Skip to main content
SpringerLink
Log in

Characterization of Kurtz Randomness by a Differentiation Theorem

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Brattka, Miller and Nies (2012) showed that some major algorithmic randomness notions are characterized via differentiability. The main goal of this paper is to characterize Kurtz randomness by a differentiation theorem on a computable metric space. The proof shows that integral tests play an essential part and shows that how randomness and differentiation are connected.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Besicovitch, A.S.: A general form of the covering principle and relative differentiation of additive functions. Proc. Camb. Philos. Soc. 41(2), 103–110 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosserhoff, V.: Notions of probabilistic computability on represented spaces. J. Univers. Comput. Sci. 14(6), 956–995 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Brattka, V.: Computability over topological structures. In: Cooper, S.B., Goncharov, S.S. (eds.) Computability and Models, pp. 93–136. Kluwer Academic, New York (2003)

    Chapter  Google Scholar 

  4. Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: New Computational Paradigms, pp. 425–491 (2008)

    Chapter  Google Scholar 

  5. Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability (2012, submitted)

  6. Demuth, O.: The differentiability of constructive functions of weakly bounded variation on pseudo numbers. Comment. Math. Univ. Carol. 16(3), 583–599 (1975)

    MathSciNet  MATH  Google Scholar 

  7. Downey, R., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  8. Freer, C., Kjos-Hanssen, B., Nies, A.: Computable aspects of Lipshitz functions. In preparation

  9. Gács, P.: Uniform test of algorithmic randomness over a general space. Theor. Comput. Sci. 341, 91–137 (2005)

    Article  MATH  Google Scholar 

  10. Gács, P., Hoyrup, M., Rojas, C.: Randomness on computable probability spaces—a dynamical point of view. Theory Comput. Syst. 48(3), 465–485 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Galatolo, S., Hoyrup, M., Rojas, C.: A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties. Theor. Comput. Sci. 410(21–23), 2207–2222 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galatolo, S., Hoyrup, M., Rojas, C.: Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Inf. Comput. 208(1), 23–41 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoyrup, M., Rojas, C.: Computability of probability measures and Martin-Löf randomness over metric spaces. Inf. Comput. 207(7), 830–847 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kurtz, S.A.: Randomness and genericity in the degrees of unsolvability. PhD thesis, University of Illinois at Urbana-Champaign (1981)

  15. Lebesgue, H.: Leçons sur l’Intégration et la Recherche des Fonctions Primitives. Gauthier-Villars, Paris (1904)

    Google Scholar 

  16. Lebesgue, H.: Sur l’intégration des fonctions discontinues. Ann. Sci. Éc. Norm. Super. 27, 361–450 (1910)

    MathSciNet  MATH  Google Scholar 

  17. Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Graduate Texts in Computer Science. Springer, New York (2009)

    Google Scholar 

  18. Nies, A.: Computability and Randomness. Oxford University Press, London (2009)

    Book  MATH  Google Scholar 

  19. Pathak, N., Rojas, C., Simpson, S.G.: Schnorr randomness and the Lebesgue Differentiation Theorem. Proc. Am. Math. Soc. (2012, to appear)

  20. Schröder, M.: Admissible representations for probability measures. Math. Log. Q. 53(4–5), 431–445 (2007)

    MATH  Google Scholar 

  21. Tiser, J.: Differentiation theorem for Gaussian measures on Hilbert space. Trans. Am. Math. Soc. 308(2), 655–666 (1988)

    MathSciNet  MATH  Google Scholar 

  22. Vovk, V., Vyugin, V.: On the empirical validity of the Bayesian method. J. R. Stat. Soc. B 55(1), 253–266 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Weihrauch, K.: Computable Analysis: An Introduction. Springer, Berlin (2000)

    MATH  Google Scholar 

  24. Weihrauch, K., Grubba, T.: Elementary computable topology. J. Univers. Comput. Sci. 15(6), 1381–1422 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks Jason Rute and André Nies for the useful comments. The author also appreciates the anonymous referees for their careful reading and many comments. This work was partly supported by GCOE, Kyoto University and JSPS KAKENHI 23740072.

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kenshi Miyabe

Authors
  1. Kenshi Miyabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenshi Miyabe.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Miyabe, K. Characterization of Kurtz Randomness by a Differentiation Theorem. Theory Comput Syst 52, 113–132 (2013). https://doi.org/10.1007/s00224-012-9422-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-012-9422-3

Keywords

Search

Navigation