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Yet one of the characteristics of the kinds of models based on simple programs that I have developed in this book is that they do appear successfully to capture the computational capabilities of a wide range of systems in nature and elsewhere.
So what this means is that systems one uses to make predictions cannot be expected to do computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we might try to predict.
And so, for example, I suspect that it does not take a cellular automaton nearly as complicated as the one on page 767 for it to be an NP-complete problem to determine whether initial conditions exist that lead to particular behavior.
Indeed, in an extreme case it might even be possible to do the analog of what has been done, say, in the computation of symbolic integrals, and to set up some kind of uniform procedure for finding a proof of essentially any short theorem.
Yet all sorts of non-living systems—from crystals to flames—also do this.
But doing this has immediately excluded many of the systems that I have studied in this book—or for that matter that occur in nature.
The number of elements does end up increasing in this particular example, but only by a fixed amount at each step.
Yet despite occasional optimism, traditional approaches do not make this seem close at hand.
Given the first two axioms (commutativity and associativity) it turns out that no shorter third axiom will work in this case (though ones such as f[g[f[a, g[f[a, b]]]], g[g[b]]] b of the same size do work).
And it then becomes somewhat difficult to understand why different regions of spacetime seem to behave so similarly—and do not, for example, seem to depend on the details of their coordinates.