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

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 ΔPk-immune set in ΣPk; 2. for any k>, the existence of a Σk-simple set; 3. for any k>, the existence of a ΔPk-immune set in a relativized polynomial-time hierarchy for which ΣPk=ΠPk≠ΔPk; 4. for any k>1, the existence of a Σpk-1-immune set in a relativized polynomial-time hierarchy for which ΣPk=ΔPk;.