Relativizing complexity classes with sparse oracles

Abstract

Baker, Gill, and Solovay constructed sparse sets A and B such that P(A) ≠ NP(A) and NP(B) ≠ co-NP(B). In contrast to their results, we prove that P = NP if and only if for every tally language T, P(T) = NP( T), and that NP = co-NP if and only if for every tally language T, NP(T) = co-NP(T). We show that the polynomial hierarchy collapses if and only if there is a sparse set S such that the polynomial hierarchy relative to S collapses. Similar results are obtained for several other complexity classes. © 1986, ACM. All rights reserved.