Finding a logical formula that separates positive and negative examples given in the form of labeled data items is fundamental in applications such as concept learning, reverse engineering of database queries, and generating referring expressions. In this paper, we investigate the existence of a separating formula for incomplete data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and three important fragments thereof: the description logic $\mathcal{ALCI}$, the guarded fragment, and the two-variable fragment. We consider several forms of separability that differ in the treatment of negative examples and in whether or not they admit the use of additional helper symbols to achieve separation. We characterize separability in a model-theoretic way, compare the separating power of the different languages, and determine the computational complexity of separability as a decision problem.

中文翻译：

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本体下不完全数据的逻辑可分性

在概念学习、数据库查询的逆向工程和生成引用表达式等应用中，找到一个逻辑公式来区分以标记数据项形式给出的正例和反例。在本文中，我们研究了存在本体的情况下不完整数据的分离公式的存在。无论是本体语言还是分离语言，我们都专注于一阶逻辑及其三个重要的片段：描述逻辑$\mathcal{ALCI}$、保护片段和二变量片段。我们考虑了几种不同形式的可分离性，它们在处理反例以及它们是否允许使用额外的辅助符号来实现分离方面有所不同。我们以模型理论的方式描述可分离性，