  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.
