There is another way to use the cartesian product to create a composite
automaton. Begin with the 
automaton whose transition rule is
. Define, for pairs such as 
(
is exclusive
or),
The left part of the pair simply remembers the right part from the old generation while the right part combines the new and old values of the cell. Thus both items of information are always present and can be extracted if needed. The inverse rule, whose structure is similar, follows the same procedure.
States a and e are of no consequence in defining the transition,
serving simply as a one-generation memory which will be forgotten by
the second generation. Nevertheless their presence is essential,
likewise the fact that 
must ignore them.
As with the cartesian product, we have a 
automaton with the
square of the number of states and the same width. Still another way to
induce an automaton in the cartesian product would be to omit the
exclusive or:
from which the previous generation with respect to 
can always
be recovered, but not with respect to 
. By contrast all of
's previous generations can be recovered, but it must not be
thought that the same applies to 
, even though it participates
in the definition of 
.
For example, suppose that 
, which is not reversible at
all. Then 
, indeed reversible, but
of no help whatsoever for obtaining 
.