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 language in our newly introduced class. 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 $L$ is $\mathsf{ACC}^0(m)$-hard (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 Bojańczyk and Walukiewicz), and use Green's relations.

翻译：我们研究哪些可见推后语言（VPL）属于复杂度类$\mathsf{AC}^0$，以及如何有效判定这一问题。我们的贡献在于引入了一类特殊的单轮VPL，称为中间VPL，针对该类语言提出的问题完全不明朗：据我们所知，学术共同体尚未发现我们所引入的新类中任一语言是否属于$\mathsf{AC}^0$的包含性或非包含性证据。我们的主要结论表明：存在一种算法，给定一个可见推后自动机，能够正确输出以下结果之一：其语言属于$\mathsf{AC}^0$；输出某个$m\geq 2$使得$L$是$\mathsf{ACC}^0(m)$-难的（这意味着$L$不属于$\mathsf{AC}^0$）；或输出一个有限不交并的中间VPL集合，且$L$与这些集合在常数深度意义下等价。在后一种情形中，还可有效计算$k,l\in\mathbb{N}_{>0}$且$k\neq l$，使得具体中间VPL $L(S\rightarrow\varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$可常数深度归约为语言$L$。基于其特殊性质，我们推测要么所有中间VPL都属于$\mathsf{AC}^0$，要么全都不属于。作为主要结论的推论，当输入语言是可见计数器语言时，我们的算法能有效判定其是否属于$\mathsf{AC}^0$——因此，我们的主要结论推广了Krebs等人的结果，该结果指出给定可见计数器语言（在良匹配词限制下）是否属于$\mathsf{AC}^0$是可判定的。在证明中，我们重新审视了与森林代数（由Bojańczyk和Walukiewicz引入）密切相关的Ext-代数（由Czarnetzki等人引入），并使用了Green关系。