Derandomizing from Random Strings

TL;DR

This paper demonstrates that BPP can be reduced to the set of Kolmogorov random strings using a non-adaptive approach, revealing new structural properties of R_K and its implications for derandomization.

Contribution

It introduces a non-adaptive reduction from BPP to R_K and establishes that initial segments of R_K are incompressible, advancing understanding of randomness and complexity.

Findings

BPP is truth-table reducible to R_K

Initial segments of R_K are incompressible by recursive means

High Kolmogorov-complexity strings as advice are not significantly more useful than random strings

Abstract

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.