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Constructive dimension and Turing degrees. (English) Zbl 1183.68281
Summary: 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 \(\dim_{H}(R)\geq (\dim_{H}(S)/\dim_{P}(S)) - \epsilon \), for arbitrary \(\epsilon >0\). Furthermore, if \(\dim_{P}(S)>0\), then \(\dim_{P}(R)\geq 1 - \epsilon \). 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.

MSC:
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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