I am guessing there isn’t going to be a super direct analog of Fourier transforms that we just missed, but perhaps there is a way to generalize these things further.

There is of course also the question of what objective such an “external definition” should satisfy.

Do we want to have a conjectural correspondence between the “internal” and “external” definitions that implies a lot of interesting conditional implications?

Or maybe a proven correspondence that can serve as a pathway to proving unconditional results?

Or maybe a research program of proving both the correspondence and the implications?

]]>The questions he raises seem possibly related to computational notions of entropy, though I am not yet sure if there is a precise connection. ]]>

So maybe the dual of a complexity class will be a set of functions from languages in the class to some simple target space, maybe just {0,1}. But what is a “nice function”? An efficiently computable one? How efficiently? Can you do this in such a way that if you take the double dual you go back to the original complexity class?

]]>Descriptive complexity theory does have some of this flavor, but I think it is not as radically different viewpoint ,and hence does not have as far reaching applications, as to the dream version of an external definition.

]]>Also, can you explain the difference between what you’re looking for and something like descriptive complexity?

]]>