Presentations of K-Trivial Reals and Kolmogorov Complexity

  • Frank Stephan
  • Guohua Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


For given real α ∈ {0,1} ∞ , a presentation V of α is a prefix-free and recursively enumerable subset of {0,1}* such that \(\alpha = \Sigma_{\sigma\epsilon\nu}2^{-|\sigma|}\). So, α has a presentation iff α is a left-r.e. real.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambos-Spies, K., Jockusch, C., Shore, R., Soare, R.: An algebraic decomposition of the recursively enumerable degrees and classes equal to the promptly simple degrees. Transactions of the American Mathematical Society 281, 109–128 (1984)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arslanov, M.M.: On some generalizations of the Fixed-Point Theorem. Soviet Mathematics (Iz. VUZ), Russian 25(5), 9–16 (1981); English translation 25(5), 1–10 (1981)Google Scholar
  3. 3.
    Arslanov, M.M.: M-reducibility and fixed points. Complexity problems of mathematical logic, Collection of scientific Works, Russian, Kalinin, pp. 11-18 (1985)Google Scholar
  4. 4.
    Calude, C., Hertling, P., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin’s Ω number. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 596–606. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Chaitin, G.: A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery 22, 329–340 (1975) (reprinted in [6])Google Scholar
  6. 6.
    Chaitin, G.: Information, Randomness & Incompleteness, 2nd edn. Series in Computer Science, vol. 8. World Scientific, River Edge (1990)MATHGoogle Scholar
  7. 7.
    Cutland, N.: Computability, an introduction to recursive function theory. Cambridge University Press, Cambridge (1980)MATHGoogle Scholar
  8. 8.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Heidelberg (in preparation)Google Scholar
  9. 9.
    Downey, R., Hirschfeldt, D., Nies, A., Stephan, F.: Trivial reals. In: Proceedings of the 7th and 8th Asian Logic Conferences, pp. 103–131. World Scientific, River Edge (2003)CrossRefGoogle Scholar
  10. 10.
    Downey, R., LaForte, G.: Presentations of computably enumerable reals. Theoretical Computer Science (to appear)Google Scholar
  11. 11.
    Kjos-Hanssen, B., Merkle, W., Stephan, F.: Kolmogorov complexity and the Recursion theorem (2005) (manuscript)Google Scholar
  12. 12.
    Nies, A.: Lowness properties and randomness. Advances in Mathematics (to appear)Google Scholar
  13. 13.
    Odifreddi, P.: Classical recursion theory, vol. 1. North-Holland, Amsterdam (1989); vol. 2. Elsevier, Amsterdam (1999)Google Scholar
  14. 14.
    Soare, R.: Recursively enumerable sets and degrees. Springer, Heidelberg (1987)Google Scholar
  15. 15.
    Solovay, R.: Draft of a paper (or series of papers) on Chaitin’s work, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, p. 215 (1975) (unpublished manuscript)Google Scholar
  16. 16.
    Wang, Y.: Randomness and Complexity. PhD Dissertation, University of Heidelberg (1996)Google Scholar
  17. 17.
    Wu, G.: Prefix-free languages and initial segments of computably enumerable degrees. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 576–585. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Frank Stephan
    • 1
  • Guohua Wu
    • 2
    • 3
  1. 1.School of ComputingNational University of SingaporeSingapore
  2. 2.School of Mathematics, Statistics and Computer ScienceVictoria University of WellingtonWellingtonNew Zealand
  3. 3.School of Physical and Mathematical SciencesNayang Technological UniversitySingapore

Personalised recommendations