The Complexity of Definability by Open First-Order Formulas

Carlos Areces; Miguel Campercholi; Daniel Penazzi; Pablo Ventura

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arXiv · 1904.04637

The Complexity of Definability by Open First-Order Formulas

Carlos Areces·Miguel Campercholi·Daniel Penazzi·Pablo Ventura

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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.

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Carlos Areces, Miguel Campercholi, Daniel Penazzi, Pablo Ventura. 2019-04-09. The Complexity of Definability by Open First-Order Formulas. https://arxiv.org/abs/1904.04637

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