also transmitted our compressed version of S1-2. If that’s the case, then our interlocutor can, by following along with R, reconstruct the missing data and fill in the gaps in his signal. This, of course, requires more transmission overall—we have to transmit the bitmap and the pattern-encoding—but in some cases, this might well be worth the cost (for instance, in cases where there is a tremendous amount of latency between signal transmission and signal reception, so asking to have specific digits repeated is prohibitively difficult). This is in fact very close to how the Transmission-Control Protocol (TCP) works to ensure that the vast amount of data being pushed from computer to computer over the Internet reaches its destination intact.
Ok, but how does this bear on our problem? Next, consider the blanks in the information our interlocutor receives not as errors or miscommunication, but simply as unobserved cases. What our interlocutor has, in this case, is a partial record of S1-2; just as before, he’s missing some of the bits, but rather than resulting from an error in communication, this time we can attribute the information deficit to the fact that he simply hasn’t yet looked at the missing cases. Again, we can construct a similar solution—if he knows R, then just by looking at the bits he does have, then our interlocutor can make a reasonable guess as to what the values of his unobserved bits might be. It’s worth pointing out here that, given enough observed cases, our interlocutor need not have learned of R independently: he might well be able to deduce that it is the pattern underlying the data points he has, and then use that deduction to generate an educated guess about the value of missing bits. If an observer is clever, then, he can use a series of measurements on part of his data-set to ground a guess about a pattern that holds in that data set, and then use that pattern to ground a guess about the values of unmeasured parts of the data set.
At last, then, we’re in a position to say what it is that separates S3 from S0 such that it is
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