Chapter 3 Description Logics [chapter]

Franz Baader, Ian Horrocks, Ulrike Sattler
2008 Foundations of Artificial Intelligence  
In this chapter we will introduce description logics, a family of logic-based knowledge representation languages that can be used to represent the terminological knowledge of an application domain in a structured way. We will first review their provenance and history, and show how the field has developed. We will then introduce the basic description logic ALC in some detail, including definitions of syntax, semantics and basic reasoning services, and describe important extensions such as
more » ... roles, number restrictions, and concrete domains. Next, we will discuss the relationship between description logics and other formalisms, in particular first order and modal logics; the most commonly used reasoning techniques, in particular tableau, resolution and automata based techniques; and the computational complexity of basic reasoning problems. After reviewing some of the most prominent applications of description logics, in particular ontology language applications, we will conclude with an overview of other aspects of description logic research, and with pointers to the relevant literature. Description Logics This description employs the Boolean constructors conjunction ( ), which is interpreted as set intersection, disjunction ( ), which is interpreted as set union, and negation (¬), which is interpreted as set complement, as well as the existential restriction constructor (∃r.C), and the value restriction constructor (∀r.C). An individual, say Bob, belongs to ∃married.Doctor if there exists an individual that is married to Bob (i.e., is related to Bob via the married role) and is a doctor (i.e., belongs to the concept Doctor). Similarly, Bob belongs to ∀hasChild.(Doctor Professor) if all his children (i.e., all individuals related to Bob via the hasChild role) are either doctors or professors. Concept descriptions can be used to build statements in a DL knowledge base, which typically comes in two parts: a terminological and an assertional one. In the terminological part, called the TBox, we can describe the relevant notions of an application domain by stating properties of concepts and roles, and relationships between them-it corresponds to the schema in a database setting. In its simplest form, a TBox statement can introduce a name (abbreviation) for a complex description. For example, we could introduce the name HappyMan as an abbreviation for the concept description from above: HappyMan ≡ Human ¬Female (∃married.Doctor) (∀hasChild.(Doctor Professor)). More expressive TBoxes allow the statement of more general axioms such as ∃hasChild.Human Human, which says that only humans can have human children. Note that, in contrast to the abbreviation statement from above, this statement does not define a concept. It just constrains the way in which concepts and roles (in this case, Human and hasChild) can be interpreted. Obviously, all the knowledge we have described in our example could easily be represented by formulae of first-order predicate logic (see also Section 3.3). The variable-free syntax of description logics makes TBox statements easier to read than the corresponding first-order formulae. However, the main reason for using DLs rather than predicate logic is that DLs are carefully tailored such that they combine interesting means of expressiveness with decidability of the important reasoning problems (see below). The assertional part of the knowledge base, called the ABox, is used to describe a concrete situation by stating properties of individuals-it corresponds to the data in a database setting. For example, the assertions HappyMan(BOB), hasChild(BOB, MARY), ¬Doctor(MARY) state that Bob belongs to the concept HappyMan, that Mary is one of his children, and that Mary is not a doctor. Modern DL systems all employ this kind of restricted ABox formalism, which basically can be used to state ground facts. This differs from the use of the ABox in the early DL system KRYPTON [38], where ABox statements could be arbitrary first-order formulae. The underlying idea was that the ABox could then be used to represent knowledge that was not expressible in the restricted TBox formalism of KRYP-TON, but this came with a cost: reasoning about ABox knowledge required the use of a general theorem prover, which was quite inefficient and could lead to non-termination of the reasoning procedure. Modern description logic systems provide their users with reasoning services that can automatically deduce implicit knowledge from the explicitly represented knowledge, and
doi:10.1016/s1574-6526(07)03003-9 fatcat:wa2lwv7ywrhvpnk3emfqv2pgyu