桥接深度的计算概念
Bridging Computational Notions of Depth
AI总结：
本文研究序列深度概念与深度Π⁰₁类的关系，证明Π⁰₁类成员均为序深度（蕴含强深度）、强深度序列集合可忽略，且强深度的两类变体等价于弱深度。
摘要：
本文研究了序列深度概念之间的关系，即Bennett的强深度与弱深度概念，以及由作者受Levin先前工作启发而提出的深度Π⁰₁类。研究的第一项主要结果表明，Π⁰₁类的每个成员都是序深度的，该性质蕴含强深度。由此结果，我们基于可计算性理论和算法随机性中研究的性质，得到了强深度序列的新例子。我们进一步证明，并非每个强深度序列都是深度Π⁰₁类的成员。第二项主要结果表明，强深度序列的集合是可忽略的，这等价于用某个随机神谕计算出强深度序列的概率为0，该性质也为每个深度Π⁰₁类所共有。最后，我们证明，用先验复杂度和单调复杂度给出的强深度变体等价于弱深度。
英文摘要：
In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $Π^0_1$ classes, introduced by the authors and motivated by previous work of Levin. For the first main result of the study, we show that every member of a $Π^0_1$ class is order-deep, a property that implies strong depth. From this result, we obtain new examples of strongly deep sequences based on properties studied in computability theory and algorithmic randomness. We further show that not every strongly deep sequence is a member of a deep $Π^0_1$ class. For the second main result, we show that the collection of strongly deep sequences is negligible, which is equivalent to the statement that the probability of computing a strongly deep sequence with some random oracle is 0, a property also shared by every deep $Π^0_1$ class. Finally, we show that variants of strong depth, given in terms of a priori complexity and monotone complexity, are equivalent to weak depth.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
