Introduction to Coalgebra

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

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.

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

Bisimilar linear systems

Preface: Volume 59, Issue 4

The computational card (c-cards for short) project is a study and realization of an educational tool based on playing cards. C-cards are an educational tool to introduce children 8 to 10 (or older) to the concept of computation, seen as manipulation of symbols. The game provides teachers and learners with a physical, tangible metaphor for exploring core concepts of computer science, such as deterministic and probabilistic state machines, frequencies and probability distributions, and the central elements of Shannon's information theory, like information, communication, errors and error detection. Our idea is implemented both with paper cards and by an editor/simulator software (a prototype realized in javascript). We also designed the structure of a course in (theoretical) computer science, based on c-cards, and we will test it this summer.

The correspondence between partial metrics and semivaluations

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.

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).

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

Preface: Volume 81

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”.

Computer science “result of the decade” found?

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.

The papers in this volume were presented at the 7th Innovations in Theoretical Computer Science (ITCS 2016) conference, sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT). The conference was held at the Massachusetts Institute of Technology in Cambridge, MA, USA, January 14-16, 2016. 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, introducing new techniques, or making novel connections between existing areas and ideas. The conference format is single-session with ample time for discussion, to promote the exchange of ideas between different areas of theoretical computer science and with other disciplines. The call for papers welcomed all submissions, whether aligned with current theory of computation research directions or deviating from them. 145 submissions were received. Of these, the program committee selected 40 papers. I would like to thank authors of all submissions for their interest in ITCS. The program committee consisted of 23 members (plus the chair): Nikhil Bansal, Eindhoven; Eli Ben-Sasson, Technion; Bernard Chazelle, Princeton; Moritz Hardt, Google; Yuval Ishai, Technion; Brendan Juba, Washington U., St. Louis; Adam Kalai, MSR; Anna Karlin, U. Washington; Sanjeev Khanna, U. Penn.; Valerie King, U. Victoria; Katrina Ligett, Caltech; Ruta Mehta, Georgia Tech.; Rotem Oshman, Tel Aviv; Rafael Pass, Cornell; Anup Rao, U. Washington; Omer Reingold, Samsung Research America; Aaron Roth, U. Penn.; Rocco Servedio, Columbia; Yaron Singer, Harvard; Adam Smith, Penn. State; Greg Valiant, Stanford; Thomas Vidick, Caltech; and Nisheeth Vishnoi, EPFL. I wish to express my sincere thanks to them for agreeing to join the committee and then for investing time and effort the merits of the submissions. The many subreviewers who assisted the reviewing process deserve acknowledgment as well. The local organizations were undertaken by Shafi Goldwasser (MIT & Weizmann). I'd like to thank Shafi, for this service as also for her service as steering committee chair of ITCS while ITCS remains in its early stages! The submissions and program committee deliberations were conducted using the platform generated by Shai Halevi. I would like to thank Shai for generously contributing his time to the running of the system and helping me work with it! I would additionally like to thank IACR for hosting the server. I also thank Tim Roughgarden (Program Chair of ITCS 2015) for his help. Finally, I would like to thank all the presenters and the audience at ITCS and hope it continues to be a unique experience.