A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and non-deterministic double-exponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NP-completeness, self-reducibility, interactive proofs, program checking, sparse sets,