Some Natural Structures which Fail to Possess a Sound and Decidable Hoare-Like Logic for their While-Programs

With the term Hoare-Zike lugic we have in mind some proof system designed for the formal manipulation of assertions about the partial correctnesls of program texts witR respect to a tied interpretation A for the programming language. Stated simply, and informally, our aim in this paper is to exhibit some familiar algebraic structures A over which any sound Honre-like logic for the partial correctness of while-program computations in A will possess some unfamiliar structural properties. From this exercise follows somewhat stronger incompleteness results than those first reported in Cook [lo] and in Wand [25] for Hoare’s original system about whileprograms. And, as we shall make clear in a moment, these results in turn address some sharply defined issues in the theoretical literature to do with the comple:tity of the programming language in the design of a Hoare logic. Our point of departure is Hoare’s proof system as it is formailly co&tutr!d for while-programs in [lo]. We take it for granted that the reader is familiar wizh the papers Hoare [l3], Coo& [lo] and Wand [25]: with these prerequisites, or the invaluable survey paper Apt [lj, we can discuss ow examples in more technical terms. Let A be any relational structure and let -TKp be ttie class of all w destined to compute functions on A. On choosing the first-order logical lang?~~:~ge L as assertion language, and applying a definition of thy semantics Y’ of VV.9 to interpretation A, one may identify the study of partial correctness for %V computations over A as the study of a