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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...
Reference Counting as a Computational Interpretation of Linear Logic
 Journal of Functional Programming
, 1996
"... 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 o ..."
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Cited by 35 (0 self)
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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 referencecounting interpretation of the primitives that the language draws from the rules for the `of course' operation. Our semantics is `lowlevel' 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 runtime 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...
A Model for Syntactic Control of Interference
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 1993
"... 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; ..."
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Cited by 22 (4 self)
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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.
The power of linear functions
 PROCEEDINGS OF THE 15TH EACSL CONFERENCE ON COMPUTER SCIENCE LOGIC (CSL’06), VOLUME 4207 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... The linear lambda calculus is very weak in terms of expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with Booleans, natural numbers and a linear iterator. We show properties of this linear version of Gödel’s System T and study th ..."
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Cited by 10 (6 self)
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The linear lambda calculus is very weak in terms of expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with Booleans, natural numbers and a linear iterator. We show properties of this linear version of Gödel’s System T and study the class of functions that can be represented. Surprisingly, this linear calculus is extremely expressive: it is as powerful as System T.
Lehman Brothers Corporation
, 1995
"... 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 ..."
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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 �high� level � 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.
On Linear Types and Imperative Update
, 1997
"... of Dissertation August, 1997 The rift between imperative and functional programming is one of the oldest in computing. Imperative languages provide for greater speed and efficiency, especially through the use of destructive update of variables. Functional computation is often regarded as being much ..."
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of Dissertation August, 1997 The rift between imperative and functional programming is one of the oldest in computing. Imperative languages provide for greater speed and efficiency, especially through the use of destructive update of variables. Functional computation is often regarded as being much cleaner mathematically, and is supportive of simple, rigorous reasoning principles. However, the absence of destructive update (along with problems with garbage collection, etc.) in functional languages results in a loss of speed and efficiency. Additionally, functional languages are conceptually indirect and cumbersome when used to express naturally imperative computation. The prospect of unifying these diverse forms of programming, to combine the best features from each, is a continuing concern in programming languages. This thesis investigates the extent to, and the sense in which, programming languages based on linear logic are simultaneously imperative and purely functional, and to wha...