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Excluded states for Rule 22

As an example, consider Rule 22. For brevity, label the nodes A, C, D, F. They are linked by zeroes and ones according to the following table:

The links are not evenly distributed between zeroes and ones; indeed there are five zeroes and three ones so there just couldn't be a balance. In this case the discrepancy shows up at node F, which will surely cause trouble.

Figure gif shows the connections between the subsets of the initial subset {A C D F}, and the resulting subset diagram. Note that the penultimate row, containing the unit classes, almost reflects the de Bruijn diagram, except where the imbalance in links results in a link descending to the empty set or rising to a higher level (which can only be to the row of doublets, for a binary rule).


Figure: The subset diagram (exit links) for Rule 22. 

It seems that all sixteen subsets have to be used to complete the analysis. It also appears that both 10101001 and its reverse, 10010101, lead directly to the null set, so they are both excluded words; the shortest ones that can be found for Rule 22.

Harold V. McIntosh