Deciding Semantic Finiteness of Pushdown Processes and First-Order Grammars w.r.t. Bisimulation Equivalence

The problem if a given configuration of a pushdown automaton (PDA) is bisimilar with some (unspecified) finite-state process is shown to be decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic popping epsilon-steps are allowed, i.e. to the model for which Senizergues showed an involved procedure deciding bisimilarity (FOCS 1998). Such a procedure is here used as a black-box part of the algorithm. For deterministic PDA the regularity problem was shown decidable by Valiant (JACM 1975) but the decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Goeller, FSTTCS 2012)

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

Checking whether two pushdown automata with restricted silent actions are weakly bisimilar was shown decidable by S\'enizergues (1998, 2005). We provide the first known complexity upper bound for this famous problem, in the equivalent setting of first-order grammars. This ACKERMANN upper bound is optimal, and we also show that strong bisimilarity is primitive-recursive when the number of states of the automata is fixed