Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

Davie, George

doi:10.1016/j.ipl.2004.06.016

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Layerwise Computability1

Kučera [67] and Gács1

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2023

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Cited by 3 publications

(2 citation statements)

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“…Given a Martin-Löf µ-random sequence x, one can only compute approximations of d µ (x) from below, but no upper bound on it can be computed. The fundamental idea behind the notion of layerwise computability is that having such an upper bound provides much additional information about x that can be used to perform algorithmic tasks that would be impossible without this information (this idea had been exploited earlier by Davie [Dav01,Dav04]).…”

Section: Layerwise Computabilitymentioning

confidence: 99%

Algorithmic Randomness

Hoyrup

2020

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Section: Layerwise Computabilitymentioning

confidence: 99%

Algorithmic Randomness

Hoyrup

2020

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“…Theorem 10.27 (Davie [28]). A set A is 1-random iff there is a constant c such that for each p, if the p-th computable sequence of intervals I 1 , I 2 , .…”

Section: Kučera [67] and Gácsmentioning

confidence: 99%

Calibrating Randomness

Downey¹,

Hirschfeldt²,

Nies³

et al. 2006

Bull. symb. log.

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We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set.

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Randomness Quality and Trade-Offs for CA Random String Generators

Martin

2023

Lecture Notes in Computer Science

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Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

Cited by 3 publications

References 10 publications

Algorithmic Randomness

Algorithmic Randomness

Calibrating Randomness

Randomness Quality and Trade-Offs for CA Random String Generators

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