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.

翻译：我們研究哪些可視化下推語言 (VPL) 屬於複雜度類 $\mathsf{AC}^0$，以及如何有效判定此問題。我們的貢獻在於引入一類特殊的單轉向 VPL（稱為中間 VPL），對這類語言而言，上述問題完全不明確：據我們所知，學術界尚未知曉任何中間 VPL 是否包含於 $\mathsf{AC}^0$ 或不被包含。我們的主要結果表明，存在一個演算法，給定一個可視化下推自動機後，能正確輸出以下三者之一：其語言屬於 $\mathsf{AC}^0$；輸出某個 $m\geq 2$ 使得 $\mathsf{MOD}_m$ 可常數深度歸約至 $L$（隱含 $L$ 不屬於 $\mathsf{AC}^0$）；或者輸出一個有限不交並的中間 VPL，使得 $L$ 與之常數深度等價。在後者情況下，還可有效計算出 $k,l\in\mathbb{N}_{>0}$ 且 $k\neq l$，使得具體的中間 VPL $L(S\rightarrow\varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$ 可常數深度歸約至語言 $L$。基於其特殊性質，我們推測所有中間 VPL 要麼全部屬於 $\mathsf{AC}^0$，要麼全都不屬於。作為主要結果的推論，我們得到：若輸入語言是可視化計數器語言，則演算法可有效判定其是否在 $\mathsf{AC}^0$ 中——因此我們的主要結果推廣了 Krebs 等人的結論，即給定可視化計數器語言（限制於良匹配詞）時，可判定其是否屬於 $\mathsf{AC}^0$。在證明中，我們重新審視了所謂的 Ext-代數（由 Czarnetzki 等人引入，與森林代數密切相關，後者由 Bojańczyk 和 Walukiewicz 引入），並運用了 Green 關係。