# Conservative Extensions in Guarded and Two-Variable Fragments

## TL;DR

This paper explores the decidability and complexity of conservative extensions in fragments of first-order logic, showing undecidability in most fragments and decidability with high complexity in a specific intersection.

## Contribution

It establishes the boundaries of decidability for conservative extensions in guarded and two-variable first-order logic fragments.

## Key findings

- Conservative extensions are undecidable in FO fragments containing FO$^2$ or GF.
- Decidable and 2 ext{ExpTime}-complete in GF$^2$ (intersection of FO$^2$ and GF).

## Abstract

We investigate the decidability and computational complexity of (deductive) conservative extensions in fragments of first-order logic (FO), with a focus on the two-variable fragment FO$^2$ and the guarded fragment GF. We prove that conservative extensions are undecidable in any FO fragment that contains FO$^2$ or GF (even the three-variable fragment thereof), and that they are decidable and 2\ExpTime-complete in the intersection GF$^2$ of FO$^2$ and GF.