Martin-Löf random and PA-complete sets

Stephan, Frank; Chatzidakis, Zoé; Koepke, Peter; Pohlers, Wolfram

doi:10.1017/9781316755723.016

Cited by 31 publications

(34 citation statements)

References 8 publications

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“…A distinctive feature of Kučera's work has always been that the theory of Martin-Löf random degrees is developed in parallel to the theory of PA degrees, with the techniques in the two topics being intrinsically connected. A definitive result about the relation between the PA degrees and the Martin-Löf random degrees (extending previous work of Kučera) was shown by Frank Stephan in [Ste06] and says that a PA degree is Martin-Löf random iff it computes the halting problem. As discussed in [Ste06] this result strongly suggests a dichotomy of the Martin-Löf random degrees to the ones which contain a lot of information (they compute the halting problem) and the ones which are computationally weak, in the sense that they are not PA.…”

supporting

confidence: 58%

“…A definitive result about the relation between the PA degrees and the Martin-Löf random degrees (extending previous work of Kučera) was shown by Frank Stephan in [Ste06] and says that a PA degree is Martin-Löf random iff it computes the halting problem. As discussed in [Ste06] this result strongly suggests a dichotomy of the Martin-Löf random degrees to the ones which contain a lot of information (they compute the halting problem) and the ones which are computationally weak, in the sense that they are not PA. In Section 2 we reveal another connection between these classes of degrees: every PA degree is the least upper bound of two Martin-Löf random degrees.…”

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confidence: 58%

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The importance of Π₁⁰ classes in effective randomness

Barmpalias

Lewis²,

2010

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confidence: 58%

supporting

confidence: 58%

The importance of Π₁⁰ classes in effective randomness

Barmpalias

Lewis²,

2010

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“…However, this is only a partial answer to the question. A theorem of Stephan's [34] establishes a dichotomy between two kinds of random sets. On the one hand, those randoms that compute H 1 , the complete ones, pass all the relevant statistical tests (the effective null classes) not because they are in some way typical, but because their strong information content allows them mimic typical sets.…”

mentioning

confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

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Abstract. Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute H 1 .A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turing degrees. At some level of randomness all interesting interactions cease. Computably enumerable sets are each definable and, in a sense, they are as effective as any noncomputable set can be. As a consequence, the lower and upper cones of noncomputable c.e. sets are definable null sets, and thus if a set is "sufficiently" random, it cannot compute, nor be computed by, a noncomputable c.e. set. However, the most studied notion of algorithmic randomness, Martin-Löf randomness, is not strong enough to support this argument, and in fact, significant interactions between Martin-Löf random sets and c.e. sets occur. The study of these interactions has lead to a number of surprising results that show a remarkably robust relationship between Martin-Löf random sets and the class of K-trivial sets. Interestingly, the significant interaction occurs "at the boundaries": the Martin-Löf random sets in question are close to being non-random (in that they fail fairly simple statistical tests), and K-trivial c.e. sets are close to being computable.The following theorem resolves one of the main open questions in algorithmic randomness, and further strengthens the relationship between the Martin-Löf random sets and the K-trivial sets. Theorem 1. There is an incomplete Martin-Löf random set that computes every K-trivial set.This theorem is essentially a corollary of two recent results, both proved in 2012: the first by Bienvenu, Greenberg, Kučera, Nies and Turetsky [3]; and the second by Day and Miller [10]. In the remainder of this announcement, we will explain the background to the problem behind this theorem, and indicate the main ideas used in the proof.In this announcement, by "random" we will henceforth mean Martin-Löf random. We will give the full definition shortly, but essentially, an element X of Cantor space is Martin-Löf random if it is not an element of a particular kind of effectively

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“…Stephan proved that any Martin-Löf random degree that is also a PA degree must Turing compute ∅ [21]. (Recall that a degree is PA if it computes a complete extension of Peano arithmetic.)…”

Section: Definition 12 ([1])mentioning

confidence: 99%

Difference randomness

Franklin

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper, we define new notions of randomness based on the difference hierarchy. We consider various ways in which a real can avoid all effectively given tests consisting of n-r.e. sets for some given n. In each case, the n-r.e. randomness hierarchy collapses for n ≥ 2. In one case, we call the resulting notion difference randomness and show that it results in a class of random reals that is a strict subclass of the Martin-Löf random reals and a proper superclass of both the Demuth random and weakly 2-random reals. In particular, we are able to characterize the difference random reals as the Turing incomplete Martin-Löf random reals. We also provide a martingale characterization for difference randomness.

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Martin-Löf random and PA-complete sets

Cited by 31 publications

References 8 publications

The importance of Π₁⁰ classes in effective randomness

The importance of Π₁⁰ classes in effective randomness

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Difference randomness

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Martin-Löf random and PA-complete sets

Cited by 31 publications

References 8 publications

The importance of Π10 classes in effective randomness

The importance of Π10 classes in effective randomness

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Difference randomness

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The importance of Π₁⁰ classes in effective randomness

The importance of Π₁⁰ classes in effective randomness