An Improved Algorithm for Coarse-Graining Cellular Automata

12/22/2020

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by   Yerim Song, et al.

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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 ≤ 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.