Relationships between NP-sets, Co-NP-sets, and P-sets relative to random oracles

Abstract

In the present paper we prove that relative to random oracle A (with respect to the uniform measure) the following three assertions hold: (1) there is a pair of disjoint NP A -sets which are separable by no P A - set, (2) there is a pair of disjoint Co-NP A -sets which are separable by no P A -set and (3) there is an infinite Co-NP A -set having no infinite NP A -subset 1 Introduction Many important problems in Complexity theory remain open. The most known one is whether the classes P and NP are equal. It is also unknown if the class NP coincides with the class Co-NP and if NP " Co-NP = P: In the paper [1] it was shown that all these problems have no relativizable solutions. More exactly, oracles A and B were constructed such that P A = NP A (and, therefore, P A = NP A = NP A " Co-NP A ) and NP B 6= Co-NP B (and, therefore, P B 6= Co-NP B ). Using the same technique one can construct an oracle C for which NP C " Co-NP C 6= P C . As the rela..