Hoare calculi for higher — type control siructures and their completeness in the sense of cook

Abstract

We show that the Hoare Calculus for the language of finitely typed λ-terms, introduced in Go 85, is complete in the sense of Cook, i. e. complete for expressive interpretations, if the first-order theory of the interpretation is supplied as oracle. This is the classical notion of completeness for Hoare Calculi [Ap 81]. In Go 85 completeness in the sense of Cook was only proved with respect to Herbrand definable interpretations, which is much easier because of the characterizations in Ur 83, and amounts essentially to completeness for arithmetic and finite interpretations.

The completeness result above implies that we get — in a uniform manner as demonstrated in Go 87 — Hoare Calculi which are complete in the sense of Cook for imperative languages with control structures ranging from normal recursive procedures with local variables (i. e. variables declared by new x) to recursive procedures and functions of higher type (without global variables [Cl 79]). No such calculi were known for languages with higher type procedures and functions, hence we solve positively a question in TrHaMe 83.