Ergodicity of some classes of cellular automata subject to noise

TL;DR

This paper investigates how small random perturbations cause various classes of cellular automata to become ergodic, effectively losing memory of initial states, with implications for their stability and computational reliability.

Contribution

It provides a comprehensive analysis showing that multiple classes of cellular automata are highly unstable under noise, leading to ergodicity, using diverse mathematical techniques.

Findings

Cellular automata become ergodic with any positive noise level.

Different classes of CA exhibit instability due to noise.

Multiple mathematical methods are used to prove ergodicity.

Abstract

Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise. We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.