Asymptotic Behaviour and Ratios of Complexity in Cellular Automata

TL;DR

This paper investigates the long-term behavior of one-dimensional cellular automata, revealing that complex rules tend to dominate as the rule space expands, using entropy and complexity measures to analyze their distribution.

Contribution

It introduces a formal analysis of asymptotic and limit behaviors of cellular automata, connecting Wolfram's classification with Kolmogorov complexity.

Findings

Complex rules become more prevalent with larger rule spaces.

Both Shannon's entropy and Kolmogorov complexity indicate increasing complexity dominance.

A formal limit function for Wolfram's classification is proposed.

Abstract

We study the asymptotic behaviour of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighbourhood range and number of symbols (or colours), and how different behaviour is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioural fractions. We also advance a notion of asymptotic and limit behaviour for individual rules, both over initial conditions and runtimes, and we provide a formalisation of Wolfram's classification as a limit function in terms of Kolmogorov complexity.