Dimension Characterizations of Complexity Classes

Abstract.

We use derandomization to show that sequences of positive pspace-dimension − in fact, even positive $\Delta ^{p}_{k}$-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 $\Delta ^{p}_{3}$-dimension is positive, then ${\rm BPP} \subseteq {\rm P} ^S$ 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, Promise-BPP = dimalmost-P-Sep, and AM = dimalmost-NP, that refine previously known results on almost-classes.