## Schnorr trivial sets and truth-table reducibility

Citations: | 2 - 1 self |

### BibTeX

@MISC{Franklin_schnorrtrivial,

author = {Johanna N. Y. Franklin and Frank Stephan},

title = {Schnorr trivial sets and truth-table reducibility},

year = {}

}

### OpenURL

### Abstract

Abstract In this paper, we give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations enable us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte. 1 Introduction One of the major achievements in the study of Martin-L"of randomness is Nies's discovery [14] that the following conditions on a set A coincide. * A is low for Martin-L"of randomness, that is, every set that is Martin-L"of random

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Citation Context ...spect to truth-table reducibility. The interested reader is referred to the textbooks of Odifreddi [16, 17], Rogers [18] and Soare [21] for background information on recursion theory. Li and Vit'anyi =-=[10]-=-, Schnorr [19] and Downey, Hirschfeldt, Nies and Terwijn [4] provide an overview of algorithmic randomness. However, before we begin, we recall the martingale equivalents of Schnorr randomness and rec... |

884 |
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Citation Context ... Schnorr trivial sets cannot be characterized as bases of Schnorr randomness with respect to truth-table reducibility. The interested reader is referred to the textbooks of Odifreddi [16, 17], Rogers =-=[18]-=- and Soare [21] for background information on recursion theory. Li and Vit'anyi [10], Schnorr [19] and Downey, Hirschfeldt, Nies and Terwijn [4] provide an overview of algorithmic randomness. However,... |

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Citation Context ...owed in the same paper that these sets form an ideal with respect to Turing reducibility. Schnorr [19, 20] introduced an alternative definition of randomness, as he felt that Martin-L"of's definition =-=[11]-=- was too restrictive. For example, Martin-L"of random sets do not exist in any incomplete r.e. Turing degree [1], while Schnorr random sets exist in every high r.e. Turing degree [15]. It is possible ... |

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(Show Context)
Citation Context ... we show that the Schnorr trivial sets cannot be characterized as bases of Schnorr randomness with respect to truth-table reducibility. The interested reader is referred to the textbooks of Odifreddi =-=[16, 17]-=-, Rogers [18] and Soare [21] for background information on recursion theory. Li and Vit'anyi [10], Schnorr [19] and Downey, Hirschfeldt, Nies and Terwijn [4] provide an overview of algorithmic randomn... |

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Citation Context ...hat is, A is Turing reducible to a set which is Martin-L"of random relative to A. Furthermore, Nies showed in the same paper that these sets form an ideal with respect to Turing reducibility. Schnorr =-=[19, 20]-=- introduced an alternative definition of randomness, as he felt that Martin-L"of's definition [11] was too restrictive. For example, Martin-L"of random sets do not exist in any incomplete r.e. Turing ... |

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Citation Context ...on that B is Schnorr trivial must be wrong. This result can be used to give an alternative proof of a result of Franklin [5]. This result shows, together with the basis theorems of Jockusch and Soare =-=[8]-=-, that there is a Schnorr trivial set of low Turing degree and one of hyperimmune-free Turing degree. Here we say that a set A has low Turing degree if and only if A0 jT K and has hyperimmune-free Tur... |

21 |
Some generalizations of a fixed-point theorem
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Citation Context ...uced an alternative definition of randomness, as he felt that Martin-L"of's definition [11] was too restrictive. For example, Martin-L"of random sets do not exist in any incomplete r.e. Turing degree =-=[1]-=-, while Schnorr random sets exist in every high r.e. Turing degree [15]. It is possible to define trivial sets not only for Martin-L"of randomness but also for Schnorr randomness [3, 5, 6]. This notio... |

18 |
Classical Recursion Theory, Volume II
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(Show Context)
Citation Context ... we show that the Schnorr trivial sets cannot be characterized as bases of Schnorr randomness with respect to truth-table reducibility. The interested reader is referred to the textbooks of Odifreddi =-=[16, 17]-=-, Rogers [18] and Soare [21] for background information on recursion theory. Li and Vit'anyi [10], Schnorr [19] and Downey, Hirschfeldt, Nies and Terwijn [4] provide an overview of algorithmic randomn... |

13 |
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Citation Context ...al by Theorem 3.2 (c). Here, we will say that A has high Turing degree if and only if A0 >=T K0. This is equivalent to the existence of an A-recursive function that dominates every recursive function =-=[13]-=-. Jockusch and Stephan [9] showed that there are cohesive sets that do not have a high Turing degree. Recall that an infinite set A is cohesive if there is no r.e. set W such that both W " A and W " (... |

10 | On Schnorr and computable randomness, martingales and machines
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(Show Context)
Citation Context ....e. Turing degree [1], while Schnorr random sets exist in every high r.e. Turing degree [15]. It is possible to define trivial sets not only for Martin-L"of randomness but also for Schnorr randomness =-=[3, 5, 6]-=-. This notion turned out to be incompatible with Turing reducibility, since the Schnorr trivials are not closed downward under this reduction [3]. Furthermore, when defining lowness for Schnorr random... |

9 | Lowness properties of reals and randomness
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(Show Context)
Citation Context ...but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte. 1 Introduction One of the major achievements in the study of Martin-L"of randomness is Nies's discovery =-=[14]-=- that the following conditions on a set A coincide. * A is low for Martin-L"of randomness, that is, every set that is Martin-L"of random unrelativized is also Martin-L"of random relative to A. * A is ... |

4 | André Nies and Sebastiaan A. Terwijn. Calibrating randomness - Downey, Hirschfeldt |

3 |
Hyperimmune-free degrees and Schnorr triviality
- Franklin
- 2008
(Show Context)
Citation Context ...this reduction [3]. Furthermore, when defining lowness for Schnorr randomness using Turing reducibility, only the Schnorr trivial sets of hyperimmune-free Turing degree are low for Schnorr randomness =-=[7]-=-. Here, as well, Turing reducibility does not seem to be appropriate. In this paper, the problem of finding results analogous to those of Nies is reconsidered. Our main contribution is to consider tru... |

3 |
Randomness, relativization and Turing degrees. The Journal of Symbolic Logic
- Nies, Stephan, et al.
- 2005
(Show Context)
Citation Context ..."of's definition [11] was too restrictive. For example, Martin-L"of random sets do not exist in any incomplete r.e. Turing degree [1], while Schnorr random sets exist in every high r.e. Turing degree =-=[15]-=-. It is possible to define trivial sets not only for Martin-L"of randomness but also for Schnorr randomness [3, 5, 6]. This notion turned out to be incompatible with Turing reducibility, since the Sch... |

2 |
Calude and Andr'e Nies, Chaitin \Omega numbers and strong reducibilities
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- 1997
(Show Context)
Citation Context ...among all possible values of the string B(0)B(1) . . . B(u(f (n))). Now Remark 3.3 can be used to show that A is Schnorr random. The second result is a generalization of the result of Calude and Nies =-=[2]-=- that the halting problem is not truth-table reducible to any Martin-L"of random set. Theorem 6.2. There is a partial recursive {0, 1}-valued functionswhose domain is dense simple such that no set A w... |

2 |
Aspects of Schnorr randomness
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- 2007
(Show Context)
Citation Context ....e. Turing degree [1], while Schnorr random sets exist in every high r.e. Turing degree [15]. It is possible to define trivial sets not only for Martin-L"of randomness but also for Schnorr randomness =-=[3, 5, 6]-=-. This notion turned out to be incompatible with Turing reducibility, since the Schnorr trivials are not closed downward under this reduction [3]. Furthermore, when defining lowness for Schnorr random... |

2 | Algorithmic Randomness
- Mihailovi'c
- 2007
(Show Context)
Citation Context ...tion 2.1 below. We then discuss which of these notions is most suitable for relativization. The main elements of the proof of the proposition are two aspects of the martingale: the "savings property" =-=[12]-=- and how the performance bound of the martingale is introduced. Recall that a recursive martingale mg is a recursive function mapping strings in{ 0, 1}* to nonnegative recursive real numbers that sati... |

1 |
trivial reals. A construction. The Archive for Mathematical Logic
- Schnorr
(Show Context)
Citation Context ...ere we say that a set A has low Turing degree if and only if A0 jT K and has hyperimmune-free Turing degree if and only if every A-recursive function is majorized by a recursive one [13]. Theorem 5.5 =-=[6]-=-. There is a \Pi 01-class of Schnorr trivial sets with no recursive members. Proof. Let A be a maximal set. By Sacks's Splitting Theorem, there is a partial recursive {0, 1}-valued functionswith domai... |

1 |
A cohesive set which is not high. Mathematical Logic Quarterly
- Jockusch, Stephan
- 1993
(Show Context)
Citation Context ..., we will say that A has high Turing degree if and only if A0 >=T K0. This is equivalent to the existence of an A-recursive function that dominates every recursive function [13]. Jockusch and Stephan =-=[9]-=- showed that there are cohesive sets that do not have a high Turing degree. Recall that an infinite set A is cohesive if there is no r.e. set W such that both W " A and W " (N - A) are infinite. The n... |