## Computably enumerable sets in the Solovay and the strong weak truth table degrees (2005)

Venue: | in New Computational Paradigms: First Conference on Computability in Europe, CiE 2005 |

Citations: | 6 - 5 self |

### BibTeX

@INPROCEEDINGS{Barmpalias05computablyenumerable,

author = {George Barmpalias},

title = {Computably enumerable sets in the Solovay and the strong weak truth table degrees},

booktitle = {in New Computational Paradigms: First Conference on Computability in Europe, CiE 2005},

year = {2005},

pages = {8--12},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential geometry. Yu and Ding showed that the relevant degree structure restricted to the c.e. reals has no greatest element, and asked for maximal elements. We answer this question for the case of c.e. sets. Using a doubly non-uniform argument we show that there are no maximal elements in the sw degrees of the c.e. sets. We note that the same holds for the Solovay degrees of c.e. sets. 1

### Citations

488 | Recursively Enumerable Sets and Degrees - Soare - 1987 |

316 |
Classical Recursion Theory
- Odifreddi
- 1989
(Show Context)
Citation Context ...e : ΦA e �= W1 ∨ Ψ A e �= W2. Each N ′ e will occupy the odd numbers of an interval [2ce +1, 2ce+1 +1) of N and use them as diagonalization witnesses. So the e-th requirement will have ce+1 − ce := k =-=(5)-=- numbers available for each of W1, W2, from 2ce + 1 on. To find a k sufficiently big to guarantee the success of this diagonalization ripple we consider the rectification resources of A below [2(ce+1 ... |

28 | Randomness and reducibility
- Downey, Hirschfeldt, et al.
- 2004
(Show Context)
Citation Context ...ability theory background; knowledge of algorithmic randomness is not essential but can be useful. For definitions, motivation and history of related notions as the Solovay degrees we refer mainly to =-=[1]-=- and secondly to [4]. Studying relative randomness, Downey, Hirschfeldt and LaForte [2] found Solovay reducibility insufficient, especially as far as non-c.e. reals are concerned. One of the two new m... |

13 | Randomness, computability and density
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...ed into W by this strategy will be because of numbers appearing in A ∩ (ℓe−1, ℓe] after stage s0. By the first step of the strategy Ne, at s0 we have |A ↾ (ℓe + e)| + |A ∩ (ℓe−1, ℓe]| < |(ℓe−1, ℓe]|. =-=(3)-=- Strategy Q can enumerate into W ∩ (ℓe−1, ℓe] no more than the first |A[s0] ∩ (ℓe−1, ℓe]| elements of (ℓe−1, ℓe]. Also, no other strategy apart from Ne can enumerate numbers of this interval into W. S... |

10 | Computability theory and differential geometry
- Soare
(Show Context)
Citation Context ...h that Γ B = A and the use of this computation on any argument n is bounded by n + c. The special case when c = 0 gives a stronger reducibility which was used by Soare, Nabutovsky and Weinberger (see =-=[7]-=-) in applying computability theory to differential geometry. We remind the definition of a c.e. real. Definition 2. A real number is computably enumerable (c.e.) if it is the limit of a computable inc... |

7 |
Some recent progress in algorithmic randomness
- Downey
- 2004
(Show Context)
Citation Context ...round; knowledge of algorithmic randomness is not essential but can be useful. For definitions, motivation and history of related notions as the Solovay degrees we refer mainly to [1] and secondly to =-=[4]-=-. Studying relative randomness, Downey, Hirschfeldt and LaForte [2] found Solovay reducibility insufficient, especially as far as non-c.e. reals are concerned. One of the two new measures for relative... |

4 |
Randomness and Reducibility. Mathematical foundations of computer science
- Downey, Hirschfeldt, et al.
- 2001
(Show Context)
Citation Context ...sw degrees present other difficulties (as the lack of join operator, see below) but they are nevertheless very interesting to study from a wider perspective. Moreover, Downey, Hirschfeldt and LaForte =-=[2]-=- noticed that as far as the computably enumerable sets are concerned, the sw degrees coincide with the Solovay degrees. So we also show that the Solovay degrees of c.e. sets have no maximal element.s2... |

2 |
There is no sw-complete c.e
- Yu, Ding
(Show Context)
Citation Context ... as domination) and strong weak truth table reducibility coincide on the c.e. sets. But, as we see below, this is not true for the c.e. reals. Yu and Ding proved the following Theorem 1. (Yu and Ding =-=[6]-=-) There is no sw-complete c.e. real. By a ‘uniformization’ of their proof they got two c.e. reals which have no c.e. real sw-above them. Hence Corollary 1. (Downey, Hirschfeldt, LaForte [2]) The struc... |