Deciding Word Problems of Semigroups using Finite State Automata

We explore natural classes of finitely generated semigroups that have word problem decidable by synchronous or asynchronous two-tape finite state automata. Synchronous two-tape automata decide regular word problems and asynchronous automata decide rational word problems. We argue that asynchronous two-tape automata are a more suitable choice than synchronous two-tape automata and show that the word problem being rational is invariant under change of generating set and some basic algebraic constructions. We also examine some algebraic properties of semigroups with rational word problem.