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44
Discontinuity and the Lambek Calculus
, 1994
"... This paper is concerned with the treatment of discontinuous constituency within Categorial Grammar ..."
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Cited by 13 (2 self)
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This paper is concerned with the treatment of discontinuous constituency within Categorial Grammar
The Girard Translation Extended with Recursion
 In Proceedings of Computer Science Logic
, 1995
"... This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of ..."
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This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec calculus into terms of the linear rec calculus is induced via the extended CurryHoward isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...
! 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
Memoisation for Glue Language Deduction and Categorial Parsing
, 1998
"... The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number ..."
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Cited by 8 (1 self)
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The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number of deduction methods for multiplicative linear logic are known, none of them are tabular meth ods, which bring a substantial efficiency gain by avoiding redundant computation (c.f. chart methods in CFG parsing): this paper presents such a method, and discusses its use in relation to the above applications.
Explicit Substitution Internal Languages for Autonomous and *Autonomous Categories
 In Proc. Category Theory and Computer Science (CTCS'99), Electron
, 1999
"... We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show tha ..."
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Cited by 7 (2 self)
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We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the standard simplytyped calculus with surjective pairing is...
Linear Logic, Comonads and Optimal Reductions
 Fundamentae Informaticae
, 1993
"... The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a ..."
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The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the calculus. In particular, we relate the two "brackets" in [GAL92a] to the two operations associated with the comonad "!" of Linear Logic. The rewriting rules can be then understood as a "local implementation" of naturality laws, that is as the broadcasting of some information from the output to the inputs of a term, following its connected structure. 1 Introduction More than fifteen years ago, L'evy [Le78] proposed a theoretical notion of optimality for calculus normalization. Roughly speaking, a reduction technique is optimal if it is able to profit of all the sharing expressed in initial term, avoiding useless duplications. For a long time, no implementation was able to achieve L'evy's performance (see [Fie90] for a quick survey). People started already to doubt of the existence of optimal evaluators, when Lamping and Kathail independently found a solution [Lam90,Ka90]...
A New Framework for Declarative Programming
, 2001
"... We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure. ..."
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Cited by 6 (3 self)
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We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure.
Mixing Modes of Linguistic Description in Categorial Grammar
, 1995
"... Recent work within the field of Categorial Grammar has seen the development of approaches that allow different modes of logical behaviour to be displayed within a single system, something corresponding to making available differing modes of linguistic description. Earlier attempts to achieve this go ..."
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Recent work within the field of Categorial Grammar has seen the development of approaches that allow different modes of logical behaviour to be displayed within a single system, something corresponding to making available differing modes of linguistic description. Earlier attempts to achieve this goal have employed modal op erators called structural modalities, whose use presents a number of problems. I propose an alternative approach, involving co existence and interrelation of different sublogics, that eliminates the need for structural modalities, whilst maintaining the de scriptive power they provide.
Type Theories for Autonomous and *Autonomous Categories: I. Type Theories and Rewrite Systems  II. Internal Languages and Coherence Theorems
, 1998
"... We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian clos ..."
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Cited by 5 (4 self)
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We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new letconstructs, one for each of the three type constructors ?;\Omega and (, and a Parigotstyle ¯abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)fragmen...
Synthetic domain theory and models of linear Abadi & Plotkin logic
, 2005
"... Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can r ..."
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Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPLstructure. This adds to the evidence that the notion of LAPLstructure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1