string:

S0: 00001000011111

The numbers in this string are not consistent with our hypothesis that all the numbers in the sequence at large are generated by R. Does this mean that we’ve failed in our goal of identifying a pattern, though? Not necessarily. Why not?

There’s another important question that we’ve been glossing over in our discussion here: for a pattern in some data to be genuine must it also be global[1]? That is, for us to say reasonably that R describes the sequence S, must R describe the sequence S everywhere? Here’s all the data we have now:

S0-2: 000010000111111100010101100010101100010101

It is clear that we can no longer say that R (or indeed any single pattern at all) is the pattern generating all of S. This is not at all the same thing as saying that we have failed to identify a pattern in S simpliciter, though. Suppose that we have some reason to be particularly interested in what’s going on in a restricted region of S: the region S1-2. If that’s the case, then the fact that R turns out not to hold for the totality of S might not trouble us at all; identifying a universal pattern would be sufficient for predicting what sequence of numbers will show up in S1-2, but it is by no means necessary. If all we’re interested in is predicting the sequence in a particular region of S, identifying a pattern that holds only[2] in that region is no failure at all, but rather precisely

1. The sense of 'global' here is the computer scientist's sense—a global pattern is one that holds over the entirety of the data set in question.
2. Of course, it might not be true that R holds only in S1-2. It is consistent with everything we’ve observed about S so far to suppose that the sub-set S0 and the sub-set S1-2 might be manifestations of an over-arching pattern, of which R is only a kind of component, or sub-pattern.

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