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Sewing-Machine comments on The prior of a hypothesis does not depend on its complexity - Less Wrong
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I argued here that it means that an AI can't compute its own prior. That argument is about all mathematical statements. If we just restrict to boolean combinations on a finite sample space then I think a similar argument shows that an AI can't compute its own prior in polynomial time, unless P = NP.
Edit: Wait, that last might be boneheaded. For given inputs we can compute Boolean expressions pretty efficiently, I think we just can't efficiently determine whether they're tautologies.
That depends on what statements are "expressible" by the AI. If it can use quantification ("this boolean formula is true for some/all inputs"), computing the prior becomes NP-hard. An even trickier case: imagine you program the AI with ZFC and ask it for its prior about the continuum hypothesis? On the other hand, you may use clever programming to avoid evaluating prior values that are difficult or impossible to evaluate, like Monte Carlo AIXI does.
Far out.