Examples

In this section, we present some examples of the matrix methods developed in sections 5 and 6 for detecting and classifying $(4,1/2)$ reversible one dimensional cellular automata. These methods were implemented in the system RLCAU that calculates $(k,1/2)$ reversible one dimensional cellular automata using block permutations.

Every cellular automaton in the following examples has an hexadecimal number that identifies each one of these cellular automata. This hexadecimal number is calculated taking the evolution rule, sorting it in descending lexicografical order and dividing this sort in pairs of two neighborhoods, every pair has associated an unique hexadecimal symbol depending of the evolution of its neighborhoods. In this way we have $8$ pairs of two neighborhoods, then we have an hexadecimal number of $8$ symbols identifying every evolution rule in $(4,1/2)$ one dimensional cellular automata.

We shall use a matrix for representing the evolution rule of a $(4,1/2)$ one dimensional cellular automaton. In this matrix, the indices represent partial neighborhoods, thereby the positions of the elements are complete neighborhoods. Every element represents the evolution of every neighborhood. We also use the system NXLCAU [McI90] for developing these examples.