What Is Fuzzy Logic and Fuzzy Ontology?

Robert Fullér

KnowMobile National Workshop, October 30, 2008, Helsinki.

Thursday, March 4, 2010 Web Ontology Language

• The Web Ontology Language (OWL) is a family of knowledge representation languages for authoring ontologies, and is endorsed by the World Wide Web Consortium. This family of languages is based on two semantics: OWL DL and OWL Lite semantics that are based on Description Logics.

Thursday, March 4, 2010 Description logic

• Description logics (DL) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. Today description logic has become a cornerstone of the Semantic Web for its use in the design of ontologies.

Thursday, March 4, 2010 Semantic Web • The Semantic Web is an evolving extension of the World Wide Web in which the semantics of information and services on the web is deﬁned, making it possible for the web to understand and satisfy the requests of people and machines to use the web content. The Web is considered as a universal medium for data, information, and knowledge exchange.

Thursday, March 4, 2010 Ontology

• An ontology consists of a hierarchical description of important classes (or concepts) in a particular domain, along with the description of the properties (of the instances) of each concept. • Web content is then annotated by relying on the concepts deﬁned in a speciﬁc domain ontology. • An ontology is a specification of conceptualization. • A conceptualization is an abstract, simplified view of the world that we wish to represent for some purpose. Every knowledge base, knowledge-based system, or knowledge- level agent is committed to some conceptualization, explicitly or implicitly.

Thursday, March 4, 2010 Thursday, March 4, 2010 Thursday, March 4, 2010 Thursday, March 4, 2010 Concept lattice for concepts of the formal context.

Source: W.C. Cho and D. Richards, Ontology construction and concept reuse with formal concept analysis for improved web document retrieval, Web Intelligence and Agent Systems: An international journal 5 (2007) 109–126

Thursday, March 4, 2010 The general Recall and Precision are inversely relate.

Thursday, March 4, 2010 Source: W.C. Cho and D. Richards, Ontology construction and concept reuse with formal concept analysis for improved web document retrieval, Web Intelligence and Agent Systems: An international journal 5 (2007) 109–126

Thursday, March 4, 2010 Description Logics with fuzzy Domain

• Web Ontology Language Description Logics (OWL DL) becomes less suitable in domains in which the concepts to be represented have not a precise definition. As we have to deal with Web content, it is easily verified that this scenario is, unfortunately, likely the rule rather than an exception. • For instance, just consider the case we would like to build an ontology about flowers. Then we may encounter the problem of representing concepts like “Candia is a creamy white rose with dark pink edges to the petals”, “Jacaranda is a hot pink rose”, “Calla is a very large, long white flower on thick stalks”. As it becomes apparent such concepts hardly can be encoded into OWL. • As it becomes apparent such concepts hardly can be encoded into OWL DL, as they involve fuzzy or vague concepts, like “creamy”, “dark”, “hot”, “large” and “thick”, for which a clear and precise definition is impossible.

Thursday, March 4, 2010 Fuzzy Ontology

• A fuzzy ontology is a quintuple F =< I, C,T,N,X > where • I is the set of individuals (objects), also called instances of the concepts. • C is a set of fuzzy concepts (or classes - cf. in OWL - of individuals, or categories, or types). Each concept is a fuzzy set on the domain of instances. • The set of entities of the fuzzy ontology is defined by E = C ∪ I. • T denotes the fuzzy taxonomy relations among the set of concepts C. It organizes concepts into sub-(super-)concept tree structures. The taxonomic relationship T (i, j ) indicates that the child j is a conceptual speciﬁcation of the parent i with a certain degree. • N denotes the set of non-taxonomy fuzzy associative relationships that relate entities across tree structures, for example:

- Naming relationships, describing the names of concepts - Locating relationships, describing the relative location of concepts - Functional relationships, describing the functions (or properties) of concepts • X is the set of axioms expressed in a proper logical language, i.e., predicates that constrain the meaning of concepts, individuals, relationships and functions.

Thursday, March 4, 2010 A fuzzy ontology scheme.

Source: David Tudor Parry: Fuzzy Ontology and Intelligent Systems for Discovery of Useful Medical Information, Ph.D. Thesis, Auckland University of Technology, 2005.

Thursday, March 4, 2010 Fuzzy Ontology Generation for Semantic Web. C = {”Document,” ”Research Area”}

Fuzzy formal context can also be represented as a cross-table as shown in Table 1.

The context has three objects representing three documents, D1,D2,D3.

It also has three attributes Data Mining, Clustering and Fuzzy Logic representing three research topics.

The relationship between an object and an attribute is represented by a membership value in [0,1].

An α-cut can be set to eliminate relations that have low membership values.

Source: Quan Thanh Tho, Siu Cheung Hui, Automatic Fuzzy Ontology Generation for Semantic Web, IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 18, NO. 6, JUNE 2006

Thursday, March 4, 2010 A fuzzy ontology for wine

Source: Silvia Calegari and Davide Ciucci, INTEGRATING FUZZY LOGIC IN ONTOLOGIES, 8th International Conference on Enterprise Information Systems, 2006.

Illustration of a non-taxonomic relation. Name of fuzzy relation: Taste Name of instance: cabernet Name of property: dry

Some possible values:

Taste(cabernet, dry)=0.7 ⇔ Cabernet has a dry taste with value 0.7 Taste(cabernet, very_dry)=0.7×0.7 ⇔ Cabernet has a very dry taste with value 0.49 Taste(cabernet, not_dry)=1-0.7 ⇔ Cabernet has not a dry taste with value 0.3

Thursday, March 4, 2010