On Enumeration of Motion Representable Two-Dimensional Two-State Number-Conserving Cellular Automata

We study two-dimensional two-state number-conserving cellular automata (NCCAs). Although one-dimensional NCCAs are known to be characterized by motion representations, there is no such result in the general case of two-dimension. We first enumerate two-state 2 × 2 and 2 × 3 rectangular neighborhood NCCAs by a local function based exhaustive searching. We then propose an algorithm to enumerate motion representable two-state rectangular neighborhood NCCAs. The results agree with the above 2 × 2 and 2 × 3 cases. We conjecture that the algorithm can enumerate two-state rectangular neighborhood NCCAs.