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 |$\boldsymbol{\mathcal{L}}$|⁠, the |$\boldsymbol{\mathcal{L}}$|-definability problem for finite structures takes as an input a finite structure |$\boldsymbol{A}$| and a target relation |$T$| over the domain of |$\boldsymbol{A}$| and determines whether there is a formula of |$\boldsymbol{\mathcal{L}}$| whose interpretation in |$\boldsymbol{A}$| 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 |$\textrm{coW}[1]$|-complete for every vocabulary |$\tau $| with at least one, at least binary, relation.

中文翻译：

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开放式一阶公式的可定义性的复杂性

在本文中，我们正式定义和研究具有相等性的开放式一阶公式（即无量词的一阶公式）的可定义性问题的计算复杂性。给定逻辑| $ \ boldsymbol {\ mathcal {L}} $ |⁠，| $ \ boldsymbol {\ mathcal {L}} $$ 有限结构的可定义性问题以有限结构| $ \ boldsymbol {A} $ |作为输入。和目标关系| $ T $ | | $ \ boldsymbol {A} $ |的域中 并确定是否存在公式| $ \ boldsymbol {\ mathcal {L}} $ | 其解释在| $ \ boldsymbol {A} $ | 与| $ T $ |⁠相符。我们表明，对于开放式一阶公式（简称开放定义性），此问题的复杂性是coNP完全的。我们还研究了问题的参数复杂度，并证明了如果目标关系的大小和奇异性| $ T $ | 用作参数，则开放定义为| $ \ textrm {coW} [1] $ | -每个词汇的完整| $ \ tau $ | 与至少一个，至少是二元关系。