Logical Separability of Labeled Data Examples under Ontologies

TL;DR

This paper studies the conditions under which logical formulas can separate positive and negative labeled data examples within ontologies, focusing on various logical fragments and their computational properties.

Contribution

It provides model-theoretic characterizations of separability, compares the expressive power of different languages, and analyzes the complexity of deciding separability.

Findings

Model-theoretic characterizations of separability variants

Comparison of separating power across logical languages

Complexity analysis of separability decision problems

Abstract

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.