Equivalence of NC^k and AC^k-1 closures of NP and other classes

Gottlob, (1993, in "Proceedings, 34th IEEE Symp. on Found. Comput. Sci.," pp. 42-- 51), showed that any set recognized by polynomial-size, log-depth trees with queries to SAT is p tt -reducible to NP. Based on his technique, it is shown for any set A and any k 1, that NC k (A) ` AC k\Gamma1 (R NP ctt (A)), where R NP ctt (A) is the NP ctt -closure of A. As a consequence, it is shown for any class C that is closed under NP ctt -reductions, such as NP and C= P, and for any k 1, that NC k (C) = AC k\Gamma1 (C), which resolves a question that has been left open for a long time. 1 Introduction Resource-bounded reducibility is one of the most important concepts in complexity theory. Resource-bounded reducibilities are classified into two types: (1) time- and/or space-bounded reducibilities (Ladner, Lynch, and Selman, 1975; Ladner and Lynch, 1976; Cook, 1971) which are defined in terms of Turing machines, and (2) size- and/or depth-bounded reducibilities (Furst, Saxe, and..