Solutions to twisted word equations and equations in virtually free groups

Abstract

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in $\mathsf{PSPACE}$. Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in $\mathsf{PSPACE}$ an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or "small enough") finite monoid, then our algorithms can be implemented in $\mathsf{NSPACE}(n^2\log n)$, that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and S\'enizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.