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|Title:||Constructive dimension and turing degrees|
|Source:||Bienvenu, L., Doty, D., Stephan, F. (2009-08). Constructive dimension and turing degrees. Theory of Computing Systems 45 (4) : 740-755. ScholarBank@NUS Repository. https://doi.org/10.1007/s00224-009-9170-1|
|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 dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R)≥(dimH(S)/dimP(S))-ε, 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 Turing degrees. A lower bound of dimH(S)/dimP(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, dimH(S)=dimP(S)) such that dimH(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. © 2009 Springer Science+Business Media, LLC.|
|Source Title:||Theory of Computing Systems|
|Appears in Collections:||Staff Publications|
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