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

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.