A Note on the Differences of Computably Enumerable Reals

Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

References

  1. [ASWZ00]
    Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complex. 16(4), 676–690 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. [BDG10]
    Barmpalias, G., Downey, R., Greenberg, N.: Working with strong reducibilities above totally \(\omega \)-c.e. and array computable degrees. Trans. Am. Math. Soc. 362(2), 777–813 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. [Ben88]
    Bennett, C.H.: Logical depth and physical complexity. In: The Universal Turing Machine: A Half-Century Survey, pp. 227–257. Oxford University Press (1988)Google Scholar
  4. [BLP16]
    Barmpalias, G., Lewis-Pye, A.: Differences of halting probabilities. arXiv:1604.00216 [cs.CC], April 2016
  5. [CHKW01]
    Calude, C., Hertling, P., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin \(\Omega \) numbers. Theor. Comput. Sci. 255(1–2), 125–149 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. [CN97]
    Calude, C., Nies, A.: Chaitin \(\Omega \) numbers, strong reducibilities. J. UCS 3(11), 1162–1166 (1997)MathSciNetMATHGoogle Scholar
  7. [Coo71]
    Cooper, S.B.: Degrees of Unsolvability. PhD thesis, Leicester University (1971)Google Scholar
  8. [Dem75]
    Demuth, O.: On constructive pseudonumbers. Comment. Math. Univ. Carolinae 16, 315–331 (1975). In RussianMathSciNetGoogle Scholar
  9. [DH10]
    Downey, R.G., Hirshfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefGoogle Scholar
  10. [DHN02]
    Downey, R.G., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31, 1169–1183 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. [DWZ04]
    Downey, R.G., Guohua, W., Zheng, X.: Degrees of d.c.e. reals. Math. Log. Q. 50(4–5), 345–350 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. [FSW06]
    Figueira, S., Stephan, F., Guohua, W.: Randomness and universal machines. J. Complex. 22(6), 738–751 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. [KS01]
    Kučera, A., Slaman, T.: Randomness and recursive enumerability. SIAM J. Comput. 31(1), 199–211 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. [LV97]
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Graduate Texts in Computer Science, 2nd edn. Springer, New York (1997)CrossRefMATHGoogle Scholar
  15. [Ng06]
    Ng, K.-M.: Some properties of d.c.e. reals and their Degrees. M.Sc. thesis, National University of Singapore (2006)Google Scholar
  16. [Nie09]
    Nies, A.: Computability and Randomness. Oxford University Press, 444 pp. (2009)Google Scholar
  17. [Rai05]
    Raichev, A.: Relative randomness and real closed fields. J. Symb. Log. 70(1), 319–330 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. [Sol75]
    Solovay, R.: Handwritten manuscript related to Chaitin’s work. IBM Thomas J. Watson Research Center, Yorktown Heights, 215 pages (1975)Google Scholar

Copyright information

© Springer International Publishing AG 2017

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

Personalised recommendations