Kobayashi compressibility

George Barmpalias; Rodney G. Downey

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arXiv · 1608.00692

Kobayashi compressibility

George Barmpalias·Rodney G. Downey

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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.

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George Barmpalias, Rodney G. Downey. 2016-08-02. Kobayashi compressibility. https://arxiv.org/abs/1608.00692

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