Nondeterminism in Logics of Programs

Abstract

We investigate the principles underlying reasoning about nondeterministic programs, and present a logic to support this kind of reasoning. Our logic, an extension of dynamic logic ([22] and [12]), subsumes most existing first-order logics of nondeterministic programs, including that developed by Dijkstra based on the concept of weakest precondition. A significant feature is the strict separation between the two kinds of nonterminating computations: infinite computations and failures. The logic has a Tarskian truth-value semanics, an essential prerequisite to establishing completeness of axiomatizations of the logic. We give an axiomatization for flowchart (regular) programs that is complete relative to arithmetic in the sense of Cook. Having a satisfactory tool at hand, we turn to the clarification of the concept of the total correctness of nondeterministic programs, providing in passing, a critical evaluation of the widely used "predicate transformer" approach to the definition of programming constructs, initiated by Dijkstra [5]. Our axiom system supplies a complete axiomatization of wp.