Volume 10010 of the series Lecture Notes in Computer Science pp 623632
A Note on the Differences of Computably Enumerable Reals
 George BarmpaliasAffiliated withState Key Lab of Computer Science, Institute of Software, Chinese Academy of SciencesSchool of Mathematics, Statistics and Operations Research, Victoria University of Wellington Email author
 , Andrew LewisPyeAffiliated withDepartment of Mathematics, Columbia House, London School of Economics
Abstract
We show that given any noncomputable leftc.e. real \(\alpha \) there exists a leftc.e. real \(\beta \) such that \(\alpha \ne \beta +\gamma \) for all leftc.e. reals and all rightc.e. reals \(\gamma \). The proof is nonuniform, the dichotomy being whether the given real \(\alpha \) is MartinLöf 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 leftc.e. real. We do not know if there is a uniform proof of this fact.
 Title
 A Note on the Differences of Computably Enumerable Reals
 Book Title
 Computability and Complexity
 Book Subtitle
 Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday
 Pages
 pp 623632
 Copyright
 2017
 DOI
 10.1007/9783319500621_37
 Print ISBN
 9783319500614
 Online ISBN
 9783319500621
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 10010
 Series ISSN
 03029743
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing AG
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 Editors

 Adam Day ^{(13)}
 Michael Fellows ^{(14)}
 Noam Greenberg ^{(15)}
 Bakhadyr Khoussainov ^{(16)}
 Alexander Melnikov ^{(17)}
 Frances Rosamond ^{(18)}
 Editor Affiliations

 13. Victoria University of Wellington
 14. University of Bergen
 15. Victoria University of Wellington
 16. University of Auckland
 17. Massey University
 18. University of Bergen
 Authors

 George Barmpalias ^{(19)} ^{(20)}
 Andrew LewisPye ^{(21)}
 Author Affiliations

 19. State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
 20. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington, New Zealand
 21. Department of Mathematics, Columbia House, London School of Economics, Houghton Street, London, WC2A 2AE, UK
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