Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences

TL;DR

This paper demonstrates that combining two independent infinite sequences with positive randomness can effectively produce a new sequence with a higher Kolmogorov complexity rate, approaching maximal randomness.

Contribution

It introduces a uniform effective procedure that transforms two independent sequences into one with near-maximal randomness rate, a novel approach in Kolmogorov complexity theory.

Findings

The procedure works with sequences having positive but small randomness rates.

The output sequence's randomness rate can be made arbitrarily close to 1.

The transformation is a truth-table reduction.

Abstract

The randomness rate of an infinite binary sequence is characterized by the sequence of ratios between the Kolmogorov complexity and the length of the initial segments of the sequence. It is known that there is no uniform effective procedure that transforms one input sequence into another sequence with higher randomness rate. By contrast, we display such a uniform effective procedure having as input two independent sequences with positive but arbitrarily small constant randomness rate. Moreover the transformation is a truth-table reduction and the output has randomness rate arbitrarily close to 1.