Strong jump-traceability II: K-triviality

Downey, Rodney G.; Greenberg, Noam

doi:10.1007/s11856-011-0217-z

Cited by 10 publications

(7 citation statements)

References 21 publications

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“…K-trivial set is not strongly jump traceable. In later papers such as [3,10], strong jump traceability was studied in great depth. For general background, see Section 10.13 of the excellent book [4], and also Section 8.5 of [22].…”

Section: Definition 23 ( [22]mentioning

confidence: 99%

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Calibrating the complexity of $\Delta^{0}_{2}$ sets via their changes

Nies

2013

Proceedings of the 12th Asian Logic Conference

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The computational complexity of a ∆ 0 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Löf random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate.

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Section: Definition 23 ( [22]mentioning

confidence: 99%

“…The purpose of this work is to provide a unifying background for results in the papers [3,[6][7][8]10,11,13]. It contains many new observations on the amount of changes of knights and peasants, and how they relate.…”

Section: Introductionmentioning

confidence: 99%

Calibrating the complexity of $\Delta^{0}_{2}$ sets via their changes

Nies

2013

Proceedings of the 12th Asian Logic Conference

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“…A much weaker notion is jump traceability, where one merely requires that there is a c.e. trace for J A with some computable bound h. There is a perfect class of sets that are jump traceable as shown in [37, 8.4.4], while each SJT set is ∆ 0 2 by [7].…”

Section: Strongly Jump Traceable Setsmentioning

confidence: 99%

“…sets SJT is more complicated than the K-trivials as a class, even though its members are closer to being computable. Recent research of Downey and Greenberg [7] shows that in fact each SJT set (c.e. or not) is K-trivial.…”

Section: Superlowmentioning

confidence: 99%

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Interactions of Computability and Randomness

Nies

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Martin–Löf reducibility and cost functions

Greenberg,

Miller,

Nies

et al. 2023

Isr. J. Math.

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Martin—Löf (ML)-reducibility compares the complexity of K-trivial sets of natural numbers by examining the Martin—Löf random sequences that compute them. One says that a K-trivial set A is ML-reducible to a K-trivial set B if every ML-random computing B also computes A. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets intended to capture ML-random sequences. We show that for every cost function there is a c.e. set that is ML-complete among the sets obeying it. We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω given by a computable set of bit positions. This leads to a new characterisation of strong jump-traceability.

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Strong jump-traceability II: K-triviality

Abstract: Abstract. We show that every strongly jump-traceable set is K-trivial. Unlike other results, we do not assume that the sets in question are computably enumerable.

Cited by 10 publications

References 21 publications

Calibrating the complexity of $\Delta^{0}_{2}$ sets via their changes

Calibrating the complexity of $\Delta^{0}_{2}$ sets via their changes

Interactions of Computability and Randomness

Martin–Löf reducibility and cost functions

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