On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree

David J. Webb
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Abstract

We prove various results connected together by the common thread of computability theory. First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension. We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how this notion of $\Pi ^0_1$ -immunity is connected to other immunity notions, and construct $\Pi ^0_1$ -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a $\Pi ^0_1$ -immune set. Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random. Abstract prepared by David J. Webb. E-mail: dwebb@math.hawaii.edu URL: https://arxiv.org/pdf/2209.05659.pdf
论算法维数、免疫和梅德韦杰夫度的新概念
我们证明了由可计算性理论的共同线索联系在一起的各种结果。首先,我们研究了一种新的算法维数的概念——不可逃避维数,它位于有效豪斯多夫维数和包装维数之间。我们还研究了它的推广,得到了图灵度嵌入到维数概念中的方法。然后,我们研究了在前面的研究过程中产生的一个新的可计算理论免疫的概念,即没有共枚举子集的自然数集。我们证明了$\Pi ^0_1$ -免疫的概念如何与其他免疫概念联系起来,并在整个高/低和Ershov层次中构造$\Pi ^0_1$ -免疫实数。我们还研究了那些不能计算或不能共枚举$\Pi ^0_1$免疫集的度。最后,我们讨论了最近发现的将Kolmogorov-Loveland随机输入转换为Martin-Löf随机输出的真值表约简,该约简利用了这样一个KL-random的至少一半本身是ML-random的事实。我们证明,没有更好的算法依赖于这个事实,在某种意义上,没有正的,线性的,或有界的真值表约简做到了这一点。我们也将这些结果推广到从无限多个输入输出随机性的问题,其中只有一些是随机的。David J. Webb撰写的摘要。电子邮件:dwebb@math.hawaii.edu URL: https://arxiv.org/pdf/2209.05659.pdf
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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