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Indeed, the rules for this cellular automaton are in some respects much simpler than for even a rather basic linear congruential generator. Yet the sequences it produces seem perfectly random, and do not suffer from any of the problems that are typically found in linear congruential generators.

So why do linear congruential generators not produce better randomness? Ironically, the basic reason is also the reason for their popularity. The point is that unlike the rule 30 cellular automaton that we discussed above, linear congruential generators are readily amenable to detailed mathematical analysis. And as a result, it is possible for example to guarantee that a particular generator will indeed have a maximal repetition period.

Almost inevitably, however, having such a maximal period implies a certain regularity. And in fact, as we shall see later in this book, the very possibility of any detailed mathematical analysis tends to imply the presence of at least some deviations from perfect randomness.

But if one is not constrained by the need for such analysis, then as we saw in the cellular automaton example above, remarkably simple rules can successfully generate highly random behavior.

And indeed the existence of such simple rules is crucial in making it plausible that the general mechanism of intrinsic randomness generation can be widespread in nature. For if the only way for intrinsic randomness generation to occur was through very complicated sets of rules, then one would expect that this mechanism would be seen in practice only in a few very special cases.

But the fact that simple cellular automaton rules are sufficient to give rise to intrinsic randomness generation suggests that in reality it is rather easy for this mechanism to occur. And as a result, one can expect that the mechanism will be found often in nature.

So how does the occurrence of this mechanism compare to the previous two mechanisms for randomness that we have discussed?

The basic answer, I believe, is that whenever a large amount of randomness is produced in a short time, intrinsic randomness generation is overwhelmingly likely to be the mechanism responsible.

We saw in the previous section that random details of the initial conditions for a system can lead to a certain amount of randomness in

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