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A Note on the Differences of Computably Enumerable Reals

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Computability and Complexity

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

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.

Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, and the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative No. 2010Y2GB03. Additional support was received by the CAS and the Institute of Software of the CAS. Partial support was also received from a Marsden grant of New Zealand and the China Basic Research Program (973) grant No. 2014CB340302. Lewis-Pye was supported by a Royal Society University Research Fellowship.

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Authors and Affiliations

  1. State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    George Barmpalias

  2. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington, New Zealand

    George Barmpalias

  3. Department of Mathematics, Columbia House, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Andrew Lewis-Pye

Authors
  1. George Barmpalias
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  2. Andrew Lewis-Pye
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Correspondence to George Barmpalias .

Editor information

Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

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Barmpalias, G., Lewis-Pye, A. (2017). A Note on the Differences of Computably Enumerable Reals. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_37

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  • DOI: https://doi.org/10.1007/978-3-319-50062-1_37

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

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