Sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density

TL;DR

This paper examines how noise affects the density classification ability of a specific cellular automaton, revealing that its performance degrades with noise and that its noisy version becomes ergodic with algebraically decaying relaxation times.

Contribution

It provides a numerical assessment of the noise sensitivity of a composite cellular automaton and characterizes the ergodic behavior of its noisy variant.

Findings

Performance degrades with increasing noise levels

Noisy automaton becomes ergodic with algebraic relaxation times

Cannot outperform the noisy Gacs-Kurdyumov-Levin automaton under noise

Abstract

We investigate the sensitivity of the composite cellular automaton of H. Fuk\'{s} [Phys. Rev. E 55, R2081 (1997)] to noise and assess the density classification performance of the resulting probabilistic cellular automaton (PCA) numerically. We conclude that the composite PCA performs the density classification task reliably only up to very small levels of noise. In particular, it cannot outperform the noisy Gacs-Kurdyumov-Levin automaton, an imperfect classifier, for any level of noise. While the original composite CA is nonergodic, analyses of relaxation times indicate that its noisy version is an ergodic automaton, with the relaxation times decaying algebraically over an extended range of parameters with an exponent very close (possibly equal) to the mean-field value.