Kobayashi compressibility

TL;DR

Kobayashi compressibility offers a uniform framework to characterize various randomness notions through Turing reductions, connecting it to current research in algorithmic randomness.

Contribution

This work demonstrates that Kobayashi compressibility can define multiple randomness notions purely via Turing reductions, clarifying its relevance in algorithmic randomness.

Findings

Kobayashi compressibility defines Martin-Loef randomness via Turing reductions.

It characterizes strong and Kurtz randomness through weak truth-table and truth-table reductions.

The paper discusses and extends Kobayashi's original results on computably enumerable sets.

Abstract

Kobayashi introduced a uniform notion of compressibility of infinite binary sequences in terms of relative Turing computations with sub-identity use of the oracle. Kobayashi compressibility has remained a relatively obscure notion, with the exception of some work on resource bounded Kolmogorov complexity. The main goal of this note is to show that it is relevant to a number of topics in current research on algorithmic randomness. We prove that Kobayashi compressibility can be used in order to define Martin-Loef randomness, a strong version of finite randomness and Kurtz randomness, strictly in terms of Turing reductions. Moreover these randomness notions naturally correspond to Turing reducibility, weak truth-table reducibility and truth-table reducibility respectively. Finally we discuss Kobayashi's main result from his 1981 technical report regarding the compressibility of computably enumerable sets, and provide additional related original results.