Optimal redundancy in computations from random oracles

TL;DR

This paper introduces a new coding method in algorithmic information theory that achieves the optimal logarithmic redundancy in computations from random oracles, significantly improving previous bounds and establishing the best possible redundancy rate.

Contribution

The paper presents a novel coding technique that attains the optimal logarithmic redundancy in computations from random oracles, surpassing prior methods.

Findings

Redundancy can be reduced to logarithmic scale, which is optimal.

Previous bounds were n log n and sqrt(n) log n, now improved to logarithmic.

Redundancy r log n is achievable if and only if r > 1.

Abstract

A classic result in algorithmic information theory is that every infinite binary sequence is computable from a Martin-Loef random infinite binary sequence. Proved independently by Kucera and Gacs, this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n + g(n) bits of the random oracle, then g is the redundancy of the computation. Kucera implicitly achieved redundancy n log n while Gacs used a more elaborate block-coding procedure which achieved redundancy sqrt(n) log n. Different approaches to coding such as the one by Merkle and Mihailovic have not improved this redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. Our redundancy bound is exponentially smaller than the previously best known bound and is known to be the best possible. It follows that redundancy r log n in computation from a random oracle is possible for every stream, if and only if r > 1.