Strong separations of the polynomial hierarchy with oracles: constructive separations by immune and simple sets

Abstract

In this paper we show that the techniques introduced by Furst (1984), which connected oracle separation results for the relativized polynomial-time hierarchy to the problem of proving lower bounds for constant-depth circuits, and the subsequent probabilistic arguments introduced by Yao (1985), Hastad (1986), and Ko (1989), in order to prove the existence of relativized polynomial-time hierarchies with different structures, can be adapted for resolving the main problems related to the existence of immune and simple sets in the relativized polynomial-time hierarchy. In particular, we construct oracles which witness: 1. for any k>, the existence of a Δ P k -immune set in Σ P k ; 2. for any k>, the existence of a Σ k -simple set; 3. for any k>, the existence of a Δ P k -immune set in a relativized polynomial-time hierarchy for which Σ P k =Π P k ≠Δ P k ; 4. for any k>1, the existence of a Σ p k−1 -immune set in a relativized polynomial-time hierarchy for which Σ P k =Δ P k ;