# Density classification performance and ergodicity of the Gacs-Kurdyumov-Levin cellular automaton model IV

## TL;DR

This study evaluates the density classification performance and ergodic properties of Gacs-Kurdyumov-Levin's four-state cellular automaton model IV, revealing its similarities to model II and signs of a phase transition under noise.

## Contribution

It provides the first detailed numerical analysis of model IV's classification ability and ergodic behavior, highlighting its performance and phase transition characteristics.

## Key findings

- Model IV performs comparably to model II in density classification.
- Additional states slow convergence but slightly improve classification accuracy.
- Signs of an ergodic-nonergodic phase transition under noise were observed.

## Abstract

Almost four decades ago, Gacs, Kurdyumov, and Levin introduced three different cellular automata to investigate whether one-dimensional nonequilibrium interacting particle systems are capable of displaying phase transitions and, as a by-product, introduced the density classification problem (the ability to classify arrays of symbols according to their initial density) in the cellular automata literature. Their model II became a well known model in theoretical computer science and statistical mechanics. The other two models, however, did not receive much attention. Here we characterize the density classification performance of Gacs, Kurdyumov, and Levin's model IV, a four-state cellular automaton with three absorbing states---only two of which are attractive---, by numerical simulations. We show that model IV compares well with its sibling model II in the density classification task: the additional states slow down the convergence to the majority state but confer a slight advantage in classification performance. We also show that, unexpectedly, initial states diluted in one of the nonclassifiable states are more easily classified. The performance of model IV under the influence of noise was also investigated and we found signs of an ergodic-nonergodic phase transition at some small finite positive level of noise, although the evidences are not entirely conclusive. We set an upper bound on the critical point for the transition, if any.