Hausdorff Dimension and Oracle Constructions

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 Φ, then the set of oracles relative to which Φ 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 ∩ coNP^A relative to a Hausdorff dimension 1 set of oracles.

Theoretical Computer Science, 355(3):382-388, 2006.
[PostScript | PDF | DOI link]
Technical Report TR04-072, Electronic Colloquium on Computational Complexity, 2004.