In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $\Pi^0_1$ 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 $\Pi^0_1$ 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 $\Pi^0_1$ 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 $\Pi^0_1$ class. Finally, we show that variants of strong depth, given in terms of a priori complexity and monotone complexity, are equivalent to weak depth.

翻译：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，以及由作者引入并受Levin先前工作启发的深度$\Pi^0_1$类。针对研究的第一个主要结果，我们证明了每个$\Pi^0_1$类的成员都是序深的，这一性质蕴含强深度。基于此结果，我们获得了基于可计算性理论和算法随机性所研究性质的新强深度序列示例。此外，我们进一步证明并非每个强深度序列都是深度$\Pi^0_1$类的成员。针对第二个主要结果，我们证明了强深度序列集合是可忽略的，这等价于用某个随机谕示计算强深度序列的概率为0的陈述，该性质也为每个深度$\Pi^0_1$类所共享。最后，我们展示了以先验复杂度和单调复杂度给出的强深度变体与弱深度等价。