## A random degree with strong minimal cover (2006)

Venue: | the Bulletin of the London Mathematical Society |

Citations: | 4 - 4 self |

### BibTeX

@INPROCEEDINGS{Lewis06arandom,

author = {Andrew E. M. Lewis},

title = {A random degree with strong minimal cover},

booktitle = {the Bulletin of the London Mathematical Society},

year = {2006}

}

### OpenURL

### Abstract

We show that there exists a Martin-Löf random degree which has a strong minimal cover. 1.

### Citations

473 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
(Show Context)
Citation Context ...TRONG MINIMAL COVER Page 3 of 9 In what follows all notation and terminology will be standard unless explicitly stated otherwise. For background in computability theory we refer the reader to [2] and =-=[8]-=-. 2. ProofofTheorem1.2 We consider the Cantor space 2 ω and denote the standard measure on 2 ω by μ. Λ⊆ 2 <ω is said to be downward closed if, whenever τ ∈ Λ, all initial segments of τ are in this set... |

93 | Computability and Randomness
- Nies
- 2009
(Show Context)
Citation Context ...y. Certainly if a computably enumerable (c.e.) set A is not ML-cuppable then it is K-trivial and Nies has shown that there exist non-computable c.e. sets which are not ML-cuppable. Theorem 1.1. (Nies =-=[AN]-=-) Let Y ∈ ∆0 2 be Martin-Löf random. There exists a non-computable c.e. (in fact promptly simple) set A such that for each Martin-Löf random set Z: Y ≤T A ⊕ Z ⇒ Y ≤T Z. Barmpalias [GB] has shown that ... |

28 |
classes and complete extensions of
- Kucera, Measure
- 1985
(Show Context)
Citation Context ...ctions of sets of minimal and of hyperimmune-free degree, perhaps the strongest result on the negative side of Yates’ question is the result of Kučera that the PA degrees satisfy the cupping property =-=[4]-=-, implying that there exists a hyperimmune-free degree with no strong minimal cover. Since all PA degrees are FPF, this theorem combines with Theorem 1.3 to suggest the following question. Question 1.... |

18 |
Π 0 1 -classes and complete extensions of PA, in: Recursion Theory Week (Oberwolfach
- Kučera, Measure
- 1984
(Show Context)
Citation Context ...ions of sets of minimal and of hyperimmune-free degree, perhaps the strongest result on the negative side of Yates’ question is the result of Kučera’s that the PA degrees satisfy the cupping property =-=[AK]-=-, implying that there exists a hyperimmune-free degree with no strong minimal cover. Since all PA degrees are FPF, this theorem combines with theorem 1.3 to suggest the following question.sA RANDOM DE... |

8 |
Computability Theory, Chapman
- Cooper
- 2004
(Show Context)
Citation Context ...E WITH STRONG MINIMAL COVER Page 3 of 9 In what follows all notation and terminology will be standard unless explicitly stated otherwise. For background in computability theory we refer the reader to =-=[2]-=- and [8]. 2. ProofofTheorem1.2 We consider the Cantor space 2 ω and denote the standard measure on 2 ω by μ. Λ⊆ 2 <ω is said to be downward closed if, whenever τ ∈ Λ, all initial segments of τ are in ... |

8 |
A fixed point free minimal degree
- Kumabe, Lewis
(Show Context)
Citation Context ... 3 Question 1.2. Do all FPF degrees satisfy the cupping property, or do they at least fail to have a strong minimal cover? A positive answer would have been very exciting. Given Kumabe’s construction =-=[MK]-=- of a FPF minimal degree we would then have a negative solution to Yates’ question together with a complete characterization of the hyperimmune-free degrees which have a strong minimal cover as those ... |

6 | A basis theorem for Π0 1 classes of positive measure and jump inversion for random reals
- Downey, Miller
(Show Context)
Citation Context ... have to consider in defining Pj+1 is that, for some A ∈Pj, itmaybe the case that Ψj(A) is partial. It is a familiar technique in dealing with Π 0 1 classes of positive measure, see for example [4] or=-=[3]-=-, that we may take a Π 0 1 subclass P ′ j ⊆Pj which is of positive measure and such that the intersection of P ′ j with any other Π0 1 class is either empty or is of positive measure.sA RANDOM DEGREE ... |

6 |
Π0 1 classes, strong minimal covers and hyperimmune-free degrees
- Lewis
(Show Context)
Citation Context ...degrees below and including a. Theorem 1.2. There exists a Martin-Löf random degree which has a strong minimal cover. This paper may also be seen as a continuation of the investigation carried out in =-=[6]-=- regarding an old question of Yates. Question 1.1 (Yates). Does every minimal degree have a strong minimal cover? Recall that A ⊆ ω is of hyperimmune-free degree if for every f �T A there exists a com... |

5 | Random non-cupping revisited
- Barmpalias
(Show Context)
Citation Context ...7]). Let Y ∈ Δ 0 2 be Martin-Löf random. There exists a noncomputable computably enumerable (in fact promptly simple) set A such that for each Martin-Löf random set Z, Y �T A ⊕ Z ⇒ Y �T Z. Barmpalias =-=[1]-=- has shown that in the statement of Theorem 1.1 we may remove the condition that Y should be Martin-Löf random. We call a degree a Martin-Löf random if it contains a set which is Martin-Löf random. Le... |

4 |
A fixed point free minimal degree’, unpublished
- Kumabe
- 1993
(Show Context)
Citation Context ...n. Question 1.2. Do all FPF degrees satisfy the cupping property, or do they at least fail to have a strong minimal cover? A positive answer would have been very exciting. Given Kumabe’s construction =-=[5]-=- ofan FPF minimal degree we would then have a negative solution to Yates’ question together with a complete characterization of the hyperimmune-free degrees which have a strong minimal cover as those ... |