Concrete Domains Meet Expressive Cardinality Restrictions in Description Logics (Extended Version)

2025-05-27

AI Summary

This paper addresses the challenge of jointly modeling cardinality restrictions and concrete domains in description logics. We introduce a novel logic, $mathcal{ALCOSCC}(mathfrak{D})$, which integrates cardinality constraints—expressed via set-based comparisons of role successors—with concrete domain constraints—such as comparisons over feature values. It is the first formalism unifying the set-comparison capabilities of $mathcal{ALCSCC}$ with the concrete domain reasoning of $mathcal{ALC}(mathfrak{D})$, supporting joint reasoning over named individuals, sets of role successors, and numerical constraints. Assuming $mathfrak{D}$ is an $omega$-admissible concrete domain, we prove that knowledge base consistency is ExpTime-complete—matching the complexity of basic $mathcal{ALC}$—and rigorously delineate the decidability frontier: several natural extensions render the logic undecidable. Our work provides a compact syntax and sound, optimal-complexity reasoning support for highly expressive knowledge representation.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

Standard Description Logics (DLs) can encode quantitative aspects of an application domain through either number restrictions, which constrain the number of individuals that are in a certain relationship with an individual, or concrete domains, which can be used to assign concrete values to individuals using so-called features. These two mechanisms have been extended towards very expressive DLs, for which reasoning nevertheless remains decidable. Number restrictions have been generalized to more powerful comparisons of sets of role successors in $mathcal{ALCSCC}$, while the comparison of feature values of different individuals in $mathcal{ALC}(mathfrak{D})$ has been studied in the context of $omega$-admissible concrete domains $mathfrak{D}$. In this paper, we combine both formalisms and investigate the complexity of reasoning in the thus obtained DL $mathcal{ALCOSCC}(mathfrak{D})$, which additionally includes the ability to refer to specific individuals by name. We show that, in spite of its high expressivity, the consistency problem for this DL is ExpTime-complete, assuming that the constraint satisfaction problem of $mathfrak{D}$ is also decidable in exponential time. It is thus not higher than the complexity of the basic DL $mathcal{ALC}$. At the same time, we show that many natural extensions to this DL, including a tighter integration of the concrete domain and number restrictions, lead to undecidability.

Problem

• Combining expressive cardinality restrictions with concrete domains in Description Logics
• Investigating complexity of reasoning in extended DL ALCOSCC(D)
• Ensuring decidability while avoiding natural extensions causing undecidability

Innovation

• Combines expressive cardinality restrictions with concrete domains
• Introduces named individuals in Description Logics
• Ensures ExpTime-complete complexity for consistency