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the system from some particular initial condition will ever generate a specific arrangement of cell colors—or whether it will yield a pattern that is, say, ultimately repetitive or ultimately nested.

And if one asks whether any initial conditions exist that lead, for example, to a pattern that does not die out, then this too will in general be undecidable—though in a sense this is just an immediate consequence of the fact that given a particular initial condition one cannot tell whether or not the pattern it produces will ever die out.

But what if one just looks at possible sequences—as might be used for initial conditions—and asks whether any of them satisfy some constraint? Even if the constraint is easy to test it turns out that there can again be undecidability. For there may be no limit on how far one has to go to be sure that out of the infinite number of possible sequences there are really none that satisfy the constraint.

The pictures on the facing page show a simple example of this. The idea is to pick a set of pairs of upper and lower blocks, and then to ask whether there is any sequence of such pairs that satisfies the constraint that the upper and lower strings formed end up being in exact correspondence.

When there are just two kinds of pairs it turns out to be quite straightforward to answer this question. For if any sequence is going to satisfy the constraint one can show that there must already be a sequence of limited length that does so—and if necessary one can find this sequence by explicitly looking at all possibilities.

But as soon as there are more than two pairs things become much more complicated, and as the pictures on the facing page demonstrate, even with very short blocks remarkably long and seemingly quite random sequences can be required in order to satisfy the constraints.

And in fact I strongly suspect that even with just three pairs there is already computational irreducibility, so that in effect the only way to answer the question of whether the constraints can be satisfied is explicitly to trace through some fraction of all arbitrarily long sequences—making this question in general undecidable.

And indeed whenever the question one has can somehow involve looking at an infinite number of steps, or elements, or other things, it

From Stephen Wolfram: A New Kind of Science [citation]