Tightening the Complexity of Equivalence Problems for Commutative Grammars

Given two finite-state automata, are the Parikh images of the languages they generate equivalent? This problem was shown decidable in coNEXP by Huynh in 1985 within the more general setting of context-free commutative grammars. Huynh conjectured that a Π2 upper bound might be possible, and Kopczyński and To established in 2010 such an upper bound when the size of the alphabet is fixed. The contribution of this paper is to show that the language equivalence problem for regular and context-free commutative grammars is actually coNEXP-complete. In addition, our lower bound immediately yields further coNEXP-completeness results for equivalence problems for regular commutative expressions, reversal-bounded counter automata and communication-free Petri nets. Finally, we improve both lower and upper bounds for language equivalence for exponent-sensitive commutative grammars. 1998 ACM Subject Classification F.1.1 Models of Computation