Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

Abstract

Checking whether two pushdown automata with restricted silent actions are weakly bisimilar was shown decidable by Sénizergues (1998Sénizergues ( , 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.

The parametric complexity of bisimulation equivalence of normed pushdown automata

“…The restriction on silent actions is a decidability border since Jančar and Srba showed that weak bisimilarity between general PDA is undecidable [11]. Yin et al proved that branching bisimilarity between PDA is also undecidable [12].…”

Bisimulation Finiteness of Pushdown Systems Is Elementary

“…A central ingredient to this semi-decision procedure is an oracle call to test the equivalence of pushdown systems, the latter itself being an intricate problem whose decidability has been proven by Sénizergues [27]. Only recently an Ackermannian upper bound for bisimilarity of pushdown system has been proven by Jančar and Schmitz [17]; ACKER-MANN-hardness is only known to hold in the presence of deterministic ε-popping rules [13], whereas without ε-rules the problem is nonelementary [2]. Coming back to bisimulation finiteness of pushdown systems, the oracle calls to a bisimulation equivalence check for pushdown systems is the inherent bottleneck of Jančar's approach.…”