Advertisement

The Kolmogorov complexity of real numbers

  • Ludwig Staiger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1684)

Abstract

We consider for a real number α 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 α 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [CH94]
    J.-Y. Cai and J. Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. J. Comput. System Sci. 49 (1994) 3, 605–619.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Ca94]
    C. Calude, Information and Randomness. An Algorithmic Perspective. Springer-Verlag, Berlin, 1994.zbMATHGoogle Scholar
  3. [CC96]
    C. Calude and C. Câmpeanu, Are Binary Codings Universal?, Complexity 1 (1996) 15, 47–50.Google Scholar
  4. [CJ94]
    C. Calude and H. Jürgensen, Randomness as an invariant for number representations. In: Results and Trends in Theoretical Computer Science (H. Maurer, J. Karhumäki, G. Rozenberg, Eds.), Lecture Notes in Comput. Sci., Vol. 812, Springer-Verlag, Berlin, 1994, 44–66.Google Scholar
  5. [Cs59]
    J.W.S. Cassels, On a problem of Steinhaus about normal numbers. Colloquium Math. 7 (1959), 95–101.zbMATHMathSciNetGoogle Scholar
  6. [Ch87]
    G. J. Chaitin, Information, Randomness, & Incompleteness. Papers on Algorithmic Information Theory. World Scientific, Singapore, 1987.zbMATHGoogle Scholar
  7. [Da74]
    R.P. Daley, The extent and density of sequences within the minimal-program complexity hierarchies. J. Comput. System Sci. 9 (1974), 151–163.zbMATHMathSciNetGoogle Scholar
  8. [Fa90]
    K.J. Falconer, Fractal Geometry. Wiley, Chichester, 1990.zbMATHGoogle Scholar
  9. [He96]
    P. Hertling, Disjunctive ω-words and Real Numbers, Journal of Universal Computer Science 2 (1996) 7, 549–568.MathSciNetGoogle Scholar
  10. [HW98]
    P. Hertling and K. Weihrauch, Randomness Spaces. in: Automata, Languages and Programming, Proc. 25th Int. Colloq. ICALP’98, (K. G. Larsen, S. Skyum and G. Winskel, Eds.), Lecture Notes in Comput. Sci., Vol. 1443, Springer-Verlag, Berlin, 1998, 796–807.Google Scholar
  11. [JT88]
    H. Jürgensen and G. Thierrin, Some structural properties of ω-languages. in: Sbornik 13th Nat. School “Applications of Mathematics in Technology”, Sofia, 1988, 56–63.Google Scholar
  12. [LV93]
    M. Li and P.M.B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications. Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  13. [MS94]
    W. Merzenich and L. Staiger, Fractals, dimension, and formal languages. RAIRO-Inform. Thèor. 28 (1994) 3-4, 361–386.zbMATHMathSciNetGoogle Scholar
  14. [Ox71]
    J.C. Oxtoby, Measure and Category. Springer-Verlag, Berlin 1971.zbMATHGoogle Scholar
  15. [Ry86]
    B.Ya. Ryabko, Noiseless coding of combinatorial sources, Hausdorff dimension and Kolmogorov complexity. Problems of Information Transmission 22 (1986) 3, 170–179.zbMATHMathSciNetGoogle Scholar
  16. [Sc60]
    W.M. Schmidt, On normal numbers. Pac. J. Math. 10 (1960), 661–672.zbMATHGoogle Scholar
  17. [St93]
    L. Staiger, Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. 103 (1993) 2, 159–194. Preliminary version in: “Fundamentals of Computation Theory” (J. Csirik, J. Demetrovics and F. Gècseg Eds.), Lecture Notes in Comput. Sci., No. 380, Springer-Verlag, Berlin 1989, 334–343.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [St98]
    L. Staiger, A tight upper bound on Kolmogorov complexity and uniformly optimal prediction. Theory of Computing Systems 31 (1998), 215–229.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [We92]
    K. Weihrauch, The Degrees of Discontinuity of some Translators Between Representations of the Real Numbers. Informatik-Bericht 129, FernUniversität Hagen 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Ludwig Staiger
    • 1
  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany

Personalised recommendations