Randomness extraction and asymptotic Hamming distance

We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1

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)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0, then dim_P(R) >= 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.Comment: The version of this paper appearing in Theory of Computing Systems, 45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to insufficient care with the choice of delta. This version modifies that proof to fix the error

Constructive dimension and weak truth-table degrees

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

Computability Theory (hybrid meeting)

Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work