Theory of Computing Systems

, Volume 52, Issue 1, pp 113–132 | Cite as

Characterization of Kurtz Randomness by a Differentiation Theorem

  • Kenshi Miyabe


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.


Differentiability The differentiation theorem Kurtz randomness Integral test Computable metric space 



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.


  1. 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) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bosserhoff, V.: Notions of probabilistic computability on represented spaces. J. Univers. Comput. Sci. 14(6), 956–995 (2008) MathSciNetMATHGoogle Scholar
  3. 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) CrossRefGoogle Scholar
  4. 4.
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: New Computational Paradigms, pp. 425–491 (2008) CrossRefGoogle Scholar
  5. 5.
    Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability (2012, submitted) Google Scholar
  6. 6.
    Demuth, O.: The differentiability of constructive functions of weakly bounded variation on pseudo numbers. Comment. Math. Univ. Carol. 16(3), 583–599 (1975) MathSciNetMATHGoogle Scholar
  7. 7.
    Downey, R., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  8. 8.
    Freer, C., Kjos-Hanssen, B., Nies, A.: Computable aspects of Lipshitz functions. In preparation Google Scholar
  9. 9.
    Gács, P.: Uniform test of algorithmic randomness over a general space. Theor. Comput. Sci. 341, 91–137 (2005) MATHCrossRefGoogle Scholar
  10. 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) MathSciNetMATHCrossRefGoogle Scholar
  11. 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) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Galatolo, S., Hoyrup, M., Rojas, C.: Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Inf. Comput. 208(1), 23–41 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hoyrup, M., Rojas, C.: Computability of probability measures and Martin-Löf randomness over metric spaces. Inf. Comput. 207(7), 830–847 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kurtz, S.A.: Randomness and genericity in the degrees of unsolvability. PhD thesis, University of Illinois at Urbana-Champaign (1981) Google Scholar
  15. 15.
    Lebesgue, H.: Leçons sur l’Intégration et la Recherche des Fonctions Primitives. Gauthier-Villars, Paris (1904) Google Scholar
  16. 16.
    Lebesgue, H.: Sur l’intégration des fonctions discontinues. Ann. Sci. Éc. Norm. Super. 27, 361–450 (1910) MathSciNetMATHGoogle Scholar
  17. 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. 18.
    Nies, A.: Computability and Randomness. Oxford University Press, London (2009) MATHCrossRefGoogle Scholar
  19. 19.
    Pathak, N., Rojas, C., Simpson, S.G.: Schnorr randomness and the Lebesgue Differentiation Theorem. Proc. Am. Math. Soc. (2012, to appear) Google Scholar
  20. 20.
    Schröder, M.: Admissible representations for probability measures. Math. Log. Q. 53(4–5), 431–445 (2007) MATHGoogle Scholar
  21. 21.
    Tiser, J.: Differentiation theorem for Gaussian measures on Hilbert space. Trans. Am. Math. Soc. 308(2), 655–666 (1988) MathSciNetMATHGoogle Scholar
  22. 22.
    Vovk, V., Vyugin, V.: On the empirical validity of the Bayesian method. J. R. Stat. Soc. B 55(1), 253–266 (1993) MathSciNetMATHGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer, Berlin (2000) MATHGoogle Scholar
  24. 24.
    Weihrauch, K., Grubba, T.: Elementary computable topology. J. Univers. Comput. Sci. 15(6), 1381–1422 (2009) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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