Constructive dimension and weak truth-table degrees

Abstract

submitted to Theory of Computing SystemsThis 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 $\dim(S)$ and constructive packing dimension \Dim(S) is weak truth-table equivalent to a sequence $R$ with \dim(R) \geq \dim(S) / \Dim(S) - \epsilon, for arbitrary $\epsilon > 0$. Furthermore, if \Dim(S) > 0, then \Dim(R) \geq 1 - \epsilon. The reduction thus serves as a \emph{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 \dim(S) / \Dim(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 \emph{regular} sequence $S$ (that is, \dim(S) = \Dim(S)) such that $\dim(S) > 0$, 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 \emph{universal} constructive Hausdorff dimension extractor, and that \emph{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