Bisimilarity of Pushdown Automata is Nonelementary

Abstract

Abstract-Given two pushdown automata, the bisimilarity problem asks whether the infinite transition systems they induce are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIMEhardness, which was the best previously known lower bound for this problem. Our lower bound result holds for normed pushdown automata as well.

Equivalences of Pushdown Systems Are Hard

“…As already mentioned, this result is derived by a finer look at Stirling's approach [26], where only a primitive recursive upper bound is claimed. The TOWER-hardness for RT-PDA is, in fact, a slight extrapolation of the result presented in [2]. The authors only claim nonelementary complexity but their proof can be adjusted to show TOWER-hardness in the sense of [19].…”

“…Language equivalence of deterministic pushdown automata (DPDA) was shown to be decidable by Sénizergues (1997Sénizergues ( , 2001Stirling (2002) then showed that the problem is primitive recursive.Sénizergues (1998, 2005) also generalized his proof to show decidability of bisimulation equivalence of (nondeterministic) PDA where ε-rules can be only deterministic and popping; this problem was shown to be nonelementary by Benedikt, Göller, Kiefer, and Murawski (2013), even for PDA with no ε-rules.Here we refine Stirling's analysis and show that DPDA equivalence is in TOWER, i.e., in the "least" nonelementary complexity class. The basic proof ideas remain the same but the presentation and the analysis are simplified, in particular by using a first-order term framework.…”

“…Sénizergues (1998, 2005) also generalized his proof to show decidability of bisimulation equivalence of (nondeterministic) PDA where ε-rules can be only deterministic and popping; this problem was shown to be nonelementary by Benedikt, Göller, Kiefer, and Murawski (2013), even for PDA with no ε-rules.…”

Bisimilarity in Fresh-Register Automata

“…Moving to FRAs, the first obstacle we face is that actual configurations contain the full history of names and have therefore unbounded size. For bisimulation purposes, though, keeping track of the whole 5 The latter condition above is slightly different but equivalent to that used in [27]. In loc.…”

“…Since VPDRA(SF ) are a particularly weak variant, this result implies undecidability for all PDRA considered in [18]. In contrast, for finite alphabets, bisimilarity of pushdown automata is known to be decidable [24] but non-elementary [5] and, in the visibly pushdown case, EXPTIME-complete [25].…”