## Constructive dimension and weak truth-table degrees (2007)

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Venue: | In Computation and Logic in the Real World - Third Conference of Computability in Europe. Springer-Verlag Lecture Notes in Computer Science #4497 |

Citations: | 13 - 3 self |

### BibTeX

@INPROCEEDINGS{Bienvenu07constructivedimension,

author = {Laurent Bienvenu and David Doty and Frank Stephan},

title = {Constructive dimension and weak truth-table degrees},

booktitle = {In Computation and Logic in the Real World - Third Conference of Computability in Europe. Springer-Verlag Lecture Notes in Computer Science #4497},

year = {2007},

publisher = {Springer}

}

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### Abstract

Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truth-table degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truth-table equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.

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Citation Context ... most c [8, 17]. 4sWe say S is truth-table (tt) reducible to R via M and we write S <=tt R via M , if S <=T R via M and, for all R0 2 C, M (R0) is defined; i.e., M is total with respect to the oracle =-=[18]-=-. Bounded reductions and wtt reductions are incomparable. However, every tt reduction is a wtt reduction. Given \Deltas2 {tt, wtt, bT}, define S <=\DeltasR, S j\DeltasR, span\Delta (S) and deg\Delta (... |

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Citation Context ...≡ p T . Theorem 2.10 ([4]). For all ɛ > 0 and all S ∈ C such that dimP(S) > 0, there exists R ≡ p T S such that dimP(R) ≥ 1 − ɛ. In fact, Theorem 2.10 holds for any resource-bounded packing dimension =-=[7]-=- defined by Turing machines allowed at least polynomial space, which includes constructive packing dimension as a special case, thus proving a spectrum of zero-one packing dimension laws for various d... |

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Citation Context ...pment. Furthermore, Lutz [8, Section 6] reviews early results that anticipated the effectivization of ⋆ Corresponding author.sHausdorff dimension. Constructive Hausdorff dimension was defined by Lutz =-=[8]-=- to study effective dimension at the level of computability theory. Intuitively, given an infinite binary sequence S – interpreted as a language or decision problem – the constructive Hausdorff dimens... |

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Citation Context ...ts by considered the interaction between constructive Hausdorff dimension and constructive packing dimension [1], a dual quantity that is a constructive effectivization of classical packing dimension =-=[22, 21]-=-, another widely-studied fractal dimension. The constructive packing dimension dimP(S) of a sequence S always obeys 0 ≤ dimH(S) ≤ dimP(S) ≤ 1, with each inequality tight in the strong sense that there... |

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Citation Context ...utability theory. Intuitively, given an infinite binary sequence S – interpreted as a language or decision problem – the constructive Hausdorff dimension dimH(S) of S is a real number in the interval =-=[0,1]-=- indicating the density of algorithmic randomness of the sequence. The constructive Hausdorff dimension of a class C of sequences is the supremum of the dimensions of individual sequences in C. For ma... |

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Citation Context ...omputing S[0 . . n−1], all results of the present paper still hold.sConstructive Dimension. Lutz [8] gives an introduction to the theory of constructive dimension. We use Mayordomo’s characterization =-=[10]-=- of the constructive dimensions of sequences. For all S ∈ C, the constructive Hausdorff dimension and the constructive packing dimension of S are respectively defined as dimH(S) = lim inf n→∞ C(S[0 . ... |

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Citation Context ...ts by considered the interaction between constructive Hausdorff dimension and constructive packing dimension [1], a dual quantity that is a constructive effectivization of classical packing dimension =-=[22, 21]-=-, another widely-studied fractal dimension. The constructive packing dimension dimP(S) of a sequence S always obeys 0 ≤ dimH(S) ≤ dimP(S) ≤ 1, with each inequality tight in the strong sense that there... |

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Citation Context ...ructive dimension. Theorem 2.1 (Nies and Reimann [11]). For every rational number α with 0 < α < 1, there exists a sequence S ∈ C such that, for all wtt reductions M, dimH(M(S)) ≤ dimH(S) = α. Ryabko =-=[16, 17]-=- discovered the next theorem. Theorem 2.2 (Ryabko [16, 17]). For all S ∈ C and δ > 0, there exists R ∈ C and Nd ∈ OTM such that 1. S ≤T R via Nd and R ≤T S. 2. ρ − Nd (S, R) ≤ dimH(S) + δ. The followi... |

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Citation Context ... concepts and notations formally. Notation. We refer the reader to the textbooks of Li and Vitányi [6] for an introduction to Kolmogorov complexity and algorithmic information theory and of Odifreddi =-=[13]-=- and Soare [19] for an introduction to computability theory. Although we follow mainly the notation in these books, we nevertheless want to remind the reader on the following definitions, either for t... |

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Citation Context ...at no single Turing reduction can be a universal constructive Hausdorff dimension extractor. Keywords: constructive dimension, weak truth-table, extractor, degree, randomness 1 Introduction Hausdorff =-=[5]-=- initiated the study of dimension as a general framework to define the size of subsets of metric spaces. Recently this framework had been effectivized; Lutz [9] gives an overview of this historical de... |

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Citation Context ...o truth-table degrees. Stephan [20] showed that there is a relativized world in which there exists a wtt degree of constructive Hausdorff dimension between 1 1 4 and 2 . Furthermore, Nies and Reimann =-=[11]-=- obtained a non-relativized variant of this result and constructed, for each rational α between 0 and 1, a wtt degree of constructive Hausdorff dimension α. Doty [3] attempted positive results by cons... |

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Citation Context ...ctive dimensions of arbitrary Turing degrees. Doty [4] obtained stronger results for other effective dimensions such as computable dimension and various time and space bounded dimensions [10]. Zimand =-=[29]-=- has shown that, given two independent sequences with positive constructive Hausdorff dimension, a truth-table reduction with access to both sequences suffices to compute a sequence with constructive ... |

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Citation Context ...ructive dimension. Theorem 2.1 (Nies and Reimann [11]). For every rational number α with 0 < α < 1, there exists a sequence S ∈ C such that, for all wtt reductions M, dimH(M(S)) ≤ dimH(S) = α. Ryabko =-=[16, 17]-=- discovered the next theorem. Theorem 2.2 (Ryabko [16, 17]). For all S ∈ C and δ > 0, there exists R ∈ C and Nd ∈ OTM such that 1. S ≤T R via Nd and R ≤T S. 2. ρ − Nd (S, R) ≤ dimH(S) + δ. The followi... |

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Citation Context ...ay S is weak truth-table (wtt) reducible to R via M and we write S <=wtt R via M , if S <=T R via M and there is a computable function q : N ! N such that, for all n 2 N, #(M R, S[0 . . n-1]) <= q(n) =-=[6]-=-. We say S is bounded Turing (bT) reducible to R via M (or simply bounded reducible) and we write S <=bT R via M , if S <=T R via M and there exists a constant c 2 N such that, for every n 2 N, the nu... |

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Citation Context ...he best- and worst-case compression ratios as M decompresses R into S. Note that 0 ≤ ρ − M (S, R) ≤ ρ+ M (S, R) ≤ ∞. The following lemma is useful when one wants to compose two reductions: Lemma 1.1. =-=[2]-=- Let S, S ′ , S ′′ ∈ C and M1, M2 ∈ OTM such that S ′ ≤T S via M1 and S ′′ ≤T S ′ via M2. There exists M ∈ OTM such that S ′′ ≤T S via M and: ρ + M (S′′ , S) ≤ ρ + M2 (S′′ , S ′ )ρ + M1 (S′ , S). ρ − ... |

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Citation Context ...e dimension. Theorem 2.1 (Nies and Reimann [15]). For every rational number ff with 0 < ff < 1, there exists a sequence S 2 C such that, for all wtt reductions M , dimH(M (S)) <= dimH(S) = ff. Ryabko =-=[21, 22]-=- discovered the next theorem. Theorem 2.2 (Ryabko [21, 22]). For all S 2 C and ffi > 0, there exists R 2 C and Nd 2 OTM such that 1. S <=T R via Nd and R <=T S. 2. ae-Nd (S, R) <= dimH(S) + ffi. The f... |

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Citation Context ...t there are many-one and bounded truth-table degrees with constructive Hausdorff dimension strictly between 0 and 1. Later Reimann and Slaman [15] extended this result to truth-table degrees. Stephan =-=[20]-=- showed that there is a relativized world in which there exists a wtt degree of constructive Hausdorff dimension between 1 1 4 and 2 . Furthermore, Nies and Reimann [11] obtained a non-relativized var... |

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Citation Context ...simply bounded reducible) and we write S <=bT R via M , if S <=T R via M and there exists a constant c 2 N such that, for every n 2 N, the number of bits of R queried when computing S[n] is at most c =-=[8, 17]-=-. 4sWe say S is truth-table (tt) reducible to R via M and we write S <=tt R via M , if S <=T R via M and, for all R0 2 C, M (R0) is defined; i.e., M is total with respect to the oracle [18]. Bounded r... |

3 |
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Citation Context ...that the zero-one law for the constructive packing dimension of Turing and wtt lower spans and degrees also follows from the following theorem due to Fortnow, Hitchcock, Pavan, Vinodchandran and Wang =-=[4]-=-, giving a polynomial-time extractor for constructive packing dimension. For R, S ∈ C, write R ≤ p T S if R ≤T S via an OTM that, on input n, runs in time polynomial in n, and similarly for ≡ p T . Th... |

3 |
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Citation Context ...ee, randomness 1 Introduction Hausdorff [5] initiated the study of dimension as a general framework to define the size of subsets of metric spaces. Recently this framework had been effectivized; Lutz =-=[9]-=- gives an overview of this historical development. Furthermore, Lutz [8, Section 6] reviews early results that anticipated the effectivization of ⋆ Corresponding author.sHausdorff dimension. Construct... |

3 |
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Citation Context ...emum of the dimensions of individual sequences in C. For many classes C of interest in computability theory, the problem of determining the constructive Hausdorff dimension of C remains open. Reimann =-=[14]-=- investigated in particular whether there are degrees of fractional constructive Hausdorff dimension. Stated in terms of individual sequences, Reimann asked which reducibilities (such as Turing, many-... |

3 |
Dimension und "ausseres Mass
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Citation Context ...annot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension. 1 Introduction Hausdorff =-=[7]-=- initiated the study of dimension as a general framework to define the size of subsets of metric spaces. Recently this framework had been effectivized; Lutz [12] gives *Laboratoire d'Informatique Fond... |

3 | Randomness, relativization and Turing degrees. The Journal of Symbolic Logic - Nies, Stephan, et al. - 2005 |

2 | Dimension extractors
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Citation Context .... Furthermore, Nies and Reimann [11] obtained a non-relativized variant of this 2 result and constructed, for each rational α between 0 and 1, a wtt degree of constructive Hausdorff dimension α. Doty =-=[3]-=- attempted positive results by considered the interaction between constructive Hausdorff dimension and constructive packing dimension [1], a dual quantity that is a constructive effectivization of cla... |