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16
Notions of Computation and Monads
, 1991
"... The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with ..."
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Cited by 734 (15 self)
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The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from calues to values) that may jeopardise the applicability of theoretical results, In this paper we introduce calculi. based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation.
Computational LambdaCalculus and Monads
, 1988
"... The calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with terms. However, if one goes further and uses fijconversion to prove equivalence of programs, then a gross simplification 1 is introduced, that may jeopardise the ..."
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Cited by 445 (6 self)
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The calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with terms. However, if one goes further and uses fijconversion to prove equivalence of programs, then a gross simplification 1 is introduced, that may jeopardise the applicability of theoretical results to real situations. In this paper we introduce a new calculus based on a categorical semantics for computations. This calculus provides a correct basis for proving equivalence of programs, independent from any specific computational model. 1 Introduction This paper is about logics for reasoning about programs, in particular for proving equivalence of programs. Following a consolidated tradition in theoretical computer science we identify programs with the closed terms, possibly containing extra constants, corresponding to some features of the programming language under consideration. There are three approaches to proving equivalence of programs: ffl T...
The Lazy Lambda Calculus
 Research Topics in Functional Programming
, 1990
"... Introduction The commonly accepted basis for functional programming is the calculus; and it is folklore that the calculus is the prototypical functional language in puri ed form. But what is the calculus? The syntax is simple and classical; variables, abstraction and application in the pure cal ..."
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Cited by 239 (3 self)
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Introduction The commonly accepted basis for functional programming is the calculus; and it is folklore that the calculus is the prototypical functional language in puri ed form. But what is the calculus? The syntax is simple and classical; variables, abstraction and application in the pure calculus, with applied calculi obtained by adding constants. The further elaboration of the theory, covering conversion, reduction, theories and models, is laid out in Barendregt's already classical treatise [Bar84]. It is instructive to recall the following crux, which occurs rather early in that work (p. 39): Meaning of terms: rst attempt The meaning of a term is its normal form (if it exists). All terms without normal forms are identi ed. This proposal incorporates such a simple and natural interpretation of the calculus as
HasCASL: Towards Integrated Specification and Development of Functional Programs
, 2002
"... The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an exe ..."
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Cited by 24 (10 self)
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The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an executable subset in order to facilitate rapid prototyping. We lay out the design of HasCasl, a higher order extension of the algebraic specification language Casl that is geared towards precisely this purpose. Its semantics is tuned to allow program development by specification refinement, while at the same time staying close to the settheoretic semantics of first order Casl. The number of primitive concepts in the logic has been kept as small as possible; we demonstrate how various extensions to the logic, in particular general recursion, can be formulated within the language itself.
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding &quot;good &quot; quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the &quot;best &quot; regular category (called its regular completion) that embeds it. The second assigns to
Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory
 In Proc. 17th IEEE Symposium on Logic in Computer Science
, 2003
"... This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown ..."
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Cited by 8 (2 self)
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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic ZermeloFraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambdacalculus with callbyvalue operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domaintheoretic models.
Two constructive embeddingextension theorems with applications to continuity principles and to BanachMazur computability
 Mathematical Logic Quarterly
"... We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor ..."
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Cited by 6 (1 self)
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We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally noncompact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between “continuity principles ” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to R are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally noncompact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally noncompact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursiontheoretic setting, then, for any such space X, there exists a BanachMazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers.
Paracategories I: Internal Paracategories and Saturated Partial Algebras
 Comp. Sci
, 2002
"... Based on the monoid classi er , we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a freemonoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an ..."
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Cited by 5 (1 self)
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Based on the monoid classi er , we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a freemonoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras . We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via inclusions and Kleene morphisms.
An Equational Notion of Lifting Monad
 TITLE WILL BE SET BY THE PUBLISHER
, 2003
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category ..."
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Cited by 4 (1 self)
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1
A Simple Adequate Categorical Model for PCF
 In Proceedings of Third International Conference on Typed Lambda Calculi and Applications
, 1997
"... Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for ..."
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Cited by 4 (0 self)
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Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for PCF in the sense that we will introduce categorical axioms enabling an adequate semantics to be given. We assume the presence of certain "bottom" maps with the role of being the interpretation of nonterminating terms, but the orderstructure is left out. This is different from previous approaches where some kind of ordertheoretic structure has been considered as part of an adequate categorical model for PCF. We take the point of view that partiality is the fundamental notion from which orderstructure should be derived, which is corroborated by the observation that our categorical model induces an ordertheoretic model for PCF in a canonical way.