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Constructive Dimension and Turing Degrees

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Abstract

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. 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 Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing 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, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale de Marseille, Université de Provence, 39 rue Joliot-Curie, 13453, Marseille Cedex 13, France

    Laurent Bienvenu

  2. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    David Doty

  3. School of Computing and Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Frank Stephan

Authors
  1. Laurent Bienvenu
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  2. David Doty
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  3. Frank Stephan
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Correspondence to Laurent Bienvenu.

Additional information

F. Stephan was supported in part by NUS research grants Nos. R252-000-212-112 and R252-000-308-112.

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Bienvenu, L., Doty, D. & Stephan, F. Constructive Dimension and Turing Degrees. Theory Comput Syst 45, 740–755 (2009). https://doi.org/10.1007/s00224-009-9170-1

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  • DOI: https://doi.org/10.1007/s00224-009-9170-1

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