The $\mathsf{AC}^0$-Complexity Of Visibly Pushdown Languages
Stefan Göller, Nathan Grosshans
TL;DR
This work investigates which visibly pushdown languages (VPLs) lie in AC^0 and develops an effective procedure to decide this for DVPA-generated languages. The authors introduce intermediate VPLs—quasi-counterfree one-turn VPLs generated by vertically visible grammars with a weakly length-synchronous but not length-synchronous context set—and prove a trichotomy: a given DVPA language is either in AC^0, ACC^0(m)-hard for some m, or constant-depth equivalent to a finite union of intermediate VPLs. The analysis relies on Ext-algebras, their morphisms, and Green’s relations to relate language recognition to finite algebraic structures, enabling: (i) decidability of quasi-aperiodicity, (ii) length-synchronicity properties, and (iii) a constructive decomposition into intermediate VPLs in the non-synchronous case. The results generalize prior work on visibly counter languages and offer a framework suggesting a broader path to AC^0 decidability for VPLs, with concrete corollaries and effective procedures. The paper leaves open whether all intermediate VPLs are either entirely in AC^0 or entirely outside, hinting at deeper structural dichotomies in the circuit complexity of VPLs.
Abstract
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.