Using Transition Systems to Formalize Ideas from Vedānta

Bisimilar linear systems

ABSTRACT:(Labelled) transition systems are relatively common in theoretical computer science, chiefly as vehicles for operational semantics. The first part of this paper constructs a hierarchy of canonical transition systems and associated maps, aiming to give a strongly extensional theory of transition systems, where any two points with equivalent behaviors are identified. The cornerstone of the development is a notion of convergence in arbitrary transition systems, generalizing the idea of finite (n‐step) approximations to a given point. In particular, our canonical transition systems are also uniform spaces.The resulting hierarchy has very rich combinatorial (and topological) structure, and a lot of the first part of the paper is devoted to its study. We also discuss fixed points in this framework.In the second part of the paper, we show how to obtain a model of set theory with Aczel's Anti‐Foundation Axiom (AFA) from canonical transition systems constructed earlier. We study further the structure of the model thus obtained, and also give a few more abstract results, concerning consistency and independence in the presence of AFA.

Preface: Volume 59, Issue 4

Preface

In the last two decades, modal and description logics have provided a theoretical framework for important applications in many areas of computer science. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In this chapter we show how efficient Boolean reasoning techniques have been imported, used and integrated into reasoning tools for modal and description logics. To this extent, we focus on modal logics, and in particular mainly on K(m). In particular, we provide some background in modal logics; we describe a basic theoretical framework and we present and analyze the basic tableau-based and DPLL-based techniques; we describe optimizations and extensions of the DPLL-based procedures; we introduce the automata-theoretic/OBDD-based approach; finally, we present the eager approach.

The area of coalgebra has emerged within theoretical computer science with a unifying claim: to be the mathematics of computational dynamics. It combines ideas from the theory of dynamical systems and from the theory of state-based computation. Although still in its infancy, it is an active area of research that generates wide interest. Written by one of the founders of the field, this book acts as the first mature and accessible introduction to coalgebra. It provides clear mathematical explanations, with many examples and exercises involving deterministic and non-deterministic automata, transition systems, streams, Markov chains and weighted automata. The theory is expressed in the language of category theory, which provides the right abstraction to make the similarity and duality between algebra and coalgebra explicit, and which the reader is introduced to in a hands-on manner. The book will be useful to mathematicians and (theoretical) computer scientists and will also be of interest to mathematical physicists, biologists and economists.

An introduction to (first-order) Ehrenfeucht-Fraïssé games is presented, and three applications in theoretical computer science are discussed. These are concerned with the expressive power of first-order logic over graphs, formal languages definable in first-order logic, and modal logic over labelled transition systems.

Aim/Purpose: The purpose of this study is to identify how Advanced level Mathematics and Mathematics course units offered at university level do impact on the academic performance of theoretical Computer Science course units. Background: In Sri Lankan state universities, students have been enrolled only from the Physical Science stream to do a degree program in Computer Science. In addition to that, universities have been offering some course units in Mathematics to provide the required mathematical maturity to Computer Science undergraduates. Despite of this it is observed that the failure rates in fundamental theoretical Computer Science course units are much higher than other course units offered in the general degree program every year. Methodology : Academic records comprised of all 459 undergraduates from three consecutive batches admitted to the degree program in Computer Science from a university were considered for this study. Contribution: This study helps academics in identifying suitable curricula for Mathematics course units to improve students’ performance in theoretical Computer Science courses. Findings: Advanced level Mathematics does not have any significant effect on the academic performance of theoretical Computer Science course units. Even though all Mathematics course units offered were significantly correlated with academic performance of every theoretical Computer Science course unit, only the Discrete Mathematics course unit highly impacted on the academic performance of all three theoretical Computer Science course units. Further this study indicates that the academic performance of female undergraduates is better than males in all theoretical Computer Science and Mathematics course units. Future Research: Identifying other critical success factors contributing to the students’ academic performance of the theoretical Computer Science through empirical studies

Topological Cones: Foundations for a Domain Theoretical Semantics Combining Probability and Nondeterminism

This volume contains the proceedings of the 28th international conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), organized under the auspices of the Indian Association for Research in Computing Science (IARCS).

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f(x1, ..., xn) is a multilinear low-degree polynomial with low influences then the distribution of f(B1, ..., Bn) is close (in various senses) to the distribution of f(G1, ..., Gn), where Bi ∈R {-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans--Williamson algorithm for MAXCUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (X1, ..., Xn), (Y1, ..., Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: the variables X1, ..., Xn are independent, as are Y1, ..., Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (X1, ..., Xn) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ([EQUATION]) which consists of all vectors (x1, ..., xn) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos--Ko--Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, ..., Xn) is the uniform distribution on a slice ([EQUATION]) and (Y1, ..., Yn) consists either of n independent Ber(p) random variables, or of n independent N(p, p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler--Safra structural theorem, and a stability version of the t-intersecting Erdos--Ko--Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

: Modern modal logic begins with the work of C.I. Lewis early on in the present century. We can think of Lewis thinking to himself as follows: Well, I can't analyze the notions of metaphysical or logical possibility and necessity; but I can sure formulate alternative axiomatizations of such notions. I can then compare and contrast such axiomatic systems and see what I learn. Thus were born the Lewis Systems, S1-S5, axiomatizing increasingly strong conceptions of necessity. Another 40 or so years went by before the purely axiomatic approach was properly systematized and rendered fit for human consumption. In current lore, a certain axiomatic system, K, is central. The standard presentation of K consists of infinitely many axioms plus one axiom scheme and two rules of inference... It may be, then, that to take epistemic/doxastic logics seriously, one must both be working from within that conceptualization of cognitive states according to which they are either essentially or importantly language involving and, further, conceive of the language(s) in question on the model of standard formal languages, as consisting, that is, of eternal sentences only. This could be taken as an argument to the effect that the proper home of epistemic/doxastic logic is theoretical computer science -- precisely the locus of its greatest current vitality.

Telemail, an electronic mail system for the Univac 1110, was designed and implemented by the Academic Computer Center at the University of Wisconsin with support from Edunet. At present, there are approximately two hundred users including several organized groups: Educom staff and university liasons; Theory-Net researchers in theoretical computer science; a USDA sponsored research project; and a group of experts on mental retardation services/materials/literature. Most participants in the above groups access Telemail via Telenet, a commercial packet switching network. In addition, the system is used by computer center staff and researchers at the university.The Theory-Net project, sponsored by NSF, involves the use of Telemail by a group of about thirty researchers in theoretical computer science at fifteen universities. The goal is to assess the value of an electronic mail system in facilitating interaction (and thereby research productivity) in a small research community. To date a number of Joint research projects have made use of the system. In addition, the 1978 Sigact Symposium Program Committee communicated via Telemail, The next phase of the project involves a substantial increase in the number of participants. Individuals from small schools having only one or two researchers in theoretical computer science will be given high priority.

The papers in this volume were presented at the 5th Innovations in Theoretical Computer Science (ITCS 2014) conference, sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT). The conference was held in Princeton, New Jersey, USA, January 11--14, 2014. ITCS (previously known as ICS) seeks to promote research that carries a strong conceptual message, for instance, introducing a new concept or model, opening a new line of inquiry within traditional or cross-interdisciplinary areas, or introducing new techniques or new applications of known techniques). The call for papers welcomed all submissions, whether aligned with current theory of computation research directions or deviating from them. Altogether 116 submissions were received worldwide. Of these the program committee selected 48 papers. The accepted papers cover a wide range of topics in theoretical computer science, including algorithms, complexity, cryptography, learning, data privacy, quantum, physical and biological computing and relations between computing and social sciences. In addition to the selected papers the committee invited Professor Peter Winkler of Dartmouth to give an evening talk and we are grateful for his acceptance. Another evening was devoted to "Graduating Bits" - short talks by recent graduates. The program committee consisted of 24 members (plus the chair): Deeparnab Chakrabarty (Microsoft Research India), Timothy Chan (University of Waterloo), Costis Daskalakis (MIT), Yuval Emek (ETH and Technion), Kousha Etessami (University of Edinburgh), Yuval Filmus (University of Toronto and Simons Institute, Berkeley), Arpita Ghosh (Cornell University), Monika Henzinger (University of Vienna), Sandy Irani (University of California Irvine), Michael nKearns (University of Pennsylvania), Lap Chi Lau (The Chinese University of Hong Kong), Nati Linial (Hebrew University of Jerusalem), Kobbi Nissim (Ben-Gurion University), Rasmus Pagh (IT University of Copenhagen), Shubhangi Saraf (Rutgers University), Ola Svensson (EPFL), Vinod Vaikuntanathan (University of Toronto and MIT), Jan Vondrak (IBM Almaden Research Center), Manfred Warmuth (University of California, Santa Cruz), Daniel Wichs (Northeastern University), Udi Wieder (Microsoft Research SVC), Ryan Williams (Stanford University), Ronald de Wolf (CWI and University of Amsterdam), David Xiao (CNRS and Université Paris 7). I wish to express my admiration for their hard work of reading, evaluating and debating the merits of the submissions. The many individuals who assisted the reviewing process as subreviewers and extended the expertise of the committee deserve acknowledgments as well.

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.