Axiomatic Definability and Completeness for Recursive Programs

Albert R. Meyer, John C. Mitchell

Abstract: The termination assertion p<S>q means that whenever the formula p is true, there is an execution of the possibly nondeterministic program S which terminates in a state in which q is true. Termination assertions are more tractable technically than the partial correctness assertions usually treated in the literature. Termination assertions are studied for a programming language which includes local variable declarations, calls to undeclared global procedures, and nondeterministic recursive procedures with call-by-address and call-by-value parameters. By allowing formulas p and q to place conditions on global procedures, we provide a method for reasoning about programs with calls to global procedures based on hypotheses about procedure input-output behavior. The set of first-order termination assertions valid over all interpretations is completely axiomatizable without reference to the theory of any interpretation. Although uninterpreted assertions have limited expressive power, the set of valid termination assertions defines the semantics of recursive programs in the sense of Meyer and Halpern [10]. Thus the axiomatization constitutes an axiomatic definition of the semantics of recursive programs.