Finding All Solutions of Equations in Free Groups and Monoids with Involution

TL;DR

This paper introduces a PSPACE algorithm using recompression techniques to find all solutions of equations in free groups and monoids with involution, accommodating rational constraints and simplifying existing proofs.

Contribution

It extends the recompression technique to handle involution and rational constraints, providing a unified PSPACE algorithm for solving equations in free groups and monoids.

Findings

Algorithm describes all solutions in exponential-sized finite graph

Decidable in PSPACE whether the solution set is finite

Simplifies proofs of PSPACE complexity for word equations

Abstract

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented emph{recompression} technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is general enough to accommodate both extensions. In the end, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.