NondeterministicNC1Computation

Abstract

We define the counting classes #NC1,GapNC1,PNC1, andC=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⊆#L, thatPNC1⊆L, and thatC=NC1⊆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 ofACC0fromMOD-PHand that ofTC0from the counting hierarchy. Moreover, we obtain that if dlogtime-uniformity and logspace-uniformity forAC0coincide then the polynomial time hierarchy equalsPSPACE.