Completeness of Hoare Logic over Nonstandard Models

The nonstandard approach to program semantics has successfully resolved the completeness problem of Floyd-Hoare logic. The known versions of nonstandard semantics, the Hungary semantics and axiomatic semantics, are so general that they are absent either from mathematical elegance or from practical usefulness. The aim of this paper is to exhibit a not only mathematically elegant but also practically useful nonstandard semantics. A basic property of computable functions in the standard model $N$ of Peano arithmetic $PA$ is $\Sigma_1$-definability. However, the functions induced by the standard interpretation of while-programs $S$ in nonstandard models $M$ of $PA$ are not always arithmetical. The problem consists in that the standard termination of $S$ in $M$ uses the finiteness in $N$, which is not the finiteness in $M$. To this end, we shall give a new interpretation of $S$ in $M$ such that the termination of $S$ uses $M$-finiteness, and the functions produced by $S$ in all models of $PA$ have the uniform $\Sigma_1$-definability. Then we define, based on the new semantics of while-programs, a new semantics of Hoare logic in nonstandard models of $PA$, and show that the standard axiom system of Hoare logic is sound and complete w.r.t. the new semantics. It will be established, in $PA$, that the Hungary semantics and axiomatic semantics coincide with the new semantics of while-programs. Moreover, various comparisons with the previous results, usefulness of the nonstandard semantics, and remarks on the completeness issues are presented.