K-Triviality, Oberwolfach Randomness, and Differentiability

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F12%3A10126358" target="_blank" >RIV/00216208:11320/12:10126358 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.mfo.de/scientific-programme/publications/owp/2012/OWP2012_17.pdf" target="_blank" >http://www.mfo.de/scientific-programme/publications/owp/2012/OWP2012_17.pdf</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

K-Triviality, Oberwolfach Randomness, and Differentiability

Popis výsledku v původním jazyce

We show that a Martin-Lof random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390-410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.

Název v anglickém jazyce

K-Triviality, Oberwolfach Randomness, and Differentiability

Popis výsledku anglicky

We show that a Martin-Lof random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390-410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

IN - Informatika

OECD FORD obor

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Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2012

Kód důvěrnosti údajů

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

Název periodika

Oberwolfach Preprints

ISSN

1864-7596

e-ISSN

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Svazek periodika

2012

Číslo periodika v rámci svazku

17

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

40

Strana od-do

1-40

Kód UT WoS článku

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EID výsledku v databázi Scopus

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