Description Logics

Description logics (DL) are a decidable fragment of first-order logic, whose development has been driven by the need to increase expressivity of knowledge representation formalisms, while still maintaining decidabil- ity [BAADERand NUTT2003]. Description logics provide formal semantics that allow to specify precise meaning of the concepts in a domain. Earlier results on DL have been utilized for ontological modeling in the Semantic Web, forming also the basis of the Web Ontology Language (OWL) [BECHHOFERet al. 2004]. An extensive introduction of DL is presented in the work of Baader et al. [BAADERand NUTT2003]. We restrict to an overview of the key features of DL, presenting the most basic fragment called ALC, and the description logic SHOIN (D) that is used in our work. SHOIN (D) is the logic formalism under- pinning OWL.

There are three main building blocks in DL: individuals, concepts, and roles. The symbols referred to as constants in first-order logic are referred to as individuals in DL. Concepts in a DL correspond to unary predicates denoting sets of individuals, whereas roles correspond to binary predicates, i.e. relations between individuals. The state of a domain is represented by a DL ontology, which is a set of DL state- ments (axioms) that are true in that state. There are two types of axioms in DL: terminological axioms and assertional axioms. The set of all terminological ax- ioms form the TBox (short for terminology box), whereas the set of assertional axioms form the ABox (short for assertion box, which asserts statements about a given terminology). As such, concepts and roles are separated from individuals, but TBox and ABox altogether compose the knowledge base (KB).

We adhere to the common notation in which capital letters A, B denote atomic concepts, C, D denote complex descriptions of concepts, R, S denote atomic roles, and c, d are individuals. Complex concepts and complex roles can be built by using concept constructors, which include the universal concept (>), bottom concept (⊥), atomic negation (¬), intersection (u), value restrictions (∀R.C), and existential quantification (∃R.C).The supported concept and role constructors in a specific DL determine its expressive power [GROSOFet al. 2003].

2.1. KNOWLEDGE REPRESENTATION FORMALISMS

Description Logic ALC

A simple description logic with a basic set of language constructs and low expressivity is the Attributive Language with Complements ALC [SCHMIDT-SCHAUSSand SMOLKA1991]. Table 2.2 illustrates the syntax and semantics of the concept constructors in ALC given the interpretation

(∆I, I). We also give in Table2.2a short description for each constructor.

Constructor Syntax Semantics Description

Universal Concept > ∆I set of all individuals

Bottom Concept ⊥ ∅ empty set

Intersection C u D CI∩ DI set of individuals that are both in CI and DI

Union C t D CI∪ DI set of individuals that

are in CI or DI

Complement ¬C ∆I− CI _{set of individuals that}

are not in CI Existential Restriction ∃R.C {c | ∃(c, d) ∈

RI∧ d ∈ CI}

set of individuals that are related via the role

RI to an individual in

CI

Universal Restriction ∀R.C {c | ∀(c, d) ∈

RI ⇒ d ∈ CI}

individuals that are re- lated via the role RI only to individuals in

CI

Table 2.2: Constructs in ALC description logic

A DL knowledge base consists of a set of assertional axioms and a set of termino- logical axioms. In Table2.3, we summarize the syntax and semantics of the axioms that can be expressed in ALC. In the table, we also give a short informal descrip- tion for each axiom. Subsumption in DL is defined as an inclusion axiom A v B, read as “A is subsumed by B” (or “B subsumes A”), which is also interpreted as

B is a subclass of A.

An important feature that can be integrated in description logics is the inclusion of built-in predicates modeling concrete properties with values from a fixed domain, e.g. integers or strings. As such, DLs may be extended with concrete domains that

TBox Axiom Syntax Semantics Description

Concept Equivalence C ≡ D CI = DI C is equivalent to D,

i.e. every individual in CI is also an indi- vidual in DI, and vice versa

Role Equivalence R ≡ S RI = SI R is equivalent to S,

i.e. every pair of in- dividuals in the set RI also belongs to the set

SI, and vice versa Concept Subsumption C v D CI ⊆ DI C is a subconcept of

D, i.e. every individ-

ual in the set CIis also an individual in DI Role Subsumption R v S RI ⊆ SI R is a subrole of S, i.e.

every pair of individu- als in RI belongs also to the set SI

ABox Axiom Syntax Semantics Description

Concept Assertion C(c) cI ∈ CI c is an instance of con-

cept C, i.e. cIis in the set CI

Role Assertion R(c, d) (cI, dI) ∈ RI c is related via R to d

Individual Equivalence c ≡ d cI = dI cI and dIare the same individual

Individual Inequivalence c 6= d cI 6= dI _cI _{and d}I _{are different}

individuals Table 2.3: Axioms in ALC description logic

can be used to build complex concepts, for example the following axiom describes adults as humans whose age is at least 18: Adult ≡ Human u ∃age. ≥18.

When this feature is integrated in a logic, its name is extended accordingly with the symbol D. Hence, the description logic ALCD integrates concrete domains D in the basic ALC. The feature of concrete domains is useful for the practical applications of DL.

2.1. KNOWLEDGE REPRESENTATION FORMALISMS

Description Logic SHOIN (D)

A more expressive description logic, which is also the formalism underlying the ontology languages in the Web, is SHOIN (D) description logic. The naming also reflects its expressive power. Specifically, SHOIN (D) extends ALC with features of role transitivity, inverse roles, role hierarchy, nominals, cardinality re- strictions, and datatypes.

Constructor Syntax Semantics

Qualified (atleast) Restriction ≥ nR {c | #{d|(c, d) ∈ RI_{} ≥ n}}

Qualified (atmost) Restriction ≤ nR {c | #{d|(c, d) ∈ RI_{} ≤ n}}

Enumeration {c₁, . . . , cn} {cI1, . . . , cIn}

Inverse Role R− {(d, c) | (c, d) ∈ RI_}

Role Transitivity T rans(R) RI = (RI)+

Table 2.4: Constructs in SHOIN (D) description logic in addition to ALC constructs

The constructor for SHOIN (D) are those used in ALC (Table2.2) extended with additional constructors that we present in Table2.4.