Constructive Dimension and Weak Truth-Table Degrees

This 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 dimH(S) and constructive packing dimension dimP(S) is weak truth-table equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S)− ǫ, for arbitrary ǫ > 0. Furthermore, if dimP(S) > 0, then dimP(R) ≥ 1− ǫ. 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 wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(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 regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S) > 0, the wtt degree of S has Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive dimension extractor.