Equivalence. For what the result implies is that in many kinds of systems particular rules can be found that achieve universality and thus show the same level of computational sophistication. But the result says nothing about whether such rules are somehow typical, or are instead very rare and special.

And in practice, almost without exception, the actual rules that have been established to be universal have tended to be quite complex. Indeed, most often they have in effect been engineered out of all sorts of components that are direct idealizations of various elaborate structures that exist in practical digital electronic computers.

And on the basis of traditional intuition it has almost always been assumed that this is somehow inevitable, and that in order to get something as sophisticated as universality there must be no choice but to set up rules that are themselves special and sophisticated.

One of the dramatic discoveries of this book, however, is that this is not the case, and that in fact even extremely simple rules can be universal. Indeed, from our discussion in the previous chapter, we already know that among the 256 very simplest possible cellular automaton rules at least rule 110 and three others like it are universal.

And my strong suspicion is that this is just the beginning, and that in time a fair fraction of other simple rules will also be shown to be universal. For one of the implications of the Principle of Computational Equivalence is that almost any rule whose behavior is not obviously simple should ultimately be capable of achieving the same level of computational sophistication and should thus in effect be universal.

So far from universality being some rare and special property that exists only in systems that have carefully been built to exhibit it, the Principle of Computational Equivalence implies that instead this property should be extremely common. And among other things this means that universality can be expected to occur not only in many kinds of abstract systems but also in all sorts of systems in nature.

And as we shall see in this chapter, this idea already has many important and surprising consequences. But still it is far short of what the full Principle of Computational Equivalence has to say.

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