Abstract
In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a classK, (1) reducibility notions of sets toK under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set inK is already inK. From these results, we derive some observations on the reducibility notions to C=P and NP.
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