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.
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Chen, H. Inverse NP Problems. comput. complex. 17, 94–118 (2008). https://doi.org/10.1007/s00037-008-0240-6
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DOI: https://doi.org/10.1007/s00037-008-0240-6