Description Logics

objects such as the age, the weight, or the name of a person, and the comparison of these concrete properties. Unfortunately, in their unrestricted form, concrete domains can have dramatic effects on the decidability and computational complexity of the underlying DL [17, 115]. For this reason, a more restricted form of concrete domain, known as datatypes [101], is often used in practice. The nominal constructor allows us to use individual names also within concept descriptions: if a is an individual name, then {a} is a concept, called a nominal, which is interpreted by a singleton set. Using the individual Turing, we can describe all those computer scientists that have met Turing by CScientist � ∃hasMet.{Turing}. The so-called “one-of” constructor extends the nominal constructor to a finite set of individuals. In the presence of disjunction, it can, however, be expressed using nominals: {a1, . . . , an} is equivalent to {a1}� · · ·� {an}. The presence of nominals can have dramatic effects on the complexity of reasoning [159]. An additional comment on the naming of DLs is in order. Recall that the name given to a particular DL usually reflects its expressive power, with letters expressing the constructors provided. For expressive DLs, starting with the basic DL �L would lead to quite long names. For this reason, the letter S is often used as an abbreviation for the “basic” DL consisting of �LC extended with transitive roles (which in the �L naming scheme would be called �LCR+). 2 The letter H represents subroles (role Hierarchies), O represents nominals (nOminals), I represents inverse roles (Iinverse), N represent number restrictions (Number), and Q represent qualified number restrictions (Qualified). The integration of a concrete domain/datatype is indicated by appending its name in parenthesis, but sometimes a “generic” D is used to express that some concrete domain/datatype has been integrated. The DL corresponding to the OWL DL ontology language includes all of these constructors and is therefore called SHOIN (D). 2The use of S is motivated by the close connection between this DL and the modal logic S4. 144 3. Description Logics 3.3 Relationships with other Formalisms In this section, we discuss the relationships between DLs and predicate logic, and between DLs and Modal Logic. This is intended for readers who are familiar with these logics; those not familiar with these logics might want to skip the following subsection(s), since we do not introduce modal or predicate logic here—we simply use standard terminology. Here, we only describe the relationship of the basic DL �LC and some of its extensions to these other logics (for a more detailed analysis, see [33] and Chapter 4 of [14]). 3.3.1 DLs and Predicate Logic Most DLs can be seen as fragments of first-order predicate logic, although some provide operators such as transitive closure of roles or fixpoints that require second-order logic [33]. The main reason for using Description Logics rather than general first-order predicate logic when representing knowledge is that most DLs are actually decidable fragments of first-order predicate logic, i.e., there are effective procedures for deciding the inference problems introduced above. Viewing role names as binary relations and concept names as unary relations, we define two translation functions, πx and πy , that inductively map �LC-concepts into first order formulae with one free variable, x or y: πx(A) = A(x), πy(A) = A(y), πx(C �D) = πx(C) ∧ πx(D), πy(C �D) = πy(C) ∧ πy(D), πx(C �D) = πx(C) ∨ πx(D), πy(C �D) = πy(C) ∨ πy(D), πx(∃r.C) = ∃y.r(x, y) ∧ πy(C), πy(∃r.C) = ∃x.r(y, x) ∧ πx(C), πx(∀r.C) = ∀y.r(x, y)⇒ πy(C), πy(∀r.C) = ∀x.r(y, x)⇒ πx(C). Given this, we can translate a TBox T and an ABox � as follows, where ψ[x/a] denotes the formula obtained from ψ by replacing all free occurrences of x with a: