P-Optimal Proof Systems for Each Set in NP but no Complete Disjoint NP-pairs Relative to an Oracle
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Consider the following conjectures: - DisjNP: there exist no many-one complete disjoint NP-pairs. - SAT: there exist P-optimal proof systems for SAT. Pudlák [Pud17] lists several conjectures (among these, DisjNP and SAT) and asks for new equivalences or oracles that separate corresponding relativized conjectures. We partially answer this question by constructing an oracle relative to which - no many-one complete disjoint NP-pairs exist - and each problem in NP has a P-optimal proof system, i.e., there is no relativizable proof for DisjNPSAT. Since Khaniki [Kha19] constructs an oracle showing that there exists no relativizable proof for the converse implication, the conjectures DisjNP and SAT are independent in a relativized way. In a similar way, our oracle shows that DisjNP and TFNP as well as DisjNP and DisjCoNP are also independent in a relativized way, where TFNP is the conjecture that TFNP has complete elements with respect to polynomial reductions and DisjCoNP is the conjecture that there exist no many-one complete disjoint coNP-pairs.
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