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.


Binary String Recursive Function Computable Function Kolmogorov Complexity Recursive Approximation 
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.


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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHGoogle Scholar
  7. 7.
    Cutland, N.: Computability, an introduction to recursive function theory. Cambridge University Press, Cambridge (1980)zbMATHGoogle 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