Ground Reducibility is EXPTIME-Complete We prove that ground reducibility is EXPTIME-complete Ground reducibility of a term t w.r.t. a term rewriting system R expresses that all ground instances (instances without variables) of t are reducible by R. This property is fundamental in automating inductive proofs in equational theories without constructors [10] . It is also related to sufficient completeness in algebraic specifications (see e.g. [12] ). Roughly, it expresses that all cases have been covered by R and that t will be reducible for any inputs. Many papers have been devoted to decision of ground reducibility. Let us report a brief history of the milestones, starting only in 1985 with the general case.. Ground reducibility was first shown decidable by D. Plaisted [14] . The algorithm is however quite complex: a tower of 9 exponentials though there is no explicit complexity analysis in the paper. D. Kapur et al. gave another decidability proof [12] which is conceptually simpler, though still very complicated, and whose complexity is a tower of 7 exponentials in the size of R, t. More precisely, they show that checking the reducibility of all ground instances of t can be reduced to checking the reducibility of all ground instances of t of depth smaller than N(R) where N(R) is a tower of 5 exponentials in the size of R. A third proof was. proposed by E. Kounalis in [13] . The result is generalized to co-ground reducibility and the expected complexity is 5 exponentials, though there is no explicit complexity analysis in the paper. These three algorithms use combinatorial arguments and some "pumping property": if there is a deep enough irreducible instance of t, then there is also a smaller instance which is also irreducible. This yielded the idea of making explicit the pumping argument as a pumping lemma in some tree language. In support of this idea, when both t and the left members of R are linear, i.e. each variable appears only once, then the set of reducible instances of t is accepted by a finite tree automaton [8] . Hence the set of irreducible ground instances is also accepted by a tree automaton, by complement. This easily gives a simply exponential algorithm in the linear case. (As we will see this algorithm is optimal).. H. Comon expressed first the problem of ground reducibility as an emptiness problem for some tree language [3] . He also gave a decision proof whose complexity is even worse than the former ones. A.-C Caron, J.-L. Coquidé and M. Dauchet proved a very beautiful result in 1993 [2, 5] , enlighting the pumping properties and their difficulty. They actually show a more general result: the first-order theory of unary encompassment predicates is decidable. And it turns out that ground reducibility can be expressed as a simple formula in this logic. Their technique consists in associating an automaton with each formula, in the spirit of Büchi's and Rabin's method. The kind of automata which is appropriate here is what they call reduction automata, a particular case of automata with constraints introduced by M. Dauchet in 1981. Such tree automata have the ability to check for equality or disequality of some subtrees before applying a transition rule. In general, emptiness of languages recognized by such automata is undecidable. However, when we only allow a fixed number of equality tests on each computation branch, then emptiness becomes decidable. Unfortunately, their result does not give any information about possible efficient algorithms. The complexity which results from their proof is not better than Plaisted's bound. We tried to specialize the tree automata technique for ground reducibility and we got in this way a triple exponential bound [4] . This is better than previous methods, but still far from the lower bound.. The problem in all works about ground reducibility is that they give a bound on the depth of a minimal irreducible instance of t (or a minimal term accepted by the automaton).. However, after establishing carefully such an upper bound, they use a brute-force algorithm, checking the reducibility of all terms of depth smaller than the bound, which increases the complexity by a double exponential.. We use here a different approach. We still rely on automata with disequality constraints. However, we do not try to give a bound on the depth of an accepted term. Rather, we show a stronger result: with an appropriate notion of minimality, a minimal term accepted by the automaton contains at most an exponential number of distinct subterms. To prove this, we use a generalization of pumping to arbitrary replacements for which the term is decreasing according to some well chosen well founded ordering. With a few more ingredients, this yields an algorithm for deciding the emptiness of an automaton with disequality constraints which runs in polynomial time w.r.t. the number of states and in exponential time w.r.t. the size of the constraints.. On the other hand, we show that ground reducibility of t w.r.t. R can be reduced to the emptiness problem for an automaton A with disequality constraints whose number of states is an exponential in the size of R and t and whose constraints are polynomial in size. Altogether, we have a simply exponential algorithm for ground reducibility.. This result is optimal since ground reducibility is EXPTIME-hard, already for linear rewrite systems and linear t. A O(2 n log n ) lower bound was proved by Kapur et al [11] . We give here a simple proof of EXPTIME-hardness: the computations of alternating Turing machines with polynomially bounded space can be encoded (in polynomial time) into ground reducibility. Hence, quite surprisingly, ground reducibility is not (at least in theory) harder in the general case than in the linear case.. In section 2 we recall the definition of automata with disequality constraints. In section 3, we show how to construct an automaton with disequality constraints whose emptiness is equivalent to the ground reducibility of t w.r.t. R and we analyse carefuly the complexity of such a construction. Section 4 is devoted to pumpings lemmas for automata with disequality constraints. These lemmas are applied in section 5 to derive an optimal algorithm which checks the emptiness of (the language recognized by) an automaton with disequality constraints. Finally, we study the lower bound of ground reducibility in section 6.. Some proofs are long and technical, hence not included in the present abstract. Except for a small combinatorial argument due to B. Reed, the complete proofs can be found in F. Jacquemard's thesis [9] (in French). A complete version of the paper is also available on the Web (http://www.mpi-sb.de/ florent/art/). We proved that ground reducibility is EXPTIMEcomplete for both the linear and the non-linear case. This closes a pending question. However, we do not claim that this result in itself gives any hint on how to implement a ground reducibility test. As we have seen, it is not tractable in general. A possible way to implement these techniques as efficiently as possible was suggested in [1] . On the average, some algorithms may behave well. In any case, we claim that tree automata help both in theory and in practice.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O


Factoring rational polynomials over the complexes We give NC algorithms for determining the number and degrees of the absolute factors (factors irreducible over the complex numbers C) of a multivariate polynomial with rational coefficients. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polyne mial are its degree d, coefficient length c, number of variables ra, and for sparse polynomials, the number of non-zero coefficients s. For the general case, we give a random (Monte-Carlo) NC algorithm in these input measures. If n is fixed, or if the polynomial is dense, we give a deterministic NC algorithm. The algorithm also works in random NC for polynomials represented by straight-line programs, provided the polynomial can be evaluated at integer points in NC. Finally, we discuss a method for obtaining an approximation to the coefficients of each factor whose running time is polynomial in the size of the original (dense) polynomial.These methods rely on the fact that the connected components of a complex hypersurface P(zl, . . . , zn) = 0 minus its singular points correspond to the absolute factors of P. Factoring polynomials is an important problem in symbolic computation with applications as diverse as theorem proving and computer-aided design. Methods for factoring polynomials with rational coefficients over the rational numbers are wellknown. [Lenstra 82, Kaltofen 85b ] establish that factoring polynomials in a fixed number of indeterminates over the field of rational numbers Q is in polynomial time.. However, factoring polynomials over C differs from factoring over Q. For example, x2 + 2y2 is irreducible over Q. However, x2 + 2y2 = (x + fiiy)(z -v%y) when factored over C. This example illustrates one difficulty in factoring over C. The coefficients in an exact factorization over C must be represented symbolically (possibly by polynomials of high degree).. Work on factoring rational polynomials over C has not been as extensive as that of factorization over Q. [Noether 22, Davenport 81, Heintz 811 each give methods that require time exponential in the degree of the input polynomial.. [DiCrescenzo 84,Duval 871 give geometric methods of factorization based on algebraic geometry. [Kaltofen SSa] describes an NC method for testing whether a rational polynomial is irreducible over C, The method involves computing approximate roots and their corresponding minimum polynomials. The first polynomial time algorithm for factoring over C seems to have been [Chistov 831. However, it has remained an open problem whether computing the number of factors, irreducible over C, of a rational polynomial is in NC.. Given a polynomial P with rational coefficients, the input size is measured by number of variables n, degree d, coefficient size c, and number of non-zero coefficients 8. We show that the general problem of computing number and degrees of the factors is in random NC in these measures, in the Monte-Carlo sense (definitely fast, probably correct). If the num-ber of variables is fixed, or if the polynomial P is dense, we give a deterministic NC solution. Finally, if the polynomial is represented as a straight-line program of length p our algorithm runs in random NC plus the time to evaluate the polynomial at an integer point. By the parallelization result of Valiant et al. [Valiant 831, program of size p and degree d can be converted into an equivalent program of polynomial size, and polylogarithmic depth in d and p, which can therefore be evaluated in NC. However, the conversion itself is not in NC, and seems intrinsically sequentiai, because of constant evaluation which is P-complete. So we cannot run our algorithm in random NC for straight-line program polynomials unless we are given a program of low depth.. Finally, we discuss a method for obtaining an approximation to the coefficients of each factor whose running time is polynomial if the polynomial P ia dense. Previous methods of factoring have typically relied on an algebraic approach.. We take a geometric approach, relying on the fact, described in section two, that the number of connected components of a complet hypersurface P(z1,. . . , zn) = 0 minus its singular points is precisely the number of factors, irreducible over C, of P(zl, . . . ,z,). In section three, we describe a fast parallel method for reducing the factorization problem for P(aI, . . . , zn) to the bivariate case. In section four, we describe a fast parallel method for determining the number of connected components of P(z1, zz) minus its singular points. This computation can be done using the sign sequences associated with various Sturm sequences.

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Symmetry detection by generalized complex (GC) moments: a close-form solution Abstract-This paper presents a unified method for detecting both reflection-symmetry and rotation-symmetry of 2D images based on an identical set of features, i.e., the first three nonzero generalized complex (GC) moments. This method is theoretically guaranteed to detect all the axes of symmetries of every 2D image, if more nonzero GC moments are used in the feature set. Furthermore, we establish the relationship between reflectional symmetry and rotational symmetry in an image, which can be used to check the correctness of symmetry detection. This method has been demonstrated experimentally using more than 200 images. SYMMETRY, as one of the basic features of shapes and objects, has been studied extensively in the computer vision area. Symmetrical descriptions of shapes and the detection of symmetrical features of objects are very useful in guiding shape matching, model-based object matching and object recognition [1] , [2] , [3] , [4] . Symmetrical information is also useful in robotics for recognition, inspection, grasping, and reasoning [5] . Usually, symmetry is considered as a binary feature, i.e., an object is either symmetric or not symmetric. For quantifying the symmetry of objects, symmetry can also be a continuous feature [7] .. Rotational symmetry and reflectional symmetry are two commonly studied types of symmetries. A 2D image is said to be reflection-symmetric if it is invariant to reflection with respect to one or more straight lines, denoted as reflection-symmetric axes. A 2D image is said to be order K rotationally symmetric, if it is invariant under rotation of 2p/K radians about the center of mass of the image and K is the largest integer. This image is usually called rotationally symmetric image with K folds (K-RSI). The orientations of these K folds are defined in this paper by K fold axes. The following overview will clarify that none of the existing techniques could provide a complete solution to the problem in the sense that no algorithm yet exists for detecting all the reflection-symmetric axes and rotationally symmetric folds (or fold axes) of a 2D image.. Much of the research literature is concerned with reflectionsymmetry detection. Labonte et al. [11] considered the problem of detecting global bilateral symmetry. Zielke et al. [12] only looked at vertical or near-vertical symmetry axes in the image for car-following. Atallah [8] detected the axis of reflectional symmetry by first determining the centroid position and then using a string pattern matching technique, which considered all possible lines passing through the centroid. However, this method is only suitable for planar figures made up of segments, circles, points, etc. Marloa [6] presented an algorithm for finding the number and positions of the symmetry axes of a symmetric or almost-symmetric planar image. This technique required the evaluation of rational functions and addressed the detection of rotational symmetry. Sun [28] used orientation histograms for detecting reflection symmetry in gray-level image, but it can only detect simple bilateral symmetry.. The orientation detection methods, i.e., GPA [18] , FPA [19] , and UPA [20] , can only determine the rotational symmetries of very simple objects. Pei and Lin [13] introduced a modified Fourier descriptor to determine the fold number of RSI, based on the observation that the first nonzero Fourier coefficient is located on the fundamental frequency. However, this observation is not always true for certain types of images, i.e., the three images shown in Fig. 1 , their Fourier coefficients at their own fold number are zeros. Similar assumption was also implicitly suggested and used by Sun [17] . Lin [14] detected the fold number using a simple mathematical property. However, the detection of fold axes was not totally solved, which makes the technique unable to detect the fold axes of any of the images shown in Fig. 1 .. Masuda et al. [9] described a method of extracting both rotational and reflectional symmetries by performing correlation with the rotated and reflected images. This method incurs high computational cost and memory requirements, since all possible transformations have to be tried. Sun and Sherrah [10] used the extended images for 3D symmetry detection. They formulated the symmetry detection problem as a correlation of the Gaussian image. However, different processing and representation methods were needed, respectively, for reflectional and rotational symmetry detection, which obviously increased computation. Furthermore, this method is unable to determine whether the input image is symmetric or not.. The above review clearly shows that most of the existing methods can determine either reflectional symmetry or rotational symmetry. Even for those methods which are able to detect both reflectional and rotational symmetry, they rely on different processes. Currently, there is no method which uses an identical set of features to detect these two forms of symmetries. Furthermore, no method is guaranteed to detect all the symmetric axes of every 2D image. In this paper, we have proposed a unified method of using an identical set of features, i.e., nonzero GC moments, to detect all reflectional and fold axes of every 2D image. In practice, all the reflection-symmetric axes and the fold axes of almost every image can be determined by the first three nonzero GC moments and the proposed method can be used for the retrieval of complex geometrical shapes such as trademark images [27] . In this paper, we have presented a unified method for detecting symmetries of every image. Unlike previous approaches that require different methods to detect different types of symmetries, here we combine the detection of both reflectional-symmetry and rotational-symmetry into a single process by using the same set of GC moment features derived from the image. In practice, once the three nonzero GC moments have been calculated, the symmetry axes for reflectional and rotational symmetries can be efficiently determined. We have applied the techniques to over 200 images in our image database. Experimental results show that the proposed method can detect both reflectional symmetry and rotational symmetry of every image. Since GC moments extract global features of the image, our GC moments-based method also has the ability to detect symmetries of almost-symmetric image, even though the image is not exactly symmetric. Furthermore, the derived relationship between reflectional symmetry and rotational symmetry can be used to check the correctness of symmetry detection.. As a conclusion, the main contributions of this paper can be summarized as follows:. 1) The reflection-symmetric axes are actually the intersection of all axes sets defined by all the nonzero GC moments. 2) The fold number can be determined by calculating the biggest common factor of all orders making GC moments nonzero.. 3) The fold axes can be defined by both the phase and the order of the combined moment, which is generated from all the nonzero GC moments. 4) The relationship between reflectional symmetry and rotational symmetry has been derived. 5) Since GC moments extract global features of the image, the proposed method is able to detect the symmetries of almostsymmetric image. In the experiments above, all the test images are actually gray scale images, however they appear to be binary since their gray-level values are close to either black or white. In fact, the proposed techniques are suitable to detecting symmetries for gray scale images. To further illustrates this point, Fig. 10 shows two examples.. Weighting factor (r p ) in the GC moment definition is very important, which makes us select special orders p for special images. For example, p = 8 was used for Fig. 7b , p = 1 and p = 3 were used for Fig. 8c and Fig. 8d , respectively, and p = 3 and p = 1 were used for Fig. 9f and Fig. 9h , respectively. Since GC moments are not invariant to affine transform, we first make the affine-transformed object compact by using the Compact Algorithm in [15] , before applying GC moments to detect its symmetries. Additionally, there exist methods for detecting affine symmetries based on affine-invariant features defined on the contour of the object [26] .. Theoretically, our technique is applicable to object with higher degree of symmetries. However, it depends on the resolutions of the image due to the use of global features. For our case, the image size is 128 128 and we can detect the degree of symmetries up to K = 20. For our technique, the computation cost is mainly in the calculation of GC moments. No other computation is required besides logical checking. Also, the speed of our technique depends on the images themselves. Simple image needs less time compared with more complex images. The computation time for calculating one GC moment using a Pentium 233 PC is 0.05 sec. The robustness of our algorithm against random noises and errors is dependent on the robustness of computing the GC moments in the presence of noises. We have studied the robustness of GC moments computation in the presence of Gaussian noise [15] and shown that GC moments, as global features, are relatively robust to noises. As to uneven lighting effects, i.e., a symmetric object with one side darker than the other side, our technique can endure only a small amonut of nonuniform illuminations due to the use of global features. We have applied our algorithm to the almost-symmetric images, i.e., inexact images, in Subsection 5.3. These examples show that our algorithm can also detect inexact symmetry.

B I O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The Workshop on Internet Topology (WIT) Report Internet topology analysis has recently experienced a surge of interest in computer science, physics, and the mathematical sciences. However, researchers from these different disciplines tend to approach the same problem from different angles. As a result, the field of Internet topology analysis and modeling must untangle sets of inconsistent findings, conflicting claims, and contradicting statements.On May 10-12, 2006, CAIDA hosted the Workshop on Internet topology (WIT). By bringing together a group of researchers spanning the areas of computer science, physics, and the mathematical sciences, the workshop aimed to improve communication across these scientific disciplines, enable interdisciplinary crossfertilization, identify commonalities in the different approaches, promote synergy where it exists, and utilize the richness that results from exploring similar problems from multiple perspectives.This report describes the findings of the workshop, outlines a set of relevant open research problems identified by participants, and concludes with recommendations that can benefit all scientific communities interested in Internet topology research.

O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O


Analyzing change in spatial data by utilizing polygon models Analyzing change in spatial data is critical for many applications including developing early warning systems that monitor environmental conditions, epidemiology, crime monitoring, and automatic surveillance. In this paper, we present a framework for the detection and analysis of patterns of change; the framework analyzes change by comparing sets of polygons. A contour clustering algorithm is utilized to obtain polygon models from spatial datasets. A set of change predicates is introduced to analyze changes between different models which capture various types of changes, such as novel concepts, concept drift, and concept disappearance. We evaluate our framework in case studies that center on ozone pollution monitoring, and on diagnosing glaucoma from visual field analysis. With advances in data acquisition technologies, huge amount of spatial data are collected every day. Detecting and analyzing changes in such data is of particular importance and has many applications including developing early warning systems that monitor environmental or weather conditions, epidemiology, crime monitoring, automatic surveillance, emergency first responders' coordination for natural disasters. Addressing this need, we introduce a framework that includes methodologies and tools to discover and analyze change patterns in spatial data. Change patterns capture how the most recent data differ from the data model established from the historical data.. The main challenges for developing systems that automatically detect and analyze change in spatial datasets include:. 1. The development of a formal framework that characterizes different types of change patterns 2. The development of a methodology to detect change patterns in spatial datasets 3. The capability to find change patterns in regions of arbitrary shape and granularity 4. The development of scalable change pattern discovery algorithms that are able to cope with large data sizes and large numbers of patterns In this paper, we address a problem of change analysis in spatial data; a framework that detects and analyzes the patterns of change is proposed. We introduce a change analysis method that operates on cluster models generated by density and interpolation functions and a contour clustering algorithm. A set of change predicates is introduced that is capable of defining change of concepts in spatial data such as concept drift, novel concept, concept disappearance, and local rearrangement.. A proposed change analysis is conducted as follows. We introduce a framework for change analysis that utilizes polygons as cluster models to capture changes in spatial datasets between different time frames. The proposed framework have been applied to two real-world datasets to analyze the changes in the visual fields and the ozone concentrations. The framework consists of three modules: Cluster Generation, Model Generation, and Change Analysis. The Cluster Generation module creates contour clusters from newly arriving data using density or interpolation functions and a contour clustering algorithm that operates on those functions. The polygon models are updated in the Model Generation module to reflect changes in data that occur over time. The Change Analysis module uses sets of change predicates, which can be viewed as queries that detect and analyze changes through polygon operations. In general, the change analysis tool is highly generic and supports arbitrary sets of change predicates as long as they operate on sets of polygons. Our prototype implementation uses the Java-based Cougar^2 framework [2] , [5] ; however, we are beginning to re-implement the change analysis tool as a subcomponent of a spatial database system.. The case studies show that our framework can capture various kinds of changes in spatial datasets. We plan to implement a tool for optometrists to study various change patterns that are associated with different stages in the progression of glaucoma. The tool aims to assist the optometrists to better understand the development and progression of the disease. Our ultimate vision of this project is to use the tool to help diagnose glaucoma and to provide an expert system to assist optometry students in learning about the different stages of the disease.. Moreover, we plan to integrate our change analysis framework and change report generators into early warning systems so that alarms are raised and change reports are generated automatically when critical events are discovered. To accomplish this goal, efficient data-driven algorithms that integrate change analysis tools into early warning systems have to be developed. Recently, several techniques have been proposed to detect and analyze change patterns in different types of data. In work by Asur et al. [1] , a technique for mining evolutionary behavior of interaction graphs is proposed. Yang et al. [17] proposed a technique to discover the evolution of spatial objects. Association analysis is used to detect changes. In [7] , Jiang and Worboys propose a technique to detect topological changes in sensor networks. The topological changes that can be discovered include region appearance and disappearance, region merging and splitting, and hole formation and elimination. A framework for tracking cluster transition in data streams is proposed by Spiliopoulou et al in [13] . The proposed framework can detect external and internal transitions of clusters. Kifer et al. [11] present a method for the detection and estimation of change in data streams. The proposed algorithm reduces the problem of analyzing continuously streaming data to analyzing two static sample sets: a reference window and a sliding window containing current points in the data streams. The algorithm compares discrepancies between two windows by means of analyzing probability distributions.. In summary, the work in [17] and [7] is similar to our work in that they focus on spatial data. Unlike [17] which uses association analysis, our work utilizes cluster analysis. Similar to our technique, the techniques in [1] [17], [7] , [13] , and [6] analyze changes in data but require that the identity of objects must be known or restrict analysis to objects that are characterized by nominal attributes. In contrast, our work is applicable for data with unknown object identity and for datasets that contain numerical attributes.. There are many papers that address the problems of concept drift or novelty detection but the Spinosa et al. [14] are the first that tackles the problem of novelty detection in the presence of concept drift. The paper introduces a cluster-based technique that can detect novel concepts as well as deal with concept drift in data streams; cluster models are created by k-means clustering algorithm. Our framework, on the other hand, relies on contour clustering algorithm to create polygon models; moreover, polygons in our approach do not need to be convex. An advantage of our work over [14] is that our change predicates allow discovery of not only novelty and concept drift but also other types of change patterns.

O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O


Verified Bytecode Verification and Type-Certifying Compilation This article presents a type certifying compiler for a subset of Java and proves the type correctness of the bytecode it generates in the proof assistant Isabelle. The proof is performed by defining a type compiler that emits a type certificate and by showing a correspondence between bytecode and the certificate which entails welltyping. The basis for this work is an extensive formalization of the Java bytecode type system, which is first presented in an abstract, lattice-theoretic setting and then instantiated to Java types. This paper provides an in-depth analysis of type systems in compilation, by taking the Java source language and Java bytecode as examples and showing that the bytecode resulting from compiling a type correct source program yields type correct bytecode.. We do not cover all language constructs of Java and neglect some subtleties, in particular exceptions and the jump-subroutine mechanism, while otherwise using a faithful model of Java and the Java Virtual Machine (JVM). It is an advance of this work over previous investigations of this kind that the definitions and proofs have been done entirely within the Isabelle verification assistant, resulting in greater conceptual clarity, as far as notation is concerned, and a more precise statement of theorems and proofs than can be achieved with pencil-and-paper formalizations (see Section 7 for a discussion).. Type correctness of bytecode produced by our compiler, comp, is proved by having a type compiler, compTp, emit a type certificate and showing that this certificate is a correct type of the code, in a sense to be made precise. This type certificate is related to (even though not identical with) what would be inferred by a bytecode verifier (BCV). Transmitting such a certificate along with bytecode and then checking its correctness is an attractive alternative to full bytecode verification, in particular for devices with restricted resources such as smart cards. The idea of using separate type certificates is not novel (see the concept of "lightweight bytecode verification" [RR98, KN01, Ros02] ); however, we are not aware of a Java compiler other than ours which explicitly generates them.. Apart from this potential application, compilation of types, in analogy to compilation of code, gives insight into type systems of programming languages and how they are related. Incompatibilities discovered in the source and bytecode type systems of Java [SS01] demonstrate the need for such a study. Even though these inconsistencies do not arise in the language subset we examine, we hope to cover larger fragments with the same techniques as presented below.. The work described here is part of a larger effort aiming at formalizing diverse aspects of the Java language, such as its operational and axiomatic semantics [Ohe01] , its bytecode type system and bytecode verifier [Kle03] and the correctness (in the sense of preservation of semantics) of a compiler [Str02a] .. This article extends on a previous paper [Str02b] by providing an in-depth exposition of the bytecode type system and by presenting large parts of the formalization of the type compiler and a detailed discussion of the type preservation proof. As far as we are aware, ours is the first fully formal treatment covering these aspects of Java in one comprehensive model.. In the following, we will first summarize the most important concepts of our Java and JVM formalization (Section 2). The bytecode type system and its relation to bytecode verification are further elaborated in Section 3. We define the code compiler comp in Section 4, the type compiler compTp in Section 5. The type correctness statement for generated code and a detailed discussion of the proof follow in Section 6. Section 7 concludes with a discussion of related and future work.. Due to space limitations, we can only sketch our formalization. The full Isabelle sources are available from http://isabelle.in.tum.de/verificard/. Given the above correctness theorem, the question arises whether semantically correct code could be type-incorrect. Quite abstractly, note that a type system always imposes a constraint on a language, thus marking even "valid" programs as type-incorrect. And indeed, the empirical evidence given in [SS01] shows that there is a mismatch between the Java source and bytecode type systems: Code containing a try . . . finally statement is accepted by a standard Java typechecker, compiled to semantically equivalent bytecode, but then rejected by the bytecode verifier. The deeper reason is that source code typechecker and bytecode verifier have different notions of when variables have been "definitely assigned" [BGJS00] . Since our restricted language fragment does not contain the above-mentioned construct, we cannot reproduce this problem.. Still, there are sufficient sources of potential bytecode type errors. For example, different branches leading to a jump target could generate operand stacks of different heights -possibly an innocuous situation if not all of the operand stack is used in the sequel (see Figure 11) . However, such code is rejected by the bytecode verifier. In this paper, we have defined a type certifying compiler and shown the type correctness of the code it generates. Even though the definitions are given in the proof assistant Isabelle, we can convert them to executable ML code using Isabelle's extraction facility [BN00, BS03] .. Our encoding of the Java source language owes much to the formalization on paper in [DE99] , which has also been the basis for an alternative formalization, including a type soundness proof, in the Declare system [Sym99] . Both differ from our definition in that they use a small-step operational semantics. Each approach has particular merits: A small-step semantics is suitable for modelling non-terminating computations and concurrency; type soundness can be defined in terms of a defensive machine, as for the JVM in Section 2.3.3. However, it is often clumsy to handle: For stating a compiler correctness theorem, we would have needed a complex bisimulation relation between source and target states. In [PV98] , Posegga and Vogt look at bytecode verification from a model checking perspective. They transform a given bytecode program into a finite state machine and check type safety, which they phrase in terms of temporal logic, by using an off-the-shelf model checker. Basin, Friedrich, and Gawkowski [BFG02] use Isabelle/HOL, µJava, and the abstract BCV framework [Nip01] to prove the model checking approach correct.. Working towards a verified implementation in Specware, Qian, Goldberg and Coglio have specified and analyzed large portions of the bytecode verifier [CGQ98,CGQ00]. Goldberg [Gol98] rephrases and generalizes the overly concrete description of the BCV given in the JVM specification [LY99] as an instance of a generic data flow framework. Qian [Qia99] specifies the BCV as a set of typing rules, a subset of which was proved correct formally by Pusch [Pus99] . Qian [Qia00] also proves the correctness of an algorithm for turning his type checking rules into a data flow analyzer.. Stata and Abadi [SA98] were the first to specify a type system for a subset of Java bytecode. They focused on the problem of bytecode subroutines. The typing rules they use are clearer and more precise than the JVM specification, but they accept fewer safe programs.. Freund and Mitchell [FM98,FM99,Fre00] develop typing rules for increasingly large subsets of the JVM, including exception handling, object initialization, and subroutines. They do not look at compilation.. Leroy [Ler01, Ler03] gives a very good overview on bytecode verification, and proposes a polyvariant data flow analysis in the BCV to solve the notorious subroutine problem. Coglio [Cog01, Cog02] provides an even simpler analysis for handling subroutines in the BCV. The most recent version of our BCV [Kle03] uses this scheme as basis for the formalization of subroutines in µJava.. Compiler correctness proofs have for a long time been an active research area, starting with pencil-and-paper proofs for a simple expression language [MP67] , and more recently using diverse specification formalisms such as Z [Ste98] and verification systems such as ACL2 [You89, Goe00] , HOL [Cur93] and PVS [DV01] . Little attention is given to preservation of type correctness, which is not surprizing since the source language (such as Lisp) or the target language only have a weak type system. The Verifix project [GZ99] has attempted to develop an appropriate compiler correctness criterion for finite resources (such as memory) and nondeterministic programs. Neither of these is a problem in our case. In particular, our source and target language are abstracted over the same memory model.. In recent years, compilation with types has been the subject of intense study, which however has mostly ignored the aspect of general compiler correctness and instead focused on the preservation of certain safety properties. The source languages are mostly functional, having ML-like [Mor95, SA95] type systems or even stronger ones such as System F [MWCG99] . The purpose is to exploit types for a program analysis that allows for more efficient closure conversion [WDMT97] or that avoids boxing polymorphic variables. In [LST02] , Java is compiled not to bytecode, but to a functional intermediate language which can also be used as the target of functional programming languages [LST03] .. Since compilation is a multi-stage process involving several intermediate languages, well-typing of programs has to be preserved during compilation. The type correctness statement is mostly proved on paper. Critical questions such as naming of bound variables and α-convertibility are often glossed over, even though there is good evidence that proofs of typing properties of lambda calculi become quite demanding once these details are taken into account [NN99] .. The extensive pencil-and-paper formalization of Java using Abstract State Machines in [SSB01] is complementary to ours: whereas the ASM formalization is much more complete with respect to language features, the proofs are less detailed, and some of the underlying proof principles are unclear, such as, for example, extending inductive proofs "modularly" to deal with new language constructs.

O B I O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


About priority encoding transmission Abstract-Recently, Albanese et al. introduced priority encoding transmission (PET) for sending hierarchically organized messages over lossy packet-based computer networks [1]. In a PET system, each symbol in the message is assigned a priority which determines the minimal number of codeword symbols that is required to recover that symbol. This note revisits the PET approach using tools from network information theory. We first outline that priority encoding transmission is intimately related with the broadcast erasure channel with degraded message set. Using the information spectrum approach, we provide an informational characterization of the capacity region of general broadcast channels with degraded message set. We show that the PET inequality has an information-theoretical counterpart: The inequality defining the capacity region of the broadcast erasure channel with degraded message sets. Hence the PET approach which consists in time-sharing and interleaving classical erasure-resilient codes achieves the capacity region of this channel. Moreover, we show that the PET approach may achieve the sphere packing exponents. Finally, we observe that on some simple nonstationary broadcast channels, time-sharing may be outperformed. The impact of memory on the optimality of the PET approach remains elusive. The quality of packet voice and image on the Internet has been mediocre due, in part, to congestion-induced packet losses. From the end-user viewpoint, the Internet can actually be modeled as an erasure channel acting over the large input alphabet formed by IP packets. On an erasure channel, each input symbol is either faithfully transmitted or erased, independently from its value. The output alphabet contains the input alphabet plus a special symbol denoting erasure. Although retransmission upon request, automatic repeat request (ARQ) has traditionally been the way to turn computer networks into reliable channels, the delay requirement of multimedia applications eliminates the possibility of retransmission and renews the interest for forward error correcting (FEC) [2] . Potential users of FEC also have to take into account that standard multimedia compression techniques [4] introduce a hierarchical structure in the information source.. The priority encoding transmission (PET) approach has been motivated by the search for robust multicasting of digital video sequences conforming to the MPEG standard [1] . In a first approximation, the different sorts of frames constituting a GOP (group of pictures in the MPEG methodology [4] ) are assumed to form independent sources (this would be true if compression were perfect). When using PET, the source information is protected in such a way that even receivers undergoing high loss rates can reconstruct essential parts of the source flow (for example, I-frames), while receivers undergoing lower loss rates can reconstruct most of the flow.. This note outlines the connection between the pragmatically motivated PET approach and the broadcast channel with degraded Manuscript received October 21, 1997; revised September 22, 1999. This research was supported in part by CNET under Grant 96 1-B 212, and by Esprit Working Group RAND II.. The authors are with the LRI, CNRS UMR 8623, Bât 490, Université Paris-Sud, 91405 Orsay, France (e-mail: {bouchero; salamat}@lri.fr.).. Communicated by F. R. Kschischang, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(00)01689-8.. message set described in multiuser information theory [5] . In such a broadcast channel, one transmitter tries to send k k k independent messages m 0 ; 111 m k k k01 (corresponding to different priority levels). to k k k receivers (enjoying different reception conditions). The messages are multiplexed by a channel encoder into a sequence of input symbols. x, x = f(m 0 11 1m k k k01 ). The i i ith receiver (0 i i i < k k k) gets a corrupted version y i i i of the input x, and tries to reconstruct messages m 0 11 1m i i i . At least from an intuitive viewpoint, the task faced by priority encoding transmission and coping with broadcast channels are similar. To elaborate further, let us recall more precisely the PET code description.. A PET code over alphabet X , with message length m, code length n, and nondecreasing priority function is a pair of mappings f (encoder) from X m to X n and (decoder) from (X [ feg) n to X m [ frejectg such that if w 0 is obtained from f(w) by erasing at most n 0 (i) symbols, then (w 0 ) coincides with w at least on the first i symbols.. The values in the range of a priority function are called the priority levels. In the sequel, the i i ith level of the priority function is denoted by. The rate (resp., normalized rate) of a code represents the fraction of information bits (resp., symbols) per symbol. Formally, the rate R (resp., normalized rateR) of a code of length n, with M codewords over alphabet X is . Inequality (1) puts combinatorial limits on finite sets of fixed-length words. When restricted, for example, to the case of one priority level, it coincides with the Singleton bound:R 0 0 =n. Note that the latter bound cannot be achieved by codes of arbitrary length (cf., for example, [1, Theorem 6.1]). On a given alphabet, arbitrarily long codes satisfying the PET inequality do not exist and thus cannot be used to achieve arbitrarily reliable transmission on lossy broadcast channels. Moreover, the PET approach advocates a very simple method to design broadcast codes. Packets are divided into slots, each slot size defining an alphabet. For each priority level, apply a good point-topoint erasure-resilient code over the small alphabet corresponding to the slot size and then interleave the resulting codewords in the packets. This is a version of the engineering approach called time-sharing in [6] . This approach is known to be nonoptimal on many broadcast channels. Assessing the PET approach from an information-theoretical viewpoint amounts to first checking whether the capacity region of broadcast erasure channels with degraded message sets can be exhausted using priority encoding transmission. In the affirmative, the second natural question is: Does error probability decline as fast as possible when PET is used?. We first use the information spectrum approach [7] to provide an informational characterization of the capacity region of general broadcast channels with degraded message set (DMS) in Section II. This characterization illustrates the relevance to general broadcast channels with memory of superposition codes as introduced in [6] . Then we show in Section III that the PET approach achieves the capacity region of memoryless broadcast erasure channels with DMS and, moreover, that it may achieve the sphere-packing exponent of this channel. The last sec-0018-9448/00$10.00 © 2000 IEEE tion shows that on some simple nonstationary broadcast erasure channels, time-sharing may be outperformed. The information spectrum approach has been proposed recently to handle systems with general dependencies [7] . It relies on the notion of liminf in probability: p 0 lim inf n!1 X n is defined as the supremum of such that lim sup n!1 Pr fXn < g = 0. Given two random The rate-tuple (R i i . Sequential decoding of convolutional codes and tree codes ([1]- [5] , etc.) is a useful decoding technique wherein the average number of computations performed is linear in block length as compared to an exponential number of computations for the maximum-likelihood decoder. A vast majority of the literature on sequential decoding deals with memoryless channels. A few papers, (for example, [6] , [7] ) extend the sequential decoding technique to a class of channels with memory, namely, finite-state channels. In this work we show that the sequential decoding technique can be used on timing channels (for example, [8] and [9] ). Interestingly, this timing channel is a channel with memory and cannot be described within the class of finite-state channels.. Specifically, we want to transmit information reliably through a single-server queue [8] , [9] , at rates below half the capacity, but with manageable decoding complexity. In [8] - [10] , a decoding technique for block codes was described where the number of computations is exponential in n, the number of packets. By imposing a tree structure on the codes and using the sequential decoding technique, we save on computations at the expense of the rate at which information is reliably transmitted. This work is perhaps a first step in the direction of finding good codes for communication over timing channels.. There are many versions of the sequential decoding technique. The basic idea behind the Fano algorithm [3] is to move forward in the de- coding tree so long as we seem to be (based on a metric) on the right track. Once the metric falls below a certain threshold, we backtrack and explore other paths, possibly changing the value of the threshold to account for the changed circumstances. The stack algorithm [4] , [5] , extends the node with the highest metric at each stage, until the end of the tree is reached. There is a relation between the number of computations in both these algorithms. We are interested in finding bounds on the average number of computations before proceeding one step forward in the correct path. The difficulty with analyzing the performance of the sequential decoding technique for communication systems with memory is the following. When comparing two paths that are the same up to a certain node, the choice of one or the other depends on the branches common to both paths in a way that is typically difficult to handle. For memoryless channels, however, the metric that determines this choice can be selected so that the choice does not depend on the common branches.. We can also get over this difficulty for timing channels. We show that the first mbranches can be summed up by one quantity that lends itself to a simple analysis. Our proof is based on the proof in [2] for multiple-access channels, restricted to single-user channels. Burke's output theorem for an M=M=1 queue plays an important role in determining a suitable metric. The main contributions of this work are the choice of this metric, and a simple analytical artifice (used earlier in [8] in a different context) that shows how the elegant technique in [2] can be modified to prove the existence of a good tree code for this system with memory. Section II introduces the problem in the appropriate notation and states the result. Section III contains the proof. We conclude with a brief discussion in Section IV.

O B I I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Dynamic prediction of communication flow using social context In this paper, we develop a temporally evolving representation framework for context that can efficiently predict communication flow in social networks between a given pair of individuals. The problem is important because it facilitates determining social and market trends as well as efficient information paths among people. We describe communication flow by two parameters: the intent to communicate and communication delay. To estimate these parameters, we design features to characterize communication and social context. Communication context refers to the attributes of current communication. Social context refers to the patterns of participation in communication (information roles) and the degree of overlap of friends between two people (strength of ties). A subset of optimal features of the communication and social context is chosen at a given time instant using five different feature selection strategies. The features are thereafter used in a Support Vector Regression framework to predict the intent to communicate and the delay between a pair of individuals. We have excellent results on a real world dataset from the most popular social networking site, www.myspace.com. We observe interestingly that while context can reasonably predict intent, delay seems to be more dependent on the personal contextual changes and other latent factors characterizing communication, e.g. 'age' of information transmitted and presence of cliques among people. In this paper we develop a temporally evolving representation framework for context that can efficiently predict communication flow in social networks between a given pair of individuals. The problem is important because prediction of communication flow can help organizations determine their experts or knowledge points, monitor the dynamics of effective information paths, help targeted advertising as well as facilitate understanding community evolution in social networks.. There has been prior work on computational modeling of information diffusion [6, 10] . In [6] the authors focus on analyzing the text in blog posts and use an epidemic disease propagation model for determining information diffusion. In [10] , the authors present an early adoption based information flow model useful for recommendation systems. The authors in [9] provide simple models for the onset of epidemic behavior in diseases. Some prior work [3] also focused on determining the effect due to heterogeneous consumers on diffusion dynamics in small world networks. There has also been prior work on analysis of emails of software developers [1] , to understand the relationship between the email activities and the software roles.. However, there are several limitations of prior work. The primary observation is threefold. Firstly, communication semantics being dynamic [4] , context plays an important role in the exchange of messages in any communication. However context has not been exploited comprehensively in prior work. Secondly, the traditional approach to prediction of information flow has only been focused on modeling current context between the people. In our previous work [2] , we use information about a person's neighboring social network, relationship between messages and identity of recipient to model these interactions. However, communication patterns can be affected by the habitual and network properties that are acquired over time as well. Thirdly, the factors affecting communication between a pair of individuals evolve over time. From psychological studies [8] , we know that when Alice sends messages to Bob on a certain topic, the messages are influenced by Alice's current shortterm memory; besides attributes of the current context: e.g. the topic at hand (topic context), the messages exchanged (recipient context) as well as the response received from Alice's local social network (neighborhood context).. The main contribution of this work is an effective prediction framework for determining communication flow dynamically. We know from [2] that context is central in explaining communication flow between people. In [2] , communication flow is described by two parameters: the intent to communicate (the probability that a person Alice would communicate with another person Bob) and communication delay (the time taken for Alice to send a message to Bob). In this work, we design features to model social context additionally which can enhance the prediction of communication flow. Social context refers to the patterns of participation in communication (information roles) and the degree of overlap of friends between two people (strength of ties). A subset of optimal features of both aspects of context is chosen at a given time instant using five different feature selection strategies. They are deployed in a Support Vector Regression framework for predicting the Permission to make digital or hard copies of all or part 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. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. intent to communicate and delay between a pair of individuals. We have excellent results on a real world dataset from the most popular social networking site, www.myspace.com. We observe that modeling social context is key to determining communication flow. We also notice qualitatively that intent is more affected due to contextual dynamics than delay. Delay seems to be more dependent on other latent factors characterizing communication, e.g. 'age' of information transmitted and presence of cliques among people.. The rest of the paper is organized as follows. In the next two sections, we discuss the communication and social context. In section 4 we present the prediction framework. Section 5 discusses the dataset followed by experimental results. We conclude by discussing the major contributions and directions towards future work in section 6. In this paper we developed a temporally evolving representation framework for context that can predict communication flow in social networks efficiently between a given pair of individuals. We 1 The values of the delays range from zero to 13 days.. described communication flow using two parameters based on our prior work: intent to communicate and delay. We designed features to model communication and social context and deployed them for predicting the intent to communicate and delay in transmission in a Support Vector Regression framework. For capturing temporal dynamics, a set of optimal features is selected at a given time instant using feature selection techniques. We observe qualitatively that intent is more affected due to contextual dynamics than delay. Delay seems to be more dependent on other latent factors characterizing communication, e.g. 'age' of information, cliquey-ness among communicators etc.. There are several interesting directions to future work. There might be latent factors involved which cannot be accounted for by our model of context: e.g, moods, sentiments, changes in social relationships, location, work habits etc. To handle these dynamics, we would like to explore modeling context as a Markov decision process and predict the intent and the delay more effectively.

O O O B I O B I O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Combinatorial bounds and characterizations of splitting authentication codes We present several generalizations of results for splitting authentication codes by studying the aspect of multi-fold security. As the two primary results, we prove a combinatorial lower bound on the number of encoding rules and a combinatorial characterization of optimal splitting authentication codes that are multi-fold secure against spoofing attacks. The characterization is based on a new type of combinatorial designs, which we introduce and for which basic necessary conditions are given regarding their existence.for each 1 ≤ m ≤ t, and j 1 , . . . , j t are mutually distinct.Example 1 A splitting 2-design is a splitting BIBD. As a simple example, take as point set 2, 3, 4, 5, 6, 7, 8, 9} and as block setThis gives a 2-(9, 9, 4 = 2 × 2, 1) splitting design (see [17, Example 5.1]).178 Cryptogr. Commun. (2010) 2:173-185 Example 2 A 3-(10, 15, 6 = 2 × 3, 1) splitting design can be obtained (via a computer search) by taking as point set 2, 3, 4, 5, 6, 7, 8, 9, 0} and as block set{7, 0}} B 10 = {{2, 4}, {5, 6}, {7, 9}} B 11 = {{2, 5}, {4, 7}, {3, 0}} B 12 = {{2, 9}, {6, 8}, {3, 0}} B 13 = {{2, 0}, {4, 5}, {6, 8}} B 14 = {{3, 7}, {4, 6}, {8, 0}} B 15 = {{3, 9}, {5, 7}, {6, 0}}.We prove some basic necessary conditions for the existence of splitting designs: Authenticity is one of the fundamental components in cryptography and information security. Typically, communicating parties would like to be assured of the authenticity of information they obtain via potentially insecure channels. Concerning unconditional (theoretical) authenticity, authentication codes can be used to minimize the possibility of an undetected deception. Their initial study appears to be that of Gilbert et al. [9] . A more general and systematic theory of authenticity was developed by Simmons [22, 23] . This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via a Heisenberg grant (Hu954/4) and a Heinz Maier-Leibnitz Prize grant (Hu954/5). We have given a combinatorial lower bound on the number of encoding rules for splitting authentication codes that are multi-fold secure against spoofing attacks. Moreover, we have provided a combinatorial characterization of those codes that attain these bounds. Our characterization was based on a new type of combinatorial designs, which we introduced and for which basic necessary conditions regarding their existence were given. For future research, at least two directions would be of interest:. 1. Construction of multi-fold secure splitting authentication codes: Using Theorem 8, this asks for constructing t-(v, b , l = cu, 1) splitting design for t > 2. We remark that in the case when t = 2, various combinatorial constructions have been obtained recently in [7, 8, 30] via recursive and direct constructions by the method of differences. 2. Including the aspect of perfect secrecy: Is it possible to give a characterization of optimal splitting authentication codes that are multi-fold secure against spoofing and simultaneously achieve perfect secrecy in the sense of Shannon? For the case of multi-fold secure authentication codes without splitting such a result has been established lately in [10] (cf. Theorem 4).

O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O


Efficient Belief Propagation for Higher-Order Cliques Using Linear Constraint Nodes Belief propagation over pairwise-connected Markov random fields has become a widely used approach, and has been successfully applied to several important computer vision problems. However, pairwise interactions are often insufficient to capture the full statistics of the problem. Higher-order interactions are sometimes required. Unfortunately, the complexity of belief propagation is exponential in the size of the largest clique. In this paper, we introduce a new technique to compute belief propagation messages in time linear with respect to clique size for a large class of potential functions over real-valued variables. We discuss how this technique can be generalized to still wider classes of potential functions at varying levels of efficiency. Also, we develop a form of nonparametric belief representation specifically designed to address issues common to networks with higher-order cliques and also to the use of guaranteed-convergent forms of belief propagation.To illustrate these techniques, we perform efficient inference in graphical models where the spatial prior of natural images is captured by 2 Â 2 cliques. This approach shows significant improvement over the commonly used pairwise-connected models, and may benefit a variety of applications using belief propagation to infer images or range images, including stereo, shape-from-shading, image-based rendering, segmentation, and matting. In the past few decades, the application of probabilistic models for solving computer vision problems has lead to significant advances. Many of these probabilistic models can be simplified by factoring large probability distributions using graphical models. Unfortunately, statistical inference in arbitrary factorized distributions is still NP-hard [1] . Recently, the method of loopy belief propagation has been shown to produce excellent results in several real-world computer vision problems [2] [3] [4] [5] [6] [7] . However, this method has some drawbacks. The most serious is that the running time of belief propagation is exponential in the size of the largest graph clique. This means that for problems with many labels or real-valued variables, graphical representations are typically limited to pairwise interactions between variables. Unfortunately, for many problems in computer vision, pairwise interactions fail to capture the rich statistical distribution of the problem domain. For example, natural images exhibit rich higher-order statistics that cannot be captured by pairwise-connected Markov random fields (MRFs). In Section 3, we introduce a new technique to compute belief propagation messages in time linear with respect to clique size that works for a large class of potential functions without resorting to approximation. In Section 7, we further improve on the efficiency and performance of belief propagation by presenting a nonparametric, particle-like representation of belief propagation messages that is simultaneously compatible with higher-order non-pairwise interactions and also with recent extensions to belief propagation that guarantee convergence [8] .. These advancements allow us to efficiently solve inference problems that were previously unavailable to belief propagation. In Section 8, we show that a prior model of natural images using 2 Â 2 MRF cliques strongly outperforms pairwise-connected models. The ability to use more accurate models of image or range image priors has the potential to significantly aid the performance of several computer vision applications, including stereo [3] , photometric stereo [4] , shape-from-shading [9] , image-based rendering [10] , segmentation, and matting [7] . In this paper, we have introduced a way to efficiently perform belief propagation over large graph cliques, reducing computation from OðM N Þ to OðNM 2 Þ for a wide variety of potential functions. We have shown how these methods can be generalized in several ways to benefit a larger subclass of potential functions. Additionally, we have developed methods for representing belief propagation messages for continuous variables that remain computationally efficient for highly connected graphs, convergent variants of belief propagation, and the use of the computational shortcuts introduced in this paper. These message representations allow discret- All values are given as logpðĨ jĨ N Þ Â 10 À5 , wherepðĨ jĨ N Þ is given in Eq. (97). The values given for the Berkeley Suite images show the mean unnormalized log likelihood for the 10 images from the Berkeley segmentation database [52] used in [5] . The five denoising algorithms shown here all seek to optimize the same equation (i.e. Eq. (97) using the 2 Â 2 FoE model). In each case, belief propagation significantly outperforms gradient descent. Thus, in addition to producing denoised images with less error, belief propagation does a better job at finding the optimum values of the FoE probability model. This means that the improvement in performance is not due to peculiarities of the FoE model. Also note that, according to the model, the denoised images computed using belief propagation have greater likelihood than the original image. This suggests that improving the model is more important than improving the method of optimization. Table 5 Results as in Table 4 ization error to be minimized while at the same time preserving computational efficiency. The techniques introduced in this paper open up a wealth of powerful, higher-order statistical models for inference using belief propagation methods that would previously have been intractable. Belief propagation is a promising framework of optimization for these models, because it often outperforms gradient-based optimization methods by exploiting factorizations, and by performing optimization within the much larger search space of single-variate marginals, which is less prone to local extrema. Computer vision in particular stands to benefit greatly from higher-order statistical models due to the complex statistical structure of natural images and underlying image properties. In particular, enabling belief propagation to exploit non-pairwise statistical relationships in images such as the Fields of Experts models, and to take advantage of overcomplete representations of a state-space such as multi-resolution representations, may prove especially useful in computer vision.. To illustrate our techniques, we applied belief propagation to the problem of performing statistical inference over higher-order spatial priors of images. When these spatial priors are applied to image denoising, we demonstrated a significant performance increase over belief propagation in pairwise models and also over gradient descent based methods for higher-order models. At the same time, a sizeable speed increase was shown over previous belief propagation approaches to higher-order graph cliques. The ability to exploit higher-order spatial priors using belief propagation may be of great benefit to a number of computer vision tasks that seek to infer images or range images in ambiguous, uncertain circumstances, including stereo [3] , photometric stereo [4] , shapefrom-shading [9] , image-based rendering [10] , segmentation, and matting [7] .

O B I O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Reduction of stochastic parity to stochastic mean-payoff games Abstract. A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as parity objectives, and mean-payoff (or long-run average) objectives. These games lie in NP ∩ coNP. We present a polynomial time Turing reduction of stochastic parity games to stochastic mean-payoff games. Graph games. A stochastic graph game [Con92] is played on a directed graph with three kinds of states: player-1, player-2, and probabilistic states. At player-1 states, player 1 chooses a successor state; at player-2 states, player 2 chooses a successor state; at probabilistic states, a successor state is chosen according to a given probability distribution. The outcome of playing the game forever is an infinite path through the graph. If there are no probabilistic states, we refer to the game as a 2-player graph game; otherwise, as a 2 1 / 2 -player graph game.. Parity objectives. The theory of graph games with ω-regular winning conditions is the foundation for modeling and synthesizing reactive processes with fairness constraints. In the case of 2 1 / 2 -player graph games, the two players represent a reactive system and its environment, and the probabilistic states represent uncertainty. The parity objectives provide an adequate model, as the fairness constraints of reactive processes are ω-regular, and every ω-regular winning condition can be specified as a parity objective [Tho97] . The solution problem for a 2 1 / 2 -player game with parity objective Φ asks for each state s, for the maximal probability with which player 1 can ensure the satisfaction of Φ if the game is started from s (this probability is called the value of the game at s). An optimal strategy for player 1 is a strategy that enables player 1 to win with that maximal probability. The existence of pure memoryless optimal strategies for 2 1 / 2 -player games with parity objectives was established in [CJH04] (a pure memoryless strategy chooses for each player-1 state a unique successor state). The existence of pure memoryless optimal strategies implies that the solution problem for 2 1 / 2 -player games with parity objectives lies in NP ∩ coNP.. Mean-payoff objectives. An important class of quantitative objectives is the class of mean-payoff (or long-run average) objectives. In case of mean-payoff objectives there is a real-valued reward at each state and the payoff of player 1 for a play is the long-run average of the rewards appearing in the play. The objective of player 1 is to maximize the long-run average, and values are defined in a similar way as for parity objectives. In 2 1 / 2 -player games with mean-payoff objectives pure memoryless optimal strategies exist [LL69] . Again, the existence of pure memoryless optimal strategies implies that the solution problem for 2 1 / 2 -player games with mean-payoff objectives lies in NP ∩ coNP.. Our result. We present a polynomial time Turing reduction of 2 1 / 2 -player parity games to 2 1 / 2 -player mean-payoff games for computation of values. Similar reduction was known for the special case of 2-player games [Jur98] . As a consequence of our reduction all algorithms for 2 1 / 2 -player mean-payoff games [FV97, Put94] can now be used for 2 1 / 2 -player parity games.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O