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Normally, to find the rule that is suitable to our need, we have to generate all possible rules and apply each rule on a seed and check whether a rule produces a desired pattern. This method is exhaustive and computationaly intensive.

Is there a way to derive the rule that would generate the desired pattern?

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    $\begingroup$ With "pattern" do you mean a (single) configuration or an entire time-state diagram? $\endgroup$ – dkaeae Feb 6 at 10:06
  • $\begingroup$ I mean entire time state diagram after applying the rule, 'n' times. $\endgroup$ – Miner Feb 7 at 18:09
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One way to achieve what you describe would be to encode your problem as a SAT instance, with clauses decribing the desired initial and final pattern and the state transitions under the (fully or partly unknown) evolution rule, and then let a generic SAT solver look for a solution.

In fact, at least for certain classes of rules (specifically, two-state isotropic rules on the Moore neighborhood, a class that includes e.g. Conway's Game of Life), you won't even need to implement this yourself, since there is an existing program called LLS that can do exactly that. While a full tutorial on how to use LLS is beyond the scope of this answer, one of the things it can do is take a partial rule string (i.e. a description of the state transition rule, with some transitions left unspecified) and a list of grids fully or partially describing the evolution of a pattern over several generations, and search for a solution that fills in the unspecified bits of the transition rule and/or the pattern.

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