Solution sets for equations over free groups are EDT0L languages

TL;DR

This paper proves that the set of all solutions to equations over free groups can be effectively represented as an EDT0L language, using a novel construction that combines recompression techniques and solutions to linear Diophantine equations.

Contribution

It introduces a new method to characterize solution sets over free groups as EDT0L languages, improving complexity bounds and integrating advanced techniques.

Findings

Solution sets form an effectively constructible EDT0L language.

The approach encodes solutions using a finite directed graph.

Complexity is improved to NSPACE(n log n).

Abstract

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Je\.z, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here.