Optimal bounds for single-source Kolmogorov extractors

TL;DR

This paper establishes the optimal bounds for how much a finite set of computable transformations can increase the randomness rate of strings, providing tight bounds and translating the problem into combinatorics on hypergraphs.

Contribution

It introduces a precise, tight bound on the increase of randomness rate achievable with a finite set of transformations, improving previous loose bounds.

Findings

The maximum increase from rate α to β with k transformations is β < kα/(1+(k-1)α).

The bounds are proven to be tight and optimal.

The problem is translated into combinatorics on hypergraphs for the proof.

Abstract

The rate of randomness (or dimension) of a string $\sigma$ is the ratio $C(\sigma)/|\sigma|$ where $C(\sigma)$ is the Kolmogorov complexity of $\sigma$. While it is known that a single computable transformation cannot increase the rate of randomness of all sequences, Fortnow, Hitchcock, Pavan, Vinodchandran, and Wang showed that for any $0<\alpha<\beta<1$, there are a finite number of computable transformations such that any string of rate at least $\alpha$ is turned into a string of rate at least $\beta$ by one of these transformations. However, their proof only gives very loose bounds on the correspondence between the number of transformations and the increase of rate of randomness one can achieve. By translating this problem to combinatorics on (hyper)graphs, we provide a tight bound, namely: Using $k$ transformations, one can get an increase from rate $\alpha$ to any rate $\beta < k\alpha/(1+(k-1)\alpha)$, and this is optimal.