Lempel-Zip Complexity Reference

TL;DR

This paper reviews LZW-based complexity metrics, discussing their definitions, normalizations, and relation to descriptive length and entropy rate, providing reference facts and insights into their properties.

Contribution

It offers a comprehensive reference for LZW-derived complexity metrics, clarifying their definitions, normalizations, and connections to information theory.

Findings

Defines LZW-based complexity metrics and their normalizations

Relates LZW complexity to descriptive length and entropy rate

Provides reference facts and clarifications for LZW complexity measures

Abstract

The aim of this note is to provide some reference facts for LZW---mostly from Thomas and Cover \cite{Cover:2006aa} and provide a reference for some metrics that can be derived from it. LZW is an algorithm to compute a Kolmogorov Complexity estimate derived from a limited programming language that only allows copy and insertion in strings (not Turing complete set). Despite its delightful simplicity, it is rather powerful and fast. We then focus on definitions of LZW derived complexity metrics consistent with the notion of descriptive length, and discuss different normalizations, which result in a set of metrics we call $\rho_0$, $\rho_1$ and $\rho_2$, in addition to the Description Length $l_{LZW}$ and the Entropy Rate.