On the entailment problem for DL-Lite$_{core}$ ontologies and conjunctive queries with negation

TL;DR

The paper proves that entailment problems for DL-Lite$_{core}$ ontologies augmented with inequalities or with safe negation remain undecidable, even for unions of Boolean conjunctive queries. The authors reduce from the word problem for finitely generated semigroups (Thue systems) by constructing specialized ontologies and queries that simulate Thue rewrites within DL-Lite$_{core}$ semantics, using canonical and slot structures, perfection concepts, and carefully designed witnesses. They show undecidability for four formulations: ${\mathcal O}^{\neq} \models \psi$, ${\mathcal O}^{\neg} \models \phi$, and UCQ variants ${\mathcal O} \models \Psi$, ${\mathcal O} \models \Phi$, with nuanced discussions of UNA and PCWA; a parallel treatment extends the results to finite entailment. The work provides a unified framework for translating semigroup word problems into DL-Lite ontology entailment, highlighting the fundamental limits of query answering in this description logic under these extensions. Overall, the paper establishes strong undecidability results and clarifies how semantic assumptions influence entailment in DL-Lite$_{core}$ with negation or inequalities.

Abstract

We show that the entailment problem, for a given entailment problem for DL-Lite$_{core}$ ontology, and given conjunctive query with inequalities, is undecidable. We also show that this problem remains undecidable if conjunctive queries with safe negation are considered instead of conjunctive queries with inequalities.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 7
• Corollary 9
• Lemma 10
• Lemma 14
• Lemma 18
• Lemma 22