Fitting Description Logic Ontologies to ABox and Query Examples

TL;DR

This paper investigates the problem of finding description logic ontologies that fit given positive and negative examples of ABoxes and queries, providing complexity results for various logic and query types.

Contribution

It offers effective characterizations and complexity classifications for the ontology fitting problem in LC and LCI with different query languages.

Findings

Fitting problems are NP-complete for AQs and CQs.

Fitting problems are 2EXPTIME-complete for CQs and UCQs.

Results hold for both LC and LCI.

Abstract

We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form $(\mathcal{A},q)$ with $\mathcal{A}$ an ABox and $q$ a Boolean query, we seek an ontology $\mathcal{O}$ that satisfies $\mathcal{A} \cup \mathcal{O} \vDash q$ for all positive examples and $\mathcal{A} \cup \mathcal{O}\not\vDash q$ for all negative examples. We consider the description logics $\mathcal{ALC}$ and $\mathcal{ALCI}$ as ontology languages and a range of query languages that includes atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). For all of the resulting fitting problems, we provide effective characterizations and determine the computational complexity of deciding whether a fitting ontology exists. This problem turns out to be ${\scriptsize CO}NP$ for AQs and full CQs and $2E{\scriptsize XP}T{\scriptsize IME}$-complete for CQs and UCQs. These results hold for both $\mathcal{ALC}$ and $\mathcal{ALCI}$.