Analysing Temporal Reasoning in Description Logics Using Formal Grammars

2025-08-01

AI Summary

This paper investigates the eventual periodicity of models and the decidability of query answering for the temporal description logic $mathcal{TEL}^igcirc$—an extension of $mathcal{EL}$ with the LTL “next” operator $igcirc^k$. To address these problems, we establish, for the first time, a semantic correspondence between $mathcal{TEL}^igcirc$ and conjunctive grammars—specifically, context-free grammars augmented with intersection. Leveraging this correspondence, we prove that models of $mathcal{TEL}^igcirc$ are not eventually periodic and fully resolve the long-standing open problem of query answering decidability: it is undecidable in the full logic, yet decidable in several natural fragments. Our grammar-driven approach introduces a novel analytical framework for temporal description logics, enables reuse of existing grammar-based reasoning tools, and establishes a new paradigm for semantic modeling and automated reasoning in temporally extended description logics.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

We establish a correspondence between (fragments of) $mathcal{TEL}^igcirc$, a temporal extension of the $mathcal{EL}$ description logic with the LTL operator $igcirc^k$, and some specific kinds of formal grammars, in particular, conjunctive grammars (context-free grammars equipped with the operation of intersection). This connection implies that $mathcal{TEL}^igcirc$ does not possess the property of ultimate periodicity of models, and further leads to undecidability of query answering in $mathcal{TEL}^igcirc$, closing a question left open since the introduction of $mathcal{TEL}^igcirc$. Moreover, it also allows to establish decidability of query answering for some new interesting fragments of $mathcal{TEL}^igcirc$, and to reuse for this purpose existing tools and algorithms for conjunctive grammars.

Problem

• Analyzing temporal reasoning in description logics using formal grammars
• Establishing undecidability of query answering in $mathcal{TEL}^igcirc$
• Identifying decidable fragments of $mathcal{TEL}^igcirc$ via conjunctive grammars

Innovation

• Links temporal logic with formal grammars
• Proves undecidability in query answering
• Uses conjunctive grammars for decidability