A (2,1) counterexample

Symbolic matrices can also be used to prove that a rule is not invertible; consider Rule 30 once more:

0.45em

With little effort one can establish that

0.35em

for n>1. Given that the configuration will always have some ancestors in which 1 remains 1 and others in which 0 becomes 1, there is no neighborhood length for which the ancestral cell is uniquely determined. Therefore Rule 30 cannot be reversed, in spite of the fact that every configuration has a unique ancestor whenever its external environment is specified.

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