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Logic Programming in a Fragment of Intuitionistic Linear Logic
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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Cited by 306 (40 self)
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottomup search for a cutfree proof contexts, represented as the lefthand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goaldirected interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
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 and a c ..."
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Cited by 103 (7 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...
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
"... ..."
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 53 (8 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.
Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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Cited by 33 (6 self)
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
Unique Factorisation Lifting Functors and Categories of LinearlyControlled Processes
 Mathematical Structures in Computer Science
, 1999
"... We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce ..."
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Cited by 7 (2 self)
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We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce and study a notion of pathlinearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearlycontrolled processes (viz., processes controlled by pathlinearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their interrelations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 4 (0 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
A Logical Calculus for Polynomialtime Realizability
 Journal of Methods of Logic in Computer Science
, 1991
"... A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduct ..."
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Cited by 3 (0 self)
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A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduction In [4], a restricted notion of realizability is introduced, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized by realizers that give a measure of the resources required to establish the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all free variables universally quantified simultaneously. (There is a difference between the quantifiers 8h¸; ji and 8¸8j.) And third, the realizers code polynomialtime ("ptime") functions, rather than arbitrary recurs...
Graded Multicategories of Polynomialtime Realizers (Extended Abstract)
 Department of Mathematics Department of Mathematics and Computer Science McGill University John Abbott College Monash University 805 Sherbrooke St
, 1989
"... Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered ..."
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Cited by 1 (1 self)
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Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered, giving a notion of "dependant grades", which is also sound. Furthermore, we define a notion of closed graded multicategory, and show how the structure of polynomialtime realizers has that structure. 0 Introduction In [4], a restricted notion of realizability is defined, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized only by realizers that express a reason for the "truth" (or provability) of the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all fre...
Classical lambek logic
 TABLEAUX'95: Proceedings of the 4th International Workshop on Theorem Proving with Analytic Tableaux and Related Methods, number 918 in LNCS
, 1995
"... Abstract. We discuss different options for twosided sequent systems of noncommutative linear logic and prove a restricted form of cut elimination. By “classical Lambek logic ” we denote a sequent system with sequences of propositional formulas on the right and left side of the sequent sign, which h ..."
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Abstract. We discuss different options for twosided sequent systems of noncommutative linear logic and prove a restricted form of cut elimination. By “classical Lambek logic ” we denote a sequent system with sequences of propositional formulas on the right and left side of the sequent sign, which has no structural rule except cut. We credit this logic to J. Lambek since he was the first to investigate Gentzensystems without structural rules — originally in an intuitionistic setting, i.e. with not more than one formula in the succedent of a sequent, and motivated by linguistic considerations (see [4]). From the point of view of linear logic classical Lambek logic can be considered as pure (i.e., without exponentials) noncommutative (i.e., without the structural rules of exchange) classical (i.e., multiple succedent) linear propositional logic. This is the starting point of Abrusci’s [1] paper. Abrusci presents a sequent calculus together with a semantics in terms of phase spaces. By proving completeness he gives a semantic justification of the sequent system. Independently, under the heading “bilinear logic ” Lambek himself has studied this system (see [6]) based