processes that are not obviously simple can be viewed as computations of equivalent sophistication.

And although at first this statement might seem vague and perhaps almost inconsequential, we will see in the course of this chapter that in fact it has many very specific and dramatic implications.

One might have assumed that among different processes there would be a vast range of different levels of computational sophistication. But the remarkable assertion that the Principle of Computational Equivalence makes is that in practice this is not the case, and that instead there is essentially just one highest level of computational sophistication, and this is achieved by almost all processes that do not seem obviously simple.

So what might lead one to this rather surprising idea? An important clue comes from the phenomenon of universality that I discussed in the previous chapter and that has been responsible for much of the success of modern computer technology. For the essence of this phenomenon is that it is possible to construct universal systems that can perform essentially any computation—and which must therefore all in a sense be capable of exhibiting the highest level of computational sophistication.

The most familiar examples of universal systems today are practical computers and general-purpose computer languages. But in the fifty or so years since the phenomenon of universality was first identified, all sorts of types of systems have been found to be able to exhibit universality. Indeed, as I showed in the previous chapter, it is possible for example to get universality in cellular automata, Turing machines, register machines—or in fact in practically every kind of system that I have considered in this book.

So this implies that from a computational point of view even systems with quite different underlying structures will still usually show a certain kind of equivalence, in that rules can be found for them that achieve universality—and that therefore can always exhibit the same level of computational sophistication.

But while this is already a remarkable result, it represents only a first step in the direction of the Principle of Computational

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