Book Chapter10.1017/CBO9780511629167.005
Array nonrecursive degrees and genericity
Rodney G. Downey,Carl G. Jockusch,Michael Stob +2 more
- 01 Apr 1996
- pp 93-104
69
TL;DR: It is shown that the upward closure of the pb-generic degrees is the set of a.n.r. degrees, which is intermediate between 1-genericity and 2-Genericity, and thus to certain low degrees.
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Abstract: A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a 00 > (a 0 0) 0 (i.e. a 2 GL 2) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.). Many results for GL 2 degrees extend to the a.n.r. degrees and thus to certain low degrees. A new notion of genericity (called pb-genericity) is introduced which is intermediate between 1-genericity and 2-genericity. It is shown that the upward closure of the pb-generic degrees is the set of a.n.r. degrees.
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Citations
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Partition Theorems and Computability Theory
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Lowness for weakly 1-generic and kurtz-random
Frank Stephan,Liang Yu +1 more
- 15 May 2006
TL;DR: It is shown that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree, which answers negatively a recent question on the characterization of these sets.
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Mass problems and hyperarithmeticity
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Automorphisms of the lattice of Π₁⁰ classes; perfect thin classes and anc degrees
TL;DR: In particular, the authors showed that the collection of perfect thin classes (a notion which is definable in the lattice of Π 0 1 classes) forms an orbit in the topology of the hyperhypersimplicity, and a degree is anc iff it contains a perfect thin class.
References
Array nonrecursive sets and multiple permitting arguments
Rodney G. Downey,Carl G. Jockusch,Michael Stob +2 more
- 01 Jan 1990
TL;DR: A class of r.
79
Kolmogorov Complexity and Instance Complexity of Recursively Enumerable Sets
TL;DR: In this paper, the Kolmogorov complexity and instance complexity of recursively enumerable (re) sets are studied and the Turing degrees of sets which attain this bound are characterized.
57
Complementation in the Turing degrees
Theodore A. Slaman,John R. Steel +1 more
TL;DR: It is shown that a 1-generic complement for each set of degree between 0 and 0′ can be found uniformly and just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above ∅′.
51
The strong anticupping property for recursively enumerable degrees
TL;DR: The recent paper by Ambos-Spies, Jockusch, Shore and Soare describes a general theoretical framework for cupping and capping below 0' which seems likely to be useful in a wider context and extends Theorem 2 of [3], proved using a finite injury construction in 9(< 0').
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