Recursive Programs and Denotational Semantics in Absolute Logics of Programs

Abstract It has been proposed by Meyer, Kfouri and others to extend the so-called Nonstandard Logics of Programs approach (NLP henceforth) to recursive programs and denotational semantics in some natural way. The key step in doing NLP is selecting a semantics for programs (which, in the present paper, is denotational semantics) and then forming its KPU-absolute version. Absoluteness is a set theoretic property of logics. If a logic L is not absolute, then some formulas of L do not speak about the models of L , but about something else. If a logic has this strange property, then it is impossible to find a complete inference system for it, and this because of purely “administrative” reasons, which might not correspond to any of the inherent properties of the original situation we are modeling (in devising this logic etc.). This “administrative” incompleteness is only one of the undesirable side effects of nonabsoluteness. In NLP, the purpose of constructing and studying absolute versions of logics is aimed at, among other things, the removal of these undesirable properties. The absolute version of denotational semantics will turn out to be based on 2-sorted first order logic, or more concretely, on the usual nonstandard or weak second order logic. Some of the most relevant features of this logic are the ability to compare “realistic” existing program verification methods, create new, more powerful and still “realistic” ones, completeness etc., to mention only a very few.