Abstract
Kobayashi [21] introduced a uniform notion of compressibility of infinite binary sequences X in terms of relative Turing computations with sub-identity use of the oracle. Given f:N→N we say that X is f-compressible if there exists Y such that for each n we compute X↾n using at most the first f(n) bits of the oracle Y. 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-Löf 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 [21] regarding the compressibility of computably enumerable sets, and provide additional related original results.
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