Kobayashi compressibility

Kobayashi [Kob81] introduced a uniform notion of compressibility of infinite binary sequences X in terms of relative Turing computations with sub-linear 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 can uniformly compute X ↾n from oracle Y ↾ f (n). 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 [Kob81] regarding the compressibility of computably enumerable sets, and provide additional related original results. George Barmpalias State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand. E-mail: [email protected] Web: http://barmpalias.net Rodney G. Downey School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand. E-mail: [email protected] Web: http://homepages.ecs.vuw.ac.nz/∼downey Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, as well as the Chinese Academy of Sciences (CAS) and the Institute of Software of the CAS.