Languages that are easier than their proofs
Richard Beigel, Mihir Bellare, Joan Feigenbaum, S. Goldwasser · 2002
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 nondeterministic doubleexponential 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. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....