We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs.We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair, an oracle relative to which optimal proof systems exist, hence complete pairs exist, but no pair is NP-hard, and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.
Razborov [Raz94] proved that existence of an optimal proof system implies existence of a many-one complete disjoint Messner, and Torán [KMT03] defined a stronger form of many-one reduction and claimed to improve Razborov's result by showing under the same assumption that there is a strongly many-one complete disjoint NP-pair. Here we show that the two results are equivalent. More generally, 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 NPpair 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 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 DisjNP has a truth-table-complete disjoint NP-pair, but has no manyone-complete disjoint NP-pair.
We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P NP[1] = P NP [2] , then the polynomial-time hierarchy collapses to S p 2 ⊆ Σ p 2 ∩ Π p 2 . Even showing that the hierarchy collapsed to Σ p 2 remained open prior to this paper.
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