Classification of cellular automata through texture analysis

Construction of phase diagram for elementary cellular automata by behavior of s-step transition function

It is conjectured that the four classes of elementary cellular automata classified by Wolfram are related to phase transitions; however, the structure of the phase diagram is yet to be clarified. Hence, we propose two parameters that are calculated using the s-step transition function of elementary cellular automata. We calculate these two quantities based on a second-order approximation of the transition function. The phase diagram of these two parameters can separate the rules into four Wolfram classes. Based on this phase diagram, it is clear that the rules of class 4 are located on the phase boundary between the regions of classes 2 and 3.