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Differences of halting probabilities

Barmpalias, George and Lewis-Pye, Andrew (2017) Differences of halting probabilities. Journal of Computer and System Sciences, 89. pp. 349-360. ISSN 0022-0000

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Identification Number: 10.1016/j.jcss.2017.06.002


We study the differences of Martin-Löf random left-c.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 Martin-Löf random left-c.e. real for each positive rational q>r and a Martin-Löf random right-c.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 Martin-Löf random left-c.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 prefix-free 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 Martin-Löf 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
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Additional Information: © 2017 Elsevier Inc.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 20 Jun 2017 13:28
Last Modified: 22 May 2024 07:42

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