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I was looking over stephen wolfram's work mainly about Elementary Cellular Automata, i see that for all rules with N steps {i.e 200 steps} they show the same pattern {i,e rule 90 running on 20 steps and 100 steps are the same, like Here}.

Unlike rule 110 which shows different behavior when N = 20, 100 or 200, i know that it is universal but does that mean if a cellular automaton structure does not halt {continue to show different pattern when changing N steps} does that means it is always universal ?

Raphael
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1 Answers1

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Your problem is formalized in research as follows: given a CA, is it "Turing complete"? This has been proved for a limited set of CAs including rule 110 in the early 2000s by Cook in a notable proof. it was probably first proved for a CA for the game of life by Conway in the 1970s. There are empirical ways of analyzing CAs that can rule out Turing completeness, and "signs/ circumstantial evidence" that a CA might be Turing complete, but no CA can be proved Turing complete or not from a finite number of example cases.

TMs can execute an arbitrary number of steps and then halt, so that is a characteristic required of the CA simulation. But there is no standard way of defining whether a CA "halts" and thats up to the universality mapping. There is currently no general theory of CA←→TM mappings. There are different mappings in the literature and Cooks mapping is also called "weakly Turing complete" because it requires infinitely recurring patterns specified on the CA input.

There is no general way to find whether particular CAs and rules are universal. This is essentially equivalent to the halting problem in general.

Unfortunately Wikipedia does not highlight Turing completeness in CAs. Here are two useful alternative references.

wythagoras
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