Complexity and Ramsey Largeness of Sets of Oracles Separating Complexity Classes

TL;DR

This paper investigates the complexity and size of sets of oracles that separate various computational complexity classes, proposing a new topological approach and analyzing their descriptive complexity within the Borel hierarchy.

Contribution

It introduces a novel Ellentuck topology-based notion of largeness for oracle sets and explores their implications for class separations and descriptive complexity.

Findings

The set of oracles separating NP and co-NP is not small in the new topology.

Similar non-smallness results hold for PSPACE vs PH and NP vs BQP separations.

The descriptive complexity of these oracle classes is precisely characterized in the Borel hierarchy.

Abstract

We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP with probability 1. This hypothesis was quickly disproven in several ways, most famously in 1992 with the result that IP equals 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 NP and co-NP is not small, and obtain similar results for the separation of PSPACE from PH along with the separation of NP from BQP. We demonstrate that this version of the hypothesis turns it into a sufficient condition for unrelativized relationships, at least in the three cases considered here. Second, we example the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.