Abstract:
Bennett and Gill (1981) proved that P^A != NP^A relative to a random oracle A, or in other words, that the set O_[P=NP] = { A | P^A = NP^A } has Lebesgue measure 0. In contrast, we show that O_[P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity theoretic statement Phi, then the set of oracles relative to which Phi holds has Hausdorff dimension 1. We give several other applications including proofs that the polynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that P^A != NP^A intersect coNP^A relative to a Hausdorff dimension 1 set of oracles.