Comparing DNR and WWKL

Ambos-Spies, Klaus; Kjos-Hanssen, Bjørn; Lempp, Steffen; Slaman, Theodore A.

doi:10.2178/jsl/1102022212

Cited by 48 publications

(182 citation statements)

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“…But then, since ψ( e, n ) ↓ and ϕ (1),B e extends ψ, it follows that ϕ (1),B e,s ( e, n ) ↑, i.e., f (n) > s. We now see that, for all but finitely many n, if n ∈ 0 ′ then n ∈ 0 ′ f (n) . Thus 0 ′ ≤ T f ≤ T B.…”

Section: A Theorem Of Niesmentioning

confidence: 67%

“…This proves the lemma. For refinements, see Jockusch [14,Proposition 3] and Ambos-Spies/Kjos-Hanssen/Lempp/Slaman [1] and Simpson [31].…”

Section: Some Mass Problem Inequalitiesmentioning

confidence: 99%

“…We can find a random B < T 0 ′ such that 0 ′ is low-forrandom relative to B. r e,n,i = µ({X | ϕ (1),X e,s ( e, n ) ≃ i}) and define ψ( e, n ) ≃ i chosen so as to minimize r e,n,i . Note that ψ is partial recursive, and ψ( e, n ) ↓ if and only if n ∈ 0 ′ .…”

Section: A Theorem Of Niesmentioning

confidence: 99%

“…Define f (n) = least s such that ϕ (1),B e,s ( e, n ) ↓. Since ϕ (1),B e is total, f is total and ≤ T B.…”

Section: A Theorem Of Niesmentioning

confidence: 99%

“…Define f (n) = least s such that ϕ (1),B e,s ( e, n ) ↓. Since ϕ (1),B e is total, f is total and ≤ T B. Since B is random, it follows by Solovay's Lemma that B / ∈ V e,n for all but finitely many n. In other words, for all but finitely many n, if n ∈ 0…”

Section: A Theorem Of Niesmentioning

confidence: 99%

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Mass problems and almost everywhere domination

Simpson

2007

Mathematical Logic Qtrly

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We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR × AED, MLR ∩ AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω . Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1, 1) and inf(b2, 1) and inf(b3, 1) belong to Pw and that 0 < inf(b1, 1) < inf(b2, 1) < inf(b3, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1, 1) and inf(b3, 1) but not inf(b2, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′ . In order to make this paper more self-contained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.

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Section: A Theorem Of Niesmentioning

confidence: 67%

“…This proves the lemma. For refinements, see Jockusch [14,Proposition 3] and Ambos-Spies/Kjos-Hanssen/Lempp/Slaman [1] and Simpson [31].…”

Section: Some Mass Problem Inequalitiesmentioning

confidence: 99%

Section: A Theorem Of Niesmentioning

confidence: 99%

“…Define f (n) = least s such that ϕ (1),B e,s ( e, n ) ↓. Since ϕ (1),B e is total, f is total and ≤ T B.…”

Section: A Theorem Of Niesmentioning

confidence: 99%

Section: A Theorem Of Niesmentioning

confidence: 99%

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Mass problems and almost everywhere domination

Simpson

2007

Mathematical Logic Qtrly

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Separating principles below

Flood

Towsner

2016

Math. Log. Quart.

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In this paper, we study the Ramsey‐type weak Kőnig's Lemma, written sans-serifRWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an ω‐model satisfying one principle T1 but not another T2. The technique often allows one to translate a “one step” construction (building an instance of T2 along with a collection of solutions to each computable instance of T1) into an ω‐model separation (building a computable instance of T2 together with a Turing ideal where T1 holds but this instance has no solution). We illustrate this translation by separating d-DNR from sans-serifDNR (reproving a result of Ambos‐Spies, Kjos‐Hanssen, Lempp, and Slaman), and then apply this technique to separate sans-serifRWKL from sans-serifDNR (which has been shown separately by Bienvenu, Patey, and Shafer).

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Kolmogorov Complexity and the Recursion Theorem

2006

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Abstract. Several classes of diagonally non-recursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0, 1}-valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal C-complexity among the strings of length n. A ≥ T K iff A can compute a function F such that F (n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem.

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Comparing DNR and WWKL

Abstract: Abstract. In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak König's Lemma).

Cited by 48 publications

References 5 publications

Mass problems and almost everywhere domination

Mass problems and almost everywhere domination

Separating principles below

Kolmogorov Complexity and the Recursion Theorem

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