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An Oracle with no UP-Complete Sets, but NP = PSPACE

Authors David Dingel , Fabian Egidy , Christian Glaßer

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Author Details

David Dingel
  • Julius-Maximilians-Universität Würzburg, Germany
Fabian Egidy
  • Julius-Maximilians-Universität Würzburg, Germany
Christian Glaßer
  • Julius-Maximilians-Universität Würzburg, Germany

Funding

  • Egidy, Fabian: supported by the German Academic Scholarship Foundation.

Acknowledgements

We thank the referees for their helpful feedback. Special thanks to the anonymous reviewer who pointed us to related literature and brought several open questions to our attention.

Cite AsGet BibTex

David Dingel, Fabian Egidy, and Christian Glaßer. An Oracle with no UP-Complete Sets, but NP = PSPACE. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.50

Abstract

We construct an oracle relative to which NP = PSPACE, but UP has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [J. Hartmanis and L. A. Hemachandra, 1988] and one by Ogiwara and Hemachandra [Ogiwara and Hemachandra, 1991]. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudlák [P. Pudlák, 2017], e.g., the implications UP ⟹ CON^𝖭 and SAT ⟹ TFNP are false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that TFNP-complete problems exist, while at the same time SAT has no p-optimal proof systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Complexity theory and logic
Keywords
  • Computational Complexity
  • Promise Classes
  • Complete Sets
  • Oracle Construction

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