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Pebble-Depth
arXiv - CS - Computational Complexity Pub Date : 2020-09-24 , DOI: arxiv-2009.12225
Liam Jordon and Philippe Moser

In this paper we introduce a new formulation of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of strings from the perspective of finite-state transducers and pebble transducers. Our notion of pebble-depth satisfies the three fundamental properties of depth: i.e. easy sequences and random sequences are not deep, and the existence of a slow growth law. We also compare pebble-depth to other depth notions based on finite-state transducers, pushdown compressors and the Lempel-Ziv $78$ compression algorithm. We first demonstrate how there exists a normal pebble-deep sequence even though there is no normal finite-state-deep sequence. We next build a sequence which has a pebble-depth level of roughly $1$, a pushdown-depth level of roughly $1/2$ and a finite-state-depth level of roughly $0$. We then build a sequence which has pebble-depth level of roughly $1/2$ and Lempel-Ziv-depth level of roughly $0$.

中文翻译:

卵石深度

在本文中,我们介绍了一种基于卵石传感器的 Bennett 逻辑深度的新公式。这个概念是基于从有限状态换能器和卵石换能器的角度来看弦的最小长度描述复杂性之间的差异来定义的。我们的卵石深度概念满足深度的三个基本属性:即易序列和随机序列不深,存在缓慢增长规律。我们还将卵石深度与基于有限状态传感器、下推压缩器和 Lempel-Ziv $78$ 压缩算法的其他深度概念进行比较。我们首先演示如何存在正常的卵石深序列,即使没有正常的有限状态深序列。我们接下来构建一个序列,其卵石深度水平约为 1 美元,大约 1/2 美元的下推深度水平和大约 0 美元的有限状态深度水平。然后,我们构建了一个序列,其卵石深度级别约为 1/2 美元,Lempel-Ziv 深度级别约为 0 美元。
更新日期:2020-09-28
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