Borel complexity and Ramsey largeness of sets of oracles separating complexity classes

Abstract

We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, $P\ne \mathrm{NP}$ with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that $\mathrm{IP}=\mathrm{PSPACE}$, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating $\mathrm{NP}$ and $\mathrm{co}-\mathrm{NP}$ is not small, and obtain similar results for the separation of $\mathrm{PSPACE}$ from $\mathrm{PH}$ along with the separation of $\mathrm{NP}$ from $\mathrm{BQP}$. We also show that the set of oracles equating $\mathrm{IP}$ with $\mathrm{PSPACE}$ is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.