Turing incomparability in Scott sets

Kučera, Antonín; Slaman, Theodore A.

doi:10.1090/s0002-9939-07-08871-5

Cited by 15 publications

(11 citation statements)

References 24 publications

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“…H. Friedman and McAllister independently asked the following question: if S is a Scott set and X ∈ S is not computable, does there necessarily exist a Y ∈ S such that X | T Y ? Kučera and Slaman [10] have recently given a positive answer to this question using Theorem 2.1.…”

Section: Introductionmentioning

confidence: 99%

Using random sets as oracles

Hirschfeldt

Nies

Stephan

2007

Journal of the London Mathematical Society

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Let R be a notion of algorithmic randomness for individual subsets of N. A set B is a base for R randomness if there is a Z T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets, and discuss several consequences of this result. On the other hand, the bases for computable randomness include every Δ 0 2 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, an n-c.e. set is a base for computable randomness if and only if it is Turing incomplete.

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Section: Introductionmentioning

confidence: 99%

Using random sets as oracles

Hirschfeldt

Nies

Stephan

2007

Journal of the London Mathematical Society

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“…K-trivial sets provide an injury-free solution to Post's problem. A more surprising application was given by Kučera and Slaman [25], who used K-trivial sets in their proof that no Scott set of Turing degrees is "hourglass-like". That is, for every noncomputable real in the Scott set there is a Turing incomparable real.…”

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confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

Bull. symb. log

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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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“…For instance, K-trivial reals allow us to solve Post's problem "naturally" (well, reasonably naturally), without the use of a priority argument, or indeed without the use of requirements. Also, Kučera and Slaman [19] have used this class to solve a longstanding question in computable model theory, namely given a noncomputable Y in a Scott set S, there exists in S an element X Turing incomparable with Y .…”

Section: Introductionmentioning

confidence: 99%

The sixth lecture on algorithmic randomness

Downey¹

2009

Logic Colloquium 2006

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This paper follows on from the author's Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on Martin-Löf lowness and triviality. We present a hopefully user-friendly account of the decanter method, and discuss recent results of the author with Peter Cholak and Noam Greenberg concerning the class of strongly jump traceable reals introduced by Figueira, Nies and Stephan.

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Turing incomparability in Scott sets

Abstract: Abstract. For every Scott set F and every nonrecursive set X in F , there is a Y ∈ F such that X and Y are Turing incomparable.

Cited by 15 publications

References 24 publications

Using random sets as oracles

Using random sets as oracles

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

The sixth lecture on algorithmic randomness

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