# New relations and separations of conjectures about incompleteness in the finite domain

## TL;DR

This paper explores new relationships and separations among conjectures about proof complexity and incompleteness in finite domains, including conditional results and oracle constructions that distinguish various conjectures.

## Contribution

It establishes new relations between proof complexity conjectures, investigates p-optimal proof systems under certain collapses, and constructs oracles to separate key conjectures in relativized worlds.

## Key findings

- Proved new relations between proof complexity conjectures.
- Established conditional independence results for strong theories.
- Constructed oracles demonstrating separations of proof complexity conjectures.

## Abstract

Our main results are in the following three sections: 1. We prove new relations between proof complexity conjectures that are discussed in \cite{pu18}. 2. We investigate the existence of p-optimal proof systems for $\mathsf{TAUT}$, assuming the collapse of $\cal C$ and $\sf N{\cal C}$ (the nondeterministic version of $\cal C$) for some new classes $\cal C$ and also prove new conditional independence results for strong theories, assuming nonexistence of p-optimal proof systems. 3. We construct two new oracles ${\cal V}$ and ${\cal W}$. These two oracles imply several new separations of proof complexity conjectures in relativized worlds. Among them, we prove that existence of a p-optimal proof system for $\mathsf{TAUT}$ and existence of a complete problem for $\mathsf{TFNP}$ are independent of each other in relativized worlds which was not known before.