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The idea of a totalistic rule is to take the new color of each cell to depend only on the average color of neighboring cells, and not on their individual colors.
But what does it take to establish that such incompleteness will actually occur in a specific system?
For it takes only a finite calculation to check the solution, and this very calculation can always in effect be thought of as a proof.
For while it is common, say, to take a problem in geometry and reformulate it as a problem in algebra, this is almost always done just by setting up a direct
However, this may still take a lot of computational effort.
This feature of decrement-jump instructions may seem like a detail, but in fact it is crucial—for it is what makes it possible for our register machines to take different paths depending on values in registers through the programs they are given.
If one guesses a key it will normally take a time polynomial in n to check whether the key is correct, and thus the problem of cryptanalysis is in the class known in theoretical computer science as NP or non-deterministic polynomial time (see page 1142 ).
And probably there will be many features that are actually quite easy to take far beyond the originals.
But even with an additive rule and nested behavior, the period depends on quantities like MultiplicativeOrder[2, n] , which probably take more like n steps to evaluate.
But in practice very different shapes can probably have almost identical energies, so that in as much as a given protein always takes on the same shape this must be associated with the dynamics of the process by which the protein folds when it is assembled.