Separating Data Examples by Description Logic Concepts with Restricted Signatures

TL;DR

This paper investigates the problem of separating positive and negative data examples using description logic concepts with restricted signatures, analyzing the complexity and relationships to other logical notions like Craig interpolants and conservative extensions.

Contribution

It introduces a framework for data separation with signature restrictions, compares the expressive power of different languages, and establishes complexity results linking separability to known logical concepts.

Findings

Weak separability relates to conservative extensions.

Strong separability coincides with Craig interpolants.

Both conservative extensions and weak separability in ALCO are 3ExpTime-complete.

Abstract

We study the separation of positive and negative data examples in terms of description logic concepts in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols that can be used for separation, and we admit individual names in that signature. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated and we distinguish between separation with and without helper symbols. Within this framework, we compare the separating power of different languages and investigate the complexity of deciding separability. While weak separability is shown to be closely related to conservative extensions, strongly separating concepts coincide with Craig interpolants, for suitably defined encodings of the data and ontology. This enables us to transfer known results from those fields to separability. Conversely, we obtain original results on separability that can be transferred backward. For example, rather surprisingly, conservative extensions and weak separability in ALCO are both 3ExpTime-complete.