Kobayashi compressibility

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. George Barmpalias State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. E-mail: barmpalias@gmail.com Web: http://barmpalias.net Rodney G. Downey School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. E-mail: rod.downey@vuw.ac.nz Web: http://homepages.ecs.vuw.ac.nz/∼downey Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, grant no. D1101130. Additional support was received by the Chinese Academy of Sciences (CAS) and the Institute of Software of the CAS. Downey was supported by the Marsden Fund of New Zealand. The authors wish to thank the anonymous referees for various suggestions and corrections.

[1]  Song-Ju Kim.  Randomness and Complexity in “Nano-Chaos” , 2015 .

[2]  Rodney G. Downey,et al.  Bounded Randomness , 2012, Computation, Physics and Beyond.

[3]  A. Church.  On the concept of a random sequence , 1940 .

[4]  Donald W. Loveland,et al.  A Variant of the Kolmogorov Concept of Complexity , 1969, Information and Control.

[5]  Antonín Kucera,et al.  Randomness and Recursive Enumerability , 2001, SIAM J. Comput..

[6]  Bakhadyr Khoussainov,et al.  Recursively enumerable reals and Chaitin Ω numbers , 1998 .

[7]  Stephanie Reid,et al.  On Kurtz randomness , 2004, Theor. Comput. Sci..

[8]  George Barmpalias,et al.  Computing halting probabilities from other halting probabilities , 2017, Theor. Comput. Sci..

[9]  George Barmpalias,et al.  Analogues of Chaitin's Omega in the computably enumerable sets , 2013, Inf. Process. Lett..

[10]  Santiago Figueira,et al.  Randomness and halting probabilities , 2006, Journal of Symbolic Logic.

[11]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.

[12]  Karol Habart.  Bounds in Weak Truth-Table Reducibility , 1991, Notre Dame J. Formal Log..

[13]  George Barmpalias,et al.  Kolmogorov complexity and computably enumerable sets , 2011, Ann. Pure Appl. Log..

[14]  Guohua Wu,et al.  Anti-Complex Sets and Reducibilities with Tiny Use , 2011, The Journal of Symbolic Logic.

[15]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[16]  Karol Habart.  Randomness and Turing Reducibility Restraints , 1990, CSL.

[17]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[18]  Kohtaro Tadaki,et al.  Chaitin Ω numbers and halting problems , 2009 .

[19]  A. Nies.  Computability and randomness , 2009 .

[21]  Nan Fang,et al.  Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega , 2016, J. Comput. Syst. Sci..

[22]  Hava T. Siegelmann,et al.  Computational power of neural networks: a characterization in terms of Kolmogorov complexity , 1997, IEEE Trans. Inf. Theory.

[23]  Karol Habart.  Bounds in the Turing Reducibility of Functions , 1992, Math. Log. Q..

[24]  Martin Kummer.  Kolmogorov Complexity and Instance Complexity of Recursively Enumerable Sets , 1996, SIAM J. Comput..

[25]  Kojiro Kobayashi.  Sigma^0_n-Complete Properties of Programs and Martin-Löf Randomness , 1993, Inf. Process. Lett..

[26]  Adam R. Day.  Process and truth-table characterisations of randomness , 2012, Theor. Comput. Sci..

[27]  Claus-Peter Schnorr,et al.  Process complexity and effective random tests , 1973 .

[28]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.