Prefix-Free Languages and Initial Segments of Computably Enumerable Degrees

  • Guohua Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)


We study prefix-free presentations of computably enumerable reals. In 2, Calude et. al. proved that a real a is c.e. if and only if there is an infinite, computably enumerable prefix-free set V such that \( \alpha = \sum _{\sigma \in V} 2^{ - \left| \sigma \right|} \). Following Downey and LaForte 5, we call V a prefixfree presentation of a. Each computably enumerable real has a computable presentation. Say that a c.e. real a is simple if each presentation of a is computable. Downey and LaForte 5 proved that simple reals locate on every jump class. In this paper, we prove that there is a noncomputable c.e. degree bounding no noncomputable simple reals. Thus, simple reals are not dense in the structure of computably enumerable degrees.


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  1. 1.
    C. Calude, Information Theory and Randomness, An Algorithmic Perspective, Springer-Verlag, Berlin, 1994.Google Scholar
  2. 2.
    C. Calude, P. Hertling, B. Khoussainov, Y. Wang, Recursively enumerable reals and Chaitin’s O number, in STACS’ 98, Springer Lecture Notes in Computer Science, Vol. 1373 (1988), 596–606.CrossRefGoogle Scholar
  3. 3.
    G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press, Cambridge, 1987. (Third Print, 1990).Google Scholar
  4. 4.
    R. Downey, D. Hirschfeldt, A. Nies, Randomness, computability, and density, to appear.Google Scholar
  5. 5.
    R. Downey, G. LaForte, Presentations of computably enumerable reals, Theoretical Computer Science, to appear.Google Scholar
  6. 6.
    A. Kučera, T. Slaman, Randomness and recursive enumerability, SIAM Journal on Computing, to appear.Google Scholar
  7. 7.
    M. Li, P. Vitanyi, Kolmogorov Complexity and Its Applications, Springer-Verlag, New York, Berlin, Heidelberg, 1993.zbMATHGoogle Scholar
  8. 8.
    R.I. Soare, Recursion theory and Dedekind cuts, Transactions of American Mathematical Society 140 (1969), 271–294.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R.I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin 1987.Google Scholar
  10. 10.
    E. Spector, Nicht konstrucktive beweisbare sätze der analysics, Journal of Symbolic Logic 14 (1949), 145–158.CrossRefMathSciNetGoogle Scholar
  11. 11.
    A.M. Turing, On computable numbers with an application to the Entscheidungs problem, Proceedings of American Mathematical Society 42 (1936-7), 230–265; a correction, ibid. 43 (1937), 544-546.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Guohua Wu
    • 1
  1. 1.School of Mathematical and Computing SciencesWellingtonNew Zealand

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