The Complexity of Definability by Open First-Order Formulas
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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 L, the L-Definability Problem for finite structures takes as input a finite structure A and a target relation T over the domain of A, and determines whether there is a formula of L whose interpretation in 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 coW[1]-complete for every vocabulary τ with at least one, at least binary, relation.
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