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Cartesian effect categories are Freydcategories
, 2009
"... Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are sideeffects. In this paper Cartesian e ..."
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Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are sideeffects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freydcategories and Haskell’s Arrows. It is proved that a Cartesian effect category is a Freydcategory where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions. Keywords. Categorical logic, computational effects, monads, Freydcategories, premonoidal categories, Arrows, sequential product, effect categories, Cartesian effect categories.
A Relational Account of CallbyValue Sequentiality
 IN: PROC. 12TH SYMP. LOGIC IN COMPUTER SCIENCE
, 1999
"... We construct a model for FPC, a purely functional, sequential, callbyvalue language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract. ..."
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Cited by 13 (2 self)
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We construct a model for FPC, a purely functional, sequential, callbyvalue language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.
A Categorytheoretic characterization of functional completeness
, 1990
"... . Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a ..."
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Cited by 8 (1 self)
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. Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a such that f(x 1 ,...,x n ) = (a . x 1 . ... . x n ). Combinatory Logic is the simplest typefree language which is functionally complete. In a sound categorytheoretic framework the constant a above may be considered as an "abstract gödelnumber" for f, when gödelnumberings are generalized to "principal morphisms", in suitable categories. By this, models of Combinatory Logic are categorically characterized and their relation is given to lambdacalculus models within Cartesian Closed Categories. Finally, the partial recursive functionals in any finite higher type are shown to yield models of Combinatory Logic. ________________ (+) Theoretical Computer Science, 70 (2), 1990, pp.193211. A p...
The HasCasl prologue: categorical syntax and semantics of the partial λcalculus
 COMPUT. SCI
, 2006
"... We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we ..."
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Cited by 6 (4 self)
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We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λtheories. Building on these results, we define (settheoretic) notions of intensional Henkin model and syntactic λalgebra for Moggi’s partial λcalculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λalgebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higherorder logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the modeltheoretic equivalence result to the semantics of HasCasl and its relation to firstorder Casl.
The Logic of the Partial λCalculus With Equality
"... We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a ..."
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We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter being unique choice. We give a linguistic proof of the generalization of the fundamental theorem of toposes to pcccs with equality; type theoretically, one thus obtains that the partial λcalculus with equality encompasses a MartinLöfstyle dependent type theory. This work forms part of the semantical foundations for the higher order algebraic specification language HasCasl.
Partial Morphisms in Categories of Effective Objects
, 1996
"... This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the seco ..."
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This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the second part we dene the (partial) cartesian closed category GEN of generalized numbered sets, prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos. Introduction By data type one usually means a set of objects of the same kind, suitable for manipulation by a computer program. Of course, computers actually manipulate formal representations of objects. The purpose of the mathematical semantics of programming languages, however, is to characterize data types (and functions on them) in a way which is independent of any specic representation mechanism. So the objects one deals with are mostly elements of structures borrowed fro...