# Knowledge Representation in Description Logic. 1. Introduction Description logic denotes a family of knowledge representation formalisms that model the

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<ul><li>Slide 1</li></ul><p>Knowledge Representation in Description Logic Slide 2 1. Introduction Description logic denotes a family of knowledge representation formalisms that model the application domain by defining the relevant concepts of the domain and then using these concepts to specify properties of objects and individuals occurring in the domain (Baader and Nutt 2003). As the name implies, research on description logic emphasizes a careful formalization of the notions involved, and a preoccupation with precisely defined reasoning techniques. Slide 3 Note that we prefer the singular form, description logic, rather than the plural form, description logics, in spite of the fact that we are talking about a family of formalisms. Description logic received renewed attention recently because it provides a formal framework for the Web ontology language OWL. Slide 4 Indeed, several constructs that OWL introduces cannot be properly appreciated without at least a superficial knowledge of description logic. Furthermore, some of the ontology tools, notably Protege, offer a user interface based on notions that description logic supports. The emphasis is on the knowledge representation features of description logic. Slide 5 The history of description logic goes back to an earlier discussion about knowledge representation formalisms that flourished in the 1980s. At the heart of the discussion was the categorization of such formalisms into two groups: non-logic based and logic-based formalisms. The non-logic-based formalisms reflect cognitive notions and claim to be closer to ones intuition and, therefore, easier to comprehend. Slide 6 Such formalisms include semantic networks, frames, and rule-based representations. However, most of them lack a consistent semantics and adopt ad hoc reasoning procedures, which leads to systems that exhibit different behavior, albeit supporting virtually identical languages. Slide 7 The second category includes those formalisms that are variants of first-order logic. They reflect the belief that first-order logic is sufficient to describe facts about the real world. Because they borrow the basic syntax, semantics, and proof theory of first-order logic, formalisms in this second category have a solid foundation. Slide 8 Semantic networks and frames were later given a formal semantics by mapping them to first-order logic. Moreover, different features of these formalisms correspond to distinct fragments of first-order logic, supported by specialized reasoning techniques with quite different complexity (Brachman and Levesque 1985). Slide 9 In other words, the full power of first-order logic was not necessarily required to achieve an adequate level of expressiveness, as far as knowledge representation is concerned. As a result of this last observation, research on the socalled terminological systems began. Recently, the term Description Logic (DL) was adopted to emphasize the importance of the underlying logical system. Slide 10 From this perspective, knowledge representation systems can be characterized as pre-DL systems, DL systems, and current generation DL systems. The ancestor of DL systems, KL-ONE, introduced most of the key notions and marked the transition from semantic networks to well-founded terminological systems. Slide 11 In general, pre-DL systems were mainly concerned with concept representation schemes and classification algorithms. DL systems were inspired by theoretical research on the complexity of reasoning in description logic. Systems such as CLASSIC (Brachman et al. 1991) favored efficient reasoning techniques by adopting a description logic with limited expressive power. Slide 12 At the other extreme, systems such as BACK (Nebel and van Luck 1988) emphasized expressiveness and reasoning efficiency, sacrificing completeness (roughly, there were true sentences that the systems could not prove). Current generation DL systems, of which FACT (Horrocks 1998; 2003) and RACER (Haarslev and Moller 2001) are good examples, use optimized reasoning techniques to deal with expressive varieties of description logic and yet retain completeness. Slide 13 2. An Informal Example The following requirements largely shaped the development of description logic: The description language should support the notions of: Atomic concepts (denoting sets of individuals) Atomic roles (denoting binary relations between individuals) Constants (denoting individuals) Slide 14 The description language should include constructors to define: Complex concepts (denoting sets of individuals) Complex roles (denoting binary relations between individuals) Axioms (defining new concepts or imposing restrictions on the concepts) Assertions (expressing facts about individuals) Slide 15 The reasoning techniques should cover at least: Concept subsumption (a concept is a subconcept of another concept) Concept instantiation (an individual is an instance of a concept) Slide 16 We use the following constructions of description logic to describe our example, where C and D are complex concepts, R is an atomic role, and a and b are constants denoting individuals: Slide 17 Slide 18 The intuitive meaning of all these constructs is immediate, except for R.C and R.C, which are given special attention in the examples that follow. Consider an alphabet consisting of the following atomic concepts, atomic roles, and constants (together with their intended interpretation): Slide 19 Slide 20 Note that, strictly speaking, we cannot guarantee that H relates books to authors, and that P relates books to the countries where they were published. We can only say that H and P relate individuals to individuals, which is intrinsic to the semantics of description logic. Slide 21 A complex concept, or a concept description, is an expression that constructs a new concept out of other concepts. We illustrate how to define complex concepts that are gradually more sophisticated, using the above alphabet and the intended interpretation for the atomic concepts and atomic roles (the sets B, A, C, E, H, and P). Slide 22 That is, in the explanations that follow each example, we use the intended interpretation or intended semantics of the symbols in the alphabet. The first two examples use just the simple constructs C and C D: (1) EuroCountry (the set of individuals, not necessarily countries, that are not European countries) Observe that negation is always with respect to the set of all individuals, hence the intuitive explanation in (1). (2) Country EuroCountry (the set of countries that are not European countries) Slide 23 Note that, to define the set of countries that are not European countries, we circumscribed negation to the set of countries in (2). The next examples involve the more sophisticated constructs R.C and R.C: (3) hasAuthor. (the set of individuals, not necessarily books, that have no known author) Slide 24 Recall that H is a binary relation between individuals that represents the intended interpretation of hasAuthor. The complex concept in (3) denotes the set S of individuals such that, for each s in S, if H relates s to an individual b, then b belongs to the empty set (the intended interpretation of ). Because the empty set has no individuals, S is the set of individuals that H relates to no individual. That is, S is the set of individuals for which H is undefined, hence the explanation in (3). Slide 25 (4) publishedIn.EuroCountry (the set of individuals, not necessarily books, published in some European country and perhaps elsewhere) Also recall that P is a binary relation between individuals that represents the intended interpretation of publishedIn. The complex concept in (4) denotes the set T of individuals that P relates to some individual in E. Slide 26 However, note that (4) does not guarantee that, given an individual t in T, P relates t only to individuals in E. That is, T is the set of individuals, not necessarily books, that P relates to some individual in E and perhaps to other individuals not in E, hence the intuitive explanation in (4). Slide 27 (5) publishedIn.EuroCountry (the set of individuals, not necessarily books, published only in European countries or not published at all) The complex concept in (5) denotes the set U of individuals such that, for each u in U, if P relates u to an individual e, then e is in E. However, note that, by definition, U will also include any individual e such that P does not relate e to any individual, hence the intuitive explanation in (5). Slide 28 (6) publishedIn.EuroCountry publishedIn.EuroCountry (the set of individuals, not necessarily books, published in European countries, and only in European countries) The complex concept in (6) denotes the set V of individuals that P relates to some individual in E, and only to individuals in E. Slide 29 Therefore, it correctly constructs the set of individuals that are indeed published, and only published in European countries. Finally, note that (6) does not guarantee that the country of publication is unique. Slide 30 (7) Book hasAuthor. (the set of books that have no known author) (8) Book publishedIn.EuroCountry publishedIn.EuroCountry (the set of books published in European countries, and only in European countries) Slide 31 A definition is an axiom that introduces a new defined concept with the help of complex concepts. For example, the axioms below define the concepts nonEuroCountry, anonymousBook, nonAnonymousBook, EuroBook, and nonEuroBook: Slide 32 (9) nonEuroCountry Country EuroCountry (the concept of non-European countries is defined as those countries that are not European countries) (10) AnonymousBook Book hasAuthor. (the concept of anonymous books is defined as those books that have no known author) Slide 33 (11) nonAnonymousBook Book AnonymousBook (the concept of nonanonymous books is defined as those books that are not anonymous) (12) EuroBook Book publishedIn.EuroCountry publishedIn.EuroCountry (the concept of European books is defined as those books that are published in European countries) Slide 34 (13) nonEuroBook Book EuroBook (the concept of non-European books is defined as those books that are not European books) Note that the expression in (2) is a complex concept, whereas that in (9) is a definition that introduces a new concept, nonEuroCountry. Slide 35 Definition (11) introduces a new defined concept, nonAnonymousBook, with the help of the defined concept AnonymousBook and the atomic concept Book. Similar observations apply to the other axioms. An inclusion is an axiom that just imposes a restriction on the world being modeled, indicating that a concept is subsumed by another concept. Slide 36 An example of an inclusion is: Slide 37 Without assertion (15), we cannot correctly state that the constant The Description Logic Handbook denotes an individual which is indeed an instance of the concept Book. Likewise, assertion (23) guarantees that The Description Logic Handbook and Franz Baader denote individuals that are related by hasAuthor. Similar observations apply to the other assertions. Slide 38 A knowledge base is a set of axioms and assertions, written using a specific language. The terminology, or TBox, of the knowledge base consists of the set of axioms that define new concepts. The world description, assertional knowledge, or ABox of the knowledge base consists of the set of assertions. The TBox expresses intentional knowledge, which is typically stable, whereas the ABox captures extensional knowledge, which changes as the world evolves. Slide 39 For example, the axioms and assertions in (9) to (28) can be organized as a knowledge base, which we call BOOKS, where the TBox consists of definitions (9) to (13) and the inclusion (14), and the ABox contains the assertions in (15) to (28). Slide 40 We now informally exemplify how to deduce concept subsumptions and concept instantiations from the BOOKS knowledge base. We stress that the examples are just indicative of what can be proved, but not of how the DL proof procedures operate. We first prove that every country can be classified as either European or nonEuropean, but not both. Slide 41 (29) nonEuroCountry EuroCountry (30) Country EuroCountry nonEuroCountry The inclusion (29) follows directly from (9) and is equivalent to saying that no individual is both a European country and a non- European country. To prove (30), we establish the following sequence of equivalent complex concepts. Slide 42 We may likewise prove that every book is either anonymous or nonanonymous, but not both, using just (10) and (11). In this case, the definitions (10) and (11) already guarantee that AnonymousBook and nonAnonymousBook are subsumed by Book. Slide 43 That is, no inclusion similar to (14) is required. More precisely, we can prove that: (32) nonAnonymousBook AnonymousBook (33) Book AnonymousBook nonAnonymousBook Slide 44 The inclusion (32) follows directly from (11). To prove (33), we establish the following sequence of equivalent complex concepts. Slide 45 Finally and omitting the details, we can also prove that: (35) nonEuroBook EuroBook from (13) (36) Book EuroBook nonEuroBook from (12), (13) Slide 46 We now turn to examples of concept instantiation. Suppose we want to prove that: (37) nonAnonymousBook(Principia Mathematica) (Principia Mathematica is an instance of nonAnonymousBook) Slide 47 Slide 48 Note that, to derive (39) from (38), we used the law, and to derive (39) from (38), the law. In general, the reasoning techniques that DL systems implement should be able to solve several inference problems. Slide 49 3. The Family of Attributive Languages 3.1 Concept Descriptions Description languages differ by the collection of constructors they offer to define concept descriptions. Following Baader and Nutt (2003), we introduce in this section the syntax and semantics of the family of attributive languages, or family. Slide 50 An attributive language is characterized by an alphabet consisting of a set of atomic concepts, a set of atomic roles, and the special symbols T and , respectively called the universal concept and the bottom concept. Slide 51 The set of concept descriptions of is inductively defined as follows. (i) Any atomic concept and the universal and bottom concepts are concept descriptions (ii) If A is an atomic concept, C and D are concept descriptions, and R is an atomic role, then the following expressions are concept descriptions. A (atomic negation) C D (intersection) R.C (value restriction) R.T (limited existential quantification) Slide 52 The other classes of languages of the family maintain the same definition of alphabet, but expand the set of concept descriptions to include expressions of the one of the forms. Slide 53 (iii) If C and D are concept descriptions, R is an atomic role, and n is a positive integer, then the following expressions are concept descrip...</p>