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Term Assignment for Intuitionistic Linear Logic
, 1992
"... 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) and has two important properties which they lack. Th ..."
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Cited by 53 (9 self)
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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) 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 define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cutelimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.
A dialecticalike model of linear logic
 In Proc. Conf. on Category Theory and Computer Science, LNCS 389
, 1989
"... The aim of this work is to define the categories GC, describe their categorical structure and show they are a model of Linear Logic. The second goal is to relate those categories to the Dialectica categories DC, cf.[DCJ, using different functors for the exponential “of course”. It is hoped that this ..."
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Cited by 27 (6 self)
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The aim of this work is to define the categories GC, describe their categorical structure and show they are a model of Linear Logic. The second goal is to relate those categories to the Dialectica categories DC, cf.[DCJ, using different functors for the exponential “of course”. It is hoped that this categorical model of Linear Logic should help us to get a better understanding of the logic, which is, perhaps, the first nonintuitionistic constructive logic. This work is divided in two parts, each one with 3 sections. The first section shows that GC is a monoidal closed category and describes bifunctors for tensor “0”, internal horn “[—, —]“, par “u”, cartesian products “& “ and coproducts “s”. The second section defines linear negation as a contravariant functor obtained evaluating the internal horn bifunctor at a “dualising object”. The third section makes explicit the connections with Linear Logic, while the fourth introduces the comonads used to model the connective “of course”. Section 5 discusses some properties of these cornonads and finally section 6 makes the logical connections once more. This work grew out of suggestions of J.Y. Girard at the AMSConference on Categories, Logic and Computer Science in Boulder 1987, where I presented my earlier work on the Dialectica categories, hence the title. Still on the lines of given credit where it is due, I would like to say that Martin Hyland, under whose supervision this work was written, has been a continuous source of ideas and inspiration. Many heartfelt thanks to him. 1. The main definitions We start with a finitely complete category C. Then to describe GC say that its objects are relations on objects of C, that is monics A ~ U x X, which we usually write as (U ~ X). Given two such objects, (U ~ X) and (V L Y), which we call simply A and B, a morphism from A to B consists of a pair of maps in C, f: U — * V and F 4 Y —+ X, such that a pullback condition is satisfied, namely that where (~~)_1 represents puilbacks. (U x F) 1 (o~) ~ (f x Y) 1 (/3), (1) 342 Using diagrams, we say (f,F) is a morphism in GC if there is a (unique) map in ~, k: A ’ —~B ’ making the triangle commute: a~I Ia
Linear lambdaCalculus and Categorical Models Revisited
, 1992
"... this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor,\Omega , linear implication, \Gammaffi, and the logical operator "exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We us ..."
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Cited by 22 (0 self)
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this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor,\Omega , linear implication, \Gammaffi, and the logical operator "exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We use capital Greek letters \Gamma; \Delta for sequences of formulae and A; B for single formulae. The Exchange rule simply allows the permutation of assumptions. The `! rules' have been given names by other authors. ! L\Gamma1 is called Weakening , ! L\Gamma2 Contraction, ! L\Gamma3 Dereliction and (! R ) Promotion
Specifying Interaction Categories
, 1997
"... We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, ..."
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Cited by 11 (2 self)
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We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, as well as some new ones, can be built starting from some simple familiar categories. Using the presented constructions, interaction categories can be analysed without fixing a set of axioms, merely in terms of the way in which they are specified  just like algebras are analysed in terms of equations and relations, independently on abstract characterisations of their varieties.
! and ?  Storage as tensorial strength
, 1996
"... this paper were produced with the help of the diagram macros of F. Borceux. Blute, Cockett, & Seely 2 ..."
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Cited by 9 (5 self)
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this paper were produced with the help of the diagram macros of F. Borceux. Blute, Cockett, & Seely 2
Some Algebraic Laws for Spans (and Their Connections With MultiRelations)
 Proceedings of RelMiS 2001, Workshop on Relational Methods in Software. Electronic Notes in Theoretical Computer Science, n.44 v.3, Elsevier Science (2001
, 2001
"... This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. O ..."
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Cited by 9 (3 self)
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This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. Our results nd analogous counterparts in (and are partly inspired by) the theory of relational algebras, thus our paper also shed some light on the relationship between (co)spans and the categories of (multi)relations and of equivalence relations. And, since (co)spans yields an intuitive presentation in terms of dynamical system with input and output interfaces, our results introduce an expressive, twofold algebra that can serve as a specication formalism for rewriting systems and for composing software modules and open programs. Key words: Spans, multirelations, monoidal categories, system specications. Introduction The use of spans [1,6] (and of the dual notion of cospans) have been...
Tail Recursion Through Universal Invariants
 Theoretical Computer Science
, 1993
"... Tail recursive constructions suggest a new semantics for datatypes, which allows a direct match between specifications and tail recursive programs. The semantics focusses on loops, their fixpoints, invariants and convergence. Convergent models of the natural numbers and lists are examined in detail, ..."
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Cited by 6 (4 self)
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Tail recursive constructions suggest a new semantics for datatypes, which allows a direct match between specifications and tail recursive programs. The semantics focusses on loops, their fixpoints, invariants and convergence. Convergent models of the natural numbers and lists are examined in detail, and, under very mild conditions, are shown to be equivalent to the corresponding initial algebra models. 1 Introduction Tail recursion is a central feature of program construction because of its efficiency, but is usually assigned a secondary place in semantics, which is dominated by primitive recursion as expressed through initial algebras. The success of this approach is testimony to the ease with which we can use initial algebras to specify functions, and their theoretical power. The difficulty is that whenever such a specification is to be translated into code there remains the need to optimise it, often by conversion into tail recursive form. Conversely, it is not at all easy to provi...
2008) Classical and quantum structures
"... In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to ..."
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Cited by 6 (2 self)
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In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of ”quantization ” of a classical theory to ”classicization ” of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantumclassical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing inbetween controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classicalquantum interaction. 1
Induction, Coinduction, and Adjoints
, 2002
"... We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and supp ..."
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Cited by 3 (2 self)
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We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(X) of the functor K(Y ) = X G(Y ) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y ) = X + F (Y ) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.