Two-dimensional rotation-symmetric number-conserving cellular automata

We present a novel method to study two-dimensional rotation-symmetric number conserving multi-state cellular automata with the von Neumann neighborhood with radius one. This method enables a succinct and easy enumeration in all cases examined so far in literature, i.e., cellular automata with at most five states. Moreover, it allows to find all such cellular automata with six and seven states, while so far, even enumerating six-state rules was beyond the reach of computing machines. Such enumeration allows us to revisit some unresolved questions in the field. Furthermore, we give some rough estimates of the asymptotic growth of the number of such cellular automata with n states, as n tends to infinity. The results are obtained for finite square grids with periodic boundary conditions, but they are also valid in the case of the infinite square grid. (c) 2021 Elsevier Inc. All rights reserved.