Inverse NP Problems

Abstract.

The inverse problem relative to a verifier V of proofs of membership for a NP language is the problem of deciding, given a set π of proofs, whether or not there exists a string x having exactly π as its set of proofs. In this paper, we study the complexity of inverse problems.

We develop a new notion of reduction which allows one to compare the complexity of inverse problems. Using this notion, we classify as coNP-complete the inverse problems for the “natural” verifiers of many NP-complete problems. We also show that the inverse complexity of a verifier for a language L cannot be predicted solely from the complexity of L, but rather, is highly dependent upon the choice of verifier used to accept L. In this context, a verifier with a Σ ^p₂ -complete inverse problem is exhibited, giving a new and natural example of a Σ ^p₂ -complete problem.