The abuse of structural rules may have damaging complexity effects.

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...

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

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Vol. 2 (4:3) 2006, pp. 1–44 www.lmcs-online.org

We reformulate Danos contractibility criterion in terms of a sort of unification. As for term unification, a direct implementation of the unification criterion leads to a quasi-linear algorithm. Linearity is obtained after observing that the disjoint-set union-find at the core of the unification criterion is a special case of union-find with a real linear time solution. Keywords: linear logic, data structures. 1 Introduction A multiplicative proof net is a graph representation of a multiplicative linear logic derivation [5]. 1 The proof net N corresponding to the derivation P is a (directed) hypergraph with a link (i.e., an hyperedge) for each rule and a vertex for each formula occurrence s.t. every link of N connects the active formulas of the corresponding rule of P. For instance, let P be a cut-free derivation using atomic axioms only and ending with the sequent ` A; the corresponding proof net N is the syntax tree of the formula A, plus a set of connections between pairs of oc...

Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reduction process is local and asynchronous, and all the operations are made explicit, including discarding and copying of data. Our aim is to bridge the gap between the above formalisms by showing how to understand interaction nets in a term rewriting framework. This allows us to transfer results from one paradigm to the other, deriving syntactical properties of interaction nets from the (well-studied) properties of term rewriting systems; in particular concerning termination and modularity. Keywords: term rewriting, interaction nets, termination, modularity. 1 Introduction Term rewriting systems provide a general framework for specifying and reasoning about computation. They can be regarde...

In its most general meaning, a Boolean category is to categories what a Boolean algebra is to posets. In a more specific meaning a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category proof nets as a particularly well-behaved example of a Boolean category.

Abstract. We propose a translation of a finitary (that is, replicationfree) version of the monadic localised pi-calculus into the purely exponential part of promotion-free differential interaction nets. This embedding is a simulation of reduction. Since the introduction of Linear Logic by Girard in 1986, it was clear to many logicians and computer scientists that some deep connection between this new logical setting and concurrency should show up. This impression has been enforced by the introduction of interaction nets by Lafont [1], where reduction is given by a purely local and asynchronous interaction. There is an apparent contradiction between non-determinism and the Curry-Howard approach to computation. Indeed, one of the main properties that one expects from a well-behaved proof system is not only that it possesses a cutelimination procedure, but also that this procedure enjoys a confluence property similar to the Church-Rosser property of the λ-calculus. But confluence is a way of expressing determinism in a rewriting setting, so that being able to represent

Useful type inference must be faster than normalization. Otherwise, you could check safety conditions by running the program. We analyze the relationship between bounds on normalization and type inference. We show how the success of type inference is fundamentally related to the amnesia of the type system: the nonlinearity by which all instances of a variable are constrained to have the same type.