A Theory for Game Theories

Electronic Notes in Theoretical Computer Science, 2009

We study the algebraic structure of a programming language with higher-order store, in the style of ML references. Instead of working directly on the operational semantics of the language, we consider its fully abstract game semantics defined by Abramsky, Honda and McCusker one decade ago. This alternative description of the language is nice and conceptual, except on one significant point: the interactive behavior of the higher-order memory cell is reflected in the model by a strategy cell whose definition remains slightly enigmatic. The purpose of our work is precisely to clarify this point, by providing a neat algebraic definition of the strategy. This conceptual reconstruction of the memory cell is based on the idea that a general reference behaves essentially as a linear feedback (or trace operator) in an ambient category of Conway games and strategies. This analysis leads to a purely axiomatic proof of soundness of the model, based on a natural refinement of the replication modality of tensor logic.

Lecture Notes in Computer Science, 1999

We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H * , the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

We briefly present a new representation theory for game semantics which is very concrete: instead of playing in an arena game in which P plays the innocent strategy given by a term, the same game is played out over (a souped up version of) the abstract syntax tree of the term itself. The plays that are thus traced out are called traversals. More abstractly, traversals are the justified sequences that are obtained by performing parallel-composition less the hiding. After stating and explaining a number of Path-Traversal Correspondence Theorems, we present a tool for game semantics based on the new representation.

SpringerBriefs in Philosophy, 2015

A brief examination of the most recent literature in logic shows that a host of research in this area studies the interface between games, logic and epistemology. These studies provide the basis for ongoing enquiries in the history and philosophy of logic, going from the Indian, the Greek, the Arabic, the Obligationes of the Middle Ages to the most contemporary developments in the fields of theoretical computer science, computational linguistics, artificial intelligence, social sciences and legal reasoning. In fact, a dynamic turn, as Johan van Benthem puts it, is taking place where the epistemic aspects of inference are linked with game theoretical approaches to meaning. 1 This turn came about in the 1960's when Paul Lorenzen and Kuno Lorenz developed dialogical logic-inspired by Wittgenstein's language games and mathematical game theory-and when some time later Jaakko Hintikka combined gametheoretical semantics with epistemic (modal) logic. If we had to pinpoint a date, it would be 1958 with Lorenzen's talk 2 "Logik und Agon". However, these two approaches to logic-the dialogical one and the one based on Hintikka's GTS-springing from a dynamic reading of the epistemic conception of logic, disregarded a major advance precisely in the epistemic approach to logic, namely, the development by Per Martin-Löf of Constructive Type Theory (CTT)with the sole exception of the pioneering paper of Aarne Ranta [1988]. 3 This framework, providing a type theoretical development of the Curry-Howard-isomorphism 1 New results in linear logic by J.-Y. Girard at the interface between mathematical game theory and proof theory on the one hand and argumentation theory and logic on the other resulted in the work of, among others, S.

Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies -that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of "atomic" strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way. † This work has been supported by the ANR Invariants algébriques des systèmes informatiques (INVAL). Physical address:Équipe PPS, CNRS and Université Paris 7, 2 place Jussieu, case 7017,

1997

Mathematical models of computation, from the-calculus to PRAM's, have played a key role in the development of software technology, from programming languages to the design and analysis of algorithms. The rst generation of models of computation (c. 1950 {1980) were functional in character; that is, they abstracted the behaviour of a program as the computation of a function.

1993

this paper an alternative description of the game semantics for the untyped lambda calculus is given. More precisely, we introduce a finitary description of lambda terms. This description turns out to be equivalent to a particular game denotational semantics of the lambda calculus. Introduction

1998

Abstract A games model of a programming language with higher-order store in the style of ML-references is introduced. The category used for the model is obtained by relaxing certain behavioural conditions on a category of games previously used to provide fully abstract models of pure functional languages. The model is shown to be fully abstract by means of factorization arguments which reduce the question of definability for the language with higher-order store to that for its purely functional fragment

Foundations of Software Science and Computation Structures, 2018

The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten. We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation. Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism, and still get a category-this partial hiding remains sound up to weak bisimulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences.

In this paper we present a fully abstract game model for the pure lazy λ-calculus, i.e. the lazy λ-calculus without constants. In order to obtain this result we introduce a new category of games, the monotonic games, whose main characteristic consists in having an order relation on moves.

Annals of Pure and Applied Logic, 1997

I present a semantics for the language of first order additive-multiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sen-tences=games, sentences=resources or sentences=problems, where "truth" means existence of an effective winning (resource-using, problem-solving) strategy. The paper introduces a decidable first order logic ET in the above language and gives a proof of its soundness and completeness (in the full language) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic. The semantics presented here is very similar to Blass's game semantics (A.Blass, "A game semantics for linear logic", APAL, 56). Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability (via equality to ET), which was a major problem left open in Blass's paper. The reader needs to be familiar with classical (but not necessarily linear) logic and arithmetic.

2003

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.

2015

A brief examination of the most recent literature in logic shows that a host of research in this area studies the interface between games, logic and epistemology. These studies provide the basis for ongoing enquiries in the history and philosophy of logic, going from the Indian, the Greek, the Arabic, the Obligationes of the Middle Ages to the most contemporary developments in the fields of theoretical computer science, computational linguistics, artificial intelligence, social sciences and legal reasoning. In fact, a dynamic turn, as Johan van Benthem puts it, is taking place where the epistemic aspects of inference are linked with game theoretical approaches to meaning. 1 This turn came about in the 1960's when Paul Lorenzen and Kuno Lorenz developed dialogical logic-inspired by Wittgenstein's language games and mathematical game theory-and when some time later Jaakko Hintikka combined gametheoretical semantics with epistemic (modal) logic. If we had to pinpoint a date, it would be 1958 with Lorenzen's talk 2 "Logik und Agon". However, these two approaches to logic-the dialogical one and the one based on Hintikka's GTS-springing from a dynamic reading of the epistemic conception of logic, disregarded a major advance precisely in the epistemic approach to logic, namely, the development by Per Martin-Löf of Constructive Type Theory (CTT)with the sole exception of the pioneering paper of Aarne Ranta [1988]. 3 This framework, providing a type theoretical development of the Curry-Howard-isomorphism 1 New results in linear logic by J.-Y. Girard at the interface between mathematical game theory and proof theory on the one hand and argumentation theory and logic on the other resulted in the work of, among others, S.

Construction, 2010

This paper aims at studying relations between proof systems and games in a given logic and at analyzing what can be the interest and limits of a game formulation as an alternative semantic framework for modelling proof search and also for understanding relations between logics. In this perspective, we firstly study proofs and games at an abstract level which is neither related to a particular logic nor adopts a specific focus on their relations. Then, in order to instantiate such an analysis, we describe a dialogue game for intuitionistic logic and emphasize the adequateness between proofs and winning strategies in this game. Finally, we consider how games can be seen to provide an alternative formulation for proof search and we stress on the possible mix of logical rules and search strategies inside games rules. We conclude on the merits and limits of the game semantics as a tool for studying logics, validity in these logics and some relations between them.

Theoretical Computer Science, 2008

We present a type assignment system that provides a finitary interpretation of lambda terms in a game semantics model. Traditionally, type assignment systems describe the semantic interpretation of terms in domain theoretic models. Quite surprisingly, the type assignment system presented in this paper is very similar to the traditional ones, the main difference being the omission of the subtyping rules.