This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon and Moser, notions of finite-state-depth and pushdown-depth were examined and compared. These were based on finite-state transducers and information lossless pushdown compressors respectively. Unfortunately a full separation between the two notions was not established. This paper introduces a new formulation of pushdown-depth based on restricting how fast a pushdown compressor's stack can grow. This improved formulation allows us to do a full comparison by demonstrating the existence of sequences with high finite-state-depth and low pushdown-depth, and vice-versa. A new notion based on the Lempel-Ziv `78 algorithm is also introduced. Its difference from finite-state-depth is shown by demonstrating the existence of a Lempel-Ziv deep sequence that is not finite-state deep and vice versa. Lempel-Ziv-depth's difference from pushdown-depth is shown by building sequences that have a pushdown-depth of roughly $1/2$ but low Lempel-Ziv depth, and a sequence with high Lempel-Ziv depth but low pushdown-depth. Properties of all three notions are also discussed and proved.

中文翻译：

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下推和 Lempel-Ziv 深度

本文扩展了现有的 Bennett 逻辑深度并介绍了新的表述。在 Jordon 和 Moser 先前发表的工作中，对有限状态深度和下推深度的概念进行了检查和比较。它们分别基于有限状态传感器和信息无损下推压缩器。不幸的是，这两个概念之间没有完全分离。本文介绍了一种新的下推深度公式，该公式基于限制下推压缩器堆栈的增长速度。这种改进的公式使我们能够通过证明具有高有限状态深度和低下推深度的序列的存在来进行全面比较，反之亦然。还介绍了基于 Lempel-Ziv '78 算法的新概念。它与有限状态深度的区别通过证明非有限状态深度的 Lempel-Ziv 深度序列的存在来显示，反之亦然。Lempel-Ziv 深度与下推深度的差异通过构建下推深度约为 1/2 美元但 Lempel-Ziv 深度较低的序列以及具有高 Lempel-Ziv 深度但下推深度较低的序列来显示。还讨论并证明了所有三个概念的性质。