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A Categorical Semantics of Quantum Protocols
 In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS‘04), IEEE Computer Science
"... Quantum information and computation is concerned with the use of quantummechanical systems to carry out computational and informationprocessing tasks [16]. In the few short years that this approach has been studied, a ..."
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Cited by 258 (46 self)
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Quantum information and computation is concerned with the use of quantummechanical systems to carry out computational and informationprocessing tasks [16]. In the few short years that this approach has been studied, a
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...
Linear Logic, Autonomous Categories and Cofree Coalgebras
 In Categories in Computer Science and Logic
, 1989
"... . A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products an ..."
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Cited by 130 (8 self)
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. A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a closed involution. Girard's exponential operator, ! , is a cotriple on G which carries the canonical comonoid structure on A with respect to cartesian product to a comonoid structure on !A with respect to tensor product. This makes the Kleisli category for ! cartesian closed. 0. INTRODUCTION. In "Linear logic" [1987], JeanYves Girard introduced a logical system he described as "a logic behind logic". Linear logic was a consequence of his analysis of the structure of qualitative domains (Girard [1986]): he noticed that the interpretation of the usual conditional ")" could be decomposed into two more primitive notions, a linear conditional "\Gammaffi" and a unary operator "!" (called "of cours...
Autonomous categories and linear logic
 Mathematical Structures in Computer Science
, 1991
"... The subject of linear logic has recently become very important in theoretical computer science. It is apparent that the ∗autonomous categories studied at length in [Barr, 1979] are a model for a large fragment of linear logic, although not quite for the whole thing. Since the main reference is out ..."
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Cited by 106 (9 self)
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The subject of linear logic has recently become very important in theoretical computer science. It is apparent that the ∗autonomous categories studied at length in [Barr, 1979] are a model for a large fragment of linear logic, although not quite for the whole thing. Since the main reference is out of print and since large parts of that volume
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
SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 80 (9 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semiabelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar nonabelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...
A typed calculus of synchronous processes
 In Proceedings of IEEE Symposium on Logic in Computer Science
, 1995
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Fair Games and Full Completeness for Multiplicative Linear Logic without the MIXRule
, 1993
"... We introduce a new category of finite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll wh ..."
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Cited by 45 (4 self)
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We introduce a new category of finite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll which satisfies the property that every (historyfree, uniformly) winning strategy is the denotation of a unique cutfree proof net. Abramsky and Jagadeesan first proved a result of this kind and they refer to this property as full completeness. Our result differs from theirs in one important aspect: the mixrule, which is not part of Girard's Linear Logic, is invalidated in our model. We achieve this sharper characterization by considering fair games. A finite, fair game is specified by the following data: ffl moves which Player can play, ffl moves which Opponent can play, and ffl a collection of finite sequences of maximal (or terminal) positions of the game which are deemed to be fair. N...
Equilogical Spaces
, 1998
"... It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relation ..."
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Cited by 41 (12 self)
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It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this categoryin contradistinction to Top 0 is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the KleeneKreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1