Appendix D Formal Models for Artificial Intelligence Methods: Foundations of Description Logics

Description logics is the family of formal systems that are based on mathematical logic and are used for inferring in ontologies. As we mentioned in Chap. 7 , ontology is a model of a conceptual knowledge concerning a specific application domain. In 1979 Patrick J. Hayes discussed in [ 130 ] a possible use of First-Order Logic (FOL) semantics for Minsky frame systems. Since the beginning of the 1980s many

description logics (AL, FL − ,FL 0 , ALC , etc.) have been developed. In general, each of these logics can be treated as a certain subset of FOL. 12 So, one can ask: “Why do we not use just FOL for representing ontologies?”. There are two reasons pointed out as an answer to this question in the literature. First of all, the use of FOL without some restrictions does not allow us to take into account the structural nature of ontologies. (And a structural aspect of ontologies is vital in an inference procedure.) Secondly, we demand an effective inference procedure. Therefore, description logics are defined on the basis of decidable subsets of FOL. 13

In 1991 Manfred Schmidt-Schauß and Gert Smolka defined in [ 266 ] one of the most popular description logics, namely logic ALC. Its syntax and semantics is introduced in the first section, whereas a formal notion of knowledge base defined

with this logic is presented in the second section [ 15 , 266 ].

D.1 Syntax and Semantic of Logic ALC

In Chap. 7 we introduced basic elements useful for defining structural models of knowledge representation, i.e.: objects, concepts, and roles representing relations between objects. We discriminate additionally atomic concepts relating to basic (elementary) notions of a given domain. Let us introduce these elements in a formal way.

12 There are also description logics based on second-order logic. 13 FOL is not decidable in general.

268 Appendix D: Formal Models for Artificial Intelligence Methods … Definition D.1 Let N C be a set of atomic concept names, N R

be a set of role names, N O

be a set of object names. A triple (N C , N R , N O ) is called a signature. Instead of atomic concept name, role name, object name we will say atomic

concept, role, object . Definition D.2 Let (N C , N R , N O )

be a signature. A set of (descriptions of) ALC- concepts is the smallest set defined inductively as follows.

1. The following constructs are ALC-concepts: (a) ⊤, the universal concept,

(b) ⊥, the empty concept,

2. If C and D are ALC-concepts, R ∈ N R , then the following constructs are ALC- concepts:

(a) C ⊓ D, (b) C ⊔ D, (c) ¬C, (d) ∀R · C, (e) ∃R · C.

Before we introduce a formal characterization of semantics of logic ALC, we in- terpret elements defined above in an intuitive way. The universal notion corresponds to the whole domain that an ontology is constructed for, whereas the empty concept represents a concept that has no instances. Elements defined in points 2(a), 2(b), and 2(c) correspond to the intersection of two concepts, the union of two concepts, and the complement of a concept, respectively. A universal quantification, 2(d), determines a set of objects, for which all the relations with the help of a role R concern objects that fall within a concept C. An existential quantification, 2(e), determines a set of objects that are at least once in a relation represented by a role R with an object that falls within a concept C.

Definition D.3 Let (N C , N R , N O )

be a signature. An interpretation is a pair

I I ), where

I is a nonempty set called the domain of I,

such that for each C and D being ALC-concepts, R ∈ N R , the following conditions hold:

I 1(a) (⊤) I , 1(b) (⊥) I = ∅,

2(a) (C ⊓ D) I I =C I ∩D ,

Appendix D: Formal Models for Artificial Intelligence Methods … 269 2(b) (C ⊔ D) I I =C I ∪D ,

2(c) (¬C) I I \C I ,

2(d) (∀R · C) I I I ( x, y)

I ∈R I ⇒y∈C },

2(e) (∃R · C) I I I ( x, y)

I ∈R I ∧y∈C }.

C I (r I ) is called the extension of the concept C (the role r) in the interpretation I. If x ∈ C I , then x is called an instance (object) of the notion C in the interpretation I. Additionally, it is assumed that a concept C is included in a concept D, denoted

C I I the following condition holds: C I ⊆D I .

D.2 Definition of Knowledge Base in Logic ALC

In Chap. 7 we defined a knowledge base as a system (structure) of frames consisting of class frames and object frames. A set of class frames constitutes a terminological knowledge, and a set of object frames corresponds to knowledge about specific objects belonging to a domain. These sets are defined in logic ALC with the help of notions of TBox (Terminological part of knowledge base) and ABox (Assertional part of knowledge base ), respectively. Both TBox and ABox contain knowledge in the form of axioms. Axioms of TBox are defined with a general concept inclusion.

Definition D.4

A general concept inclusion is of the form C ⊑ D, where C, D are ALC -concepts.

A TBox is a finite set of general concept inclusions. An interpretation I is a model of a general concept inclusion C ⊑ D, if C I ⊆D I .

An interpretation I is a model of a TBox T , if it is a model of every general concept inclusion T .

C ⊑ D and D ⊑ C is denoted as C ≡ D. An axiom of a TBox can be of the form of a definition, i.e., A ≡ D, where A is a unique concept name. Now, we characterize an ABox. It can contain axioms of two types. The first type relates to assertions describing a fact that an object is an instance of a given concept, i.e., an object belongs to a given class, denoted C(a), for example: Polish(John- Kowalski) . The second type includes statements representing a fact that a pair of objects constitutes an instance of a role, denoted R(a, b), for example: Married- couple(John-Kowalski, Mary-Kowalski) . Let us formalize our considerations.

Definition D.5 An assertional axiom is of the form C(a) or R(a, b), where C is an ALC -concept, R ∈ N R , a, b ∈ N O .

An ABox is a finite set of assertional axioms. An interpretation I is called a model of an assertional axiom C(a) iff a I ∈C I . An interpretation I is called a model of an assertional axiom R(a, b) iff (a I , b I )

∈R I . An interpretation I is called a model of an ABox A iff I is a model of every

assertional axiom of A.

270 Appendix D: Formal Models for Artificial Intelligence Methods … At the end of this appendix, let us introduce a formal definition of a knowledge

base constructed with the help of logic ALC. Definition D.6

A pair K = (T , A), where T is a TBox, A is an ABox is called a knowledge base . An interpretation I is a model of a knowledge base K iff I is a model of T , and

I is a model of A.

Appendix D Formal Models for Artificial Intelligence Methods: Foundations of Description Logics