构造维数与图灵度
Constructive Dimension and Turing Degrees
摘要：
本文研究了图灵度的构造性Hausdorff维数和packing维数。主要结果是：任意构造Hausdorff维数dim_H（S）和构造packing维数dim_P（S）的无穷序列S都与一个满足dim_H（R）<=（dim_H（S）/ dim_P（S））-dim的序列R是图灵等价的，其中任意dim> 0.此外，如果dim_P（S）> 0，则dim_P（R）>= 1 -n。因此，减少作为一个 * 随机性提取器 *，增加了S的算法随机性，如建设性的尺寸。这一结果的一些应用揭示了图灵度的建设性维度。证明了dim_H（S）/ dim_P（S）的下界对任意序列S的图灵度成立.给出了图灵度的构造性填充维数的一个已知的零-一定律的新证明。证明了对于任意正则序列S（即dim_H（S）= dim_P（S）），使得dim_H（S）> 0，S的图灵度具有构造性Hausdorff维数，且packing维数等于1.最后，证明了没有一个图灵约简可以是普适的构造性Hausdorff维数抽取器，并且有界图灵约简不能抽取构造性Hausdorff维数.我们还展示了序列弱真值表和有界图灵减少不同的能力，以提取尺寸。
英文摘要：
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.
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