The Kolmogorov complexity of real numbers

We consider for a real number ff the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other. More precisely, we show that the complexity of the length l. logr b prefix of the base r expansion of ff is the same (up to an additive constant) as the logr b-fold complexity of the length l prefix of the base b expansion of α. Then we use this fact to derive complexity theoretic proofs for the base independence of the randomness of real numbers and for some properties of Liouville numbers.

[1]  Ludwig Staiger,et al.  Kolgomorov Complexity and Hausdorff Dimension , 1993, Inf. Comput..

[2]  Cristian S. Calude,et al.  Are binary codings universal? , 1996, Complex..

[3]  Ludwig Staiger,et al.  Kolmogorov Complexity and Hausdorff Dimension , 1989, FCT.

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

[5]  F. Takens.  Measure and category , 1988 .

[6]  Klaus Weihrauch,et al.  Randomness Spaces , 1998, ICALP.

[7]  S. S. Pillai,et al.  On normal numbers , 1939 .

[8]  Robert P. Daley.  The Extent and Density of Sequences within the Minimal-Program Complexity Hierarchies , 1974, J. Comput. Syst. Sci..

[9]  Ludwig Staiger,et al.  Fractals, Dimension, and Formal Languages , 1993, RAIRO Theor. Informatics Appl..

[10]  J. W. S. Cassels,et al.  On a problem of Steinhaus about normal numbers , 1959 .

[11]  Cristian S. Calude,et al.  Randomness as an Invariant for Number Representations , 1994, Results and Trends in Theoretical Computer Science.

[12]  S. Krantz.  Fractal geometry , 1989 .

[13]  K. Weihrauch.  The Degrees of Discontinuity of some Translators Between Representations of the Real Numbers , 1992 .

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

[15]  Peter Hertling.  Disjunctive omega-Words and Real Numbers , 1996, J. Univers. Comput. Sci..

[16]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[17]  Cristian S. Calude.  Information and Randomness , 1994, Monographs in Theoretical Computer Science An EATCS Series.

[18]  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 .

[19]  Gregory J. Chaitin,et al.  Information, Randomness & Incompleteness - Papers on Algorithmic Information Theory - Second Edition , 1997 .

[20]  Ludwig Staiger,et al.  A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction , 1998, Theory of Computing Systems.

[21]  Juris Hartmanis,et al.  On Hausdorff and Topological Dimensions of the Kolmogorov Complexity of the Real Line , 1994, J. Comput. Syst. Sci..

[22]  Peter Hertling,et al.  Disjunctive w-words and Real Numbers , 1996 .