A general method to construct oracles realizing given relationships between complexity classes

doi:10.1016/0304-3975(95)00161-1

Theoretical Computer Science

Volume 157, Issue 2, 5 May 1996, Pages 227-258

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Abstract

We present a method to prove oracle theorems of the following type. Let K₁, …, k_2n and L₁, …, L_2m be complexity classes. The method provides a general framework for constructing an oracle A such that K_{2i − 1}^A ≠ K_2i^A for i = 1,h.,n and L_{2j − 1}^A ≠ L_2j^A for j = 1,…,m. Using this method we obtain several results of this kind. The hardest of them is the existence of an oracle A such that P^A ≠ NP^A, P^A = BPP^A and both Co-NP^A-sets and NP^A-sets are P^A-separable. We exhibit also two theorems that cannot be proved by this method.

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¹: This research was in part supported by a grant from the American Mathematical Society.

²: This research was in part supported by the grant MQT000 from the International Science Foundation, a NAS/NRC COBASE grant, and NSF grant CCR-8957604. Work done in part while visiting the University of Rochester.

³: All the results in this paper are joint results by both authors. The text was written by the second author.