Abstract
Disjoint NP-pairs are pairs ( A , B ) of nonempty, disjoint sets in NP. We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C , and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs ( A , B ) and ( C , D ) is a Turing reduction with the additional property that if the input belongs to A ∪ B , then all queries belong to C ∪ D . We prove under the reasonable assumption that UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs ( A , B ) and ( C , D ) such that ( A , B ) is truth-table reducible to ( C , D ), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DistNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.
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