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DanielLC comments on Completeness of simulations - Less Wrong Discussion
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Your actions are compact and continuous, thanks to the laws of physics. If the GLUT outputs a prediction in a way that is also compact and continuous, either because it follows the laws of physics or you just programmed it that way, then there's at least one fixed point where he'll take the action he sees. Text output is discrete, and thus this wouldn't work, but there are other ways of doing this. For example, you could show him video of what he'll do.
It says that it's not necessarily possible to make a self-fulfilling prophesy. You can predict what someone will do without input from you, but you may or may not be able to make an accurate prediction given that you tell them the prediction.
For those interested: Brouwer fixed-point theorem
I don't see compactness of the set of possible predictions.
I'd use the Schauder fixed-point theorem so that you don't have to worry as much about what space you use.
One way to define the prediction space would be to have it predict the state of the universe immediately after the prediction is stated. Since anything after that is a function of that point in time, it's sufficient. Every particle is within the distance light would have moved in that time. Each particle has at most the energy of the entire universe plus the maximum energy output of the GLUT response (I'm assuming it's somehow working from outside this universe). This includes some impossible predictions, but that just means that they're not in the range. They're still in the domain. Just take the Cartesian products of the positions and momentums of the particles, and you end up in a Euclidean space.
If you want to take quantum physics into account, you'd need to use the quantum configuration space for each prediction. You only need to worry about everything in the light cone of the prediction again. You define the distance between each configuration space as the integral of the square of the magnitude of the difference of the quantum waveform at each point. Since the integral of the square of the magnitude of each waveform adds to one, it's bounded. Since it's always exactly one, you get a sphere, which isn't convex, but that's just the range. You just take the convex hull, a ball, as the domain.
The actual output of the GLUT may or may not be compact depending on how you display the output.
I don't understand why that follows; can you elaborate?
Additionally, "at least one fixed point" seems distinct from "we can construct X for all situations".
A sufficiently nice function is mathematically guaranteed to have at least one fixed point where f(x) = x. We need to make some assumptions to make it nice enough, but once we do that, we just set x as the hypothetical GLUT output, and f(x) to the GLUT output of the subject's reaction to x, and we know there's some value of x where the GLUT output of the reaction is the same as what the subject is reacting to.
f is the situation, and the fixed point is what you're calling X. Also, I'm not sure if there's a method to construct X. We just know it exists. You can't even just check every value of X, because that only works if it's discrete, which means it's not sufficiently nice.
Thanks for the elaboration; this is a very interesting point that I wasn't aware of. But it does seem to rely on the function having the same domain as its range, which presumably is one of the assumptions going into the niceness. It is not clear to me, although perhaps I'm just not thinking it through, that "future movements of quarks" is the same as "symbols to be interpreted as future movements of quarks".
You could think of it as x is the GLUT output, f(x) is the subject's response, and g(f(x)) is the GLUT's interpretation of the subject's response. f maps from GLUT output to subject response, and g maps from subject response to GLUT output. f and g don't have fixed points, because they don't have the same domain and range. f∘g, however, maps from GLUT output to GLUT output, so it has the same domain and range. I was just calling it f, but this way it might be less confusing.