We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass et al.

An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.

Introduction If we consider the interpretation of proofs as programs, say in intuitionistic logic, the question of equality between proofs becomes crucial: The syntax introduces meaningless distinctions whereas the (denotational) semantics makes excessive identifications. This question does not have a simple answer in general, but it leads to the notion of proof-net, which is one of the main novelties of linear logic. This has been already explained in [Gir87] and [GLT89]. The notion of interaction net introduced in [Laf90] comes from an attempt to implement the reduction of these proof-nets. It happens to be a simple model of parallel computation, and so it can be presented independently of linear logic, as in [Laf94]. However, we think that it is also useful to relate the exact origin of interaction nets, especially for readers with some knowledge in linear logic. We take this opportunity to give a survey of the theory of proof-nets, including a new proof of the sequentializ

Girard's execution formula (given in [Gir88a]) is a decomposition of usual fi-reduction (or cut-elimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a -term or a net, as the sum of maximal paths on the -term/net that are not cancelled by the algebra L (as was done in [Dan90, Reg92]). It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the -term/net. Remarkably, the proof puts to use for the first time the interpretation of -terms/nets as operators on the Hilbert space. 1 Presentation -Calculus is simple but not completely convincing as a real machine-language. Real machine instructions have a fixed run-time; a fi-reduction step does not. Some implementations do map fi-reductions into sequences of real elementary steps (as in environment machines for example) but they use a global time t...

This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and object-oriented programming and some other applications of LL, like semantics of negation in LP, non-monotonic issues in AI planning, etc. Although the overview covers pretty much the state-of-the-art in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...

We introduce a new class of higher order rewriting systems, called Interaction Systems (IS's). IS's come from Lafont's (Intuitionistic) Interaction Nets [Lafont 1990] by dropping the linearity constraint. In particular, we borrow from Interaction Nets the syntactical bipartitions of operators into constructors and destructors and the principle of binary interaction. As a consequence, IS's are a subclass of Klop's Combinatory Reduction Systems [Klop 1980] where the Curry-Howard analogy still "makes sense". Destructors and constructors respectively corresponds to left and right logical introduction rules, interaction is cut and reduction is cut-elimination. Interaction Systems have been primarily motivated by the necessity of extending the practice of optimal evaluators for -calculus [Lamping 1990, Gonthier et al. 1992a] to other computational constructs as conditionals and recursion. In this paper we focus on the theoretical aspects of optimal reductions. In particular, we ge...

This paper is the continuation of the author 's work on interaction nets, inspired by Girard's proof nets for linear logic, but no preliminary knowledge of these topics is required for its reading. Introduction

Lamping's optimal graph reduction technique for the -calculus is generalized to a new class of higher order rewriting systems, called Interaction Systems. Interaction Systems provide a nice integration of the functional paradigm with a rich class of data structures (all inductive types), and some basic control flow constructs such as conditionals and (primitive or general) recursion. We describe a uniform and optimal implementation, in Lamping's style, for all these features. The paper is the natural continuation of [3], where we focused on the theoretical aspects of optimal reductions in Interaction Systems (family relation, labeling, extraction). 1 Introduction At the end of 70's, L'evy fixed the theoretical performance of what should be considered as an optimal implementation of the -calculus. The optimal evaluator should always keep shared those redexes in a -expression that have a common origin (e.g. that are copies of a same redex). For a long time, no implementation achieved L'...

Confluence of orthogonal rewriting systems can be proved using the Finite Developments Theorem. We present, in a general setting, several adaptations of this proof method for obtaining confluence of `not quite' orthogonal systems. 1. Introduction Rewriting as studied here is based on the analogy: rewriting = substitution + rules. This analogy is useful since it enables a clearcut distinction between the `designer' defined substition process, i.e. management of resources, and the `user' defined rewrite rules, of rewriting systems. For example, application of the `user' defined term rewriting rule 2 \Theta x ! x + x to the term 2 \Theta 3 gives rise to the duplication of the term 3 in the result 3 + 3. How this duplication is actually performed (for example, using sharing) depends on the `designer's' implementation of substitution. This decomposition has been shown useful in [OR94, Oos94] in the case of first-order term rewriting systems (TRSs, [DJ90, Klo92]) and higher-order term r...

An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cut-free proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspace-hardness of imall. It exploits several proof-theoretic properties of intuitionistic implication that analyze the use of resources in iil proofs. Linear logic is a refinement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard's words [12], "linear logic is a logic behind logic." A convenient way to present linear logic is by modifying the traditional Gentzen-style sequent calculus axiomatization of classical logic (see, e.g., [15, 22]). The modification may be briefly described in three steps. The first step is to remove two structural rules, contraction and weakening, which manipulate the use of hypotheses and conclusi...