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Unnamed comments on Completeness of simulations - Less Wrong Discussion
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Comments (31)
There is not a general solution.
Suppose that the human is just a mathematical function, where the input is a real number x and his output is a real number y. The person is the function y=f(x); and (in the non-self-referential case) an accurate GLUT just predicts: if the person is given the number x, then he will give you the number y.
In the second case, where the person is told what the GLUT says, he is now a function of two variables h(x,y)=z. The person is given the number x, and told that the GLUT predicts that he will output y, and then he outputs z. In order for the GLUT to be an accurate predictor we must have y=z, but that is not possible in general. For instance, consider h(x,y)=y+1; the person always outputs a number that is 1 more than what the GLUT said he would output.
The problem that the GLUT-maker faces is that she knows the human's function, h(x,y)=z, and she must create a GLUT function, y=g(x), such that g(x)=h(x,g(x)) for all x. I presume that there is math out there which says what features h(x,y) must have in order for this to be possible, but I don't know it.
What about Exists y: h(x,y) = y? If this holds, take Forall x: g(x) = y and we are done.
Edit: to be more general, we can let y depend on x, which gives Forall x: Exists y(x): h(x,y(x)) = y(x), but that is actually the original statement with the letter y standing in place of the original g. I doubt there is anything more to say about such functions, at least in general.