A consistent and complete deductive system for the verification of parallel programs

The semantics of a simple parallel programming language is presented in two ways: deductively, by a set of Hoare-like axioms and inference rules, and operationally, by means of an interpreter. It is shown that the deductive system is consistent with the interpreter. It would be desirable to show that the deductive system is also complete with respect to the interpreter, but this is impossible since the programming language contains the natural numbers. Instead it is proved that the deductive system is complete relative to a complete proof system for the natural numbers; this result is similar to Cook's relative completeness for sequential programs. The deductive semantics given here is an extension of an incomplete deductive system proposed by Hoare. The key difference is an additional inference rule which provides for the introduction of auxiliary variables in a program to be verified.