Solutions of Word Equations Over Partially Commutative Structures
This paper investigates the satisfiability and finiteness problems for systems of word equations with recognizable constraints over the free partially commutative monoid $M(A,I)$ and its quotient group $G(A,I)$. Methodologically, it characterizes solution sets as EDT0L languages and provides an explicit construction of these languages within NSPACE$(n log n)$. Building on this, a unified algorithm is devised that simultaneously decides both satisfiability and finiteness—each in NSPACE$(n log n)$. Notably, the long-standing finiteness problem, previously known to be PSPACE-complete, is resolved at the significantly lower complexity NSPACE$(n log n)$. Moreover, the solution sets are described in a tight, computable, and structurally explicit form. The approach integrates techniques from combinatorial group theory, EDT0L language theory, space-bounded computation, and partial commutation algebraic modeling.