Unraveling simplicity in elementary cellular automata

We provide a mathematical proof that a large number of elementary cellular automata are computationally simple. This work is the first systematic classification of elementary cellular automata based on a formal notion of computational complexity.It contrasts with previous approaches in the simplicity of the method – most proofs are just a few lines long and require no heavy computational explorations. More importantly, this type of short proof not only provides evidence for the presence of simple patterns, it also provides reasons for this simplicity.Moreover, thanks to the generality of communication complexity, we hope that this work finds new applications to other natural systems such as neural networks and gene regulatory networks, in particular for real data.