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Blocking transformations

There are no doubt countless ways of embedding an automaton in another whose neighborhoods are constructed differently, or which has a different state set; but one mapping which might be among the first to be tried out would be to group several cells together and treat them as a single unit. It is not as useful a procedure as it might seem at first because there are necessarily redundancies in the evolutionary rules. This is because cells which lie beyond the range of interaction after the grouping cannot influence the evolution, so that blocked neighborhoods in which they differ must have the same rule of evolution. Of course there is a small possibility that this might turn out to be an advantage going in the opposite direction, whereby the number of states might be reduced by dividing the cells into pairs.

For simplicity, suppose that a automaton undergoes an evolution described by the following sequences of cells

Then the transition rule for the blocked automaton is defined by

Conversely, any automaton for which a coding of its states could be exhibited satisfying this rule could be regarded as a automaton by splitting its states.

Harold V. McIntosh