Neighborhood matrix
To form complex symbolic expressions, the language of regular expressions is
convenient, whose operations are union, concatenation, and iteration. Another operation
which is sometimes useful is the overlapping (whose symbol could be 
) of two strings; if the words ax and yb each consist of
letters a and b, joined to the shorter words x and y, then:
(
is the empty word).
With such notation a symbolic de Bruijn matrix
could be defined; the symbolic form of the matrix shown above would be 0.30em
dots now represent the empty word. Such matrices would be multiplied by using 
to multiply individual matrix elements. Subsequently
the advantage of substituting images --- such as the evolved cell --- for the symbolic
neighborhoods will be seen. The full neighborhood matrix is rarely used, but it serves to
illustrate very explicitly the involvement of the row indices and the column indices with
the individual matrix elements.
Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx