Constructive Dimension and Weak Truth-Table Degrees

  • Laurent Bienvenu
  • David Doty
  • Frank Stephan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

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 \({\rm dim_H}({\it R}) \geq {\rm dim_H}(S) / {\rm dim_P}(S) - \epsilon\), for arbitrary ε> 0. Furthermore, if dimP(S) > 0, then dimP(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 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.

Keywords

constructive dimension weak truth-table extractor degree randomness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Laurent Bienvenu
    • 1
  • David Doty
    • 2
  • Frank Stephan
    • 3
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille Cedex 13France
  2. 2.Department of Computer Science, Iowa State University, Ames, IA 50011USA
  3. 3.School of Computing and Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543Republic of Singapore

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