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