Nondeterministic NC1 Computation

We deene the counting classes #NC 1 , GapNC 1 , PNC 1 and C = NC 1. We prove that boolean circuits, algebraic circuits, programs over non-deterministic nite automata, and programs over constant integer matrices yield equivalent deenitions of the latter three classes. We investigate closure properties. We observe that #NC 1 #L, that PNC 1 L, and that C = NC 1 L. Then we exploit our nite 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 TC 0 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.ported by the Alexander von Humboldt Foundation under a Feodor Lynen scholarship.