Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one

• Published in: Physical review. E 2023-02, Vol.107 (2-1), p.024211-024211, Article 024211
• Main Authors: Wolnik, Barbara, Nenca, Anna, Dzedzej, Adam, De Baets, Bernard
• Format: Article
• Language: English
• ISSN: 2470-0045; 2470-0053
• DOI: 10.1103/PhysRevE.107.024211

We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two properties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of the neighborhood by 90^{∘}) and number conservation (i.e., the sum of all the cell states is conserved upon every update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0,1,...,k-1}. In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate cellular automata should not only focus on those that have {0,1,...,k-1} as a state set.