Solutions of Word Equations over Partially Commutative Structures

Abstract

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ its quotient group (for example, a right-angled Artin or Coxeter group). We show that for any system of word equations over $M(A,I)$ with recognizable constraints, the solution set - in $M(A,I)$ or in $G(A,I)$ - is an EDT0L language. It is given by an NFA $\mathcal{A}$ recognizing endomorphisms over some extended monoid. Furthermore, if the input size is $n$, then the automaton $\mathcal{A}$ can be constructed effectively by an NSPACE$(n\log n)$-transducer. As a consequence, both Satisfiability (whether the system admits a solution) and Finiteness (whether the solution set is infinite) are decidable in NSPACE$(n \log n)$. For a natural subclass of constraints, we conjecture that these problems are NP-complete.