# Coherent randomness tests and computing the $K$-trivial sets

Laurent Bienvenu; Noam Greenberg; Antonín Kučera; André Nies; Dan Turetsky

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 4, page 773-812
- ISSN: 1435-9855

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topBienvenu, Laurent, et al. "Coherent randomness tests and computing the $K$-trivial sets." Journal of the European Mathematical Society 018.4 (2016): 773-812. <http://eudml.org/doc/277230>.

@article{Bienvenu2016,

abstract = {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 that 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.},

author = {Bienvenu, Laurent, Greenberg, Noam, Kučera, Antonín, Nies, André, Turetsky, Dan},

journal = {Journal of the European Mathematical Society},

keywords = {coherent randomness tests; $K$-trivial sets; coherent randomness tests; -trivial sets},

language = {eng},

number = {4},

pages = {773-812},

publisher = {European Mathematical Society Publishing House},

title = {Coherent randomness tests and computing the $K$-trivial sets},

url = {http://eudml.org/doc/277230},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Bienvenu, Laurent

AU - Greenberg, Noam

AU - Kučera, Antonín

AU - Nies, André

AU - Turetsky, Dan

TI - Coherent randomness tests and computing the $K$-trivial sets

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 4

SP - 773

EP - 812

AB - 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 that 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.

LA - eng

KW - coherent randomness tests; $K$-trivial sets; coherent randomness tests; -trivial sets

UR - http://eudml.org/doc/277230

ER -

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