Every Sequence is Decompressible from a Random One

TL;DR

This paper demonstrates that any infinite sequence can be computed from a Martin-Lof random sequence with a query ratio matching its constructive dimension, establishing an optimal compression ratio and offering a new characterization of this dimension.

Contribution

It extends previous results by showing the optimal query ratio for Turing reductions from random sequences to any sequence, linking it to the sequence's constructive dimension.

Findings

Every sequence is reducible to a random sequence with optimal query ratio.

The query ratio equals the constructive dimension of the sequence.

Provides a new characterization of constructive dimension via Turing reduction compression ratios.

Abstract

Kucera and Gacs independently showed that every infinite sequence is Turing reducible to a Martin-Lof random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a Martin-Lof 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. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduction compression ratios.