Quantum information and computation is concerned with the use of quantum-mechanical systems to carry out computational and information-processing tasks [16]. In the few short years that this approach has been studied, a

There are many situations in logic, theoretical computer science, and category theory where two binary operations---one 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 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], Jean-Yves 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...

. 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 Introduction Some of the best things in life are free; and some are not. Truth is free. Having proved a theorem, you may use this proof as many times as you wish, at no extra cost. Food, on the other hand, has a cost. Having baked a cake, you may eat it only once. If traditional logic is about truth, then linear logic is about food. In traditional logic, if a fact is used to conclude another fact, the first fact is still available. For instance, given that A implies B and given A, one may deduce both A and B. In symbols, this is written as the judgement A ! B; A ` A \Theta B (i) where A ! B is read `A implies B', and A \Theta B is read `...

We propose a typed calculus of synchronous processes based on the structure of interaction categories. Our aim has been to develop a calculus for concurrency that is canonical in the sense that the typed -calculus is canonical for functional computation. We show strong connections between syntax, logic and semantics, analogous to the familiar correspondence between the typed -calculus, intuitionistic logic and cartesian closed categories. 1 Introduction T ypes are fundamental to the study of functional computation, for both theoretical and practical reasons. On the foundational side there are elegant connections between the typed -calculus, intuitionistic logic and cartesian closed categories, leading to the Propositions as Types paradigm [14] and the development of categorical logic [9,17]. From a practical point of view, compile-time type reconstruction is a boon to the programmer in languages such as Standard ML and Haskell. Turning to concurrency, the situation is much less sati...

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 (history-free, uniformly) winning strategy is the denotation of a unique cut-free 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 mix-rule, 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...

Linear logic is a resource-aware logic that is based on an analysis of the classical proof rules of contraction (copying) and weakening (throwing away). In this paper we study the decision problem for the multiplicative fragment of linear logic without quantifiers or propositions: the constant-only case. We show that this fragment is np-complete. Earlier work by Max Kanovich showed that propositional multiplicative linear logic is np-complete. With Natarajan Shankar, the first author developed a simplified proof for the propositional case. The structure of this simplified proof is utilized here with a new encoding which uses only constants. The end product is the somewhat surprising result that simply evaluating expressions in true, false, and, and or in multiplicative linear logic (\Omega , , 1, and ?) is np-complete. By conservativity results not proven here, the np-hardness of larger fragments of linear logic follows. 1 Introduction When Girard introduced linear logic [7], he bro...

The notion of semi-abelian 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 abelian-group and module theory. In modern terms, semi-abelian 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 semi-abelian 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 non-abelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...

This thesis concerns rewriting in the typed -calculus. Traditional categorical models of typed -calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for -terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed -calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...

It is well known that one can build models of full higher-order 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 category|in 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 Kleene-Kreisel 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