In this chapter, we introduce and explain the basic notions of Description Logic, including syntax, semantics and reasoning services, and we explain how the latter are used in applications. The concept language of the DL ALCIn this section, we will describe the central notions of Description Logic first on an intuitive level and then on a more precise level. As a running example, we use the domain of university courses and teaching, and we will use a conceptualisation given informally, in graphical form, in Figure 2.1. Please note that this is one way of viewing university teachingwhich might be very different from the reader's way of viewing it. Also, as it is an informal representation, different readers may interpret arrows in different ways; that is, our representation does not come with a well-defined semantics that would inform us in an unambiguous way how to interpret the different arrows. 1 In the next sections, we will describe our way of viewing university teaching in a DL knowledge base, thereby establishing some constraints on the meaning of terms like "Professor" and "teaches" used in Figure 2.1 and throughout this section.In Description Logic, we assume that we want to describe some abstraction of some domain of interest, and that this abstraction is populated by elements. 2 We use three main building blocks to describe these elements:• Concepts represent sets of elements and can be viewed as unary pred-1 Our graphical representation looks somewhat similar to an extended ER diagram, for which such a well-defined semantics has been specified [Che76, CLN94]. 2 We have chosen the term "elements" rather than "individuals" or "objects" to prevent the reader from making false assumptions.
OWL DL, a new W3C ontology language recommendation, is based on the expressive description logic SHOIN . Although the ontology consistency problem for SHOIN is known to be decidable, up to now there has been no known "practical" decision procedure, that is, a goal-directed procedure that is likely to perform well with realistic ontology derived problems. We present such a decision procedure for SHOIQ, a slightly more expressive logic than SHOIN , extending the well-known algorithm for SHIQ, which is the basis for several highly successful implementations.
Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles, role hierarchies, and qualifying number restrictions. Early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSpace. Finally, we investigate the limits of decidability for this family of DLs.
As applications of description logics proliferate, efficient reasoning with knowledge bases containing many assertions becomes ever more important. For such cases, we developed a novel reasoning algorithm that reduces a SHIQ knowledge base to a disjunctive datalog program while preserving the set of ground consequences. Queries can then be answered in the resulting program while reusing existing and practically proven optimization techniques of deductive databases, such as join-order optimizations or magic sets. Moreover, we use our algorithm to derive precise data complexity bounds: we show that SHIQ is data complete for NP, and we identify an expressive fragment of SHIQ with polynomial data complexity.
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