Dimension Characterizations of Complexity Classes

Abstract

We use derandomization to show that sequences of positive pspace-dimension – in fact, even positive Δ$^{\rm p}_{\rm k}$-dimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose Δ$^{\rm p}_{\rm 3}$-dimension is positive, then BPP ⊆ P^S and, moreover, every BPP promise problem is P^S-separable. We prove analogous results at higher levels of the polynomial-time hierarchy.

The dimension-almost-class of a complexity class $\mathcal{C}$, denoted by dimalmost-$\mathcal{C}$, is the class consisting of all problems A such that $A \in \mathcal{C}^S$ for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.