Constructive Dimension and Turing Degrees.

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. [ABSTRACT FROM AUTHOR]