## Fuzzy Description Logics and the Semantic Web (2005)

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Citations: | 114 - 25 self |

### BibTeX

@MISC{Straccia05fuzzydescription,

author = {Umberto Straccia and Benjamin C. Pierce and David N. Turner},

title = {Fuzzy Description Logics and the Semantic Web},

year = {2005}

}

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### Abstract

nd (universal child.Human 7 . is given in terms of an the domain (a non-empty set) an interpretation function that maps: (class) a (property) a an element of Interpretation extended to concept expressions: = = = = = \ = {x = {x 8 and . mapping to FOL: introduce unary an atomic binary a . Translate follows x) = x) = false t(A, x) ## A(x) x) = x) x) ## x) t(C, x) = t(#R.C, x) = y) t(#R.C, x) = 9 Knowledge . DL Knowledge Base is a A#, a TBox containing general inclusion axioms of the ("concept C"), i# definitions are of the (equiv A) concept definitions are of the Sometimes, a TBox can contain primitive and concept definitions only, where no atom can be defined more than once and no recursion is allowed complexity changes dramatically a ABox containing assertions of the

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Citation Context ...e “creamy”, “dark”, “hot”, “large” and “thick”, for which a clear and precise definition is not possible. 2 The problem to deal with imprecise concepts has been addressed several decades ago by Zadeh =-=[31]-=-, which gave bird in the meanwhile to the so-called fuzzy set and fuzzy logic theory and a huge number of real life applications exists. Unfortunately, despite the popularity of fuzzy set theory, rela... |

1910 | O.: The Semantic Web
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Citation Context ... capture the meaning of the most popular features of structured representation of knowledge. Nowadays, DLs have gained even more popularity due to their application in the context of the Semantic Web =-=[4, 15]-=-. Semantic Web has recently attracted much attention both from academia and industry, and is widely regarded as the next step in the evolution of the World Wide Web. It aims at enhancing content on th... |

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522 | From SHIQ and RDF to OWL: The making of a web ontology language
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Citation Context ... capture the meaning of the most popular features of structured representation of knowledge. Nowadays, DLs have gained even more popularity due to their application in the context of the Semantic Web =-=[4, 15]-=-. Semantic Web has recently attracted much attention both from academia and industry, and is widely regarded as the next step in the evolution of the World Wide Web. It aims at enhancing content on th... |

496 | Metamathematics of Fuzzy Logic - Hájek - 1998 |

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Citation Context ... K satisfies E. According to [16], the entailment and subsumption problem can be reduced to knowledge base satisfiability problem, for which decision procedures and reasoning tools exists (e.g. RACER =-=[11]-=- and FACT [14]). Example 1. Let us consider the following excerpt of a simple ontology (TBox T ) about cars, with empty RBox (R = ∅): Car ⊑ (= 1 maker) ⊓ (= 1 passanger) ⊓ (= 1 speed) (= 1 maker) ⊑ Ca... |

270 | A scheme for integrating concrete domains into concept languages
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Citation Context ...ical SHOIN (D). We present its syntax and semantics. Interesting features are that concrete domains are fuzzy and entailment and subsumption relationships may hold to some degree in the unit interval =-=[0, 1]-=-. 1 Introduction In the last decade a substantial amount of work has been carried out in the context of Description Logics (DLs) [2]. DLs are a logical reconstruction of the so-called framebased knowl... |

263 | Using an expressive description logic: FaCT or fiction
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Citation Context .... According to [16], the entailment and subsumption problem can be reduced to knowledge base satisfiability problem, for which decision procedures and reasoning tools exists (e.g. RACER [11] and FACT =-=[14]-=-). Example 1. Let us consider the following excerpt of a simple ontology (TBox T ) about cars, with empty RBox (R = ∅): Car ⊑ (= 1 maker) ⊓ (= 1 passanger) ⊓ (= 1 speed) (= 1 maker) ⊑ Car ⊤ ⊑ ∀maker.M... |

260 |
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Citation Context ... YoungPerson = Person ⊓ ∃age.Young (2) will denote a young person. Furthermore, by referring to Example 1, we may define the concept of sports car as the concept SportsCar = Car ⊓ ∃speed.very(High) , =-=(3)-=- where very is a concept modifier and High is a fuzzy concrete predicate over the domain of speed expressed in kilometers per hour and may be defined as High(x) = min(1, 0.004x). Similarly, we may rep... |

210 | Reducing OWL entailment to Description Logic satisfiability
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Citation Context ...[15]. Although several XML and RDF syntaxes for OWL-DL exist, in this paper we use the traditional description logic notation. For explicating the relationship between OWL DL and DLs syntax, see e.g. =-=[15, 16]-=-. The purpose of this section is to make the paper selfcontained. More importantly it helps in understanding the differences between classical SHOIN (D) and fuzzy SHOIN (D). The reader confident with ... |

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Citation Context ...e evolution of the World Wide Web. It aims at enhancing content on the World Wide Web with meta-data, enabling agents (machines or human users) to process, share and interpret Web content. Ontologies =-=[10]-=- play a key role in the Semantic Web and major effort has been put by the Semantic Web community into this issue. Informally, an ontology consists of a hierarchical description of important concepts i... |

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Citation Context ... not satisfiable . Unfortunately, from a computational point of view, no calculus exists yet checking satisfiability of fuzzy SHOIN (D) knowledge bases. [13, 29] report a calculus for the case of ALC =-=[24]-=- (with concept constructors ⊤, ⊥, ¬, ⊓, ⊔, ∀, ∃) with modifiers and simple TBox, with min, max and →KD connectives. No indication for the BDB problem is given. [25, 26] reports a calculus for ALC and ... |

166 | Reasoning Within Fuzzy Description Logics
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Citation Context ...cation operator i. 3.2 Fuzzy SHOIN (D) In this section we give syntax and semantics of fuzzy SHOIN (D), using the fuzzy operators defined in the previous section. We generalize the semantics given in =-=[13, 26, 29]-=-. Syntax. We have seen that SHOIN (D) allows to reason with concrete data types, such as strings and integers using so-called concrete domains. In our fuzzy approach, concrete domains may be based on ... |

159 | Description logics with concrete domains|a survey
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Citation Context ...der confident with the SHOIN (D) terminology may skip directly to Section 3. Syntax. SHOIN (D) allows to reason with concrete data types, such as strings and integers using so-called concrete domains =-=[1, 18, 20, 21]-=-. A concrete domain D is a pair 〈∆D, ΦD〉, where ∆D is an interpretation domain and ΦD is the set of concrete domain predicates d with a predefined arity n and an interpretation dD ⊆ ∆n D . For instanc... |

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Citation Context ...embership of being the individual a a member of the fuzzy set C. In the following, we present first some preliminaries on fuzzy set theory (for a more complete and comprehensive presentation see e.g. =-=[7]-=-) and then define fuzzy SHOIN (D). 3.1 Preliminaries on fuzzy set theory Let X be a countable crisp set and let A be a fuzzy subset of X, with membership function µA(x), or simply A(x) ∈ [0, 1], x ∈ X... |

70 | A description logic for vague knowledge
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Citation Context ...cation operator i. 3.2 Fuzzy SHOIN (D) In this section we give syntax and semantics of fuzzy SHOIN (D), using the fuzzy operators defined in the previous section. We generalize the semantics given in =-=[13, 26, 29]-=-. Syntax. We have seen that SHOIN (D) allows to reason with concrete data types, such as strings and integers using so-called concrete domains. In our fuzzy approach, concrete domains may be based on ... |

61 | The Complexity of Reasoning with Concrete Domains
- Lutz
- 2001
(Show Context)
Citation Context ...der confident with the SHOIN (D) terminology may skip directly to Section 3. Syntax. SHOIN (D) allows to reason with concrete data types, such as strings and integers using so-called concrete domains =-=[1, 18, 20, 21]-=-. A concrete domain D is a pair 〈∆D, ΦD〉, where ∆D is an interpretation domain and ΦD is the set of concrete domain predicates d with a predefined arity n and an interpretation dD ⊆ ∆n D . For instanc... |

59 |
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Citation Context ..., relative little work has been carried out in extending DLs towards the representation of imprecise concepts, notwithstanding DLs can be considered as a quite natural candidate for such an extension =-=[5, 6, 13, 23, 25, 26, 27, 29, 30]-=- (see also [9], Chapter 6). In this paper we consider a fuzzy extension of SHOIN(D), the corresponding DL of the ontology description language OWL DL, and present its syntax and semantics. The main fe... |

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Citation Context ..., relative little work has been carried out in extending DLs towards the representation of imprecise concepts, notwithstanding DLs can be considered as a quite natural candidate for such an extension =-=[5, 6, 13, 23, 25, 26, 27, 29, 30]-=- (see also [9], Chapter 6). In this paper we consider a fuzzy extension of SHOIN(D), the corresponding DL of the ontology description language OWL DL, and present its syntax and semantics. The main fe... |

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47 | Transforming fuzzy description logics into classical description logics, in
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Citation Context ...ith min, max and →KD connectives. No indication for the BDB problem is given. [25, 26] reports a calculus for ALC and simple TBox, with min, max and →KD connectives and addresses the BDB problem and, =-=[27]-=- shows how the satisfiability problem and the BDB problem can be reduced to classical ALC and, thus, can be resolved by means of a tools like FACT and RACER. However, despite these negative results, r... |

26 | Generalizing quantification in fuzzy description logics
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Citation Context ...to rely on [18]. Another direction is in extending fuzzy SHOIN (D) with fuzzy quantifiers, where the ∀ and ∃ quantifiers are replaced with fuzzy quantifiers like most, some, usually and the like (see =-=[23]-=- for a preliminary work in this direction). This allows to define concepts like TopCustomer = Customer ⊓ (Usually)buys.ExpensiveItem ExpensiveItem = Item ⊓ ∃price.High . Here, the fuzzy quantifier Usu... |

25 | A fuzzy description logic with hedges as concept modifiers
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Citation Context ...cation operator i. 3.2 Fuzzy SHOIN (D) In this section we give syntax and semantics of fuzzy SHOIN (D), using the fuzzy operators defined in the previous section. We generalize the semantics given in =-=[13, 26, 29]-=-. Syntax. We have seen that SHOIN (D) allows to reason with concrete data types, such as strings and integers using so-called concrete domains. In our fuzzy approach, concrete domains may be based on ... |

24 | A Treatise on Many-Valued Logics. Research Studies - Gottwald - 2001 |

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13 | Some complexity results on fuzzy description logics
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Citation Context ..., relative little work has been carried out in extending DLs towards the representation of imprecise concepts, notwithstanding DLs can be considered as a quite natural candidate for such an extension =-=[5, 6, 13, 23, 25, 26, 27, 29, 30]-=- (see also [9], Chapter 6). In this paper we consider a fuzzy extension of SHOIN(D), the corresponding DL of the ontology description language OWL DL, and present its syntax and semantics. The main fe... |

10 |
Fuzzy description logics with concrete domains
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- 2005
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Citation Context ...ow the satisfiability problem and the BDB problem can be reduced to classical ALC and, thus, can be resolved by means of a tools like FACT and RACER. However, despite these negative results, recently =-=[28]-=- reports a calculus for ALC(D) whenever the connectives, the modifiers and the concrete fuzzy predicates are representable as a bounded Mixed Integer Program. For instance, Lukasiewicz logic satisfies... |

9 | Making fuzzy description logics more expressive - Hájek |

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Citation Context ...out in extending DLs towards the representation of imprecise concepts, notwithstanding DLs can be considered as a quite natural candidate for such an extension [5, 6, 13, 23, 25–27, 29, 30] (see also =-=[9]-=-, Chapter 6). In this paper we consider a fuzzy extension of SHOIN (D), the corresponding DL of the ontology description language OWL DL, and present its syntax and semantics. The main feature of fuzz... |

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Citation Context .... ALC with concrete domains and arbitrary t-norm, co-norm, negation and residuum as implication. We are investigating the possibility to use the methods developed in the context of Many-Valued Logics =-=[12]-=-, which seem to particularly well-suited to our context. These procedures have then to be combined with a procedure to deal with fuzzy concrete domains, for which we plan to rely on [18]. Another dire... |

4 | An overview of fuzzy quantifiers (I). Interpretations” and “An overview of fuzzy quantifiers (II). Reasoning and applications - Liu, Kerre - 1998 |

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