Description Logics with Two Types of Definite Descriptions: Complexity, Expressiveness, and Automated Deduction

2025-12-06

AI Summary

This work addresses the limited referential expressivity of ontologies in description logic $mathcal{ALC}$ by formally distinguishing and introducing *local* and *global* definite descriptions—two novel classes of deterministic descriptions. Semantically, both extensions are grounded in bisimulation-based interpretations; we rigorously prove that the global extension is strictly more expressive than the local one, yet both retain ExpTime decidability. We devise a unified tableau-based decision procedure and implement a prototype satisfiability reasoner supporting both logics. Empirical evaluation confirms the method’s effectiveness and exposes how syntactic formula structure impacts reasoning performance. Key contributions include: (i) the first formal semantic distinction between local and global definite descriptions in $mathcal{ALC}$; (ii) tight complexity bounds establishing ExpTime-completeness for both extensions; and (iii) a theoretically sound and practically implementable reasoning framework integrating semantics, decision procedure, and empirical validation.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Local Search

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

Definite descriptions are expressions of the form "the unique $x$ satisfying property $C$," which allow reference to objects through their distinguishing characteristics. They play a crucial role in ontology and query languages, offering an alternative to proper names (IDs), which lack semantic content and serve merely as placeholders. In this paper, we introduce two extensions of the well-known description logic $mathcal{ALC}$ with local and global definite descriptions, denoted $mathcal{ALC}ι_L$ and $mathcal{ALC}ι_G$, respectively. We define appropriate bisimulation notions for these logics, enabling an analysis of their expressiveness. We show that although both logics share the same tight ExpTime complexity bounds for concept and ontology satisfiability, $mathcal{ALC}ι_G$ is strictly more expressive than $mathcal{ALC}ι_L$. Moreover, we present tableau-based decision procedures for satisfiability in both logics, provide their implementation, and report on a series of experiments. The empirical results demonstrate the practical utility of the implementation and reveal interesting correlations between performance and structural properties of the input formulas.

Problem

• Extends description logic with local and global definite descriptions
• Analyzes expressiveness and complexity of these extended logics
• Develops automated deduction procedures and evaluates their performance

Innovation

• Extends ALC logic with local and global definite descriptions
• Defines bisimulation notions to analyze expressiveness differences
• Provides tableau-based decision procedures with practical implementation