for example purely repetitive patterns. But the crucial point is that as one looks at cellular automata with progressively greater computational capabilities, one will eventually pass the threshold of universality. And once past this threshold, the set of computations that can be performed will always be exactly the same.

One might assume that by using more and more sophisticated underlying rules, one would always be able to construct systems with ever greater computational capabilities. But the phenomenon of universality implies that this is not the case, and that as soon as one has passed the threshold of universality, nothing more can in a sense ever be gained.

In fact, once one has a system that is universal, its properties are remarkably independent of the details of its construction. For at least as far as the computations that it can perform are concerned, it does not matter how sophisticated the underlying rules for the system are, or even whether the system is a cellular automaton, a Turing machine, or something else. And as we shall see, this rather remarkable fact forms the basis for explaining many of the observations we made in Chapter 3, and indeed for developing much of the conceptual framework that is needed for the new kind of science in this book.

The Rule 110 Cellular Automaton

In previous sections [4, 5, 6] I have shown that a wide variety of different kinds of systems can in principle be made to exhibit the phenomenon of universality. But how complicated do the underlying rules need to be in a specific case in order actually to achieve universality?

The universal cellular automaton that I described earlier in this chapter had rather complicated underlying rules, involving 19 possible colors for each cell, depending on next-nearest as well as nearest neighbors. But this cellular automaton was specifically constructed so as to make its operation easy to understand. And by not imposing this constraint, one might expect that one would be able to find universal cellular automata that have at least somewhat simpler underlying rules.

Fairly straightforward modifications to the universal cellular automaton shown earlier in this chapter allow one to reduce the number

Exportable Images for This Page:

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