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that no fixed point is reached, and instead there is exponential growth in total size—with apparently rather random internal behavior.

Other combinators yield still more complicated behavior—sometimes with overall repetition or nesting, but often not.

There are features of combinators that are not easy to capture directly in pictures. But from pictures like the ones on the facing page it is rather clear that despite their fairly simple underlying rules, the behavior of combinators can be highly complex.

And while issues of typical behavior have not really been studied before, it has been known that combinators are universal almost since the concept of universality was first introduced in the 1930s.

One way that we can now show this is to demonstrate that combinators can emulate rule 110. And as the pictures on the next page illustrate, it turns out that just repeatedly applying the combinator expression below reproduces successive steps in the evolution of rule 110.

There has in the past been no overall context for understanding universality in combinators. But now what we have seen suggests that such universality is in a sense just associated with general complex behavior.

Yet we saw in Chapter 3 that there are symbolic systems with rules even simpler than combinators that still show complex behavior. And so now I suspect that these too are universal.

And in fact wherever one looks, the threshold for universality seems to be much lower than one would ever have imagined. And this is one of the important basic observations that led me to formulate the Principle of Computational Equivalence that I discuss in the next chapter.

Captions on this page:

A combinator expression that corresponds to the operation of doing one step of rule 110 evolution.

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