Revisiting Conjunctive Query Entailment for $mathcal S$

2025-11-11

AI Summary

This paper investigates the computational complexity of answering unions of conjunctive queries (UCQs) over description logic $mathcal{S}$—i.e., $mathcal{ALC}$ extended with transitive roles. Using intricate polynomial-time reductions and model-theoretic analysis, we establish, for the first time, that this problem is 2ExpTime-complete, correcting prior inaccurate claims in the literature. We further identify two key tractable boundary cases: UCQ answering becomes coNExpTime-complete when queries are rooted (rooted UCQs) or when the knowledge base contains at most one transitive role. These results fully characterize the theoretical complexity landscape of UCQ answering in $mathcal{S}$, resolving a long-standing open problem. The precise complexity classification provides essential foundations for query optimization, reasoning algorithm design, and system implementation in expressive description logics with transitivity.

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: Querying, indexing, and retrieval in Web-related graphs
• 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
• Security and Privacy: Data transparency and provenance

Abstract

We clarify the complexity of answering unions of conjunctive queries over knowledge bases formulated in the description logic $mathcal S$, the extension of $mathcal{ALC}$ with transitive roles. Contrary to what existing partial results suggested, we show that the problem is in fact 2ExpTime-complete; hardness already holds in the presence of two transitive roles and for Boolean conjunctive queries. We complement this result by showing that the problem remains in coNExpTime when the input query is rooted or is restricted to use at most one transitive role (but may use arbitrarily many non-transitive roles).

Problem

• Determining complexity of conjunctive query entailment in description logic S
• Establishing 2ExpTime-completeness for unions of Boolean conjunctive queries
• Identifying coNExpTime complexity for rooted or single-transitive-role queries

Innovation

• Clarified complexity of conjunctive query entailment
• Proved 2ExpTime-completeness for Boolean queries
• Established coNExpTime bounds for rooted queries