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So what about the cellular automata that we discussed earlier in this book? What kinds of computations can they perform?

At some level, any cellular automaton—or for that matter, any system whatsoever—can be viewed as performing a computation that determines what its future behavior will be.

But for the cellular automata that I have discussed in this section, it so happens that the computations they perform can also conveniently be described in terms of traditional mathematical notions.

And this turns out to be possible for some of the cellular automata that I discussed earlier in this book. Thus, for example, as shown below, rule 94 can effectively be described as enumerating even numbers. Similarly, rule 62 can be thought of as enumerating numbers that are multiples of 3, while rule 190 enumerates numbers that are multiples of 4. And if one looks down the center column of the pattern it produces, rule 129 can be thought of as enumerating numbers that are powers of 2.

But what kinds of computations are cellular automata like the ones on the right performing? If we compare the patterns they produce to the patterns we have seen so far in this section, then immediately we suspect that we cannot describe these computations by anything as simple as saying, for example, that they generate primes.

So how then can we ever expect to describe these computations? Traditional mathematics is not much help, but what we will see is that there are a collection of ideas familiar from practical computing that provide at least the beginnings of the framework that is needed.

Captions on this page:

Examples of simple cellular automata whose evolution corresponds to computations that can easily be described in traditional mathematical terms. In analogy to the previous page, the positions of white cells at the bottom of the rule 94 picture correspond to even numbers, on the left in rule 62 to multiples of 3, in rule 190 to multiples of 4, and in the center column of rule 129 to powers of 2.

Examples of cellular automata that have simple underlying rules but whose overall behavior does not seem to correspond to computations with any kind of simple description in standard mathematical or other terms.

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