Optimal Proof Systems for Complex Sets are Hard to Find

Abstract

We provide the first evidence for the inherent difficulty of finding complex sets with optimal proof systems. For this, we construct oracles $O_1$ and $O_2$ with the following properties, where $\mathrm{RE}$ denotes the class of recursively enumerable sets and $\mathrm{NQP}$ the class of sets accepted in non-deterministic quasi-polynomial time. - $O_1$: No set in $\mathrm{PSPACE} \setminus \mathrm{NP}$ has optimal proof systems and $\mathrm{PH}$ is infinite - $O_2$: No set in $\mathrm{RE} \setminus \mathrm{NQP}$ has optimal proof systems and $\mathrm{NP} \neq \mathrm{coNP}$ Oracle $O_2$ is the first relative to which complex sets with optimal proof systems do not exist. By oracle $O_1$, no relativizable proof can show that there exist sets in $\mathrm{PSPACE} \setminus \mathrm{NP}$ with optimal proof systems, even when assuming an infinite $\mathrm{PH}$. By oracle $O_2$, no relativizable proof can show that there exist sets outside $\mathrm{NQP}$ with optimal proof systems, even when assuming $\mathrm{NP} \neq \mathrm{coNP}$. This explains the difficulty of the following longstanding open questions raised by Kraj\'i\v{c}ek and Pudl\'ak in 1989, Sadowski in 1997, K\"obler and Messner in 1998, and Messner in 2000. - Q1: Are there sets outside $\mathrm{NP}$ with optimal proof systems? - Q2: Are there arbitrarily complex sets outside $\mathrm{NP}$ with optimal proof systems? Moreover, relative to $O_2$, there exist arbitrarily complex sets $L \notin \mathrm{NQP}$ having almost optimal algorithms, but none of them has optimal proof systems. This explains the difficulty of Messner's approach to translate almost optimal algorithms into optimal proof systems.