Not a complete answer, but here’s commentary from a ffdn review of Chapter 14:
Kevin S. Van Horn 7/24/10 . chapter 14 Harry is jumping to conclusions when he tells McGonagall that the Time-Turner isn’t even Turing computable. Time travel simulation is simply a matter of solving fixed-point equation f(x) = x. Here x is the information sent back in time, and f is a function that maps the information received from the future to the information that gets sent back in time. If a solution exists at all, you can find it to any desired degree of accuracy by simply enumerating all possible rational values of x until you find one that satisfies the equation. And if f is known to be both continuous and have a convex compact range, then the Brouwer fixed-point theorem guarantees that there will be a solution.
So the only way I can see that simulating the Time-Turner wouldn’t be Turing computable would be if the physical laws of our universe give rise to fixed-point equations that have no solutions. But the existence of the Time-Turner then proves that the conditions leading to no solution can never arise.
I got the impression that what “not Turing-computable” meant is that there’s no way to only compute what ‘actually happens’; you have to somehow iteratively solve the fixed-point equation, maybe necessarily generating experiences (waves hands confusedly) corresponding to the ‘false’ timelines.
Wow. That’s really cool, thank you. Upvoted you, jeremysalwen and Nornagest. :)
Could you also explain why the HPMoR universe isn’t Turing computable? The time-travel involved seems simple enough to me.
Not a complete answer, but here’s commentary from a ffdn review of Chapter 14:
I got the impression that what “not Turing-computable” meant is that there’s no way to only compute what ‘actually happens’; you have to somehow iteratively solve the fixed-point equation, maybe necessarily generating experiences (waves hands confusedly) corresponding to the ‘false’ timelines.
Sounds rather like our own universe, really.
There’s also the problem of an infinite number of possible solutions.
The number of solutions is finite but (very, very, mind-bogglingly) large.
Ah. It’s math.
:) Thanks.