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"
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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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I combined it with a descendant for space that was mostly developed in the 90s... RCC, region connection calculus, that did for 2d space what allen did for time. It's a useful representation for abstract spatial reasoning.
There was a result that axiomatized RCC (and I think something similar would hold for TIC) as regular hausdorff (t3) spaces. So it shares the logical properties of that. You're doing probabilistic reasoning with objects in T3.

