A Note on the Differences of Computably Enumerable Reals

  • George Barmpalias
  • Andrew Lewis-Pye
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)


We show that given any non-computable left-c.e. real \(\alpha \) there exists a left-c.e. real \(\beta \) such that \(\alpha \ne \beta +\gamma \) for all left-c.e. reals and all right-c.e. reals \(\gamma \). The proof is non-uniform, the dichotomy being whether the given real \(\alpha \) is Martin-Löf  random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability \(\Omega _U\) of U is not a translation of the halting probability \(\Omega _V\) of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.


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Authors and Affiliations

  1. 1.State Key Lab of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematics, Statistics and Operations ResearchVictoria University of WellingtonWellingtonNew Zealand
  3. 3.Department of MathematicsColumbia House, London School of EconomicsLondonUK

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