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A note on the differences of computably enumerable reals

Barmpalias, George and Lewis-Pye, Andrew (2017) A note on the differences of computably enumerable reals. In: Day, Adam, Fellows, Michael, Greenberg, Noam, Khoussainov, Bakhadyr, Melnikov, Alexander and Rosamond, Frances, (eds.) Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday. Lecture Notes in Computer Science. Springer International (Firm), Cham, Switzerland, pp. 623-632. ISBN 9783319500614

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Identification Number: 10.1007/978-3-319-50062-1_37


We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α≠β+γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.

Item Type: Book Section
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Additional Information: © 2017 Springer International Publishing AG
Divisions: Mathematics
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 13 Jul 2016 16:24
Last Modified: 01 Jun 2024 16:03
Projects: 1000 Talents Program for Young Scholars, President’s International Fellowship Initiative No. 2010Y2GB03, No. 2014CB340302, Marsden grant of New Zealand
Funders: Chinese Government, Chinese Academy of Sciences, China Basic Research Program (973), New Zealand

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