Abstract

Cellular automata (CA) can be viewed as maps in the space of probability measures. Such maps are normally infinitely-dimensional, and in order to facilitate investigations of their properties, especially in the context of applications, finite-dimensional approximations have been proposed. The most commonly used one is known as the local structure theory, developed by H. Gutowitz et al. in 1987. In spite of the popularity of this approximation in CA research, examples of rigorous evaluations of its accuracy are lacking. In an attempt to fill this gap, we construct a local structure approximation for rule 14, and study its dynamics in a rigorous fashion, without relying on numerical experiments. We then compare the outcome with known exact results.

Highlights

  • One-dimensional elementary cellular automata (CA) can be viewed as maps in the space of probability measures over bi-infinite binary sequences

  • Suppose that we start with a large set of initial configurations drawn from a certain distribution

  • We can say that the CA rule transforms the initial probability measure into some other measure, and when we apply the local rule again and again, we obtain a sequence of measures, to be called the orbit of the initial measure

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Summary

Introduction

One-dimensional elementary cellular automata (CA) can be viewed as maps in the space of probability measures over bi-infinite binary sequences (to be called configurations). The goal of this paper is to provide an example of a CA rule for which some block probabilities are known exactly, and for which local structure equations can be analyzed rigorously, without relying exclusively on numerical iterations. Since block probabilities of length 3 are known for this rule, we will construct local approximation of level 3 and investigate its dynamics by simple numerical iterations, but by finding invariant manifolds at the fixed point and determining the nature of the flow on these manifolds.

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