But the point is that all the evidence I have so far suggests that for any class 4 rule such a construction will eventually turn out to be possible.

So what kinds of rules show class 4 behavior?

Among the 256 so-called elementary cellular automata that allow only two possible colors for each cell and depend only on nearest neighbors, the only clear immediate example is rule 110—together with rules 124, 137 and 193 obtained by trivially reversing left and right or black and white. But as soon as one allows more than two possible colors, or allows dependence on more than just nearest neighbors, one immediately finds all sorts of further examples of class 4 behavior.

In fact, as illustrated in the pictures on the facing page, it is sufficient in such cases just to use so-called totalistic rules in which the new color of a cell depends only on the average color of cells in its neighborhood, and not on their individual colors.

In two dimensions class 4 behavior can occur with rules that involve only two colors and only nearest neighbors—as shown on page 249. And indeed one example of such a rule is the so-called Game of Life that has been popular in recreational computing since the 1970s.

The strategy for demonstrating universality in a two-dimensional cellular automaton is in general very much the same as in one dimension. But in practice the comparative ease with which streams of localized structures can be made to cross in two dimensions can reduce some of the technical difficulties involved. And as it turns out there was already an outline of a proof given even in the 1970s that the Game of Life two-dimensional cellular automaton is universal.

Returning to one dimension, one can ask whether among the 256 elementary cellular automata there are any apart from rule 110 that show even signs of class 4 behavior. As we will see in the next section, one possibility is rule 54. And if this rule is in fact class 4 then it is my expectation that by looking at interactions between the localized structures it supports it will in the end—with enough effort—be possible to show that it too exhibits the phenomenon of universality.

Exportable Images for This Page:

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