Dimension Extractors and Optimal Decompression

TL;DR

This paper introduces dimension extractors that enhance the effective randomness of infinite binary sequences and explores their implications for optimal decompression via Turing reductions, linking randomness and computational complexity.

Contribution

It presents new algorithms for increasing the effective dimension of sequences and characterizes effective dimensions through optimal Turing-based decompression methods.

Findings

Constructive dimension can be increased using dimension extractors.

Every sequence is reducible to a Martin-Loef random sequence with optimal compression ratio.

New characterizations of effective dimensions via Turing decompression are established.

Abstract

A *dimension extractor* is an algorithm designed to increase the effective dimension -- i.e., the amount of computational randomness -- of an infinite binary sequence, in order to turn a "partially random" sequence into a "more random" sequence. Extractors are exhibited for various effective dimensions, including constructive, computable, space-bounded, time-bounded, and finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Loef random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. The extractors and decompressors that are developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions.