Conjunctive Query Answering for the Description Logic SHIQ (1111.0049v1)

Abstract: Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, conjunctive query answering over DL knowledge bases is only poorly understood if transitive roles are admitted in the query. In this paper, we consider unions of conjunctive queries over knowledge bases formulated in the prominent DL SHIQ and allow transitive roles in both the query and the knowledge base. We show decidability of query answering in this setting and establish two tight complexity bounds: regarding combined complexity, we prove that there is a deterministic algorithm for query answering that needs time single exponential in the size of the KB and double exponential in the size of the query, which is optimal. Regarding data complexity, we prove containment in co-NP.

• The paper presents a decision procedure for answering unions of conjunctive queries in SHIQ using canonical models and structured query rewriting.
• It introduces a deterministic algorithm with single exponential combined complexity and co-NP data complexity to manage transitive role challenges.
• The rewriting techniques transform complex cyclic queries into decidable fragments, enhancing practical ontology-based reasoning applications.

Conjunctive Query Answering for the Description Logic \SHIQ

The paper by Glimm et al. addresses the challenge of conjunctive query (CQ) answering within the framework of \SHIQ, an expressive Description Logic (DL). The work extends our understanding of CQ handling when transitive roles are involved both in queries and in DL knowledge bases (KBs). The paper's core contribution is proving the decidability of answering unions of conjunctive queries (UCQs) in \SHIQ, and it establishes tight complexity bounds for this problem, offering a significant advancement in DL query handling.

Complexity and Decidability Insights

The authors provide robust complexity results, delineating two key bounds:

1. Combined Complexity: The proposed deterministic algorithm for CQ answering exhibits a complexity that is single exponential in the size of the KB and double exponential in the size of the query. This result is optimal and places the problem in the 2\ExpTime-complete class, emphasizing the inherent difficulty of dealing with transitive roles in conjunction with large CQs.
2. Data Complexity: For data complexity, where the focus is primarily on the size of the ABox, the problem is shown to be co-\NPclass-complete. This implies that for practical purposes, when the ABox is large compared to the TBox and query, the performance impact remains within a feasible computational boundary.

Canonical Models and Query Rewriting

Glimm et al. employ the concept of canonical models and forest bases as foundational structures that enable CQ entailment checking. The concept of forest bases — interpretations where elements form a tree-like structure — provides a basis for unraveling the implicit cycles and transitive relationships in models to reveal hidden query matches.

The paper introduces a structured rewriting process:

1. Collapsing: Adds equality atoms to simplify the query.
2. Split Rewriting: Addresses transitive shortcuts between elements.
3. Loop and Forest Rewriting: Breaks down cyclic structures into manageable tree-like structures.
4. Conceptual Rolling-up: Transforms tree structures into concept atoms that translate into equivalent logical constructs within the TBox framework.

By these transformations, more intricate queries can be reduced to simpler, decidable fragments, either grounded queries or expressible as singular concept atoms.

Practical and Theoretical Implications

The implications of this work are notable for fields relying on DLs, such as semantics, ontology-based data access, and AI. By confirming decidability and establishing efficient algorithms for CQ answering in \SHIQ, the research directly impacts the scalability and applicability of reasoning engines like FaCT++, KAON2, and Pellet in complex, real-world applications.

Speculatively, this research paves the way for extending such decision procedures to even more expressive DLs, such as \SHOIQ, further enabling robust CQ answering capabilities in ontologies underpinning web technologies like OWL.

In conclusion, the paper by Glimm et al. marks a significant step in DL research by resolving critical open questions concerning the practical application of DLs in querying and reasoning, thus expanding the computational effectiveness of semantic technologies.