Solution Sets for Equations over Free Groups are EDT0L Languages

Abstract

We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been open for some years [9, 10], apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star _{1 \le i \le s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by [8], which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper [5]. For example, we improve the complexity from quadratic nondeterministic space in [5] to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: $\mathsf {NSPACE}(n\log n$). The conjectured complexity is $\mathsf {NP}$, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured $\mathsf {NP}$.

Research supported by the Australian Research Council FT110100178 and the University of Newcastle G1301377. The first author was supported by a Swiss National Science Foundation Professorship FN PP00P2-144681/1. The first and third authors were supported by a University of Neuchâtel Overhead grant in 2013.

Manfred Kudlek has the distinction of being the only person to have attended all ICALP conferences during his lifetime. He worked on Lindenmayer systems, visited Kyoto several times, and taught the second author that bikes are the best means of transport inside Kyoto.