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BenjaminFox comments on An Attempt at Logical Uncertainty - Less Wrong
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Hi Bejnamin! Have some links you might already have read.
For an introduction to the topic, you can't go wrong with the scholarly literature: Gaifman's "Reasoning with limited resources and assigning probabilities to arithmetical statements."
I will also happily recommend my own posts on the subject, beginning at the very beginning, here.
For breadth, you might also check out Abram Demski's description, which is good to think about though unsatisfying to me. There's also some discussion on lesswrong and a MIRI document uses a slightly modified version, somewhere.
Anyhow, I won't cotton to any method of assigning a logical probability that takes longer than just brute-forcing the right answer. For this particular problem I think a bottom-up approach is what you want to use.
I appreciate the links. I haven't read Gaifman's paper before, so I'll go ahead and read that.
I see the sentiment there, and that too is a valid approach. That said, after trying to use the bottom-up approach many times and failing, and after seeing others fail using bottom-up approaches, I think that if we can at least build a nonconstructive top-down theory, that would be a starting point. After all, Solomonoff Induction is completely top down, yet it's a very powerful theoretical tool.
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Well, care to explain what I did wrong?
One nonconstructive (and wildly uncomputable) approach to the problem is this one: http://www.hutter1.net/publ/problogics.pdf