Logical Separability of Labeled Data Examples under Ontologies
This work addresses a fundamental problem in concept learning and knowledge graphs, but it is incremental as it extends separability analysis to various logical fragments and ontologies.
The paper investigates the existence of logical formulas that separate positive and negative labeled data examples under ontologies, focusing on first-order logic fragments like description logic and guarded fragments. It provides model-theoretic characterizations, compares separation power across languages, and analyzes computational complexity.
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, generating referring expressions, and entity comparison in knowledge graphs. In this paper, we investigate the existence of a separating formula for data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and the following important fragments thereof: the description logic $\mathcal{ALCI}$, the guarded fragment, the two-variable fragment, and the guarded negation fragment. For separation, we also consider (unions of) conjunctive queries. 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. Our main results are model-theoretic characterizations of (all variants of) separability, the comparison of the separating power of different languages, and the investigation of the computational complexity of deciding separability.