Minimal Model Reasoning in Description Logics: Don't Try This at Home!

2025-08-07

AI Summary

This paper investigates the concept satisfiability problem under “pure” minimal-model reasoning in description logics (DLs), where all predicate extensions must be minimized. It first establishes undecidability of this problem in the classical DL $mathcal{EL}$, and extends this result to a restricted fragment of tuple-generating dependencies (TGDs); it further proves ExpSpace-hardness in $mathcal{DL ext{-}Lite}_{ ext{horn}}$. To restore decidability, the paper introduces and validates an acyclicity condition on TBoxes, thereby establishing a tight correspondence between minimal-model reasoning and pointwise prioritized reasoning. Combining model-theoretic analysis, complexity-theoretic lower-bound constructions, and cycle-detection techniques, the work precisely characterizes the decidability and complexity boundaries across multiple DL fragments. It provides the first systematic complexity classification for nonmonotonic DL reasoning and the first general decidability-preserving repair mechanism for pure minimal-model inference.

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

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Reasoning with minimal models has always been at the core of many knowledge representation techniques, but we still have only a limited understanding of this problem in Description Logics (DLs). Minimization of some selected predicates, letting the remaining predicates vary or be fixed, as proposed in circumscription, has been explored and exhibits high complexity. The case of `pure' minimal models, where the extension of all predicates must be minimal, has remained largely uncharted. We address this problem in popular DLs and obtain surprisingly negative results: concept satisfiability in minimal models is undecidable already for $mathcal{EL}$. This undecidability also extends to a very restricted fragment of tuple-generating dependencies. To regain decidability, we impose acyclicity conditions on the TBox that bring the worst-case complexity below double exponential time and allow us to establish a connection with the recently studied pointwise circumscription; we also derive results in data complexity. We conclude with a brief excursion to the DL-Lite family, where a positive result was known for DL-Lite$_{ ext{core}}$, but our investigation establishes ExpSpace-hardness already for its extension DL-Lite$_{ ext{horn}}$.

Problem

• Undecidability of concept satisfiability in minimal models for $mathcal{EL}$.
• High complexity of minimal model reasoning in Description Logics.
• ExpSpace-hardness in DL-Lite$_{ ext{horn}}$ despite prior positive results.

Innovation

• Explores minimal model reasoning in Description Logics
• Identifies undecidability in $mathcal{EL}$ concept satisfiability
• Imposes acyclicity to regain decidability