very short computation that tests whether that position lies outside the center triangular region of the pattern. And in a class 4 cellular automaton such as rule 110 one can readily shortcut the process of evolution for at least a limited number of steps in places where there happen to be only a few well-separated localized structures present.

And indeed in general almost any regularities that we manage to recognize in the behavior of a system will tend to reflect some kind of computational reducibility in this behavior.

If one views the pattern of behavior as a piece of data, then as we discussed in Chapter 10 regularities in it allow a compressed description to be found. But the existence of a compressed description does not on its own imply computational reducibility. For any system that has simple rules and simple initial conditions—including for example rule 30—will always have such a description.

But what makes there be computational reducibility is when only a short computation is needed to find from the compressed description any feature of the actual behavior.

And it turns out that the kinds of compressed descriptions that can be obtained by the methods of perception and analysis that we use in practice and that we discussed in Chapter 10 all essentially have this property. So this is why regularities that we recognize by these methods do indeed reflect the presence of computational reducibility.

But as we saw in Chapter 10, in almost any case where there is not just repetitive or nested behavior, our normal powers of perception and analysis recognize very few regularities—even though at some level the behavior we see may still be generated by extremely simple rules.

And this supports the assertion that beyond perhaps some small superficial amount of computational reducibility a great many systems are in the end computationally irreducible. And indeed this assertion explains, at least in part, why our methods of perception and analysis cannot be expected to go further in recognizing regularities.

But if behavior that we see looks complex to us, does this necessarily mean that it can exhibit no computational reducibility? One way to try to get an idea about this is just to construct patterns

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