Further oracles separating conjectures about incompleteness in the finite domain

Dose, Titus

doi:10.1016/j.tcs.2020.09.040

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2022

2023

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Cited by 2 publications

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“…Our oracle supports this estimation, as it rules out relativizable proofs for "CON ⇒ NP ∩ coNP" and "TFNP ⇒ NP ∩ coNP." By Dose [Dos20c;Dos20a], the same holds for the converse implications. Overall, we recognize a strong independence between NP ∩ coNP and all remaining hypotheses: (i) There does not exist a relativizable proof for NP ∩ coNP, even if we simultaneously assume all remaining hypotheses.…”

Section: Introductionmentioning

confidence: 70%

See 1 more Smart Citation

Oracle with $\mathrm{P=NP\cap coNP}$, but no Many-One Completeness in UP, DisjNP, and DisjCoNP

Ehrmanntraut¹,

Egidy²,

Glaßer³

2022

Preprint

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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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Section: Introductionmentioning

confidence: 70%

“…(ii) There exists a relativizable proof for the implication NP ∩ coNP ⇒ CON ∨ SAT [KMT03]. But there does not exist a relativizable proof showing that NP ∩ coNP implies one of the remaining hypotheses [Dos20c;Dos20a].…”

Section: Introductionmentioning

confidence: 99%