Two oracles that force a big crunch

Abstract.

We construct an oracle A such that ${\rm NEXP}^A = {\rm P^{NP}}^A$. The construction of this oracle answers a long standing open question first posed by Heller, and unsuccessfully attacked many times since. For the first construction of the oracle, we present a new type of injury argument that we call “resource-bounded injury.” In the special case of the construction of this oracle, a tree method can be used to transform unbounded search into exponentially bounded, hence recursive, search. This transformation of the construction can be interleaved with another construction so that relative to the new combined oracle also ${\rm P = UP = NP \cap coNP}$. This leads to the curious situation where LOW(NP) = P, but LOW(P^NP) = NEXP, and the complete $\le^p_m$-degree for P^NP collapses to a single p-isomorphism type.