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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.
Reviewer: Reviewer (Berlin)

##### MSC:
 68Q05 Models of computation (Turing machines, etc.) (MSC2010)
##### Keywords:
constructive dimension; Turing; extractor; degree; randomness
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##### References:
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