# P-Optimal Proof Systems for Each NP-Complete Set but no Complete Disjoint NP-Pairs Relative to an Oracle

## TL;DR

This paper constructs an oracle demonstrating that certain proof complexity conjectures, including the non-existence of complete disjoint NP-pairs and P-optimal proof systems for NP-complete sets, cannot be proven relativistically.

## Contribution

It provides a relativized separation showing that these conjectures are independent of relativizable proof techniques.

## Key findings

- An oracle where disjoint NP-pairs are not complete.
- An oracle where NP does not have P-optimal proof systems.
- An oracle where NP∩coNP lacks complete problems.

## Abstract

Pudl\'ak [Pud17] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: - $\mathsf{DisjNP}$: The class of all disjoint NP-pairs does not have many-one complete elements. - $\mathsf{SAT}$: NP does not contain many-one complete sets that have P-optimal proof systems. - $\mathsf{UP}$: UP does not have many-one complete problems. - $\mathsf{NP}\cap\mathsf{coNP}$: $\text{NP}\cap\text{coNP}$ does not have many-one complete problems. As one answer to this question, we construct an oracle relative to which $\mathsf{DisjNP}$, $\neg \mathsf{SAT}$, $\mathsf{UP}$, and $\mathsf{NP}\cap\mathsf{coNP}$ hold, i.e., there is no relativizable proof for the implication $\mathsf{DisjNP}\wedge \mathsf{UP}\wedge \mathsf{NP}\cap\mathsf{coNP}\Rightarrow\mathsf{SAT}$. In particular, regarding the conjectures by Pudl\'ak this extends a result by Khaniki [Kha19].