Relative enumerability in the difference hierarchy

Arslanov, M. M.; LaForte, Geoffrey; Slaman, Theodore A.

doi:10.2307/2586839

Cited by 24 publications

(17 citation statements)

References 10 publications

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“…We will construct ω-c. e. sets so that they are c. e. in Lachlan's set M for L. The theorem follows from the following result from [2]. Proposition 1 (Arslanov, LaForte and Slaman [2]) .…”

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There is no low maximal d. c. e. degree – Corrigendum

Arslanov

Cooper

2004

Mathematical Logic Qtrly

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The purpose of this short paper is to clarify and correct the proof of the main result contained in [1]: namely, that there exists no low maximal d. c. e. degree. There we gave a simple proof obtained as an immediate corollary of the following posited extension of the Robinson Splitting Theorem ([1, Theorem 1.7]: For any c. e. set A,Denis Hirschfeldt (private communication) was the first to notice a problem with the particular application of the Recursion Theorem in the proof of this result, one which does not occur in the original Robinson proof [5].We present below the following reformulation of the use of the Recursion Theorem sufficient to correct the proof of our main result (the non-existence of a low maximal d. c. e. 1) ), via a degree-theoretic extension of the Robinson Splitting Theorem:Theorem For any d. c. e. degree l, any c. e. degree a, if l is low and l < a, then there are d. c. e. degrees a 0 , a 1 such that l < a 0 , a 1 < a and a 0 ∪ a 1 = a. P r o o f. Let L be a d. c. e. set of low degree. Let L = L 0 − L 1 for some c. e. sets L 0 , L 1 such that L 0 ⊃ L 1 . Let f be a 1 − 1 computable function such that L 0 = {f (x) : x ∈ ω}, and M = f −1 (L 1 ). Then M is c. e. and M ≤ T L. (M is called Lachlan's set for L.)Given a c. e. set A, and a d. c. e. set L, assume that L < T A and L has low degree. First we construct ω-c. e. sets A 0 , A 1 to satisfy the following requirements:where i = 0, 1, e ∈ ω, and {Φ e : e ∈ ω} is a standard list of all partial computable (p. c.) functionals Φ. Let a = deg T (A), l = deg T (L) and a i = deg T (A i ⊕ L) for i = 0, 1. Then a i is an ω-c. e. degree, for each i = 0, 1. By the R-requirement, a 0 ∪ a 1 = a, and by the S-requirements, l ≤ a i < a, i ∈ {0, 1}. Then by the S-and R-requirements we will have l < a i .We will construct ω-c. e. sets so that they are c. e. in Lachlan's set M for L. The theorem follows from the following result from [2].Proposition 1 (Arslanov, LaForte and Slaman [2]) . Let A and C be sets such that C is c. e., A is c. e. in C, C ≤ T A, and A is ω-c. e. Then deg(A) is d.

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“…We will construct ω-c. e. sets so that they are c. e. in Lachlan's set M for L. The theorem follows from the following result from [2]. Proposition 1 (Arslanov, LaForte and Slaman [2]) .…”

mentioning

confidence: 89%

“…Proposition 1 (Arslanov, LaForte and Slaman [2]) . Let A and C be sets such that C is c. e., A is c. e. in C, C ≤ T A, and A is ω-c. e. Then deg(A) is d. c. e.…”

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confidence: 99%

There is no low maximal d. c. e. degree – Corrigendum

Arslanov

Cooper

2004

Mathematical Logic Qtrly

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“…But this is not true: Theorem 3.29. (Arslanov, LaForte, and Slaman [12]) There exists a dc.e. set D such that, for every n 3, there exists a set A n which is simultaneously D-CEA and (n + 1)-c.e., yet fails to be of n-c.e.…”

Section: 22mentioning

confidence: 99%

“…(Cooper and Yi [25] for n = 2; Arslanov, LaForte, and Slaman [12] for n > 2) For any c.e. degree x and any n-c.e.…”

Section: Soare and Stobmentioning

confidence: 99%

The Ershov Hierarchy

Arslanov

2011

Computability in Context

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“…множеств в случае n < ω. В работах [4] и [5] была установлена интересная связь понятий n-вычислимой перечислимости и относительной перечислимости, а именно, была доказана следующая теорема. иерархии Ершова в этом случае.…”

Section: математические заметкиunclassified

Относительная Перечислимость В Иерархии Ершова

Batyrshin¹

2008

Mat. Zametki

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Relative enumerability in the difference hierarchy

Abstract: We show that the intersection of the class of 2-REA degrees with that of the ω-r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy.

Cited by 24 publications

References 10 publications

There is no low maximal d. c. e. degree – Corrigendum

There is no low maximal d. c. e. degree – Corrigendum

The Ershov Hierarchy

Относительная Перечислимость В Иерархии Ершова

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