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

Authors Anton Ehrmanntraut , Fabian Egidy , Christian Glaßer

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Anton Ehrmanntraut
  • Julius-Maximilians-Universität Würzburg, Germany
Fabian Egidy
  • Julius-Maximilians-Universität Würzburg, Germany
Christian Glaßer
  • Julius-Maximilians-Universität Würzburg, Germany

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Anton Ehrmanntraut, Fabian Egidy, and Christian Glaßer. Oracle with P = NP ∩ coNP, but No Many-One Completeness in UP, DisjNP, and DisjCoNP. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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 [P. Pudlák, 2017], 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Proof complexity
  • computational complexity
  • promise classes
  • proof complexity
  • complete sets
  • oracle construction


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