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

TL;DR

This paper constructs an oracle where P equals NP intersect coNP, yet there are no many-one complete sets in UP, disjoint NP pairs, or disjoint coNP pairs, highlighting limitations of certain completeness notions.

Contribution

It provides the first oracle demonstrating P=NP∩coNP without many-one complete sets in key classes, addressing an open problem in incompleteness research.

Findings

No many-one complete sets in UP under the oracle.

No many-one complete disjoint NP pairs.

No many-one complete disjoint coNP pairs.

Abstract

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