Qualitative relativizations of complexity classes

摘要

The principal result of this paper is a “positive relativization” of the open question “P = ?NP ⌢ co-NP.” That is, the nondeterministic polynomial time-bounded oracle Turing machine endowed with designated accepting states and with designated rejecting states is considered, and suitable restrictions R of this device are developed such that P = NP ⌢ co-NP if and only if for every oracle D, P(D) = NPR(D), where NPR(D) is the class of languages LϵNP(D) that are accepted by oracle machines operating with restrictions R. Positive relativizations are obtained for the P = ? U ⌢ co - U and U = ? NP questions also, where U is the class of languages L in NP accepted by nondeterministic machines that operate in polynomial time and that have for each input at most one accepting computation. The restrictions developed here are “qualitative” in the sense that they restrict the form and pattern of access to the oracle. In contrast, a previous paper [3] developed quantitative relativizations-the number of distinct queries to the oracle is restricted-but no quantitative positive relativization of P = ? NP ⌢ co-NP is known.