The multiplication of ordinary symbolic matrices follows the algebra of regular expressions; that is, multiplication concatenates symbols, union (addition) offers alternatives, the empty set acts like zero.
Moreover the power of a matrix M contains all the possible words of length n, distributed throughout the matrix according to the overlap of the indices of constituent letters as chains are built up to length n. All this overlap in is no longer explicit, yet its matrix elements are still indexed by their initial and final partial neighborhoods.
To track the evolution of a configuration consider the following variant of Eq. 8, which describes the evolution of individual neighborhoods:
which can be decomposed into the sum
As an example, for Rule 22 0.30em
decomposable into the two matrices 0.30em
Altogether, we see that 0 has five counterimages, 1 only three. From the product , 0.30em
the string 00 is seen to have seven counterimages, and so on for all the other configurations.
Harold V. McIntosh