Mechanizing Programming Logics in Higher Order Logic

Abstract

Formal reasoning about computer programs can be based directly on the semantics of the programming language, or done in a special purpose logic like Hoare logic. The advantage of the first approach is that it guarantees that the formal reasoning applies to the language being used (it is well known, for example, that Hoare’s assignment axiom fails to hold for most programming languages). The advantage of the second approach is that the proofs can be more direct and natural.

In this paper, an attempt to get the advantages of both approaches is described. The rules of Hoare logic are mechanically derived from the semantics of a simple imperative programming language (using the HOL system). These rules form the basis for a simple program veri­fier in which verification conditions are generated by LCF-style tactics whose validations use the derived Hoare rules. Because Hoare logic is derived, rather than postulated, it is straightforward to mix semantic and axiomatic reasoning. It is also straightforward to combine the con­structs of Hoare logic with other application-specific notations. This is briefly illustrated for various logical constructs, including termina­tion statements, VDM-style ‘relational’ correctness specifications, weak­est precondition statements and dynamic logic formulae.

The theory underlying the work presented here is well known. Our contribution is to propose a way of mechanizing this theory in a way that makes certain practical details work out smoothly.