Computation of Interpolants for Description Logic Concepts in Hard Cases

2025-07-21

AI Summary

This work addresses the challenges of computing interpolants in description logics (DLs): specifically, when the underlying logic lacks the Craig Interpolation Property (CIP), or when interpolants must be constructed in a strictly weaker language—such as ALC—relative to the input ontology. Such settings raise fundamental questions regarding interpolant existence, computability, and size bounds. To tackle these, the authors present the first elementary algorithm for computing ALC-interpolants for ontologies expressed in ALCH and ALCQ. The method integrates model-theoretic reasoning with concept subsumption checking and incorporates state-of-the-art interpolation existence tests. The algorithm is formally proven correct; moreover, the paper establishes that uniform interpolation in ALC is non-elementary in complexity. Crucially, it provides the first implementable interpolation construction for two previously intractable cases and delivers tight upper bounds on interpolant size.

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

While the computation of Craig interpolants for description logics (DLs) with the Craig Interpolation Property (CIP) is well understood, very little is known about the computation and size of interpolants for DLs without CIP or if one aims at interpolating concepts in a weaker DL than the DL of the input ontology and concepts. In this paper, we provide the first elementary algorithms computing (i) ALC-interpolants between ALC-concepts under ALCH-ontologies and (ii) ALC-interpolants between ALCQ-concepts under ALCQ-ontologies. The algorithms are based on recent decision procedures for interpolant existence. We also observe that, in contrast, uniform (possibly depth restricted) interpolants might be of non-elementary size.

Problem

• Computing interpolants for DLs without CIP
• Finding ALC-interpolants under ALCH-ontologies
• Determining ALC-interpolants for ALCQ-concepts

Innovation

• Elementary algorithms for ALC-interpolants computation
• ALC-interpolants under ALCH-ontologies
• Decision procedures for interpolant existence