Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors

TL;DR

This paper establishes a symmetry property for Kolmogorov complexity of random strings and determines the minimal advice needed for extracting high-quality randomness from a single source.

Contribution

It proves a strong symmetry of information relation for Kolmogorov complexity and tight bounds on advice required for randomness extraction.

Findings

Symmetry of information holds for all pairs of n-bit random strings.

O(1) advice is insufficient for extracting randomness rate 1.

More than constant advice is necessary for effective randomness extraction.

Abstract

We prove a strong Symmetry of Information relation for random strings (in the sense of Kolmogorov complexity) and establish tight bounds on the amount on nonuniformity that is necessary for extracting a string with randomness rate 1 from a single source of randomness. More precisely, as instantiations of more general results, we show: (1) For all n-bit random strings x and y, x is random conditioned by y if and only if y is random conditioned by x, and (2) while O(1) amount of advice regarding the source is not enough for extracting a string with randomness rate 1 from a source string with constant random rate, \omega(1) amount of advice is. The proofs use Kolmogorov extractors as the main technical device.