A categorical interpretation of partial function logic and Hoare logic

Abstract

In this paper we define a categorical interpretation of the first-order Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the well-known notion of (first-order) 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 fibers. 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. Further new results of the paper include a formulation of intuitionistic Hoare logic, and Hoare logic with partial functions.