Pebble-Depth

Abstract: 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 prefixes of infinite sequences 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 type result. 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 that there exists a normal pebble-deep sequence even though there is no normal finite-state-deep sequence. We then show that there exists a sequence which has pebble-depth level of roughly $1/2$ and Lempel-Ziv-depth level of roughly $0$. Finally we show the existence of a sequence which has a pebble-depth level of roughly $1$ and a pushdown-depth level of roughly $1/2$.