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>> (thus far, the biggest obstacle to proving P != NP has been the difficulty of proving anything about lower bounds on runtime that isn't directly tied to spac

>> (thus far, the biggest obstacle to proving P != NP has been the difficulty of proving anything about lower bounds on runtime that isn't directly tied to space consumption) I probably don't understand the proofs of these bounds - it seems to me though, that any lower bound without some relationship to the time or space to input the problem into some kind of computer, implies a belief that an instance might be solvable in-place, in some physical system. I have some doubts about that ....

Jweb_Guru 10y ago
To be clearer, a trivial lower bound is something like "the input is of size n and for any n-1 length yes-instance you can find n-length extensions that are no-instances, so you have to check all n elements." It doesn't tell us anything interesting about the worst-case problem complexity. What's been surprising is how hard it is to make meaningful lower-bound statements stronger than that (well, not surprising to people who work on this stuff for a living, I guess). But that doesn't make it more likely that P = NP, really, except in the sense that we (obviously) haven't characterized the entire space of problems in P yet. To more specifically address your "in-place" concerns, you can often formulate the problem in terms of circuit complexity--circuits are inherently parallel, so minimum circuit depth is a good proxy for a required number of steps, and it's also generally agreed that if you can design a circuit to do something it's physically realizable (even though that's arguably not always true, especially in cases with very wide fanout). So a (somewhat) promising direction has been to try to find circuit complexity lower bounds for problems, especially nontrivial ones.