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33
Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respect ..."
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Cited by 140 (20 self)
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There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a "linearization" of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules) and can be strengthened in a natural way to generate  autonomous categories. We also point out that this "linear" notion of distributivity is virtually orthogonal to the usual notion as formalized by distributive categories. 0 Introduction There are many situations in logic, theoretical co...
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.
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...
A Linear Specification Language for Petri Nets
, 1991
"... This paper defines a category GNet with object set all Petri nets. A morphism in GNet from a net N to a net N ’ gives a precise way of simulating every evolution of N by an evolution of N’. We exhibit a morphism from a simple message handler to one with error—correction, showing that the more refine ..."
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Cited by 29 (2 self)
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This paper defines a category GNet with object set all Petri nets. A morphism in GNet from a net N to a net N ’ gives a precise way of simulating every evolution of N by an evolution of N’. We exhibit a morphism from a simple message handler to one with error—correction, showing that the more refined message handler can simulate any behaviour ofits simple counterpart. The existence of such a morphism proves the correctness of the refinement. Earlier work [BroPO, BG9O, BG] defined a modular theory of elementary Petri nets based on de Paiva’s Dialectica categorical models of linear logic. We here modify her construction, defining categories MNC which model intuitionistic linear logic [GL87]. GNet arises naturally from MNSOt, inheriting the structure which models linear logic. This more general framework has several advantages over our previous one. The theory is simplified, we obtain precise results about morphisms as simulations, relating them to CCS, and we obtain a natural extension to marked nets. The linear connectives are modelled in GNet by net combinators. Being functonal, these combinators opt are such that, if each N~is refined by a net N~,then op~(Ns Nm) is refined by opt(N ~ N~,,).We show that the operation of restriction also has this property, and thus (in the language of algebraic specification) our notion ofrefinement composes horizontally with respect to the linear connectives and restriction. Furthermore, our notion ofrefinement composes vertically because it corresponds to categorical morphisms. These properties of our notion of refinement are precisely those required to develop an algebra of nets in which complex nets can be built from smaller components, and refined in a modular and compositional way. We illustrate our approach with an extended example, analogous to Mllner’s Jobshop example. 1
Higher Dimensional Automata Revisited
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2000
"... ..."
Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics
 Theoretical Computer Science
, 1998
"... The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soun ..."
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Cited by 16 (2 self)
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The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambdacalculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 13 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at
Spaces: Complementarity and Uncertainty in Rational Mechanics
, 1994
"... 1 Introduction to Chu spaces 1.1 Basic notions A Boolean Chu space A = (X, =, A) consists of two sets X and A and a binary relation  = ⊆ X × A from X to A. We call the elements x, y,... of X states or opens, and the elements a, b,... of A points, propositions, or events. We ..."
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Cited by 12 (0 self)
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1 Introduction to Chu spaces 1.1 Basic notions A Boolean Chu space A = (X, =, A) consists of two sets X and A and a binary relation  = ⊆ X × A from X to A. We call the elements x, y,... of X states or opens, and the elements a, b,... of A points, propositions, or events. We