Counting Short Trajectories in Elementary Cellular Automata using the Transfer Matrix Method

TL;DR

This paper introduces a transfer matrix method to quantitatively analyze the dynamics of elementary cellular automata, linking entropy measures to Wolfram's classification and enabling exact computation of short trajectory counts.

Contribution

The paper adapts the transfer matrix method to compute the entropy of configurations leading to attractors in ECAs, providing a new quantitative tool for analyzing their global dynamics.

Findings

Class 1 rules rapidly reach maximal entropy for stationary states.

Class 2 rules quickly approach maximal entropy for certain cycle lengths.

Class 3 rules exhibit low or zero entropy that saturates quickly.

Abstract

Elementary Cellular Automata (ECAs) exhibit diverse behaviours often categorized by Wolfram's qualitative classification. To provide a quantitative basis for understanding these behaviours, we investigate the global dynamics of such automata and we describe a method that allows us to compute the number of all configurations leading to short attractors in a limited number of time steps. This computation yields exact results in the thermodynamic limit (as the CA grid size grows to infinity), and is based on the Transfer Matrix Method (TMM) that we adapt for our purposes. Specifically, given two parameters $(p, c)$ we are able to compute the entropy of all initial configurations converging to an attractor of size $c$ after $p$ time-steps. By calculating such statistics for various ECA rules, we establish a quantitative connection between the entropy and the qualitative Wolfram classification scheme. Class 1 rules rapidly converge to maximal entropy for stationary states ($c=1$) as $p$ increases. Class 2 rules also approach maximal entropy quickly for appropriate cycle lengths $c$, potentially requiring consideration of translations. Class 3 rules exhibit zero or low finite entropy that saturates after a short transient. Class 4 rules show finite positive entropy, similar to some Class 3 rules. This method provides a precise framework for quantifying trajectory statistics, although its exponential computational cost in $p+c$ restricts practical analysis to short trajectories.