$\mathrm{P}$-Optimal Proof Systems for Each $\mathrm{coNP}$-Complete Set and no Complete Problems in $\mathrm{NP}\cap\mathrm{coNP}$ Relative to an Oracle

TL;DR

This paper constructs an oracle where all coNP-complete sets have P-optimal proof systems, and the intersection of NP and coNP lacks complete problems, advancing understanding of proof complexity and problem completeness.

Contribution

It provides a new oracle construction demonstrating coNP-complete sets have P-optimal proof systems and no complete problems in NP∩coNP, addressing open questions in proof complexity.

Findings

Existence of an oracle with P-optimal proof systems for coNP-complete sets

No complete problems in NP∩coNP relative to the oracle

Advancement in proof complexity and problem completeness understanding

Abstract

We build on a working program initiated by Pudl\'ak [Pud17] and construct an oracle relative to which each $\mathrm{coNP}$-complete set has $\mathrm{P}$-optimal proof systems and $\mathrm{NP}\cap\mathrm{coNP}$ does not have complete problems.