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The general completeness problem of Hoare logic relative to the standardmodel $N$ of Peano arithmetic has been studied by Cook, and it allows for theuse of arbitrary arithmetical formulas as assertions. In practice, theassertions would be simple arithmetical formulas, e.g. of a low level in thearithmetical hierarchy. In addition, we find that, by restricting inputs to$N$, the complexity of the minimal assertion theory for the completeness ofHoare logic to hold can be reduced. This paper further studies the completenessof Hoare Logic relative to $N$ by restricting assertions to subclasses ofarithmetical formulas (and by restricting inputs to $N$). Our completenessresults refine Cook's result by reducing the complexity of the assertiontheory.

On Completeness Results of Hoare Logic Relative to the Standard Model