In the 1950's Friedberg and Muchnik independently showed there existed computably enumerable but non-computable sets that are strictly weaker than the halting problem. How about a polynomial-time version? We have some natural sets that are good candidates like Factoring and Graph Isomorphism but no proofs that these sets lie in-between P and NP. Any proof would imply P≠NP and Ladner, in a theorem that now bears his name, shows that P≠NP is the only assumption you need.
Ladner shows that if P≠NP there exists an A such that
- A is not in P,
- A is in NP, and
- A is not NP-complete.
Ladner's proof works by blowing holes in SAT using a clever looking-back technique to keep the set in NP. In the end it is a little unsatisfying because from the viewpoint of any fixed length, the set is either NP-complete or easy on that length. Impagliazzo's proof tries to get around this by slowing down the reduction but his proof still leaves large gaps of easily computable inputs. But until we learn how to show P≠NP we won't have any other method for proving the existence of incomplete sets.