Investigating Emergence by Coarse Graining Elementary Cellular Automata

We extend coarse graining of cellular automata to investigate aspects of emergence. From the total coarse graining approach introduced by Israeli and Goldenfeld, Coarsegraining of cellular automata, emergence, and the predictability of complex systems, Phys. Rev. E, 2006, we devise partial coarse graining, and show qualitative differences in the results of total and partial coarse graining. Mutual information is used to show objectively how coarse grainings are related to the identification of emergent structure. We show that some valid coarse grainings have high mutual information, and are thus good at identifying and predicting emergent structures. We also show that the mapping from lower to emergent levels crucially affects the quality emergence.