Maybe it would be possible to create some kind of equilibrium/fixed-point based AI that, for everything it knows, it knows that it knows, knows that it knows that it knows, and so on.
Then again, perhaps not, because PPAD is much harder than P; so just getting the AI to maintain the perfect self-knowledge invariant as it learns more things could be intractable. There might also be halting problem reductions if the AI is sufficiently powerful, although I'm not sure.
Then again, perhaps not, because PPAD is much harder than P; so just getting the AI to maintain the perfect self-knowledge invariant as it learns more things could be intractable. There might also be halting problem reductions if the AI is sufficiently powerful, although I'm not sure.