Randomness and reducibility

How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as measures of relative randomness, as embodied in the concept of initial-segment complexity. One such reducibility, called domination or Solovay reducibility, was introduced by Solovay [34], and has been studied by Calude, Hertling, Khoussainov, and Wang [8], Calude [3], Kucera and Slaman [22], and Downey, Hirschfeldt, and Nies [15], among others. Solovay reducibility has proved to be a powerful tool in the study of randomness of effectively presented reals. Motivated by certain shortcomings of Solovay reducibility, which we will discuss below, we introduce two new reducibilities and study, among other things, the relationships between these various measures of relative randomness.

[1]  R. Solomonoff.  A PRELIMINARY REPORT ON A GENERAL THEORY OF INDUCTIVE INFERENCE , 2001 .

[2]  Cristian S. Calude,et al.  Degree-Theoretic Aspects of Computably Enumerable Reals , 1998 .

[3]  Cristian S. Calude,et al.  Mathematics: Randomness everywhere , 1999, Nature.

[4]  André Nies,et al.  Randomness, Computability, and Density , 2001, STACS.

[5]  L. Levin,et al.  THE COMPLEXITY OF FINITE OBJECTS AND THE DEVELOPMENT OF THE CONCEPTS OF INFORMATION AND RANDOMNESS BY MEANS OF THE THEORY OF ALGORITHMS , 1970 .

[6]  Cristian S. Calude.  Information and Randomness: An Algorithmic Perspective , 1994 .

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

[8]  R. M. Solovay.  On Random R, E. Sets , 1977 .

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

[10]  Cristian S. Calude,et al.  Chaitin Numbers and Strong Reducibilities , 1997 .

[11]  Cristian Claude,et al.  Information and Randomness: An Algorithmic Perspective , 1994 .

[12]  Manuel Lerman,et al.  Computability Theory and Its Applications , 2000 .

[13]  H. G. Rice,et al.  Recursive real numbers , 1954 .

[14]  Rodney G. Downey.  Some Computability-Theoretical Aspects of Reals and Randomness , 2002 .

[15]  Rodney G. Downey,et al.  Presentations of computably enumerable reals , 2002, Theor. Comput. Sci..

[16]  Chun-Kuen Ho,et al.  Relatively Recursive Reals and Real Functions , 1994, Theor. Comput. Sci..

[17]  Cristian S. Calude,et al.  Program-Size Complexity of Initial Segments and Domination Reducibility , 1999, Jewels are Forever.

[18]  André Nies,et al.  Trivial Reals , 2002, CCA.

[19]  Ker-I Ko,et al.  On the Continued Fraction Representation of Computable Real Numbers , 1986, Theor. Comput. Sci..

[20]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[21]  Cristian S. Calude,et al.  Chaitin Omega Numbers and Strong Reducibilities , 1997, J. Univers. Comput. Sci..

[22]  Klaus Weihrauch,et al.  Weakly Computable Real Numbers , 2000, J. Complex..

[23]  Cristian S. Calude.  A characterization of c.e. random reals , 2002, Theor. Comput. Sci..

[24]  Cristian S. Calude,et al.  Recursively enumerable reals and Chaitin Ω numbers , 2001, Theoretical Computer Science.

[25]  R. Soare.  Recursive theory and Dedekind cuts , 1969 .

[26]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

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

[28]  G. Chaitin.  Incompleteness theorems for random reals , 1987 .

[29]  Elvira Mayordomo,et al.  A Kolmogorov complexity characterization of constructive Hausdorff dimension , 2002, Inf. Process. Lett..

[30]  Cristian S. Calude,et al.  Recursively Enumerable Reals and Chaitin Omega Numbers , 1998, STACS.

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

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

[33]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[34]  N. Costa,et al.  Non-Classical Logics, Model Theory and Computability , 1980 .

[35]  A. Troelstra.  Models and computability , 1973 .

[36]  Klaus Ambos-Spies,et al.  Randomness in Computability Theory , 2000 .

[37]  Robert I. Soare,et al.  Cohesive sets and recursively enumerable Dedekind cuts , 1969 .

[38]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[39]  Barbara F. Csima,et al.  Applications of computability theory to prime models and differential geometry , 2003 .

[40]  G. Paun,et al.  Jewels are Forever , 1999, Springer Berlin Heidelberg.

[41]  Jack H. Lutz,et al.  Gales and the Constructive Dimension of Individual Sequences , 2000, ICALP.

[42]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

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