Computability and Complexity pp 623-632 | Cite as

# A Note on the Differences of Computably Enumerable Reals

Chapter

First Online:

## 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

- [ASWZ00]Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complex.
**16**(4), 676–690 (2000)MathSciNetCrossRefMATHGoogle Scholar - [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 - [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
- [BLP16]Barmpalias, G., Lewis-Pye, A.: Differences of halting probabilities. arXiv:1604.00216 [cs.CC], April 2016
- [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 - [CN97]Calude, C., Nies, A.: Chaitin \(\Omega \) numbers, strong reducibilities. J. UCS
**3**(11), 1162–1166 (1997)MathSciNetMATHGoogle Scholar - [Coo71]Cooper, S.B.: Degrees of Unsolvability. PhD thesis, Leicester University (1971)Google Scholar
- [Dem75]Demuth, O.: On constructive pseudonumbers. Comment. Math. Univ. Carolinae
**16**, 315–331 (1975). In RussianMathSciNetGoogle Scholar - [DH10]Downey, R.G., Hirshfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefGoogle Scholar
- [DHN02]Downey, R.G., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput.
**31**, 1169–1183 (2002)MathSciNetCrossRefMATHGoogle Scholar - [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 - [FSW06]Figueira, S., Stephan, F., Guohua, W.: Randomness and universal machines. J. Complex.
**22**(6), 738–751 (2006)MathSciNetCrossRefMATHGoogle Scholar - [KS01]Kučera, A., Slaman, T.: Randomness and recursive enumerability. SIAM J. Comput.
**31**(1), 199–211 (2001)MathSciNetCrossRefMATHGoogle Scholar - [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
- [Ng06]Ng, K.-M.: Some properties of d.c.e. reals and their Degrees. M.Sc. thesis, National University of Singapore (2006)Google Scholar
- [Nie09]Nies, A.: Computability and Randomness. Oxford University Press, 444 pp. (2009)Google Scholar
- [Rai05]Raichev, A.: Relative randomness and real closed fields. J. Symb. Log.
**70**(1), 319–330 (2005)MathSciNetCrossRefMATHGoogle Scholar - [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