The AC0-Complexity of Visibly Pushdown Languages

We study the question of which visibly pushdown languages (VPLs) are in the complexity class AC 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 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 exactly one of the following: that its language L is in AC(0), some m >= 2 such that MODm, (the words over {0, 1} having a number of l's divisible by m) is constant-depth reducible to L (implying that L is not in AC(0)), 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, 1 is an element of N->0 with k not equal 1 such that the concrete intermediate VPL L(S -> epsilon vertical bar ac(k-1) lSb(1) vertical bar ac(t-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 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 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 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 Bojaficzyk and Walukiewicz), and use Green's relations.