## Array Nonrecursive Degrees and Genericity (1996)

Venue: | London Mathematical Society Lecture Notes Series 224 |

Citations: | 26 - 7 self |

### BibTeX

@INPROCEEDINGS{Downey96arraynonrecursive,

author = {Rod Downey and Carl G. Jockusch and Michael Stob},

title = {Array Nonrecursive Degrees and Genericity},

booktitle = {London Mathematical Society Lecture Notes Series 224},

year = {1996},

pages = {93--105},

publisher = {University Press}

}

### Years of Citing Articles

### OpenURL

### 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 (i.e. a 2 GL 2 ) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.).

### Citations

503 |
Recursively enumerable sets and degrees
- Soare
- 1987
(Show Context)
Citation Context ...e the pb-generic sets as a convenient tool to study the cupping properties of a.n.r. degrees. (For background on genericity see, for example, [7] or [9].) Our notation is standard and follows that of =-=[14]-=-. We are indebted to Peter Fejer and Martin Kummer for helpful corrections and suggestions. Our new definition of array nonrecursiveness is based on domination properties of functions. We first recall... |

29 |
Array nonrecursive sets and multiple permitting arguments
- Downey, Jockusch, et al.
- 1990
(Show Context)
Citation Context ...uced 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. 1. Introduction This paper is a sequel to =-=[5]-=-, which was a study of certain recursively enumerable sets called array nonrecursive sets. A number of characterizations of the degrees of these sets were obtained, showing that they are precisely tho... |

24 | Kolmogorov complexity and instance complexity of recursively enumerable sets - Kummer - 1996 |

21 |
Double jumps of minimal degrees
- Posner
- 1978
(Show Context)
Citation Context ...the other hand, we will show that a.n.r. degrees have many of the properties of GL 2 degrees. For example, we will show that every a.n.r. degree a bounds a 1-generic degree (extending Jockusch-Posner =-=[8]-=-) and that every recursive lattice with distinct least and greatest elements can be embedded in D (a) preserving the lattice operations and the least and greatest elements (extending Fejer [6]). Downe... |

21 | Complementation in the Turing degrees - Slaman, Steel - 1989 |

15 | Retraceable sets - Dekker, Myhill - 1958 |

12 |
Degrees of generic sets. In Recursion theory: its generalisation and applications (Proc. Logic Colloq
- Jockusch
- 1979
(Show Context)
Citation Context ...rather than multiple permitting arguments as in [5]. We use the pb-generic sets as a convenient tool to study the cupping properties of a.n.r. degrees. (For background on genericity see, for example, =-=[7]-=- or [9].) Our notation is standard and follows that of [14]. We are indebted to Peter Fejer and Martin Kummer for helpful corrections and suggestions. Our new definition of array nonrecursiveness is b... |

10 | of unsolvability complementary between recusively enumerable degrees - Cooper, Degrees - 1972 |

8 |
The strong anti-cupping property for recursively enumerable degrees
- Cooper
- 1989
(Show Context)
Citation Context ...ontrivially cupped up to all higher degrees. On the other hand, it is not true that every 1-generic degree can be nontrivially cupped up to all higher degrees. This follows from the theorem of Cooper =-=[2]-=- and Slaman-Steel [13, Theorem 3.1] that there are r.e. degrees a; c with 0 ! a ! c such that no degree b ! c satisfies a [ b = c and the fact that every non-zero r.e. degree bounds a 1-generic degree... |

5 | Embedding lattices with top preserved below non-GL 2 degrees
- Fejer
- 1989
(Show Context)
Citation Context ...ich is a.n.r. in the sense of [5] is the sup of a minimal pair of (possibly non-r.e.) degrees. Below, this result is extended to a.n.r. degrees in general and also (applying the construction of Fejer =-=[6]-=-) to a more general class of lattices. (Another extension to a.n.r. degrees in general will be given in Theorem 2.5.) Theorem 2.2. Let a be any a.n.r. degree. Then any recursively presented lattice wi... |

4 |
Randomness and Genericity in the Degrees of Unsolvability, Ph
- Kurtz
(Show Context)
Citation Context ...2 and the fact that 0 0 is a.n.r. that there is a pb-generic degree ds0 0 . Clearly d is not 2-generic, as any 2-generic degree is 1-generic relative to 0 0 and hence not below 0 0 .sWork of S. Kurtz =-=[11]-=- gives further examples of 1-generic degrees which are not pb-generic. Specifically, Kurtz [11, Theorem 4.1] showed that for almost every degree b (in the sense of the measure on degrees induced by th... |

3 |
Array nonrecursive degrees and lattice embeddings of the diamond
- Downey
- 1993
(Show Context)
Citation Context ...and that every recursive lattice with distinct least and greatest elements can be embedded in D (a) preserving the lattice operations and the least and greatest elements (extending Fejer [6]). Downey =-=[4]-=- previously showed that every r.e. degree which is a.n.r. in the sense of [5] is the supremum of a minimal pair of (not necessarily r.e.) degrees, and our result extends this with a much easier proof.... |

2 |
A recursively enumerable degree that is not the top of a diamond in the Turing degrees
- Slaman
(Show Context)
Citation Context ...over ! !! .) The definition of f is quite involved, but makes it clear that f has a natural recursive approximation which changes at most 5 times on each argument. Thus f wtt K, so F wtt K.sT. Slaman =-=[12]-=- proved that there is a nonzero r.e. degree b which is not the sup of a minimal pair of (not necessarily r.e.) degrees. Downey [4, x1] then answered a question raised by Slaman by showing that there i... |

1 |
Degrees of generic sets, This Volume
- Kumabe
(Show Context)
Citation Context ...than multiple permitting arguments as in [5]. We use the pb-generic sets as a convenient tool to study the cupping properties of a.n.r. degrees. (For background on genericity see, for example, [7] or =-=[9]-=-.) Our notation is standard and follows that of [14]. We are indebted to Peter Fejer and Martin Kummer for helpful corrections and suggestions. Our new definition of array nonrecursiveness is based on... |