An Improved Algorithm for Coarse-Graining Cellular Automata

TL;DR

This paper presents a more efficient algorithm for coarse-graining cellular automata, enabling systematic exploration of larger supercell sizes and advancing understanding of emergent phenomena in complex systems.

Contribution

A novel backtracking-based algorithm significantly improves coarse-graining of cellular automata, allowing analysis of larger supercell sizes than previous methods.

Findings

Enabled exploration of supercell sizes up to N=7

Systematic analysis of cellular automata coarse-grainings

Improved computational efficiency over prior algorithms

Abstract

In studying the predictability of emergent phenomena in complex systems, Israeli & Goldenfeld (Phys. Rev. Lett., 2004; Phys. Rev. E, 2006) showed how to coarse-grain (elementary) cellular automata (CA). Their algorithm for finding coarse-grainings of supercell size $N$ took doubly-exponential $2^{2^N}$-time, and thus only allowed them to explore supercell sizes $N \leq 4$. Here we introduce a new, more efficient algorithm for finding coarse-grainings between any two given CA that allows us to systematically explore all elementary CA with supercell sizes up to $N=7$, and to explore individual examples of even larger supercell size. Our algorithm is based on a backtracking search, similar to the DPLL algorithm with unit propagation for the NP-complete problem of Boolean Satisfiability.