Nondeterministic NC1 computation

We define the counting classes #NC1, GapNC(1), PNC1, and C=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1 subset of or equal to #L, that PNC1 subset of or equal to L, and that C=NC1 subset of or equal to L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages, Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations of ACC(0) from MOD-PH and that of TC0 from the counting hierarchy. Moreover, we obtain that if dlogtime-uniformity and logspace-uniformity for AC(0) coincide then the polynomial time hierarchy equals PSPACE. (C) 1998 Academic Press.