Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential The overlap operator in lattice QCD requires the computation of the sign function of a matrix, which is non-Hermitian in the presence of a quark chemical potential. In previous work we introduced an Arnoldi-based Krylov subspace approximation, which uses long recurrences. Even after the deflation of critical eigenvalues, the low efficiency of the method restricts its application to small lattices. Here we propose new short-recurrence methods which strongly enhance the efficiency of the computational method. Using rational approximations to the sign function we introduce two variants, based on the restarted Arnoldi process and on the two-sided Lanczos method, respectively, which become very efficient when combined with multishift solvers. Alternatively, in the variant based on the two-sided Lanczos method the sign function can be evaluated directly. We present numerical results which compare the efficiencies of a restarted Arnoldi-based method and the direct two-sided Lanczos approximation for various lattice sizes. We also show that our new methods gain substantially when combined with deflation. While this paper discusses new numerical methods that are expected to be useful in a large number of applications, the main motivation for these new methods comes from quantum chromodynamics (QCD) formulated on a discrete space-time lattice. QCD is the fundamental theory of the strong interaction. Being a non-Abelian gauge theory, it is notoriously difficult to deal with. Lattice QCD is the only systematic non-perturbative approach to compute observables from the theory, and it is amenable to numerical simulations. The main object relevant for our discussion is the Dirac operator, for which there exist several formulations that differ on the lattice but are supposed to give the same continuum limit when the lattice spacing is taken to zero. We are focusing on the overlap Dirac operator D ov [1, 2] , which is the cleanest formulation in terms of lattice chiral symmetry [3, 4] but very expensive in terms of the numerical effort it requires. Trying to improve algorithms dealing with the overlap operator is an active field of research, and even small improvements can have an impact on the large-scale lattice simulations that are being run by the lattice QCD collaborations worldwide.. The overlap operator is essentially given by the sign function of its kernel, which we assume is the usual Hermitian Wilson operator H W = γ 5 D W (see [5] for the notation). On the lattice, this operator is represented by a sparse matrix, and on current production lattices the dimension of this matrix can be as large as 10 8 ∼ 10 9 . The main numerical effort lies in the inversion of the overlap operator, which is done by iterative methods and requires the repeated application of the sign function of H W on a vector. At zero chemical potential µ, H W is Hermitian, and many sophisticated methods have been developed for this case (see, e.g., [6] ). However, one can also study QCD at nonzero quark chemical potential (or, equivalently, density), which is relevant for many physical systems such as neutron stars, relativistic heavy ion collisions, or the physics of the early universe. The overlap operator has been generalized to this case [5, 7] . While the result is formally similar to the one at µ = 0, it is in fact more complicated since H W becomes a non-Hermitian matrix, of which we need to compute the sign function. This case is much less studied and the focus of the present paper, which is a natural continuation of earlier work [8] . For simplicity we will still refer to H W = γ 5 D W as the "Hermitian" Wilson operator.. In mathematical terms, we investigate the computation of f (A)b, where A ∈ C n×n is non-Hermitian and f is a general function defined on the spectrum of A such that the extension of f to matrix arguments is defined. For a simple definition of matrix functions we assume that A is diagonalizable and let A = RΛR −1 be the eigendecomposition with R ∈ C n×n and diagonal Λ containing the eigenvalues λ 1 , . . . , λ n ∈ C. Then the matrix evaluation of f is defined as. Accordingly, if b = Ry ∈ C n is a vector expressed in terms of the eigenvectors, then. For a thorough treatment of matrix functions see [9] ; a compact overview is given in [10] . The case f = sign will be of particular interest. We use the standard definition sign z = sign Re(z) for z ∈ C [9] , which in the physics case we are considering was also shown to follow from the domain-wall formalism [7] . If A is large and sparse, f (A) is too costly to compute, whereas f (A)b can still be obtained in an efficient manner via a Krylov subspace method.. The foundation for any Krylov subspace method is the computation of an appropriate basis for the Krylov subspace K k (A, b) = span{b, Ab, . . . , A k−1 b}. For Hermitian matrices an orthonormal basis can be built with short recurrences using the Lanczos process. For non-Hermitian matrices the corresponding process, which again computes an orthonormal basis, is known as the Arnoldi process. It requires long recurrences and is usually summarized via the Arnoldi relation. Here,. n×k is the matrix which contains the computed basis vectors (the Arnoldi vectors), H k = V k † AV k is the upper Hessenberg matrix containing the recurrence coefficients h i j , and e k denotes the k-th unit vector of C k . H k being upper Hessenberg reflects the fact that the computation of the next Arnoldi vector v k+1 results in a long recurrence since the projection of Av k on all previous vectors v 1 , . . . , v k has to be subtracted from Av k . Long recurrences slow down computation and increase storage requirements, and thus become inefficient or even infeasible if k, the dimension of the Krylov subspace, becomes large. This is the reason why in this paper we investigate two ways to circumvent this problem for non-Hermitian matrices, i.e., restarts of the Arnoldi process and the use of the twosided Lanczos process. We will consider these two methods in combination with a rational function approximation to f . In the case of two-sided Lanczos, we will also consider a direct evaluation of the function. This paper is organized as follows. In Section 2 we describe the two alternatives just mentioned to obtain short recurrences. In Section 3 we address several aspects of the important issue of deflation. Section 4 contains the descriptions of four different short-recurrence algorithms to compute the sign function, all of which use the preferred method of LR deflation. In Section 5 we discuss the choice of the rational function to approximate the sign function. Our numerical results are presented in Section 6, and conclusions are drawn in Section 7. At nonzero chemical potential, the overlap Dirac operator contains the sign function of the Wilson operator H W = γ 5 D W , which is non-Hermitian. The by far most expensive part when applying the overlap Dirac operator to a field 4 The plots in Figure 2 are for a single configuration per volume. One might ask to what extent this configuration is typical. In the present context the main difference between configurations lies in the magnitude of their smallest Dirac eigenvalues. The removal of the latter by deflation makes the configuration typical. 5 Note that the cost of deflation, i.e., the cost to compute the m critical eigenvalues and eigenvectors, is not included in these figures (and in the figures below) because it only needs to be paid once for each A. In the case of lattice QCD, sign(A)b has to be computed for many different b in an iterative inverter. One should then choose m such that the total run time, including the cost of deflation (which strongly depends on the details of A), is minimized. However, this optimization issue is not the focus of the current paper. 6 Note that FOM-LR applied to Eq. (30) works with A 2 , so we actually have C n = 2M n . 7 This is not necessarily so for the small lattices, where the superlinear convergence of Krylov subspace methods might become noticeable. vector b -the standard step in any iterative solver for the overlap Dirac operator -is the computation of the action of sign(H W ) on b. As a step towards developing computationally feasible methods for the dynamical simulation of overlap fermions at nonzero chemical potential, we proposed in this paper several short-recurrence Krylov subspace methods to efficiently compute sign(H W )b. One class of methods is based on restarts of the Arnoldi process and requires a precise rational approximation for the sign function on the (complex) spectrum of the Wilson operator. This means that we need to have information on the location of the spectrum in the complex plane and that we have to adapt the number of poles in the rational approximation accordingly. The storage requirements for these methods depend on the restart value, a parameter which has to be tuned to be optimal, and the number of poles in the rational approximation. Storage does not depend on the number of iterations to be performed.. The other class of methods relies on the two-sided Lanczos process. We can use a rational function approximation, in which case the comments made in the previous paragraph apply as well, except that there is no restart. Alternatively, the sign function can be evaluated directly. In that case, if a two-pass strategy is used, the storage requirements are minimal; otherwise storage increases linearly with the number of iterations. No a priori knowledge on the spectrum is required. If the number of iterations to be performed gets large, the work spent in evaluating the sign function of the projected operator, which is represented by a tridiagonal matrix, becomes decisive in terms of computational cost. Therefore, the methods based on a rational function approximation were faster in the numerical experiments that we performed on lattices with sizes ranging from 4 4 to 10 4 . However, fast methods are currently being developed to compute the sign of the projected tridiagonal matrix, which will speed up the direct two-sided Lanczos method substantially [26] .. For both classes of methods the deflation of critical eigenvalues is an important ingredient towards efficiency. We showed that LR deflation is to be preferred to Schur deflation.

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


A Markovian Approach to Multipath Data Transfer in Overlay Networks Abstract-The use of multipath routing in overlay networks is a promising solution to improve performance and availability of Internet applications, without the replacement of the existing TCP/IP infrastructure. In this paper, we propose an approach to distribute data over multiple overlay paths that is able to improve Quality of Service (QoS) metrics, such as the data transfer time, loss, and throughput. By using the Imbedded Markov Chain technique, we demonstrate that the system under analysis, observed at specific instants, possesses the Markov property. We therefore cast the data distribution problem into the Markov Decision Process (MDP) framework, and design a computationally efficient algorithm named Online Policy Iteration (OPI), to solve the optimization problem on the fly. The proposed approach is applied to the problem of multipath data distribution in various wired/wireless network scenarios, with the objective of minimizing the data transfer time as well as the delay and losses. Through both intensive ns-2 simulations with data collected from real heterogeneous networks and experiments over real networks, we show the superior performance of the proposed traffic control mechanism in comparison with two classical schemes, that are Weighted Round Robin and Join the Shortest Queue. In this paper, we studied the problem of high performance multipath data transfers in the context of multipath routing overlay networks. We proposed a control mechanism based on Markov Decision Process to distribute traffic over multiple overlay paths while optimizing QoS metrics such as the transfer time, delay, and losses. The optimal control problem was formulated as a Markov Decision Process, and a computationally efficient algorithm, named Online Policy Iteration, was used to optimally transfer the traffic by solving the formulated MDP online. Performance of the proposed control mechanism was evaluated and compared with the classical multipath schemes including Weighted Round Robin and Join the Shortest Queue in both simulation and over a real testbed. The results indicate that the proposed control mechanism significantly outperforms classical multipath schemes in all considered scenarios. Antonio Pescapé received the MS Laurea degree (summa cum laude) in computer engineering and the PhD degree in computer engineering and systems from the University of Napoli Federico II. He is an assistant professor at the Department of Computer Engineering and Systems of the University of Napoli Federico II, Italy. As member of COMICS research group at University of Napoli Federico II and ITeM Lab of CINI, Italy, he is a principal investigator for several national and international research projects. His research interests include the networking field with focus on models and algorithms for Internet traffic, network measurements and management of heterogeneous IP networks, and network security. He has coauthored more than 80 journal and conference publications. He is a senior member of the IEEE, and he has been serving on several technical program committees of the IEEE and the ACM conferences (IEEE Globecom, IEEE ICC, IEEE WCNC, IEEE HPSR,. . . ). He also serves as a editorial board member of IEEE Survey and Tutorials and was guest editor for the Special Issue of Computer Networks on "Traffic classification and its applications to modern networks." In [18] , the authors overview the main motivations, challenges, and techniques for multipath routing at different levels (packet, flow) and in different operating scenarios (intradomain, interdomain) whereas the work presented in [37] is concerned with the experimental analysis of the possible path diversity (i.e., the number of routes available to transmit a packet between two hosts in a network) in ISP networks. Optimal multipath data transfers on overlay networks have been studied in [14] , [16] , [36] , [42] . In [36] , Tao et al. demonstrated the potential of multiple overlay paths for improving Internet performance. They developed a path switching mechanism, that subsequently selects a potentially less congested path among several available paths to transmit data. The path selection is based on path congestion states, predicted by a Markov model, that is trained with data collected through active measurements. A drawback of this approach is the overhead caused by active measurements and the accuracy of the path model. Moreover, in many cases, sending data always to the less congested path is not the optimal policy. In [42] , a multipath controller is proposed that randomly chooses a transmission path following a throughput-proportional selection scheme. This path selection scheme is a randomized version of the well-known Weighted Round Robin (WRR) scheme [27] where the weights, in this case, are the path throughputs. A key issue of this scheme is how to accurately obtain the path throughput, by means of online measurements. However, we later show that, even with the correctly estimated path throughput, WRR is unlikely to be an optimal scheme for QoS metrics, such as loss, delay and jitter. At transport layer, the work [17] proposes an enhanced TCP scheme which works just above the transport layer opening different TCP connections toward the same host, whereas modifications to STCP aimed at supporting the effective use of multiple independent paths are proposed in [19] . In wireless scenarios, the work [12] presents an algorithm, borrowed from wireless scheduling problems, to distribute packets on different available paths while minimizing delay, maximizing throughput, and respecting the split proportions assigned by the routing algorithm. In [38] , an analysis of path diversity at packet level in IEEE 802.11 scenarios with multiplechannel transmission is conducted, whereas the problem of multipath routing in ad-hoc networks is analyzed in [26] . In the context of data replication and collection, the authors of [14] and [16] studied the problem of multipath data transfer, where the objective was to minimize the makespan, i.e., the duration of the longest transfer. Both studies formulated the problem by using graph theory, where the graph is the overlay network topology, and the constraints are the path capacities. However, this approach has a number of potential issues, as admitted by the same authors [14] . First, the graph theory formulation assumed that the time required to transfer a unit of data through a link is fixed, which is unrealistic: in real networks, in the presence of background traffic, this transfer time is varying. Second, since end-to-end delay of a path is not always proportional to its capacity, using path capacities as constraints to minimize the makespan may not lead to an optimal solution. In such a dynamic environment, an approach based on a dynamic optimization framework like Markov Decision Processes (MDPs) is more appropriate. Markov Decision Processes [34] are a powerful mathematical framework for making decisions in a dynamic environment that exhibits Markov property. 1 In this framework, the control environment is modeled as a system of states. The system changes its state according to some state transition probability, which is a function of actions, i.e., decisions, taken by the decision maker. Outcome of each action is assessed by associating it with a reward. The goal of the decision maker is to maximize some function of rewards. Given an MDP, a decision is made by mapping an observed system state to an action according with a predefined policy. Once a decision policy is determined, actions are taken instantly without any extra computation. Therefore, MDPs can be used for time critical decision tasks, such as network traffic control.

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


On developmental mental architectures This paper presents a computational theory of developmental mental architectures for artificial and natural systems, motivated by neuroscience. The work is an attempt to approximately model biological mental architectures using mathematical tools. Six types of architecture are presented, beginning with the observation-driven Markov decision process as Type-1. From Type-1 to Type-6, the architecture progressively becomes more complete toward the necessary functions of autonomous mental development. Properties of each type are presented. Experiments are discussed with emphasis on their architectures. r 2007 Published by Elsevier B.V. This paper deals with the architecture level of a biologically inspired system that is capable of autonomous mental development (AMD). By AMD, we mean an agent, natural or artificial, improves its mental (perceptual, cognitive, behavioral and motivational) capabilities through autonomous interactions with its environments. Mental architecture is a challenging and important subject, but there have been relatively few systematic (agent-wise) investigations. Further, the working of the mind is hard to understand without understanding AMD. A mathematically rigorous investigation of mental architecture is necessary in order to achieve this understanding.. A system that senses and acts is called an agent. In the machine learning community, the agent architecture has been a very important and extensively studied subject. Supervised and reinforcement learning, based on the Markov decision Process (MDP) architecture (single-or multi-level), enables a robot (an embodied agent) to learn autonomously while the environment (including humans) provides labels for supervised learning [28] or rewards for reinforcement learning [14, 34] . The model is for a general purpose, in that given an appropriate task, a human programmer can establish the correspondence between every state (node) and the designed task-specific concept, before the initialization and learning of MDP take place. However, the MDP architecture, as explained in the following sections, has fundamental limitations that prevent it from being effective for the developmental robots described in [47] . A major limitation of the MDP model is its non-developmental nature, in the sense that the meanings of the states (or nodes) are fixed (manually designed instead of observation driven).. Additional general-purpose architectures have been proposed by researchers in artificial intelligence and psychology. Major remarkable ones include Soar proposed by Laird et al. [17] , ACT-R by Anderson [3] , the architecture by Albus [1] , and the subsumption architecture by Brooks [6] . Soar and ACT-R incorporated many useful concepts that are necessary for human intelligence. Albus' architecture outline was motivated by neural architecture and gives a temporal view of a complex mental architecture. The subsumption architecture is a biologically motivated architecture component, motivated by motor cortex. It provides a way to decompose the architecture according to behaviors.. The architecture models discussed above do not directly address perception, such as vision and audition. Neither do the models that aim to model human performance, e.g., error rate (see, e.g., EPIC by Meyer and Kieras [20, 21] ) as opposed to actual mental capabilities. A common limitation of these architectures is in the use of symbolic representation in internal representation. Inspired by the biological brain (e.g., cortical representation) and its development, no task-specific symbolic representation will be used in the proposed developmental architectures. Distributed (i.e., vector), receptor-and effector-derived numeric representation is used through the proposed series of mental architectures.. Perceptual architecture is still an unsolved, important, yet very challenging problem. Neisser [23] pointed out that any model of vision that is based on spatial computational parallelism alone is doomed to failure. He proposed a twostage visual process which consists of a pre-attentive phase followed by an attentive phase. Feldman and Ballard [9] proposed a ''100-step rule:'' A biologically plausible algorithm for immediate vision (one that does not involve slower deliberate thinking) can require no more than 100 steps. John Tsotsos' study [36] on the complexity of immediate vision proposed a coarse architecture for a biologically motivated general purpose vision system (for immediate vision).. All the above architectures are nondevelopmental in the sense that the information processor is not generated through real-time interactions with the environment. In other words, a developmental architecture requires not only a specification of processors and their interconnections, but also their online, incremental, automatic generation from real-time experience.. Recently, there has been an onset of efforts on computational studies of AMD (e.g., the Workshop on Development and Learning [48] , the series of International Conferences on Development and Learning, and the International Workshops on Epigenetic Robotics). Although many of the published studies have not reached fully autonomous incremental online development, these studies are very important for understanding AMD by higher animals and the realization of artificial AMD by robots. Darwin V robot [2] , for example, had plastic association between feature extractors and motors, although other parts are not plastic. A reason for this status quo is the lack of studies on developmental mental architecture. This paper deals with this important issue.. The paper presents a series of architectures, from the simplest Type 1 to the full Type 6 where every module is plastic (developmental). The paper does not describe our developmental algorithms that use such architectures, but cites the related publications that specify the full detail for every module. The paper also discusses some of our experimental systems that use the architecture models described here. The computational models proposed here are inspired by networked neurons, cortical regions and their development. The correspondence of the architecture components with brain regions is discussed, but an exact provable correspondence is not possible at the current state-of-the-art.. It is difficult to be both biologically inspired and mathematically grounded, which the work presented here strives to achieve. Like the architecture of MDP [28, 14, 34] , the presentation of the series of mental architectures is in mathematical terms, which is necessary for a clear and rigorous understanding of the related concepts. The brain operates quantitatively (e.g., the cortex is a function); and mathematics is a powerful tool to model quantitative relationships. It is also necessary to derive the mathematical properties of the corresponding architecture so that the power and the limitation of the architecture are better understood.. The work reported here is a theoretical, computational, and experimental step toward the goal of understanding mental architecture. This paper does not intend to solve all the problems and answer all the questions about this subject. However, it is for the first time, as far as the author knows, a systematic presentation of a series of biologically inspired mental architectures in rigorous mathematical terms. Many of the architecture concepts and properties presented here are new (e.g., Types 3 to 6).. In the following sections, I first introduce a series of architectures, from simple to complex, along with the associated properties. Then, in Section 9, I propose a series of new modes of learning that are necessary for understanding various modes of development as well as their differences with traditional machine learning modes. In Section 10, the presented architectures are related to our experimental systems. This is the first over-arching mental architecture that deals with autonomous development for every major module, the attention selector T (early processing), the regressor (later processing), motor mapping (motor cortex), and the motivational system. Although the major modules of the architecture, such as the sensory mapping (T) [51] and the regressor [12, 44] , are not presented here, these modules and their automatic development have been studied and simulated.. The observation-driven MDP (Type-1) is more suited for autonomous mental development than the traditional MDP. This paper provides Type-1-Type-5 (DOSASE MDP) architectures, up to Type-6 (multilevel DOSASE MDP). A single DOSASE MDP can perform nonassociative learning, classical conditioning, instrumental conditioning, and general purpose reasoning and planning.. The proposed series of mental architectures might be useful for further understanding neuroscience and behavioral evidence and may inform more detailed models of cognitive development. (e)-(g) are composite actions achieved by transferring and chaining some or all of the basic ones. Fig. 13 . The SAIL robot learned longer and more complex composite skills through transfer-and-chaining based on previously learned simpler and shorter skills, while it was interacting with human trainers in real-time. We have systematically introduced six types of architectures. Although the order at which new capabilities are. According to this new kind of classification, all learning types using a non-developmental learning method correspond to Types 4-7, this is because the task-specific representation is handcrafted after the task is given. AMD uses Types 0-3.. We first consider traditional machine learning Types 4-7, where the internal states are hand-crafted based on a given task, such as the non-ODMDP. The traditional supervised learning corresponds to Type 6 and the traditional reinforcement learning corresponds to Type 5. The operational mode of type 4 is typically called test, not learning. Type 7 has been rare in machine learning.. Next, we consider developmental learning Types 0-3, where the internal states are not hand-crafted based on any tasks, since tasks are not given during the time of programming (or before the birth). Types 3 and 2 enable the teacher to manually guide the effector of the learner, e.g., the teacher holds the hand of a child and guides it to teach the child how to draw. With Type 3, the teacher also gives physical rewards to the learner, e.g., giving him candy. However, types 3 and 2 are too tedious for autonomous learning, since it requires a human teacher to specify exact output in real time. Therefore, it is possible only for a limited time for some small number of effectors. Furthermore, internal actions cannot be taught using type 3 or 2, since there is no way for a teacher to directly impose an internal action.. Type 1, state-autonomous, effector-autonomous reinforcement learning, enables the teacher to teach internal behaviors. For example, when the robot looks at the teacher, the teacher presses its ''good'' button, which reinforces the attention selection behavior. However, Type 1 learning is not very effective for more general settings. First, the number of all possible actions to explore is simply too large to try them all. Second, there is the credit assignment problem. For example, when the teacher gives the baby a candy, is the reward due to the fact that the baby looks at the teacher or that she is simply being quiet? The following new mode of learning can drastically reduce the complexity of autonomous learning, for both internal and external actions. Definition 16 (Communicative learning). Communicative learning is achieved through the Type-0 learning mode. However, Type-0 learning mode requires two processes, which can be interwoven through development: (1) Grounded language acquisition (using any mode of learning). (2) Teaching using an acquired language (verbal, written, sign, etc.).

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Application of network calculus to general topologies using turn-prohibition Abstract-Network calculus is known to apply in general only to feedforward routing networks, i.e., networks where routes do not create cycles of interdependent packet flows. In this paper, we address the problem of using network calculus in networks of arbitrary topology. For this purpose, we introduce a novel algorithm, called turn-prohibition (TP), that breaks all the cycles in a network and, thus, prevents any interdependence between flows. We prove that the TP-algorithm prohibits the use of at most 1/3 of the total number turns in a network, for any network topology. Using analysis and simulation, we show that the TP-algorithm significantly outperforms other approaches for breaking cycles, such as the spanning tree and up/down routing algorithms, in terms of network utilization and delay bounds. Our simulation results also show that the network utilization achieved with the TP-algorithm is within a factor of two of the maximum theoretical network utilization, for networks of up to 50 nodes of degree four. Thus, in many practical cases, the restriction of network calculus to feed-forward routing networks may not represent a significant limitation. Computational Complexity: The worst-case computational complexity of the TP-algorithm is polynomial in N , more precisely O(N 2 d), where d represents the maximal degree of any node in the network. This complexity is certainly reasonable, as long as the network topology does not change too often.. A derivation of this complexity is as follows. At each step of the algorithm, one node is deleted from further consideration. Thus, the algorithm consists of at most N steps. At each of these steps the following computations are performed: 1. A node of minimal degree is selected, an operation of complexity O(N ).. 2. The components of connectivity are determined. Using a spanning tree construction algorithm, the complexity of this operation is O(Nd).

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A pragmatic approach to area coverage in hybrid wireless sensor networks Success of Wireless Sensor Networks (WSN) largely depends on whether the deployed network can provide desired area coverage with acceptable network lifetime. This paper seeks to address the problem of determining the current coverage achieved by the non-deterministic deployment of static sensor nodes and subsequently enhancing the coverage using mobile sensors. We identify three key elements that are critical for ensuring effective area coverage in Hybrid WSN: (i) determining the boundary of the target region and evaluating the area coverage (ii) locating coverage holes and maneuvering mobile nodes to fill these voids, and (iii) maintaining the desired coverage over the entire operational lifetime of the network. We propose a comprehensive solution that addresses all of the aforementioned aspects of the area coverage, called MAPC (mobility assisted probabilistic coverage). MAPC is a distributed protocol that operates in three distinct phases. The first phase identifies the boundary nodes using the geometric right-hand rule. Next, the static nodes calculate the area coverage and identify coverage holes using a novel probabilistic coverage algorithm (PCA). PCA incorporates realistic sensing coverage model for range-based sensors. The second phase of MAPC is responsible for navigating the mobile nodes to plug the coverage holes. We propose a set of coverage and energy-aware variants of the basic virtual force algorithm (VFA). Finally, the third phase addresses the problem of coverage loss due to faulty and energy depleted nodes. We formulate this problem as an Integer Linear Program (ILP) and propose practical heuristic solutions that achieve similar performance as that of the optimal ILP solution. A guiding principle in our design process has been to ensure that the MAPC can be readily implemented in real-world applications. We implemented the boundary detection and PCA algorithm (i.e., Phase I) of the MAPC protocol on off-the-shelf sensor nodes and results show that the MAPC can successfully identify boundary nodes and accurately determine the area coverage in the presence of real radio irregularities observed during the experiments. Extensive simulations were carried out to evaluate the complete MAPC protocol and the results demonstrate that MAPC can enhance and maintain the area coverage, while reducing the total energy consumption by up to 70% as compared with the basic VFA. Providing adequate sensing coverage of the target area is of paramount importance for proper functioning of a WSN. In theory, this can be achieved by placing sensor nodes at predetermined locations in the target area. However, in hostile, and harsh environments such as enemy territory in battlefields, and vast forest-lands, the sensor nodes cannot be carefully deployed according to a pre-determined regular Figure 1 illustrates the building blocks of the MAPC protocol. The protocol works in three distinct phases. Phase I and II are executed during the initial deployment of a WSN, while phase III is executed over the entire operational lifetime of the network. Phase I consists of two sub-tasks; boundary estimation and coverage estimation. Detection of nodes lying on the outer boundary of a network is essential for self-organization of a sensor network. From a coverage perspective, boundary nodes circumscribe the target area that needs to be adequately covered. We propose a distributed boundary estimation algorithm based on the geometric right-hand rule. Boundary estimation results in the identification of virtual boundary formed by the static nodes lying on the perimeter of the deployed network. After the target region has been defined, all the static nodes in the network estimate the area coverage. The widely used binary detection model assumes that the sensing coverage of a sensor node is uniform in all directions, often represented by a unit disk. However, in reality, the sensing capabilities of a sensor are often affected by environmental factors. In particular, for range-based sensor modalities such as acoustics, radio, etc, the signal strength of the triggering signal decays as a function of distance. In an effort to employ more realistic models in the computation of area coverage, we propose the probabilistic coverage algorithm (PCA), that takes into account the probabilistic sensing behavior of sensor nodes.. Mobile nodes are introduced in the topology in phase II. They are assumed to be initially concentrated at one or more points on the boundary of the target area. Phase II of MAPC is responsible for managing the iterative relocation of the mobile sensors to meet two objectives. The primary goal is to fill the coverage holes identified by the static nodes in Phase I. The secondary goal is to uniformly spread the additional mobile sensors in the target area, so that these nodes can replace any faulty or dead sensor nodes during the operational lifetime of the network. Further, given that the mobile nodes expend significant energy in maneuvering, the energy expenditure during the relocation process should be minimized. We propose a set of coverage and energy aware variants of virtual force algorithm (VFA) to meet these objectives.. The goal of the third phase of MAPC is to ensure that the area coverage is maintained during the operational lifetime of WSN. This pre-empts any possible loss of coverage due to nodes depleting their batteries (proactive maintenance) or due to dead sensors, e.g., nodes with hardware failure or physical damage (reactive maintenance). Phase II seeks to first identify such coverage holes and then mobilize the mobile nodes to fill the same. In this paper, we have modeled the coverage maintenance problem as an integer linear program (ILP), and have proposed heuristic solutions that perform close to the optimal. This paper has proposed a comprehensive three phase protocol, MAPC, for ensuring area coverage employing a hybrid sensor network. MAPC utilizes mobile sensor nodes to ensure that adequate area coverage is maintained from deployment through the lifetime of a WSN. To our knowledge, no such comprehensive solution has been presented in the literature that addresses the deployment, coverage enhancement, and coverage maintenance issues for a hybrid sensor network in its entirety.. The primary contribution of this work is a pragmatic approach to sensor coverage and maintenance that we hope would lower the technical barriers to its field deployment. A distinguishing feature is that most of the assumptions made in the MAPC protocol are realistic e.g., practical boundary estimation, coverage calculations based on realistic sensing model, and use of movement triggering thresholds based on real radio characteristics.. First, we proposed a practical boundary estimation scheme with the objective of identifying the nodes lying on the outer boundary of a deployed network. The scheme is distributed and is executed only by a subset of all deployed nodes. The boundary information is utilized by static nodes during coverage estimation and mobile nodes during the coverage enhancement phase of the MAPC protocol. The performance of the boundary estimation scheme has been studied both through discrete event simulations and conducting experiments on real motes.. Second, we proposed a novel coverage estimation algorithm, PCA, that evaluates the area coverage in WSN. PCA takes into account the realistic variations in sensing behavior associated with range based sensors and evaluates probability of successful sensing at discrete intervals around a sensor location. The algorithm takes advantage of the sensing contributions from neighbors for accurate estimation of a node's area coverage. The performance of the PCA has been investigated by simulation and experiments.. Results show that PCA can accurately estimate the area coverage in a region.. Next, a set of integrated deployment and coverage enhancement algorithms have been proposed that are based on VFA. The algorithms use movement triggering thresholds that are based on real radio characteristics. Our evaluations show that, for different types of initial deployments, the proposed movement algorithms result in substantial energy savings, 60-70% as compared to basic virtual force-based algorithms, a major challenge for the success of a WSN.. Finally, this work addressed the problem of coverage maintenance in WSN by considering coverage loss due to damaged and energy depleted nodes. The coverage maintenance problem has been classified into two distinct categories: proactive and reactive maintenance. The coverage maintenance problem has been formulated as an ILP and implementable heuristics are developed that perform close to optimal solutions.. The evaluation results demonstrate that MAPC can enhance and maintain the area coverage by efficiently moving mobile sensor nodes to strategic positions in the uncovered area. To validate our simulations results, we ran experiments with Tmote-sky devices using the TinyOS operating system. We started by conducting preliminary calibration experiments to ascertain the effect of radio antenna orientation and the height above ground on radio range of Tmote-sky motes. The aim is to find a realistic value of the Tmote's transmission range in the given outdoor environment.

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Automatic sampling with the ratio-of-uniforms method Applying the ratio-of-uniforms method for generating random variates results in very efficient, fast, and easy-to-implement algorithms. However parameters for every particular type of density must be precalculated analytically. In this article we show, that the ratio-ofuniforms method is also useful for the design of a black-box algorithm suitable for a large class of distributions, including all with log-concave densities. Using polygonal envelopes and squeezes results in an algorithm that is extremely fast. In opposition to any other ratio-ofuniforms algorithm the expected number of uniform random numbers is less than two. Furthermore, we show that this method is in some sense equivalent to transformed density rejection. There exists a large amount of literature on generation methods for standard continuous distributions; see, for example, Devroye [1986] . These algorithms are often especially designed for a particular distribution and tailored to the features of each density. However, in many situations the application of standard distributions is not adequate for a Monte Carlo simulation. Besides sheer brute force inversion (that is, tabulate the distribution function at many points), several universal methods for large classes of distributions have been developed to avoid the design of special algorithms for these cases. Some of these methods are either very slow (e.g., Devroye [1984] ) or need a slow setup step and large tables (e.g., Ahrens and Kohrt [1981] , Marsaglia and Tsang [1984] , and Devroye [1986] ).. Recently two more efficient methods have been proposed. The transformed density rejection by Gilks and Wild [1992] and Hörmann [1995] is an acceptance/rejection technique that uses the concavity of the transformed density to generate a hat function automatically. The user only needs to provide the probability density function and perhaps the (approximate) location of the mode. A table method by Ahrens [1993] also is an acceptance/rejection method, but uses a piecewise constant hat. A region of immediate acceptance makes the algorithm fast when a large number of constant pieces is used. The tail region of the distribution is treated separately. In Ahrens [1995] the algorithm is modified to use a piecewise constant hat such that the area below each piece is the same. Thus, generation is simplified, but the algorithm requires more adjustments for the setup for each distribution.. The ratio-of-uniforms method introduced by Kinderman and Monahan [1977] is another flexible method that can be adjusted to a large variety of distributions. It has become a popular transformation method to generate nonuniform random variates, since it results in exact, efficient, fast, and easy-to-implement algorithms. Typically these algorithms have only a few lines of code (e.g., Barabesi [1993] gives a survey and examples of Fortran codes for several standard distributions). It is based on the following theorem.. THEOREM 1 [KINDERMAN AND MONAHAN 1977] . Let X be a random variable with density function f͑x͒ ϭ g͑x͒ ր ͵ g͑x͒dx, where g͑x͒ is a positive integrable function with support ͑x 0 , x 1 ͒ not necessarily finite. If ͑V, U ͒ is uniformly distributed in. then X ϭ V ր U has probability density function f͑x͒.. For sampling random points uniformly distributed in Ꮽ g rejection from a convenient enveloping region g is used. The basic form of the ratio-ofuniforms method is given by Algorithm rou.

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A posteriori error estimates for finite volume approximations of elliptic equations on general surfaces In this paper, we present a residual-based a posteriori error estimate for the finite volume discretization of steady diffusion-convection-reaction equations defined on general surfaces in R 3 , which are often implicitly represented as level sets of smooth functions. Reliability and efficiency of the proposed a posteriori error estimator are rigorously proved. Numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator. In the area of numerical solution of partial differential equations (PDEs), a posteriori error estimators are computable quantities in terms of the approximate solutions, and provide a reliable and efficient measurement for the errors of the discrete solution without knowing the exact solution. Reliability is often referred to that the true error can be bounded from above by the error estimator and efficiency implies that the true error is also locally bounded from below by the error estimator. A posteriori error estimates have played a very important role in adaptive meshes generation and algorithm designing for numerical PDEs. Theoretical and systematical studies of a posteriori error estimators for finite element approximation began in the late of 1970s [5] , and since then a vast number of literatures gradually appeared, see [2, 4, 6, 7, 9, 11, 14, 29, 37] and the references cited therein. We would like to point out that elegant analysis on residual-based a posteriori estimates of finite volume approximations for elliptic equations can be found in [1, 13] .. Numerical solution of PDEs defined on smooth surfaces (or manifolds) in R 3 recently attracted a lot of attentions due to its applications in various area, such as surface diffusion, global and local geophysical flows, ice formation, brain warping and so on [27, 32, 33, 34] . Thus it is of interest to investigate useful a posteriori error estimates for these type of problems. Some of a priori error analysis of second-order and fourth-order problems have been done for finite element methods [22, 23, 24] and finite volume methods [8, 20, 19] . Recently, a posteriori error estimates of finite element methods for discretizing the Laplace-Beltrami operator on surfaces were rigorously analyzed in [3, 15, 28] while similar study for finite volume methods is still lacking as far as we know. In this paper, we will rigorously derive a residual-based explicit a posteriori error estimator (in the sense of energy norm) for the finite volume discretization of the elliptic equations defined on a smooth surface which is represented as the zero level set of a signed distance function d to the surface.. The paper is organized as follows: in Section 1.1, we will briefly review the model problem defined on surfaces and define some notations to facilitate our analysis. Then the finite volume discretization of the problem is given in Section 2. In Section 3, we will derive a residual-based a posteriori estimate for the discretization in terms of the approximate solution, and prove its reliability and efficiency. Some numerical experiments are included in Section 4, to verify the theoretical results. In addition, we also numerically demonstrate that the a posteriori error estimator is quite robust, i.e., the constants in the a posteriori estimates are almost uniform across all test problems. Finally, concluding remarks are given in Section 5. In this paper, we derive a residual-based explicit a posteriori error estimate for the finite volume discretization of elliptic partial differential equations defined on a smooth surface in R 3 . We rigorously prove both the reliability and the efficiency of the proposed error estimator and verified the theoretical results through numerical examples. The numerical results also demonstrate the robustness of the error estimator. The on-going and future works involve studying similar a posteriori error estimators on finite volume approximation for higher-order and time-dependent problems defined on surfaces.

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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Coordination mechanisms for selfish scheduling Abstract. In machine scheduling, a set of n jobs must be scheduled on a set of m machines. Each job i incurs a processing time of pij on machine j and the goal is to schedule jobs so as to minimize some global objective function, such as the maximum makespan of the schedule considered in this paper. Often in practice, each job is controlled by an independent selfish agent who chooses to schedule his job on machine which minimizes the (expected) completion time of his job. This scenario can be formalized as a game in which the players are job owners; the strategies are machines; and the disutility to each player in a strategy profile is the completion time of his job in the corresponding schedule (a player's objective is to minimize his disutility). The equilibria of these games may result in larger-than-optimal overall makespan. The ratio of the worstcase equilibrium makespan to the optimal makespan is called the price of anarchy of the game. In this paper, we design and analyze scheduling policies, or coordination mechanisms, for machines which aim to minimize the price of anarchy (restricted to pure Nash equilibria) of the corresponding game. We study coordination mechanisms for four classes of multiprocessor machine scheduling problems and derive upper and lower bounds for the price of anarchy of these mechanisms. For several of the proposed mechanisms, we also are able to prove that the system converges to a pure Nash equilibrium in a linear number of rounds. Finally, we note that our results are applicable to several practical problems arising in networking. With the advent of the Internet, large-scale autonomous systems have become increasingly common. These systems consist of many independent and selfish agents all competing to share a common resource like bandwidth in a network or processing power in a parallel computing environment. Practical settings range from smart routing among stub autonomous systems (AS) in the Internet [14] to selfish user association in wireless local area networks [15] . In many systems of this kind, it is infeasible to impose some centralized control upon the users.. Rather, a centralized authority can only design protocols a priori and hope that the independent and selfish choices of the users-given the rules of the protocolcombine to create socially desirable results.. This approach, termed mechanism design has received considerable attention in the recent literature (see, for example, [22] ). A common goal is to design system-wide rules which, given the selfish decisions of the users, maximize the total social welfare. The degree to which these rules approximate the social welfare in a worst-case equilibrium is known as the price of anarchy of the mechanism, and was introduced in 1999 by Koutsoupias and Papadimitriou [18] in the context of selfish scheduling games (see below for details). This seminal paper spawned a series of results which attempt to design mechanisms with minimum price of anarchy for a variety of games. One approach for achieving this goal [4, 7, 12] is to impose economic incentives upon users in the form of tolls; i.e., the disutility of a user is affected by a monetary payment to some central authority for the use of a particular strategy such as a route in a network. Another approach [3, 17, 23, 27] assumes that the central authority is able to enforce particular strategies upon some fraction of the users and thus perhaps utilize an unpopular resource. This action of the central authority is called a Stackelberg strategy.. A drawback of the above approaches is that many of the known algorithms assume global knowledge of the system and thus have high communication complexity. In many settings, it is important to be able to compute mechanisms locally. A third approach, which we follow here, is called coordination mechanisms, first introduced by Koutsoupias and Nanavati [6] . A coordination mechanism is a local policy that assigns a cost to each strategy s, where the cost of s is a function of the agents who have chosen s. Consider, for example, the selfish scheduling game in which there are n jobs owned by independent agents, m machines, and a processing time p ij for job i on machine j. Each agent selects a machine on which to schedule its job with the objective of minimizing its own completion time. The social objective is to minimize the maximum completion time. A coordination mechanism [6] for this game is a local policy, one for each machine, that determines how to schedule jobs assigned to that machine. It is important to emphasize that a machine's policy is a function only of the jobs assigned to that machine. This allows the policy to be implemented in a completely distributed fashion.. Coordination mechanisms are closely related to local search algorithms. A local search algorithm iteratively selects a solution "close" to the current solution which improves the global objective. It selects the new solution from among those within some search neighborhood of the current solution. Given a coordination mechanism, we can define a local search algorithm whose search neighborhood is the set of best responses for each agent. Similarly, given a local search algorithm, it is sometimes possible to define a coordination mechanism whose pure strategy equilibria are local optima with respect to the search neighborhood. The locality gap of the search neighborhood, or approximation factor of the local search algorithm, is precisely the price of anarchy of the corresponding coordination mechanism and vice versa. In particular, designing new coordination mechanisms may lead to the discovery of a new local search algorithms for a particular problem.. In this paper, we are primarily interested in the properties of pure strategy Nash equilibria 4 for the selfish scheduling games we define. A pure strategy Nash equilibrium is an assignment of jobs to machines such that no job has a unilateral incentive to switch to another machine. Although a non-cooperative game always has a mixed strategy equilibrium [21] , it may in general not have a pure strategy equilibrium. However, by restricting ourselves to pure strategies, we are able to bound the rate of convergence of the mechanism. That is, if the jobs, starting from an arbitrary solution, iteratively play their best response strategies, how long does it take for the mechanism to reach a pure Nash equilibrium? Guarantees of this sort are important for bounded price-of-anarchy mechanisms to be applicable in a practical setting.. Preliminaries. In machine scheduling, there are n jobs must be processed on m machines. Job i has processing time p ij on machine j. A schedule µ is a function mapping each job to a machine. The makespan of a machine j in schedule µ. The goal is to find a schedule µ which minimizes the maximum makespan, C max = max j M j . Different assumptions regarding the relationship between processing times yield different scheduling problems, of which we consider the following four: (i) Identical machine scheduling (P ||C max ) in which p ij = p ik = p i for each job i and machines j and k; (ii) Uniform or related machine scheduling (Q||C max ) in which p ij = pi sj , where p i is the load of job i and s j ≤ 1 is the speed of machine j; (iii) machine scheduling for restricted assignment or bipartite machine scheduling (B||C max ) in which each job i can be scheduled on a subset S i of machines, i.e., p ij is equal to p i if j ∈ S i and is equal to ∞ otherwise; and (iv) unrelated machine scheduling (R||C max ) in which the processing times p ij are arbitrary positive numbers.. In selfish scheduling, each job is owned by an independent agent whose goal is to minimize the completion time of his own job. To induce these selfish agents to take globally near-optimal actions, we introduce the notion of a coordination mechanism [6] . A coordination mechanism is a set of scheduling policies, one for each machine. A scheduling policy for a machine j takes as input a set S ⊆ {p 1j , . . . , p nj } of jobs on machine j along with their processing times on machine j and outputs an ordering in which they will be scheduled. The policy is run locally at a machine, and so does not have access to information regarding the global state of the system (the set of all jobs, for instance, or the processing times of the jobs on other machines). A coordination mechanism defines a game in which there are n agents. An agent's strategy set is the set of possible machines {1, ..., m}. Given a strategy profile, the disutility of agent i is the (expected) completion time of job i in the schedule defined by the coordination mechanism. We study four coordination mechanisms. In the ShortestFirst and LongestFirst policies, we sequence the jobs in non-decreasing and non-increasing order of their processing times, respectively. In the Randomized policy, we process the jobs in a random order. 5 In the Makespan policy, we process all jobs on the same machine in parallel, and so the completion time of a job on machine j is the makespan of machine j.. We are interested in the properties of the solution imposed by the selfish strategies of the agents in a pure Nash equilibrium, in particular, the price of anarchy. In this setting, the price of anarchy is the worst-case ratio (over all instances) of the maximum makespan in a (pure) Nash equilibrium to the optimal makespan. For each policy and each scheduling problem, we prove upper and lower bounds on the price of anarchy. Some of these bounds are already known as the approximation factor of some local search algorithms or the price of anarchy in some selfish load balancing games. All bounds, known and new, are summarized in Table 1 .. Related work. The Makespan policy [8, 13, 18] is perhaps the best known policy among the above policies. Czumaj and Vocking [8] give tight results on the price of anarchy for the Makespan policy for mixed Nash equilibria and Q||C max . Gairing et al. [13] study the Makespan policy for B||C max and give a polynomialtime algorithm for computing a pure Nash equilibrium with makespan at most twice the optimal makespan. They also give a tight bound for the price of anarchy of the Makespan policy for pure Nash equilibria and B||C max . Azar et al. [1, 2] proved that the greedy list scheduling algorithm is an O(log m)-approximation algorithm for Q||C max and B||C max . Their proof can be used to bound the price of anarchy of the LongestFirst and ShortestFirst polices for B||C max and Q||C max .. Coordination mechanism design was introduced by Koutsoupias and Nanavati [6] . In their paper, they analyzed the LongestFirst policy for P ||C max and also studied a selfish routing game. As we have mentioned before, the price of anarchy for coordination mechanisms is closely related to the approximation factor of local search algorithms. The speed of convergence and the approximation factor of local search algorithms for scheduling problems were studied in several papers [9-11, 16, 24, 25, 28] . Vredeveld surveyed some of the results on local search algorithms for scheduling problems in his thesis [28] . The jump model in his survey [28] is similar to the Makespan policy. Moreover, the push model in his survey is related to the LongestFirst policy. It is known that the jump local search algorithm is an Θ( √ m)-approximation for Q||C max [24, 28] . Cho and Sahni [24] showed that the approximation factor of the shortest-first greedy algorithm is not better than log m for Q||C max . Ibarra and Kim [16] analyzed several greedy algorithms for R||C max . In particular, they proved that the shortest-first greedy algorithm is an m-approximation for R||C max . Davis and Jaffe [9] showed that the approximation factor of this greedy algorithm is at least log m. Our lower bound example for the ShortestFirst and the LongestFirst policy is the same as the example in [9] . Davis and Jaffe [9] also gave a √ m-approximation for R||C max . The best known approximation factor for R||C max is given by a 2-approximation algorithm due to Lenstra, Shmoys and Tardos [19] . Even-Dar et al. [10] considered the convergence time to Nash equilibria for variants of the selfish scheduling problem. In particular, they studied the Makespan policy and bounded the number of required steps to reach a pure Nash equilibrium.. Our Contribution. We give almost tight bounds for the price of anarchy of pure Nash equilibria for the Randomized, the ShortestFirst, and the LongestFirst policies. We give a proof that the price of anarchy of any deterministic coordination mechanism (including ShortestFirst and LongestFirst) for Q||C max and B||C max is at most O(log m). This result is also implied in the results of Azar et al [1, 2] on the approximation factor of the greedy list scheduling algorithm for Q||C max and B||C max . We also prove that any coordination mechanism based on a universal ordering (such as ShortestFirst and LongestFirst) has a price of anarchy Ω(log m) for B||C max . Moreover, we show that the price of anarchy of the LongestFirst policy for Q||C max is at most 2 − 2 m . In addition, we analyze the Randomized policy for R||C max machine scheduling and prove a bound of Θ(m) for the price of anarchy of this policy. We further study the convergence and existence of pure Nash equilibria for the ShortestFirst and LongestFirst policies. In particular, we show that the mechanism based on the ShortestFirst policy is a potential game, and thus any sequence of best responses converges to a Nash equilibrium after at most n rounds. We also prove fast convergence of the LongestFirst policy for Q||C max and B||C max . We have studied abstract scheduling games where the disutility of each player is its completion time. We note that our results can be applied in many practical network settings. Our results can be directly applied to the Internet setting [26] where there is a set of selfish clients, each of whom must choose a server from a set of servers. Each client tries to minimize its latency, or job completion time; the social welfare is the total system-wide latency. Similar problems arise in the wireless network setting. For example, the basic fairness and load balancing problem in wireless LANs [15] is reduced to the problem of unrelated parallel machine scheduling. Centralized algorithms have been designed for this problem in [15] . We hope to use ideas from our coordination mechanisms to design decentralized algorithms for this problem. In the third generation wireless data networks, the channel quality of a client is typically time-varying [5] ; it would be interesting to study coordination mechanisms in this context given that users may exhibit selfish behavior.. Theoretically, the most interesting open problem in this paper is to find coordination mechanisms with constant price of anarchy for B||C max and R||C max . We have shown that this cannot be achieved by any deterministic coordination mechanism which uses a common tie-breaking rule. Another problem left open in this paper is the existence of pure Nash equilibria for the Randomized policy for R||C max .. Through-out this paper, we assumed all information regarding job processing times was public knowledge. A new direction considered in [6] and [20] is the design of coordination mechanisms in a private information setting, i.e. where a job's processing time is a private value. In such a setting, it would be nice if a coordination mechanism incentivizes jobs to announce their true processing times. The only constant-factor price of anarchy for Q||C max and the best factor for P ||C max in our paper are achieved using the LongestFirst policy, but this policy is not truthful. In particular, jobs can artificially inflate their length (by inserting empty cycles, perhaps) and as a result actually decrease their disutility. A truthful coordination mechanism with a constant price of anarchy in these settings would be an interesting result.

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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B 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 B 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 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 B I I 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 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 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 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 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 O O O O O O O O O O O B I 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B 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 B I 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 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 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 B 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 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 B I O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O B I 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 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 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Parallel search for maximum satisfiability The predominance of multicore processors has increased the interest in developing parallel Boolean Satisfiability (SAT) solvers. As a result, more parallel SAT solvers are emerging. Even though parallel approaches are known to boost performance, parallel solvers developed for Boolean optimization are scarce.This dissertation proposes parallel search algorithms for Maximum Satisfiability (MaxSAT) and introduces PWBO, the first parallel solver for MaxSAT. PWBO can use two different strategies for parallel search. The first strategy performs a portfolio approach by searching on the lower and upper bound values of the optimal solution using different encodings of cardinality constraints for each thread. The second strategy splits the search space considering different upper bound values of the optimal solution for each thread.As others parallel solvers, PWBO suffers from non-deterministic behavior, i.e. several runs of the same solver can lead to different solutions. This is a clear downside for applications that require solving the same problem instance more than once. Therefore, we also present the first deterministic parallel MaxSAT solver that ensures reproducibility of results. Finally, we also propose partitioning techniques to improve sequential MaxSAT algorithms.

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 O O O O O O O O O O O O O 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 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


Determination of the Rank of an Integration Lattice The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangula lattice form) of the corresponding dual lattice Λ ⊥ . We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B. We start with background material on lattice rules. An s-dimensional lattice is an infinite set of points in R s that is closed under addition and subtraction and has no limit points. The unit lattice Λ 0 comprises all points λ = (λ 1 , λ 2 , . . . , λ s ), all of whose components λ i are integers. An integration lattice Λ is a lattice that contains Λ 0 as a sublattice, that is, Λ ⊇ Λ 0 . An s-dimensional lattice rule Q(Λ) is a quadrature rule for [0, 1) s that assigns equal weight to each point of the specified integration lattice Λ that lies in [0, 1) s , that is,. 2:01 p.m., March 27, 2007 where N(Λ) is the number of abscissas.. The reciprocal (or dual) lattice Λ ⊥ corresponding to Λ is conventionally defined by. Λ ⊥ plays a major role in the theory of lattice rules. The discretization error of Q(Λ) can be expressed in a Fourier series whose nonzero elements are related to Λ ⊥ [Ly89] , [SJ94] . Developing that result leads to a criterion for the trigonometric degree of Q(Λ) in terms of the properties of Λ ⊥ [CL01] .. An s × s matrix A is termed a generator matrix of Λ when Λ comprises precisely all points of the form. where a i is the ith row of A. When Λ is generated by A, Λ ⊥ may be generated by B = (A T ) −1 . When Λ is an integration lattice, Λ ⊥ is an integer lattice; that is, all the components of x ∈ Λ ⊥ are integers, and all the elements of B are integers. Moreover, the abscissa count of the rule is. This is also termed the order of Λ ⊥ .. It is known [Sc86] that any two generator matrices, B and B , that generate the same lattice Λ ⊥ are related by B = UB , where U is a unimodular matrix. Moreover, B can always be chosen to be in the form given by the next definition.

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


Declarative Semantics for Active Rules In this paper we analyze declarative deterministic and non-deterministic semantics for active rules. In particular we consider several (partial) stable model semantics, previously defined for deductive rules, such as well-founded, max deterministic, unique total stable model, total stable model, and maximal stable model semantics. The semantics of an active program AP is given by first rewriting it into a deductive program LP, then computing a model M defining the declarative semantics of LP and, finally, applying 'consistent' updates contained in M to the source database. The framework we propose permits a natural integration of deductive and active rules and can also be applied to queries with function symbols or to queries over infinite databases. Active databases is an emerging technology combining techniques from databases, expert systems and artificial intelligence. The main peculiarity of this technology is the support for automatic 'triggering' of rules in response to events. Automatic triggering of rules can be useful in different areas such as integrity constraint maintenance, update of materialized views, knowledge bases and expert systems etc. (Widom & Ceri, 1996) .. Active rules follow the so called Event-Condition-Action (ECA) paradigm; rules autonomously react to events occurring on the data, by evaluating a data dependent condition, and by executing a reaction whenever the condition is true. Active rules consist of three parts: Event (which causes the rule to be triggered), Condition (which is checked when the rule is triggered) and Action (which is executed when the rule is triggered and the condition is true). Thus, the semantics of a single active rule is that the rule reacts to a given event, tests a condition, and performs a given action. However, understanding the behavior of active rules, especially in the case of rules which interact with one another, is very difficult, and often the actions performed are not the expected ones. A very important issue in active databases is the development of tools which help in the design of programs with clear and intuitive semantics (Ceri & Fraternali, 1997) .. The semantics of active rules are usually given in terms of execution models, which specify how and when rules will be applied. However, execution models are not completely satisfactory since their behavior is not always clear and could result in nonterminating computations. This is shown by the following example. In this paper we have analyzed declarative semantics for active rules. The advantages of using declarative semantics are that termination is guaranteed for function free queries, confluence is also guaranteed by using confluent semantics and the semantics is well defined and intuitive. Furthermore, the framework proposed in this paper permits a natural integration of deductive and active rules and can also be applied to queries with function symbols or to queries over infinite databases. We have also shown that semantics which reduce the set of unknown literals have high complexity (and also high expressive power). Further work should be devoted to investigating the relation between non-deterministic and procedural semantics; in both cases different computations of the same program on the same database could result in different outcomes. Now we can show the above containment by induction on the fixpoint operator W P .. Step 0. I 1 = I 2 = ∅.. Step i > 0.. Consider two interpretations I 1 and I 2 for P 1 and P 2 , respectively, such that I 1 ⊆ (I 2 ∪ K I2 ). We must show that W(I 1 ) ⊆ (W(I 2 ) ∪ K W(I2) ).. Consider, first, the containment T P1 (I 1 ) ⊆ T P2 (I 2 ) ∪ K W(I2) . For each ground predicate a(t ) ∈ I 1 there are two possibilities:

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Proof-term synthesis on dependent-type systems via explicit substitutions Abstract. Typed λ-terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the Curry-Howard isomorphism relates proof trees with typed λ-terms. The proofs-as-terms principle can be used to check a proof by type checking the λ-term extracted from the complete proof tree. However, proof trees and typed λ-terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependent-type systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as meta-variables are first-class objects.

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 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