## A lower cone in the wtt degrees of non-integral effective dimension (2006)

Venue: | In Proceedings of IMS workshop on Computational Prospects of Infinity |

Citations: | 21 - 2 self |

### BibTeX

@INPROCEEDINGS{Nies06alower,

author = {André Nies and Jan Reimann},

title = {A lower cone in the wtt degrees of non-integral effective dimension},

booktitle = {In Proceedings of IMS workshop on Computational Prospects of Infinity},

year = {2006}

}

### OpenURL

### Abstract

ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truth-table reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wtt-lower cone of effective dimension r. 1.

### Citations

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Citation Context ... | σ ) = λ(C ∩ [σ ])2 |σ | . For all unexplained notions from computability theory we refer to any standard textbook such as [14] or [19], for details on Kolmogorov complexity, the reader may consult =-=[7]-=-; [13]æwill provide background on the use of measure theory, especially martingales, in the theory of algorithmic randomness. In the proof of our main result we will use so-called Kraft-Chaitin sets. ... |

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Citation Context ...measurable C ⊆ 2 ω , recall that the conditional probability is λ(C | σ ) = λ(C ∩ [σ ])2 |σ | . For all unexplained notions from computability theory we refer to any standard textbook such as [14] or =-=[19]-=-, for details on Kolmogorov complexity, the reader may consult [7]; [13]æwill provide background on the use of measure theory, especially martingales, in the theory of algorithmic randomness. In the p... |

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Citation Context ...) = λ(C ∩ [σ ])2 |σ | . For all unexplained notions from computability theory we refer to any standard textbook such as [14] or [19], for details on Kolmogorov complexity, the reader may consult [7]; =-=[13]-=-æwill provide background on the use of measure theory, especially martingales, in the theory of algorithmic randomness. In the proof of our main result we will use so-called Kraft-Chaitin sets. A Kraf... |

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Citation Context ... has dimension at least r. This implies, for any rational r, the existence of a wtt-lower cone of effective dimension r. 1. INTRODUCTION Since the introduction of effective dimension concepts by Lutz =-=[8, 9]-=-, considerable effort has been put into studying the effective or resource-bounded dimension of objects occurring in computability or complexity theory. However, up to now there are basically only thr... |

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Citation Context ...wise. Then, using Theorem 1, it is easy to see that dim 1 H Xr = r. This technique can be refined to obtain sets of effective dimension s, where 0 ≤ s ≤ 1 is any �0 2-computable real number (see e.g. =-=[10]-=-). (2) Given a Bernoulli measure µp with bias p ∈ Q ∩ [0, 1], the effective dimension of any set that is Martin-Löf random with respect to µp equals the entropy of the measure H(µp) = −[log p + log(1 ... |

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Citation Context ...er words, the effective dimension of an individual set equals its lower asymptotic entropy. In the following, we will use K (A) to denote lim inf K (A ↾ n)/n. Theorem 1 was first explicitly proved in =-=[11]-=-, but much of it is already present in earlier works on Kolmogorov complexity and Hausdorff dimension, such as [17] or [20]. Examples for effective dimension. As mentioned in the introduction, there a... |

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Citation Context ...upper cone has classical Hausdorff dimension 1 (and hence effective dimensionA LOWER CONE IN THE WTT DEGREES OF NON-INTEGRAL EFFECTIVE DIMENSION 4 1, too). This contrasts a classical result by Sacks =-=[18]-=- which shows that the Turing upper cone of a set has Lebesgue measure zero unless the set is recursive. As regards lower cones and degrees, the situation is different. First, using coding at very spar... |

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Citation Context ... (A) to denote lim inf K (A ↾ n)/n. Theorem 1 was first explicitly proved in [11], but much of it is already present in earlier works on Kolmogorov complexity and Hausdorff dimension, such as [17] or =-=[20]-=-. Examples for effective dimension. As mentioned in the introduction, there are mainly three types of examples of sets of non-integral effective dimension. (1) If 0 < r < 1 is rational, let Zr = {⌊n/r... |

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Citation Context ...sion of the real number �s = ∑ |σ | − 2 s σ ∈dom(U) has effective dimension s. This was shown by Tadaki [22]. 2.1. Effective Dimension of cones and degrees. Fundamental results by Gacs [2] and Kučera =-=[6]-=- showed that every set is Turing reducible to a Martin-Löf random one. Since a Martin-Löf random set has effective dimension 1, it follows from (1) that every Turing upper cone is of effective dimensi... |

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Citation Context ...ll use K (A) to denote lim inf K (A ↾ n)/n. Theorem 1 was first explicitly proved in [11], but much of it is already present in earlier works on Kolmogorov complexity and Hausdorff dimension, such as =-=[17]-=- or [20]. Examples for effective dimension. As mentioned in the introduction, there are mainly three types of examples of sets of non-integral effective dimension. (1) If 0 < r < 1 is rational, let Zr... |

47 |
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(Show Context)
Citation Context ...oes (or any other computable set). The second example comprises all sets which are random with respect to a Bernoulli distribution on Cantor space. Here Lutz transferred a classic result by Eggleston =-=[1]-=- to show that if µ is a (generalized) Bernoulli measure, then the effective dimension of a Martin-Löf µ-random set coincides with the entropy H(µ) of the measure µ. Finally, the third example is a par... |

47 |
Classical Recursion Theory, volume 125
- Odifreddi
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(Show Context)
Citation Context ...or each measurable C ⊆ 2 ω , recall that the conditional probability is λ(C | σ ) = λ(C ∩ [σ ])2 |σ | . For all unexplained notions from computability theory we refer to any standard textbook such as =-=[14]-=- or [19], for details on Kolmogorov complexity, the reader may consult [7]; [13]æwill provide background on the use of measure theory, especially martingales, in the theory of algorithmic randomness. ... |

43 |
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(Show Context)
Citation Context ...he binary expansion of the real number �s = ∑ |σ | − 2 s σ ∈dom(U) has effective dimension s. This was shown by Tadaki [22]. 2.1. Effective Dimension of cones and degrees. Fundamental results by Gacs =-=[2]-=- and Kučera [6] showed that every set is Turing reducible to a Martin-Löf random one. Since a Martin-Löf random set has effective dimension 1, it follows from (1) that every Turing upper cone is of ef... |

42 |
Gales and the constructive dimension of individual sequences
- Lutz
(Show Context)
Citation Context ... has dimension at least r. This implies, for any rational r, the existence of a wtt-lower cone of effective dimension r. 1. INTRODUCTION Since the introduction of effective dimension concepts by Lutz =-=[8, 9]-=-, considerable effort has been put into studying the effective or resource-bounded dimension of objects occurring in computability or complexity theory. However, up to now there are basically only thr... |

30 |
Randomness and computability: open questions
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(Show Context)
Citation Context ...f dimension r. This result was independently announced by Hirschfeldt and Miller [3]. The case of Turing reducibility seems much more difficult and remains a major open problem in the field (see also =-=[12]-=-). Notation. Our notation is fairly standard. 2 ω denotes Cantor space, the set of all infinite binary sequences. We identify elements of 2 ω with subsets of the natural numbers N by means of the char... |

17 |
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(Show Context)
Citation Context ... three types of examples mentioned above compute a Martin-Löf random set, albeit for different reasons. It is obvious that any diluted set Xr computes a Martin-Löf random sequence. Furthermore, Levin =-=[24]-=- and independently Kautz [4] showed that any sequence which is random with respect to a computable probability measure on 2ω (which includes the Bernoulli measures µp with rational bias) computes a Ma... |

11 |
Computability and fractal dimension. Doctoral dissertation, Universität
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(Show Context)
Citation Context ...e Turing equivalent to a Martin-Löf random set. Therefore, one cannot use them to obtain Turing cones of non-integral dimension. However, when restricted to many-one reducibility, Reimann and Terwijn =-=[15]-=- showed that the lower cone of a Bernoulli random set cannot contain a set of higher dimension than the random set it reduces to, thereby obtaining many-one lower cones of non-integral effective dimen... |

11 | Complexity and randomness - Terwijn - 2003 |

7 |
Degrees of Random Sequences
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(Show Context)
Citation Context ...ioned above compute a Martin-Löf random set, albeit for different reasons. It is obvious that any diluted set Xr computes a Martin-Löf random sequence. Furthermore, Levin [24] and independently Kautz =-=[4]-=- showed that any sequence which is random with respect to a computable probability measure on 2ω (which includes the Bernoulli measures µp with rational bias) computes a Martin-Löf random set. Finally... |

6 |
A generalization of Chaitin’s halting probability � and halting selfsimilar sets
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(Show Context)
Citation Context ..., then the effective dimension of a Martin-Löf µ-random set coincides with the entropy H(µ) of the measure µ. Finally, the third example is a parameterized version of Chaitin’s � introduced by Tadaki =-=[22]-=-. An obvious question is whether there exist examples of non-integral effective dimension among classes of central interest to computability theory, such as cones or degrees. It is interesting to note... |

4 |
Hausdorff dimension and weak truth-trable reducibility
- Stephan
- 2005
(Show Context)
Citation Context ...the random set it reduces to, thereby obtaining many-one lower cones of non-integral effective dimension. But the proof does not transfer to weaker reducibilities. Using a different approach, Stephan =-=[21]-=- was able to construct an oracle relative to which there exists a wtt-lower cone of positive effective dimension at most 1/2. In this paper we construct, for an arbitrary rational number r, a wtt-lowe... |