  We consider a many-sorted variant of Japaridze's polymodal provability logic
$\mathsf{GLP}$. In this variant, which is denoted $\mathsf{GLP}^\ast$,
propositional variables are assigned sorts $\alpha \leq \omega$, where
variables of finite sort $n < \omega$ are interpreted as $\Pi_{n+1}$-sentences
of the arithmetical hierarchy, while those of sort $\omega$ range over
arbitrary ones. We prove that $\mathsf{GLP}^\ast$ is arithmetically complete
with respect to this interpretation. Moreover, we relate $\mathsf{GLP}^\ast$ to
its one-sorted counterpart $\mathsf{GLP}$ and prove that the former inherits
some well-known properties of the latter, like Craig interpolation and PSpace
decidability. We also study a positive variant of $\mathsf{GLP}^\ast$ which
allows for an even richer arithmetical interpretation---variables are permitted
to range over theories rather than single sentences. This interpretation in
turn allows the introduction of a modality that corresponds to the full uniform
reflection principle. We show that our positive variant of $\mathsf{GLP}^\ast$
is arithmetically complete.
