Kolmogorov-Loveland Randomness and Stochasticity

  • Wolfgang Merkle
  • Joseph Miller
  • André Nies
  • Jan Reimann
  • Frank Stephan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3404)

Abstract

One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as Kolmogorov-Loveland (or KL) randomness, where an infinite binary sequence is KL-random if there is no computable non-monotonic betting strategy that succeeds on the sequence in the sense of having an unbounded gain in the limit while betting successively on bits of the sequence. Our first main result states that every KL-random sequence has arbitrarily dense, easily extractable subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. We show that this splitting property does not characterize KL-randomness by constructing a sequence that is not even computably random such that every effective split yields subsequences that are 2-random, hence are in particular Martin-Löf random.

A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence. Our second main result asserts that every KL-stochastic sequence has constructive dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α< 1 with respect to prefix-free Kolmogorov complexity. This improves on a result by Muchnik, who has shown a similar implication where the premise requires that such compressible prefixes can be found effectively.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambos-Spies, K., Kučera, A.: Randomness in computability theory. In: Computability theory and its applications, Boulder, CO. Contemp. Math., vol. 257, pp. 1–14. Amer. Math. Soc., Providence (2000)Google Scholar
  2. 2.
    Buhrman, H., van Melkebeek, D., Regan, K.W., Sivakumar, D., Strauss, M.: A generalization of resource-bounded measure, with application to the BPP vs. EXP problem. SIAM J. Comput. 30, 576–601 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, J.Y., Hartmanis, J.: On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. J. Comput. System Sci. 49, 605–619 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Downey, R., Hirschfeldt, D., Nies, A., Terwijn, S.: Calibrating randomness. Bulletin of Symbolic Logic (to appear) Google Scholar
  5. 5.
    Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its applications. Graduate Texts in Computer Science. Springer, New York (1997)MATHGoogle Scholar
  6. 6.
    Lutz, J.H.: Gales and the constructive dimension of individual sequences. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 902–913. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inform. Process. Lett. 84, 1–3 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Merkle, W.: The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences. J. Symbolic Logic 68, 1362–1376 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Merkle, W., Miller, J., Nies, A., Reimann, J., Stephan, F.: Kolmogorov-Loveland randomness and stochasticity. Annals of Pure and Applied Logic (to appear)Google Scholar
  11. 11.
    Muchnik, A.A., Semenov, A.L., Uspensky, V.A.: Mathematical metaphysics of randomness. Theoret. Comput. Sci. 207, 263–317 (1998)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Odifreddi, P.: Classical recursion theory. North-Holland, Amsterdam (1989)MATHGoogle Scholar
  13. 13.
    Reimann, J.: Computability and fractal dimension. Doctoral Dissertation, Universität Heidelberg, Heidelberg, Germany (2004)Google Scholar
  14. 14.
    Ryabko, B.Y.: Coding of combinatorial sources and Hausdorff dimension. Sov. Math. Dokl. 30, 219–222 (1984)MATHGoogle Scholar
  15. 15.
    Ryabko, B.Y.: Noiseless coding of combinatorial sources, Hausdorff dimension and Kolmogorov complexity. Probl. Information Transmission 22, 170–179 (1986)MATHMathSciNetGoogle Scholar
  16. 16.
    Ryabko, B.Y.: Private communication (April 2003)Google Scholar
  17. 17.
    Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. 103, 159–194 (1993)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Staiger, L.: A tight upper bound on Kolmogorov complexity and uniformly optimal prediction. Theory of Computing Systems 31, 215–229 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Uspensky, V.A., Semenov, A.L., Shen, A.K.: Can an (individual) sequence of zeros and ones be random? Russian Math. Surveys 45, 121–189 (1990)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Van Lambalgen, M.: Random sequences. Doctoral Dissertation, Universiteit van Amsterdam, Amsterdam, Netherlands (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Wolfgang Merkle
    • 1
  • Joseph Miller
    • 2
  • André Nies
    • 3
  • Jan Reimann
    • 1
  • Frank Stephan
    • 4
  1. 1.Universität HeidelbergHeidelbergGermany
  2. 2.Indiana UniversityBloomingtonUSA
  3. 3.University of AucklandAucklandNew Zealand
  4. 4.National University of SingaporeSingapore

Personalised recommendations