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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 Logical View Of Concurrent Constraint Programming
, 1995
"... . Concurrent Constraint Programming (CCP) has been the subject of growing interest as the focus of a new paradigm for concurrent computation. Like logic programming it claims close relations to logic. In fact CCP languages are logics in a certain sense that we make precise in this paper. In recent ..."
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Cited by 23 (4 self)
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. Concurrent Constraint Programming (CCP) has been the subject of growing interest as the focus of a new paradigm for concurrent computation. Like logic programming it claims close relations to logic. In fact CCP languages are logics in a certain sense that we make precise in this paper. In recent work it was shown that the denotational semantics of determinate concurrent constraint programming languages forms a fibred categorical structure called a hyperdoctrine, which is used as the basis of the categorical formulation of firstorder logic. What this shows is that the combinators of determinate CCP can be viewed as logical connectives. In this paper we extend these ideas to the operational semantics of such languages and thus make available similar analogies for a much broader variety of languages including indeterminate CCP languages and concurrent blockstructured imperative languages. CR Classification: F3.1, F3.2, D1.3, D3.3 Key words: Concurrent constraint programming, simula...
Classical categories and deep inference
"... Deep inference is a prooftheoretic notion in which proof rules apply arbitrarily deeply inside a formula. We show that the essense of deep inference is the bifunctorality of the connectives. We demonstrate that, when given an inequational theory that models cutreduction, a deep inference calculus ..."
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Cited by 6 (0 self)
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Deep inference is a prooftheoretic notion in which proof rules apply arbitrarily deeply inside a formula. We show that the essense of deep inference is the bifunctorality of the connectives. We demonstrate that, when given an inequational theory that models cutreduction, a deep inference calculus for classical logic (SKSg) is a categorical model of the classical sequent calculus LK in the sense of Führmann and Pym. We uncover a mismatch between this notion of cutreduction and the usual notion of cut in SKSg. Viewing SKSg as a model of the sequent calculus uncovers new insights into the Craig interpolation lemma and intuitionistic provablility.
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ii COPYRIGHT ..."
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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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...
An institutional view on categorical logic and the CurryHowardTaitisomorphism
"... We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number ..."
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We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.
Quotients over Minimal Type Theory
"... Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid à la Bishop we build a mo ..."
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Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid à la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.
Indexed Categories and BottomUp Semantics of Logic Programs
, 2001
"... We propose a categorical framework which formalizes and extends the syntax, operational semantics and declarative model theory of a broad range of logic programming languages. A program is interpreted in an indexed category in such a way that the base category contains all the possible states wh ..."
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We propose a categorical framework which formalizes and extends the syntax, operational semantics and declarative model theory of a broad range of logic programming languages. A program is interpreted in an indexed category in such a way that the base category contains all the possible states which can occur during the execution of the program (such as global constraints or type information), while each ber encodes the logic at each state.