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Equivalence is concerned not only with the computational sophistication of complete systems but also with the computational sophistication of specific processes that occur within systems.

And when one says that a particular system is universal what one means is that it is possible by choosing appropriate initial conditions to make the system perform computations of essentially any sophistication. But from this there is no guarantee that the vast majority of initial conditions—including perhaps all those that could readily arise in nature—will not just yield behavior that corresponds only to very simple computations.

And indeed in the proof of the universality of rule 110 in the previous chapter extremely complicated initial conditions were used to perform even rather simple computations.

But the Principle of Computational Equivalence asserts that in fact even if it comes from simple initial conditions almost all behavior that is not obviously simple will in the end correspond to computations of equivalent sophistication.

And certainly there are all sorts of pictures in this book that lend support to this idea. For over and over again we have seen that simple initial conditions are quite sufficient to produce behavior of immense complexity, and that making the initial conditions more complicated typically does not lead to behavior that looks any different.

Quite often part of the reason for this, as illustrated in the pictures on the facing page, is that even with a single very simple initial condition the actual evolution of a system will generate blocks that correspond to essentially all possible initial conditions. And this means that whatever behavior would be seen with a given overall initial condition, that same behavior will also be seen at appropriate places in the single pattern generated from a specific initial condition.

So this suggests a way of having something analogous to universality in a single pattern instead of in a complete system. The idea would be that a pattern that is universal could serve as a kind of directory of possible computations—with different regions in the pattern giving results for all possible different initial conditions.

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