## Lowness Properties of Reals and Hyper-Immunity (2003)

Venue: | In Wollic 2003 |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{René03lownessproperties,

author = {Benjamín René and Benjamín René Callejas Bedregal and André Nies},

title = {Lowness Properties of Reals and Hyper-Immunity},

booktitle = {In Wollic 2003},

year = {2003},

pages = {84},

publisher = {Elsevier}

}

### Years of Citing Articles

### OpenURL

### Abstract

Ambos-Spies and Kucera [1, Problem 4.5] asked if there is a non-computable set A which is low for the computably random reals. We show that no such A is of hyper-immune degree. Thus, each g T A is dominated by a computable function. Ambos-Spies and Kucera [1, Problem 4.8] also asked if every S-low set is S 0 -low. We give a partial solution to this problem, showing that no S-low set is of hyper-immune degree.

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Citation Context ...posal was widely accepted, but still was deficient from a statistical point of view. Kolmogoroff intended to define randomness in terms of his complexity notion. This motivated Per Martin-Löf in 1966=-= [5] to-=- propose a new formal definition for algorithmic randomness without the statistical problem of Church’s concept. Martin Löf randomness is very restrictive since the tests are r.e. rather than compu... |

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Citation Context ...z c○2003 Published by Elsevier Science B. V.sBedregal and Nies imposed to capture better the intuitive notion of randomness. This leads to a hierarchy of notions of randomness. In 1940, Alonso Churc=-=h [2] s-=-uggested that intuitive randomness should be defined as algorithmic randomness proposing a formal notion of computable randomness. Church’s proposal was widely accepted, but still was deficient from... |

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Citation Context ... A is hyper-immune free if each total function, recursive in A, is majorized by a recursive function, that is, if for each g ≤T A there exists a computable function f such that for all x, g(x) ≤ f=-=(x) [6]. A s-=-et A is low for random, if each random real X is already random relative to A (that is, X passes all A-recursive enumerable test, where an A-recursive enumerable test is a set U ⊆ ω×2 <ω 4 which ... |

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Citation Context .... The formalization of the intuitive notion of randomness has also motivated several mathematicians and computer scientist to study the subject. Because randomness in an absolute sense does not exist =-=[1], -=-some restrictions must be 1 Email: bedregal@dimap.ufrn.br 2 Email: andre@cs.auckland.ac.nz c○2003 Published by Elsevier Science B. V.sBedregal and Nies imposed to capture better the intuitive notion... |

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Citation Context ...uences of 0’s and 1’s will be called reals and are identified with sets of natural numbers. A lowness property of a real says that, in some sense, it has a low computational power when used as ora=-=cle [7]. -=-For instance, a real B is low for random if each Martin-Löf random real is already MartinLöf random relative to B, i.e. using B as an oracle does not help to detect regularities in any random set. W... |

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Citation Context ...[k] is finite and the function mapping k into the canonical index of T [k] is computable. The concept of martingales, proposed by P. Levy, has been widely applied in the study of stochastic processes =-=[4]-=-, learning [10] and randomness [13]. A martingale allows us to calculate the gambling-account of a player who always tries to predict the next value of a function [10]. The idea is that martingales ca... |

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Citation Context ...ping k into the canonical index of T [k] is computable. The concept of martingales, proposed by P. Levy, has been widely applied in the study of stochastic processes [4], learning [10] and randomness =-=[13]-=-. A martingale allows us to calculate the gambling-account of a player who always tries to predict the next value of a function [10]. The idea is that martingales capture betting strategies to predict... |

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Citation Context ...e sense of C. An oracle A is hyper-immune if there is a computable function g relative to A, g ≤T A, which is not dominated by any recursive function and is hyperimmune free if it is not hyper-immun=-=e [10]. -=-In other words, an oracle A is hyper-immune free if each total function, recursive in A, is majorized by a recursive function, that is, if for each g ≤T A there exists a computable function f such t... |