  We study the question of which visibly pushdown languages (VPLs) are in the
complexity class $\mathsf{AC}^0$ and how to effectively decide this question.
Our contribution is to introduce a particular subclass of one-turn VPLs, called
intermediate VPLs, for which the raised question is entirely unclear: to the
best of our knowledge our research community is unaware of containment or
non-containment in $\mathsf{AC}^0$ for any intermediate VPL.
  Our main result states that there is an algorithm that, given a visibly
pushdown automaton, correctly outputs either that its language is in
$\mathsf{AC}^0$, outputs some $m\geq 2$ such that $\mathsf{MOD}_m$ is
constant-depth reducible to $L$ (implying that $L$ is not in $\mathsf{AC}^0$),
or outputs a finite disjoint union of intermediate VPLs that $L$ is
constant-depth equivalent to. In the latter case one can moreover effectively
compute $k,l\in\mathbb{N}_{>0}$ with $k\not=l$ such that the concrete
intermediate VPL $L(S\rightarrow\varepsilon\mid a c^{k-1} S b_1\mid
ac^{l-1}Sb_2)$ is constant-depth reducible to the language $L$. Due to their
particular nature we conjecture that either all intermediate VPLs are in
$\mathsf{AC}^0$ or all are not. As a corollary of our main result we obtain
that in case the input language is a visibly counter language our algorithm can
effectively determine if it is in $\mathsf{AC}^0$ -- hence our main result
generalizes a result by Krebs et al. stating that it is decidable if a given
visibly counter language is in $\mathsf{AC}^0$ (when restricted to well-matched
words).
  For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et
al.), which are closely related to forest algebras (introduced by Boja\'nczyk
and Walukiewicz), and use Green's relations.
