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RolfAndreassen comments on [HPMOR][Possible Spoilers] Gedankenexperiment: Time Turner Meta-Informational Relativity - Less Wrong
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For Dumbledore, sure. Are the scenarios really indistinguishable for the whatever-it-is that keeps histories consistent and, presumably, enforces the six-hour rule? If not, Dumbledore has an obvious means of distinguishing between the scenarios: Attempt to travel six hours into the past, in steps of one hour, precommitting to telling someone what Amelia told him. In scenario B, the third hop will fail; in A, only the sixth.
This is precisely what I meant when I mentioned the empirical side information detector. The "informational point of view of Dumbledore" is "whatever-it-is that keeps histories consistent," and the indistinguishability only has to come into play in the local context of whenever Dumbledore uses the time turner. In the way I've envisioned it to work, Dumbledore can only use your algorithm to detect leaked information or side-information that was available to him which he might not be aware of.