We study the question of which visibly pushdown languages (VPLs) are in the complexity class AC⁰ 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 AC⁰ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language L is in AC⁰, some m ≥ 2 such that MODₘ (the words over {0,1} having a number of 1’s divisible by m) is constant-depth reducible to L (implying that L is not in AC⁰), or a finite disjoint union of intermediate VPLs that L is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute k,l ∈ ℕ_{> 0} with k≠l such that the concrete intermediate VPL L(S → ε ∣ ac^{k-1}Sb₁ ∣ ac^{l-1}Sb₂) is constant-depth reducible to the language L. Due to their particular nature we conjecture that either all intermediate VPLs are in AC⁰ 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 AC⁰ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in AC⁰ (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 Bojańczyk and Walukiewicz), and use Green’s relations.