Cyclic Behaviour of 3-State 3-Neighborhood Cellular Automata

Abstract This work studies the problem of cycle structures of 3-state 3-neighborhood uniform cellular automata under both the null and periodic boundary conditions. Here, the target rules are studied on the aspect of non self replicating RMTs and we classify these rules broadly in two categories: rules with one non self replicat- ing RMT and rules with more than one non self replicating RMTs. The former category ensures the generation of strictly single length cycle cellular automata. An effcient scheme of evaluating the cycle structures of these CAs is also reported. Also, the property of isomorphism is found in the CAs designed by the rules of this category. The latter category brings the notion of multi-length cycles in CA and the minimum count of non-self-replicating RMTs is figured out for multi-length cycle generation. Besides, the cycle structures are estimated for the CAs with at least two non self replicating RMTs.