Advertisement

Initial Segment Complexities of Randomness Notions

  • Rupert Hölzl
  • Thorsten Kräling
  • Frank Stephan
  • Guohua Wu
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 323)

Abstract

Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m and all n > f(m) it holds that C(A(0)A(1)...A(n)) ≤ n − m.

In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strongly random, Kurtz random relative to K, PA-incomplete Martin-Löf random and strongly Kurtz random; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n > f(m) it holds that C(A(0)A(1)...A(n)) ≤ n − m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a K-recursive function; this solves an open problem of Nies.

In addition to this, characterisations of a similar style are also given for Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.

Keywords

Initial Segment Recursive Function Kolmogorov Complexity Strong Randomness Peano Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bienvenu, L., Merkle, W.: Reconciling Data Compression and Kolmogorov Complexity. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 643–654. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Calude, C.S.: Information and Randomness. In: An Algorithmic Perspective, 2nd edn., Springer, Heidelberg (2002)Google Scholar
  3. 3.
    Chaitin, G.J.: A theory of program size formally identical to information theory. Journal of the ACM 22, 329–340 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Downey, R., Nies, A., Weber, R., Yu, L.: Lowness and \(\Pi_2^0\) Nullsets. Journal of Symbolic Logic 71, 1044–1052 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Downey, R., Griffiths, E.: On Schnorr randomness. Journal of Symbolic Logic 69, 533–554 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gács, P.: Every sequence is reducible to a random one. Information and Control 70, 186–192 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kučera, A.: Measure, \(\Pi^0_1\)-classes and complete extensions of PA. In: PPSN 1996. LNM, vol. 1141, pp. 245–259. Springer, Heidelberg (1985)Google Scholar
  8. 8.
    Kurtz, S.A.: Randomness and genericity in the degrees of unsolvatibility. PhD dissertation, University of Illinois at Urbana-Champaign (1981)Google Scholar
  9. 9.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  10. 10.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Miller, J.S.: Every 2-random real is Kolmogorov random. Journal of Symbolic Logic 69, 907–913 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Miller, J.S.: The K-degrees, low for K degrees and weakly low for K oracles. Notre Dame Journal of Formal Logic (to appear)Google Scholar
  13. 13.
    Nies, A.: Computability and Randomness. Oxford Science Publications (2009)Google Scholar
  14. 14.
    Nies, A., Stephan, F., Terwijn, S.A.: Randomness, relativization and Turing degrees. Journal of Symbolic Logic 70, 515–535 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Odifreddi, P.: Classical Recursion Theory I. North-Holland, Amsterdam (1989)Google Scholar
  16. 16.
    Odifreddi, P.: Classical Recursion Theory II. Elsevier, Amsterdam (1999)zbMATHGoogle Scholar
  17. 17.
    Osherson, D., Weinstein, S.: Recognizing the strong random reals. The Review of Symbolic Logic 1, 56–63 (2008)CrossRefGoogle Scholar
  18. 18.
    Williams, H.C. (ed.): CRYPTO 1985. LNCS, vol. 218. Springer, Heidelberg (1971)Google Scholar
  19. 19.
    Schnorr, C.-P.: Process complexity and effective random tests. Journal of Computer and System Sciences 7, 376–388 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar
  21. 21.
    Stephan, F.: Martin-Löf Random and PA-complete Sets. In: Proceedings of ASL Logic Colloquium 2002. ASL Lecture Notes in Logic, vol. 27, pp. 342–348 (2006)Google Scholar
  22. 22.
    Stephan, F., Wu, G.: Presentations of K-Trivial Reals and Kolmogorov Complexity. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 461–469. Springer, Heidelberg (2005)Google Scholar

Copyright information

© IFIP 2010

Authors and Affiliations

  • Rupert Hölzl
    • 1
  • Thorsten Kräling
    • 1
  • Frank Stephan
    • 2
  • Guohua Wu
    • 3
  1. 1.Institut für InformatikUniversität HeidelbergHeidelbergGermany
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore
  3. 3.Division of Mathematical Sciences, School of Physical and Mathematical Sciences, College of ScienceNanyang Technological UniversitySingapore

Personalised recommendations