DOCSLIB.ORG

On Complexity of Delp Through Game Semantics∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Complexity of Delp Through Game Semantics∗ On Complexity of DeLP through Game Semantics∗ Laura A. Cecchi Pablo R. Fillottrani and Guillermo R. Simari Depto. Ciencias de la Computacion´ - Fa.E.A. Depto. de Cs. e Ing. de la Comp. Universidad Nacional del Comahue Universidad Nacional del Sur Buenos Aires 1400 Av. Alem 1253 (8300) Neuquen´ - ARGENTINA (8000) Bah´ıa Blanca - ARGENTINA [email protected] {prf,grs}@cs.uns.edu.ar Abstract on a dialectical analysis where arguments for and against Defeasible Logic Programming (DeLP) is a general argumen- a literal interact in order to determine whether this literal tation based system for knowledge representation and reason- is believed by a reasoning agent. The semantics GS is a ing. Its proof theory is based on a dialectical analysis where declarative trivalued game-based semantics for DeLP that arguments for and against a literal interact in order to deter- links game-semantics (Abramsky & McCusker 1997) and mine whether this literal is believed by a reasoning agent. The model-theory. Soundness and completeness of GS with re- semantics GS is a declarative trivalued game-based semantics spect to DeLP proof theory have been proved (Cecchi & for DeLP that is sound and complete for DeLP proof theory. Simari 2004). Complexity theory is an important tool for comparing differ- Complexity theory is an important tool for comparing dif- ent formalism and for helping to improve implementations ferent formalism, and for helping to improve implementa- whenever it is possible. In this work we address the prob- tions whenever it is possible. For this reason, it is important lem of studying the complexity of some important decision to analyze the computational complexity and the expressive problems in DeLP. Thus, we characterize the relevant deci- sion problems in the context of DeLP and GS, and we define power of DeLP. The former tells us how difficult it is to an- data and combined complexity for DeLP. Since DeLP com- swer a query, while the latter gives a precise characterization putes every argument from a set of defeasible rules, it is of of the concepts that are definable as queries. central importance to analyze the complexity of two decision Even thought complexity for nonmonotonic reasoning problems. The first one can be defined as “Is a set of defea- systems has been studied in depth for several formalisms sible rules an argument for a literal under a defeasible logic such us default logic, autoepistemic logic, circumscription, program?”. We prove that this problem is P-complete. The abduction and logic programming (Cadoli & Schaerf 1993; second decision problem is “Does there exist an argument for Dantsin et al. 2001) until recently not many complexity re- a literal under a defeasible logic program?”. We prove that sults for argumentation systems have been reported. this problem is in NP. Furthermore, we study data complex- This situation can be explained in part by the fact that, ity of query answering in the context of DeLP. As far as we know, data complexity has not been introduced in the context historically, implementations of argumentation systems have of argumentation systems. been limited to areas with no real time response restriction (see (Verheij 1998; Gordon & Karacapilidis 1997)). Re- cently, however, several applications have been developed, and implemented using argumentation systems related, for KEYWORDS: Argumentation Systems, Defeasible Rea- instance, with multiagent systems and web search (Atkin- soning, Logic Programming, Game-based Semantics, son, Bench-Capon, & Mc Burney 2004; Chesnevar˜ & Ma- Complexity guitman 2004a; 2004b; Bassiliades, Antoniou, & Vlahavas 2004). Scalability and robustness of such approaches heav- ily depend on the computational properties of the underly- Introduction ing algorithms. It is hence crucial to study these properties Defeasible Logic Programming (DeLP) is a general argu- in order to expand the application fields of argumentation mentation based tool for knowledge representation and rea- systems. soning (Garc´ıa & Simari 2004)1. Its proof theory is based Different computational complexity results (Dimopoulos, ∗ Nebel, & Toni 2002; Bench-Capon 2003; Amgoud & Cayrol This research was partially supported by Secretar´ıa General 2002; Dunne & Bench-Capon 2002) have been presented on de Ciencia y Tecnolog´ıa of the Universidad Nacional del Sur, by the Universidad Nacional del Comahue (Proyecto de Investigacion´ argumentation abstract framework (Bondarenko et al. 1997; 04/E062), by Agencia Nacional de Promocion´ Cient´ıfica y Tec- Dung 1995), based on admissibility and preferability seman- nologica´ (PICT 2002 No. 13096, PICT 2003 15043, PAV 076) and tics. However, those results do not apply directly to DeLP, by the National Research Council (CONICET), ARGENTINA. because its semantics are quite different. Another notable 1The interested reader can find an on-line interpreter for DeLP study of the computational complexity of defeasible systems in http://lidia.cs.uns.edu.ar/DeLP has been done in (Maher 2001). But, defeasible theory an- denoted as L, is defined as follows: L = ∼A, if L is an alyzed in this work greatly differs from DeLP in several atom, otherwise if L is a negated atom, L = A. Let X be points, such as knowledge representation (facts and strict a set of literals, X is the set of the complement of every rule, defeasible and defeaters rules) and their proof theory. member in X. When measuring the complexity of evaluating queries in a specific language, we distinguish between several kinds Definition 1 A strict rule is an ordered pair, denoted of complexity according to (Vardi 1982; Papadimitriou & “Head ← Body”, where “Head” is a ground literal, and Yannakakis 1997; Dantsin et al. 2001). Data complexity “Body” is a finite set of ground literals. A strict rule is the complexity of evaluating a specific query in the lan- with head L0 and body {L1,...Ln, n > 0} is written as guage, when the query is fixed, and we study the complexity L0 ← L1,...Ln. If body is the empty set, then we write of applying this query to arbitrary databases; the complex- L0., and the rule is called a Fact. A defeasible rule is an ity is thus given as a function of the size of the database. ordered pair, denoted “Head —≺ Body”, where “Head” is Program or Expression complexity appears when a specific a ground literal, and “Body” is a finite, non-empty set of database is fixed, and we study the complexity of applying ground literals. A defeasible rule with head L0 and body queries represented by arbitrary expressions in the language; {L1,...Ln, n > 0} is written as L0 —≺ L1,...Ln. the complexity is given as a function of the length of the ex- A defeasible logic program P, abbreviated de.l.p., is a set pression. Combined complexity considers both query and of strict rules and defeasible rules. We will distinguish the database instance as input variables. subsets ΠF of facts, ΠR of strict rules, Π = ΠF ∪ ΠR and In this work we are concerned with the study of complex- the subset ∆ of defeasible rules. ity of some important decision problems of DeLP. The sys- tem and its asociated game semantics GS are analyzed intro- Intuitively, whereas Π is a set of certain and exception-free ducing relevant decision problems in relation to the possible knowledge, ∆ is a set of defeasible knowledge, i.e., tentative query answers. information that could be used, whenever nothing is posed Since DeLP builds the arguments from a defeasible logic against it. program results of central importance to consider and eval- By definition a de.l.p. may be an infinite set of strict and uate two questions: “is a set of defeasible rules an argument defeasible rules but for complexity analysis we restrict our- for a literal under a defeasible logic program?” which has selves to finite defeasible logic programs. been proved to be P-complete, and does there exit an argu- We denote by Lit the set of all the ground literals that ment for a literal under a defeasible logic program? which can be generated considering the underlying signature of a + has been proved to be in NP. de.l.p. an we denote by Lit the set of all the atoms in Lit. We define data, expression and combined complexity in DeLP proof theory is based on developments in non the context of DeLP, in order to evaluate the efficiency of monotonic argumentation systems (Pollock 1987; Simari & DeLP implementations. In particular, we study data com- Loui 1992). An argument for a literal L is a minimal subset plexity of query answering to assess DeLP applications over of ∆ that together with Π consistently entails L. The no- database technologies. As far as we know data complexity tion of entailment corresponds to the usual SLD derivation has not been introduced in the context of argumentation sys- used in logic programming, performed by backward chain- tems. ing on both strict and defeasible rules, where negated atoms The paper is structured as follows. In the following sec- are treated as a new atom in the underlying signature. Thus, tion we briefly outline the fundamentals of DeLP, and de- an agent can explain a literal L, throughout this argument. scribe the declarative game-based semantics GS. Then, we In order to determine whether a literal L is supported from discuss DeLP through GS semantics pointing out the deci- a de.l.p. a dialectical tree for L is built. An argument for L sion problems that are of central importance, and we define represents the root of the dialectical tree, and every other data, expression and combined complexity in the context of node in the tree is a defeater argument against its parent. At DeLP. Afterwards, we give complexity results on the exis- each level, for a given a node we must consider all the argu- tence of an argument for a literal L under a defeasible logic ments against that node.
Load more

Recommended publications
  • Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation
    1 Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation Patrick Cousota aD´epartement d’Informatique, Ecole´ Normale Sup´erieure, 45 rue d’Ulm, 75230 Paris cedex 05, France, [email protected], http://www.di.ens.fr/~cousot We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step seman- tics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational se- mantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conser- vative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these seman- tics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski fixpoint approximation transfer theorems. Contents 1 Introduction 2 2 Abstraction of Fixpoint Semantics 3 2.1 Fixpoint Semantics ............................... 3 2.2 Fixpoint Semantics Approximation ...................... 4 2.3 Fixpoint Semantics Transfer .......................... 5 2.4 Semantics Abstraction ............................. 7 2.5 Fixpoint Semantics Fusion ........................... 8 2.6 Fixpoint Iterates Reordering .......................... 8 3 Transition/Small-Step Operational Semantics 9 4 Finite and Infinite Sequences 9 4.1 Sequences .................................... 9 4.2 Concatenation of Sequences .......................... 10 4.3 Junction of Sequences ............................. 10 5 Maximal Trace Semantics 10 2 5.1 Fixpoint Finite Trace Semantics .......................
    [Show full text]
  • A Game Theoretical Semantics for a Logic of Formal Inconsistency
    A game theoretical semantics for a logic of formal inconsistency CAN BA¸SKENT∗, Department of Computer Science, University of Bath, BA2 7AY, England. Downloaded from https://academic.oup.com/jigpal/article/28/5/936/5213088 by guest on 25 September 2020 PEDRO HENRIQUE CARRASQUEIRA∗∗, Center for Logic, Epistemology and the History of Science (CLE), Institute of Philosophy and the Humanities (IFCH), State University of Campinas (UNICAMP), Campinas 13083-896, Brazil. Abstract This paper introduces a game theoretical semantics for a particular logic of formal inconsistency called mbC. Keywords: Game theoretical semantics, logic of formal inconsistency, mbC, non-deterministic runs, oracles 1 Motivation Paraconsistent logics are the logics of inconsistencies. They are designed to reason about and with inconsistencies and generate formal systems that do not get trivialized under the presence of inconsistencies. Formally, a logic is paraconsistent if the explosion principle does not hold—in paraconsistent systems, there exist some formulas ϕ, ψ such that ϕ, ¬ϕ ψ. Within the broad class of paraconsistent logics, Logics of Formal Inconsistency (LFIs, for short) present an original take on inconsistencies. Similar to the da Costa systems, LFIs form a large class of paraconsistent logics and make use of a special operator to control inconsistencies [8, 9]. Another interesting property of LFIs is that they internalize consistency and inconsistency at the object level. To date, the semantics for LFIs has been suggested using truth tables and algebraic structures. What we aim for in this paper is to present a game theoretical semantics for a particular logic within the class of LFIs. By achieving this, we attempt at both presenting a wider perspective for the semantics of paraconsistency and a broader, more nuanced understanding of semantic games.
    [Show full text]
  • Game Semantics of Homotopy Type Theory
    Game Semantics of Homotopy Type Theory Norihiro Yamada [email protected] University of Minnesota Homotopy Type Theory Electronic Seminar Talks (HoTTEST) Department of Mathematics, Western University February 11, 2021 N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 1 / 19 Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation). Proofs/objects as computations (e.g., succ : : max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies. Introduction Background: MLTT vs. HoTT N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 Proofs/objects as computations (e.g., succ : : max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies. Introduction Background: MLTT vs. HoTT Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation). N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 MLTT as a foundation of constructive maths.
    [Show full text]
  • The Game Semantics of Game Theory
    The game semantics of game theory Jules Hedges We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by n ≥ 0 noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a com- pact closed category of ‘computable open games’ by replacing the underly- ing dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the traced cartesian category of directed-complete partial orders. 1 Introduction Although the mathematics of games shares a common ancestor in Zermelo’s work on backward induction, it split early on into two subjects that are essentially disjoint: game theory and game semantics. Game theory is the applied study of modelling real-world interacting agents, for example in economics or artificial intelligence. Game semantics, by contrast, uses agents to model – this time in the sense of semantics rather than mathematical modelling – situations in which a system interacts with an environment, but neither would usually be thought of as agents in a philosophical sense. On a technical level, game semantics not only restricts to the two-player zero-sum case, but moreover promotes one of the players to be the Player, and demotes the other to mere Opponent.
    [Show full text]
  • Game Theoretical Semantics for Paraconsistent Logics
    Game Theoretical Semantics for Paraconsistent Logics Can Ba¸skent Department of Computer Science, University of Bath, England [email protected], www.canbaskent.net/logic 1 Introduction Game theoretical semantics suggests a very intuitive approach to formal seman- tics. The semantic verification game for classical logic is played by two players, verifier and falsifier who we call Heloise and Abelard respectively. The goal of Heloise in the game is to verify the truth of a given formula in a given model whereas for Abelard it is to falsify it. The rules are specified syntactically based on the form of the formula. During the game, the given formula is broken into subformulas step by step by the players. The game terminates when it reaches the propositional literals and when there is no move to make. If the game ends up with a propositional literal which is true in the model in question, then Heloise wins the game. Otherwise, Abelard wins. Conjunction is ssociated with Abelard, disjunction with Heloise. That is, when the main connective is a conjunction, it is Abelard’s turn to choose and make a move, and similarly, disjunction yields a choice for Heloise. The negation operator switches the roles of the players: Heloise becomes the falsifier, Abelard becomes the verifier. The major result of this approach states that Heloise has a winning strategy if and only if the given formula is true in the given model. The semantic verification game and its rules are shaped by classical logic and consequently by its restrictions. In this work, we first observe how the verification games change in non-classical, especially propositional paraconsistent logics, and give Hintikka-style game theoretical se- mantics for them.
    [Show full text]
  • A Game Theoretical Semantics for Logics of Nonsense
    A Game Theoretical Semantics for Logics of Nonsense Can Bas¸kent Department of Computer Science, Middlesex University, London, UK [email protected] Logics of non-sense allow a third truth value to express propositions that are nonsense. These logics are ideal formalisms to understand how errors are handled in programs and how they propagate throughout the programs once they appear. In this paper, we give a Hintikkan game semantics for logics of non-sense and prove its correctness. We also discuss how a known solution method in game theory, the iterated elimination of strictly dominated strategies, relates to semantic games for logics of nonsense. Finally, we extend the logics of nonsense only by means of semantic games, developing a new logic of nonsense, and propose a new game semantics for Priest’s Logic of Paradox. Keywords Game semantics, logics of nonsense, logic of paradox, iterated elimination of strictly dominated strategies. 1 Introduction Logics of nonsense allow a third truth value to express propositions that are nonsense. The initial mo- tivation behind introducing nonsensical propositions was to capture the logical behavior of semantic paradoxes that were thought to be nonsensical. As argued by Ferguson, nonsense is “infectious” [8]: “Meaninglessness [...] propagates through the language; that a subformula is meaningless entails that any complex formula in which it finds itself is likewise meaningless.” Logics of nonsense, for this reason, are intriguing. Nonsense appears in various contexts. In mathematics, the expression “1/x” is considered mean- ingless when x = 0. Certain approaches to truth disregard paradoxes of self-reference and exclude them as meaningless.
    [Show full text]
  • Semantics of Interaction Samson Abramsky
    Semantics of Interaction Samson Abramsky Abstract The “classical” paradigm for denotational semantics models data types as domains, i.e. structured sets of some kind, and programs as (suitable) functions between do- mains. The semantic universe in which the denotational modelling is carried out is thus a category with domains as objects, functions as morphisms, and composition of morphisms given by function composition. A sharp distinction is then drawn between denotational and operational semantics. Denotational semantics is often referred to as “mathematical semantics” because it exhibits a high degree of math- ematical structure; this is in part achieved by the fact that denotational semantics abstracts away from the dynamics of computation—from time. By contrast, op- erational semantics is formulated in terms of the syntax of the language being modelled; it is highly intensional in character; and it is capable of expressing the dynamical aspects of computation. The classical denotational paradigm has been very successful, but has some definite limitations. Firstly, fine-structural features of computation, such as se- quentiality, computational complexity, and optimality of reduction strategies, have either not been captured at all denotationally, or not in a fully satisfactory fashion. Moreover, once languages with features beyond the purely functional are con- sidered, the appropriateness of modelling programs by functions is increasingly open to question. Neither concurrency nor “advanced” imperative features such as local references have been captured denotationally in a fully convincing fashion. This analysis suggests a desideratum of Intensional Semantics, interpolating between denotational and operational semantics as traditionally conceived. This should combine the good mathematical structural properties of denotational se- mantics with the ability to capture dynamical aspects and to embody computational intuitions of operational semantics.
    [Show full text]
  • Algorithmic Game Semantics and Component-Based Verification
    Algorithmic Game Semantics and Component-Based Verification Samson Abramsky, Dan R. Ghica, Andrzej S. Murawski, C.-H. Luke Ong Oxford University Computing Laboratory ABSTRACT and strategies on games. Strategies can be seen as certain kinds of We present a research programme dedicated to the application of highly-constrained processes, hence they admit the same kind of Game Semantics to program analysis and verification. We high- automata-theoretic representations central to model checking and light several recent theoretical results and describe a prototypical allied methods in computer-assisted verification. Moreover, games software modeling and verification tool. The distinctive novel fea- and strategies naturally form themselves into rich mathematical tures of the tool are its ability to handle open programs and the structures which yield very accurate models of advanced high-level fact that the models it produces are observationally fully abstract. programming languages, as the various full abstraction results show. These features are essential in the modeling and verification of soft- Thus the promise of this approach is to carry over the methods of ware components such as modules. Incidentally, these features also model checking (see e.g. [10]), which has been so effective in the lead to very compact models of programs. analysis of circuit designs and communications protocols, to much more structured programming situations, in which data-types as well as control flow are important. 1. INTRODUCTION AND BACKGROUND A further benefit of the algorithmic approach is that by embody- Game Semantics has emerged as a powerful paradigm for giving ing game semantics in tools, and making it concrete and algorith- semantics to a variety of programming languages and logical sys- mic, it should become more accessible and meaningful to practi- tems.
    [Show full text]
  • DIALECTICAL GAMES in the ACADEMY Benoît Castelnérac† & Mathieu Marion‡
    ARGUING FOR INCONSISTENCY: DIALECTICAL GAMES IN THE ACADEMY Benoît Castelnérac† & Mathieu Marion‡ SOCRATES: So now please do whichever of these you like: either ask questions or answer them. Gorgias 462b This paper is part of a larger research programme concerning dialectical games played in Ancient Greece, their origin, their rules, their influence on philosophy, and their role in the origin of logic. By dialectical games, we mean the sort of games exemplified in the dialogues of Plato, with Socrates as the main character trying to drive his opponent into an elenchus, and for which Aristotle wrote later on his textbook, the Topics – there are of course other minor sources. These are games where, typically, the proponent of a thesis, say, A is asked a series of questions by an opponent – the role usually assumed by Socrates in Plato’s dialogues – whose task is to show that, in holding A, the proponent will contradict some other beliefs that he also happens to hold, i.e., if successful, the opponent shows that A is part of an inconsistent set of premises. In other words, these games can be described as games of ‘consistency management’.1 We wish merely to offer here the preliminary groundwork for a formalization of these games, through a listing and brief discussion of their apparent rules. We will provide a snapshot of games that were played in the mid-4th century BC, therefore at a time when both Plato and Aristotle where at the Academy. Within the context of our research programme this period should be seen as a terminus ante quem, from which one could begin the historical search, going back to Parmenides and Zeno.
    [Show full text]
  • Game Semantics
    Game Semantics Samson Abramsky University of Edinburgh Department of Computer Science James Clerk Maxwell Building Edinburgh EH9 3JZ Scotland email: [email protected] Guy McCusker St John’s College Oxford OX1 3JP England email: [email protected] 1 Introduction The aim of this chapter is to give an introduction to some recent work on the appli- cation of game semantics to the study of programming languages. An initial success for game semantics was its use in giving the first syntax-free descriptions of the fully abstract model for the functional programming language PCF [1, 14, 31]. One goal of semantics is to characterize the “universe of discourse” implicit in a programming language or a logic. Thus for a typed, higher-order functional programming language such as PCF, one may try to characterize “what it is to be a PCF-definable functional”. Well established domain-theoretic models [12, 35] provide sufficiently rich universes of functionals to interpret languages such as PCF, but in fact they are too rich; they include functionals, even “finitary” ones (defined over the booleans, say), which are not definable in PCF. Moreover, by a remarkable recent result of Ralph Loader [25], this is not an accident; this result (technically the undecidability of observation equivalence on finitary PCF) implies that no effective characterization of which functionals are definable in PCF (even in finitary PCF) can exist. Thus in particular a model containing all and only the PCF-definable functionals cannot be effectively presentable. However, rather than focussing on the functionals in extenso, we may instead seek to characterize those computational processes which arise in computing the functionals.
    [Show full text]
  • Nominal Game Semantics
    Full text available at: http://dx.doi.org/10.1561/2500000017 Nominal Game Semantics Andrzej S. Murawski University of Warwick Nikos Tzevelekos Queen Mary University of London Boston — Delft Full text available at: http://dx.doi.org/10.1561/2500000017 Foundations and Trends R in Programming Languages Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is A. S. Murawski and N. Tzevelekos. Nominal Game Semantics. Foundations and Trends R in Programming Languages, vol. 2, no. 4, pp. 191–269, 2015. R This Foundations and Trends issue was typeset in LATEX using a class file designed by Neal Parikh. Printed on acid-free paper. ISBN: 978-1-68083-107-8 c 2016 A. S. Murawski and N. Tzevelekos All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The ‘services’ for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged.
    [Show full text]
  • Intensionality, Definability and Computation
    Chapter 1 Intensionality, Definability and Computation Samson Abramsky Abstract We look at intensionality from the perspective of computation. In par- ticular, we review how game semantics has been used to characterize the sequential functional processes, leading to powerful and flexible methods for constructing fully abstract models of programming languages, with appli- cations in program analysis and verification. In a broader context, we can regard game semantics as a first step towards developing a positive theory of intensional structures with a robust mathematical structure, and finding the right notions of invariance for these structures. Keywords Computation, intentionality, extensionality, definability, game semantics. 1.1 Introduction Our aim in this paper is to give a conceptual discussion of some issues con- cerning intensionality, definability and computation. Intensionality remains an elusive concept in logical theory, but actually becomes much more tangi- arXiv:1705.04744v1 [cs.LO] 12 May 2017 ble and, indeed, inescapable in the context of computation. We will focus on a particular thread of ideas, leading to recent and ongoing work in game se- mantics. Technical details will be kept to a minimum, while ample references will be provided to the literature. Department of Computer Science, University of Oxford Wolfson Building Building, parks Road, Oxford OX1 3QD, U.K. e-mail: [email protected] 1 2 Samson Abramsky It is a pleasure and a privilege to contribute to a volume in honour of Johan van Benthem. Johan has been a friend and an inspiring and support- ive colleague for well over a decade. He has encouraged me to broaden my intellectual horizons and to address wider conceptual issues.
    [Show full text]