## Lowness for weakly 1-generic and Kurtz-random (2006)

Venue: | in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006 |

Citations: | 8 - 2 self |

### BibTeX

@INPROCEEDINGS{Stephan06lownessfor,

author = {Frank Stephan and Liang Yu},

title = {Lowness for weakly 1-generic and Kurtz-random},

booktitle = {in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006},

year = {2006},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We prove that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1-generic is also low for Kurtz-random. 1

### Citations

1747 | An Introduction to Kolmogorov Complexity and its Applications
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(Show Context)
Citation Context ...-000-054-123 of Singapore, NSF of China No. 10471060 and No. 10420130638.sNotation 3. We follow the standard notation. We list some notations below. For other terminology, we refer the reader to [1], =-=[6]-=-, [10] and [11]. In this paper, a real means an element in Cantor space {0, 1} ω . By identifying subsets of natural numbers with their characteristic function, we obtain that reals and subsets of nat... |

490 |
Recursively Enumerable Sets and Degrees
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Citation Context ...f Singapore, NSF of China No. 10471060 and No. 10420130638.sNotation 3. We follow the standard notation. We list some notations below. For other terminology, we refer the reader to [1], [6], [10] and =-=[11]-=-. In this paper, a real means an element in Cantor space {0, 1} ω . By identifying subsets of natural numbers with their characteristic function, we obtain that reals and subsets of natural numbers ar... |

316 |
Classical Recursion Theory
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(Show Context)
Citation Context ...054-123 of Singapore, NSF of China No. 10471060 and No. 10420130638.sNotation 3. We follow the standard notation. We list some notations below. For other terminology, we refer the reader to [1], [6], =-=[10]-=- and [11]. In this paper, a real means an element in Cantor space {0, 1} ω . By identifying subsets of natural numbers with their characteristic function, we obtain that reals and subsets of natural n... |

165 |
Algorithmic Randomness and Complexity
- Downey, Hirschfeldt
(Show Context)
Citation Context ...R-146-000-054-123 of Singapore, NSF of China No. 10471060 and No. 10420130638.sNotation 3. We follow the standard notation. We list some notations below. For other terminology, we refer the reader to =-=[1]-=-, [6], [10] and [11]. In this paper, a real means an element in Cantor space {0, 1} ω . By identifying subsets of natural numbers with their characteristic function, we obtain that reals and subsets o... |

82 | Lowness properties and randomness
- Nies
(Show Context)
Citation Context ...s has been studied by lots of people and is one of the main topics in the theory of algorithmic randomness. We first summarize some known facts. Theorem 1. Let x be a set of natural numbers. 1. (Nies =-=[8]-=-) x is low for 1-randomness iff x is H-trivial iff x is low for Ω and ∆2. 2. (Nies [8]) x is low for recursively random iff x is recursive. 3. (Terwijn and Zambella [12]; Kjos-Hanssen, Nies and Stepha... |

53 |
Randomness and Genericity in the Degrees of Unsolvability
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- 1981
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Citation Context ...he degree of x is hyperimmune-free and each 1-generic real is weakly 1-x-generic. 4s4. The degree of x is hyperimmune-free and not dnr. Proof. Obviously, the first statement implies the second. Kurtz =-=[5]-=- showed that every hyperimmune degree contains a weakly 1-generic real and thus the second statement implies the third. Proposition 10 below proves that the third statement implies the fourth. The imp... |

48 | Kolmogorov complexity and the recursion theorem
- Kjos-Hanssen, Merkle, et al.
(Show Context)
Citation Context ...egree. Note that every x of hyperimmune-free degree is non-high. One can combine results of Kjos-Hanssen and Merkle to the following theorem. Theorem 7 (Kjos-Hanssen; Merkle, Kjos-Hanssen and Stephan =-=[4]-=-). Let x be not high. Then the following are equivalent: 1. x is not dnr; 2. x is not autocomplex, that is, there is no f ≤T x such that C(x ↾ m) ≥ n whenever m ≥ f(n); 3. for every g ≤T x there is a ... |

24 | Array nonrecursive degrees and genericity
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Citation Context ...hich is neither Turing complete nor low2 has this property. Result (b) can be obtained by considering sets which are generic for “very strong array forcing” as considered by Downey, Jockusch and Stob =-=[2, 9]-=-; as Kjos-Hanssen pointed out to the authors, those sets are neither autocomplex nor r.e. traceable nor do they have hyperimmune Turing degree. An application of the following result would be that the... |

21 |
and André Nies. Randomness and computability: open questions
- Miller
(Show Context)
Citation Context ...atical branches (or theoretical computer sciences). In this paper, we answer the following conjecture which was raised by several people. Conjecture 2 (Downey, Griffiths and Reid [3], Miller and Nies =-=[7]-=-, Yu [13]). Is a set x low for weakly 1-generic iff x is recursively traceable? Is x low for Kurtz-random iff x is recursively traceable? We refute the conjecture. Further, we obtain a characterizatio... |

13 | Lowness for the class of Schnorr random reals
- Kjos-Hanssen, Nies, et al.
- 2005
(Show Context)
Citation Context ...x is low for 1-randomness iff x is H-trivial iff x is low for Ω and ∆2. 2. (Nies [8]) x is low for recursively random iff x is recursive. 3. (Terwijn and Zambella [12]; Kjos-Hanssen, Nies and Stephan =-=[9]-=-) x is low for Schnorr-random iff x is recursively traceable. 4. (Greenberg, Miller and Yu [13]) x is low for 1-generic iff x is recursive. From the theorem above, we see that there are some deep conn... |

8 |
Terwijn and Domenico Zambella. Computational randomness and lowness
- Sebastiaan
(Show Context)
Citation Context ...et of natural numbers. 1. (Nies [8]) x is low for 1-randomness iff x is H-trivial iff x is low for Ω and ∆2. 2. (Nies [8]) x is low for recursively random iff x is recursive. 3. (Terwijn and Zambella =-=[12]-=-; Kjos-Hanssen, Nies and Stephan [9]) x is low for Schnorr-random iff x is recursively traceable. 4. (Greenberg, Miller and Yu [13]) x is low for 1-generic iff x is recursive. From the theorem above, ... |

4 |
On Kurtz randomness. Theoret
- Downey, Griffiths, et al.
(Show Context)
Citation Context ...eory and other mathematical branches (or theoretical computer sciences). In this paper, we answer the following conjecture which was raised by several people. Conjecture 2 (Downey, Griffiths and Reid =-=[3]-=-, Miller and Nies [7], Yu [13]). Is a set x low for weakly 1-generic iff x is recursively traceable? Is x low for Kurtz-random iff x is recursively traceable? We refute the conjecture. Further, we obt... |

1 |
Lowness for genericity. Archive for Mathematical Logic
- Yu
(Show Context)
Citation Context ... for recursively random iff x is recursive. 3. (Terwijn and Zambella [12]; Kjos-Hanssen, Nies and Stephan [9]) x is low for Schnorr-random iff x is recursively traceable. 4. (Greenberg, Miller and Yu =-=[13]-=-) x is low for 1-generic iff x is recursive. From the theorem above, we see that there are some deep connections between computability theory and other mathematical branches (or theoretical computer s... |