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ygert comments on Harry Potter and the Methods of Rationality discussion thread, part 20, chapter 90 - Less Wrong
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A lack of a "reasonable" completion with that property I agree with. But one could easily construct a computable completion. Specifically, the null completion. In other words, everything that that we don't see and is irrelevant to the story simply does not exist. (Until or unless it does at a future point have an effect on the story.)
In fact, you could argue that this completion is the "real" one: Until Eliezer includes something into the story, how can we say that it exists?
Harry's universe may not be Turing computable in the absolute sense assuming that arbitrary time travel is possible, but with even minor limits you can come up with algorithms that largely work, or will work most of the time.
As an example, run the simulation forward taking snapshots at every point until a backward looking event occurs. Take the snapshots of the two time periods and brute force search for a solution (any solution) that can link the two time periods together without breaking constraints. If a solution is found, throw all the intermediate snapshots away and replace them with the found solution. Otherwise, keep the existing data and fail the time travel event in some fashion.
My understanding is that it is possible to find solutions to these kinds of problems (otherwise we wouldn't know and busy beaver numbers.) It's just not possible to find them via some general, easily computable algorithm.
This could explain the six-hour limit on Time-Turners - that's the maximum lookback the Atlantis algorithm allows.