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But with almost any general classification scheme there are inevitably borderline cases which get assigned to one class by one definition and another class by another definition. And so it is with cellular automata: there are occasionally rules like those in the pictures below that show some features of one class and some of another.

But such rules are quite unusual, and in most cases the behavior one sees instead falls squarely into one of the four classes described above.

So given the underlying rule for a particular cellular automaton, can one tell what class of behavior the cellular automaton will produce?

In most cases there is no easy way to do this, and in fact there is little choice but just to run the cellular automaton and see what it does.

But sometimes one can tell at least a certain amount simply from the form of the underlying rule. And so for example all rules that lie in the first two columns on page 232 can be shown to be unable ever to produce anything besides class 1 or class 2 behavior.

In addition, even when one can tell rather little from a single rule, it is often the case that rules which occur next to each other in some sequence have similar behavior. This can be seen for example in the pictures on the facing page. The top row of rules all have class 1 behavior. But then class 2 behavior is seen, followed by class 4 and then class 3. And after that, the remainder of the rules are mostly class 3.

The fact that class 4 appears between class 2 and class 3 in the pictures on the facing page is not uncommon. For while class 4 is above class 3 in terms of apparent complexity, it is in a sense intermediate

Captions on this page:

Rare examples of borderline cellular automata that do not fit squarely into any one of the four basic classes described in the text. Different definitions based on different specific properties will place these cellular automata into different classes. The rules shown are totalistic ones involving nearest neighbors and three possible colors for each cell. The first rule can be either class 2 or class 4, the second class 3 or 4, the third class 2 or 3 and the fourth class 1, 2 or 3.

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