This question asks about a corner case of NP classes. From Wikipedia, NP is defined as:
the set of all decision problems for which the instances where the answer is "yes" have efficiently verifiable proofs of the fact that the answer is indeed "yes"
NP-Hard is defined informally as the set of problems that are
at least as hard as the hardest problems in NP
The hardest problems in NP are NP-complete, whose best known solutions run in exponential time.
My question is, are there any decision problems that aren't as hard as NP-complete problems and do not have an efficient verifier?
I know that it cannot be any problem in P, because P is entirely contained in NP. And it probably cannot have exponential complexity, because that would likely move it to NP-complete or NP-hard. I'm thinking if such a problem exists, it would probably have to have sub-exponential complexity. Perhaps someone knows of a sub-exponential complexity problem that lacks an efficient verifiable proof. Either that or someone can tell me that it's been proven that there are no non-NP, non-NP-hard decision problems. Or that it hasn't been proven either way.
Thanks.
