Journal of Artificial Intelligence Research 31 (2008) 157-204 Submitted 06/07; published 01/08
Conjunctive Query Answering for the Description Logic SHIQ
Birte Glimm [email protected] Ian Horrocks [email protected] Oxford University Computing Laboratory, UK Carsten Lutz [email protected] Dresden University of Technology, Germany Ulrike Sattler [email protected] The University of Manchester, UK
Abstract Conjunctive queries play an important role as an expressive query language for Descrip- tion 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 knowl- edge 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.
1. Introduction
Description Logics (DLs) are a family of logic based knowledge representation formalisms (Baader, Calvanese, McGuinness, Nardi, & Patel-Schneider, 2003). Most DLs are fragments of First-Order Logic restricted to unary and binary predicates, which are called concepts and roles in DLs. The constructors for building complex expressions are usually chosen such that the key inference problems, such as concept satisfiability, are decidable and preferably of low computational complexity. A DL knowledge base (KB) consists of a TBox, which contains intensional knowledge such as concept definitions and general background knowledge, and an ABox, which contains extensional knowledge and is used to describe individuals. Using a database metaphor, the TBox corresponds to the schema, and the ABox corresponds to the data. In contrast to databases, however, DL knowledge bases adopt an open world semantics, i.e., they represent information about the domain in an incomplete way. Standard DL reasoning services include testing concepts for satisfiability and retrieving certain instances of a given concept. The latter retrieves, for a knowledge base consisting of an ABox A and a TBox T , all (ABox) individuals that are instances of the given (possibly complex) concept expression C, i.e., all those individuals a such that T and A entail that a is an instance of C. The underlying reasoning problems are well-understood, and it is known that the combined complexity of these reasoning problems, i.e., the complexity measured in the size of the TBox, the ABox, and the query, is ExpTime-complete for SHIQ (Tobies,