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...

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'...

. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1

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There are two quite different possibilities for implementing linear head reduction in -calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former. Keywords: -calculus, abstract machines, geometry of interaction, reversible computations. 1 Introduction Notation. We denote the application of U to V by (U)V , e.g., the Church integer ¯ 2 will be fx (f)(f)x. Linear head reduction. But what is exactly linear head reduction, to begin with. It is a variant of head reduction, where one substitutes at each step the leftmost occurrence of c fl1996 Elsevier Science B. V. Danos & Regnier variable whenever it is engaged into a redex, as in: (f (f )(f)x)y y ! (f(y y)(f)x)y y ! (f(y (f )x)(f)x)y y ! (f(y (y y)...

The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the -calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a "local implementation" of naturality laws, that is as the broadcasting of some information from the output to the inputs of a term, following its connected structure. 1 Introduction More than fifteen years ago, L'evy [Le78] proposed a theoretical notion of optimality for -calculus normalization. Roughly speaking, a reduction technique is optimal if it is able to profit of all the sharing expressed in initial term, avoiding useless duplications. For a long time, no implementation was able to achieve L'evy's performance (see [Fie90] for a quick survey). People started already to doubt of the existence of optimal evaluators, when Lamping and Kathail independently found a solution [Lam90,Ka90]...

This note defines a new graphical local calculus, directed virtual reductions. It is designed to compute Girard's execution formula EX, an invariant of closed functional evaluation obtained from the "geometry of interaction" interpretation of -calculus [4]. The calculus is obtained by synchronizing another graphical local calculus presented in "local and asynchronous beta-reduction": virtual reductions [3]. This synchronization makes it easier to mechanize than general virtual reductions. In undirected virtual reductions the consistency of the computation is insured by an algebraic mechanism called the bar. This mechanism in general induces correction terms of any order. The directed virtual reduction has been designed to keep those terms at order one. A further synchronization, the combustion strategy will even wipe out first order correction terms. Applied to sharing graphs, the combustion strategy yields Lamping's optimal graphical calculus as presented in [1]. But more efficient op...

Since the rebirth of -calculus in the late sixties, three major theoretical investigations of fi-reduction were undertaken: 1) L'evy's analysis of families of redexes (and the associated concept of labeled reductions) ; 2) Lamping's graph-reduction algorithm; 3) Girard 's geometry of interaction. All these three studies happened to make crucial (if not always explicit) use of a notion of path. Namely and respectively: legal, consistent and regular paths. Now, these three different notions stand in no obvious relation at first sight. In this paper we prove they are equivalent. 1 Introduction Let us first survey the three different notions of paths which we want to prove are equivalent. L'evy took hold in [12] of the difficult notion of two redexes being created in the "same" way during a reduction (in which case they were said to belong to the same "family"). Then he labeled terms and made beta reductions act on labels so that two redexes were in the same family iff they had the same ...