A Note on the Differences of Computably Enumerable Reals
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.
- [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
- [Coo71]Cooper, S.B.: Degrees of Unsolvability. PhD thesis, Leicester University (1971)Google 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
- [Sol75]Solovay, R.: Handwritten manuscript related to Chaitin’s work. IBM Thomas J. Watson Research Center, Yorktown Heights, 215 pages (1975)Google Scholar