Rules spanning three generations

The next rule depends upon three generations of evolution, following the same convention as before:

 

Time reversal cannot be achieved as before by a rule with a two cell neighborhood, because the new state is independent of both b and d. Again supposing a cell to the right of the first two, all three generating the evolved pair

the lack of b prevents recovery of , while the lack of i prevents releasing d from the combination in which it is bound (admittedly, a degenerate would help) in order to recover . Not even can be obtained, for the lack of g; even less information is available going further afield.

However, a third cell in the second generation, involving yet a fourth cell from the first generation, would have the form wherein the value of x is immaterial, but i combines with f to release d from revealing . Altogether, we have a reversible rule which requires an (three cell neighborhood) rule for its reversion. Altogether, the inverse rule appears to be

 

Evidently considerable ingenuity can be exercised in devising reversible automata, but it would be desirable to have criteria of necessity or sufficiency for the process. One significant aspect of the formulas shown so far is their uniformity in the respect that every state has the same number of counterimages as any other. Additionally, the rules are entirely local; even though the size of the neighborhood of an inverse rule may be different from that of the original rule, it is fixed; moreover distant boundary conditions do not affect it.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx