Pushdown and Lempel-Ziv Depth

TL;DR

This paper introduces new formulations of logical depth, compares pushdown, finite-state, and Lempel-Ziv depths, and demonstrates sequences with contrasting depth properties to clarify their differences.

Contribution

It presents a new unary-stack pushdown depth formulation and compares it with finite-state and Lempel-Ziv depths, establishing clear distinctions among these complexity measures.

Findings

Sequences with high finite-state depth and low pushdown depth exist.

Sequences with high Lempel-Ziv depth and low pushdown depth exist.

Properties of all three depth notions are characterized and proved.

Abstract

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 unary-stack pushdown compressors. 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 level 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.