An Oracle with no $\mathrm{UP}$-Complete Sets, but $\mathrm{NP}=\mathrm{PSPACE}$
David Dingel, Fabian Egidy, Christian Glaßer
TL;DR
This work constructs an oracle $\Phi$ relative to which $\mathrm{NP}^{\Phi} = \mathrm{PSPACE}^{\Phi}$ while $\mathrm{UP}^{\Phi}$ has no polynomial-time many-one complete sets. The construction uses a PSPACE-hard encoding $Z^{\Phi}$ of QBF$^{\Phi}$ and witness languages $W_i^{\Phi}$ to force noncompleteness, balancing UP-separations with NP=PSPACE. Consequences for Pudlák's program include refuting several implications such as $\mathsf{UP} \Rightarrow \mathsf{CON}^{\mathsf{N}}$ and showing that $\mathsf{CON}$ and $\mathsf{SAT}$ become equivalent under the oracle, with $\mathrm{NP} = \mathrm{coNP}$ in this world. The findings also illustrate that TFNP-complete problems can exist even when SAT lacks p-optimal proof systems, highlighting limits of current methods in proof complexity. Overall, the results demonstrate that strong separations among foundational conjectures can be achieved in oracle worlds, informing the landscape of computational and proof complexity.
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 Pudlák [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.