# Oracle with P=NP∩ coNP, but no Many-One Completeness in UP, DisjNP, and DisjCoNP

We construct an oracle relative to which P = NP∩coNP, but there are no many-one complete sets in UP, no many-one complete disjoint NP-pairs, and no many-one complete disjoint coNP-pairs. This contributes to a research program initiated by Pudlák [Pud17], which studies incompleteness in the finite domain and which mentions the construction of such oracles as open problem. The oracle shows that 𝖭𝖯∩𝖼𝗈𝖭𝖯 is indispensable in the list of hypotheses studied by Pudlák. Hence one should consider stronger hypotheses, in order to find a universal one.

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