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

TL;DR

This paper constructs an oracle where NP equals PSPACE, yet unambiguous polynomial-time problems lack many-one complete sets, providing new insights into classical conjectures and proof system separations.

Contribution

It introduces an oracle demonstrating NP=PSPACE without UP having complete sets, combining previous oracle properties and addressing open questions about proof systems and problem completeness.

Findings

NP equals PSPACE relative to the oracle.

UP has no many-one complete sets in the oracle model.

Existence of TFNP-complete problems with no p-optimal proof systems.

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.