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

Text
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## Abstract

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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Official URL: | https://www.journals.elsevier.com/journal-of-compu... |

Additional Information: | © 2017 Elsevier Inc. |

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: | 11 Oct 2017 11:46 |

URI: | http://eprints.lse.ac.uk/id/eprint/81819 |

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