Computability and Complexity

Volume 10010 of the series Lecture Notes in Computer Science pp 623-632


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 Lewis-PyeAffiliated withDepartment of Mathematics, Columbia House, London School of Economics

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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.