Subsumption in $\mathcal{FL}_{\bot \mathit{reg}}$ with TBoxes Is in ExpTime

2026-05-13

AI Summary

This study investigates the computational complexity of concept inclusion problems in the description logic $\mathcal{FL}_{\bot \mathit{reg}}$ and its sublanguages, distinguishing between settings with and without a TBox. For languages lacking concept negation, it establishes for the first time that the concept inclusion problem is ExpTime-complete in the presence of a TBox, while it is PSpace-complete without one. The key technical contribution lies in the introduction of parity pushdown games as a novel reduction framework, bridging description logic reasoning with formal language theory to derive these tight complexity bounds. This approach not only yields precise complexity classifications for $\mathcal{FL}_{\bot \mathit{reg}}$ and $\mathcal{FL}_{\mathit{reg}}$, but also provides a new paradigm for analyzing decision problems in related logical systems.

Abstract

Description logics (DL) are a family of formal languages for representing and reasoning about structured knowledge in terms of concepts and their relationships. A central reasoning problem in DL is concept subsumption. Although this problem has been widely studied, important open problems remain for certain logics. The expressive power of DLs depends on the constructors available for building complex concepts. In this work, we investigate subsumption in the restricted logic $\mathcal{FL}_{\bot \mathit{reg}}$ and its related fragments $\mathcal{FL}_\mathit{reg}$, $\mathcal{FL}_\bot$, and $\mathcal{FL}_0$. These logics support value restrictions over role names, where the subscript $\bot$ denotes the presence of the empty concept and ${reg}$ denotes the use of regular expressions over roles. None of these logics includes concept negation. We show that deciding subsumption between two concept descriptions in $\mathcal{FL}_{\bot \mathit{reg}}$ and $\mathcal{FL}_\mathit{reg}$ is PSpace-complete. When subsumption is considered with respect to a TBox (i.e., a set of axioms), the complexity increases to ExpTime-complete. Our results are obtained via a novel reduction to parity pushdown games.

Problem

• subsumption
• description logics
• TBox
• computational complexity
• ExpTime

Innovation

• subsumption
• description logic
• TBox
• ExpTime-completeness
• parity pushdown games