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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)

Abstract

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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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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