We present a subsystem of second order Linear Logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice-versa.

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In this paper we study a notion of operational equivalence for interaction nets, following the recent success of applying methods based on bisimulation to functional and object oriented programming languages. We set up notions of contextual equivalence and bisimilarity and show that they coincide. A coinduction principle then gives a simple and robust way of showing when two interaction nets are contextually equivalent. We include several examples to demonstrate the usefulness of the approach, in particular for optimizing interaction nets. 1 Introduction One of the most fundamental notions in programming languages is that of program equivalence: when can one program fragment be replaced by another. A notion of equivalence should be substitutive so that programs remain equivalent in all contexts. Amongst other applications, this facilitates proving properties of programs, and gives a sound basis for program optimization. Interaction nets provide an interesting new perspective as both a...

Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. -calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semi-simple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for -calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...

Many calculi for describing interactive behaviour involve names, name-abstraction and name-restriction. Milner's reflexive action calculi provide a framework for exploring such calculi. It is based on names and name-abstraction. We introduce an alternative framework, the symmetric action calculi, based on names, co-names and name-restriction (or hiding). Name-abstraction is intepreted as a derived operator. The symmetric framework conservatively extends the reflexive framework. It allows for a natural intepretation of a variety of calculi: we give interpretations for the -calculus, the I -calculus and a variant of the fusion calculus. We then give a combinatory version of the symmetric framework, in which name-restriction also is expressed as a derived operator. This combinatory account provides an intermediate step between our non-standard use of names in graphs, and the more standard graphical structure arising from category theory. To conclude, we briey illustrate the connection ...

We give a hypergraph rewriting semantics for the polyadic π-calculus, based on rewriting rules equivalent to those in the double-pushout approach. The structural congruence of the π-calculus is replaced by hypergraph isomorphism. The correctness of the encoding from the graph-based notation into π-calculus can be shown by using an intermediate notation, so-called name-based graph terms, which form a bridge from graphs with explicit connections (by fusing nodes) to process calculi with implicit connections (by common channel names).

The purpose of this paper is to demonstrate how Lafont’s interaction combinators, a system of three symbols and six interaction rules, can be used to encode linear logic. Specifically, we give a translation of the multiplicative, exponential and additive fragments of linear logic together with a strategy for cut-elimination which can be faithfully simulated. Finally, we show briefly how this encoding can be used for evaluating �-terms. In addition to offering a very simple, perhaps the simplest, system of rewriting for linear logic and the �-calculus, the interaction net implementation that we present has been shown by experimental testing to offer a good level of sharing, in terms of the number of cut-elimination steps (resp. ¬reduction steps). In particular it performs better than all extant finite systems of interaction nets.

This paper presents a simple implementation of the -calculus in the interaction net paradigm. It is based on a two-fold translation. -terms are coded (for duplication) or decoded (for execution) and reduction is achieved by switching between these two states: decoding corresponds to head reduction and encoding to left reduction

Sharing graphs are the structures introduced by Lamping to implement optimal reductions of lambda calculus. Gonthier's reformulation of Lamping's technique inside Geometry of Interaction, and Asperti and Laneve's work on Interaction Systems have shown that sharing graphs can be used to implement a wide class of calculi. Here, we give a general characterization of sharing graphs independent from the calculus to be implemented. Such a characterization rests on an algebraic semantics of sharing graphs exploiting the methods of Geometry of Interaction. By this semantics we can de ne an unfolding partial order between proper sharing graphs, whose minimal elements are unshared graphs. The least-shared instance of a sharing graph is the unique unshared graph that the unfolding partial order associates to it. The algebraic semantics allows to prove that we can associate a semantical read-back to each unshared graph and that such a read-back can be computed

We define an observational equivalence for Lafont’s interaction combinators, which we prove to be the least discriminating non-trivial congruence on total nets (nets admitting a deadlock-free normal form) respecting reduction. More interestingly, this equivalence enjoys an internal separation property similar to that of Böhm’s Theorem for the λ-calculus.