the universal systems I have discussed—and must thus violate the Principle of Computational Equivalence.

For as I will describe in more detail later in this chapter, an ordinary universal system cannot in any finite number of steps guarantee to be able to tell whether, say, there is any pattern of black and white squares that satisfies some constraint of the type I discussed at the end of Chapter 5. Yet traditional mathematical models often in effect imply that systems in nature can do things like this.

But I explained at the end of Chapter 7 this is presumably just an idealization. For while in simple cases complicated molecules may for example arrange themselves in configurations that minimize energy, the evidence is that in more complicated cases they typically do not. And in fact, what they actually seem to do is instead to explore different configurations by an explicit process of evolution that is quite consistent with the Principle of Computational Equivalence.

One of the features of cellular automata and most of the other computational systems that I have discussed in this book is that they are in some fundamental sense discrete. Yet traditional mathematical models almost always involve continuous quantities. And this has in the past often been taken to imply that systems in nature are able to do computations that are somehow fundamentally more sophisticated than standard computational systems.

But for several reasons I do not believe this conclusion.

For a start, the experience has been that if one actually tries to build analog computers that make use of continuous physical processes they usually end up being less powerful than ordinary digital computers, rather than more so.

And indeed, as I have discussed several times in this book, it is in many cases clear that the whole notion of continuity is just an idealization—although one that happens to be almost required if one wants to make use of traditional mathematical methods.

Fluids provide one obvious example. For usually they are thought of as being described by continuous mathematical equations. But at an underlying level real fluids consist of discrete particles. And this means that whatever the mathematical equations may suggest, the actual