Randomness extraction in computability theory

TL;DR

This paper investigates the extraction rate of Turing functionals translating between different notions of randomness, analyzing various classes of extraction procedures inspired by classical and modern techniques.

Contribution

It introduces a unified analysis of multiple classes of randomness extraction procedures and determines the conditions for achieving optimal extraction rates.

Findings

Identifies input randomness levels guaranteeing extraction rate attainment.

Calculates extraction rates for sufficiently random input sequences.

Generalizes classical randomness extraction methods to a computability-theoretic framework.

Abstract

In this article, we study a notion of the extraction rate of Turing functionals that translate between notions of randomness with respect to different underlying probability measures. We analyze several classes of extraction procedures: a first class that generalizes von Neumann's trick for extracting unbiased randomness from the tosses of a biased coin, a second class based on work of generating biased randomness from unbiased randomness by Knuth and Yao, and a third class independently developed by Levin and Kautz that generalizes the data compression technique of arithmetic coding. For the first two classes of extraction procedures, we identify a level of algorithmic randomness for an input that guarantees that we attain the extraction rate along that input, while for the third class, we calculate the rate attained along sufficiently random input sequences.