Barmpalias, George and Lewis-Pye, Andrew and Li, Angsheng
(2017)
*Pointed computations and Martin-Löf randomnesss.*
Computability.
ISSN 2211-3568
(In Press)

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## Abstract

Schnorr showed that a real X is Martin-Löf random if and only if K(X �n) ≥ n − c for some constant c and all n, where K denotes the prefix-free complexity function. Fortnow (unpublished) and Nies, Stephan and Terwijn [NST05] observed that the condition K(X �n) ≥ n−c can be replaced with K(X �rn ) ≥ rn −c, for any fixed increasing computable sequence (rn), in this characterization. The purpose of this note is to establish the following generalisation of this fact. We show that X is Martin-Löf random if and only if ∃c ∀n K(X �rn ) ≥ rn − c, where (rn) is any fixed pointedly X-computable sequence, in the sense that rn is computable from X in a self-delimiting way, so that at most the first rn bits of X are queried in the computation of rn. On the other hand, we also show that there are reals X which are very far from being Martin-Löf random, but for which there exists some X-computable sequence (rn) such that ∀n K(X �rn ) ≥ rn.

Item Type: | Article |
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Official URL: | http://www.computability.de/journal/ |

Additional Information: | © 2017 IOS Press |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 17 Mar 2017 15:44 |

Last Modified: | 05 Sep 2017 09:14 |

Projects: | D1101130, 2014CB340302 |

Funders: | 1000 Talents Program for Young Scholars, Chinese Academy of Sciences, Institute of Software of the CAS, Royal Society University Research Fellowship, National Basic Science Program (973 Program Group) |

URI: | http://eprints.lse.ac.uk/id/eprint/69854 |

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