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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...
Partial Hyperdoctrines: Categorical Models for Partial Function Logic and Hoare Logic
, 1993
"... In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important ..."
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In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial (first order) hyperdoctrine is a different notion of morphism between the fibres. We also use this partial hyperdoctrine to give a model for Beeson's Partial Function Logic such that (a version of) his axiomatization is complete w.r.t. this model. This shows the usefulness of the notion independent of its intended use as a model for Hoare logic.