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 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 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 [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (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.

Equations in groups that are virtually direct products

“…The Diophantine Search Problem was solved in free groups by the work of Makanin and Razborov [19], and descriptions of the solutions are possible via Makanin-Razborov diagrams, or as EDT0L formal languages [3]. Then [7] shows the solubility of equations with rational constraints in virtually free groups, and [4] the solubility of equations with q.i. embeddable rational constraints in hyperbolic groups.…”

EDT0L solutions to equations in group extensions

“…Equations with rational constraints have also attracted a large amount of attention; the addition of constraints allows for a certain level of control on what each of the variables can be. The fact that systems of equations in free and virtually free groups have EDT0L solutions was also shown to be true if rational constraints are added ( [4], [11]). In hyperbolic groups, the addition of rational constraints was shown to preserve the fact that systems of equations have EDT0L solutions if the rational constraints were quasi-isometrically embedded [5].…”

“…Solutions to systems of equations in right-angled Artin groups were then shown to be EDT0L by Diekert, Jeż and Kufleitner [12]. Virtually free groups [11], hyperbolic groups [5], and virtually abelian groups [19] followed later.…”

The complexity of solution sets to equations in hyperbolic groups

“…Let G be a hyperbolic group with finite symmetric generating set S. In this paper we show that any system of equations in G has solutions which, when written as shortlex representatives over S, admit a particularly simple description as formal languages, and moreover, this description can be given in very low space complexity. Our work combines the geometric results for determining the satisfiability of equations in hyperbolic groups of Rips, Sela, Dahmani and Guirardel [12,39], with recent tools developed in theoretical computer science which give PSPACE algorithms for solving equations in semigroups and groups [7,14,15,16,17,28,29].…”