Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

TL;DR

This paper demonstrates that for a specific number-conserving cellular automaton with a blocking word, a local structure approximation accurately predicts steady-state probabilities, providing insight into why finite-dimensional approximations work well.

Contribution

The paper provides a rigorous proof that a local structure approximation accurately predicts steady states for a particular number-conserving CA with a blocking word, explaining the approximation's effectiveness.

Findings

Exact computation of block probabilities for the CA rule

Local structure approximation matches steady-state probabilities

Number conservation and blocking word are key factors

Abstract

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely-dimensional system. While it is well known that this approximation works surprisingly well for some cellular automata, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.