In a pattern like the one obtained from rule 30 above different computations are presumably not arranged in any such straightforward way. But I strongly suspect that even though it may be quite impractical to find particular computations that one wants, it is still the case that essentially any possible computation exists somewhere in the pattern.
Much as in the case of universality for complete systems, however, the Principle of Computational Equivalence does not just say that a sophisticated computation will be found somewhere in a pattern produced by a system like rule 30. Rather, it asserts that unless it is obviously simple essentially any behavior that one sees should correspond to a computation of equivalent sophistication.
And in a sense this can be viewed as providing a new way to define the very notion of computation. For it implies that essentially any piece of complex behavior that we see corresponds to a kind of lump of computation that is at some level equivalent.
It is a little like what happens in thermodynamics, where all sorts of complicated microscopic motions are identified as corresponding in some uniform way to a notion of heat.
But computation is both a much more general and much more powerful notion than heat. And as a result, the Principle of Computational Equivalence has vastly richer implications than the laws of thermodynamics—or for that matter, than essentially any single collection of laws in science.
The Validity of the Principle
With the intuition of traditional science the Principle of Computational Equivalence—and particularly many of its implications—might seem almost absurd. But as I have developed more and more new intuition from the discoveries in this book so I have become more and more certain that the Principle of Computational Equivalence must be valid.
But like any principle in science with real content it could in the future always be found that at least some aspect of the Principle of Computational Equivalence is not valid. For as a law of nature the principle could turn out to disagree with what is observed in our