Two-way Parikh Automata

Abstract

Parikh automata extend automata with counters whose values can only be tested at the end of the computation, with respect to membership into a semi-linear set. Parikh automata have found several applications, for instance in transducer theory, as they enjoy decidable emptiness problem.In this paper, we study two-way Parikh automata. We show that emptiness becomes undecidable in the non-deterministic case. However, it is PSpace-C when the number of visits to any input position is bounded and the semi-linear set is given as an existential Presburger formula. We also give tight complexity bounds for the inclusion, equivalence and universality problems. Finally, we characterise precisely the complexity of those problems when the semi-linear constraint is given by an arbitrary Presburger formula.1 E. Filiot is a research associate of F.R.S.-FNRS. He is supported by the ARC Project Transform Fédération Wallonie-Bruxelles and the FNRS CDR project J013116F. 2 S. Guha is supported by the ARC project "Non-Zero Sum Game Graphs: Applications to Reactive Synthesis and Beyond" ( Fédération Wallonie-Bruxelles) 3 N Mazzocchi is a PhD student funded by a FRIA fellowship from the F.R.S.-FNRS.