Dynamical Equivalence of Two-Dimensional Cellular Automata Under Dihedral Symmetry

The identification of dynamically equivalent rules is a fundamental problem in the mathematical theory of cellular automata. This paper develops a group-theoretic framework for two-dimensional cellular automata on finite square lattices based on the action of the dihedral group [Formula: see text]. The induced action partitions the rule space into symmetry-induced equivalence classes under both null and periodic boundary conditions. It is proved that rules within the same class generate topologically conjugate cellular automata with isomorphic state transition diagrams. Consequently, all dynamical invariants preserved under topological conjugacy are identical within each class, implying that the essential dynamical behavior of a class can be characterized by any one of its representative rules. The proposed framework reduces the effective rule space by [Formula: see text]–[Formula: see text], providing a symmetry-based approach for reducing the complexity of analyzing two-dimensional cellular automata while preserving their underlying dynamical structures.