Solutions of Word Equations over Partially Commutative Structures

TL;DR

This paper characterizes the solution sets of word equations over partially commutative structures as EDT0L languages, providing effective methods for decision problems and extending classical results to new algebraic contexts.

Contribution

It introduces a structural description of solutions as EDT0L languages and develops efficient algorithms for satisfiability and finiteness problems over these structures.

Findings

Solution sets are EDT0L languages recognized by an NFA.

Decidability of satisfiability and finiteness in NSPACE(n log n).

Complexity bounds for decision problems, including PSPACE-completeness and potential NP-completeness.

Abstract

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.