Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
An introduction to Kolmogorov complexity and its applications
An introduction to Kolmogorov complexity and its applications
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Randomness and Recursive Enumerability
SIAM Journal on Computing
Presentations of computably enumerable reals
Theoretical Computer Science
Recursively Enumerable Reals and Chaitin Omega Numbers
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
Presentations of K-trivial reals and kolmogorov complexity
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
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We study prefix-free presentations of computably enumerable reals. In [2], Calude et. al. proved that a real α is c.e. if and only if there is an infinite, computably enumerable prefix-free set V such that α = Σσ∈V 2-|σ|. Following Downey and LaForte [5], we call V a prefixfree presentation of α. Each computably enumerable real has a computable presentation. Say that a c.e. real α is simple if each presentation of α 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.