(I don't know if there's an analogously strong result in control theory—my vague recollection is that an optimality theorem hasn't been proved rigorously but it's generally taken as true. would know!)
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Not sure what you're asking. Kalman filters are a standard part of control toolkits, and they're "optimal" in the sense of being the optimal LQG solution for linear time invariant systems with gaussian noise. They rest on something called the "certainty equivalence principle"
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Thanks, yeah, I was being vague. Kalman filters are optimal for the thing they are optimal for, and then there’s various broader classes of problems for which analogous things are provably optimal (?), and you’d like a strong result for a very broad class that isn’t done…
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That is exactly correct. Under some nice conditions, a noisy signal can be used in place of a non-noisy signal and you'll still get an optimal solution. LQG is a weird but practically important special case that reduces to a simpler case.
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😂
It's not entirely faking it since general control theory is just nonlinear stochastic ODE theory applied to engineering, so if you know basic grad level math, your intuitions will be roughly right
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To your larger point re: expressivity of logics, control theory can be viewed as a "hack" where we work with rare islands of tractability where probabilistic systems behave as simply as first-order predicate logic systems despite not being generally reduiable to them
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In general, I think the correct logic for control theory is second order... modal logic in possible worlds. Control theory rarely works with that explicitly (I did, briefly) but implicitly, that's the assumed world
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(Although my master’s thesis was on the use of a modal logic of time in robot planning, so maybe I would have :)
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heh my postdoc was also robot planning and also used modal logic along with temporal interval calculus (Allen) representations. It's the obvious tool to apply when you run into certain problems. But I only got as far as "you can represent things this way"
Whoa, I had not thought about Allen’s thing in >30 years and had completely forgotten its existence until you mentioned it!
I had no idea you did modal logic. What a weird coincidence! Mine proved a model-theoretic completeness result for one:
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:)
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