The practical importance of this phenomenon depends greatly however on how far one has to go to get to the threshold of universality.

But knowing that a system like rule 110 is universal, one now suspects that this threshold is remarkably easy to reach. And what this means is that beyond the very simplest rules of any particular kind, the behavior that one sees should quickly become as complex as it will ever be.

Remarkably enough, it turns out that this is essentially what we already observed in Chapter 3. Indeed, not only for cellular automata but also for essentially all of the other kinds of systems that we studied, we found that highly complex behavior could be obtained even with rather simple rules, and that adding further complication to these rules did not in most cases noticeably affect the level of complexity that was produced.

So in retrospect the results of Chapter 3 should already have suggested that simple underlying rules such as rule 110 might be able to achieve universality. But what the elaborate construction in the previous section has done is to show for certain that this is the case.

Class 4 Behavior and Universality

If one looks at the typical behavior of rule 110 with random initial conditions, then the most obvious feature of what one sees is that there are a large number of localized structures that move around and interact with each other in complicated ways. But as we saw in Chapter 6, such behavior is by no means unique to rule 110. Indeed, it is in fact characteristic of all cellular automata that lie in what I called class 4.

The pictures on the next page show a few examples of such class 4 systems. And while the details are different in each case, the general features of the behavior are always rather similar.

So what does this mean about the computational capabilities of such systems? I strongly suspect that it is true in general that any cellular automaton which shows overall class 4 behavior will turn out—like rule 110—to be universal.

We saw at the end of Chapter 6 that class 4 rules always seem to yield a range of progressively more complicated localized structures. And my expectation is that if one looks sufficiently hard at any

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