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A Term Calculus for Intuitionistic Linear Logic
, 1993
"... . 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 th ..."
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. 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 welltyped). We also consider term reduction arising from cutelimination 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...
A syntax for linear logic
 Presented at Conference on Mathematical Foundations of Programming Language Semantics
, 1993
"... Abstract. This tutorial paper provides an introduction to intuitionistic logic and linear logic, and shows how they correspond to type systems for functional languages via the notion of ‘Propositions as Types’. The presentation of linear logic is simplified by basing it on the Logic of Unity. An app ..."
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Cited by 89 (7 self)
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Abstract. This tutorial paper provides an introduction to intuitionistic logic and linear logic, and shows how they correspond to type systems for functional languages via the notion of ‘Propositions as Types’. The presentation of linear logic is simplified by basing it on the Logic of Unity. An application to the array update problem is briefly discussed. 1
QuasiLinear Types
, 1999
"... Linear types (types of values that can be used just once) have been drawing a great deal of attention because they are useful for memory management, inplace update of data structures, etc.: an obvious advantage is that a value of a linear type can be immediately deallocated after being used. Howeve ..."
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Cited by 72 (6 self)
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Linear types (types of values that can be used just once) have been drawing a great deal of attention because they are useful for memory management, inplace update of data structures, etc.: an obvious advantage is that a value of a linear type can be immediately deallocated after being used. However, the linear types have not been applied so widely in practice, probably because linear values (values of linear types) in the traditional sense do not so often appear in actual programs. In order to increase the applicability of linear types, we relax the condition of linearity by extending the types with information on an evaluation order and simple dataflow information. The extended type system, called a quasilinear type system, is formalized and its correctness is proved. We have implemented a prototype type inference system for the coreML that can automatically find out which value is linear in the relaxed sense. Promising results were obtained from preliminary experiments with the p...
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives ..."
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Cited by 61 (33 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
Types and subtypes for clientserver interactions
 Proceedings of the 1999 European Symposium on Programming, number 1576 in Lecture Notes in Computer Science
, 1999
"... Abstract. We define an extension of the πcalculus with a static type system which supports highlevel specifications of extended patterns of communication, such as clientserver protocols. Subtyping allows protocol specifications to be extended in order to describe richer behaviour; an implemented ..."
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Cited by 57 (6 self)
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Abstract. We define an extension of the πcalculus with a static type system which supports highlevel specifications of extended patterns of communication, such as clientserver protocols. Subtyping allows protocol specifications to be extended in order to describe richer behaviour; an implemented server can then be replaced by a refined implementation, without invalidating typecorrectness of the overall system. We use the POP3 protocol as a concrete example of this technique. 1
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 56 (10 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Linear Logic, Monads and the Lambda Calculus
 In 11 th LICS
, 1996
"... Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage ( ..."
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Cited by 45 (7 self)
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Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direct, callbyname and callbyvalue) correspond exactly to three translations, due mainly to Girard, of intuitionistic logic into intuitionistic linear logic. We also consider extending these results to languages with recursion. 1. Introduction Two of the most significant developments in semantics during the last decade are Girard's linear logic [10] and Moggi's computational metalanguage [14]. Any student of these formalisms will suspect that there are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model  a symmetric monoidal closed category with a comonad ! which satisfies some extr...
Applications of Linear Logic to Computation: An Overview
, 1993
"... 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 objectoriented programming and some other applications of LL, li ..."
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Cited by 42 (3 self)
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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 objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart 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...
CallbyName, CallbyValue, CallbyNeed, and the Linear Lambda Calculus
, 1994
"... Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second correspond ..."
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Cited by 39 (6 self)
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Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds to callbyvalue. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a callbyneed calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky, and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the CurryHoward isomorphism.