The complexity of generating and checking proofs of membership
Can one compute satisfying assignments for satisfiable Boolean formulas in polynomial time with parallel queries to NP?
Is the unique optimal clique problem (UOCLIQUE) complete for PNP[O(log n)]?
Is the unique satisfiability problem (USAT) NP hard? We define a framework that enables us to study the complexity of generating and checking proofs of membership. We connect the above three questions to the complexity of generating and checking proofs of membership for sets in NP and PNP[O(log n)]. We show that an affirmative answer to any of the three questions implies the existence of coNP checkable proofs for PNP[O(log n)] that can be generated in FP∥NP. Furthermore, we construct an oracle relative to which there do not exist coNP checkable proofs for NP that are generated in FP∥NP. It follows that relative to this oracle all of the above questions are answered negatively.
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