What Can be Efficiently Reduced to the K-Random Strings?

Abstract

We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorov-random strings R _K . We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. Among other results, we show that although for every universal machine U, there are very complex sets that are $\leq^{p}_{dtt}$-reducible to R _k ∪, it is nonetheless true that P=REC $\cap\bigcap\cup\{A:A\leq^{p}_{dtt} R_{k\cup}\}$. We also show for a broad class of reductions that the sets reducible to R _K have small circuit complexity.