Upper bounds on ideals in the computably enumerable Turing degrees

Barmpalias, George; Nies, André

doi:10.1016/j.apal.2010.12.005

Cited by 6 publications

(3 citation statements)

References 25 publications

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“…Yates showed that each Σ 0 3 ideal of R T is uniformly generated. In Barmpalias and Nies (2011), it is shown that K forms a Σ 0 3 ideal of R T , and K has a low 2 c.e. upper bound.…”

Section: Upper Boundsmentioning

confidence: 99%

Comparing the Medvedev and Turing degrees of Π⁰₁ classes

Kihara¹

2014

Math. Struct. Comp. Sci.

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Every co-c.e. closed set (Π01 class) in Cantor space is represented by a co-c.e. tree. Our aim is to clarify the interaction between the Medvedev and Muchnik degrees of co-c.e. closed subsets of Cantor space and the Turing degrees of their co-c.e. representations. Among other results, we present the following theorems: if v and w are different c.e. degrees, then the collection of the Medvedev (Muchnik) degrees of all Π01 classes represented by v and the collection represented by w are also different; the ideals generated from such collections are also different; the collections of the Medvedev and Muchnik degrees of all Π01 classes represented by incomplete co-c.e. sets are upward dense; the collection of all Π01 classes represented by K-trivial sets is Medvedev-bounded by a single Π01 class represented by an incomplete co-c.e. set; and the Π01 classes have neither nontrivial infinite suprema nor infima in the Medvedev lattice.

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“…Yates showed that each Σ 0 3 ideal of R T is uniformly generated. In Barmpalias and Nies (2011), it is shown that K forms a Σ 0 3 ideal of R T , and K has a low 2 c.e. upper bound.…”

Section: Upper Boundsmentioning

confidence: 99%

Comparing the Medvedev and Turing degrees of Π⁰₁ classes

Kihara¹

2014

Math. Struct. Comp. Sci.

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“…From the point of view of classical computability theory, the study of the ideal of the K-trivial sequences in the Turing degrees has attracted considerable attention. A number of results about the upper bounds of this ideal where established in [58,23,10], in response to [68,Questions 4.2 and 4.3]. The study of the quotient structure of the c.e.…”

Section: Triviality Notionsmentioning

confidence: 99%

Algorithmic Randomness and Measures of Complexity

Barmpalias¹

2013

Bull. symb. log

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We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

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“…This turns out to be a Σ 0 3 ideal, in the sense that the index set of its members is Σ 0 3 . Moreover by [BN11] it has a c.e. upper bound that is strictly below the degree 0 of the halting problem (moreover, by [KS09] it has a low upper bound b, which means that the halting problem relativized to b has degree 0 ).…”

Section: Introductionmentioning

confidence: 99%

EXACT PAIRS FOR THE IDEAL OF THEK-TRIVIAL SEQUENCES IN THE TURING DEGREES

Barmpalias

Downey

2014

J. symb. log.

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Abstract. The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a K-trivial degree d such that for all degrees a, b which are not K-trivial and a > d, b > d there exists a degree v which is not K-trivial and a > v, b > v. This work sheds light to the question of the definability of the K-trivial degrees in the c.e. degrees.

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Upper bounds on ideals in the computably enumerable Turing degrees

Cited by 6 publications

References 25 publications

Comparing the Medvedev and Turing degrees of Π⁰₁ classes

Comparing the Medvedev and Turing degrees of Π⁰₁ classes

Algorithmic Randomness and Measures of Complexity

EXACT PAIRS FOR THE IDEAL OF THEK-TRIVIAL SEQUENCES IN THE TURING DEGREES

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Upper bounds on ideals in the computably enumerable Turing degrees

Cited by 6 publications

References 25 publications

Comparing the Medvedev and Turing degrees of Π01 classes

Comparing the Medvedev and Turing degrees of Π01 classes

Algorithmic Randomness and Measures of Complexity

EXACT PAIRS FOR THE IDEAL OF THEK-TRIVIAL SEQUENCES IN THE TURING DEGREES

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Product

Resources

About

Comparing the Medvedev and Turing degrees of Π⁰₁ classes

Comparing the Medvedev and Turing degrees of Π⁰₁ classes