Solvability of Equations in Graph Groups Is Decidable

Solving equations in algebraic structures is a fundamental task in mathematics. Here we tackle this problem for free partially commutative monoids with involution and for graph groups, which are free groups with a partial commutation relation between generators. Basic algebraic structures involving partial commutations are free partially commutative monoids (also known as trace monoids). They were considered in combinatorics by Cartier and Foata [5] and in computer science by Keller [14] and Mazurkiewicz [20,21]. Trace monoids serve as an algebraic tool for investigating concurrent systems. Atomic actions are represented by letters and independency of actions is reflected by a partial commutation relation. If each atomic action a has an inverse a such that aa = aa = 1, then, on the algebraic level, we switch from monoids to groups. Without the cancellation law aa = aa = 1, we obtain