Threshold of complexity in 2D binary Cellular Automata found through Polynomial CNNs

The theoretical studies about the relationship between Cellular Nonlinear Networks (CNNs) and Cellular Automata (CA) have led to the definition of new concepts, such as the complexity index and the threshold of complexity, for 1D CA. However, their interpretation in 2D CA has not been investigated yet. In this paper, we show that Polynomial CNNs can provide a deep insight into 2D binary CA and allow us to introduce a rigorous definition of complexity index. Among other results, we prove that the threshold of complexity in 2D CA is the same as in 1D CA and that the well-known Game of Life is the simplest universal 2D binary Cellular Automaton.