Kobayashi compressibility

George Barmpalias, Rodney G. Downey
2017 Theoretical Computer Science  
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
more » ... 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. The compressibility of a finite binary program σ is defined in terms of the shortest program that can generate σ. This is the idea behind the theory of Kolmogorov complexity C of strings. For example, if c ∈ N then σ is c-incompressible if C(σ) ≥ |σ| − c, and similar definitions are used with respect to the prefix-free complexity K, where the underlying universal machine is prefix-free. This notion of incompressibility has a well-known extension to infinite binary streams X, where we say that X is c-incompressible if K(X n ) ≥ n − c for all n. Then the algorithmic randomness of X is often identified with the property that X is c-incompressible for some c, and coincides with the notion of Martin-Löf randomness 1 . These concepts are basic in Kolmogorov complexity, and the reader is referred to the standard textbooks [LV97, DH10] for the relevant background.
doi:10.1016/j.tcs.2017.02.029 fatcat:63iwdmnqjrcutctsg5oiikbnt4