Inducing an order on cellular automata by a grouping operation
A grouped instance of a cellular automaton (CA) is another one obtained by grouping several states into blocks and by letting interact neighbor blocks. Based on this operation a preorder ≤ on the set of one dimensional CA is introduced. It is shown that (CA,≤) admits a global minimum and that on the bottom of (CA,≤) very natural equivalence classes are located. These classes remind us the first two well-known Wolfram ones because they capture global (or dynamical) properties as nilpotency or periodicity. Non trivial properties as the undecidability of ≤ and the existence of bounded infinite chains are also proved. Finally, it is shown that (CA,≤) admits no maximum. This result allows us to conclude that, in a “grouping sense”, there is no universal CA.
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