## Lowness properties and approximations of the jump (2006)

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Venue: | Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143 |

Citations: | 21 - 10 self |

### BibTeX

@INPROCEEDINGS{Figueira06lownessproperties,

author = {Santiago Figueira and Frank Stephan},

title = {Lowness properties and approximations of the jump},

booktitle = {Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143},

year = {2006},

pages = {45--57}

}

### Years of Citing Articles

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### Citations

1690 | Kolmogorov complexity and its applications - Li, Vitanyi - 1990 |

477 | Recursively enumerable sets and degrees - Soare - 1987 |

331 | A theory of program size formally identical to information theory
- Chaitin
- 1975
(Show Context)
Citation Context ... or 4|nx| ≤ nx/2. In the second case nx − 4|nx| ≥ nx/2 and by (1), nx/2 ≤ |h(|qx|)| + O(1). So, in both cases, we have nx ≤ 2|h(|qx|)| + O(1). Lemma 16. For all x ∈ {0, 1} ∗ and d ∈ N, Proof. Chaitin =-=[6]-=- proved that |{y : C(x, y) ≤ C(x) + d}| ≤ O(d 4 2 d ). ∀d, n ∈ N |{σ : |σ| = n ∧ C(σ) ≤ C(n) + d}| ≤ O(2 d ). Let c be such that ∀x C(x) ≤ str −1 (x)+c. Consider the partial recursive function f(x, y,... |

161 | Algorithmic Randomness and Complexity - Downey, Hirschfeldt - 2010 |

119 | Computability: An Introduction to Recursive Function Theory - Cutland - 1986 |

78 | Lowness properties and randomness
- Nies
(Show Context)
Citation Context ...lowness properties in the direction of characterizing K-trivial sets. A set is K-trivial when it is highly compressible in terms of Kolmogorov complexity (see Section 2 for the formal definition). In =-=[18]-=-, Nies proved that a set is K-trivial if and only if A is low for Martin-Löf-random (i.e. each Martin-Löf-random set is already random relative to A). Terwijn and Zambella [23] defined a set A to be r... |

60 | Information-theoretic characterizations of recursive infinite strings
- Chaitin
- 1976
(Show Context)
Citation Context ...uivalent to being K-trivial. In particular, non-computable low for K sets exist. The corresponding property involving C is only satisfied by the computable sets (because it implies being C-trivial by =-=[7]-=-, which is the same as computable). The characterization of strongly jump-traceable is via a property that states that C A is very close to C, while not implying computability. We know that K-triviali... |

56 | Trivial reals - Downey, Hirschfeldt, et al. - 2003 |

53 | of a paper (or series of papers) on Chaitin’s work ... done for the most part during the period of Sept - Solovay, Draft - 1974 |

46 | Kolmogorov complexity and the recursion theorem - Kjos-Hanssen, Merkle, et al. |

36 | Soare, An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees - Ambos-Spies, Jockusch, et al. |

30 | Randomness and Complexity - Wang |

28 | Reals which compute little
- Nies
- 2002
(Show Context)
Citation Context ... Science and Mathematics, National University of Singapore, Singapore 1sThey showed that this combinatorial notion characterizes the sets that are low for Schnorr tests. This property was modified in =-=[19]-=- to jump-traceability. A set A is jump traceable if its jump at argument e, written J A (e) = {e} A (e), has few possible values. Definition 1. A uniformly r.e. family T = {T0, T1, . . .} of sets of n... |

26 | Computational randomness and lowness
- Terwijn, Zambella
(Show Context)
Citation Context ... formal definition). In [18], Nies proved that a set is K-trivial if and only if A is low for Martin-Löf-random (i.e. each Martin-Löf-random set is already random relative to A). Terwijn and Zambella =-=[23]-=- defined a set A to be recursively traceable if there is a recursive bound p such that for every f ≤T A, there is a recursive r such that for all x, |Dr(x)| ≤ p(x), and (Dr(x))x∈N is a set of possible... |

18 | Some generalizations of a fixed-point theorem - Arslanov |

12 | Classical Recursion Theory, volume 1 - Odifreddi - 1989 |

10 | Bakhadyr Khoussainov and Yongge Wang. Recursively enumerable reals and Chaitin Ω numbers, Theoretical Computer Science 255:125–149 - Calude, Hertling - 2001 |

10 | Presentations of computably enumerable reals - Downey, LaForte |

10 | Randomness and universal machines - Figueira, Stephan, et al. |

8 |
André Nies. Relativizing Chaitin’s halting probability
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...) = ψs(z) for all z �= y. Now let A be a set whose characteristic function extends ψ and which is low for Ω. Such a set A exists since ψ defines a Π 0 1 class and Downey, Hirschfeldt, Miller and Nies =-=[11]-=- showed every Π 0 1 class (of sets) has a member which is low for Ω. Reviewing the construction of ψ, condition (1) enforces that ψ is defined on the complete interval Ix if x /∈ Y and condition (2) e... |

7 |
Lowness properties of r.e. sets
- Bickford, Mills
- 1982
(Show Context)
Citation Context ...trace T such that ∀e [J A (e) ↓ ⇒ J A (e) ∈ Te]. We say that A is jump traceable via a function h if, additionally, T has bound h. Another notion studied in [19] is super-lowness, first introduced in =-=[4, 17]-=-. Definition 2. A set A is ω-r.e. iff there exists a recursive function b such that A(x) = lims→∞ g(x, s) for a recursive {0, 1}-valued g such that g(x, s) changes at most b(x) times, i.e. |{s : g(x, ... |

4 |
A refinement of low n and high n for the r.e. degrees. Zeitschrift für mathematische Logik und Grundlagen der Mathematik
- Mohrherr
- 1986
(Show Context)
Citation Context ...trace T such that ∀e [J A (e) ↓ ⇒ J A (e) ∈ Te]. We say that A is jump traceable via a function h if, additionally, T has bound h. Another notion studied in [19] is super-lowness, first introduced in =-=[4, 17]-=-. Definition 2. A set A is ω-r.e. iff there exists a recursive function b such that A(x) = lims→∞ g(x, s) for a recursive {0, 1}-valued g such that g(x, s) changes at most b(x) times, i.e. |{s : g(x, ... |

4 |
Strongly jump-traceability I: the computably enumerable case
- Cholak, Downey, et al.
(Show Context)
Citation Context ...gly jump-traceable is via a property that states that C A is very close to C, while not implying computability. By [19], K-triviality implies jump-traceability. Recently, Cholak, Downey and Greenberg =-=[9]-=- have shown that for r.e. sets A, strong jump-traceability implies K-triviality. They also prove that there is a K-trivial r.e. set that is not strongly jump-traceable. 2 Basic definitions If A is a s... |

3 | M-reducibility and fixed points, Complexity problems of mathematical logic - Arslanov - 1985 |

1 |
Relativizing Chaitin's halting probability, manuscript
- Downey, Hirschfeldt, et al.
- 2005
(Show Context)
Citation Context ...= s(z) for all z 6= y. Now let A be a set whose characteristic function extendssand which is low for \Omega . Such a set A exists sincesdefines a \Pi 01 class and Downey, Hirschfeldt, Miller and Nies =-=[11]-=- showed every \Pi 01 class (of sets) has a member which is low for \Omega . Reviewing the construction of , condition (1) enforces thatsis defined on the complete interval Ix if x /2 Y and condition (... |

1 |
Algorithmic randomness and lowness, The Journal of Symbolic Logic
- Terwijn, Zambella
- 2001
(Show Context)
Citation Context ...mal definition). In [19], Nies proved that a set is K-trivial if and only if A is low for Martin-Löf-random (that is, each Martin-Löf-random set is already random relative to A). Terwijn and Zambella =-=[24]-=- defined a set A to be recursively traceable if there is a recursive bound p such that for every f ≤T A, there is a recursive r such that for all x, |Dr(x)| ≤ p(x), and (Dr(x))x∈N is a set of possible... |