. In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content. 1 Intuitionistic Linear Logic Girard's Intuitionistic Linear Logic [3] is a refinement of Intuitionistic Logic where formulae must be used exactly once. Given this restriction the familiar logical connectives become divided into multiplicative and additive versions. Within this paper, we shall only consider the multiplicatives. Int...

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 develop formal methods for reasoning about memory usage at a level of abstraction suitable for establishing or refuting claims about the potential applications of linear logic for static analysis. In particular, we demonstrate a precise relationship between type correctness for a language based on linear logic and the correctness of a reference-counting interpretation of the primitives that the language draws from the rules for the `of course' operation. Our semantics is `low-level' enough to express sharing and copying while still being `highlevel ' enough to abstract away from details of memory layout. This enables the formulation and proof of a result describing the possible run-time reference counts of values of linear type. Contents 1 Introduction 1 2 Operational Semantics with Memory 4 3 A Programming Language Based on Linear Logic 9 4 Semantics 14 5 Properties of the Semantics 24 6 Linear Logic and Memory 27 7 Discussion 32 A Proofs of the Main Theorems 36 Acknowledgements...

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Two imperative programming language phrases interfere when one writes to a storage variable that the other reads from or writes to. Reynolds has described an elegant linguistic approach to controlling interference in which a refinement of typed calculus is used to limit sharing of storage variables; in particular, different identifiers are required never to interfere. This paper examines semantic foundations of the approach. We describe a category that has (an abstraction of) interference information built into all objects and maps. This information is used to define a “tensor” product whose components are required never to interfere. Environments are defined using the tensor, and procedure types are obtained via a suitable adjunction. The category is a model of intuitionistic linear logic. Reynolds’ concept of passive type – i.e. types for phrases that don’t write to any storage variables – is shown to be closely related, in this model, to Girard’s “of course” modality.

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