The Complexity of Definability by Open First-Order Formulas

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 $\mathcal{L}$, the $\mathcal{L}$-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 $\mathcal{L}$ 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.

17 pages