Solutions of Word Equations over Partially Commutative Structures

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size $n$ we show the structural result about the solution set of the system: the set of all solutions in $M(A,I)$ or in the group $G(A,I)$ is an EDT0L language. That is, it is given by an NFA $\mathcal{A}$ recognizing endomorphisms over some extended monoid. Moreover, $\mathcal{A}$ is effectively constructible by an NSPACE($n \log n$)-transducer. This implies that Satisfiability: `Is the system is solvable?' and Finiteness: `Are there infinitely many solutions?' can be decided in NSPACE($n \log n$). In the uniform version, these problems are PSPACE-complete, but for a suitable subclass of constraints we have more precise complexities and we conjecture that the decision problems above are NP-complete in this setting. Our results apply also to word equation over free monoids in the classical case where the involution is reading words right-to-left. This allows to specify that solutions are restricted to be palindromes.