Nondeterministic NC1 Computation

We define the counting classes #NC1, GapNC1, 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 subseteq #L and that C=NC1 subseteq 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 separation of ACC0 from ModPH as well as that of TC0 from the counting hierarchy. Moreover we obtain that dlogtime-uniformity and logspace-uniformity for AC0 coincide if and only if the polynomial time hierarchy equals PSPACE.