$\mathrm{P}\ne\mathrm{NP}$ and All Non-Empty Sets in $\mathrm{NP}\cup\mathrm{coNP}$ Have P-Optimal Proof Systems Relative to an Oracle

TL;DR

This paper constructs an oracle demonstrating that under certain conditions, P does not equal NP and all non-empty NP and coNP sets possess P-optimal proof systems, advancing understanding of proof complexity and computational class separations.

Contribution

It provides a specific oracle relative to which P≠NP and all non-empty NP∪coNP sets have P-optimal proof systems, addressing open questions in proof complexity.

Findings

Existence of an oracle where P≠NP

All non-empty NP∪coNP sets have P-optimal proof systems in this oracle

Advances understanding of proof systems relative to complexity class separations

Abstract

As one step in a working program initiated by Pudl\'ak [Pud17] we construct an oracle relative to which $\mathrm{P}\ne\mathrm{NP}$ and all non-empty sets in $\mathrm{NP}\cup\mathrm{coNP}$ have $\mathrm{P}$-optimal proof systems.