Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 533–546 | Cite as

Algorithmic randomness of continuous functions

  • George Barmpalias
  • Paul Brodhead
  • Douglas Cenzer
  • Jeffrey B. Remmel
  • Rebecca Weber
Article

Abstract

We investigate notions of randomness in the space \({{\mathcal C}(2^{\mathbb N})}\) of continuous functions on \({2^{\mathbb N}}\). A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random \({\Delta^0_2}\) continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any \({y \in 2^{\mathbb N}}\), there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.

Keywords

Computable analysis Computability Randomness 

Mathematics Subject Classification (2000)

03D28 68Q30 60D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asarin, E.A., Prokovskiy, V.: Application of Kolmogorov complexity to the dynamics of controllable systems. Automat. Telemekh. 1, 25–53 (1986)Google Scholar
  2. 2.
    Brodhead, P., Cenzer, D., Dashti, S.: Random closed sets. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) Logical Approaches to Computational Barriers. Springer Lecture Notes in Computer Science 3988, pp. 55–64 (2006)Google Scholar
  3. 3.
    Barmpalias, G., Brodhead, P., Cenzer, D., Dashti, S., Weber, R.: Algorithmic randomness of closed sets. J. Logic Comput. (2007) (to appear)Google Scholar
  4. 4.
    Cenzer, D.: \({\Pi^0_1}\) Classes, ASL Lecture Notes in Logic (2007) (to appear)Google Scholar
  5. 5.
    Cenzer, D., Remmel, J.B.: \({\Pi^0_1}\) classes. In: Ersov, Y., Goncharov, S., Marek, V., Nerode, A., Remmel, J. (eds.) Handbook of Recursive Mathematics, Vol. 2: Recursive Algebra, Analysis and Combinatorics. Elsevier Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 623–821 (1998)Google Scholar
  6. 6.
    Chaitin, G.: Information-theoretical characterizations of recursive infinite strings. Theor. Comp. Sci. 2, 45–48 (1976)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity (in preparation). Current draft available at http://www.mcs.vuw.ac.nz/~downey/
  8. 8.
    Fouche, W.: Arithmetical representations of Brownian motion. J. Symbolic Logic 65, 421–442 (2000)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Gács, P.: On the symmetry of algorithmic information. Soviet Mat. Dokl. 15, 1477–1480 (1974)MATHGoogle Scholar
  10. 10.
    Kolmogorov, A.N.: Three approaches to the quantitative defintion of information. Problems Inform. Trans. 1, 1–7 (1965)Google Scholar
  11. 11.
    Levin, L.: On the notion of a random sequence. Soviet Mat. Dokl. 14, 1413–1416 (1973)MATHGoogle Scholar
  12. 12.
    Martin-Löf, P.: The definition of random sequences. Inform. Control 9, 602–619 (1966)CrossRefGoogle Scholar
  13. 13.
    Schnorr, C.P.: A unified approach to the definition of random sequences. Math. Syst. Theory 5, 246–258 (1971)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    van Lambalgen, M.: Random Sequences. Ph.D. Dissertation, University of Amsterdam (1987)Google Scholar
  15. 15.
    Ville, J.: Étude Critique de la Notion de Collectif. Gauthier-Villars, Paris (1939)Google Scholar
  16. 16.
    von Mises, R.: Grundlagen der Wahrscheinlichkeitsrechnung. Math. Zeitschrift 5, 52–99 (1919)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • George Barmpalias
    • 1
  • Paul Brodhead
    • 2
  • Douglas Cenzer
    • 2
  • Jeffrey B. Remmel
    • 3
  • Rebecca Weber
    • 4
  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA
  3. 3.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  4. 4.Department of MathematicsDartmouth CollegeHanoverUSA

Personalised recommendations