# The Complexity of Definability by Open First-Order Formulas

## TL;DR

This paper studies the computational complexity of determining whether a relation can be defined by open first-order formulas in finite structures, establishing coNP-completeness and parameterized complexity results.

## Contribution

It formally defines the open definability problem and proves its coNP-completeness, along with its parameterized complexity classification as coW[1]-complete.

## Key findings

- Open definability is coNP-complete.
- The problem is coW[1]-complete when parameterized by size and arity of the target relation.
- Complexity results hold for any vocabulary with at least one binary relation.

## Abstract

In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the $\mathbf{\mathcal{L}}$-Definability Problem for finite structures takes as input a finite structure $\mathbf{A}$ and a target relation $T$ over the domain of $\mathbf{A}$, and determines whether there is a formula of $\mathbf{\mathcal{L}}$ whose interpretation in $\mathbf{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem, and prove that if the size and the arity of the target relation $T$ are taken as parameters then open definability is $\mathrm{coW}[1]$-complete for every vocabulary $\tau$ with at least one, at least binary, relation.