Two (2,1/2) examples
Some surjective rules are injective, others are not. As an example of the first kind, consider Rule 6, and some of the products of ancestor matrices. Recalling the definition of in Eq. 12, there is either a left or a right ancestor matrix; choosing the right handed version, when , otherwise. Accordingly,
The four pair products are
Here the detail of interest is the fact that neither the left nor the right member of the matrix elements in is consistent, nor does any such consistency arise for any of the higher order products (note that row and column indices must also be taken into account when judging consistency). Consequently one concludes that this rule does not have a general purpose inverse.
Repeating the display for Rule 12 produces
This time, there is a consistency in the left member of each ancestral pair, so that each evolved pair has a unique ancestor which can be used as the evolved state of the inverse rule, and all this quite independently of boundary conditions. Note that once consistency is found, it persists for all longer products.
Harold V. McIntosh