Probability one separation of the Boolean hierarchy

Abstract

As a natural extension of P and NP, the Boolean closure of NP has attracted much attention recently. Many interesting properties of the class BH were discussed in [CH86] and [Wag85].

In this paper, we show that a random oracle A separates the Boolean hierarchy into an infinite hierarchy, i.e.

$$P^A \ne NP^A \ne NP^A (2) \ne \cdots \ne NP^A (k) \ne \cdots \ne BH^A$$

with probability one.

The proof technique is a combination of the “sawing” argument introduced in [CH86] and the method of Bennet and Gill [BG81] to handle the base case.

Research supported by a Sage Fellowship and NSF grant DCR-8301766.