Abstract
In 1987, Blum and Impagliazzo, using techniques of Hartmanis and Hemachandra and Rackoff, showed that ifP=NPthenP(G)=NP(G)∩coNP(G)=UP(G), whereGis a generic oracle. They leave open the question as to whether these collapses occur at higher levels of the polynomial-time hierarchy, i.e.,Δpk(G)=Σpk(G)∩Πpk(G)=UΔpk(G) fork⩾2. Here we give a negative answer to these questions. In fact, we demonstrate that, relative to any generic oracleGand for everyk⩾2, there exists a tally set inUΔpk(G)∩Πpk(G) but not inΔpk(G) by showing an exponential lower bound of a certain type of families of constant-depth circuits. An immediate corollary is that generic oracles separateΣpk∩ΠpkandΔpk. We also show that related results hold for type-2 complexity.
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