Minimal entropy approximation for cellular automata

TL;DR

This paper introduces a minimal entropy approximation method for cellular automata that often surpasses local structure theory in characterizing automata behaviors, demonstrated through numerical experiments.

Contribution

The paper develops a novel minimal entropy approach for approximating cellular automata dynamics, offering an alternative to existing local structure theory.

Findings

Minimal entropy approximation can outperform local structure theory in certain cases.

Numerical evidence shows improved accuracy in characterizing cellular automata.

Density response curve for rule 26 illustrates the method's effectiveness.

Abstract

We present a method for construction of approximate orbits of measures under the action of cellular automata which is complementary to the local structure theory. The local structure theory is based on the idea of Bayesian extension, that is, construction of a probability measure consistent with given block probabilities and maximizing entropy. If instead of maximizing entropy one minimizes it, one can develop another method for construction of approximate orbits, at the heart of which is the iteration of finitely-dimensional maps, called minimal entropy maps. We present numerical evidence that minimal entropy approximation sometimes spectacularly outperforms the local structure theory in characterizing properties of cellular automata. Density response curve for elementary CA rule 26 is used to illustrate this claim.