Reductions to the set of random strings: The resource-bounded case

TL;DR

This paper explores the relationship between resource-bounded Kolmogorov randomness and complexity classes, showing that certain reductions imply containment in P/poly or PSPACE, thus advancing understanding of BPP characterization.

Contribution

It demonstrates that reductions to time-bounded Kolmogorov-random strings imply complexity class containments, refining previous approaches to characterizing BPP.

Findings

Reductions to time-t-bounded Kolmogorov random strings imply A is in P/poly.

Such reductions for any universal Turing machine imply A is in PSPACE.

Approach using unbounded Kolmogorov complexity cannot succeed without significant changes.

Abstract

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.