An Oracle with no $\mathrm{UP}$-Complete Sets, but $\mathrm{NP}=\mathrm{PSPACE}$

David Dingel; Fabian Egidy; Christian Glaßer

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arXiv · 2404.19104

An Oracle with no $\mathrm{UP}$-Complete Sets, but $\mathrm{NP}=\mathrm{PSPACE}$

David Dingel·Fabian Egidy·Christian Glaßer

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Abstract

We construct an oracle relative to which $\mathrm{NP} = \mathrm{PSPACE}$, but $\mathrm{UP}$ has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [HH88] and one by Ogiwara and Hemachandra [OH93]. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudl\'ak [Pud17], e.g., the implications $\mathsf{UP} \Longrightarrow \mathsf{CON}^{\mathsf{N}}$ and $\mathsf{SAT} \Longrightarrow \mathsf{TFNP}$ are false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that $\mathrm{TFNP}$-complete problems exist, while at the same time $\mathrm{SAT}$ has no p-optimal proof systems.

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BibTeX RIS

David Dingel, Fabian Egidy, Christian Glaßer. 2024-04-29. An Oracle with no $\mathrm{UP}$-Complete Sets, but $\mathrm{NP}=\mathrm{PSPACE}$. https://arxiv.org/abs/2404.19104

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