P-Optimal Proof Systems for Each NP-Complete Set but no Complete Disjoint NP-Pairs Relative to an Oracle

Abstract

The class of all disjoint NP-pairs does not have many-one complete elements.• SAT: NP does not contain many-one complete sets that have P-optimal proof systems.• UP: UP does not have many-one complete problems.• NP ∩ coNP: NP ∩ coNP does not have many-one complete problems.As one answer to this question, we construct an oracle relative to which DisjNP, ¬SAT, UP, and NP ∩ coNP hold, i.e., there is no relativizable proof for the implication DisjNP ∧ UP ∧ NP ∩ coNP ⇒ SAT. In particular, regarding the conjectures by Pudlák this extends a result by Khaniki [Kha19].