Classifing $({k^{}},1/2)$ reversible one dimensional cellular automata using their periodical behavior

The methods presented in section 5 could be useful for comparing the dynamical behavior of $({k^{}},1/2)$ reversible one dimensional cellular automata and thereby we can get a dynamical classification of these systems.

In particular we will use the transitive closure of the connectivity relation defined in Table 1 for analyzing periodical behavior. Take two $({k^{}},1/2)$ reversible one dimensional cellular automata and the transitive closures of their connectivity relations, then we consider that these automata belong to the same dynamical class if:

1. The number of equivalence classes is the same in both equivalence relations.
2. There exists an isomorphism from every equivalence class in one equivalence relation to another equivalence class in the other equivalence relation.

An example of equivalence classes that belong to the same dynamical class is showed in Figure 13.

Figure 13: Equivalence relations of the connectivity matrices that belong to the same dynamical class

With this process we are comparing the quantitative behavior of the transitive closure of the connectivity relations, that is, we are contrasting if the block permutations of different $({k^{}},1/2)$ reversible one dimensional cellular automata have the same periodical behavior.

This way of classifing $({k^{}},1/2)$ reversible one dimensional cellular automata is based more in an experimental and numerical approach than in a theoretical one, besides, this process is easy for calculating if the $({k^{}},1/2)$ reversible one dimensional cellular automata have not a big number of states.