A (2,1) counterexample

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


With little effort one can establish that


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.

Harold V. McIntosh