🤖 AI Summary
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.
📝 Abstract
We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups.
