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A crucial feature of these rules, however, is that they make the system behave in a way that depends sensitively on the details of its initial conditions. In the particular case shown, the rules are simply set up to shift every color one position to the left at each step.

And what this does is to make the sequence of colors taken on by any particular cell depend on the colors of cells progressively further and further to the right in the initial conditions. Insofar as the initial conditions are random, therefore, so also will the sequence of colors of any particular cell be correspondingly random.

In general, the rules can be more complicated than those shown in the example on the previous page. But the basic idea of this mechanism for randomness is that the randomness one sees arises from some kind of transcription of randomness that is present in the initial conditions.

The two mechanisms for randomness just discussed have one important feature in common: they both assume that the randomness one sees in any particular system must ultimately come from outside of that system. In a sense, therefore, neither of these mechanisms takes any real responsibility for explaining the origins of randomness: they both in the end just say that randomness comes from outside whatever system one happens to be looking at.

Yet for quite a few years, this rather unsatisfactory type of statement has been the best that one could make. But the discoveries about simple programs in this book finally allow new progress to be made.

The crucial point that we first saw on page 27 is that simple programs can produce apparently random behavior even when they are given no random input whatsoever. And what this means is that there is a third possible mechanism for randomness, which this time does not rely in any way on randomness already being present outside the system one is looking at.

If we had found only a few examples of programs that could generate randomness in this way, then we might think that this third mechanism was a rare and special one. But in fact over the past few chapters we have seen that practically every kind of simple program that we can construct is capable of generating such randomness.

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