The 𝖠𝖢^0-Complexity Of Visibly Pushdown Languages
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We concern ourselves with the question which visibly pushdown languages are in the complexity class 𝖠𝖢^0. We provide a conjectural characterization that isolates a stubborn subclass of particular one-turn visibly pushdown languages (that we call intermediate VPLs) all of which our community seems to lack tools for determining containment in 𝖠𝖢^0. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs if its language is in 𝖠𝖢^0, some m≥ 2 such that MOD_m≤_cd L (implying that L is not in 𝖠𝖢^0), or a finite disjoint union of intermediate languages L is constant-depth equivalent to. In the latter case one can moreover effectively compute k,l>0 with k≠l such that the visibly pushdown language is hard for the more concrete intermediate language L(S→ε| a c^k-1 S b_1| ac^l-1Sb_2). For our proofs we revisit so-called Ext-algebras, introduced by Czarnetzki, Krebs and Lange, which in turn are closely related to forest algebras introduced by Bojańczyk and Walukiewicz, and Green's relations.
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