Characterization of 3-State Cellular Automata with Single Length Cycles

We consider the 3-state 3-neighborhood uniform cellular automata, and study scenarios where the automata can generate one or more single length cycles. We show that self-replicating RMTs play the key role in this aspect; the greater the number of self-replicating RMTs in a rule, the more is the number of single length cycles generated by a CA with that rule. However, this is not the only criterion; this work identifies a set of distinguished RMTs which determines whether the given cellular automaton can generate single length cycle or not. The above analysis can also be used to compute the cycle structures in some special cases. Finally, we demonstrate the efficacy of our approach in pattern classification using real world dataset.