Equivalence of NC^k and AC^k-1 closures of NP and Other Classes

Gottlob (1993, in "Proceedings, 34th IEEE Symposium on Foundations of Computer Science," pp. 42-51) showed that any set recognized by polynomial-size, log-depth trees with queries to SAT is ≤ptt-reducible to NP. Based on his technique, it is shown for any set A and any k ≥ 1 that NCk (A) ⊆ ACk − 1(RNPctt(A)) where RNPctt(A) is the ≤NPctt-closure of A. As a consequence, it is shown for any class C that is closed under ≤NPctt-reductions, such as NP and C=P, and for any k ≥ 1 that NCk(C) = ACk − 1 (C), which resolves a question that has remained open for a long time.