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
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Czech description
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Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
IN - Informatics
OECD FORD branch
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Result continuities
Project
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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
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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