Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question

Abstract

We investigate relativized versions of the open question of whether every language accepted nondeterministically in polynomial time can be recognized deterministically in polynomial time. For any set X, let $\mathcal{P}^X (\text{resp. }\mathcal{NP}^X )$ be the class of languages accepted in polynomial time by deterministic (resp. nondeterministic) query machines with oracle X. We construct a recursive set A such that $\mathcal{P}^A = \mathcal{NP}^A $. On the other hand, we construct a recursive set B such that $\mathcal{P}^B \ne \mathcal{NP}^B $. Oracles X are constructed to realize all consistent set inclusion relations between the relativized classes $\mathcal{P}^X $, $\mathcal{NP}^X $, and co $\mathcal{NP}^X $, the family of complements of languages in $\mathcal{NP}^X $. Several related open problems are described.