Automatic structure

Definition
An automatic structure on a finitely generated group $$G$$ with respect to a finite generating set $$S$$ is defined as the following collection of finite state automata:


 * 1) word acceptor: This accepts, for every element of $$G$$, at least one word using letters from $$S \cup S^{-1}$$ that evaluates to that element.
 * 2) multiplier, one for each $$s \in S$$: This accepts a pair of words $$(w_1,w_2)$$ iff both $$w_1,w_2$$ are accepted by the word acceptor and $$w_1s = w_2$$.