Finite monoids and the fine structure of NC1

Recently a new connection was discovered between the parallel complexity class NC1 and the theory of finite automata, in the work of Barrington [Ba86] on bounded width branching programs. There (non-uniform) NC1 was characterized as those languages recognized by a certain non-uniform version of a DFA. Here we extend this characterization to show that the internal structures of NC1 and the class of automata are closely related. In particular, using Thérien's classification of finite monoids [Th81], we give new characterizations of the classes AC0, depth-k AC0, and ACC, the last being the AC0 closure of the mod q functions for all constant q. We settle some of the open questions in [Ba86], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [BK78], and offer a new framework for understanding the internal structure of NC1.