an overcommitment policy, i.e., selling resources beyond capacity. Setting the right overcommitment level can induce a significant cost reduction for the cloud provider, while only inducing a very low risk of violating capacity constraints. We introduce and study a model that quantifies the value of overcommitment by modeling the problem as a bin packing with chance constraints. We then propose an alternative formulation that transforms each chance constraint into a submodular function. We show that our model captures the risk pooling effect and can guide scheduling and overcommitment decisions. We also develop a family of online algorithms that are intuitive, easy to implement and provide a constant factor guarantee from optimal. Finally, we calibrate our model using realistic workload data, and test our approach in a practical setting. Our analysis and experiments illustrate the benefit of overcommitment in cloud services, and suggest a cost reduction of 1.5% to 17% depending on the provider’s risk tolerance. Key words : Bin packing, Approximation algorithms, Cloud computing, Overcommitment 1. Introduction Bin packing is an important problem with numerous applications such as hospitals, call centers, filling up containers, loading trucks with weight capacity constraints, creating file backups and more recently, cloud computing. A cloud provider needs to decide how many physical machines to purchase in order to accommodate the incoming jobs efficiently. This is typically modeled as a bin packing optimization problem, where one minimizes the cost of acquiring the physical machines subject to a capacity constraint for each machine. The jobs are assumed to arrive in an online fashion according to some vaguely specified arrival process. In addition, the jobs come with a specific requirement, but the effective job size and duration are not exactly known until after the 1 2 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints actual scheduling has occurred. In practice, job size and duration can be estimated from historical data. One straightforward way to schedule jobs is to assume that each job will fully utilize its requirement (e.g., if a job requests 32 CPU cores, the cloud provider allocates this exact amount for the job). However, there is empirical evidence, that most of the virtual machines do not use the full requested capacity. This offers an opportunity for the cloud provider to employ an overcommitment policy, i.e., to schedule sets of jobs with the total requirement exceeding the respective capacities of physical machines. On one hand, the provider faces the risk that usage exceeds the physical capacity, which can result in severe penalties (e.g., acquiring or reallocating machines on the fly, canceling and rescheduling running jobs, mitigating interventions, etc.). On the other hand, if many jobs do not fully utilize their requested resources, the provider can potentially reduce the costs significantly. This becomes even more impactful in the cloud computing market, which has become increasingly competitive in recent years as Google, Amazon, and Microsoft aim to replace private data centers. “The race to zero price” is a commonly used term for this industry, where cloud providers have cut their prices very aggressively. According to an article in Business Insider in January 2015: “Amazon Web Services (AWS), for example, has cut its price 44 times during 2009-2015, while Microsoft and Google have both decreased prices multiple times to keep up with AWS”. In January 2015, RBC Capital’s Mark Mahaney published a chart that perfectly captures this trend and shows that the average monthly cost per gigabyte of RAM, for a set of various workloads, has dropped significantly: AWS dropped prices 8% from Oct. 2013 to Dec. 2014, while both Google and Microsoft cut prices by 6% and 5%, respectively, in the same period. Other companies who charge more, like Rackspace and AT&T, dropped prices even more significantly. As a result, designing the right overcommitment policy for servers has a clear potential to increase the cloud provider profit. The goal of this paper is to study this question, and propose a model that helps guiding this type of decisions. In particular, we explicitly model job size uncertainty to motivate new algorithms, and evaluate them on realistic workloads. Our model and approaches are not limited to cloud computing and can be applied to several resource allocation problems. However, we will illustrate most of the discussions and applications using examples borrowed from the cloud computing world. Note that describing the cloud infrastructure and hardware is beyond the scope of this paper. For surveys on cloud computing, see, for example Dinh et al. (2013) and Fox et al. (2009). We propose to model the problem as a bin packing with chance constraints, i.e., the total load assigned to each machine should be below physical capacity with a high pre-specified probability. Chance constraints are a commonly used modeling tool to capture risks and constraints on random variables (Charnes and Cooper (1963)). Introducing chance constraints to several continuous optimization problems was extensively studied in the literature (see, e.g., Calafiore and El Ghaoui Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 3 (2006) and Delage and Ye (2010)). This paper is the first to incorporate capacity chance constraints in the bin packing problem, and to propose efficient algorithms to solve the problem. Using some results from distributionally robust optimization (Calafiore and El Ghaoui (2006)), we reformulate the problem as a bin packing with submodular capacity constraints. Our reformulations are exact under the assumption of independent Gaussian resource usages for the jobs. More generally, they provide an upper bound and a good practical approximation in the realistic case where the jobs’ usages are arbitrarily distributed but bounded. Using some machinery from previous work (see Goemans et al. (2009), and Svitkina and Fleischer (2011)), we show that for the bin packing problem with general monotone submodular constraints, it is impossible to find a solution within any reasonable factor from optimal (more precisely, √ N , ln(N ) where N is the number of jobs). In this paper, we show that our problem can be solved using a class of simple online algorithms that guarantee a constant factor of 8/3 from optimal (Theorem 2). This class of algorithms includes the commonly used Best-Fit and First-Fit heuristics. We also develop an improved constant guarantee of 9/4 for the online problem (Theorem 4), and a 2-approximation for the offline version (Theorem 6). We further refine our results to the case where a large number of jobs can be scheduled on each machine (i.e., each job has a small size relative to the machine capacity). In this regime, our approach asymptotically converges to a 4/3 approximation. More importantly, our model and algorithms allow us to draw interesting insights on how one should schedule jobs. In particular, our approach (i) translates to a transparent recipe on how to assign jobs to machines; (ii) explicitly exploits the risk pooling effect; and (iii) can be used to guide an overcommitment strategy that significantly reduces the cost of purchasing machines. We apply our algorithm to a synthetic but realistic workload inspired by historical production workloads in Google data centers, and show that it yields good performance. In particular, our method reduces the necessary number of physical machines, while limiting the risk borne by the provider. Our analysis also formalizes intuitions and provides insights regarding effective job scheduling strategies in practical settings. 1.1. Contributions Scheduling jobs on machines can be modeled as a bin packing problem. Jobs arrive online with some requirements, and the scheduler decides how many machines to purchase and how to schedule the jobs. Assuming random job sizes and limited machine capacities, one can formulate the problem as a 0/1 integer program. The objective is to minimize the number of machines required, subject to constraints on the capacity of each machine. In this paper, we model the capacity constraints as chance constraints, and study the potential benefit of overcommitment. The contributions of the paper can be summarized as follows. 4 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints • Formulating the overcommitment bin packing problem. We present an optimization formulation for scheduling jobs on machines, while allowing the provider to overcommit. We first model the problem as Bin Packing with Chance Constraints (BPCC). Then, we present an alternative Submodular Bin Packing (SMBP) formulation that explicitly captures the risk pooling effect on each machine. We show that the SMBP is equivalent to the BPCC under common assumptions (independent Gaussian usage distributions), and that it is distributionally robust for usages with given means and diagonal covariance matrix). Perhaps most importantly from a practical perspective, the SMBP provides an upper bound and a good approximation under generic independent distributions over bounded intervals (see Proposition 1). This last setting is most common in today’s cloud data centers, where virtual machines are sold as fixed-size units. • Developing simple algorithms that guarantee a constant factor approximation from optimal. We show that our (SMBP) problem can be solved by well-known online algorithms such as First-Fit and Best-Fit, while guaranteeing a constant factor of 8/3 from optimal (Theorem 2). We further refine this result in the case where a large number of jobs can be scheduled on each machine, and obtain a 4/3 approximation asymptotically (Corollary 1). We also develop an improved constant guarantee of 9/4 for the online problem using First-Fit (Theorem 4), and a 2 approximation for the offline version (Theorem 6). We then use our analysis to infer how one should assign jobs to machines, and show how to obtain a nearly optimal assignment (Theorem 5). • Using our model to draw practical insights on the overcommitment policy. Our approach translates to a transparent and meaningful recipe on how to assign jobs to machines by clustering similar jobs in terms of statistical information. In addition, our approach explicitly captures the risk pooling effect: as we assign more jobs to a given machine, the “safety buffer” needed for each job decreases. Finally, our approach can be used to guide a practical overcommitment strategy, where one can significantly reduce the cost of purchasing machines by allowing a low risk of violating capacity constraints. • Calibrating and applying our model to a practical setting. We use realistic workload data inspired by Google Compute Engine to calibrate our model and test our results in a practical setting. We observe that our proposed algorithm outperforms other natural scheduling schemes, and realizes a cost saving of 1.5% to 17% relative to the no-overcommitment policy. 1.2. Literature review This paper is related to different streams of literature. In the optimization literature, the problem of scheduling jobs on virtual machines has been studied extensively, and the bin packing problem is a common formulation. Hundreds of papers Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 5 study the bin packing problem including many of its variations, such as 2D packing (e.g., Pisinger and Sigurd (2005)), linear packing, packing by weight, packing by cost, online bin packing, etc. The basic bin packing problem is NP-hard, and Delorme et al. (2016) provide a recent survey of exact approaches. However, several simple online algorithms are often used in practice for largescale instances. A common variation is the problem where jobs arrive online with sizes sampled independently from a known discrete distribution with integer support and must be immediately packed onto machines upon arrival. The size of a job is known when it arrives, and the goal is to minimize the number of non-empty machines (or equivalently, minimize the waste, defined as the total unused space). For this variation, the sum-of-squares heuristic represents the state-ofthe-art. It is almost distribution-agnostic, and nearly universally optimal for most distributions by achieving sublinear waste in the number of items seen (see, Csirik et al. (2006)). In Gupta and Radovanovic (2012), the authors propose two algorithms based on gradient descent on a suitably √ defined Lagrangian relaxations of the bin packing linear program that achieve additive O( N ) waste relative to the optimal policy. This line of work bounds the expected waste for general classes of job size distribution in an asymptotic sense. Worst-case analysis of (finite, deterministic) bin packing solutions has received a lot of attention as well. For deterministic capacity constraints, several efficient algorithms have been proposed. They can be applied online, and admit approximation guarantees in both online and offline settings. The offline version of the problem can be solved using (1 + )OP T + 1 bins in linear time (de La Vega and Lueker (1981)). A number of heuristics can solve large-scale instances efficiently while guaranteeing a constant factor cost relative to optimal. For a survey on approximation algorithms for bin packing, see for example Coffman Jr et al. (1996). Three such widely used heuristics are First-Fit (FF), Next-Fit (NF) and Best-Fit (BF) (see, e.g., Bays (1977), Keller et al. (2012) and Kenyon et al. (1996)). FF assigns the newly arrived job to the first machine that can accommodate it, and purchase a new machine only if none of the existing ones can fit the new job. NF is similar to FF but continues to assign jobs from the current machine without going back to previous machines. BF uses a similar strategy but seeks to fit the newly arrived job to the machine with the smallest remaining capacity. While one can easily show that these heuristics provide a 2-approximation guarantee, improved factors were also developed under special assumptions. Dósa and Sgall (2013) provide a tight upper bound for the FF strategy, showing that it never needs more than 1.7OP T machines for any input. The offline version of the problem also received a lot of attention, and the Best-Fit-Decreasing (BFD) and First-Fit-Decreasing (FFD) strategies are among the simplest (and most popular) heuristics for solving it. They operate like BF and FF but first rank all the jobs in decreasing order of size. Dósa (2007) show that the tight bound of FFD is 11/9OP T + 6/9. 6 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Our problem differs as our goal is to schedule jobs before observing the realization of their size. In this case, stochastic bin packing models, where the job durations are modeled as random variables, are particularly relevant. Coffman et al. (1980) consider the problem and study the asymptotic and convergence properties of the Next-Fit online algorithm. Lueker (1983) considers the case where the job durations are drawn uniformly from intervals of the form [a, b], and derive a lower bound on the asymptotic expected number of bins used in an optimum packing. However, unlike this and other asymptotic results where the jobs’ sizes are known when scheduling occurs, we are interested in computing a solution that is feasible with high probability before observing the actual sizes. Our objective is to assign the jobs to as few machines as possible such that the set of jobs assigned to each machine satisfies the capacity constraint with some given probability (say 99%). In other words, we are solving a stochastic optimization problem, and studying/analyzing different simple heuristic solutions to achieve this goal. To make the difference with the worst case analysis clear, we note that the worst case analysis becomes a special case of our problem when the objective probability threshold is set to 100% (instead of 99%, or any other number strictly less than 1). The whole point of our paper is to exploit the stochastic structure of the problem in order to reduce the scheduling costs via overcommitment. In this paper, we consider an auxiliary deterministic bin packing problem with a linear cost but non-linear modified capacity constraints. In Anily et al. (1994), the authors consider general cost structures with linear capacity constraints. More precisely, the cost of a machine is assumed to be a concave and monotone function of the number of jobs in the machine. They show that the Next-Fit Increasing heuristic provides a worst-case bound of no more than 1.75, and an asymptotic worst-case bound of 1.691. The motivation behind this paper is similar to the overbooking policy for airline companies and hotels. It is very common for airlines to overbook and accept additional reservations for seats on a flight beyond the aircraft’s seating capacity1 . Airline companies (and hotels) employ an overbooking strategy for several reasons, including: (i) no-shows (several passengers are not showing up to their flight, and the airline can predict the no-show rate for each itinerary); (ii) increasing the profit by reducing lost opportunities; and (iii) segmenting passengers (charging a higher price as we get closer to the flight). Note that in the context of this paper, the same motivation of no-shows applies. However, the inter-temporal price discrimination is beyond the scope of our model. Several academic papers in operations research have studied the overbooking problem within the last forty years (see, e.g., Rothstein (1971), Rothstein (1985), Weatherford and Bodily (1992), Subramanian et al. (1999) and Karaesmen and Van Ryzin (2004)). The methodology is often based on solving 1 http://www.forbes.com/2009/04/16/airline-tickets-flights-lifestyle-travel-airlines-overbooked.html Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 7 a dynamic program incorporating some prediction of the no-show rate. In our problem, we face a large-scale bin packing problem that needs to be solved online. Rather than deciding how many passengers (jobs) to accept and at what price, cloud providers today usually aim to avoid declining any reasonable workloads at a fixed list price2 . This paper is also related to the robust optimization literature, and especially to distributionally robust optimization. The goal is to solve an optimization problem where the input parameter distribution belongs to a family of distributions that share some properties (e.g., all the distributions with the same mean and covariance matrix) and consider the worst-case within the given family (concrete examples are presented in Section 2.4). Examples of such work include: Ghaoui et al. (2003), Bertsimas and Popescu (2005), Calafiore and El Ghaoui (2006) and Delage and Ye (2010). That work aims to convert linear or convex (continuous) optimization problems with a chance constraint into tractable formulations. Our paper shares a similar motivation but considers a problem with integer variables. To the best of our knowledge, this paper is the first to develop efficient algorithms with constant approximation guarantees for the bin packing problem with capacity chance constraints. Large-scale cluster management in general is an important area of computer systems research. Verma et al. (2015) provide a full, modern example of a production system. Among the work on scheduling jobs, Sindelar et al. (2011) propose a model that also has a certain submodular structure due to the potential for sharing memory pages between virtual machines (in contrast to the risk-pooling effect modeled in this paper). Much experimental work seek to evaluate the real-world performance of bin packing heuristics that also account for factors such as adverse interactions between jobs scheduled together, and the presence of multiple contended resources (see for example Rina Panigrahy (2011) and Alan Roytman (2013)). While modeling these aspects is likely to complement the resource savings achieved with the stochastic model we propose, these papers capture fundamentally different efficiency gains arising from technological improvements and idiosyncratic properties of certain types (or combinations) of resources. In this paper, we limit our attention to the benefit and practicality of machine over-commitment in the case where a single key resource is in short supply. This applies directly to multi-resource settings if, for example, the relatively high cost of one resource makes over-provisioning the others worthwhile, or if there is simply an imbalance between the relative supply and demand for the various resources making one of the resources scarce. Structure of the paper. In Section 2, we present our model and assumptions. Then, we present the results and insights for special cases in Section 3. In Section 4, we consider the general 2 The ”spot instances” provided by Amazon and other heavily discounted reduced-availability services are notable exceptions. 8 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints case and develop a class of efficient approximation algorithms that guarantee a constant factor from optimal. In Section 5, we exploit the structure of the problem in order to obtain a nearly optimal assignment and to draw practical insights. In Sections 6 and 7, we present extensions and computational experiments using realistic data respectively. Finally, our conclusions are reported in Section 8. Most of the proofs of the Theorems and Propositions are relegated to the Appendix. 2. Model In this section, we present the model and assumptions we impose. We start by formulating the problem we want to solve, and then propose an alternative formulation. As we previously discussed, job requests for cloud services (or any other resource allocation problem) come with a requested capacity. This can be the memory or CPU requirements for virtual machines in the context of cloud computing, or job duration in more traditional scheduling problems where jobs are processed sequentially3 . We refer to Aj as the size of job j and assume that Aj is a random variable. Historical data can provide insight into the distribution of Aj . For simplicity, we first consider the offline version of the problem where all the jobs arrive simultaneously at time 0, and our goal is to pack the jobs onto the minimum possible number of machines. Jobs cannot be delayed or preempted. The methods we develop in this paper can be applied in the more interesting online version of the problem, as we discuss in Section 4. We denote the capacity of machine i by Vi . Motivated by practical problems, and in accordance with prior work, we assume that all the machines have the same capacity, i.e., Vi = V ; ∀i. In addition, each machine costs ci = c, and our goal is to maximize the total profit (or equivalently, minimize the number of machines), while scheduling all the jobs and satisfying the capacity constraints. Note that we consider a single dimensional problem, where each job has one capacity requirement (e.g., the number of virtual CPU cores or the amount of memory). Although cloud virtual machine packing may be modeled as a low-dimensional vector bin packing problem (see for example, Rina Panigrahy (2011)), one resource is often effectively binding and/or more critical so that focusing on it offers a much larger opportunity for overcommitment4 . 3 Although there is also a job duration in cloud computing, it is generally unbounded and hence, even less constrained than the resource usage from the customer’s perspective. The duration is also less important than the resource usage, since most virtual machines tend to be long-lived, cannot be delayed or pre-empted, and are paid for by the minute. In contrast, over-allocating unused, already paid-for resources can have a large impact on efficiency. 4 Insofar as many vector bin packing heuristics are actually straightforward generalizations of the FF, NF and BF rules, it will become obvious how our proposed algorithm could similarly be adapted to the multi-resource setting in Section 4, although we do not pursue this idea in this paper. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 2.1. 9 Bin packing problem For the case where Aj is deterministic, we obtain the classical deterministic bin packing problem: B = min xij ,yi s.t. N X yi i=1 N X j=1 N X Aj xij ≤ V yi ∀i (DBP) xij = 1 ∀j i=1 xij ∈ {0, 1} yi ∈ {0, 1} ∀i, j ∀i For the offline version, we have a total of N jobs and we need to decide which machines to use/purchase (captured by the decision variable yi that is equal to 1, if machine i is purchased and 0 otherwise). The solution is a B-partition of the set {1, 2, . . . , N } that satisfies the capacity constraints. The decision variable xij equals one if job j is assigned to machine i and zero otherwise. As we discussed in Section 1.2, there is an extensive literature on the DBP problem and its many variations covering both exact algorithms as well and approximation heuristics with performance bounds. The problem faced by a cloud provider is typically online in nature since jobs arrive and depart over time. Unfortunately, it is not possible to continually re-solve the DBP problem as the data is updated for both practical and computational reasons. Keeping with the majority of prior work, we start by basing our algorithms on static, single-period optimization formulations like the DBP problem, rather than explicitly modeling arrivals and departures. The next section explains how, unlike prior work, our single-period optimization model efficiently captures the uncertainty faced by a cloud provider. We will consider both the online and offline versions of our model. We remark that, while our online analysis considers sequentially arriving jobs, none of our results explicitly considers departing jobs. This is also in line with the bin-packing literature, where results usually apply to very general item arrival processes {Aj }, but it is typically assumed that packed items remain in their assigned bins. In practice, a large cloud provider is likely to be interested in a steady-state where the distribution of jobs in the systems is stable over time (or at least predictable), even if individual jobs come and go. Whereas the online model with arrivals correctly reflects that the scheduler cannot optimize to account for unseen future arrivals, it is unclear if and how additionally modeling departures would affect a system where the overall distribution of jobs remains the same over time. We therefore leave this question open. Note that several works consider bin-packing with item departures (see, e.g., Stolyar and Zhong (2015) and the references therein). In this work, the authors design a simple greedy algorithm for general packing constraints and show that it can be asymptotically optimal. 10 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 2.2. Chance constraints The DBP problem suffers from the unrealistic assumption that the jobs’ sizes Aj are deterministic. In reality, jobs’ requirements (or durations) can be highly unpredictable and quite volatile, especially from the perspective of a cloud provider with no control over the software executed in a virtual machine. Ensuring that the capacity constraints are satisfied for any realization of Aj generally yields a conservative outcome. For example, if the jobs’ true requirements are Bernoulli random variables taking on either 0.3 or 1.0 with equal probability, one needs to plan as if each job consumes a capacity of 1.0. By overcommitting resources, the provider can reduce the cost significantly. Caution is required however, since overcommitting can be very expensive if not done properly. Planning according to the expected value (in the previous simple example, 0.65), for instance, would result in capacity being too tight on many machines. Specifically, for large machines, the realized requirements could exceed capacity up to half of the time. Depending on the specific resource and the degree of violation, such performance could be catastrophic for a cloud service provider. Concretely, sustained CPU contention among virtual machines would materially affect customers’ performance metrics, whereas a shortage of available memory could require temporarily “swapping” some data to a slower storage medium with usually devastating consequences on performance. Other mitigations are possible, including migrating a running virtual machine to another host, but these also incur computational overhead for the provider and performance degradation for the customer. In the extreme case where overly optimistic scheduling results in inadequate capacity planning, there is even a stock-out risk where it is no longer possible to schedule all customers’ jobs within a data center. With this motivation in mind, our goal is to propose a formulation that finds the right overcommitment policy. We will show that by slightly overcommitting (defined formally in Section 2.3), one can reduce the costs significantly while satisfying the capacity constraints with high probability. While not strictly required by our approach, in practice, there is often an upper bound on Aj , denoted by Āj . In the context of cloud computing, Āj is the requested capacity that a virtual machine is not allowed to exceed (32 CPU cores, or 128 GB of memory, say). However, the job may end up using much less, at least for some time. If the cloud provider schedules all the jobs according to their respective upper bounds Āj , then there is no overcommitment. If the cloud provider schedules all the jobs according to some sizes smaller than the Āj , then some of the machines may be overcommitted. We propose to solve a bin packing problem with capacity chance constraints. Chance constraints are widely used in optimization problems, starting with Charnes and Cooper (1963) for linear Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 11 programs, and more recently in convex optimization (see, e.g., Nemirovski and Shapiro (2006)) and in finance (see, e.g., Abdelaziz et al. (2007)). In this case, the capacity constraints are replaced by: P N X  Aj xij ≤ V yi ≥ α, (1) j=1 where α represents the confidence level of satisfying the constraint (α = 0.999, say) and is exogenously set by the cloud provider depending on considerations such as typical job’s running time and contractual agreements. Note that when α = 1, this corresponds to the setting with no overcommitment, or in other words, to the worst-case solution that covers all possible realizations of all the Aj ’s. One of our goals is to study the trade-off between the probability of violating physical capacity and the cost reduction resulting from a given value of α. The problem becomes the bin packing with chance constraints, parameterized by α: B(α) = min N X xij ,yi s.t. yi i=1 P N X  Aj xij ≤ V yi ≥ α ∀i j=1 N X xij = 1 (BPCC) ∀j i=1 xij ∈ {0, 1} yi ∈ {0, 1} 2.3. ∀i, j ∀i Overcommitment One can define the overcommitment level as follows. Consider two possible (equivalent) benchmarks. First, one can solve the problem for α = 1, and obtain a solution (by directly solving the IP or any other heuristic method) with objective B(1). Then, we solve the problem for the desired value α < 1. The overcommitment benefit can be defined as 0 < B(α)/B(1) ≤ 1. It is also interesting to compare the two different jobs assignments. The second definition goes as follows. We define the overcommitment factor as the amount of sellable capacity divided by the physical capacity of machines in the data center, that is: P j Āj OCF (α) , P . i V yi Since we assume that all the machines have the same capacity and cost, we can write: P P Āj j Āj OCF (α) = ≥ = OCF (1). V B(α) V B(1) j 12 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Note that OCF(1) is (generally strictly) less than one, as the bin packing overhead prevents the sale of all resources5 . Then, we have: OCF (1) B(α) = . OCF (α) B(1) For illustration purposes, consider the following simple example with N = 70 1-core jobs. The jobs are independent and Bernoulli distributed with probability 0.5. In particular, the jobs are either high usage (i.e., fully utilize the 1 core), or low usage (in this case, idle). Each machine has a capacity V = 48 cores. Without overcommitting, we need 2 machines, i.e., B(1) = 2. What happens if we schedule all the jobs in a single machine? In this case, one can reduce the cost (number of machines) by half, while satisfying the capacity constraint with probability 0.9987. In other words, B(0.99) = 1. The overcommitment benefit in this simple example is clear. Our goal is to formalize a systematic way to overcommit in more complicated and realistic settings. Note that overcommitment may lead to Service Level Agreement (SLA) violations. This paper does not discuss in detail the SLAs (with some possible associated metrics), and the corresponding estimation/forecast procedures as they are usually application and resource specific. Instead, this research treats a general Virtual Machine (VM) scheduling problem. More precisely, our context is that of a cloud computing provider with limited visibility into the mix of customer workloads, and hard SLAs. While the provider does track numerous service-level indicators, they are typically monotonic in the resource usage on average (we expect more work to translate to worse performance). Therefore, we believe that it is reasonable to rely on resource utilization as the sole metric in the optimization problem. 2.4. A variant of submodular bin packing In this section, we propose an alternative formulation that is closely related to the (BPCC) problem. Under some mild assumptions, we show that the latter is either exactly or approximately equivalent to the following submodular bin packing problem: BS (α) = min xij ,yi s.t. N X yi i=1 PN j=1 µj xij + D(α) N X qP N j=1 bj xij ≤ V yi ∀i (SMBP) xij = 1 ∀j i=1 xij ∈ {0, 1} yi ∈ {0, 1} 5 ∀i, j ∀i Technical note: other production overheads such as safety stocks for various types of outages and management overheads, are generally also included in the denominator. For the purpose of this paper, we omit them. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 13 The difference between the (BPCC) and the (SMBP) problems is the way the capacity constraints are written. Here, we have replaced each chance constraint with a linear term plus a square root term. These constraints are submodular with respect to the vector x. The variable µj denotes the expected value of Aj . In what follows, we will consider different definitions of bj and D(α) in three different settings. The first two are concrete motivational examples, whereas the third one is a generalization. In each case, we formally show the relation between the (BPCC) and the (SMBP) problems. 1. Gaussian case: Assume that the random variables Aj are Gaussian and independent. In this PN case, the random variable Z = j=1 Aj xij for any given binary vector x is Gaussian, and therefore, one can use the following simplification: P N X    Aj xij ≤ V yi = P Z ≤ V yi ≥ α. j=1 For each machine i, constraint (1) becomes: N X v u N uX −1 σj2 xij ≤ V yi , µj xij + Φ (α) · t (2) j=1 j=1 where Φ−1 (·) is the inverse CDF of a normal N (0, 1), µj = E[Aj ] and σj2 = Var(Aj ). Note that we have used the fact that x is binary so that x2ij = xij . Consequently, the (BPCC) and the (SMBP) problems are equivalent with the values bj = σj2 and D(α) = Φ−1 (α). When the random variables Aj are independent but not normally distributed, if there are a large number of jobs per machine, one can apply the Central Limit Theorem and obtain a similar approximate argument. In fact, using a result from Calafiore and El Ghaoui (2006), one can extend this equivalence to any radial distribution6 . 2. Hoeffding’s inequality: Assume that the random variables Aj are independent with a finite support [Aj , Aj ], 0 ≤ Aj < Aj with mean µj . As we discussed, one can often know the value of Aj and use historical data to estimate µj and Aj (we discuss this in more detail in Section 7). Assume PN that the mean usages fit on each machine, i.e., j=1 xij µj < yi Vi . Then, Hoeffding’s inequality states that: P N X  Aj xij ≤ V yi ≥ 1 − e P 2 −2[V yi − N j=1 µj xij ] PN 2 (A −Aj ) j=1 j . j=1 Equating the right hand side to α, we obtain: PN −2[V yi − j=1 µj xij ]2 = ln(1 − α), PN b x j ij j=1 6 Radial distributions include all probability densities whose level sets are ellipsoids. The formal mathematical definition can be found in Calafiore and El Ghaoui (2006). 14 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints where bj = (Aj − Aj )2 represents the range of job j’s usage. Re-arranging the equation, we obtain for each machine i: N X v u N uX µj xij + D(α)t bj xij ≤ V yi , j=1 where in this case, D(α) = p (3) j=1 −0.5 ln(1 − α). Note that in this setting the (BPCC) and the (SMBP) problems are not equivalent. We only have that any solution of the latter is a feasible solution for the former. We will demonstrate in Section 7 that despite being very conservative, this formulation based on Hoeffding’s inequality actually yields good practical solutions. The next case is a generalization of the last two. 3. Distributionally robust formulations: Assume that the random variables Aj are independent with some unknown distribution. We only know that this distribution belongs to a family of probability distributions D. We consider two commonly used examples of such families. First, we consider the family D1 of distributions with a given mean and (diagonal) covariance matrix, µ and Σ, respectively. Second, we look at D2 , the family of generic distributions of independent random variables over bounded intervals [Aj , Aj ]. In this setting, the chance constraint is assumed to be enforced robustly with respect to the entire family D of probability distributions on A = (A1 , A2 , . . . , AN ), meaning that: inf P A∼D N X  Aj xij ≤ V yi ≥ α. (4) j=1 In this context, we have the following result. Proposition 1. Consider the robust bin packing problem with the capacity chance constraints (4) for each machine i. Then, for any α ∈ (0, 1), we have: • For the family D1 of distributions with a given mean and diagonal covariance matrix, the p robust problem is equivalent to the (SMBP) with bj = σj2 and D1 (α) = α/(1 − α). • For the family D2 of generic distributions of independent random variables over bounded inter- vals, the robust problem can be approximated by the (SMBP) with bj = (Aj − Aj )2 and D2 (α) = p −0.5 ln(1 − α). The details of the proof are omitted for conciseness. In particular, the proof for D1 is analogous to an existing result in continuous optimization that converts linear programs with a chance constraint into a linear program with a convex second-order cone constraint (see Calafiore and El Ghaoui (2006) and Ghaoui et al. (2003)). The proof for D2 follows directly from the fact that Hoeffding’s inequality applies for all such distributions, and thus for the infimum of the probability. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 15 We have shown that the (SMBP) problem is a good approximation for the bin packing problem with chance constraints. For the case of independent random variables with a given mean and covariance, the approximation is exact and for the case of distributions over independent bounded intervals, the approximation yields a feasible solution. We investigate practical settings in Section 7, and show that these approximate formulations all yield good solutions to the original problem. From now on, we consider solving the (SMBP) problem, that is repeated here for convenience: BS (α) = min xij ,yi s.t. N X yi i=1 PN j=1 µj xij + D(α) N X qP N j=1 bj xij ≤ V yi ∀i (SMBP) xij = 1 ∀j i=1 xij ∈ {0, 1} yi ∈ {0, 1} ∀i, j ∀i As discussed, the capacity constraint is now replaced by the following equation, called the modified capacity constraint: N X j=1 v u N uX µj xij + D(α)t bj xij ≤ V yi . (5) j=1 One can interpret equation (5) as follows. Each machine has a capacity V . Each job j consumes capacity µj in expectation, as well as an additional buffer to account for the uncertainty. This buffer depends on two factors: (i) the variability of the job, captured by the parameter bj ; and (ii) the acceptable level of risk through D(α). The function D(α) is increasing in α, and therefore we impose a stricter constraint as α approaches 1 by requiring this extra buffer to be larger. Equation (5) can also be interpreted as a risk measure applied by the scheduler. For each machine PN i, the total (random) load is j=1 Aj xij . If we consider that µj represents the expectation and bj qP PN N corresponds to the variance, then j=1 µj xij and j=1 bj xij correspond to the expectation and the standard deviation of the total load on machine i respectively. As a result, the right hand side of equation (5) can be interpreted as an adjusted risk utility, where D(α) is the degree of risk aversion of the scheduler. The additional amount allocated for job j can be interpreted as a safety buffer to account for the uncertainty and for the risk that the provider is willing to bear. As we discussed, this extra buffer decreases with the number of jobs assigned to the same machine. In Section 4, we develop efficient methods to solve the (SMBP) with analytical performance guarantees. 16 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 2.5. Two naive approaches In this section, we explore the limitations of two approaches that come to mind. The first attempt is to rewrite the problem as a linear integer program: the decision variables are all binary and the non-linearity in (SMBP) can actually be captured by common modeling techniques, as detailed in Appendix A. Unfortunately, solving this IP is not a viable option. Similarly as for the classical deterministic bin packing problem, solving even moderately large instances with commercial solvers takes several hours. Moreover, applying the approach to smaller, specific toy instances provides little insight about the assignment policy, and how the value of α affects the solution. Since our goal is to develop practical strategies for the online problem, we chose not to further pursue exact solutions. The second potential approach is to develop an algorithm for a more general problem: the bin packing with general monotone submodular capacity constraints. Unfortunately, using some machinery and results from Goemans et al. (2009) and Svitkina and Fleischer (2011), we next show that it is in fact impossible to find a solution within any reasonable factor from optimal. Theorem 1. Consider the bin packing problem with general monotone submodular capacity constraints for each machine. Then, it is impossible to guarantee a solution within a factor better than √ N ln(N ) from optimal. The proof can be found in Appendix A. We will show that the (SMBP) problem that we consider is more tractable as it concerns only a specific class of monotone submodular capacity constraints that capture the structure of the chance-constrained problem. In the next session, we start by addressing simple special cases in order to draw some structural insights. 3. Results and insights for special cases In this section, we consider the (SMBP) problem for some given µj , bj , N and D(α). Our goals are to: (i) develop efficient approaches to solve the problem; (ii) draw some insights on how to schedule the different jobs and; (iii) study the effect of the different parameters on the outcome. This will ultimately allows us to understand the impact of overcommitment in resource allocation problems, such as cloud computing. 3.1. Identical distributed jobs We consider the symmetric setting where all the random variables Aj have the same distribution, such that µj = µ and bj = b in the (SMBP) problem. By symmetry, we only need to find the number of jobs n to assign to each machine. Since all the jobs are interchangeable, our goal is to assign as Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 17 many jobs as possible in each machine. In other words, we want to pick the largest value of n such that the constraint (5) is satisfied, or equivalently: √ nµ + D(α) nb ≤ V [V − nµ]2 D(α)2 ≤ . nb For a given value of α, this is the largest integer smaller than: n(α) = p  1  V + 2 bD(α)2 − b2 D(α)4 + 4bD(α)2 V µ . µ 2µ (6) • For a given value of α, the number of jobs n(α) increases with V /µ. Indeed, since µ represents the expected job size, increasing the ratio V /µ is equivalent to increasing the number of ”average” jobs a machine can host. If the jobs are smaller or the machines larger, one can fit more jobs per machine, as expected. • For a given value of V /µ, n(α) is a non-increasing function of α. When α increases, it means that we enforce the capacity constraint in a stricter manner (recall that α = 1 corresponds to the case without overcommitment). As a result, the number of jobs per machine cannot increase. • For given values of α and V /µ, n(α) is a non-increasing function of b. Recall that the parameter b corresponds to some measure of spread (the variance in the Gaussian setting, and the range for distributions with bounded support). Therefore, when b increases, it implies that the jobs’ resource usage is more volatile and hence, a larger buffer is needed. Consequently, the number of jobs cannot increase when b grows. • For given values of α and V , n(α) is non-increasing with √ b/µ. The quantity √ b/µ represents the coefficient of variation of the random job size in the Gaussian case, or a similarly normalized measure of dispersion in other cases. Consequently, one should be able to fit less jobs, as the variability increases. The simple case of identically distributed jobs allows us to understand how the different factors affect the number of jobs that one can assign to each machine. In Figure 1, we plot equation (6) for an instance with A = 1, A = 0.3, µ = 0.65, V = 30 and 0.5 ≤ α < 1. The large dot for α = 1 in the figure represents the case without overcommitment (i.e., α = 1). Interestingly, one can see that when the value of α approaches 1, the benefit of allowing a small probability of violating the capacity constraint is significant, so that one can increase the number of jobs per machine. In this case, when α = 1, we can fit 30 jobs per machine, whereas when α = 0.992, we can fit 36 jobs, hence, an improvement of 20%. Note that this analysis guarantees that the capacity constraint is satisfied with at least probability α. As we will show in Section 7 for many instances, the capacity constraint is satisfied with an even higher probability. 18 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Figure 1 Parameters: A = 1, A = 0.3, µ = 0.65, V = 30 Alternatively, one can plot α as a function of n (see Figure 2a for an example with different values for V /µ). As expected, the benefit of overcommitting increases with V /µ, i.e., one can fit a larger number of jobs per machine. In our example, when V /µ = 25, by scheduling jobs according to A (i.e., α = 1, no overcommitment), we can schedule 14 jobs, whereas if we allow a 0.1% violation probability, we can schedule 17 jobs. Consequently, by allowing 0.1% chance of violating the capacity constraint, one can save more than 20% in costs. (a) Parameters: A = 1, A = 0.3, µ = 0.65 Figure 2 (b) Parameters: V = 30 Example for identically distributed jobs We next discuss how to solve the problem for the case with a small number of different classes of job distributions. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 3.2. 19 Small number of job distributions We now consider the case where the random variables Aj can be clustered in few different categories. For example, suppose standard clustering algorithms are applied to historical data to treat similar jobs as a single class with some distribution of usage. For example, one can have a setting with four types of jobs: (i) large jobs with no variability (µj is large and bj is zero); (ii) small jobs with no variability (µj is small and bj is zero); (ii) large jobs with high variability (both µj and bj are large); and (iv) small jobs with high variability (µj is small and bj is high). In other words, we have N jobs and they all are from one of the 4 types, with given values of µj and bj . The result for this setting is summarized in the following Observation (the details can be found in Appendix C). Observation 1. In the case where the number of different job classes is not too large, one can solve the problem efficiently as a cutting stock problem. The resulting cutting stock problem (see formulation (14) in Appendix C) is well studied in many contexts (see Gilmore and Gomory (1961) for a classical approach based on linear programming, or the recent survey of Delorme et al. (2016)). For example, one can solve the LP relaxation of (14) and round the fractional solution. This approach can be very useful for cases where the cloud provider have enough historical data, and when the jobs can all be regrouped into a small number of different clusters. This situation is sometimes realistic but not always. Very often, grouping all possible customer job profiles into a small number of classes, each described by a single distribution is likely unrealistic in many contexts. For example, virtual machines are typically sold with 1, 2, 4, 8, 16, 32 or 64 CPU cores, each with various memory configurations, to a variety of customers with disparate use-cases. Aggregating across these jobs is already dubious, before considering differences in their usage means and variability. Unfortunately, if one decides to use a large number of job classes, solving a cutting stock problem is not scalable. In addition, this approach requires advance knowledge of the number of jobs of each class and hence, cannot be applied to the online version of our problem. 4. Online constant competitive algorithms In this section, we analyze the performance of a large class of algorithms for the online version of problem (SMBP). We note that the same guarantees hold for the offline case, as it is just a simpler version of the problem. We then present a refined result for the offline problem in Section 6.1. 4.1. Lazy algorithms are 38 -competitive An algorithm is called lazy, if it does not purchase/use a new machine unless necessary. The formal definition is as follows. Definition 1. We call an online algorithm lazy if upon arrival of a new job, it assigns the job to one of the existing (already purchased) machines given the capacity constraints are not violated. 20 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints In other words, the algorithm purchases a new machine if and only if non of the existing machines can accomodate the newly arrived job. Several commonly used algorithms fall into this category, e.g., First-Fit, Best-Fit, Next-Fit, greedy type etc. Let OP T be the optimal objective, i.e., the minimum number of machines needed to serve all the jobs {1, 2, · · · , N }. Recall that in our problem, each job 1 ≤ j ≤ N , has two characteristics: µj and bj which represent the mean and the uncertain part of job j respectively. For a qP P set of jobs S, we define the corresponding cost Cost(S) to be j∈S µj + j∈S bj . Without loss of generality, we can assume (by normalization of all µj and bj ) that the capacity of each machine is 1 and that D(α) is also normalized to 1. We call a set S feasible, if its cost is at most the capacity limit 1. In the following Theorem, we show that any lazy algorithm yields a constant approximation for the (SBBP) problem. Theorem 2. Any lazy algorithm ALG purchases at most 83 OP T machines, where OP T is the optimum number of machines to serve all jobs. Proof. Let m be the number of machines that ALG purchases when serving jobs {1, 2, · · · , N }. For any machine 1 ≤ i ≤ m, we define Si to be the set of jobs assigned to machine i. Without loss of generality, we assume that the m machines are purchased in the order of their indices. In other words, machines 1 and m are the first and last purchased ones respectively. For any pair of machines 1 ≤ i < i0 ≤ m, we next prove that the set Si ∪ Si0 is infeasible for the qP P modified capacity constraint, i.e., j∈Si ∪S 0 µj + j∈Si ∪S 0 bj > 1. Let j be the first job assigned i i to machine i0 . Since ALG is lazy, assigning j to machine i upon its arrival time was not feasible, i.e., the set {j } ∪ Si is infeasible. Since we only assign more jobs to machines throughout the course of the algorithm, and do not remove any job, the set Si0 ∪ Si is also infeasible. In the next Lemma, we lower bound the sum of µj + bj for the jobs in an infeasible set. Lemma 1. For any infeasible set T , we have P j∈T (µj + bj ) > 34 . qP P Proof. For any infeasible set T , we have by definition j∈T µj + j∈T bj > 1. We denote qP P x = j∈T µj , and y = j∈T bj . Then, y > 1 − x. If x is greater than 1, the claim of the lemma holds trivially. Otherwise, we obtain: 1 3 3 x + y 2 > x + (1 − x)2 = x2 − x + 1 = (x − )2 + ≥ . 2 4 4 We conclude that P j∈T (µj + bj ) > 43 .  As discussed, for any pair of machines i < i0 , the union of their sets of jobs Si ∪ Si0 is an infeasible set that does not fit in one machine. We now apply the lower bound from Lemma 1 for the infeasible Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints set Si ∪ Si0 and imply P j∈Si ∪Si0 (µj + bj ) > 34 . We can sum up this inequality for all m 2
21 pairs of machines i and i0 to obtain: X X 1≤i<i0 ≤m j∈Si ∪Si0   3 m (µj + bj ) > . 4 2 We claim that the left hand side of inequality (7) is equal to (m − 1) (7) PN j=1 (µj + bj ). We note that for each job j ∈ Sk , the term µj + bj appears k − 1 times in the left hand side of inequality (7) when i0 is equal to k. In addition, this term also appears m − k times when i is equal to k. Therefore, every µj + bj appears k − 1 + m − k = m − 1 times, which is independent of the index k of the machine that contains job j. As a result, we obtain: N X (µj + bj ) > j=1   m 3m 3 . = 4(m − 1) 2 8 On the other hand, we use the optimal assignment to upper bound the sum PN j=1 (µj +bj ), and relate m to OP T . Let T1 , T2 , · · · , TOP T be the optimal assignment of all jobs to OP T machines. Since Ti is qP P P a feasible set, we have j∈Ti µj + j∈Ti bj ≤ 1, and consequently, we also have j∈Ti (µj + bj ) ≤ 1. PN POP T P Summing up for all the machines 1 ≤ i ≤ OP T , we obtain: j=1 (µj + bj ) = i=1 j∈Ti (µj + bj ) ≤ OP T . We conclude that: OP T ≥ N X j=1 This completes the proof of m < 8OP T 3 (µj + bj ) > 3m . 8 . Theorem 2 derives an approximation guarantee of 8/3 for any lazy algorithm. In many practical settings, one can further exploit the structure of the set of jobs, and design algorithms that achieve better approximation factors. For example, if some jobs are usually larger relative to others, one can incorporate this knowledge into the algorithm. We next describe the main intuitions behind the 8/3 upper bound. In the proof of Theorem 2, we have used the following two main proof techniques: P • First, we show a direct connection between the feasibility of a set S and the sum j∈S (µj + bj ). P In particular, we prove that j∈S (µj + bj ) ≤ 1 for any feasible set, and greater than 3/4 for any infeasible set. Consequently, OP T cannot be less than the sum of µj + bj for all jobs. The gap of 4/3 between the two bounds contributes partially to the final upper bound of 8/3. • Second, we show that the union of jobs assigned to any pair of machines by the lazy algorithm is an infeasible set, so that their sum of µj + bj should exceed 3/4. One can then find m/2 disjoint pairs of machines, and obtain a lower bound of 3/4 for the sum µj + bj for each pair. The fact that we achieve this lower bound for every pair of machines (and not for each machine) contributes another factor of 2 to the approximation factor, resulting to 4 3 × 2 = 83 . 22 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Note that the second loss of a factor of 2 follows from the fact that the union of any two machines forms an infeasible set and nothing stronger. In particular, all machines could potentially have a cost of 1/2 +  for a very small , and make the above analysis tight. Nevertheless, if we assume that each machine is nearly full (i.e., has Cost close to 1), one can refine the approximation factor. Theorem 3. For any 0 ≤  ≤ 0.3, if the lazy algorithm ALG assigns all the jobs to m machines such that Cost(Si ) ≥ 1 −  for every 1 ≤ i ≤ m, we have m ≤ ( 34 + 3)OP T , i.e., a ( 43 + 3) approximation guarantee. P To simplify the analysis, we denote β to be 1 − . For a set Si , we define x = j∈Si µj qP and y = j∈Si bj . Since Cost(Si ) is at least β, we have x + y ≥ β. Assuming x ≤ β, we have: Proof.  2β − 1 2 1 1 3 (µj + bj ) = x + y 2 ≥ x + (β − x)2 = x − + β − ≥ β − = − , 2 4 4 4 j∈S X i where the first equality is by the definition of x and y, the second inequality holds by x + y ≥ α, P and the rest are algebraic manipulations. For x > β, we also have j∈Si (µj + bj ) ≥ x > β > 43 − . PN PN We conclude that j=1 (µj + bj ) ≥ m × ( 34 − ). We also know that OP T ≥ j=1 (µj + bj ), which implies that m ≤ OP T 3/4− ≤ OP T ( 43 + 3) for  ≤ 0.3.  A particular setting where the condition of Theorem 3 holds is when the capacity of each machine p is large compared to all jobs, i.e., max1≤j≤N µj + bj is at most . In this case, for each machine i 6= m (except the last purchased machine), we know that there exists a job j ∈ Sm (assigned to the last purchased machine m) such that the algorithm could not assign j to machine i. This means that Cost(Si ∪ {j }) exceeds one. Since Cost is a subadditive function, we have Cost(Si ∪ {j }) ≤ Cost(Si ) + Cost({j }). We also know that Cost({j }) ≤  which implies that Cost(Si ) > 1 − . Remark 1. As elaborated above, there are two main sources for losses in the approximation factors: non-linearity of the cost function that can contribute up to 4/3, and machines being only partially full that can cause an extra factor of 2 which in total implies the 8/3 approximation guarantee. In the classical bin packing case (i.e., bj = 0 for all j), the cost function is linear, and the non-linearity losses in approximation factors fade. Consequently, we obtain that (i) Theorem 2 reduces to a 2 approximation factor; and (ii) Theorem 3 reduces to a (1 + ) approximation factor, which are both consistent with known results from the literature on the classical bin packing problem. Theorem 3 improves the bound for the case where each machine is almost full. However, in practice machines are often not full. In the next section, we derive a bound as a function of the minimum number of jobs assigned to the machines. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 4.2. 23 Algorithm First-Fit is 94 -competitive So far, we considered the general class of lazy algorithms. One popular algorithm in this class (both in the literature and in practice) is First-Fit. By exploring the structural properties of allocations made by First-Fit, we can provide a better competitive ratio of 9 4 < 83 . Recall that upon the arrival of a new job, First-Fit purchases a new machine if the job does not fit in any of the existing machines. Otherwise, it assigns the job to the first machine (based on a fixed ordering such as machine IDs) that it fits in. This algorithm is simple to implement, and very well studied in the context of the classical bin packing problem. First, we present an extension of Theorem 2 for the case where each machine has at least K jobs. Corollary 1. If the First-Fit algorithm assigns jobs such that each machine receives at least K jobs, the number of purchased machines does not exceed 4 (1 3 + 1 )OP T , K where OP T is the optimum number of machines to serve all jobs. One can prove Corollary 1 in a similar fashion as the proof of Theorem 2 and using the fact that jobs are assigned using First-Fit (the details are omitted for conciseness). For example, when K = 2 (resp. K = 5), we obtain a 2 (resp. 1.6) approximation. We next refine the approximation factor for the problem by using the First-Fit algorithm. Theorem 4. The number of purchased machines by Algorithm First-Fit for any arrival order of jobs is not more than 94 OP T + 1. The proof can be found in Appendix D. We note that the approximation guarantees we developed in this section do not depend on the factor D(α), and on the specific definition of the parameters µj and bj . In addition, as we show computationally in Section 7, the performance of this class of algorithm is not significantly affected by the factor D(α). 5. Insights on job scheduling In this section, we show that guaranteeing the following two guidelines in any allocation algorithm yields optimal solutions: • Filling up each machine completely such that no other job fits in it, i.e., making each machine’s Cost equal to 1. • Each machine contains a set of similar jobs (defined formally next). We formalize these properties in more detail, and show how one can achieve optimality by qP P satisfying these two conditions. We call a machine full if j∈S µj + j∈S bj is equal to 1 (recall that the machine capacity is normalized to 1 without loss of generality), where S is the set of jobs assigned to the machine. Note that it is not possible to assign any additional job (no matter how small the job is) to a full machine. Similarly, we call a machine -full, if the the cost is at least 1 − , 24 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints i.e., P j∈S µj + qP j∈S bj ≥ 1 − . We define two jobs to be similar, if they have the same b/µ ratio. Note that the two jobs can have different values of µ and b. We say that a machine is homogeneous, if it only contains similar jobs. In other words, if the ratio bj /µj is the same for all the jobs j assigned to this machine. By convention, we define bj /µj to be +∞ when µj = 0. In addition, we introduce the relaxed version of this property: we say that two jobs are δ-similar, if their b/µ ratios differ by at most a multiplicative factor of 1 + δ. A machine is called δ-homogeneous, if it only contains δ-similar jobs (i.e., for any pair of jobs j and j 0 in the same machine, bj /µj b0j /µ0j is at most 1 + δ). Theorem 5. For any  ≥ 0 and δ ≥ 0, consider an assignment of all jobs to some machines with two properties: a) each machine is -full, and b) each machine is δ-homogeneous. Then, the number of purchased machines in this allocation is at most OP T . (1−)2 (1−δ) The proof can be found in Appendix E. In this section, we proposed an easy to follow recipe in order to schedule jobs to machines. Each arriving job is characterized by two parameters µj and bj . Upon arrival of a new job, the cloud provider can compute the ratio rj = bj /µj . Then, one can decide of a few buckets for the different values of rj , depending on historical data, and performance restrictions. Finally, the cloud provider will assign jobs with similar ratios to the same machines and tries to fill in machines as much as possible. In this paper, we show that such a simple strategy guarantees a good performance (close to optimal) in terms of minimizing the number of purchased machines while at the same time allowing to strategically overcommit. 6. Extensions In this section, we present two extensions of the problem we considered in this paper. 6.1. Offline 2-approximation algorithm Consider the offline version of the (SMBP) problem. In this case, all the N jobs already arrived, and one has to find a feasible schedule so as to minimize the number of machines. We propose the algorithm Local-Search that iteratively reduces the number of purchased machines, and also uses ideas inspired from First-Fit in order to achieve a 2-approximation for the offline problem. Algorithm Local-Search starts by assigning all the jobs to machines arbitrarily, and then iteratively refines this assignment. Suppose that each machine has a unique identifier number. We next introduce some notation before presenting the update operations. Let a be the number of machines with only one job, A1 be the set of these a machines, and S1 be the set of jobs assigned to these machines. Note that this set changes throughout the algorithm with the update operations. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 25 We say that a job j ∈ / S1 is good, if it fits in at least 6 of the machines in the set A1 7 . In addition, we say that a machine is large, if it contains at least 5 jobs, and we denote the set of large machines by A5 . We say that a machine is medium size, if it contains 2, 3, or 4 jobs, and we denote the set of medium machines by A2,3,4 . We call a medium size machine critical, if it contains one job that fits in none of the machines in A1 , and the rest of the jobs in this machine are all good. Following are the update operations that Local-Search performs until no such operation is available. • Find a job j in machine i (i is the machine identifier number) and assign it to some other machine i0 < i if feasible (the outcome will be similar to First-Fit). • Find a medium size machine i that contains only good jobs. Let j1 , · · · , j` (2 ≤ ` ≤ 4) be the jobs in machine i. Assign j1 to one of the machines in A1 that it fits in. Since j1 is a good job, there are at least 6 different options, and the algorithm picks one of them arbitrarily. Assign j2 to a different machine in A1 that it fits in. There should be at least 5 ways to do so. We continue this process until all the jobs in machine i (there are at most 4 of them) are assigned to distinct machines in A1 , and they all fit in their new machines. This way, we release machine i and reduce the number of machines by one. • Find a medium size machine i that contains one job j that fits in at least one machine in A1 , and the rest of the jobs in i are all good. First, assign j to one machine in A1 that it fits in. Similar to the previous case, we assign the rest of the jobs (that are all good) to different machines in A1 . This way, we release machine i and reduce the number of purchased machines by one. • Find two critical machines i1 and i2 . Let j1 and j2 be the only jobs in these two machines that fit in no machine in A1 . If both jobs fit and form a feasible assignment in a new machine, we purchase a new machine and assign j1 and j2 to it. Otherwise, we do not change anything and ignore this update step. There are at most 6 other jobs in these two machines since both are medium machines. In addition, the rest of the jobs are all good. Therefore, similar to the previous two cases, we can assign these jobs to distinct machines in A1 that they fit in. This way, we release machines i1 and i2 and purchase a new machine. So in total, we reduce the number of purchased machines by one. We are now ready to analyze this Local-Search algorithm that also borrows ideas from First-Fit. We next show that the number of purchased machines is at most 2OP T + O(1), i.e., a 2-approximation. Theorem 6. Algorithm Local-Search terminates after at most N 3 operations (where N is the number of jobs), and purchases at most 2OP T + 11 machines. 7 The reason we need 6 jobs is technical, and will be used in the proof of Theorem 6. 26 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints The proof can be found in Appendix F. We conclude this section by comparing our results to the classical (deterministic) bin packing problem. In the classical bin packing problem, there are folklore polynomial time approximation schemes (see Section 10.3 in Albers and Souza (2011)) that achieve a (1 − )-approximation factor by proposing an offline algorithm based on clustering the jobs into 1/2 groups, and treating them as equal size jobs. Using dynamic programming techniques, one can solve the simplified problem with 1/2 different job sizes in time O(npoly(1/) ). In addition to the inefficient time complexity of these algorithms that make them less appealing for practical purposes, one cannot generalize the same ideas to our setting. The main obstacle is the lack of a total ordering among the different jobs. In the classical bin packing problem, the jobs can be sorted based on their sizes. However, this is not true in our case since the jobs have the two dimensional requirements µj and bj . 6.2. Alternative constraints Recall that in the (SMBP) problem, we imposed the modified capacity constraint (5). Instead, one can consider the following family of constraints, parametrized by 0.5 ≤ p ≤ 1: N X j=1 µj xij + D(α) N X bj xij p ≤ V yi , (8) j=1 Note that this equation is still monotone and submodular in the assignment vector x, and captures some notion of risk pooling. In particular, the “safety buffer” reduces with the number of jobs already assigned to each machine. The motivation behind such a modified capacity constraint lies in the shape that one wishes to impose on the term that captures the uncertain part of the job. In one extreme (p = 1), we consider that the term that captures the uncertainty is linear and hence, as important as the expectation term. In the other extreme case (p = 0.5), we consider that the term that captures the uncertainty behaves as a square root term. For a large number of jobs per machine, this is known to be an efficient way of handling uncertainty (similar argument as the central limit theorem). Note also that when p = 0.5, we are back to equation (5), and when p = 1 we have a commonly used benchmark (see more details in Section 7). One can extend our analysis and derive an approximation factor for the online problem as a function of p for any lazy algorithm. Corollary 2. Consider the bin packing problem with the modified capacity constraint (8). Then, any lazy algorithm ALG purchases at most 2 OP T f (p) machines, where OP T is the optimum number of machines to serve all jobs and f (p) is given by: 1 f (p) = 1 − (1 − p)p p −1 . The proof is in a very similar spirit as in Theorem 2 and is not repeated due to space limitations.
p PN PN Intuitively, we find parametric lower and upper bounds on in terms of j=1 bj xij . j=1 bj xij Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 27 Note that when p = 0.5, we recover the result of Theorem 2 (i.e., a 8/3 approximation) and as p increases, the approximation factor converges to 2. In Figure 3, we plot the approximation factor as a function of 0.5 ≤ p ≤ 1. Figure 3 Approximation factor 2 f (p) as a function of p Finally, one can also extend our results for the case where the modified capacity constrain is   PN PN given by: j=1 µj xij + D(α) log 1 + j=1 bj xij ≤ V yi . 7. Computational experiments In this section, we test and validate the analytical results developed in the paper by solving the (SMBP) problem for different realistic cases, and investigating the impact on the number of machines required (i.e., the cost). We use realistic workload data inspired by Google Compute Engine, and show how our model and algorithms can be applied in an operational setting. 7.1. Setting and data We use simulated workloads of 1000 jobs (virtual machines) with a realistic VM size distribution (see Table 1). Typically, the GCE workload is composed of a mix of CPU usages from virtual machines belonging to cloud customers. These jobs can have highly varying workloads, including some large ones and many smaller ones.8 More precisely, we assume that each VM arrives to the cloud provider with a requested number of CPU cores, sampled from the distribution presented in Table 1. 8 The average distribution of workloads we present in Table 1 assumes small percentages of workloads with 32 and 16 cores, and larger percentages of smaller VMs. A “large” workload may consist of many VMs belonging to a single customer whose usages may be correlated at the time-scales we are considering, but heuristics ensure these are spread across different hosts to avoid strong correlation of co-scheduled VMs. The workload distributions we are using are representative for some segments of GCE. Unfortunately, we cannot provide the real data due to confidentiality. 28 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Table 1 Example distribution of VM sizes in one Google data center Number of cores 1 2 4 8 16 32 % VMs 36.3 13.8 21.3 23.1 3.5 1.9 In this context, the average utilization is typically low, but in many cases, the utilization can be highly variable over time. Although we decided to keep a similarl VM size distribution as observed in a production data center, we also fitted parametric distributions to roughly match the mean and the variance of the measured usage. This allows us to obtain a parametric model that we could vary for simulation. We consider two different cases for the actual CPU utilization as a fraction of the requested job size: we either assume that the number of cores used has a Bernoulli distribution, or a truncated Gaussian distribution. As discussed in Section 2.4, we assume that each job j has lower and upper utilization bounds, Aj and Aj . We sample Aj uniformly in the range [0.3, 0.6], and Aj in the range [0.7, 1.0]. In addition, we uniformly sample µ0j and σj0 ∈ [0.1, 0.5] for each VM to serve as the parameters for the truncated Gaussian (not to be confused with its true mean and standard deviation, µj and σj ). For the Bernoulli case, µ0j = µj determines the respective probabilities of the realization corresponding to the lower or upper bound (and the unneeded σj0 is ignored). For each workload of 1000 VMs generated in this manner, we solve the online version of the (SMBP) problem by implementing the Best-Fit heuristic, using one of the three different variants for the values of D(α) and bj . We solve the problem for various values of α ranging from 0.5 to 0.99999. More precisely, when a new job arrives, we compute the modified capacity constraint in equation (5) for each already-purchased machine, and assign the job to the machine with the smallest available capacity that can accommodate it9 . If the job does not fit in any of the already purchased machines, the algorithm opens a new machine. We consider the three variations of the (SMBP) discussed earlier: • The Gaussian case introduced in (2), with bj = σj2 and D(α) = Φ−1 (α). This is now also an approximation to the chance constrained (BPCC) formulation since the true distributions are truncated Gaussian or Bernoulli. • The Hoeffding’s inequality approximation introduced in (3), with bj = (Aj − Aj )2 and D(α) = p −0.5 ln(1 − α). Note hat the distributionally robust approach with the family of distributions D2 is equivalent to this formulation. • The distributionally robust approximation with the family of distributions D1 , with bj = σj2 p and D1 (α) = α/(1 − α). 9
P Note that we clip the value of the constraint at the effective upper bound j xij Aj , to ensure that no trivially feasible assignments are excluded. Otherwise, the Hoeffding’s inequality-based constraint may perform slightly worse relative to the policy without over-commitment, if it leaves too much free space on the machines. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 7.2. 29 Linear benchmarks We also implement the following four benchmarks which consist of solving the classical (DBP) problem with specific problem data. First we have: • No overcommitment – This is equivalent to setting α = 1 in the (SMBP) problem, or solving the (DBP) problem with sizes Aj . Three other heuristics are obtained by replacing the square-root term in constraint (5) by a linear term, specifically we replace the constraint with: N X µj xij + D(α) j=1 N X p j=1 bj xij = N  X p  µj + D(α) bj xij ≤ V yi (9) j=1 to obtain: • The linear Gaussian heuristic that mimics the Gaussian approximation in (2). • The linear Hoeffding’s heuristic that mimics the Hoeffding’s approximation in (3). • The linear robust heuristic that mimics the distributionally robust approach with the family of distributions D1 . Notice that the linearized constraint (9) is clearly more restrictive for a fixed value of α by concavity of the square root, but we do of course vary the value of α in our experiments. We do not expect these benchmarks to outperform our proposed method since they do not capture the risk-pooling effect from scheduling jobs concurrently on the same machine. They do however still reflect different relative amounts of ”padding” or ”buffer” above the expected utilization of each job allocated due to the usage uncertainty. The motivation behind the linear benchmarks lies in the fact that the problem is reduced to the standard (DBP) formulation which admits efficient implementations for the classical heuristics. For example, the Best-Fit algorithm can run in time O(N log N ) by maintaining a list of open machines sorted by the slack left free on each machine (see Johnson (1974) for details and linear time approximations). In contrast, our implementation of the Best-Fit heuristic with the non-linear constraint (5) takes time O(N 2 ) since we evaluate the constraint for each machine when each new job arrives. Practically, in cloud VM scheduling systems, this quadratic-time approach may be preferred anyway since it generalizes straightforwardly to more complex “scoring” functions that also take into account additional factors besides the remaining capacity on a machine, such as multiple resource dimensions, performance concerns or correlation between jobs (see, for example, Verma et al. (2015)). In addition, the computational cost could be mitigated by dividing the data center into smaller “shards”, each consisting of a fraction of the machines, and then trying to assign each incoming job only to the machines in one of the shard. For example, in our experiments we found that there was little performance advantage in considering sets of more than 1000 jobs at 30 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints a time. Nevertheless, our results show that even these linear benchmarks may provide substantial savings (relative to the no-overcommitment policy) while only requiring very minor changes to classical algorithms: instead of Aj , we simply use job sizes defined by µj , bj and α. 7.3. Results and comparisons We compare the seven different methods in terms of the number of purchased machines and show that, in most cases, our approach significantly reduces the number of machines needed. We consider two physical machine sizes: 32 cores and 72 cores. As expected, the larger machines achieve a greater benefit from modeling risk-pooling. We draw 50 independent workloads each composed of 1000 VMs as described above. For each workload, we schedule the jobs using the BestFit algorithm and report the average number of machines needed across the 50 workloads. Finally, we compute the probability of capacity violation as follows: for each machine used to schedule each of the workloads, we draw 5000 utilization realizations (either from a sum of truncated Gaussian or a sum of Bernoulli distributions), and we count the number of realizations where the total CPU usage of the jobs scheduled on a machine exceeds capacity. The sample size was chosen so that our results reflect an effect that is measurable in a typical data center. Since our workloads require on the order of 100 machines each, this corresponds to roughly 50 × 100 × 5000 = 25, 000, 000 individual machine-level samples. Seen another way, we schedule 50 × 1000 = 50, 000 jobs and collect 5000 data points from each. Assuming a sample is recorded every 10 minutes, say, this corresponds to a few days of traffic even in a small real data center with less than 1000 machines10 . The sample turns out to yield very stable measurements, and defining appropriate service level indicators is application-dependent and beyond the scope of this paper, so we do not report confidence intervals or otherwise delve into statistical measurement issues. Similarly, capacity planning for smaller data centers may need to adjust measures of demand uncertainty to account for the different scheduling algorithms, but any conclusions are likely specific to the workload and data center, so we do not report on the variability across workloads. In Figure 4, we plot the average number of machines needed as a function of the probability that a given constraint is violated, in the case where the data center is composed of 72 CPU core machines. Each point in the curves corresponds to a different value of the parameter α. Without overcommitment, we need an average of over 54 machines in order to serve all the jobs. By allowing 10 The exact time needed to collect a comparable data set from a production system depends on the data center size and on the sampling rate, which should be a function of how quickly jobs enter and leave the system, and of how volatile their usages are. By sampling independently in our simulations, we are assuming that the measurements from each machine are collected relatively infrequently (to limit correlation between successive measurements), and that the workloads are diverse (to limit correlation between measurements from different machines). This assumption is increasingly realistic as the size of the data center and the length of time covered increase: in the limit, for a fixed sample size, we would record at most one measurement from each job with a finite lifetime, and it would only be correlated with a small fraction of its peers. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 31 a small chance of violation, say a 0.1% risk (or equivalently, a 99.9% satisfaction probability), we only need 52 machines for the Bernoulli usage, and 48 machines for the truncated Gaussian usage. If we allow a 1% chance of violation, we then only need 50 and 46 machines, respectively. The table of Figure 6 summarizes the relative savings, which are roughly 4.5% and 11.5% with a 0.1% risk, and roughly 8% and 14% with a 1% risk, for the Bernoulli and truncated Gaussian usages, respectively. In terms of the overcommitment factor defined in Section 2.3, the reported savings translate directly to the fraction of the final capacity that is due to overcommitment, B(1) − B(α) OCF (α) − OCF (1) = . B(1) OCF (α) Figure 4 shows that all three variations of our approach (the Gaussian, Hoeffding’s, and the distributionally robust approximations) yield very similar results. This suggests that the results are robust to the method and the parameters. The same is true for the corresponding linear benchmarks, though they perform worse, as expected. We remark that although the final performance tradeoff is nearly identical, for a particular value of the α parameter, the achieved violation probabilities vary greatly. For example, with α = 0.9 and the truncated normal distribution, each constraint was satisfied with probability 0.913 when using the Gaussian approximation, but with much higher probabilities 0.9972 and 0.9998 for the Hoeffding and robust approximations, respectively. This is expected, since the latter two are relatively loose upper bounds for a truncated normal distribution, whereas the distributions N (µj , σj ) are close approximations to the truncated Gaussian with parameters µ0j and σj0 . (This is especially true for their respective sums.) Practically, the normal approximation is likely to be the easiest to calibrate and understand in cases where the theoretical guarantees of the other two approaches are not needed, since it would be nearly exact for normally-distributed usages. In Figure 5, we repeat the same tests for smaller machines having only 32 physical CPU cores. The smaller machines are more difficult to overcommit since there is a smaller risk-pooling opportunity, as can be seen by comparing the columns of Table 6. The three variations of our approach still yield similar and significant savings, but now they substantially outperform the linear benchmarks: the cost reduction is at least double with all but the largest values of α. We highlight that with the “better behaved” truncated Gaussian usage, we still obtain a 5% cost savings at 0.01% risk, whereas the linear benchmarks barely improve over the no-overcommit case. As mentioned in Section 2.2, the value of α should be calibrated so as to yield an acceptable risk level given the data center, the workload and the resource in question. Any data center has a baseline risk due to machine (or power) failure, say, and a temporary CPU shortage is usually much less severe relative to such a failure. On the other hand, causing a VM to crash because of a memory shortage can be as bad as a machine failure from the customer’s point of view. Ultimately, 32 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints (a) Bernoulli usage Figure 4 (b) Truncated Gaussian usage Average number of 72-core machines needed to schedule a workload, versus the probability that any given machine’s realized load exceeds capacity (a) Bernoulli usage Figure 5 (b) Truncated Gaussian usage Results for 32 core machines the risk tolerance will be driven by technological factors, such as the ability to migrate VMs or swap memory while maintaining an acceptable performance. 7.4. Impact We conclude that our approach allows a substantial cost reduction for realistic workloads. More precisely, we draw the following four conclusions. • Easy to implement: Our approach is nearly as simple to implement as classical bin packing heuristics. In addition, it works naturally online and in real-time, and can be easily incorporated to existing scheduling algorithms. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Figure 6 33 Percentage savings due to overcommitment for two CPU usage distributions, using the three proposed variants of the chance constraint. The linear Gaussian benchmark is shown for comparison • Robustness: The three variations we proposed yield very similar results. This suggests that our approach is robust to the type of approximation. In particular, the uncertain term bj and the risk coefficient D(α) do not have a strong impact on the results. It also suggests that the method is robust to estimation errors in the measures of variability that define bj . • Significant cost reduction: With modern 72-core machines, our approach allows a 8-14% cost savings relative to the no overcommitment policy. This is achieved by considering a manageable risk level of 1%, which is comparable to other sources of risk that are not controllable (e.g., physical failures and regular maintenance operations). • Outperforming the benchmarks: Our proposals show a consistent marked improvement over three different “linear” benchmarks that reduce to directly apply the classical Best-Fit heuristic. The difference is most substantial in cases where the machines are small relative to the jobs they must contain, which is intuitively more challenging. Although our approach does not run in O(n log n) time, ”sharding” and (potentially) parallelization mitigate any such concerns in practice. 8. Conclusion In this paper, we formulated and practically solved the bin-packing problem with overcommitment. In particular, we focused on a cloud computing provider that is willing to overcommit when allocating capacity to virtual machines in a data center. We modeled the problem as bin packing with chance constraints, where the objective is to minimize the number of purchased machines, while satisfying the physical capacity constraints of each machine with a very high probability. We first showed that this problem is closely related to an alternative formulation that we call the SMBP (Submodular Bin Packing) problem. Specifically, the two problems are equivalent under the assumption of independent Gaussian job sizes, or when the job size distribution belongs to the distributionally robust family with a given mean and (diagonal) covariance matrix. In addition, the 34 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints bin packing problem with chance constraints can be approximated by the SMBP for distributions with bounded supports. We first showed that for the bin packing problem with general monotone submodular capacity constraints, it is impossible to find a solution within any reasonable factor from optimal. We then developed simple algorithms that achieve solutions within constant factors from optimal for the SMBP problem. We showed that any lazy algorithm is 8/3 competitive, and that the First-Fit heuristic is 9/4 competitive. Since the First-Fit and Best-Fit algorithms are easy to implement and well understood in practice, this provides an attractive option from an implementation perspective. Second, we proposed an algorithm for the offline version of the problem, and showed that it guarantees a 2-approximation. Then, we used our model and algorithms in order to draw several useful insights on how to schedule jobs to machines, and on the right way to overcommit. We convey that our method captures the risk pooling effect, as the “safety buffer” needed for each job decreases with the number of jobs already assigned to the same machine. Moreover, our approach translates to a transparent and meaningful recipe on how to assign jobs to machines by naturally clustering similar jobs in terms of statistical information. Namely, jobs with a similar ratio b/µ (the uncertain term divided by the expectation) should be assigned to the same machine. Finally, we demonstrated the benefit of overcommitting and applied our approach to realistic workload data inspired by Google Compute Engine. We showed that our methods are (i) easy to implement; (ii) robust to the parameters; and (iii) significantly reduce the cost (1.5-17% depending on the setting and the size of the physical machines in the data center). Acknowledgments We would like to thank the Google Cloud Analytics team for helpful discussions and feedback. 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SIAM Journal on Computing 40(6):1715–1737. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 37 Verma A, Pedrosa L, Korupolu MR, Oppenheimer D, Tune E, Wilkes J (2015) Large-scale cluster management at Google with Borg. Proceedings of the European Conference on Computer Systems (EuroSys) (Bordeaux, France). Weatherford LR, Bodily SE (1992) A taxonomy and research overview of perishable-asset revenue management: yield management, overbooking, and pricing. Operations Research 40(5):831–844. 38 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Appendix A: Details for Section 2.5 IP formulation By taking the square on both sides of the submodular capacity constraint (5), we obtain: V 2 yi + N X µj xij 2 − 2V yi j =1 N X µj xij ≥ D(α)2 · j =1 N X bj xij . j =1 Note that since yi is binary, we have yi2 = yi . We next look at the term: yi PN j =1 µj xij . One can linearize this term by using one of the following two methods. 1. Since yi = 1 if and only if at least one xij = 1, we have the constraint: PN j =1 xij ≤ M yi , for a large positive number M (actually, one can take M = N ). Consequently, one can remove the yi in the above term. 2. One can define a new variable tij , yi xij and add the four following constraints: Next, we look at the term: tij ≤ yi ; tij ≤ xij ; tij ≥ 0; tij ≥ xij + yi − 1. 2 N µ x . Since x2ij = xij , we remain only with the terms xij · xik for k > j. j ij j =1 P One can now define a new variable for each such term, i.e., zijk , xij · xik with the four constraints as before: zijk ≤ xij ; zijk ≤ xik ; zijk ≥ 0; zijk ≥ xij + xik − 1. The resulting formulation is a linear integer program. Note that the decision variables tij and zijk are continuous, and only xij and yi are binary. Appendix B: Proof. Proof of Theorem 1 In this proof, we make use of the submodular functions defined by Svitkina and Fleischer (2011) for load balancing problems. Denote the jobs by 1, 2, · · · , N , and for every subset of jobs S ⊆ [N ], let f (S) be the cost of the set S (i.e., the capacity cost induced by the function f ). We use two submodular functions f and f 0 (defined formally next) which are proved to be indistinguishable with a polynomial number of value oracle queries (see Lemma 5.1 of Svitkina and Fleischer (2011)). Let denote x = ln(N ). Note that Svitkina and Fleischer (2011) require x to be any parameter such that x2 dominates ln(N ) asymptotically and hence, √ 5 N x , α0 = N m0 2 , and β0 = x5 . We choose N such P that m0 takes an integer value. Define f (S) to be min{|S|, α0 }, and f 0 (S) to be min{ i min{β0 , |S ∩ Vi |}, α0 } includes the special case we are considering here. Define m0 = 0 where {Vi }m i=1 is a random partitioning of [N ] into m0 equal sized parts. Note that by definition, both set functions f and f 0 are monotone and submodular. As we mentioned, it is proved in Svitkina and Fleischer (2011) that the submodular functions f and 0 f cannot be distinguished from each other with a polynomial number of value oracle queries with high probability. We construct two instances of the bin packing problem with monotone submodular capacity constraints by using f and f 0 as follows. In both instances, the capacity of each machine is set to β0 . In the first instance, a set S is feasible (i.e., we can schedule all its jobs in a machine) if and only if f (S) ≤ β0 . By definition, f (S) is greater than β0 if |S| is greater than β0 . Therefore, any feasible set S in this first instance consists of at most β0 jobs. Consequently, in the first instance, any feasible assignment of jobs to machines requires at least N β0 machines. Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 39 We define the second instance of the bin packing problem based on the submodular function f 0 . A set S is feasible in the second instance, if and only if f 0 (S) ≤ β0 . Since f 0 (Vj ) is at most β0 for each 1 ≤ j ≤ m0 , each set Vj is a feasible set in this second instance. Therefore, we can assign each Vj to a separate machine to process all jobs, and consequently, m0 machines suffice to do all the tasks in the second instance. We note that with our parameter setting, m0 is much smaller that N β0 . We then conclude that the optimum solutions of these two instances differ significantly. We next prove the claim of Theorem 1 by using a contradiction argument. Assume that there exists a polynomial time algorithm ALG for the bin packing problem with monotone submodular capacity constraints √ with an approximation factor better than N ln(N ) . We next prove that by using ALG, we can distinguish 0 between the two set functions f and f with a polynomial number of value oracles, which contradicts the result of Svitkina and Fleischer (2011). The task of distinguishing between the two functions f and f 0 can be formalized as follows. We have value oracle access to a set function g, and we know that g is either the same as f or the same as f 0 . The goal is to find out whether g = f or g = f 0 using a polynomial number of value oracle queries. We construct a bin packing instance with N jobs, capacity constraints g, and ask the algorithm ALG to solve this instance. If ALG uses less than N β0 machines to process all jobs, we can say that g is the same as f 0 (since with capacity constraints f , there does not exist a feasible assignment of all jobs to less than N β0 machines). On the other hand, if ALG uses at least N β0 machines, we can say that g is 0 equal to f . This follows from the fact that if g was equal to f , the optimum number of machines would have √ been at most m0 . Since ALG has an approximation factor better than √ by ALG should have been less than m0 × N ln(N ) = √ 5 N x √ × N ln(N ) = N β0 N ln(N ) , the number of machines used . Therefore, using at least N β0 machines by ALG is a sufficient indicator of g being the same as f . This argument implies that an algorithm with an √ approximation factor better than N ln(N ) for the bin packing problem with monotone submodular constraints yields a way of distinguishing between f and f 0 with a polynomial number of value oracle queries (since ALG is a polynomial time algorithm), which contradicts the result of Svitkina and Fleischer (2011).  Appendix C: Details related to Observation 1 For ease of exposition, we first address the case with two job classes. Classes 1 and 2 have parameters (µ1 , b1 ) and (µ2 , b2 ) respectively. For example, an interesting special case is when one class of jobs is more predictable relative to the other (i.e., µ1 = µ2 = µ, b2 = b and b1 = 0). In practice, very often, one class of jobs has low variability (i.e., close to deterministic), whereas the other class is more volatile. For example, class 1 can represent loyal recurring customers, whereas class 2 corresponds to new customers. We assume that we need to decide the number of machines to purchase, as well as how many jobs of types 1 and 2 to assign to each machine. Our goal is to find the right mix of jobs of classes 1 and 2 to assign to each machine (note that this proportion can be different for each machine). Consider a given machine i and denote by n1 and n2 the number of jobs of classes 1 and 2 that we assign to this machine. We would like to ensure that the chance constraint is satisfied in each machine with the given parameter α. Assuming that V > n1 µ1 + n2 µ2 , we obtain: [V − n1 µ1 − n2 µ2 ]2 = D(α)2 . n 1 b1 + n 2 b2 (10) 40 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints For a given α, one can find the value of n1 as a function of n2 that satisfies equation (10): q  V − n2 µ2 1  n1 (n2 ) = + 2 b1 D(α)2 − b21 D(α)4 + 4b1 D(α)2 (V − n2 µ2 )µ1 + 4µ21 n2 b2 D(α)2 . µ1 2µ1 (11) As we discussed, an interesting special case is when both classes of jobs have the same expectation, i.e., µ1 = µ2 = µ but one type of jobs is much more predictable (i.e., smaller range or variance). In the extreme case, one can assume that class 1 jobs are deterministic, (i.e., b1 = 0). In this case, equation (11) becomes: √ V − n2 µ 2n2 b2 B n1 (n2 ) = − . (12) µ 2µ Alternatively, by directly looking at the modified capacity constraint (5) for this special case, we obtain: p V = (n1 + n2 )µ + D(α) n2 b2 . (13) Equation (13) can be interpreted as follows. Any additional job of type 1 takes µ from the capacity budget V , whereas any additional job of type 2 is more costly. The extra cost depends on both the uncertainty of the job (through b2 ) and the overcommitment policy (through D(α)). The higher is one of these two factors, the larger is the capacity we should plan for jobs of type 2 (i.e., “safety buffer”). Note that the submodular nature of constraint (5) implies that this marginal extra cost decreases with the number of jobs n2 . In other words, when n2 becomes large, each additional job of type 2 will converge to take a capacity of µ, as the central limit theorem applies. More generally, by taking the derivative of the above expression, any additional √ √ job of type 2 will take µ + 0.5D(α) b2 / n2 (where n2 here represents how many jobs of type 2 are already assigned to this machine). In Figure 7, we plot equation (12) for a specific instance with V = 30 and different values of α. As expected, if n2 = 0, we can schedule n1 = 50 jobs of class 1 to reach exactly the capacity V , no matter what is the value of α. On the other hand, for α = 0.99, if n1 = 0, we can schedule n2 = 38 jobs of class 2. As the value of n2 increases, the optimal value of n1 decreases. For a given value of α, any point (n1 , n2 ) on the curve (or below) guarantees the feasibility of the chance constraint. The interesting insight is to characterize the proportion of jobs of classes 1 and 2 per machine. For example, if we want to impose n1 = n2 in each machine, what is the optimal value for a given α? In our example, when α = 0.99, we can schedule n1 = n2 = 21 jobs. If we compare to the case without overcommitment (i.e., α = 1), we can schedule 18 jobs from each class. Therefore, we obtain an improvement of 16.67%. More generally, if the cost (or priority) of certain jobs is higher, we can design an optimal ratio per machine so that it still guarantees to satisfy the chance constraint. To summarize, for the case when V = 30, µ = 0.65, A = 1, A = 0.3 and α = 0.99, one can schedule either 50 jobs of class 1 or 38 jobs of class 2 or any combination of both classes according to equation (12). In other words, we have many different ways of bin packing jobs of classes 1 and 2 to each machine. We next consider solving the offline problem when our goal is to schedule N1 jobs of class 1 and N2 jobs of class 2. The numbers N1 and N2 are given as an input, and our goal is to minimize the number of machines denoted by M ∗ , such that each machine is assigned a pair (n1 , n2 ) that satisfies equation (11). Since n1 and n2 should be integer numbers, one can compute for a given value of α, all the feasible pairs that lie on the curve or just below (we have at most K = mini=1,2 max ni such pairs). In other words, for each k = 1, 2, . . . , K, Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints Figure 7 41 Parameters: A = 1, A = 0.3, µ = 0.65, V = 30. we compute a pair of coefficients denoted by (βk , γk ). The optimization problem becomes a cutting stock problem: M ∗ = min zk s.t. K X zk k=1 K X βk zk ≥ N1 (14) k=1 K X γ k z k ≥ N2 k=1 zk ≥ 0, integer ∀k. The decision variable zk represents the number of times we use the pair (βk , γk ), and M ∗ denotes the optimal number of machines. As a result, assuming that we have only two classes of jobs (with different µs and bs), one can solve the deterministic linear integer program in (14) and obtain a solution for problem (SMBP). Note that the above treatment can easily be extended to more than two classes of jobs. Appendix D: Proof. Proof of Theorem 4 Let n1 be the number of machines purchased by First-Fit with only a single job, and S1 be the set of n1 jobs assigned to these machines. Similarly, we define n2 to be the number of machines with at least two jobs, and S2 be the set of their jobs. The goal is to prove that n1 + n2 ≤ 49 OP T + 1. We know that any pair of jobs among the n1 jobs in S1 does not fit in a single machine (by the definition of First-Fit). Therefore, any feasible allocation (including the optimal allocation) needs at least n1 machines. In other words, we have OP T ≥ n1 . This observation also implies that the sum of µj + bj for any pair of jobs in S1 is greater than 3 4 (using Lemma 1). If we sum up all these inequalities for the different pairs of jobs in S1 , we
P have: (n1 − 1) j∈S1 (µj + bj ) > n21 43 . We note that the n1 − 1 term on the left side appears because every job
j ∈ S1 is paired with n1 − 1 other jobs in S1 , and the n21 term on the right side represents the total number P of pairs of jobs in S1 . By dividing both sides of this inequality by n1 − 1, we obtain j∈S1 (µj + bj ) > 3n8 1 . P We also lower bound j∈S2 (µj + bj ) as a function of n2 as follows. Let m1 < m2 < · · · < mn2 be the machines that have at least two jobs, and the ordering shows in which order they were purchased (e.g., 42 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints m1 was purchased first). Define Mi to be the set of jobs in machine mi . By definition of First-Fit, any job j in machine mi+1 could not be assigned to machine mi because of the feasibility constraints for any 1 ≤ i < n2 . In other words, the set of jobs Mi with any job j ∈ Mi+1 form an infeasible set. Therefore, we P have µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) > 43 . For each 1 ≤ i < n2 , one can pick two distinct jobs j1 and j2 from Mi+1 , and write the following two inequalities: X µj1 + bj1 + (µj 0 + bj 0 ) > j 0 ∈Mi 3 4 and µj2 + bj2 + X j 0 ∈Mi Summing up these two inequalities implies that: µj1 + bj1 + µj2 + bj2 + 2 3 (µj 0 + bj 0 ) > . 4 P j 0 ∈Mi (µj 0 + bj 0 ) > 32 . Since j1 and j2 are two distinct jobs in Mi+1 , we have: X (µj + bj ) + 2 X j 0 ∈Mi j∈Mi+1 3 (µj 0 + bj 0 ) > . 2 Now we sum up this inequality for different values of i ∈ {1, 2, · · · , n2 − 1} to achieve that: n2 −1 X 2 (µj + bj ) + 3 X X i=2 j∈Mi j∈M1 Pn2 P X (µj + bj ) + (µj + bj ) > j∈Mn2 3 × (n2 − 1), 2 3 2 P × (n2 − 1). This is equivalent to j∈S2 (µj + bj ) > P 1 × (n2 − 1). Combining both inequalities, we obtain: j∈S1 ∪S2 (µj + bj ) > 38 n1 + 12 (n2 − 1). On the other 2 P hand, OP T is at least j∈S1 ∪S2 (µj + bj ). We then have the following two inequalities: and consequently, we have 3 i=1 j∈Mi (µj + bj ) > OP T ≥ n1 , 1 3 OP T > n1 + (n2 − 1). 8 2 We can now multiply (15) by 1 4 and (16) by 2, and sum them up. We conclude that (15) (16) 9 OP T 4 + 1 is greater than n1 + n2 , which is the number of machines purchased by Algorithm First-Fit.  Appendix E: Proof of Theorem 5 We first state the following √ Lemma that provides a lower bound on OP T . For any a, b ≥ 0, we define the 2a+b+ b(4a+b) . function f (a, b) = 2 Lemma 2. For any feasible set of jobs S, the sum P µj bj j∈S f (µj , bj ) is at most 1. and b̄ = j∈S . Since the function f is concave with respect to both a and |S| P b, using Jensen’s inequality we have j∈S f (µj , bj ) ≤ |S|f (µ̄, b̄). Since S is a feasible set, Cost(S) = |S|µ̄ + p p |S|b̄ ≤ 1. The latter is a quadratic inequality with variable x = |S|, so that we we can derive an upper √ bound on |S| in terms of µ̄ and b̄. Solving the quadratic form µ̄X + b̄X = 1 yields: p √ − b̄ ± b̄ + 4µ̄ X= . 2µ̄ √ √ p 4µ̄+b̄− b̄ 4q 1 We then have |S| ≤ = q 2 √ . Therefore, |S| ≤ = f (µ̄, , where the equality 2µ̄ b̄) 4µ̄+b̄+b̄+2 b̄(4µ̄+b̄) (4µ̄+b̄)+ b̄ P follows by the definition of the function f . Equivalently, |S|f (µ̄, b̄) ≤ 1, and hence j∈S f (µj , bj ) ≤ 1.  Proof. Define µ̄ = j∈S P P |S| Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 43 Proof of Theorem 5. For any arbitrary allocation of jobs, applying Lemma 2 to all the machines implies PN that the number of machines is at least j =1 f (µj , bj ). So it suffices to upper bound the number of machines PN as a function of j =1 f (µj , bj ). For any given machine, we prove that the sum of f (µj , bj ) for all jobs j assigned to this machine is at least 1 − O( + δ). Consider the set of jobs S assigned to a given machine. P Similar to the proof of Lemma 2, we define µ̄ = j∈S µj |S| P and b̄ = j∈S |S| bj P . Define r = P j∈S j∈S bj µj = µ̄b̄ . We start by lower bounding f (µj , bj ) as a function of f (µj , rµj ). Recall that each machine is δ-homogeneous, i.e., for all pairs of jobs in the same machine, the ratios other. Hence, any ratio bj µj f (µj , bj ) ≥ f r is at least (1+δ ) bj µj are at most a multiplicative factor of 1 + δ away from each . Consequently, we have bj ≥ rµj 1+δ which implies that:
µj rµj
1 µj , bj ≥ f , = f (µj , rµj ) ≥ (1 − δ)f (µj , rµj ). 1+δ 1+δ 1+δ 1+δ The first and second inequalities follow from the monotonicity of the function f , the equality follows from P the definition of f , and the last inequality holds since δ ≥ 0. It now suffices to lower bound j∈S f (µj , rµj ) P so as to obtain a lower bound for j∈S f (µj , bj ). We note that for any η ≥ 0, we have f (ηµ, ηb) = ηf (µ, b) by the definition of f . Applying η = µj , µ = 1 and b = r, we obtain f (µj , rµj ) = µj f (1, r) which implies: X X f (µj , rµj ) = µj f (1, r) = f (A, B), j∈S where A = P A0 = and B 0 = µj , and B = r P µj = j∈S P bj . The last equality above follows from f (ηµ, ηb) = ηf (µ, b) √ with η = A. Recall that each machine is -full, i.e., Cost(S) ≥ 1 −  or equivalently, A + B ≥ 1 − . Let A (1−)2 j∈S B (1−)2 j∈S j∈S . Then, we have: √ √ A + B B A + ≥ ≥ 1. A + B0 = (1 − )2 1 −  1− 0 √ √ Since B = rA, we also have B 0 = rA0 , and the lower bound can be rewritten as follows: A0 + rA0 ≥ 1, which is the same quadratic form as in the proof of Lemma 2. Similarly, we prove that A0 ≥ is equal to 1 . f (1,r ) 0 0 0 0 0 4 √ 4+2r+2 r (4+r ) We also know that f (A , B ) = f (A , rA ) = A f (1, r). We already proved that A ≥ which 1 f (1,r ) , so we have f (A0 , B 0 ) ≥ 1. By definition of A0 and B 0 , we know thatf (A, B) = (1 − )2 f (A0 , B 0 ) ≥ (1 − )2 . P We conclude that the sum j∈S f (µj , bj ) ≥ (1 − δ)f (A, B) ≥ (1 − δ)(1 − )2 . As a result, for each machine the sum of f (µj , bj ) is at least (1 − δ)(1 − )2 . Let m be the number of purchased machines. Therefore, the PN sum j =1 f (µj , bj ) is lower bounded by (1 − δ)(1 − )2 m, and at the same time upper bounded by OP T . Consequently, m does not exceed Appendix F: Proof. OP T (1−δ )(1−)2 and this concludes the proof.  Proof of Theorem 6 We first prove that the algorithm terminates in a finite number of iterations. Note that all the update operations (except the first one) reduce the number of purchased machines, and hence, there are no more than N of those. As a result, it suffices to upper bound the number of times we perform the first update operation. Since we assign jobs to lower id machines, there cannot be more than N 2 consecutive first update operations. Consequently, after at most N × N 2 = N 3 operations, the algorithm has to terminate. Next, we derive an upper bound on the number of purchased machines at the end of the algorithm. Note that all the machines belong to one of the following four categories: • Single job machines, i.e., the set A1 . 44 Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints • Medium machines with only one non-good job – denoted by the set B. • Medium machines with at least two non-good jobs – denoted by the set C. • Large machines, i.e., the set A5 . Let a, b, c, and d be the number of machines in A1 , B, C, and A5 respectively. Since no update operation is possible (as the algorithm already terminated), the b non-good jobs assigned to the machines in the set B do not fit in any of the single job machines, and no pair of them fit together in a new machine. Consider these b non-good jobs in addition to the a jobs in the machines of the set A1 . No pair of these a + b jobs fit in one machine together and therefore, OP T ≥ a + b. PN We also know that OP T ≥ j =1 (µj + bj ). Next, we derive a more elaborate lower bound on OP T by writing the sum of µj + bj as a linear combination of the sizes of the sets A1 , B, C, and A5 . For each machine i, let Mi be the set of jobs assigned to this machine. Let i1 < i2 < · · · < id be the indices of machines in the set A5 , where d = |A5 |. Since we cannot perform the first update operation anymore, we can say that no P job in machine i`+1 fits in machine i` for any 1 ≤ ` < d. Therefore, µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) > 34 for any ` j ∈ Mi`+1 (using Lemma 1). We write this inequality for 5 different jobs (arbitrarily chosen) in Mi`+1 (recall that there are at least 5 jobs in this machine), and for all the values of 1 ≤ ` < d. If we sum up all these 5(d − 1) inequalities, then the right hand side would be 5(d − 1) × 43 . On the other hand, the term µj + bj for every job in these d machines appears on the left hand side at most 5 + 1 = 6 times. Therefore, by summing P P 1) 1) = 5(d− . up these inequalities, we obtain: i∈A5 j∈Mi (µj + bj ) > 34 × 5(d− 6 8 Each machine in A5 has at least 5 jobs. Therefore, the term 5 6 appears in the lower bound. With a similar argument, each machine in either B or C has at least 2 jobs and hence, this term is now replaced by 32 . The P P 1) = b−2 1 , and inequalities for every pair of machines in B and C are then: i∈B j∈Mi (µj + bj ) > 34 × 2(b− 3 P P 2(c−1) 3 = c−2 1 . i∈C j∈Mi (µj + bj ) > 4 × 3 P P Next, we lower bound i∈A1 ∪C j∈Mi (µj + bj ) as a function of a and c in order to complete the proof. Recall that each machine in C has at least two non-good jobs. If we pick one of these non-good jobs j, and a random machine i from A1 , with probability at least a−5 a , job j does not fit in machine i. This follows from the fact that a non-good job fits in at most 5 machines in A1 and hence, a random machine in A1 would P not be be able to fit job j with probability at least a−a 5 . Therefore, µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) ≥ 34 with probability at least a−5 a . We next consider two different cases depending on the value of c. If c is at least a2 , we pick a random machine in C, and two of its non-good jobs j1 and j2 arbitrarily. We also pick a random machine in A1 . For each of these two jobs, the sum µj + bj of the non-good job and the single job in the selected machine in A1 is greater than 3 4 with probability at least a−5 a . Summing up these two inequalities, we obtain: i 1hX X i a−5 3 2h X X (µj + bj ) + (µj + bj ) > × . a i∈A j∈M c i∈C j∈M a 2 1 i (17) i The left hand side of the above equation is composed of two terms. The first term is obtained through picking a random machine in A1 (i.e., with probability a1 ), and once the machine is picked, we sum up both equations so we obtain 2 a . For the second term, every machine in the set C is chosen with probability 1 c . Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints 45 When the machine is picked, we sum up on all the jobs and hence get an upper bound. As we have shown, P P (µj + bj ) ≥ c−2 1 . Combining these two inequalities leads to (using c ≥ a2 ): i∈C j∈Mi X X (µj + bj ) > i∈A1 ∪C j∈Mi a c − 1 3a 15 c a 1 a + c a 3(a − 5) × + (1 − ) × ≥ − + − − = − 4.25. 2 2a 2c 2 4 4 2 4 2 2 PN By combining the three different bounds (on A1 ∪C, B and A5 ), we obtain j =1 (µj +bj ) ≥ a+2b+c + 58d −5.875. PN Since OP T ≥ j =1 (µj + bj ), we conclude that the number of purchased machines a + b + c + d is no more than 2OP T + 11. In the other case, we have c < a2 . Note that inequality (17) still holds. However, since the coefficient 1 − 2ac becomes negative, we cannot combine the two inequalities as before. Instead, we lower bound the sum µj + bj of jobs in A1 . We know that there is no pair of a jobs in the A1 machines that fit together in one machine. P P Therefore, i∈A1 j∈Mi (µj + bj ) ≥ 38a . Next, we multiply inequality (17) by c, and combine it with this new P P lower bound on i∈A1 j∈Mi (µj + bj ), to obtain (using c < a2 ): X X i∈A1 ∪C j∈Mi (µj + bj ) > c × 3(a − 5) 2c 3a 3c 5 3a 3c 3a 3c 5 + (1 − ) × > − + − = + − . 2a a 8 2 4 8 4 8 4 4 Combining this inequality with similar ones on the sets B and A5 , we obtain OP T ≥ 3a 8 PN j =1 (µj + bj ) > + 2b + 34c + 58d − 19 . Finally, combining this with OP T ≥ a + b leads to a + b + c + d ≤ 85 OP T + 52 OP T + 19 = 8 5 2OP T + 3.75, which concludes the proof.  

The Complex Event Recognition Group Elias Alevizos Alexander Artikis Nikos Katzouris Georgios Paliouras Evangelos Michelioudakis arXiv:1802.04086v1 [] 12 Feb 2018 Institute of Informatics & Telecommunications, National Centre for Scientific Research (NCSR) Demokritos, Athens, Greece {alevizos.elias, a.artikis, nkatz, vagmcs, paliourg}@iit.demokritos.gr ABSTRACT The Complex Event Recognition (CER) group is a research team, affiliated with the National Centre of Scientific Research “Demokritos” in Greece. The CER group works towards advanced and efficient methods for the recognition of complex events in a multitude of large, heterogeneous and interdependent data streams. Its research covers multiple aspects of complex event recognition, from efficient detection of patterns on event streams to handling uncertainty and noise in streams, and machine learning techniques for inferring interesting patterns. Lately, it has expanded to methods for forecasting the occurrence of events. It was founded in 2009 and currently hosts 3 senior researchers, 5 PhD students and works regularly with under-graduate students. 1. INTRODUCTION The proliferation of devices that work in realtime, constantly producing data streams, has led to a paradigm shift with respect to what is expected from a system working with massive amounts of data. The dominant model for processing largescale data was one that assumed a relatively fixed database/knowledge base, i.e., it assumed that the operations of updating existing records/facts and inserting new ones were infrequent. The user of such a system would then pose queries to the database, without very strict requirements in terms of latency. While this model is far from being rendered obsolete (on the contrary), a system aiming to extract actionable knowledge from continuously evolving streams of data has to address a new set of challenges and satisfy a new set of requirements. The basic idea behind such a system is that it is not always possible, or even desirable, to store every bit of the incoming data, so that it can be later processed. Rather, the goal is to make sense out of these streams of data, without having to store them. This is done by defining a set of queries/patterns, continuously applied to the data streams. Each such pattern includes a set of temporal constraints and, possibly, a set of spatial constraints, expressing a composite or complex event of special significance for a given application. The system must then be efficient enough so that instances of pattern satisfaction can be reported to a user with minimal latency. Such systems are called Complex Event Recognition (CER) systems [6, 7, 2]. CER systems are widely adopted in contemporary applications. Such applications are the recognition of attacks in computer network nodes, human activities on video content, emerging stories and trends on the Social Web, traffic and transport incidents in smart cities, fraud in electronic marketplaces, cardiac arrhythmias and epidemic spread. Moreover, Big Data frameworks, such as Apache Storm, Spark Streaming and Flink, have been extending their stream processing functionality by including implementations for CER. There are multiple issues that arise for a CER system. As already mentioned, one issue is the requirement for minimal latency. Therefore, a CER system has to employ highly efficient reasoning mechanisms, scalable to high-velocity streams. Moreover, pre-processing steps, like data cleaning, have to be equally efficient, otherwise they constitute a “luxury” that a CER system cannot afford. In this case, the system must be able to handle noise. This may be a requirement, even if perfectly clean input data is assumed, since domain knowledge is often insufficient or incomplete. Hence, the patterns defined by the users may themselves carry a certain degree of uncertainty. Moreover, it is quite often the case that such patterns cannot be provided at all, even by domain experts. This poses a further challenge of how to apply machine learning techniques in order to extract patterns from streams before a CER system can actually run with them. Standard machine learning techniques are not always directly applicable, due to the size and variability of the training set. As a result, machine learning techniques must work in an online fashion. Finally, one often needs 2. COMPLEX EVENT RECOGNITION Numerous CER systems have been proposed in the literature [6, 7]. Recognition systems with a logic-based representation of complex event (CE) patterns, in particular, have been attracting attention since they exhibit a formal, declarative semantics [2]. We have been developing an efficient dialect of the Event Calculus, called ‘Event Calculus for Run-Time reasoning’ (RTEC) [4]. The Event Calculus is a logic programming formalism for representing and reasoning about events and their effects [14]. CE patterns in RTEC identify the conditions in which a CE is initiated and terminated. Then, according to the law of inertia, a CE holds at a time-point T if it has been initiated at some timepoint earlier than T , and has not been terminated in the meantime. RTEC has been optimised for CER, in order to be scalable to high-velocity data streams. A form of caching stores the results of subcomputations in the computer memory to avoid unnecessary recomputations. A set of interval manipulation constructs simplify CE patterns and improve reasoning efficiency. A simple indexing mechanism makes RTEC robust to events that are irrelevant to the patterns we want to match and so RTEC can operate without data filtering modules. Finally, a ‘windowing’ mechanism supports real-time CER. One main motivation for RTEC is that it should remain efficient and scalable in applications where events arrive with a (variable) delay from, or are revised by, the underlying sensors: RTEC can update the intervals of the already recognised CEs, and recognise new CEs, when data arrive with a delay or following revision. RTEC has been analysed theoretically, through a complexity analysis, and assessed experimentally in several application domains, including city transport and traffic management [5], activity recognition on video feeds [4], and maritime monitoring [18]. In all of these applications, RTEC has proven capable of performing real-time CER, scaling to large data streams and highly complex event patterns. 1 http://cer.iit.demokritos.gr/ 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ECcrisp MLN –EC P (CE | input data) to move beyond detecting instances of pattern satisfaction into forecasting when a pattern is likely to be satisfied in the future. Our CER group1 at the National Centre for Scientific Research (NCSR) Demokritos, in Athens, Greece, has been conducting research on CER for the past decade, and has developed a number of novel algorithms and publicly available software tools. In what follows, we sketch the approaches that we have proposed and present some indicative results. 0 3 initiation 10 initiation 20 termination time I Figure 1: CE probability estimation in the Event Calculus. The solid line concerns a probabilistic Event Calculus, such as MLN-EC, while the dashed line corresponds to a crisp (non-probabilistic) version of the Event Calculus. Due to the law of inertia, the CE probability remains constant in the absence of input data. Each time the initiation conditions are satisfied (e.g., in time-points 3 and 10), the CE probability increases. Conversely, when the termination conditions are satisfied (e.g., in timepoint 20), the CE probability decreases. 3. UNCERTAINTY HANDLING CER applications exhibit various types of uncertainty, ranging from incomplete and erroneous data streams to imperfect CE patterns [2]. We have been developing techniques for handling uncertainty in CER by extending the Event Calculus with probabilistic reasoning. Prob-EC [21] is a logic programming implementation of the Event Calculus using the ProbLog engine [13], that incorporates probabilistic semantics into logic programming. Prob-EC is the first Event Calculus dialect able to deal with uncertainty in the input data streams. For example, Prob-EC is more resilient to spurious data than the standard (crisp) Event Calculus. MLN-EC [22] is an Event Calculus implementation based on Markov Logic Networks (MLN)s [20], a framework that combines first-order logic with graphical models, in order to enable probabilistic inference and learning. CE patterns may be associated with weight values, indicating our confidence in them. Inference can then be performed regarding the time intervals during which CEs of interest hold. Like Prob-EC, MLN-EC increases the probability of a CE every time its initiating conditions are satisfied, and decreases this probability whenever its terminating conditions are satisfied, as shown in Figure 1. Moreover, in MLN-EC the domain-independent Event Calculus rules, expressing the law of inertia, may be associated with weight values, introducing probabilistic inertia. This way, the model is highly customisable, by tuning appropriately the weight values with the use of machine learning techniques, and thus achieves high predic- 1 0.9 0.8 F1 score 0.7 0.6 0.5 0.4 0.3 0.2 MLN –EC l–CRF 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Threshold Figure 2: CER under uncertainty. F1 -score of MLN-EC and linear-chain CRFs for different CE acceptance thresholds. tive accuracy in a wide range applications. The use of background knowledge about the task and the domain, in terms of logic (the Event Calculus), can make MLN-EC more robust to variations in the data. Such variations are very common in practice, particularly in dynamic environments, such as the ones encountered in CER. The common assumption made in machine learning that the training and test data share the same statistical properties is often violated in these situations. Figure 2, for example, compares the performance of MLN-EC against linear-chain Conditional Random Fields on a benchmark activity recognition dataset, where evidence is incomplete in the test set as compared to the training set. 4. EVENT PATTERN LEARNING The manual authoring of CE patterns is a tedious and error-prone process. Consequently, the automated construction of such patterns from data is highly desirable. We have been developing supervised, online learning learning tools for constructing logical representations of CE patterns, from a single-pass over a relational data stream. OSLα [16] is such a learner for Markov Logic Networks (MLNs), formulating CE patterns in the form of MLN-EC theories. OSLα extends OSL [9] by exploiting a background knowledge in order to significantly constrain the search for patterns. In each step t of the online procedure, a set of training examples Dt arrives containing input data along with CE annotation. Dt is used together with the already learnt hypothesis, if any, to predict the truth values of the CEs of interest. This is achieved by MAP (maximum a posteriori) inference. Given Dt , OSLα constructs a hypergraph that represents the space of possible structures as graph paths. Then, for all incorrectly predicted CEs, the hypergraph is searched using relational pathfinding, for clauses supporting the recognition of these CEs. The paths discovered during the search are generalised into first-order clauses. Subsequently, the weights of the clauses that pass the evaluation stage are optimised using off-the-shelf online weight learners. Then, the weighted clauses are appended to the hypothesis and the procedure is repeated for the next set of training examples Dt+1 . OLED [11] is an Inductive Logic Programming (ILP) system that learns CE patterns, in the form of Event Calculus theories, in a supervised fashion and in a single pass over a data stream. OLED constructs patterns by first encoding a positive example from the input stream into a so-called bottom rule, i.e., a most-specific rule of the form α ← δ1 ∧ . . . ∧ δn , where α is an initiation or termination atom and δ1 , . . . , δn are relational features, constructed onthe-fly, using prescriptions from a predefined language bias. These features express anything “interesting”, as defined by the language bias, that is true within the positive example at hand. A bottom rule is typically too restrictive to be useful. To learn a useful rule, OLED searches within the space of rules that θ-subsume the bottom rule, i.e., rules that involve some of the δi ’s only. To that end, OLED starts from the most-general rule—a rule with an empty body—and gradually specialises that rule by adding δi ’s to its body, using a rule evaluation function to assess the quality of each generated specialisation. OLED’s single-pass strategy is based on the Hoeffding bound [8], a statistical tool that allows to approximate the quality of a rule on the entire input using only a subset of the data. In particular, given a rule r and some of its specialisations r1 , . . . , rk , the Hoeffding bound allows to identify the best among them, with probability 1−δ and within an error margin , using only N = O( 12 ln 1δ ) training examples from the input stream. We have evaluated OLED and OSLα on real datasets concerning activity recognition, maritime monitoring, credit card fraud detection, and traffic management in smart cities [11, 16, 3, 15, 12]. We have also compared OLED and OSLα to OSL [9], XHAIL, a ‘batch’ structure learner requiring many passes over the data [19], and to hand-curated Event Calculus patterns (with optimised weight values). The results suggest that both OLED and OSLα can match the predictive accuracy of batch learners as well as that of hand-crafted patterns. Moreover, OLED and OSLα have proven significantly faster than both batch and online learners, making them more appropriate for large data streams. a EVENT FORECASTING Forecasting over time-evolving data streams is a task that can be defined in multiple ways. There is a conceptual difference between forecasting and prediction, as the latter term is understood in machine learning, where the main goal is to “predict” the output of a function on previously unseen input data, even if there is no temporal dimension. In forecasting, time is a crucial component and the goal is to predict the temporally future output of some function or the occurrence of an event. Timeseries forecasting is an example of the former case and is a field with a significant history of contributions. However, its methods cannot be directly transferred to CER, since it handles streams of (mostly) real-valued variables and focuses on forecasting relatively simple patterns. On the contrary, in CER we are also interested in categorical values, related through complex patterns and involving multiple variables. Our group has developed a method, where automata and Markov chains are employed in order to provide (future) time intervals during which a match is expected with a probability above a confidence threshold [1]. We start with a given pattern of interest, defining relations between events, in the form of a regular expression—i.e., using operators for sequence, disjunction and iteration. Our goal, besides detecting occurrences of this pattern, is also to estimate, at each new event arrival, the number of future events that we will need to wait for until the expression is satisfied, and thus a match be detected. A pattern in the form of a regular expression is first converted to a deterministic finite automaton (DFA) through standard conversion algorithms. We then construct a Markov chain that will be able to provide a probabilistic description of the DFA’s run-time behavior, by employing Pattern Markov Chains (PMC) [17]. The resulting PMC depends both on the initial pattern and on the assumptions made about the statistical properties of the input stream—the order m of the assumed Markov process. After constructing a PMC, we can use it to calculate the so-called waiting-time distributions, which can give us the probability of reaching a final state of the DFA in k transitions from now. To estimate the final forecasts, another step is required, since our aim is not to provide a single future point with the highest probability, but an interval in the form of I=(start, end ). The meaning of such an interval is that the DFA is expected to reach a final state sometime in the future between start and end with probability at least some constant threshold θfc (provided by the user). An example is shown a a start a 0 1 b 2 b b 3 4 a b b (a) Deterministic Finite Automaton, state 1. 1 Completion Probability 5. state:0 state:1 interval:3,8 state:2 state:3 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 Number of future events (b) Waiting-time distribution, state 1. Figure 3: Event Forecasting. The event pattern requires that one event of type a is followed by three events of type b. θf c = 0.5. For illustration, the x axis stops at 12 future events. in Figure 3, where the DFA in Figure 3a is in state 1, the waiting-time distributions for all of its nonfinal states are shown in Figure 3b, and the distribution, along with the forecast interval, for state 1 are shown in green. Figure 4 shows results of our implementation on two real-world datasets from the financial and the maritime domains. In the former case, the goal was to forecast a specific case of credit card fraud, whereas in the latter it was to forecast a specific vessel manoeuver. Figures 4a and 4d show precision results (the percentage of forecasts that were accurate), where the y axes correspond to different values of the threshold θfc , and the x axes correspond to states of the PMC (more “advanced” states are to the right of the axis), i.e., we measure precision for the forecasts produced by each individual state. Similarly, Figures 4b and 4e are per-state plots for spread (the length of the forecast interval), and Figures 4c and 4f are per-state plots for distance (the temporal distance between the time a forecast is produced and the start of the forecast interval). As expected, more “advanced” states produce forecasts with higher precision, smaller spread and distance. However, there are cases where we can get earlier both high precision and low spread scores (see Figures 4d and 4e). This may happen when there exist strong probabilistic dependencies in the stream, e.g., when one event type is very likely (or 0.6 60 0.4 40 20 0.2 0.8 8 0.6 6 0.4 4 2 0.2 1 11 12 13 2 21 3 4 5 6 7 0 1 11 12 13 2 21 3 4 5 6 0 0 1 11 12 13 2 21 3 State State (b) Spread. (c) Distance. 60 0.4 40 20 0.2 11 ts 13 gse 14 gsn 15 gsw 16 gss 17 gen 18 gew 19 ges 20 gee 0 0.8 300 5 6 7 250 0.6 200 150 0.4 100 0.2 50 15 Prediction Threshold 80 4 350 Prediction Threshold Prediction Threshold 5 0.2 State 0.6 9 tn 0.4 7 100 7 tw 10 0.6 (a) Precision. 0.8 3 te 0.8 0 0 0 15 Prediction Threshold 80 10 Prediction Threshold Prediction Threshold 100 0.8 0.8 10 0.6 0.4 5 0.2 0 3 te 7 tw 9 tn 11 ts 13 gse 14 gsn 15 gsw 16 gss 17 gen 18 gew 19 ges 20 gee 0 3 te 7 tw 9 tn 11 ts 13 gse 14 gsn 15 gsw 16 gss 17 gen 18 gew 19 ges 20 gee State State State (d) Precision. (e) Spread. (f) Distance. Figure 4: Event forecasting for credit card fraud management (top) and maritime monitoring (bottom). The y axes correspond to different values of the threshold θfc . The x axes correspond to states of the PMC. very unlikely) to appear, given that the last event(s) is of a different event type. Our system can take advantage of such cases in order to produce highquality forecasts early. 6. PARTICIPATION IN RESEARCH & INNOVATION PROJECTS The CER group has been participating in several research and innovation projects, contributing to the development of intelligent systems in challenging domains. SPEEDD2 (Scalable Proactive Event-Driven Decision Making) was an FP7 EUfunded project, coordinated by the CER group, that developed tools for proactive analytics in Big Data applications. In SPEEDD, the CER group worked on credit card fraud detection and traffic management [3, 15], developing formal tools for highly scalable CER [4], and pattern learning [10, 16]. REVEAL3 (REVEALing hidden concepts in social media) was an FP7 EU project that developed techniques for real-time extraction of knowledge from social media, including influence and reputation assessment. In REVEAL, the CER group developed a technique for online (single-pass) learning of event patterns under uncertainty [11]. datACRON4 (Big Data Analytics for Time Crit2 http://speedd-project.eu/ 3 http://revealproject.eu/ 4 http://www.datacron-project.eu/ ical Mobility Forecasting) is an H2020 EU project that introduces novel methods for detecting threats and abnormal activity in very large fleets of moving entities, such as vessels and aircrafts. Similarly, AMINESS5 (Analysis of Marine Information for Environmentally Safe Shipping) was a national project that developed a computational framework for environmental safety and cost reduction in the maritime domain. The CER group has been working on maritime and aviation surveillance, developing algorithms for, among others, highly efficient spatio-temporal pattern matching [18], complex event forecasting [1], and parallel online learning of complex event patterns [12]. Track & Know (Big Data for Mobility & Tracking Knowledge Extraction in Urban Areas) is an H2020 EU-funded project that will research, develop and exploit a new software framework increasing the efficiency of Big Data applications in the transport, mobility, motor insurance and health sectors. The CER team is responsible for the complex event recognition and forecasting technology that will be developed in Track & Know. 7. CONTRIBUTIONS TO THE COMMUNITY The CER group supports the research community at different levels; notably, by making avail5 http://aminess.eu/ able the proposed research methods as open-source solutions. The RTEC CER engine (see Section 2) is available as a monolithic Prolog implementation6 and as a parallel Scala implementation7 . The OLED system for online learning of event patterns (see Section 4) is also available as an open-source solution8 , both for single-core and parallel learning. OLED is implemented in Scala; both OLED and RTEC use the Akka actors library for parallel processing. The OSLα online learner (see Section 4), along with MAP inference based on integer linear programming, and various weight optimisation algorithms (Max-Margin, CDA and AdaGrad), are contributed to LoMRF9 , an open-source implementation of Markov Logic Networks. LoMRF provides predicate completion, clausal form transformation, and function elimination. Moreover, it provides a parallel grounding algorithm which efficiently constructs the minimal Markov Random Field. 8. REFERENCES [1] E. Alevizos, A. Artikis, and G. Paliouras. Event forecasting with pattern markov chains. In DEBS, pages 146–157. ACM, 2017. [2] E. Alevizos, A. Skarlatidis, A. Artikis, and G. Paliouras. Probabilistic complex event recognition: A survey. ACM Comput. Surv., 50(5):71:1–71:31, 2017. [3] A. Artikis, N. Katzouris, I. Correia, C. Baber, N. Morar, I. Skarbovsky, F. Fournier, and G. Paliouras. A prototype for credit card fraud management: Industry paper. In DEBS, pages 249–260, 2017. [4] A. Artikis, M. J. Sergot, and G. Paliouras. An event calculus for event recognition. IEEE TKDE, 27(4):895–908, 2015. [5] A. Artikis, M. Weidlich, F. Schnitzler, I. Boutsis, T. Liebig, N. Piatkowski, C. Bockermann, K. Morik, V. Kalogeraki, J. Marecek, A. Gal, S. Mannor, D. Gunopulos, and D. Kinane. Heterogeneous stream processing and crowdsourcing for urban traffic management. In EDBT, pages 712–723, 2014. [6] G. Cugola and A. Margara. Processing flows of information: From data stream to complex event processing. ACM Comput. Surv., 44(3):15:1–15:62, 2012. [7] N. Giatrakos, A. Artikis, A. Deligiannakis, and M. N. Garofalakis. Complex event recognition in the big data era. PVLDB, 6 https://github.com/aartikis/RTEC https://github.com/kontopoulos/ScaRTEC 8 https://github.com/nkatzz/OLED 9 https://github.com/anskarl/LoMRF 7 10(12):1996–1999, 2017. [8] W. Hoeffding. Probability inequalities for sums of bounded random variables. JASA, 58(301):13–30, 1963. [9] T. N. Huynh and R. J. Mooney. Online Structure Learning for Markov Logic Networks. In ECML, pages 81–96, 2011. [10] N. Katzouris, A. Artikis, and G. Paliouras. Incremental learning of event definitions with inductive logic programming. Machine Learning, 100(2-3):555–585, 2015. [11] N. Katzouris, A. Artikis, and G. Paliouras. Online learning of event definitions. TPLP, 16(5-6):817–833, 2016. [12] N. Katzouris, A. Artikis, and G. Paliouras. Parallel online learning of complex event definitions. In ILP. Springer, 2017. [13] A. Kimmig, B. Demoen, L. D. Raedt, V. Costa, and R. Rocha. On the implementation of the probabilistic logic programming language ProbLog. TPLP, 11(2-3):235–262, 2011. [14] R. A. Kowalski and M. J. Sergot. A logic-based calculus of events. New Generation Comput., 4(1):67–95, 1986. [15] E. Michelioudakis, A. Artikis, and G. Paliouras. Online structure learning for traffic management. In ILP. Springer, 2016. [16] E. Michelioudakis, A. Skarlatidis, G. Paliouras, and A. Artikis. Online structure learning using background knowledge axiomatization. In ECML, 2016. [17] G. Nuel. Pattern Markov Chains: Optimal Markov Chain Embedding through Deterministic Finite Automata. Journal of Applied Probability, 2008. [18] K. Patroumpas, E. Alevizos, A. Artikis, M. Vodas, N. Pelekis, and Y. Theodoridis. Online event recognition from moving vessel trajectories. GeoInformatica, 21(2), 2017. [19] O. Ray. Nonmonotonic abductive inductive learning. JAL, 7(3):329–340, 2009. [20] M. Richardson and P. M. Domingos. Markov logic networks. Machine Learning, 62(1-2):107–136, 2006. [21] A. Skarlatidis, A. Artikis, J. Filipou, and G. Paliouras. A probabilistic logic programming event calculus. TPLP, 15(2):213–245, 2015. [22] A. Skarlatidis, G. Paliouras, A. Artikis, and G. A. Vouros. Probabilistic Event Calculus for Event Recognition. ACM Transactions on Computational Logic, 16(2):11:1–11:37, 2015. 

Proceedings of the 2007 Winter Simulation Conference S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew, and R. R. Barton, eds. USING INTELLIGENT AGENTS TO UNDERSTAND MANAGEMENT PRACTICES AND RETAIL PRODUCTIVITY Peer-Olaf Siebers Uwe Aickelin Helen Celia Chris W. Clegg School of Computer Science, Automated Scheduling, Optimisation and Planning Research Group (ASAP) University of Nottingham Nottingham, NG8 1BB, UK Centre for Organisational Strategy, Learning & Change, Leeds University Business School University of Leeds Leeds, LS2 9JT, UK ABSTRACT Intelligent agents offer a new and exciting way of understanding the world of work. In this paper we apply agentbased modeling and simulation to investigate a set of problems in a retail context. Specifically, we are working to understand the relationship between human resource management practices and retail productivity. Despite the fact we are working within a relatively novel and complex domain, it is clear that intelligent agents could offer potential for fostering sustainable organizational capabilities in the future. The project is still at an early stage. So far we have conducted a case study in a UK department store to collect data and capture impressions about operations and actors within departments. Furthermore, based on our case study we have built and tested our first version of a retail branch simulator which we will present in this paper. 1 INTRODUCTION The retail sector has been identified as one of the biggest contributors to the productivity gap that persists between the UK, Europe and the USA (Reynolds et al. 2005). It is well documented that measures of UK retail productivity rank lower than those of countries with comparably developed economies. Intuitively, it is inevitable that management practices are inextricably linked to a company’s productivity and performance. However, many researchers have struggled to provide clear empirical evidence using more traditional research methods (for a review, see Wall and Wood 2005). Significant research has been done to investigate the productivity gap and the common focus has been to quantify its size and determine the contributing factors. Best practice guidelines have been developed and published, but there remains considerable inconsistency and uncer- tainty regarding how these are implemented and manifested in the retail work place. Siebers et al. (submitted) have conducted a comprehensive literature review of this pertinent research area linking management practices to firm-level productivity. Practices are dichotomized according to their focus, whether operationally-focused or people-focused. The authors conclude that management practices are multidimensional constructs that generally do not demonstrate a straightforward relationship with productivity variables. Empirical evidence affirms that management practices must be context specific to be effective, and in turn productivity indices must also reflect a particular organization’s activities. Currently there is no reliable and valid way to delineate the effects of management practices from other socially embedded factors. Most Operational Research (OR) methods can be applied as analytical tools once management practices have been implemented, however they are not very useful at revealing system-level effects of the introduction of specific management practices. This holds particularly when the focal interest is the development of the system over time, like in the real world. This contrasts with more traditional techniques, which allow us to identify the state of the system at a certain point in time. The overall aim of our project is to understand and predict the impact of different management practices on retail store productivity. To achieve this aim we have adopted a case study approach and integrated applied research methods to collect both qualitative and quantitative data. In summary, we have conducted four weeks of informal participant observations, forty staff interviews supplemented by a short questionnaire on the effectiveness of various management practices, and drawn upon a variety of established informational sources internal to the case study organization. Using this data, we are applying Agent-Based Modeling and Simulation (ABMS) to try to Siebers, Aickelin, Celia, & Clegg devise a functional representation of the case study departments. In this paper we will focus on the simulation side of the project. In Section 2 we summarize the literature review we have conducted to find a suitable research tool for our study. Section 3 describes the conceptualization, design and implementation of our retail branch simulator. In Section 4 we describe two experiments that we have conducted as a first step to validate our retail branch simulator. Section 5 concludes the paper and unveils our future ideas. 2 WHY AGENT-BASED SIMULATION? OR is applied to problems concerning the conduct and coordination of the operations within an organization (Hillier and Lieberman 2005). An OR study usually involves the development of a scientific model that attempts to abstract the essence of the real problem. When investigating the behavior of complex systems the choice of an appropriate modeling technique is very important. In order to be able to make a choice for our project, we reviewed the relevant literature spanning the fields of Economics, Social Science, Psychology, Retail, Marketing, OR, Artificial Intelligence, and Computer Science. Within these fields a wide variety of approaches are used which can be classified into three main categories: analytical approaches, heuristic approaches, and simulation. In many cases we found that combinations of these were used within a single model. Common combinations were ‘simulation / analytical’ for comparing efficiency of different non-existing scenarios, (e.g. Greasley 2005), and ‘simulation / analytical’ or ‘simulation / heuristic’ where analytical or heuristic models were used to represent the behavior of the entities within the simulation model (e.g. Schwaiger and Stahmer 2003). In our review we put a particular emphasis on those publications that try to model the link between management practices and productivity in the retail sector. We found a very limited number of papers that investigate management practices in retail at firm level. The majority of these papers focus on marketing practices (e.g. Keh et al. 2006). By far the most frequently used modeling technique we found being used was agent-based modeling employing simulation as the method of execution. It seems to be the natural way of system representation for these purposes. Simulation introduces the possibility of a new way of thinking about social and economic processes, based on ideas about the emergence of complex behavior from relatively simple activities (Simon 1996). While analytical models typically aim to explain correlations between variables measured at one single point in time, simulation models are concerned with the development of a system over time. Furthermore, analytical models usually work on a much higher level of abstraction than simulation models. For simulation models it is critical to define the right level of abstraction. Csik (2003) states that on the one hand the number of free parameters should be kept on a level as low as possible. On the other hand, too much abstraction and simplification might threaten the homomorphism between reality and the scope of the simulation model. There are several different approaches to simulation, amongst them discrete event simulation, system dynamics, micro simulation and agent-based simulation. The choice of the most suitable approach always depends on the issues investigated, the input data available, the level of analysis and the type of answers to be sought. Although computer simulation has been used widely since the 1960s, ABMS only became popular in the early 1990s (Epstein and Axtell 1996). ABMS can be used to study how micro-level processes affect macro-level outcomes. A complex system is represented by a collection of individual agents that are programmed to follow simple behavioral rules. Agents can interact with each other and with their environment to produce complex collective behavioral patterns. Macro behavior is not explicitly simulated; it emerges from the micro-decisions made by the individual agents (Pourdehnad et al. 2002). The main characteristics of agents are their autonomy, their ability to take flexible action in reaction to their environment and their pro-activeness depending on motivations generated from their internal states. They are designed to mimic the attributes and behaviors of their real-world counterparts. The simulation output may be potentially used for explanatory, exploratory and predictive purposes (Twomey and Cadman 2002). This approach offers a new opportunity to realistically and validly model organizational characters and their interactions, to allow a meaningful investigation of human resource management practices. ABMS is still a relatively new simulation technology and its principle application has been in academic research. With the appearance of more sophisticated modeling tools in the broader market, things are starting to change (Luck et al. 2005). Also, an ever increasing number of computer games use the ABMS approach. A detailed description of ABMS and thoughts on the appropriate contexts for ABMS versus conventional modeling techniques can be found in WSC introductory tutorial on ABMS (Macal and North 2006). Therefore we provide only a brief summary of some of our thoughts. Due to the characteristics of the agents, this modeling approach appears to be more suitable than Discrete Event Simulation (DES) for modeling human-oriented systems (Siebers 2006). ABMS seems to promote a natural form of modeling, as active entities in the live environment are interpreted as actors in the model. There is a structural correspondence between the real system and the model representation, which makes them more intuitive and easier to understand than for example a system of differential Siebers, Aickelin, Celia, & Clegg equations as used in System Dynamics. Hood (1998) emphasized that one of the key strengths of ABMS is that the system as a whole is not constrained to exhibit any particular behavior as the system properties emerge from its constituent agent interactions. Consequently assumptions of linearity, equilibrium and so on, are not needed. With regard to disadvantages there is a general consensus in the literature that it is difficult to evaluate agent-based models, because the behavior of the system emerges from the interactions between the individual entities. Furthermore, problems often occur through the lack of adequate empirical data. Finally, there is always the danger that people new to ABMS may expect too much from the models, particularly in regard to predictive ability. 3 MODEL DESIGN AND IMPLEMENTATION The strategy for our project is iterative, creating a relatively simple model and then building in more and more complexity. To begin with we have been trying to understand the particular problem domain, to generate the underlying rules currently in place. We are now in the process of building an agent based simulation model of the real system using the information gathered during our case study and will then validate our model by simulating the operation of the real system. This approach will allow us to assess the accuracy of the system representation. If the simulation provides a sufficiently good representation we are able to move to the next stage, and generate new scenarios for how the system could work using new rules. 3.1 Modelling Concepts Our case study approach and analysis has played a crucial Customer Agent role allowing us to acquire a conceptual idea of how the real system is structured. This is an important stage of the project, revealing insights into the operation of the system as well as the behavior of and interactions between the different characters in the system. We have designed the system by applying a DES approach to conceptualize and model the system, and then an agent approach to conceptualize and model the actors within the system. This method made it easier to design the model, and is possible because only the actors’ action requires an agent based approach. In terms of performance indicators, these are identical to those of a DES model. Beyond this, ABMS can offer further insights. A simulation model can detect unintended consequences, which have been referred to as ‘emergent behavior’ (Gilbert and Troitzsch 2005). Such unintended consequences can be difficult to understand because they are not defined in the same way as the system inputs; however it is critical to fully understand all system output to be able to accurately draw comparisons between the relative efficiencies of competing systems. Our conceptual ideas for the simulator are shown in Figure 1. Within our simulation model we have three different types of agents (customers, sales staff, and managers) each of them having a different set of relevant parameters. We will use probabilities and frequency distributions to assign slightly different values to each individual agent. In this way a population is created that reflects the variations in attitudes and behaviors of their real human counterparts. In terms of other inputs, we need global parameters which can influence any aspect of the system, and may for example define the number of agents in the system. With regards to the outputs we always hope to find some unforeseeable, emergent behavior on a Visual Dynamic Stochastic Simulation Model Customer Agent Shopping need, attitudes, Customer Agent demographics etc. Emergent behaviour on macro level Customer Agent Understanding about interactions of entities within the system Sales Staff Agent Sales Agent Attitudes, length of service, competencies, training etc. Identification of bottlenecks Manager Agent Leadership quality, length of service, competencies, training etc. Global Parameters Performance Measures Number of customers, sales staff, managers etc. Staff utilisation, average response time, customer satisfaction etc. Interface for User Interaction during Runtime Figure 1: Conceptual model for our simulator Siebers, Aickelin, Celia, & Clegg macro level. Having a visual representation of the simulated system and its actors will allow us to monitor and better understand the interactions of entities within the system. Coupled with the standard DES performance measures, we hope to identify bottlenecks and help to optimize the modeled system. For the conceptual design of our agents we have decided to use state charts. State charts show the different states an entity can be in and also define the events that cause a transition from one state to another. This is exactly the information we need in order to represent our agents later within the simulation environment. Furthermore, this form of graphical representation is also helpful for validating the agent design as it is easier for nonspecialists to understand. The art of modelling pivots on simplification and abstraction (Shannon 1975). A model is always a restricted copy of the real world, and we have to identify the most important components of a system to build effective models. In our case, instead of looking for components we have to identify the most important behaviours of an actor and the triggers that initiate a move from one state to another. We have developed state charts for all the relevant actors in our retail branch model. Figure 2 shows as an example the state charts for a customer agent. The transition rules have been replaced by numbers to keep the chart comprehensible. They are explained in detail in the Section 3.2. A question that can be asked is whether our agents are intelligent or not? Wooldridge (2002) states that in order to be intelligent agents need to have the following attributes: being reactive, being proactive and being social. This is a widely accepted view. Being reactive means responding to changes in the environment (in a timely manner), while being proactive means persistently pursuing goals and being social means interacting with other agents (Padgham and Winikoff, 2004). Our agents perceive a goal in that they want to either buy something or return something. For buying they have a sub goal; that they are trying to buy the right thing. If they are not sure they will ask for help. Our agents are not only reactive but also flexible, i.e. they are capable to recover from a failure of action. They have alternatives inbuilt when they are unable to perceive their goal, e.g. if they want to pay and things are not moving forward in the queue they always have the chance to leave a queue and continue with another action. They are responding in a flexible way to certain changes in their environment, in this case the length of the queue. Finally, as there is communication between agents and staff, they can also be regarded as being social. 3.2 Empirical Data Often agents are based on analytical models or heuristics and in the absences of adequate empirical data theoretical models are employed. However, for our agents we use frequency distributions for state change delays and probability distributions for decision making processes as statistical distributions are the best format to represent the data we have gathered during our case study due to their numerical nature. The case study was conducted in the Audio and Television (A&TV) and the WomensWear (WW) departments of a leading UK department store. As mentioned earlier we have conducted informal participant observations, staff interviews, and drawn upon a variety of established informational sources internal to the case study organization. Our frequency distributions are modeled as triangular distributions supplying the time that an event lasts, using the minimum, mode, and maximum duration. Our triangular distributions are based on our own observation and expert estimates in the absence of numerical data. We have collected this information from the two branches and calculated an average value for each department type, creating one set of data for A&TV and one set for WW. Table 1 lists some sample frequency distributions that we have used for modeling the A&TV department (the values presented here are slightly amended to comply with con- Customer State Chart Complaining Using Aftersales Enter Seeking help Contemplating passive/active (dummy state) Browsing Leave Queuing for help Being helped Queuing at till Being served (at till) Figure 2: Conceptual model for customer agent Siebers, Aickelin, Celia, & Clegg fidentiality restrictions). The distributions are used as exit rules for most of the states. All remaining exit rules are based on queue development, i.e. the availability of staff. situation leave browse state after … leave help state after … leave pay queue (no patience) after … min 1 3 5 mode 7 15 12 max 15 30 20 Table 1: Sample frequency distribution values The probability distributions are partly based on company data (e.g. conversion rates, i.e. the percentage of customers who buy something) and partly on informed guesses (e.g. patience of customers before they would leave a queue). As before, we have calculated average values for each department type. Some examples for probability distributions we used to model the A&TV department can be found in Table 2. The distributions make up most of the transition rules at the branches where decisions are made with what action to perceive (e.g. decision to seek help). The remaining decisions are based on the state of the environment (e.g. leaving the queue, if the queue does not get shorter quickly enough). event someone makes a purchase after browsing someone requires help someone makes a purchase after getting help probability it occurs 0.37 0.38 0.56 Table 2 – Sample probabilities Company data is available about work team numbers and work team composition, varying opening hours and peak times (to be implemented in future). Also financial data (e.g. transaction numbers and values) are available but have not been used at this stage. 3.3 Implementation Our simulation has been implemented in AnyLogic™ which is a Java™ based multi-paradigm simulation software (XJ Technologies 2007). During the implementation we have used the knowledge, experience and data gained through our case study work. The simulator can represent the following actors: customers, service staff (including cashiers, selling staff of two different training levels) and managers. Figure 3 shows a screenshot of the current customer and staff agent logic as it has been implemented in Figure 3: Customer (left) and staff (right) agent logic implementation in AnyLogic™ Siebers, Aickelin, Celia, & Clegg AnyLogic™. Boxes show customer states, arrows possible transitions and numbers satisfaction weights. Currently there are two different types of customer goals implemented: making a purchase or obtaining a refund. If a refund is granted, the customer’s goal may then change to making a new purchase, or alternatively they will leave the shop straight away. The customer agent template consists of three main blocks which all use a very similar logic. These blocks are ‘Help’, ‘Pay’ and ‘Refund’. In each block, in the first instance, customers will try to obtain service directly and if they cannot obtain it (no suitable staff member available) they will have to queue. They will then either be served as soon as the right staff member becomes available or they will leave the queue if they do not wait any longer (an autonomous decision). A complex queuing system has been implemented to support different queuing rules. In comparison to the customer agent template, the staff agent template is relatively simple. Whenever a customer requests a service and the staff member is available and has the right level of expertise for the task requested, the staff member commences this activity until the customer releases the staff member. While the customer is the active component of the simulation model the staff member is currently passive, simply reacting to requests from the customer. In future we planned to add a more pro-active role for the staff members, e.g. offering services to browsing customers. A service level index is introduced as a new performance measure. The index allows customer service satisfaction to be recorded throughout the simulated lifetime. The idea is that certain situations might have a bigger impact on customer satisfaction than others, and therefore differential weightings are assigned to events to account for this. For example, in our model if a customer starts to wait for a refund and leaves without one, then their satisfaction index decreases by 4 (see figure 3). We measure customer satisfaction in two different ways derived from these weightings; both in terms of how many customers leave the store with a positive service level index value, and the sum of all customers’ service level index values. Applied in conjunction with an ABMS approach, we expect to observe interactions with individual customer differences, variations which have been empirically linked to differences in customer satisfaction. This helps the analyst to find out to what extent customers underwent a positive or negative shopping experience. It also allows the analyst to put emphasis on different operational aspects and try out the impact of different strategies. The simulator can be initialized from an Excel™ spreadsheet and supports the simulation of the two types of departments we looked at during our case study. These differ with respect to their staffing, service provision and customer requirements, which we hope will be reflected in the simulation results. WW customers will ask for help when they know what they want whereas A&TV custom- ers will ask for help when they do not know what they want. WW makes a lot more unassisted sales than A&TV and service times are very different; in WW the average service time is a lot shorter than in A&TV. This service requirement has a differential impact on the profile of employee skills at the department level. 4 A FIRST VALIDATION OF OUR SIMULATOR To test the operation of our simulator and ascertain face validity we have designed and run 2 sets of experiments for both departments. Our case study work has helped us to identify the distinguishing characteristics of the departments, for example different customer arrival rates and different service times. In these experiments we will examine the impact of these individual characteristics on the volume of sales transactions and customer satisfaction indices. All experiments hold the overall number of staffing resources constant at 10 staff and we run the simulation for a period of 10 weeks. We have conducted 20 repetitions for every experimental condition enabling the application of rigorous statistical techniques. Each set of results are analyzed for each dependent variable using a two-way between-groups analysis of variance (ANOVA). Despite our prior knowledge of how the real system operates, we were unable to hypothesize precise differences in variable relationships, instead predicting general patterns of relationships. Indeed, ABMS is a decision-support tool and is only able to inform us about directional changes between variables (actual figures are notional). Where significant ANOVA results were found, post-hoc tests were applied where possible to investigate further the precise impact on outcome variables under different experimental conditions. During our time in the case study organization, we observed that over time the number of cashiers available to serve customers would fluctuate. In the first experiments we vary the staffing arrangement (i.e. the number of cashiers) and examine the impact on the volume of sales transactions and two levels of customer satisfaction; both customer satisfaction (how many customers leave the store with a positive service level index value) and overall satisfaction (the sum of all customers’ service level index values). In reality, we saw that allocating extra cashiers would reduce the shop floor sales team numbers, and therefore the total number of customer-facing staff in each department is kept constant at 10. We therefore predict that for each of our dependent measures: number of sales transactions (1), customer satisfaction index (2) and overall satisfaction index (3): • Ha: An increase in the number of cashiers will be linked to increases in 1, 2 and 3 to a peak level, beyond which 1, 2 and 3 will decrease. • Hb: The peak level of 1, 2 and 3 will occur with a smaller number of cashiers in A&TV than in WW. Siebers, Aickelin, Celia, & Clegg Department Cashiers A&TV WW Total 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Number of Transactions mean std. dev. 4853.50 26.38 9822.20 57.89 14279.90 96.34 14630.60 86.19 13771.85 97.06 8133.75 22.16 15810.10 56.16 25439.60 113.66 30300.70 249.30 28894.25 195.75 6493.63 1661.19 12816.15 3032.61 19859.75 5651.89 22465.65 7937.00 21333.05 7659.04 Customer Satisfaction mean std. dev. 12324.05 77.64 14762.45 81.04 17429.70 103.77 17185.00 99.09 16023.20 82.66 18508.20 88.68 22640.40 92.00 28833.10 115.65 32124.60 230.13 30475.20 176.41 15416.13 3132.55 18701.43 3990.07 23131.40 5775.35 24654.80 7566.98 23249.20 7319.32 Overall Satisfaction mean std. dev. 9366.40 563.88 19985.20 538.30 28994.80 552.60 32573.60 702.64 27916.05 574.56 17327.95 556.03 42339.10 736.61 58601.10 629.68 74233.30 570.79 76838.65 744.31 13347.18 4069.20 31162.15 11337.24 43797.95 15003.13 53403.45 21104.67 52377.35 24781.61 Table 3: Descriptives for first experiments (all to 2 d.p.) An ANOVA was run for each dependent variable, and all revealed statistically significant differences (see Table 3 for descriptive statistics). For 1 and 2, Levene’s test for equality of variances was violated (p<.05) so a more stringent significance level was set (p<.01). For 1 there were significant main effects for both department [F(1, 190) = 356441.1, p<.001] and staffing [F(4, 190) = 124919.5, p<.001], plus a significant interaction effect [F(4, 190) = 20496.37, p<.001]. Tukey’s post hoc tests for the impact of staffing revealed significant differences for every single comparison (p<.001). There is clear support for Hla. We expected this to happen because the number of normal staff available to provide customer advice will eventually reduce to the extent where there will be a detrimental impact on the number of customers making a purchase. Some customers will become impatient waiting increasingly long for service, and will leave the department without making a purchase. Hlb is not supported, the data presents an interesting contrast, in that 1 plateaus in A&TV around 3 and 4 cashiers, whereas WW benefits greatly from the introduction of a fourth cashier. Nonetheless this finding supports the thinking underlying this hypothesis, in that we expected the longer average service times in A&TV to put a greater ‘squeeze’ on customer advice with even a relatively small increase in the number of cashiers. For 2, there were significant main effects for both department [F(1, 190) = 391333.7, p<.001], and staffing [F(4, 190) = 38633.83, p<.001], plus a significant interaction effect [F(4, 190) = 9840.07, p<.001]. Post hoc tests for staffing revealed significant differences for every single comparison (p<.001). The results support both H2a and H2b. We interpret these findings in terms of A&TV’s greater service requirement, combined with the reduced availability of advisory sales staff. These factors result in a peak in purchasing customers’ satisfaction with a smaller number of cashiers (4) than in WW (5). For 3, there were significant main effects for both department [F(1, 190) = 117214.4, p<.001], and staffing [F(4, 190) = 29205.09, p<.001], plus a significant interaction effect [F(4, 190) = 6715.93, p<.001]. Tukey’s post hoc comparisons indicated significant differences between all staffing levels (p<.001). Our results support H3a for A&TV, showing a clear peak in overall satisfaction. H3a is only partially supported for WW, in that no decline in 3 is evident with up to 5 cashiers, although increasing this figure may well expose a peak because the overall satisfaction appears to be starting to plateau out. The results offer firm support in favor of H3b. The second experiment investigates employee empowerment. During our case study we observed the implementation of a new refund policy. This new policy allows cashiers to independently decide whether or not to make a refund up to the value of £50, rather than referring the authorization decision to a section manager. To simulate this practice, we vary the probability that cashiers are empowered to make refund decisions autonomously. We assess its impact in terms of two performance measures: overall customer refund satisfaction and cashier utilization (a proportion of maximum capacity). The staffing arrangement is held constant, consisting of 3 cashiers, 5 normal staff members, 1 expert staff member, and 1 section manager. We hypothesize that: • H4. Higher levels of empowerment will be linked to higher refund satisfaction. • H5. Higher levels of empowerment will be linked to greater cashier utilization. An ANOVA was run for each outcome measure (see Table 4 for descriptives). For refund satisfaction, there were significant main effects for both department [F(1, 190) = 508.73, p<.001], and empowerment [F(4, 190) = 120.46, p<.001], plus a significant interaction effect [F(4, 190) = 29.81, p<.001]. Tukey’s post hoc tests for the impact of empowerment revealed significant differences between all comparisons (p<.001), except for .00 with .75, and .25 with .50, where there were no significant differences. Siebers, Aickelin, Celia, & Clegg Department A&TV WW Total Empowerment 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Overall refund mean std. dev. 3130.70 242.58 3880.70 225.70 3876.50 181.47 3716.80 225.31 2991.60 245.15 3090.30 222.00 3116.00 266.70 3041.20 211.75 2716.80 217.79 2085.20 168.19 3110.50 230.43 3498.35 457.61 3458.85 465.61 3216.80 551.59 2538.40 503.70 Cashier utilization mean std. dev. 0.6286 0.00 0.6392 0.00 0.6488 0.00 0.6571 0.00 0.6623 0.00 0.6756 0.00 0.6737 0.00 0.6736 0.00 0.6722 0.00 0.6720 0.00 0.6521 0.02 0.6565 0.02 0.6612 0.01 0.6646 0.01 0.6672 0.01 Table 4: Descriptives for the third experiment (all to 2 d.p., except cashier utilization to 4 d.p.) The data provides support for H4 between .00 and .25 levels of empowerment. However, as empowerment increases all of the results do not support our hypothesis, and demonstrate a counterintuitive progressive decline of refund satisfaction beyond the .25 level. Both departments display the curvilinear relationship between these two variables; refund satisfaction peaks at a middling level of empowerment (.50 for A&TV, .25 for WW). These results suggest that some constraining factors are occurring at the higher levels of empowerment. This may be linked to the empowered employees adhering to a stricter refund policy (resulting in less customer satisfaction), or the empowered employees taking longer to process the transaction. For cashier utilization, there were significant main effects for both department [F(1, 190) = 2913.45, p<.001], and empowerment [F(4, 190) = 126.37, p<.001], plus a significant interaction effect [F(4, 190) = 190.64, p<.001]. Tukey’s post hoc tests for the impact of cashier utilization confirmed significant differences between all comparisons (p=.01 between .75 and 1.00, p<.001 for all others). The results support H5 for A&TV, but not for WW. In WW, empowerment is significantly inversely related to till utilization. Case study observations indicated that A&TV cashiers, like A&TV sales staff, when they have a higher level of empowerment they are motivated to work more efficiently. Empirical evidence indicated that WW cashiers may not be under the same time pressures to work more quickly, however this data goes one step further and suggests an inverse relationship. 5 CONCLUSIONS AND FUTURE DIRECTIONS In this paper we present the conceptual design, implementation and operation of a retail branch simulator used to understand the impact of management practices on retail productivity. As far as we are aware this is the first time researchers have tried to use agent-based approaches to simulate management practices such as training and empowerment. Although our simulator uses specific case studies as source of information, we believe that the general model could be adapted to other retail companies and areas of management practices that have a lot of human interaction. From what we can conclude from our current analyses, some findings are as hypothesized whereas others are more mixed. Further experimentation is required to enable rigorous statistical evolution of the outcome data and identification of statistically significant differences. Currently we are developing our agents with the intention of enhancing their intelligence and heterogeneity. For this purpose we are introducing evolution and stereotypes. The most interesting system outcomes evolve over time and many of the goals of the retail company (e.g. service standards) are also planned long term. We are introducing an evolution of entities over time, including product knowledge for staff. Moreover, the customer population pool will be fixed to monitor customer agents over time. This allows us to consider shopping experience based on long-term satisfaction scores, with the overall effect being a certain ‘reputation’ for the shop. Another interesting aspect we are currently implementing is the introduction of stereotypes. Our case study organization has identified its particular customer stereotypes through market research. It will be interesting to find out how populations of certain customer types influence sales. Overall, we believe that researchers should become more involved in this multi-disciplinary kind of work to gain new insights into the behavior of organizations. In our view, the main benefit from adopting this approach is the improved understanding of and debate about a problem domain. The very nature of the methods involved forces researchers to be explicit about the rules underlying behavior and to think in new ways about them. As a result, we have brought work psychology and agent-based modeling closer together to form a new and exciting research area. REFERENCES Csik, B. 2003. Simulation of competitive market situations using intelligent agents. Periodica Polytechnica Ser. Soc. Man. Sci. 11:83-93. Epstein, J. M., and R. Axtell. 1996. Growing artificial societies: Social Science from the bottom up. Cambridge, MA: MIT Press. Gilbert, N., and K. G. Troitzsch. 2005. Simulation for the social scientist. 2nd ed. Milton Keynes, UK: Open University Press. Greasley, A. 2005. Using DEA and simulation in guiding operating units to improved performance. Journal of the Operational Research Society. 56:727-731. Siebers, Aickelin, Celia, & Clegg Hillier, F. S., and G. J. Lieberman. 2005. Introduction to Operations Research. 8th ed. Boston, MA: McGrawHill Higher Education. Hood, L. 1998. Agent-based modeling. In Conference Proceedings: Greenhouse Beyond Kyoto, Issues, Opportunities and Challenges. Canberra, Australia. Keh, H. T., S. Chu, and J. Xu. 2006. Efficiency, effectiveness and productivity of marketing in services. European Journal of Operational Research. 170:265276. Luck, M., P. McBurney, O. Shehory, and S. Willmott. 2005. Agent technology: computing as interaction (a roadmap for agent based computing). Liverpool, UK: AgentLink. Macal, C. M., and M. J. North. 2006. Tutorial on agentbased modeling and simulation part 2: how to model with agents. In Proceedings of the 37th Winter Simulation Conference, eds. L. F. Perrone, B. G. Lawson, J. Liu, and F. P. Wieland. Monterey, CA. Padgham, L., and M. Winikoff. 2004. Developing intelligent agent systems - a practical guide. New York, NY: Wiley. Pourdehnad, J., K. Maani, H. Sedehi. 2002. System dynamics and intelligent agent-based simulation: where is the synergy? In Proceedings of the 20th International Conference of the System Dynamics Society. Palermo, Italy. Reynolds, J., E. Howard, D. Dragun, B. Rosewell, and P. Ormerod. 2005. Assessing the productivity of the UK retail sector. International Review of Retail, Distribution and Consumer Research. 15:237-280. Schwaiger, A. and B. Stahmer. 2003. SimMarket: multiagent based customer simulation and decision support for category management. In Lecture Notes in Artificial Intelligence (LNAI) 2831, eds. M. Schillo, M. Klusch, J. Muller, and H. Tianfield. 74-84. Berlin: Springer. Shannon, R. E. 1975. Systems simulation: the art and science. Englewood Cliffs, NJ: Prentice-Hall. Siebers, P.-O. 2006 Worker performance modeling in manufacturing systems simulation: proposal for an agent-based approach. In Handbook of Research on Nature Inspired Computing for Economics and Management, ed. J. P. Rennard. Hershey, PA: Idea Group Publishing Siebers, P.-O., U. Aickelin, G. Battisti, H. Celia, C. W. Clegg, X. Fu, R. De Hoyos, A. Iona, A. Petrescu, and A. Peixoto. Forthcoming. The role of management practices in closing the productivity gap. Submitted to International Journal of Management Reviews. Simon, H. A. 1996. The sciences of the artificial. 3rd ed. Cambridge, MA: MIT Press. Twomey, P., and R. Cadman. 2002. Agent-based modeling of customer behavior in the telecoms and media markets. Info - The Journal of Policy, Regulation and Strategy for Telecommunications. 4:56-63. Wall, T. D., and S. J. Wood. 2005. Romance of human resource management and business performance, and the case for big science. Human Relations. 58:429462. Wooldridge, M. J. 2002. An introduction to multiagent systems. New York, NY: Wiley. XJ Technologies. XJ Technologies - simulation software and services. Available via <www.xjtek.com> [accessed April 1, 2007]. AUTHOR BIOGRAPHIES PEER-OLAF SIEBERS is a Research Fellow in the School of Computer Science and IT at the University of Nottingham. His main research interest is the application of computer simulation to study human oriented complex adaptive systems. Complementary fields of interest include distributed artificial intelligence, biologically inspired computing, game character behavior modeling, and agent-based robotics. His webpage can be found via <www.cs.nott.ac.uk/~pos>. UWE AICKELIN is a Reader and Advanced EPSRC Research Fellow in the School of Computer Science and IT at the University of Nottingham. His research interests are mathematical modeling, agent-based simulation, heuristic optimization and artificial immune systems. see his webpage for more details <www.aickelin.com>. HELEN CELIA is a Researcher at the Centre for Organisational Strategy, Learning and Change at Leeds University Business School. She is interested in developing ways of applying work psychology to better inform the modeling of complex systems using agents. For more information visit <www.leeds.ac.uk/lubs/coslac> CHRIS W. CLEGG is a Professor of Organizational Psychology at Leeds University Business School. And the Deputy Director of the Centre for Organisational Strategy, Learning and Change. His research interests include: new technology, systems design, information and control systems, socio-technical thinking and practice; organizational change, change management, technological change; the use and effectiveness of modern management practices, innovation, productivity; new ways of working, job design, work organization. An emerging research interest is modeling and simulation. His webpage can be found via: <www.leeds.ac.uk/lubs/coslac> 

Graph partitioning and a componentwise PageRank algorithm arXiv:1609.09068v1 [] 28 Sep 2016 Christopher Engström, Division of Applied Mathematics Education, Culture and Communication (UKK), Mälardalen University, christopher.engstrom@mdh.se Sergei Silvestrov Division of Applied Mathematics Education, Culture and Communication (UKK), Mälardalen University sergei.silvestrov@mdh.se January 13, 2018 Abstract In this article we will present a graph partitioning algorithm which partitions a graph into two different types of components: the well-known ‘strongly connected components’ as well as another type of components we call ‘connected acyclic component’. We will give an algorithm based on Tarjan’s algorithm for finding strongly connected components used to find such a partitioning. We will also show that the partitioning given by the algorithm is unique and that the underlying graph can be represented as a directed acyclic graph (similar to a pure strongly connected component partitioning). In the second part we will show how such an partitioning of a graph can be used to calculate PageRank of a graph effectively by calculating PageRank for different components on the same ‘level’ in parallel as well as allowing for the use of different types of PageRank algorithms for different types of components. To evaluate the method we have calculated PageRank on four large example graphs and compared it with a basic approach, as well as our algorithm in a serial as well as parallel implementation. Contents 1 Introduction 3 2 Notation and Abbreviations 3 3 Graph concepts 3.1 PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 4 Method 9 4.1 Component finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Intermediate step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 PageRank step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1 4.4 4.3.1 CAC-PageRank algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3.2 SCC-PageRank algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 15 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Experiments 5.1 Graph description and generation . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Barabási-Albert graph generation . . . . . . . . . . . . . . . . . . . . . 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 17 17 18 6 Conclusions 23 7 Future work 23 2 1 Introduction The PageRank algorithm was initially used by S. Brinn and L. Page to rank homepages on the Internet [7]. While the original algorithm itself is already very efficient, given the sheer size and rate of growth of many real life networks there is a need for even faster methods. Much is known of the original method using a power iteration of a modified adjacency matrix such as how the damping factor c affect the condition number and convergence speed [11,13]. Many ways have been proposed in order to improve the method such as aggregating webpages that are in some way ‘close’ [12] or by excluding webpages whose rank are found to already have converged from the iteration procedure [14]. Another method to speed up the algorithm is to remove so called dangling pages (pages with no links to any other page), and then calculate their rank at the end separately [2, 16]. A similar method can also be used for root vertices (pages with no links from any other pages) [19]. A good overview over different methods for calculating PageRank can be found in [6]. The method we propose here have some similarities with the method proposed by [19], the main difference being that we work on the level of graph components rather than single vertices. Another method with a similar idea is the one proposed by [3], but we use two types of components as well as look at using different PageRank algorithms for different types of components and how different components can be calculated in parallell. We presented some parts of this work at ASMDA 2015 [8] while in this paper we improve the method further as well as implementing and presenting some results using the method. The rest of this paper is organized as follows: In Sec. 3 we define some graph concepts as well as define and prove a couple of properties of the graph partitioning which will be used later in our PageRank algorithm. In Sec. 3.1 we define the variation of PageRank that will be used as well as show how PageRank can be computed for different types of vertices or graph components. In Sec. 4 we describe our proposed algorithm to calculate PageRank, first by describing how to efficiently find the graph partitioning described earlier and next how to use this to calculate PageRank. In the same section we also verify the linear computational complexity of the algorithm as well as give some error estimates in Sec. 4.4. At the end of our paper in Sec. 5 we describe our implementation of the method as well as show some results using the method on a couple of different graphs. 2 Notation and Abbreviations The following abbreviations will be used throughout the article • SCC - strongly connected component. (see Def. 3.1) • CAC - connected acyclic component. (see Def. 3.2) • DAG - directed acyclic graph. • BFS - breadth first search. • DFS - depth first search. Explanation of notation. • We denote by G a graph with with vertex set V and edge set E, by |V | the number of vertices and |E| the number of edges. ~, ~ • Vectors are denoted by an arrow for example V u and matrices in bold capital letters such as A, M. • Each vertex in a graph has an assigned level L (see Def. 3.3), L+ denotes all vertices of level L or greater while L− denotes all vertices of level L or less. For example if A is the adjacency matrix of some graph, then AL+ ,(L−1)− denotes the submatrix of A corresponding to all rows representing vertices of level L or greater and all columns representing vertices of level L − 1 or less. 3 • Different versions of PageRank will be defined where the type is denoted inside paren~ (1) (see Def. 3.4 and Def. 3.5). thesis, for example R • Lowercase letters as subscripts denote single elements while capital letters denote a set ~ corresponding to vertex vi and of elements for example wi denoting the element of W ~ L denoting the vector of elements corresponding to all vertices of level L. W • P (vi → vj ) denotes the probability to hit vertex vj in a random walk starting at vi (see Def. 3.6). 3 Graph concepts In this work we assume all graphs to be simple directed graphs. Since only the adjacency matrix of any graph will be used, all graphs are also assumed to be unweighted. It should be noted however that it would be fairly simple to generalize for certain types of weighted graphs (such as those representing a Markov chain). We will start by defining two types of graph components, one being the well-known ‘strongly connected component’ and another we call ‘connected acyclic component’. We also define the notion of ‘level’ of a component representing the depth of the component in the underlying graph where components are represented by a single vertices. Definition 3.1. A strongly connected component (SCC) of a directed graph G is a subgraph S of G such that for every pair of vertices u, v in S there is a directed path from u to v and from v to u. In addition S is maximal in the sense that adding any other set of vertices and/or edges from G to S would break this property. Definition 3.2. A connected acyclic component (CAC) of a directed graph G is a subgraph S of G such that no vertex in S is part of any non-loop cycle in G and the underlying graph is connected. Additionally any edge in G that exists between any two vertices in S is also a part of S. A vertex in the CAC with no edge to any other edge in the CAC we call a leaf of the CAC. CACs can be seen as a connected collection of 1-vertex SCCs forming a tree. While CACs keep the property that all internal edges between vertices in the component are preserved from those in the original graph, it is not maximal in the sense that no more vertices could be added to the component as is the case for SCCs. The reason for this is that we want to be able to create a graph partitioning into components in which the underlying graph is a directed acyclic graph (DAG) in the same way as for the ordinary partition into SCCs. Definition 3.3. Consider a graph G with partition P into SCCs and CACs such that each vertex is part of exactly one component and the underlying graph created by replacing every component with a single vertex. If there is an edge between any two vertices between a pair of components then there is an edge in the same direction between the two vertices representing those two components as well. Consider the case where the underlying graph is a DAG (such as for the commonly known partitioning of a graph into SCCs). • The level LC of component C is equal to the length of the longest path in the underlying DAG starting in C. • The level Lvi of some vertex vi is defined as the level of the component for which vi belongs (Lvi ≡ LC , if vi ∈ C). We note that a SCC made up of only a single vertex is also a CAC, in our work it will be easier to consider these components CACs rather than SCCs. We also note that while a single vertex can only be part of a single SCC, it could be part of multiple CACs of different size. There is however a unique graph partitioning into SCCs and CACs as seen below. Theorem 3.1. Consider a directed graph G with a partition into SCCs. Let the underlying graph be the DAG constructed by replacing every component with a single vertex. If there is 4 an edge between any two vertices between a pair of components then there is an edge in the same direction between corresponding vertices in the underlying DAG. To each vertex in the underlying DAG we attach a level equal to the longest existing path from this vertex to any other vertex in the underlying graph. Next we start to merge SCCs consisting of a single vertex into CACs under the following conditions: • We start merging from the lowest level (vertices in the DAG with no edge to any other vertex in the DAG) and only start merging on the next level when we cannot merge any more components on the current level. • All merges are done by merging a single ‘head’ 1-vertex CAC of level L containing vertex v with all CACs of level L − 1 to which there is an edge from v. Unless v have an edge to at least one SCC (of more than 1 vertex) of level L − 1 in which case no merge is made. If a merge takes place, then the level of the new merged CAC is L − 1. Then the following holds: 1. This gives a unique partitioning of the graph into SCCs and CACs and does not depend on the order in which we apply merges of ‘head’ components on the same level. 2. This partition of SCCs and CACs can also be seen as a DAG where we attach a level to each vertex equal to the longest existing path from this vertex to any other vertex in the DAG. Remark Note that after a merge some vertices with a level higher than the one where the merge was made might get a lower level compared to before. Proof. That a directed graph can be partitioned into SCCs is a well-known and easy to show result from graph theory. Applying a level to the vertices in the DAG is nothing else than a topological ordering of the vertices in the graph, something also well-known, hence we start at the merging. Obviously all 1-vertex SCCs are also 1-vertex CACs since any vertex that is part of any (non-loop) cycle must be part of a SCC of more than one vertex. Since the head CAC of a merge is always connected with each other CAC that is part of a merge, the subgraph representing the component is connected as well. It is also still obviously acyclic since no vertex of any of the CACs is part of any cycle in G from the definition of a CAC. Adding all edges between the head and all other merged CACs also ensures that there is no missing edge between any two vertices of the new CAC. This holds since there can be no edges between any two components on the same level. Hence we can conclude that merging CACs creates a new CAC. Next we prove statement 1) that the given partitioning is unique and does not depend on the order of merges. CACs are created using a bottom-up approach and it is clear that the level of a CAC never change after its first merge. This means that the level of a CAC is uniquely defined by the level of any leaf in the CAC. All the leaves of a CAC are those 1-vertex CACs which could not be the head of any merge either because they have either no outgoing edges orthey have at least one outgoing edge to a SCC (of more than 1 vertex) of the next lower level. Assume we have done all merges with head CACs of level L. Consider a 1-vertex CAC with vertex v and level L + 1 and edges to one or more CACs but no SCC of more than one vertex of level L (or higher). From the previous argument we get that the level of any CAC linked to by v will not change from any future merges. This means that eventually v will be part of the same CAC as all vertices part of any CAC with level L to which there is an edge from v regardless of which order merges are made on level L + 1. Repeating this argument for all 1-vertex CACs of level L + 1 we get that for each such vertex v the neighboring vertices part of a CAC with a lower level for which v should be part of the same CAC is uniquely determined after all the merges on level L. Repeating this for all levels gives us a unique partitioning. This proves statement 1). Last we show statement 2) by showing that merging of components does not create any cycle in the underlying DAG created by the components. 5 Consider a merge of head CAC with vertex v, level L and an edge to each of n CACs C1 , C2 , . . . , Cn with level L − 1. Since all CACs C1 , C2 , . . . , Cn have the same level, the resulting CAC after merge can only have edges to components of level at most L − 2 since we merge it with all CACs of level L − 1 to which there is an edge from v, but do not merge if there is an edge from v to any SCC of level L − 1 and there can be no edges between any components of the same level from the initial SCC partitioning. Since initially there can be no edges from any component to another of the same or larger level and we do not create any such edges when doing a merge, we do not create any cycle in the underlying DAG. Since we only merge CACs, and merges does not create any cycles in the DAG, we have proved that the new partitioning can also be represented by a DAG where each vertex corresponds to one SCC or CAC. This proves statement 2). Corollary 3.1. If a directed graph G has a SCC partitioning with maximum level LSCC and a SCC/CAC partitioning with maximum level LSCC/CAC , then LSCC/CAC ≤ LSCC Proof. Every merge lowers the level of the ‘head’ vertex and possibly one or more components of a higher level, this makes it easy to find a graph such that LSCC/CAC < LSCC such as the 2-vertex graph with a single edge between them in one direction. Similarly we can easily find a graph for which equality holds (such as the single vertex graph). However, since doing a merge can never result in any vertex getting a higher level, LSCC/CAC ≤ LSCC . For algorithms (such as PageRank) which can be done on the components of one level at a time in parallel the SCC/CAC partitioning have the advantage over the usual SCC partitioning in that it generally creates a lower number of levels and thus a larger amount of vertices (but not necessary components) on the same level in the new partitioning on average. In effect reducing the chance of bottlenecks where we have a level with only a single or a few small components. The only components that get larger are the CACs which are acyclic apart from any loops and thus often have specialized faster methods (for example by exploiting the triangular adjacency matrix). thus increasing the size of these components is often not as much of an issue and might in fact often be beneficial instead by reducing overhead. An example of a directed acyclic graph and its SCC/CAC partitioning can be seen in Fig. 1. From this figure it is clear why we cannot merge when the ‘head’ have an edge Level 2 3 Level 1 2 1 1 Level 0 1 0 0 0 Level 0 Figure 1: Example of a graph and corresponding components from SCC/CAC partitioning of the graph (2 SCCs, 1 CAC and 1 1-vertex component). Vertex labels denote the level of each vertex if we had only partitioned the graph into SCCs (for the SCC/CAC partitioning the vertex-levels is the same as the level of corresponding component in the figure). 6 to any SCC of the next lower level. If the top (level 2) component merged with the left (level 0) component then this would have created a cycle in the underlying graph. It is also possible to see how merging some components can result in the partitioning getting a lower max-level, the SCC/CAC partitioning have only 3 levels while the SCC partitioning would need 4 levels. 3.1 PageRank PageRank was originally defined by S. Brin and L. Page as the eigenvector to the dominant eigenvalue of a modified version of the adjacency matrix of a graph [7]. ~ (1) for vertices in graph G consisting of |V | vertices is defined Definition 3.4. PageRank R as the (right) eigenvector with eigenvalue one to the matrix: M = c(A + ~g w ~ > )> + (1 − c)w~ ~ e> (1) where A is the adjacency matrix weighted such that the sum over every non-zero row is equal to one (size |V | × |V |), ~g is a |V | × 1 vector with zeros for vertices with outgoing edges and 1 for all vertices with no outgoing edges, w ~ is a |V | × 1 non-negative vector with ||w|| ~ 1 = 1, ~e is a one-vector with size |V | × 1 and 0 < c < 1 is a scalar. The original normalized version of PageRank has the disadvantage in that it is harder to compare PageRank between graphs or components, because of that we use a non-normalized version of PageRank as described in for example [9]. ~ (3) for graph G is defined as Definition 3.5. ( [9]) PageRank R X > (1) ~ (1) ~ ~ (3) = R ||W ||1 , d = 1 − ~ R cA R d (2) ~ is a non-negative weight vector such that W ~ ∝ w. where W ~ In the same work [9] we also showed that it is possible to give another equivalent definition of this non-normalized version of PageRank which will be useful later in some proofs. Definition 3.6. ( [9]) Consider a random walk on a graph G = {V, E} described by A. In each step of the random walk move to a new vertex from the current vertex by traversing a random edge from the current vertex with probability 0 < c < 1 and stop the random walk ~ (3) for a single vertex vj can be written as with probability 1 − c. Then PageRank R  (3) Rj = Wj +  X Wi P (vi → vj ) vi ∈V,vi 6=vj ∞ X ! (P (vj → vj ))k (3) k=0 where P (vi → vj ) is the probability to hit vertex vj in a random walk starting at vertex vi . This can be seen as the expected number of visits to vj if we do multiple random walks, ~. starting at every vertex a number of times described by W In [8] we showed how to calculate PageRank for the five different types of vertices defined below Definition 3.7. For the vertices of a simple directed graph we can define 5 distinct groups G1 , G2 , . . . , G5 1. G1 : Vertices with no outgoing or incoming edges. 2. G2 : Vertices with no outgoing edges and at least one incoming edge (also called dangling vertices). 3. G3 : Vertices with at least one outgoing edge, but no incoming edges (also called root vertices). 7 4. G4 : Vertices with at least one outgoing and incoming edge, but which is not part of any (non-loop) directed cycle (no path from the vertex back to itself apart from the possibility of a loop). 5. G5 : Vertices that is part of at least one non-loop directed cycle. Qing Yu et al gave a similar but slightly different definition of 5 (non distinct) groups for vertices, namely dangling and root vertices (G2 and G3 ), vertices that can be made into dangling or root vertices by recursively removing dangling or root vertices (part of G4 ) and remaining vertices (part of G4 and G5 ) [19]. Given PageRank of a vertex not part of a cycle (group 1-4), then the PageRank of other vertices can be calculated by removing the vertex and modifying the initial weight of other vertices. ~ g(3) of vertex vg where vg is not part of any non-loop cycle, Theorem 3.2. Given PageRank R the PageRank of another vertex vi from which there exist no path to vg can be expressed as   (3) Ri ∞ X  X  (3) (3)   (P (vi → vi ))k (Wj + Rg cagj )P (vj → vi ) = Wi + Rg cagi + ! (4) k=0 vj ∈V vj 6=vi ,vg where cagi is the one-step probability to go from vg to vi . The proof with minor modifications is similar to the one found in [8] where it is formulated for vertices in G3 on graphs with no loops. Proof. Consider Rg(3) from Definition. 3.6. Since we know that there is no path from vi back to vg (or vg would be part of a non-loop cycle) we know that the right hand side will be identical for all other vertices. We rewrite the influence of vg using X Rg(3) P (vg → vi ) = Rg(3) cagi + Rg(3) cagj P (vj → vi ) . (5) vj ∈V vj 6=vi ,vg We can now rewrite the left sum in Definition. 3.6: X Wi P (vi → vj ) = Rg(3) cagi + vi ∈V,vi 6=vj X (Wj + Rg(3) cagj )P (vj → vi ) (6) vj ∈V vj 6=vi ,vg which when substituted into (3) proves the theorem. It is also easy to show that any SCC can also be divided into one of the first four groups if we consider each SCC as a vertex in the underlying DAG (a SCC can never be part of a cycle). The important part of this is that it is also possible to calculate PageRank one component at a time rather than for the whole graph at once. ~ (3)+ be PageRank of all vertices belonging to components of level L Corollary 3.2. Let R L or greater. Then PageRank of a vertex vi belonging to a component of level L − 1 can be computed by (3) Ri = ∞ X ! k (P (vi → vi )) k=0        >  > X   ~ (3)+ ~ (3)+  Wi + c R ~aL+ ,i + Wj + c R ~aL+ ,j P (vj → vi ) L L   vj ∈V vj 6=vi where ~aL+ ,i is a vector containing all 1-step probabilities from vertices of level L or greater to vertex vi . 8 Proof. Follows immediately from Theorem 3.2 by replacing the rank of a single vertex with the sum of rank of all vertices belonging to components of a higher level. Those in lower level components or other components on the same level does not affect the rank since they automatically does not have any path to vi . Using Corollary 3.2 it is clear that after calculating PageRank of all vertices belonging to components of level L and above we can calculate those of level L − 1 by first changing their initial weight and then consider the component by itself. In matrix notation we can update the weight vector for all components of lower level by calculating new old ~ L−1 ~ L−1 ~ (3)+ W =W + cAL+ ,L−1 R L (7) where AL+ ,L−1 corresponds to the submatrix of A with all rows corresponding to vertices of level L or greater and all columns of level L − 1. This is essentially the same method which is used in [16] but here we have formulated it for any component instead of for dangling vertices (vertices with no outgoing edges). 4 Method The complete PageRank algorithm can be described in three main steps. 1. Component finding: Finding the SCC/CAC partitioning of the graph. 2. Intermediate step: Create relevant component matrices and weight vectors. 3. PageRank step: Calculate PageRank one level at a time and components on the same level one at a time or in parallel. In order to be able calculate PageRank for each component we obviously first need to find the components themselves, this is done in the component finding part of the algorithm where we find a SCC/CAC partitioning of the graph as well as the level of each component. By using the CAC/SCC partitioning rather than the usual SCC partitioning we reduce the risk of having very few vertices on the same level, the aim of this is to be able to avoid some of the disadvantages with some other similar methods such as the one in [15] where a large number of small levels (small diagonal blocks) increases the overhead cost [19]. This step is similar to the initial matrix reordering made by [3]. However instead of only finding a partial ordering we have modified the depth first search slightly in order to identify components that can be calculated in parallell as well as group 1-vertex components on the same level together. Another advantage is that different methods can be used for different types of components as we will see later. The component finding step is described in Sec. 4.1. In the intermediate step the data (edge list, vertex weights) need to be managed such that the individual matrices for every component can quickly and easily be extracted. This section of the code can vary a lot between implementations and is one of the main contributors of overhead in the algorithm. The SCC/CAC partitioning can easily be transformed into a permutation matrix and used to permute the graph matrix and then solve the resulting linear system, this can be seen as an alternative to the recursive reordering algorithm described in [15]. This step is described in Sec. 4.2. After the intermediate step we are ready to start calculating PageRank of the vertices. This is done one level at a time starting with the highest and modifying vertex weights between levels using (7). Components on the same level can use different methods to calculate PageRank and can either be calculated sequentially or in parallel. The PageRank step is described in Sec. 4.3. 4.1 Component finding The component finding part of the algorithm consists of finding a SCC/CAC partitioning of the graph as well as the corresponding levels of the components. Since any loops in the 9 graph have no effect on which SCC or CAC a vertex is part of, these will be ignored in the component finding step. Finding the components and their level can be done through a modified version of Tarjan’s well-known SCC finding algorithm using a depth first search [18]. For every vertex v we assign six values: • v.index containing the order in which it was discovered in the depth first search. • v.lowlink for a SCC representing the lowest index of any vertex we can reach from v, or for a CAC representing the ‘head’ vertex of corresponding component. • v.comp representing the component the vertex is part of, assigned at the end of the component finding step. • v.depth used to implement efficient merges of components. It can be removed if the extra memory is needed, but it could result in slowdown because of merges for some graphs. • v.type indicates if v is part of a SCC or a CAC (1-vertex SCCs are considered CACs). • v.level indicating the level of the component to which v belongs. Of these the first three can be seen in Tarjan’s algorithm as well, and play virtually the same role here (although in Tarjan’s the comp value can be assigned as components are created). During the depth first search each vertex v goes through three steps in order. 1. Discover: Initialize values for the vertex. 2. Explore: Visit all neighbors of v, finishing the DFS of any unvisited neighbors before going to the next. After a vertex is visited we update v.lowlink and v.level. 3. Finish: After all neighbors are visited we create a new component if appropriate. If a CAC is created we also check for and do any merge with v as head. During the discover step values are initialized after which the vertex is put on the stack (.type and .comp do not need to be initialized) D i s c o v e r ( Vertex v ) v . i n d e x := i n d e x v . l o w l i n k := i n d e x v . l e v e l := 1 v . depth := 1 i n d e x++ s t a c k . push ( v ) // add v t o t h e s t a c k end Here index is a counter starting at 1 and increasing by one for every new vertex we discover. The explore step as well works much like Tarjan’s depth first search except that it also updates the level of the vertex we are exploring. E x p l o r e ( Vertex v ) f o r each ( v , w) ∈ E i f w i s not i n i t i a l i z e d DFS(w) // D i s c o v e r (w) , E x p l o r e (w) and F i n i s h (w) end /∗ At t h i s p o i n t w i s e i t h e r i n a component a l r e a d y or b e l o n g t o t h e same SCC as v ∗/ i f w i s i n a component ( . type i s d e f i n e d ) v . l e v e l = max( v . l e v e l , w . l e v e l +1) e l s e //w b e l o n g t o t h e same SCC v . l e v e l = max( v . l e v e l , w . l e v e l ) v . l o w l i n k = min ( v . l o w l i n k , w . l o w l i n k ) end 10 end end The last step in the DFS is where we evaluate if a new component should be created and handle any merges needed with this vertex as ‘head’ component. The initial component is created in the same way as in Tarjan’s algoriithm: if v.lowlink = v.index we create a SCC by popping vertices from the stack until we pop v from the stack. F i n i s h ( Vertex v ) i f v . lowlink = v . index size = 0; do w := s t a c k . pop ( ) w. l o w l i n k = v w. l e v e l = v . l e v e l w . type = s c c merge (w, v ) // add w t o component s i z e++ while (w != v ) i f s i z e = 1 // ( one v e r t e x component ) v . type = c a c // c h e c k f o r merges f o r each ( v , w) ∈ E m = false list = [] i f w . type = s c c and w . l e v e l = v . l e v e l −1 m = false break end i f w . type = c a c and w . l e v e l = v . l e v e l −1 l i s t . add (w) m = true end end i f m = true // ( a d j u s t . l e v e l i f a merge o c c u r e d ) v . l e v e l = v . l e v e l −1 f o r each w i n l i s t merge (w, v ) end end end end end The first part is similar to Tarjan’s with some extra bookkeeping so we will focus our explanation of the second part. However in order to explain how the merges are done in constant time we start by explaining how we store the component data. The components are stored in a merge-find data structure through the .lowlink and .depth attribute. A mergefind data structure allows us to do the two operations we need: merging two components and finding a ‘head’ vertex representing a component (used in merge, and by itself later). This has the advantage that both operations can be done in constant amortized time (O(α(|V |))) as well as requiring very little memory. In Tarjan’s algorithm you don’t need the .depth values since the .comp value can be assigned while creating a component. The reason we do not do it here is because it cannot 11 be updated when doing a merge of two CACs (unless we loop through all vertices), and even if we could, it would be possible to end up with some empty components that would have to be ’cleaned up’ in one way or another later anyway. Looking at the computational complexity of the component finding step we see that discover, explore and finish are all done once for every vertex, of these discover is obviously done in constant time. During explore we will eventually have to go through all edges exactly once but all operations take only constant time. Last finish is called once for every vertex doing O(α(|V |)) work in the first half (amortized constant because we do merges rather than assigning component values directly). In the second half we will at most visit every edge once (over all vertices) doing O(α(|V |)) work. Thus in total we end up with O(|V | + |E| + |V |α(|V |) + |E|α(|V |)) ≈ O(|E|α(|V |)), if |E| > |V | , in other words linear amortized time in the number of edges. Before returning the results |V | find operations also need to be done in order to assign the .comp value for each vertex, since the find operation takes O(α(|V |)) time this takes O(|V |α(|V |)) time in total. f o r each v ∈ V h := f i n d ( v ) // f i n d head v e r t e x i n d := 1 i f h . comp i s not d e f i n e d h . comp := i n d i n d++ end v . comp := h . comp end Hence the complete component finding algorithm takes O(|E|α(n)) time which is comparable to Tarjan’s which can be implemented in O(|E|) time. We note that if the .depth value is ignored everything works but the merges are no longer guaranteed to be made in constant amortized time. If memory is a concern or if the size of the CACs are assumed to be small it might be worthwile to work without the .depth value even though merges could be slow in the worst case scenario. 4.2 Intermediate step After we have found the SCC/CAC partitioning some additional work need to be done in order to continue with the PageRank calculations effectively. The goal of the preprocessing step is to make sure that we easily and quickly can construct corresponding matrix for each component as well as sort the components. We sorted components first in order of level and second in the size of the component (both descending order) so that we can work on one component at a time starting with the largest on every level. We note that the preprocessing step can differ highly depending on implementation, hence we will only give a short comment on how we choose to do it. The goal is to have separate edge lists for edges within each component as well as edges between levels. To get this we start by sorting the first according to their level and second according to their size. This is then used to permute the edge list such that they are ordered in the same way depending on their starting vertex. It is important to note that this sorting of components or edges can both be done in linear time rather than O(n log n) as with ordinary sort. This is true since the possible values are known and bounded hence no comparisons need to be made. After the edges are sorted we go through the edge list and assign each edge to the correct component or in between level list such that corresponding component matrices can 12 be created. In this step we also merge all 1-vertex components of each level to avoid having to calculate the rank of multiple 1-vertex components of the same level. After the preprocessing step we should have the matrices (or edge lists) Mc and weight vector Vc for all components stored such that we can access them when needed. It is worth to note that in our implementation this section of the code contains much of the extra overhead needed for our method (more than the previous component finding step itself). 4.3 PageRank step Now that all preliminary work is done we can start the actual PageRank calculation where we calculate PageRank for all vertices of one level at a time (starting by the largest). The PageRank step can be described by the following steps. 1. Initiate L to the maximum level among all components. 2. For each component of level L: pick a suitable method and calculate PageRank of the component. 3. Update weight vector V for all remaining components (of lower level). 4. Decrease L by one and go to step 2 unless we have already calculating PageRank of all vertices. Depending on the type and size of the component PageRank we calculate PageRank in one of four different ways: • Component is made up of a collection of single vertex components (no internal edges apart from loops): PageRank of any such collection of 1-vertex components is simply wi the initial weight wi for vertices with no loop and for any vertex with a loop, 1 − caii where aii is the weight on the loop edge. • Component is a CAC of more than one vertex: use CAC-PageRank algorithm 4.3.1 described later. • Component is a SCC but small (for example less than 100 vertices): Calculate PageR~ =W ~ (using for example ank by directly solving the linear equation system (I − cA> )R LU factorization). • Component is a SCC and large: Use iterative method, in our case using a power series formulation. Out of these the first one is done in O(|V |) time (copy the weight vector) or O(1) if no separate vector is used for the resulting PageRank and we assume there is no loops, the second and fourth is done in O(|E|), however the coefficient in front of the first is much lower since it is guaranteed to only visit every edge once, while the iterative method needs to visit every edge in every iteration (number of which depend on error tolerance and method chosen). The third is done in O(|V |3 ) using LU factorization, however since |V | is small this is still faster than the iterative method unless the error tolerance chosen is large. We also note that only the fourth method actually depend on the error tolerance at all, every other method can be done in the same time regardless of error tolerance (down to machine precision). After we have calculated PageRank for all components on the current level we need to adjust the weight of all vertices in lower level components as shown in 7. This can be done using a single matrix-vector multiplication using the edges between the two sets of components. ~ new − = W ~ old − + M>+ ~ W − RL (L−1) (L−1) 13 L ,(L−1) This is the same kind of correction as is done in for example [2] and for the non-normalized PageRank used here in [8]. In the weight adjustment step every edge is needed once if it is an edge between components and never if it is an edge within a component. Hence if we look over the whole algorithm: every edge is visited at most twice in the DFS, then every edge that is not part of a SCC is visited exactly once more (either as part of a CAC or as an edge between components) while those that are part of a SCC are typically visited a significantly larger number of times depending on algorithm, error tolerance and convergence criterion used. Of course there is also some extra overhead that would need to be taken into consideration for a more in-depth analysis. We note that calculating PageRank for all components on a single level can be computed in parallel (hence why we sort them by their size starting by the largest). The weight adjustment can either be done in parallel for each component or as we have done here once for all components of the same level. In case there is a single very large component on a level it might be more appropriate to do it one component at a time instead to reduce the time waiting for the large component to finish. 4.3.1 CAC-PageRank algorithm The CAC-PageRank algorithm exploits the fact that there are no non-loop cycles to calculate the PageRank of the graph using a variation of Breadth first search. The algorithm starts by calculating the in-degree (ignoring any loops) of every vertex and stores it in v.degree for each vertex v, this can be done by looping through all edges once. We also keep a value v.rank initialized to corresponding value in the weight vector for each vertex. The BFS itself can be described by the following psuedocode. f o r each v ∈ V i f v . degree = 0 Queue . enqueue ( v ) while Queue . s i z e > 0 w = Queue . dequeue ( ) w . rank = w . rank ∗(1/(1 −W(w, w) ) ) // a d j u s t f o r l o o p s f o r each (w, u ) ∈ E/ (w, w) u . rank = u . rank + w . rank ∗ W(w, u ) u . d e g r e e = u . d e g r e e −1 i f w. degree = 0 Queue . enqueue (w) end end w . d e g r e e = w . d e g r e e −1 // e n s u r e w i s n e v e r enqued a g a i n end end end Here W(w,u) is the weight on the (w,u)-edge (cawu ). Note that W (w, w) = 0 if w has no loop, which gives simply multiplication by 1 when adjusting for loops. The difference between this and ordinary BFS is that we only add a vertex v to the queue once we have visited all incoming edges to v. We also loop through all vertices to ensure that we wisit each vertex once since there could be multiple vertices with no incoming edges. Usually PageRank is defined only for graphs with no loops (or by ignoring any loops that are present), this simplifies the algorithm slightly in that the loop adjustment step can be ignored. 14 Looking at the computational complexity it is easy to see that we visit every vertex and every edge once doing constant time work, hence we have the same time complexity as for ordinary BFS O(|V | + |E|) ≈ O(|E|), if |E| > |V |. While it has the same computational complexity as most numerical methods used to calculate PageRank, in practice it will often be much faster in that the coefficient in front will be smaller, especially as the error tolerance decreases. 4.3.2 SCC-PageRank algorithm If the component is a SCC, then we cannot use the previous algorithm. Instead one of many iterative methods needs to be used (unless the size of the component is small). We have chosen to calculate PageRank for SCCs using a power series as described in [2] since it is a more natural fit with the non-normalized variation of PageRank used here. The method works by iterating the following until some convergence criterion is met.  ~n+1 P ~0 P ~n = cAP ~ =W (3) ~n R = n X ~k P k=0 Although any method can be used, by using a power series we have the advantage in that we get the PageRank in the correct non-normalized form we need without the need to re-scale the result. In practice any other method such as a power iteration could be used as well by scaling the result as described in Def. 3.5. The calculation of these components could also benefit from the use of other methods designed to improve the calculation time of PageRank such as the adaptive algorithm described in [14]. Any other algorithm which depend on the presence of dangling vertices or SCCs would not be useful here however since we are working on a single SCC. The method you get by using a power series formulation have been shown to have the same or similar convergence speed as the more conventional Power method experimentally [2]. It is easy to show that this is equivalent to using the Jacobi method with initial ~ , and obviously Gauss-Seidel or a similar method using a power series vector equal to W formulation could be used and would likely provide some additional speedup. Theoretically the convergence of the Power method can be shown to be geometric depending on the two largest eigenvalues with ratio |λ2 |/|λ1 | where λ2 ≤ c [11], obviously we have a similar convergence by calculating corresponding geometric sum (geometric, bounded from above by c). The method can be described through the following pseudocode. rank = W // i n i t i a l i z e rank t o t h e w e i g h t v e c t o r mr = W; // ( i n i t i a l i z e rank added i n p r e v i o u s i t e r a t i o n do mr = M∗mr rank = rank+mr while (max(mr) ≥ t o l ) This is also the baseline method we use for comparison with our own method later. 4.4 Error estimation When looking at the error we have two different kinds of errors to consider, first errors from the iterative method used to calculate PageRank of large SCCs (depending on error ~ ). We will mainly tolerance) and second any errors because of errors in data (M or W concern ourselves with the first type which is likely to dominate unless the error tolerance is very small. 15 We start by looking at a single isolated component, if this component is a CAC or a small SCC we calculate PageRank exactly and errors can be assumed to be small as long as an appropriate method is used to solve the linear system for the small SCCs using for example LU-decomposition. For large SCCs we stop iterations after the maximum change of rank for any vertex between any two iterations is less than the error tolerance (tol). Since the rank is monotonically increasing we can be sure that the true rank is always a little higher in reality than what is actually calculated. The true rank can be described by R(3) = ∞ X ~ , where we let p ~ . Since Mk W ~k = Mk W k=0 every row sum of M is less than or equal to c, one has c|~ pk−1 | ≥ |~ pk |. This means that the maximum change in rank over all vertices in the graph after K iterations is bounded by ∞ X ck |~ pK | = k=1 c|~ pK | ≈ 5.66|~ pk |, c = 0.85 1−c This does not change if there are any edges to or from other components in the graph although the difference will be spread over a larger amount of vertices if there are edges from the component. There might also be additional additive error from components with edges to the single component we are considering. Over all vertices and all components we can estimate bounds for the total error tot over all vertices as well as the average error avg given some error tolerance tol. tot < |SCCl | · tol avg < c 1−c |SCCl | c c · tol ≤ tol |V | (1 − c) (1 − c) where |SCCl | is the number of vertices part of a ’large’ SCC (for which we need to use an iterative method) and |V | is the total number of vertices in the graph. It should be noted that this estimate is likely to be many times larger than in reality unless all the vertices have approximately the same rank. Given that PageRank for many real systems approximately follows a power distribution [5], most vertices will have orders of magnitude smaller change in rank when finally those with a very high rank have a change smaller than the error tolerance. Additionally if the graph contains some dangling vertices (vertices with no outgoing edges), then these will further reduce the error (can be seen as an increased chance to stop the random walk). 5 Experiments Our Implementation of the algorithm is done in a mixture of c and c++ for the graph search algorithms (component finding and CAC-PageRank algorithm) and Matlab for the ordinary PageRank algorithm for SCCs and weight adjustment as well as the main code gluing the different parts together. The reason to use c/c++ for some parts is that while Matlab is rather fast at doing elementary matrix operations (PageRank of SCCs and weight adjustment), it is very slow when you attempt to do for example DFS or BFS on a graph. • Component finding: Implemented in c++ as a variation of the depth first search in the boost library. The method is implemented iteratively rather than recursively (hence it can handle large graphs which could otherwise give a very large recursion depth). • CAC-PageRank algorithm: Implemented in c. • Power series PageRank algorithm: Implemented in Matlab. Used for SCCs as well as on the whole graph for comparison. • Main program: Implemented in Matlab, with c/c++ parts used through mex files. 16 5.1 Graph description and generation Many real world networks including the graph used for calculating PageRank for pages on the Internet share a number of important properties. First of all they should be scale-free, these networks are characterized by their degree distributions roughly following a power law, if k is the degree then the cumulative distribution for the degree can be written as P (k) = k−γ . In practices this means that there are a low number of very high degree vertices as well as a large amount of very low degree vertices. Secondly they should be small-world, these networks are characterized by two different properties 1) the average shortest path distance between any two vertices in the graph is small, roughly the logarithm of the number of vertices in the graph and 2) the network has a high clustering coefficient. So while the first property implies a high connectivity in the network because of the short distances, the second property says that the network should contain multiple small communities with high connectivity among themselves but low connectivity to vertices outside the own group. In order to evaluate the method we have chosen to look at five different graphs of varying properties and size. • B-A: A graph generated using the Barabási-Albert graph generation algorithm [4] with mean degree 12, after generation only some of outgoing edges are kept in order to generate a directed graph. This graph contains 1000000 vertices of which 959760 are part of a CAC and 999128 edges, there are 188010 1-vertex CACs out of 239258 CACs in total as well as 19813 SCCs. Maximum component size is 483874 vertices (CAC). A second graph where even fewer outgoing edges was kept was also used. The Barabási-Albert graph generation algorithm creates a graph which is scale-free, however it does not have the small-world property [10], the reason we still used this for one of our tests is that it makes it easy to create a scale-free graph with primary acyclic components. • Web: A graph released by Google as part of a contest in 2002 [1] part of the collection of datasets maintained by the SNAP group at Stanford University [17]. This graph contains 916428 vertices of which 399605 are part of a CAC and 5105039 edges, there are 302768 1-vertex CACs out of 321098 CACs in total as well as 12874 SCCs. Maximum component size is 434818 vertices (SCC). This graph has both the scale-free and small-world properties making it a good example of the type of graph we would be interested to calculate PageRank on in real applications. We also created two additional even larger graphs by using multiple copies of this graph: 1) the graph composed of ten disjoint copies of this graph and 2) a graph composed of ten copies of the web-graph with a small number (20) extra random edges in order to get a single very large component as is common for real-world networks. 5.1.1 Barabási-Albert graph generation The model works by selecting a starting seed graph of m0 vertices, in our case a 20×20 graph with uniformly random edges with mean degree 5. Then new vertices are added iteratively to the graph one at a time by connecting each new vertex with m ≤ m0 existing vertices with probability proportional to the number of edges already connected to each old vertex. This can be written as di pi = P d j j where di is the degree of vertex i. 17 The Barabśi-Albert model gives an undirected graph. In order to transform it into a directed graph we then went through each vertex and removed some edges. The number of edges that was kept originating from each vertex Ei,keep can be described by Ei,keep = dlog2 (Ei )e where Ei is the number of edges originally originating from vertex i. We choose m = 12 which after removal of edges gives an average (in+out)-degree of log2 (24) ≈ 4.6. After removal of vertices all vertices will have similar out-degree, but the in-degree will be similar to how it was after the original Barabśi-Albert graph generation (in-degree following a power law). 5.2 Results All experiments are performed on a computer with a quad core 2.7Ghz(core)-3.5Ghz(turbo) processor (Intel(R) Core(TM) i7-4800MQ) using Matlab R2014a with four threads on any of the parts computed in parallel. Three different methods where used 1. Calculate PageRank as a single large component as described in Sec. 4.3.2 2. Using the method described in 4 with components on the same level calculated sequentially. 3. Using the method described in 4 with components on the same level calculated in parallel. In addition for all three method any loops in the graph where ignored (as is common for PageRank). We note that the intermediate step between method 2 and 3 differs. Because of limitations in how the parallelization can be implemented we had to separate edges of the same component into their own cell-array for the parallel version, this accounts for the main difference in overhead between method 2 and 3. It should be noted that the intermediate step is not parallelized (in either version) and is something which could probably be significantly improved by implementation in another programming language. After doing the SCC/CAC partitioning of the graph and sorting all components according to their level and component size (both descending order) we can visualize the non-zero values of this new reordered adjacency matrix. The density of non-zeros before and after reordering for the Web graph can be seen in Fig. 2. Note that the diagonal lines are not single vertex components but rather a large amount of small components on the same level (hence they can be computed in parallel), for a view of some of the small components see Fig. 3. Any 1-vertex component are not colored since they have no internal edges, two large section of 1-vertex components are right before the middle large component and in the bottom right corner of the matrix. A cutoff at a density of 1.7 · 10−15 non-zero elements is used in order to avoid problems with a few very high density sections as well as maintaining the same scale in both figures. After finding the SCC/CAC partitioning large sections of zeros can clearly be seen, something which is not present in the original matrix. The single very large component in the graph is seen in the middle of the matrix, with a section of small components both above and below it. Langville and Meyer does a similar reordering by recursively reordering the vertices by putting any dangling vertices last and not considering edges to those already put last once for any further reordering of remaining vertices [15]. This effectively creates one or more CACs along with one large component. The advantage of our approach compared to this is that we can also find components above the single large component rather than combining them into a single even larger component as well as finding sets of components which can be computed in parallel. In Fig. 3 some common types of smaller components can be seen. 18 Figure 2: Non-zero values of adjacency matrix for the Web graph before and after sorting vertices according to level and component. Figure 3: Non-zero elements in parts of the bottom right diagonal ’line’ of the reordered adjacency matrix. First there are 2 CACs where the majority of the vertices link to the same or a very limited number of vertices creating a very shallow tree. The third component is also a CAC (as seen by the zero rows) have a more advanced structure, while the last one is probably a SCC. There is a large amount of CACs similar to the first 2 with the majority of the vertices linking to one or two vertices but there are also some characterized by a horizontal line representing one or a few vertices linking to a large number of dangling vertices. The total number of levels in the Web graph was 28, with the majority being located right at the very top or right after the large central component. More research would be needed to verify if this is usually the case, but if so it might be a good idea to merge some of these very small levels and calculate them as if it was a SCC in order to reduce overhead. 19 For example the first 10 levels contain just 85 vertices in total and could be merged, after the large component there is only 2-3 of these very small components with the rest having at least a hundred or so vertices hence it might not be as useful here. If no merging of 1-vertex CACs was done (using the ordinary SCC partitioning) the number of levels was increased to 34 levels instead. Since PageRank of different SCCs converge in varying amounts of iterations it is also of interest to see how the number of iterations for different components varies, as well as how it compares to the number of iterations needed by the basic algorithm where we calculate PageRank as if the graph was a single component. Number of iterations for all SCCs of more than 2 vertices of the Web graph with c = 0.85 and tol = 10−9 can be seen in Fig. 4. In Fig. Figure 4: Number of iterations needed per SCC ordered according to their level first and number of vertices second (both descending order). The dotted vertical lines denotes where one level ends and the next one starts while the horizontal line denotes the result where the whole graph is considered a single component. c = 0.85, tol = 10−9 4 we can see a couple of things, first the number of iterations over any component is less than the number of iterations that would be needed if we calculated PageRank of the graph as if it was a single huge component. The average number of iterations per edge was 148, which can be compared to the number of iterations for the graph as a single component which was 168, this gives an improvement of approximately 12%. This might look small looking at the figure, but remember that the largest components on each level are put first on their level and the size of components approximately follows a power law, hence large parts of the figure represent relatively few vertices. It should also be noted that a significant number of edges (approximately 26%) lies either between components or within CACs both of which are not counted for here since they don’t use the iterative method and are instead used only once either to modify weights between levels or as part of the DFS when calculating PageRank of CACs. The second point of interest is that there is a clear difference between components at the last level compared to those of a higher level. Any SCC on the last level is by definition a stochastic matrix (before multiplication with c) since they have no edges to any vertex in any other component, this gives a lower bound on the number of iterations equal to log tol/ log c ≈ 128 easily seen from the relation citr ≤ tol, where itr is the number of iterations. However those component of higher level are by definition a sub stochastic matrix (before multiplication with c) since there is at least one edge to some other component. This is equivalent to some vertices having a lower c value and the algorithm can therefor converge 20 faster. The third observation is that a large component generally needs more iterations than a smaller component. This makes sense if we consider that as long as most vertices in the component do not contain edges to other components, as the component grows in size at least some part of the component will behave similar to those in the last level and we thus need a larger number of iterations. For large components the estimated number of iterations is usually quite good, while for small components it usually gives a too high estimate (unless it is part of the last level). The running time in seconds for the Web graph for the three different methods for different values of error tolerance from 10−1 to 10−20 can be seen in Fig. 5a. From this it (a) (b) Figure 5: Running time needed to calculate PageRank for 3 different methods (a) depending on error tolerance using c = 0.85 and (b) depending on c between 0.5 and 0.99 using tol = 10−10 on the Web graph. is clear that our method adds a significant amount of overhead, especially the parallel one. While all three methods need a longer time if the error tolerance is smaller, our method shows a significantly smaller increase compared to the basic method. While the break even here seems to lie at around 10−5 for the sequential algorithm and at around 10−9 for the parallel algorithm because of additional overhead. If the overhead in particular for the parallell algorithm could be further reduced this breakpoint could potentially be significantly earlier. Note that because of limits in machine precision we might not have the correct rank down to the last 20 decimals at the lowest tolerance, however since we sum over successively smaller parts we still get a good approximation of the actual computation time. The running time when we let c vary between 0.5 and 0.99 with a constant error tolerance (10−10 ) can be seen in Fig. 5b. From Fig. 5b there is a clear indication that our proposed method can be significantly faster for values of c close to one. This is quite natural given that computation time of some components does not depend on c at all (CACs and small components). Even for those components that does depend on error tolerance (large SCCs) there should be a significant amount of edges out of the component meaning some rows will have a sum lower than c thus roughly simulating a lower c value. In order to get some estimation of how the result changes with the size of the graph we also did the same experiments with ten copies of this graph. The results of this can be seen in Fig. 6a. With a ten times larger graph the basic and sequential approach takes about ten times longer (sequential a little less, basic a little more), hence the relationship between the two are more or less the same but with a slightly earlier point at which they have the same performance. The parallel version however only took approximately eight times longer on the much larger graph presumably since we get many more components on the same level. This moves the breakpoint from 10−10 to 10−6 . 21 (a) (b) Figure 6: Running time needed to calculate PageRank for 3 different methods depending on error tolerance used on (a) ten copies of the Web graph and (b) ten copies of the Web graph with some extra edges added. While a ten-times copied graph might not represent a true network of that size it does have one important likeness, namely that the diameter of many real world networks (such as the world wide web) increases significantly slower than the size of the network itself. If the diameter is small it is clear that the number of levels need to be small as well, which means that as the size of the network increases we should have more and more vertices on the same level increasing the opportunity for the parallelization. By adding a couple of random edges we retain a single large component (as is the case of the original graph), the results after doing this can be seen in Fig. 6b. We do not see any significant differences between this and the previous graph, the parallel method takes slightly longer while the other two stay more or less the same. From this we see that even if we have a very large component (composed of roughly half the vertices) we can still get significant gains using our method as long as the graph is sufficiently big. Last we also did the same experiment on a graph generated by the Barabási-Albert model, the results for this graph can be seen in Fig. 7. Since this graph has a very small number (a) (b) Figure 7: Running time needed to calculate PageRank for 3 different methods depending on error tolerance used on the graph generated by the BarabásiAlbert graph generation method with (a) roughly 1 edge/vertex after removal of edges and (b) roughly 5 edges/vertex after removal of edges. of SCCs, both the sequential and parallel method barely depend on the error tolerance at all, while the basic approach seems to have similar behavior as with the previous graphs 22 with a roughly linear increase in time needed as the error tolerance decreases. However since the PageRank calculations themselves are faster for this graph (for all methods), the overhead needed represent a larger part of the total time since this does not decrease in the same way. Overall the results of our approach are promising, we have gotten better results the bigger the graph is as well as the lower the error tolerance is as compared to the basic approach. Our implementation of the parallel method had significantly more overhead than the sequential method, this is likely to change if the algorithm was implemented in full (with parallelization in mind) in for example c++ where we have more control over how it can be implemented. 6 Conclusions We have seen that by dividing the graph into components we could get improved performance of the PageRank algorithm, however it does come with a cost of increased overhead (very much depending on implementation) as well as algorithm complexity. We could see that we needed to do significantly less number of iterations overall using our method compared to the ordinary method using a power series or power iterations. This came from the fact that some edges lie between components or inside a CAC (hence needing only one iteration) as well as the fact that a larger component is more likely to need a larger number of iterations than a small one. From our experiments we can see that our proposed algorithm is more effective compared to the basic approach as the size of the graph increases, at least as long as we can keep everything in memory. More experiments would be needed for any conclusions after that although given that components can be calculated by themselves in our method (hence a lower memory requirement in the PageRank step) we expect our method to compare even better when this is the case. We could also see better performance using our method compared to the basic method if the error tolerance is small while it is generally slower because of overhead if the tolerance for errors is large. The results for the parallel method compared to the serial method was not clear, however this is likely something that can be improved significantly using a better implementation and memory handling as well as having more of an advantage when memory overall is more of a concern for even larger graphs. By calculating PageRank for different kinds of components differently we could see a large improvement for certain types of graphs such as the one generated by the B-A model where the time needed to calculate PageRank was more or less constant depending on error tolerance using our method. 7 Future work One obvious and likely next step would be to implement and try the same component finding algorithm for something else such as sparse equation solving or calculating the inverse of sparse matrices, both of which have similar behavior in that components only affect other components downwards in levels. It would also be interesting to try an improved implementation of the method by implementing it all in c++ or another ‘fast’ programming language in order to reduce some of the extra overhead incurred. Similarly it would also be interesting to see how the algorithm compares to other algorithms for extremely large graphs where memory become more of an issue. A third interesting direction would be to look at how this method interacts with other methods to calculate PageRank by using a method such as the one proposed by [14] on the largest components. 23 Another important problem is how to update PageRank after doing some changes to the graph, it would be very interesting to see how this partitioning of the graph in combination with non-normalized PageRank could potentially be used to recalculate PageRank faster than for example using the old PageRank as initial rank and doing power iterations from there. References [1] Google web graph. Google programming contest, http://snap.stanford.edu/data/ web-Google.html, 2002. [2] Fredrik Andersson and Sergei Silvestrov. The mathematics of internet search engines. Acta Appl. Math., 104:211–242, 2008. [3] Arvind Arasu, Jasmine Novak, Andrew Tomkins, and John Tomlin. Pagerank computation and the structure of the web: Experiments and algorithms. In In Proceedings of the Eleventh International Conference on World Wide Web, Alternate Poster Tracks., 2002. [4] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286(5439):509–512, 1999. [5] Luca Becchetti and Carlos Castillo. The distribution of pagerank follows a powerlaw only for particular values of the damping factor. In In Proceedings of the 15th international conference on World Wide Web, pages 941–942. ACM Press, 2006. [6] Pavel Berkhin. A survey on pagerank computing. Internet Mathematics, 2:73–120, 2005. [7] Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual web search engine. In Proceedings of the Seventh International Conference on World Wide Web 7, WWW7, pages 107–117, Amsterdam, The Netherlands, The Netherlands, 1998. Elsevier Science Publishers B. V. [8] Christopher Engström and Sergei Silvestrov. A componentwise pagerank algorithm. In Applied Stochastic Models and Data Analysis (ASMDA 2015). The 16th Conference of the ASMDA International Society, pages 186–199, 2015. [9] Christopher Engström and Sergei Silvestrov. Non-normalized pagerank and random walks on n-partite graphs. In 3rd Stochastic Modeling Techniques and Data Analysis International Conference (SMTDA2014), pages 193–202, 2015. [10] Ernesto Estrada. The Structure of Complex Networks: Theory and Applications. Oxford University Press, Inc., New York, NY, USA, 2011. [11] Taher Haveliwala and Sepandar Kamvar. The second eigenvalue of the google matrix. Technical Report 2003-20, Stanford InfoLab, 2003. [12] H. Ishii, R. Tempo, E.-W. Bai, and F. Dabbene. Distributed randomized pagerank computation based on web aggregation. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, pages 3026–3031, 2009. [13] Sepandar Kamvar and Taher Haveliwala. The condition number of the pagerank problem. Technical Report 2003-36, Stanford InfoLab, June 2003. [14] Sepandar Kamvar, Taher Haveliwala, and Gene Golub. Adaptive methods for the computation of pagerank. Linear Algebra and its Applications, 386(0):51 – 65, 2004. Special Issue on the Conference on the Numerical Solution of Markov Chains 2003. [15] Amy N. Langville and Carl D. Meyer. A reordering for the pagerank problem. SIAM J. Sci. Comput., 27(6):2112–2120, December 2005. 24 [16] Chris P. Lee, Gene H. Golub, and Stefanos A. Zenios. A two-stage algorithm for computing pagerank and multistage generalizations. Internet Mathematics, 4(4):299– 327, 2007. [17] Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014. [18] Robert Tarjan. Depth first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146–160, 1972. [19] Qing Yu, Zhengke Miao, Gang Wu, and Yimin Wei. Lumping algorithms for computing google’s pagerank and its derivative, with attention to unreferenced nodes. Information Retrieval, 15(6):503–526, 2012. 25 

Slot Games for Detecting Timing Leaks of Programs Aleksandar S. Dimovski Faculty of Information-Communication Tech., FON University, Skopje, 1000, MKD aleksandar.dimovski@fon.edu.mk In this paper we describe a method for verifying secure information flow of programs, where apart from direct and indirect flows a secret information can be leaked through covert timing channels. That is, no two computations of a program that differ only on high-security inputs can be distinguished by low-security outputs and timing differences. We attack this problem by using slot-game semantics for a quantitative analysis of programs. We show how slot-games model can be used for performing a precise security analysis of programs, that takes into account both extensional and intensional properties of programs. The practicality of this approach for automated verification is also shown. 1 Introduction Secure information flow analysis is a technique which performs a static analysis of a program with the goal of proving that it will not leak any sensitive (secret) information improperly. If the program passes the test, then we say that it is secure and can be run safely. There are several ways in which secret information can be leaked to an external observer. The most common are direct and indirect leakages, which are described by the so-called non-interference property [13, 18]. We say that a program satisfies the non-interference property if its high-security (secret) inputs do not affect its low-security (public) outputs, which can be seen by external observers. However, a program can also leak information through its timing behaviour, where an external observer can measure its total running time. Such timing leaks are difficult to detect and prevent, because they can exploit low-level implementation details. To detect timing leaks, we need to ensure that the total running time of a program do not depend on its high-security inputs. In this paper we describe a game semantics based approach for performing a precise security analysis. We have already shown in [8] how game semantics can be applied for verifying the non-interference property. Now we use slot-game semantics to check for timing leaks of closed and open programs. We focus here only on detecting covert timing channels, since the non-interference property can be verified similarly as in [8]. Slot-game semantics was developed in [11] for a quantitative analysis of Algollike programs. It is suitable for verifying the above security properties, since it takes into account both extensional (what the program computes) and intensional (how the program computes) properties of programs. It represents a kind of denotational semantics induced by the theory of operational improvement of Sands [19]. Improvement is a refinement of the standard theory of operational approximation, where we say that one program is an improvement of another if its execution is more efficient in any program context. We will measure efficiency of a program as the sum of costs associated with basic operations it can perform. It has been shown that slot-game semantics is fully abstract (sound and complete) with respect to operational improvement, so we can use it as a denotational theory of improvement to analyse programming languages. The advantages of game semantics (denotational) based approach for verifying security are several. We can reason about open programs, i.e. programs with non-locally defined identifiers. Moreover, game semantics is compositional, which enables analysis about program fragments to be combined into an Gabriele Puppis, Tiziano Villa (Eds.): Fourth International Symposium on Games, Automata, Logics and Formal Verification EPTCS 119, 2013, pp. 166–179, doi:10.4204/EPTCS.119.15 c A. S. Dimovski This work is licensed under the Creative Commons Attribution License. A. S. Dimovski 167 analysis of a larger program. Also the model hides the details of local-state manipulation of a program, which results in small models with maximum level of abstraction where are represented only visible input-output behaviours enriched with costs that measure their efficiency. All other behaviour is abstracted away, which makes this model very suitable for security analysis. Finally, the game model for some language fragments admits finitary representation by using regular languages or CSP processes [10, 6], and has already been applied to automatic program verification. Here we present another application of algorithmic game semantics for automatically verifying security properties of programs. Related work. The most common approach to ensure security properties of programs is by using security-type systems [14]. Here for every program component are defined security types, which contain information about their types and security levels. Programs that are well-typed under these type systems satisfy certain security properties. Type systems for enforcing non-interference of programs have been proposed by Volpano and Smith in [20], and subsequently they have been extended to detect also covert timing channels in [21, 2]. A drawback of this approach is its imprecision, since many secure programs are not typable and so are rejected. A more precise analysis of programs can be achieved by using semantics-based approaches [15]. 2 Syntax and Operational Semantics We will define a secure information flow analysis for Idealized Algol (IA), a small Algol-like language introduced by Reynolds [16] which has been used as a metalanguage in the denotational semantics community. It is a call-by-name λ -calculus extended with imperative features and locally-scoped variables. In order to be able to perform an automata-theoretic analysis of the language, we consider here its secondorder recursion-free fragment (IA2 for short). It contains finitary data types D: intn = {0, . . . , n − 1} and bool = {tt, ff }, and first-order function types: T ::= B | B → T, where B ranges over base types: expressions (expD), commands (com), and variables (varD). Syntax of the language is given by the following grammar: M ::=x | v | skip | diverge | M op M | M; M | if M thenM else M | while M do M | M := M |!M | newD x := v in M | mkvarD MM | λ x.M | MM where v ranges over constants of type D. Typing judgements are of the form Γ ⊢ M : T, where Γ is a type context consisting of a finite number of typed free identifiers. Typing rules of the language are standard [1], but the general application rule is broken up into the linear application and the contraction rule 1 . Γ⊢M:B→T ∆⊢N:B Γ, ∆ ⊢ MN : T Γ, x1 : T, x2 : T ⊢ M : T ′ Γ, x : T ⊢ M[x/x1 , x/x2 ] : T ′ We use these two rules to have control over multiple occurrences of free identifiers in terms during typing. Any input/output operation in a term is done through global variables, i.e. free identifiers of type varD. So an input is read by de-referencing a global variable, while an output is written by an assignment to a global variable. 1 M[N/x] denotes the capture-free substitution of N for x in M. Slot Games for Detecting Timing Leaks of Programs 168 Γ ⊢ n1 op n2 , s −→kop n, s, where n = n1 opn2 Γ ⊢ skip; skip, s −→kseq skip, s′ Γ ⊢ if tt then M1 elseM2 , s −→kif M1 , s Γ ⊢ if ff then M1 elseM2 , s −→kif M2 , s Γ ⊢ x := v′ , s ⊗ (x 7→ v) −→kasg skip, s ⊗ (x 7→ v′ ) Γ ⊢!x, s ⊗ (x 7→ v) −→kder v, s ⊗ (x 7→ v) Γ ⊢ (λ x.M)M ′ , s −→kapp M[M ′ /x], s Γ ⊢ newD x := v in skip, s −→knew skip, s Table 1: Basic Reduction Rules The operational semantics is defined in terms of a small-step evaluation relation using a notion of an evaluation context [9]. A small-step evaluation (reduction) relation is of the form: Γ ⊢ M, s −→ M ′ , s′ where Γ is a so-called var-context which contains only identifiers of type varD; s, s′ are Γ-states which assign data values to the variables in Γ; and M, M ′ are terms. The set of all Γ-states will be denoted by St(Γ). Evaluation contexts are contexts 2 containing a single hole which is used to identify the next sub-term to be evaluated (reduced). They are defined inductively by the following grammar: E ::= [−] | EM | E; M | skip; E | E op M | v op E | if E then M else M | M := E | E := v |!E The operational semantics is defined in two stages. First, a set of basic reduction rules are defined in Table 1. We assign different (non-negative) costs to each reduction rule, in order to denote how much computational time is needed for a reduction to complete. They are only descriptions of time and we can give them different interpretations describing how much real time they denote. Such an interpretation can be arbitrarily complex. So the semantics is parameterized on the interpretation of costs. Notice that we write s ⊗ (x 7→ v) to denote a {Γ, x}-state which properly extends s by mapping x to the value v. We also have reduction rules for iteration, local variables, and mkvarD construct, which do not incur additional costs. Γ ⊢ while b do M, s −→ if b then (M; while b do M) else skip, s Γ, y ⊢ M[y/x], s ⊗ (y 7→ v) −→ M ′ , s′ ⊗ (y 7→ v′ ) Γ ⊢ newD x := v in M, s −→ newD x := v′ in M ′ [x/y], s′ Γ ⊢ (mkvar D M1 M2 ) := v, s −→ M1 v, s Γ ⊢!(mkvarD M1 M2 ), s −→ M2 , s Next, the in-context reduction rules for arbitrary terms are defined as: Γ ⊢ M, s −→n M ′ , s′ Γ ⊢ E[M], s −→n E[M ′ ], s′ The small-step evaluation relation is deterministic, since arbitrary term can be uniquely partitioned into an evaluation context and a sub-term, which is next to be reduced. We define the reflexive and transitive closure of the small-step reduction relation as follows: 2 A context C[−] is a term with (several occurrences of) a hole in it, such that if Γ ⊢ M : T is a term of the same type as the hole then C[M] is a well-typed closed term of type com, i.e. ⊢ C[M] : com. A. S. Dimovski 169 ′ Γ ⊢ M, s −→n M ′ , s′ Γ ⊢ M, s n M ′ , s′ Γ ⊢ M ′ , s′ n M ′′ , s′′ ′ n ′ ′ Γ ⊢ M, s M ,s Γ ⊢ M, s n+n M ′′ , s′′ Now a theory of operational improvement is defined [19]. Let Γ ⊢ M : com be a term, where Γ is a var-context. We say that M terminates in n steps at state s, written M, s ⇓n , if Γ ⊢ M, s n skip, s′ for some state s′ . If M is a closed term and M, 0/ ⇓n , then we write M ⇓n . If M ⇓n and n ≤ n′ , we write ′ M ⇓≤n . We say that a term Γ ⊢ M : T may be improved by Γ ⊢ N : T, denoted by Γ ⊢ M & N, if and only if for all contexts C[−], if C[M] ⇓n then C[N] ⇓≤n . If two terms improve each other they are considered improvment-equivalent, denoted by Γ ⊢ M ≈ N. Let Γ, ∆ ⊢ M : T be a term where Γ is a var-context and ∆ is an arbitrary context. Such terms are called split terms, and we denote them as Γ | ∆ ⊢ M : T. If ∆ is empty, then these terms are called semiclosed. The semi-closed terms have only some global variables, and the operational semantics is defined only for them. We say that a semi-closed term h : varD | − ⊢ M : com does not have timing leaks if the initial value of the high-security variable h does not influence the number of reduction steps of M. More formally, we have: Definition 1. A semi-closed term h : varD | − ⊢ M : com has no timing leaks if ∀ s1 , s2 ∈ St({h}). s1 (h) 6= s2 (h) ∧ h : varD ⊢ M, s1 ⇒ n1 = n2 n1 skip, s1 ′ ∧ h : varD ⊢ M, s2 n2 skip, s2 ′ (1) Definition 2. We say that a split term h : varD | ∆ ⊢ M : com does not have timing leaks, where ∆ = x1 : T1 , . . . , xk : Tk , if for all closed terms ⊢ N1 : T1 , . . . , ⊢ Nk : Tk , we have that the term h : varD | − ⊢ M[N1 /x1 , . . . , Nk /xk ] : com does not have timing leaks. The formula (1) can be replaced by an equivalent formula, where instead of two evaluations of the same term we can consider only one evaluation of the sequential composition of the given term with another its copy [3]. So sequential composition enables us to place these two evaluations one after the other. Let h : varD ⊢ M : com be a term, we define M ′ to be α -equivalent to M[h′ /h] where all bound vari′ ables are suitable renamed. The following can be shown: h ⊢ M, s1 n skip, s1 ′ ∧ h′ ⊢ M ′ , s2 n skip, s2 ′ ′ iff h, h′ ⊢ M; M ′ , s1 ⊗ s2 n+n skip; skip, s1 ′ ⊗ s2 ′ . In this way, we provide an alternative definition to formula (1) as follows. We say that a semi-closed term h | − ⊢ M : T has no timing leaks if ∀ s1 ∈ St({h}), s2 ∈ St({h′ }). s1 (h) 6= s2 (h′ ) ∧ h, h′ ⊢ M; M ′ , s1 ⊗ s2 ⇒ n1 = n2 n1 skip; M ′ , s1 ′ ⊗ s2 n2 skip; skip, s1 ′ ⊗ s2 ′ (2) 3 Algorithmic Slot-Game Semantics We now show how slot-game semantics for IA2 can be represented algorithmically by regular-languages. In this approach, types are interpreted as games, which have two participants: the Player representing the term, and the Opponent representing its context. A game (arena) is defined by means of a set of moves, each being either a question move or an answer move. Each move represents an observable action that a term of a given type can perform. Apart from moves, another kind of action, called token (slot), is used to take account of quantitative aspects of terms. It represents a payment that a participant needs to pay in order to use a resource such as time. A computation is interpreted as a play-withcosts, which is given as a sequence of moves and token-actions played by two participants in turns. Slot Games for Detecting Timing Leaks of Programs 170 We will work here with complete plays-with-costs which represent the observable effects along with incurred costs of a completed computation. Then a term is modelled by a strategy-with-costs, which is a set of complete plays-with-costs. In the regular-language representation of game semantics [10], types (arenas) are expressed as alphabets of moves, computations (plays-with-costs) as words, and terms (strategies-with-costs) as regular-languages over alphabets. Each type T is interpreted by an alphabet of moves A[[T]] , which can be partitioned into two subsets of questions Q[[T]] and answers A[[T]] . For expressions, we have: Q[[expD]] = {q} and A[[expD]] = D, i.e. there are a question move q to ask for the value of the expression and values from D are possible answers. For commands, we have: Q[[com]] = {run} and A[[com]] = {done}, i.e. there are a question move run to initiate a command and an answer move done to signal successful termination of a command. For variables, we have: Q[[varD]] = {read, write(a) | a ∈ D} and A[[varD]] = D ∪ {ok}, i.e. there are moves for writing to the variable, write(a), acknowledged by the move ok, and for reading from the variable, we have a question move read, and an answer to it can be any value from D. For function types, we i +A have A[[B1 →...→Bk →B]] = ∑1≤i≤k A[[B [[B]] , where + means a disjoint union of alphabets. We will use i ]] 1 k superscript tags to keep record from which type of the disjoint union each move comes from. We denote the token-action by $ . A sequence of n token-actions $ will be written as n . For any (β -normal) term we define a regular language specified by an extended regular expression R. Apart from the standard operations for generating regular expressions, we will use some more specific operations. We define composition of regular expressions R defined over alphabet A 1 + B 2 + { $ } and S over B 2 + C 3 + { $ } as follows:   R 9oB2 S = {w s/a2 · b2 | w ∈ S, a2 · s · b2 ∈ R} where R is a set of words of the form a2 · s · b2 , such that a2 , b2 ∈ B 2 and s contains only letters from A 1 and { $ }. Notice that the composition is defined over A 1 + C 3 + { $ }, and all letters of B 2 are hidden. The shuffle operation R ⊲⊳ S generates the set of all possible interleavings from words of R and S, and the restriction operation R |A ′ (R defined over A and A ′ ⊆ A ) removes from words of R all letters from A ′. If w, w′ are words, m is a move, and R is a regular expression, define m · w a w′ = m · w′ · w, and R a w′ = {w a w′ | w ∈ R}. Given a word with costs w defined over A + { $ }, we define the underlying word of w as w† = w |{ $ } , and the cost of w as w |A = n , which we denote as | w |= n. The regular
expression for Γ ⊢ M : T is denoted [[Γ ⊢ M : T]] and is defined over the alphabet A[[Γ⊢T]] = ∑x:T ′ ∈Γ A[[Tx ′ ]] + A[[T]] + { $ }. Every word in [[Γ ⊢ M : T]] corresponds to a complete play-with-costs in the strategy-with-costs for Γ ⊢ M : T. Free identifiers x ∈ Γ are interpreted by the copy-cat regular expressions, which contain all possible computations that terms of that type can have. Thus they provide the most general closure of an open term. x,k x 1 k [[Γ, x : Bx,1 1 → . . . Bk → B ⊢ x : B1 → . . . Bk → B]] = i x ∑ q · q · ∑ ( ∑ qx,i 1 · q1 · q∈Q[[B]] 1≤i≤k q1 ∈Q[[Bi ]] ∑
∗ ai1 · ax,i 1 ) · a1 ∈A[[Bi ]] ∑ ax · a a∈A[[B]] When a first-order non-local function is called, it may evaluate any of its arguments, zero or more times, and then it can return any value from its result type as an answer. For example, the term [[Γ, x : expDx ⊢ x : expD]] is modelled by the regular expression: q · qx · ∑n∈D nx · n. The linear application is defined as: [[Γ, ∆ ⊢ M N : T]] = [[∆ ⊢ N : B1 ]] o9A 1 [[Γ ⊢ M : B1 → T]] [[B]] A. S. Dimovski 171 Since we work with terms in β -normal form, function application can occur only when the function term is a free identifier. In this case, the interpretation is the same as above except that we add the cost kapp corresponding to function application. Notice that kapp denotes certain number of $ units that are needed for a function application to take place. The contraction [[Γ, x : T x ⊢ M[x/x1 , x/x2 ] : T ′ ]] is obtained from [[Γ, x1 : T x1 , x2 : T x2 ⊢ M : T ′ ]], such that the moves associated with x1 and x2 are de-tagged so that they represent actions associated with x. To represent local variables, we first need to define a (storage) ‘cell’ regular expression cellv which imposes the good variable behaviour on the local variable. So cellv responds to each write(n) with ok, and plays the most recently written value in response to read, or if no value has been written yet then answers the read with the initial value v. Then we have:
∗ cellv = (read · v)∗ · ∑ write(n) · ok · (read · n)∗ n∈D
x [[Γ, x : varD ⊢ M]] ◦ cellxv = [[Γ, x : varD ⊢ M]] ∩ (cellxv ⊲⊳ (A[[Γ⊢B]] + $ )∗ ) |A[[varD]] x [[Γ ⊢ newD x := v in M]] = [[Γ, x : varD ⊢ M]] ◦ cellv a kvar Note that all actions associated with x are hidden away in the model of new, since x is a local variable and so not visible outside of the term. Language constants and constructs are interpreted as follows: [[v : expD]] = {q · v} [[skip : com]] = {run · done} [[diverge : com]] = 0/ [[op : expD1 × expD2 → expD′ ]] = q · kop · q1 · ∑m∈D m1 · q2 · ∑n∈D n2 ·(m op n) [[; : com1 → com2 → com]] = run · run1 · done1 · kseq · run2 · done2 · done [[if : expbool1 → com2 → com3 → com]] = run · kif · q1 · tt1 · run2 · done2 · done + run · kif · q1 · ff 1 · run3 · done3 · done [[while : expbool1 → com2 → com]] = run · (kif · q1 · tt1 · run2 · done2 )∗ · kif · q1 · ff 1 · done [[:=: varD1 → expD2 → com]] = ∑n∈D run · kasg · q2 · n2 · write(n)1 · ok1 · done [[! : varD1 → expD]] = ∑n∈D q · kder · read1 · n1 · n Although it is not important at what position in a word costs are placed, for simplicity we decide to attach them just after the initial move. The only exception is the rule for sequential composition (; ), where the cost is placed between two arguments. The reason will be explained later on. We now show how slot-games model relates to the operational semantics. First, we need to show how to represent the state explicitly in the model. A Γ-state s is interpreted as follows: k 1 [[s : varDx11 × . . . × varDxkk ]] = cellxs(x ⊲⊳ . . . ⊲⊳ cellxs(x 1) k) xk x1 , and words in [[s]] are The regular expression [[s]] is defined over the alphabet A[[varD + . . . + A[[varD 1 ]] k ]] such that projections onto xi -component are the same as those of suitable initialized cells(xi ) strategies. Note that [[s]] is a regular expression without costs. The interpretation of Γ ⊢ M : com at state s is:
[[Γ ⊢ M]] ◦ [[s]] = [[Γ ⊢ M]] ∩ ([[s]] ⊲⊳ (A[[com]] + $ )∗ ) |A[[Γ]] which is defined over the alphabet A[[com]] + { $ }. The interpretation [[Γ ⊢ M]] ◦ [[s]] can be studied more closely by considering words in which moves from A[[Γ]] are not hidden. Such words are called interaction sequences. For any interaction sequence run · t · done ⊲⊳ n from [[Γ ⊢ M]] ◦ [[s]], where t is an even-length word over A[[Γ]] , we say that it leaves the state s′ if the last write moves in each xi component are such that xi is set to the value s′ (xi ). For example, let s = (x 7→ 1, y 7→ 2), then the Slot Games for Detecting Timing Leaks of Programs 172 following interaction: run · write(5)y · oky · readx · 1x · done leaves the state s′ = (x 7→ 1, y 7→ 5). Any twomove word of the form: runxi · nxi or write(n)xi · okxi will be referred to as atomic state operation of A[[Γ]] . The following results are proved in [11] for the full ICA (IA plus parallel composition and semaphores), but they also hold for the restricted fragment of it. Proposition 1. If Γ ⊢ M : {com, expD} and Γ ⊢ M, s −→n M ′ , s′ , then for each interaction sequence i · t from [[Γ ⊢ M ′ ]] ◦ [[s′ ]] (i is an initial move) there exists an interaction i · ta · t a n ∈ [[Γ ⊢ M]] ◦ [[s]] such that ta is an empty word or an atomic state operation of A[[Γ]] which leaves the state s′ . Proposition 2. If Γ ⊢ M, s n M ′ , s′ then [[Γ ⊢ M ′ ]] ◦ [[s′ ]] ⊲⊳ n ⊆ [[Γ ⊢ M]] ◦ [[s]]. Theorem 1 (Consistency). If M, s ⇓n then ∃ w ∈ [[Γ ⊢ M]] ◦ [[s]] such that | w |= n and w† = run · done . Theorem 2 (Computational Adequacy). If ∃ w ∈ [[Γ ⊢ M]] ◦ [[s]] such that | w |= n and w† = run · done, then M, s ⇓n . w† We say that a regular expression R is improved by S, denoted as R & S, if ∀ w ∈ R, ∃ t ∈ S, such that = t† and | w |≥| t |. Theorem 3 (Full Abstraction). Γ ⊢ M & N iff [[Γ ⊢ M]] & [[Γ ⊢ N]]. This shows that the two theories of improvement based on operational and game semantics are identical. 4 Detecting Timing Leaks In this section slot-game semantics is used to detect whether a term with a secret global variable h can leak information about the initial value of h through its timing behaviour. For this purpose, we define a special command skip# which similarly as skip does nothing, but its slot-game semantics is: [[skip# ]] = {run · # · done}, where # is a new special action, called delimiter. Since we verify security of a term by running two copies of the same term one after the other, we will use the command skip# to specify the boundary between these two copies. In this way, we will be able to calculate running times of the two terms separately. Theorem 4. Let h : varD | − ⊢ M : com be a semi-closed term, and 3 R = [[k : expD ⊢ newD h := k in M; skip# ; newD h′ := k in M ′ : com]] (3) Any word of R is of the form w = w1 · # · w2 such that | w1 |=| w2 | iff M has no timing leaks, i.e. the fact (2) holds. Proof. Suppose that any word w ∈ R is of the form w = w1 · # · w2 such that | w1 |=| w2 |. Let us analyse the regular expression R defined in (3). We have: R = {run · kvar · qk · vk · w1 · kseq · # · kseq · kvar · qk · v′k · w2 · done | ′ run · w1 · done ∈ [[h ⊢ M]] ◦ cellhv , run · w2 · done ∈ [[h′ ⊢ M ′ ]] ◦ cellhv′ } for arbitrary values v, v′ ∈ D. In order to ensure that one kseq unit of cost occurs before and after the delimiter action, kseq is played between two arguments of the sequential composition as was described ′ in Section 3. Given that run · w1 · done ∈ [[h ⊢ M]] ◦ cellhv and run · w2 · done ∈ [[h′ ⊢ M ′ ]] ◦ cellhv′ for any 3 The free identifier k in (3) is used to initialize the variables h and h′ to arbitrary values from D. A. S. Dimovski 173 v, v′ ∈ D, by Computational Adequacy we have that M, (h 7→ v) ⇓|w1 | and M ′ , (h′ 7→ v′ ) ⇓|w2 | . Since | w1 |=| w2 |, it follows that the fact (2) holds. Let us consider the opposite direction. Suppose that the fact (2) holds. The term in (3) is α -equivalent to k ⊢ newD h := k in newD h′ := k in M; skip# ; M ′ . Consider [[h, h′ ⊢ M; skip# ; M ′ ]] ◦ [[(h 7→ v) ⊗ (h′ 7→ v′ )]], where v, v′ ∈ D. By Consistency, we have that ∃ w1 ∈ [[h, h′ ⊢ M]] ◦ [[(h 7→ v) ⊗ (h′ 7→ v′ )]] such that | w1 |= n and w1 leaves the state (h 7→ v1 ) ⊗ (h′ 7→ v′ ), and ∃ w2 ∈ [[h, h′ ⊢ M ′ ]] ◦ [[(h 7→ v1 ) ⊗ (h′ 7→ v′ )]] such that | w2 |= n and w2 leaves the state (h 7→ v1 ) ⊗ (h′ 7→ v′1 ). Any word w ∈ R is obtained from w1 and w2 as above (| w1 |=| w2 |), and so satisfies the requirements of the theorem. We can detect timing leaks from a semi-closed term by verifying that all words in the model in (3) are in the required form. To do this, we restrict our attention only to the costs of words in R. Example 1. Consider the term: h : var int2 ⊢ if (!h > 0) then h := !h + 1; else skip : com The slot-game semantics of this term extended as in (3) is: run · kvar · qk · 0k · kseq · # · kseq · kvar · qk · (0k · done + 1k · kder · k+ · done)
+1k · kseq · kder · k+ · # · kseq · kvar · qk · (0k · done + 1k · kder · k+ · done) This model includes all possible observable interactions of the term with its environment, which contains only the identifier k, along with the costs measuring its running time. Note that the first value for k read from the environment is used to initialize h, while the second value for k is used to initialize h′ . By inspecting we can see that the model contains the word: run · kvar · qk · 0k · kseq · # · kseq · kvar · qk · 1k · kder · k+ · done which is not of the required form. This word (play) corresponds to two computations of the given term where initial values of h are 0 and 1 respectively, such that the cost of the second computation has additional kder + k+ units more than the first one. We now show how to detect timing leaks of a split (open) term h : varD | ∆ ⊢ M : com, where ∆ = x1 : T1 , . . . , xk : Tk . To do this, we need to check timing efficiency of the following model: [[h, h′ : varD ⊢ M[N1 /x1 , . . . , Nk /xk ]; skip# ; M ′ [N1 /x1 , . . . , Nk /xk ]]] (4) at state (h 7→ v, h′ 7→ v′ ), for any closed terms ⊢ N1 : T1 , . . . , ⊢ Nk : Tk , and for any values v, v′ ∈ D. As we have shown slot-game semantics respects theory of operational improvement, so we will need to examine whether all its complete plays-with-costs s are of the form s1 · # · s2 where | s1 |=| s2 |. However, the model in (4) can not be represented as a regular language, so it can not be used directly for detecting timing leaks. Let us consider more closely the slot-game model in (4). Terms M and M ′ are run in the same context ∆, which means that each occurrence of a free identifier xi from ∆ behaves uniformly in both M and M ′ . So any complete play-with-costs of the model in (4) will be a concatenation of complete plays-with-costs from models for M and M ′ with additional constraints that behaviours of free identifiers from ∆ are the same in M and M ′ . If these additional constraints are removed from the above model, then we generate a model which is an over-approximation of it and where free identifiers from ∆ can behave freely in M and M ′ . Thus we obtain: [[h, h′ : varD ⊢ M[N1 /x1 , . . . , Nk /xk ]; skip# ; M ′ [N1 /x1 , . . . , Nk /xk ]]] ⊆ [[h, h′ : varD ⊢ M; skip# ; M ′ [N1 /x1 , . . . , Nk /xk ]]] 174 Slot Games for Detecting Timing Leaks of Programs If ⊢ N1 : T1 , . . . , ⊢ Nk : Tk are arbitrary closed terms, then they are interpreted by identity (copy-cat) strategies corresponding to their types, and so we have: [[h, h′ : varD ⊢ M; skip# ; M ′ [N1 /x1 , . . . , Nk /xk ]]] = [[h, h′ : varD, ∆ ⊢ M; skip# ; M ′ ]] This model is a regular language and we can use it to detect timing leaks. Theorem 5. Let h : varD | ∆ ⊢ M : com be a split (open) term, where ∆ = x1 : T1 , . . . , xk : Tk , and S = [[k : expD, ∆ ⊢ newD h := k in M; skip# ; newD h′ := k in M ′ : com]] (5) If any word of S is of the form w = w1 · # · w2 such that | w1 |=| w2 |, Then h : varD | ∆ ⊢ M has no timing leaks. Note that the opposite direction in the above result does not hold. That is, if there exists a word from S which is not of the required form then it does not follow that M has timing leaks, since the found word (play) may be spurious introduced due to over-approximation in the model in (5), and so it may be not present in the model in (4). Example 2. Consider the term: h : varint2 , f : expint2 f ,1 → comf ⊢ f (!h) : com where f is a non-local call-by-name function. The slot-game model for this term is as follows: run · kapp · runf · (qf ,1 · kder · readh · (0h · 0f ,1 + 1h · 1f ,1 ))∗ · donef · done Once f is called, it may evaluate its argument, zero or more times, and then it terminates successfully. Notice that moves tagged with f represent the actions of calling and returning from the function f , while moves tagged with f , 1 indicate actions of the first argument of f . If we generate the slot-game model of this term extended as in (5), we obtain a word which is not in the required form: run · kvar · qk · 0k · kapp · runf · qf ,1 · kder · 0f ,1 · donef · kseq · # · kseq · kvar · qk · 1k · kapp · runf · donef · done This word corresponds to two computations of the term, where the first one calls f which evaluates its argument once, and the second calls f which does not evaluate its argument at all. The first computation will have the cost of kder units more that the second one. However, this is a spurious counter-example, since f does not behave uniformly in the two computations, i.e. it calls its argument in the first but not in the second computation. To handle this problem, we can generate an under-approximation of the model given in (4) which can be represented as a regular language. Let h : varD | ∆ ⊢ M be a term derived without using the contraction rule for any identifier from ∆. Consider the following model: [[h, h′ : varD | ∆ ⊢ M; skip# ; M ′ ]]m = [[h, h′ : varD | ∆ ⊢ M; skip# ; M ′ ]] ∩ (deltaxT11 ,m ⊲⊳ . . . ⊲⊳ deltaxTkk ,m ⊲⊳ (A[[h,h′ :varD⊢com]] + $ )∗ ) (6) where m ≥ 0 denotes the number of times that free identifiers of function types may evaluate its arguments at most. The regular expressions deltaT,m are used to repeat zero or once an arbitrary behaviour for terms of type T, and are defined as follows. deltaexpD,0 = q · ∑n∈D n · (ε + q · n) deltacom,0 = run · done · (ε + run · done) deltavarD,0 = (read · ∑n∈D n · (ε + read · n)) + (∑n∈D write(n) · ok · (ε + write(n) · ok)) A. S. Dimovski 175 If T is a first-order function type, then deltaT,m will be a regular language only when the number of times its arguments can be evaluated is limited. For example, we have that: m deltacom1 →com,m = run · ∑ (run1 · done1 )r · done · (ε + run · (run1 · done1 )r · done) r=0 If T is a function type with k arguments, then we have to remember not only how many times arguments are evaluated in the first call, but also the exact order in which arguments are evaluated. Notice that we allow an arbitrary behavior of type T to be repeated zero or once in deltaT,m , since it is possible that depending on the current value of h an occurrence of a free identifier from ∆ to be run in M but not in M ′ , or vice versa. For example, consider the term: h : var int2 | x, y : exp int2 ⊢ newint2 z := 0 in if (!h > 0) then z := x else z := y + 1 This term has timing leaks, and the corresponding counter-example contains only one interaction with x occurred in a computation, and one interaction with y occurred in the other computation. This counterexample will be included in the model in (6), only if deltaT,m is defined as above. Let h : varD | ∆ ⊢ M be an arbitrary term where identifiers from ∆ may occur more than once in M. Let h : varD | ∆1 ⊢ M1 be derived without using the contraction for ∆1 , such that h : varD | ∆ ⊢ M is obtained from it by applying one or more times the contraction rule for identifiers from ∆. Then [[h, h′ : varD | ∆ ⊢ M; skip# ; M ′ ]]m is obtained by first computing [[h, h′ : varD | ∆1 ⊢ M1 ; skip# ; M1′ ]]m as defined in (6), and then by suitable tagging all moves associated with several occurrences of the same identifier from ∆ as described in the interpretation of contraction. We have that: [[h, h′ : varD, ∆ ⊢ M; skip# ; M ′ ]]m ⊆ [[h, h′ : varD ⊢ M[N1 /x1 , . . . , Nk /xk ]; skip# ; M ′ [N1 /x1 , . . . , Nk /xk ]]] for any m ≥ 0 and arbitrary closed terms ⊢ N1 : T1 , . . . , ⊢ Nk : Tk . In the case that ∆ contains only identifiers of base types B which do not occur in any while-subterm of M, then in the above formula the subset relation becomes the equality for m = 0. If a free identifier occurs in a while-subterm of M, then it can be called arbitrary many times in M, and so we cannot reproduce its behaviour in M ′ . Theorem 6. Let h : varD | ∆ ⊢ M be a split (open) term, where ∆ = x1 : T1 , . . . , xk : Tk , and T = [[k : expD, ∆ ⊢ newD h := k in M; skip# ; newD h′ := k in M ′ : com]]m (7) (i) Let ∆ contains only identifiers of base types B, which do not occur in any while-subterm of M. Any word of T (where m = 0) is of the form w1 · # · w2 such that | w1 |=| w2 | iff M has no timing leaks. (ii) Let ∆ be an arbitrary context. If there exists a word w = w1 · # · w2 ∈ T such that | w1 |6=| w2 |, Then M does have timing leaks. Note that if a counter-example witnessing a timing leakage is found, then it provides a specific context ∆, i.e. a concrete definition of identifiers from ∆, for which the given open term have timing leaks. 5 Detecting Timing-Aware Non-interference The slot-game semantics model contains enough information to check the non-interference property of terms along with timing leaks. The method for verifying the non-interference property is analogous to Slot Games for Detecting Timing Leaks of Programs 176 the one described in [8], where we use the standard game semantics model. As slot-game semantics can be considered as the standard game semantics augmented with the information about quantitative assessment of time usage, we can use it as underlying model for detection of both non-interference property and timing leaks, which we call timing-aware non-interference. In what follows, we show how to verify timing-aware non-interference property for closed terms. In the case of open terms, the method can be extended straightforwardly by following the same ideas for handling open terms described in Section 4. Let l : varD, h : varD′ ⊢ M : com be a term where l and h represent low- and high-security global variables respectively. We define Γ1 = l : varD, h : varD′ , Γ′1 = l′ : varD, h′ : varD′ , and M ′ is α -equivalent to M[l′ /l, h′ /h] where all bound variables are suitable renamed. We say that Γ1 | − ⊢ M : com satisfies timing-aware non-interference if ∀ s1 ∈ St(Γ1 ), s2 ∈ St(Γ′1 ). s1 (l) = s2 (l′ ) ∧ s1 (h) 6= s2 (h′ ) ∧ Γ1 ⊢ M; M ′ , s1 ⊗ s2 n1 skip; M ′ , s1 ′ ⊗ s2 ⇒ s′1 (l) = s′2 (l′ ) ∧ n1 = n2 n2 skip; skip, s1 ′ ⊗ s2 ′ Suppose that abort is a special free identifier of type comabort in Γ. We say that a term Γ ⊢ M is safe 4 iff Γ ⊢ M[skip/abort] ⊏ ∼ M[diverge/abort] ; otherwise we say that a term is unsafe. It has been shown in abort , which we [5] that a term Γ ⊢ M is safe iff [[Γ ⊢ M]] does not contain any play with moves from A[[com]] call unsafe plays. For example, [[abort : comabort ⊢ skip ; abort : com]] = run · runabort · doneabort · done, so this term is unsafe. By using Theorem 4 from Section 4 and the corresponding result for closed terms from [8], it is easy to show the following result. L = [[k : expD, k′ : expD′ , abort : com ⊢ newD l := k in newD′ h := k′ in newD l′ := !l in newD′ h′ := k′ in skip# ; M; skip# ; M ′ ; skip# ; if (!l 6=!l′ ) then abort : com]] (8) The regular expression L contains no unsafe word (plays) and all its words are of the form w = w1 · # · w2 · # · w3 · # · w4 such that | w2 |=| w3 | iff M satisfies the timing-aware non-interference property. Notice that the free identifier k in (8) is used to initialize the variables l and l′ to any value from D which is the same for both l and l′ , while k′ is used to initialize h and h′ to any values from D′ . The last if command is used to check values of l and l′ in the final state after evaluating the term in (8). If their values are different, then abort is run. 6 Application We can also represent slot-game semantics model of IA2 by using the CSP process algebra. This can be done by extending the CSP representation of standard game semantics given in [6], by attaching the costs corresponding to each translation rule. In the same way, we have adapted the verification tool in [6] to automatically convert an IA2 term into a CSP process [17] that represents its slot-game semantics. The CSP process outputted by our tool is defined by a script in machine readable CSP which can be analyzed by the FDR tool. It represents a model checker for the CSP process algebra, and in this way a range of properties of terms can be verified by calls to it. 4⊏ ∼ denotes observational approximation of terms (see [1]) A. S. Dimovski 177 $ read x[0] run read h 0 x[0] 1h 0h read x[0] done 1x[0] 0 x[0] 1x[0] $ read x[1] x[1] 0,1 read x[1] $ 0,1x[1] Figure 1: Slot-game semantics for the linear search with k=2 In the input syntax of terms, we use simple type annotations to indicate what finite sets of integers will be used to model free identifiers and local variables of type integer. An operation between values of types intn1 and intn2 produces a value of type intmax{n1 ,n2 } . The operation is performed modulo max{n1 , n2 }. In order to use this tool to check for timing leaks in terms, we need to encode the required property as a CSP process (i.e. regular-language). This can be done only if we know the cost of the worst plays (paths) in the model of a given term. We can calculate the worst-case cost of a term by generating its model, and then by counting the number of tokens in its plays. The property we want to check will be: ∑ni=0 i · # · i , where n denotes the worst-case cost of a term. To demonstrate practicality of this approach for automated verification, we consider the following implementation of the linear-search algorithm. h : varint2 , x[k] : varint2 ⊢ newint2 a[k] := 0 in newintk+1 i := 0 in while (i < k) do {a[i] :=!x[i]; i :=!i + 1; } newint2 y := !h in newbool present := ff in while (i < k && ¬present) do { if (compare(!a[i], !y)) then present := tt; i :=!i + 1; } : com The meta variable k > 0 represents the array size. The term copies the input array x into a local array a, and the input value of h into a local variable y. The linear-search algorithm is then used to find whether the value stored in y is in the local array. At the moment when the value is found in the array, the term terminates successfully. Note that arrays are introduced in the model as syntactic sugar by using existing term formers. So an array x[k] is represented as a set of k distinct variables x[0], . . . , x[k − 1] (see [6, 10] for details). Suppose that we are only interested in measuring the efficiency of the term relative to the number of compare operations. It is defined as follows compare : expint2 → expint2 → expbool, and its semantics compares for equality the values of two arguments with cost $ : [[compare : expint12 → expint22 → expbool]] = q · $ · q1 · (∑m6=n m1 · q2 · n2 · ff ) + (∑m=n m1 · q2 · n2 · tt) where m, n ∈ {0, 1}. We assume that the costs of all other operations are relatively negligible (e.g. kvar = kder = . . . = 0). Slot Games for Detecting Timing Leaks of Programs 178 We show the model for this term with k = 2 in Fig. 1. The worst-case cost of this term is equal to the array’s size k, which occurs when the search fails or the value of h is compared with all elements of the array. We can perform a security analysis for this term by considering the model extended as in (7), where m = 0. We obtain that this term has timing leaks, with a counter-example corresponding to two computations, such that initial values of h are different, and the search succeeds in the one after only one iteration of while and fails in the other. For example, this will happen when all values in the array x are 0’s, and the value of h is 0 in the first computation and 1 in the second one. We can also automatically analyse in an analogous way terms where the array size k is much larger. Also the set of data that can be stored into the global variable h and array x can be larger than {0, 1}. In these cases we will obtain models with much bigger number of states, but they still can be automatically analysed by calls to the FDR tool. 7 Conclusion In this paper we have described how game semantics can be used for verifying security properties of open sequential programs, such as timing leaks and non-interference. This approach can be extended to terms with infinite data types, such as integers, by using some of the existing methods and tools based on game semantics for verifying such terms. Counter-example guided abstraction refinement procedure (ARP) [5] and symbolic representation of game semantics model [7] are two methods which can be used for this aim. The technical apparatus introduced here applies not only to time as a resource but to any other observable resource, such as power or heating of the processor. They can all be modeled in the framework of slot games and checked for information leaks. We have focussed here on analysing the IA language, but we can easily extend this approach to any other language for which game semantics exists. Since fully abstract game semantics was also defined for probabilistic [4], concurrent [12], and programs with exceptions [1], it will be interesting to extend this approach to such programs. 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