Comparing Elementary Cellular Automata Classifications with a Convolutional Neural Network

Elementary cellular automata (ECA) are simple dynamic systems which display complex behaviour from simple local interactions. The complex behaviour is apparent in the two-dimensional temporal evolution of a cellular automata, which can be viewed as an image composed of black and white pixels. The visual patterns within these images inspired several ECA classifications, aimed at matching the automatas' properties to observed patterns, visual or statistical. In this paper, we quantitatively compare 11 ECA classifications. In contrast to the a priori logic behind a classification, we propose an a posteriori evaluation of a classification. The evaluation employs a convolutional neural network, trained to classify each ECA to its assigned class in a classification. The prediction accuracy indicates how well the convolutional neural network is able to learn the underlying classification logic, and reflects how well this classification logic clusters patterns in the temporal evolution. Results show different prediction accuracy (yet all above 85%), three classifications are very well captured by our simple convolutional neural network (accuracy above 99%), although trained on a small extract from the temporal evolution, and with little observations (100 per ECA, evolving 513 cells). In addition, we explain an unreported "pathological" behaviour in two ECAs.