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Coherent randomness tests and computing the K-trivial sets

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10325909" target="_blank" >RIV/00216208:11320/16:10325909 - isvavai.cz</a>

  • Result on the web

    <a href="http://dx.doi.org/10.4171/JEMS/602" target="_blank" >http://dx.doi.org/10.4171/JEMS/602</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.4171/JEMS/602" target="_blank" >10.4171/JEMS/602</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Coherent randomness tests and computing the K-trivial sets

  • Original language description

    We introduce Oberwolfach randomness, a notion within Demuth's framework of statistical tests with moving components; here the components' movement has to be coherent across levels. We show that a ML-random set computes all K-trivial sets if and only if it is not Oberwolfach random, and indeed there is a K-trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis, such as the Lebesgue density theorem and Doob's martingale convergence theorem. We also show that random sets which are not Oberwolfach random satisfy highness properties (such as LR-hardness) which mean they are close to computing the halting problem. A consequence of these results is that a ML-random set failing the effective version of Lebesgue's density theorem for closed sets must compute all K-trivial sets. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem of algorithmic randomness. On the other hand these results settle stronger variants of the covering problem in the negative: no low ML-random set computes all K-trivial sets, and not every K-trivial set is computable from both halves of a random set.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

  • CEP classification

    IN - Informatics

  • OECD FORD branch

Result continuities

  • Project

  • Continuities

    I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Others

  • Publication year

    2016

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Journal of the European Mathematical Society

  • ISSN

    1435-9855

  • e-ISSN

  • Volume of the periodical

    18

  • Issue of the periodical within the volume

    4

  • Country of publishing house

    DE - GERMANY

  • Number of pages

    40

  • Pages from-to

    773-812

  • UT code for WoS article

    000371990000003

  • EID of the result in the Scopus database

    2-s2.0-84961839155