Barmpalias, George and LewisPye, Andrew (2017) Differences of halting probabilities. Journal of Computer and System Sciences, 89. pp. 349360. ISSN 00220000

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Abstract
We study the differences of MartinLöf random leftc.e. reals and show that for each pair of such reals α,β there exists a unique number r>0 such that qα−β is a MartinLöf random leftc.e. real for each positive rational q>r and a MartinLöf random rightc.e. real for each positive rational q<r. Based on this result we develop a theory of differences of halting probabilities, which answers a number of questions about MartinLöf random leftc.e. reals, including one of the few remaining open problems from the list of open questions in algorithmic randomness [21]. The halting probability of a prefixfree machine M restricted to a set X is the probability that the machine halts and outputs an element of X . Becher, Figueira, Grigorieff, and Miller asked whether ΩU(X) is MartinLöf random when U is universal and X is a View the MathML source set. We apply our theory of differences of halting probabilities to give a positive answer.
Item Type:  Article 

Official URL:  https://www.journals.elsevier.com/journalofcompu... 
Additional Information:  © 2017 Elsevier Inc. 
Divisions:  Mathematics 
Subjects:  Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science 
Sets:  Departments > Mathematics 
Date Deposited:  20 Jun 2017 13:28 
Last Modified:  13 Jun 2018 23:00 
URI:  http://eprints.lse.ac.uk/id/eprint/81819 
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