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A Categorical Programming Language
, 1987
"... A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Ther ..."
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A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.
On the Relation between the λµCalculus and the Syntactic Theory of Sequential Control
"... Abstract. We construct a translation of first order λµcalculus [15] into a subtheory of Felleisen’s λccalculus [5, 6]. This translation preserves typing and reduction. Then, by constructing the inverse translation, we show that the two calculi are actually isomorphic. 1 ..."
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Abstract. We construct a translation of first order λµcalculus [15] into a subtheory of Felleisen’s λccalculus [5, 6]. This translation preserves typing and reduction. Then, by constructing the inverse translation, we show that the two calculi are actually isomorphic. 1
Denotations for classical proofs – Preliminary results –
"... This paper addresses the problem of extending the formulaeastypes principle to classical logic. More precisely, we introduce a typed lambdacalculus (λLK → ) whose inhabited types are exactly the implicative tautologies of classical logic and whose type assignment system is a classical sequent ca ..."
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This paper addresses the problem of extending the formulaeastypes principle to classical logic. More precisely, we introduce a typed lambdacalculus (λLK → ) whose inhabited types are exactly the implicative tautologies of classical logic and whose type assignment system is a classical sequent calculus. Intuitively, the terms of λLK → correspond to constructs that are highly nondeterministic. This intuition is made much more precise by providing a simple model where the terms of λLK → are interpreted as nonempty sets of (interpretations of) untyped lambdaterms. We also consider the system (λLK → + cut) and investigate the relation existing between cut elimination and reduction. Finally, we show how to extend our system in order to take conjunction, disjunction and negation into account. 1
Strong Normalization in a NonDeterministic Typed LambdaCalculus
"... Abstract. In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen’s cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is ..."
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Abstract. In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen’s cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of λLK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate prooftheoretic notion of proof reduction. This reduction relation satisfies several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argument a ̀ la Tait. 1