If one tries to use some kind of systematic procedure to test whether systems are universal then inevitably there will be three types of outcomes. Sometimes the procedure will successfully prove that a system is universal, and sometimes it will prove that it is not. But very often the procedure will simply come to no definite conclusion, even after spending a large amount of effort.

Yet in almost all such cases the Principle of Computational Equivalence asserts that the systems are in fact universal. And although almost inevitably it will never be easy to prove this in any great generality, my guess is that, as the decades go by, more and more specific rules will end up being proved to exhibit universality.

But even if one becomes convinced of the abstract fact that out of all possible rules that do not yield obviously simple behavior the vast majority are universal, this still does not quite establish the assertion made by the Principle of Computational Equivalence that rules of this kind that appear in nature and elsewhere are almost always universal.

For it could still be that the particular rules that appear are somehow specially selected to be ones that are not universal. And certainly there are all sorts of situations in which rules are constrained to have behavior that is too simple to support universality. Thus, for example, in most kinds of engineering one tends to pick rules whose behavior is simple enough that one can readily predict it. And as I discussed in Chapter 8, something similar seems to happen with rules in biology that are determined by natural selection.

But when there are no constraints that force simple overall behavior, my guess is that most rules that appear in nature can be viewed as being selected in no special way—save perhaps for the fact that the structure of the rules themselves tends to be fairly simple.

And what this means is that such rules will typically show the same features as rules chosen at random from all possibilities—with the result that presumably they do in the end exhibit universality in almost all cases where their overall behavior is not obviously simple.

But even if a wide range of systems can indeed be shown to be universal this is still not enough to establish the full Principle of Computational Equivalence. For the Principle of Computational

Exportable Images for This Page:

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