## Reasoning with very expressive fuzzy description logics

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Venue: | Journal of Artificial Intelligence Research |

Citations: | 42 - 17 self |

### BibTeX

@ARTICLE{Stoilos_reasoningwith,

author = {Giorgos Stoilos and Giorgos Stamou and Jeff Z. Pan and Vassilis Tzouvaras and Ian Horrocks},

title = {Reasoning with very expressive fuzzy description logics},

journal = {Journal of Artificial Intelligence Research},

year = {},

pages = {2007}

}

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### Abstract

It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN, as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN. 1.

### Citations

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Citation Context ...matical frameworks that are intended for modelling such types of concepts (Zimmermann, 1987; Larsen & Yager, 1993; Krishnapuram & Keller, 1992). One such a mathematical framework is fuzzy set theory (=-=Zadeh, 1965-=-). Though fuzzy extensions of various logical formalisms, like propositional, predicate or modal logics have been investigated in the past (Hajek, 1998), such a framework is is not yet well developed ... |

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Citation Context ...atisfies Ψ. 2.2 Fuzzy Sets Fuzzy set theory and fuzzy logic are widely used today for capturing the inherent vagueness (the lack of distinct boundaries of sets) that exists in real life applications (=-=Klir & Yuan, 1995-=-). The notion of a fuzzy set was first introduced by Zadeh (1965). While in classical set theory an element either belongs to a set or not, in fuzzy set theory elements belong only to a certain degree... |

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Citation Context ...ll knowledge we possess for a specific entity of our world using a knowledge representation language. A logical formalism that has gained considerable attention the last decade is Description Logics (=-=Baader, McGuinness, Nardi, & Patel-Schneider, 2002-=-a). Description Logics (DLs) are a family of class-based (concept-based) knowledge representation formalisms, equipped with well defined model-theoretic semantics (Tarski, 1956). They are characterize... |

495 | From SHIQ and RDF to OWL: The Making of a Web Ontology Language - Horrocks, Patel-Schneider, et al. - 2003 |

412 |
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Citation Context ...resent the concepts that appear in a domain of interest. For example, this is particularly the case when domain knowledge is inherently imprecise or vague. Concepts like that of a “near” destination (=-=Berners-Lee et al., 2001-=-), a “highQuality” audio system (McGuiness, 2003), “many” children, a “faulty” reactor (Horrocks & Sattler, 1999), “soon” and many more, require special modelling features. In the past many applicatio... |

411 |
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Citation Context ...ch a mathematical framework is fuzzy set theory (Zadeh, 1965). Though fuzzy extensions of various logical formalisms, like propositional, predicate or modal logics have been investigated in the past (=-=Hajek, 1998-=-), such a framework is is not yet well developed for Description Logics and much research work needs to be done. More precisely, there is the need for reasoning in very expressive fuzzy Description Lo... |

319 | Practical reasoning for expressive description logics
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Citation Context ..., b〉 : R, called role assertions, or of the form a ̸ . = b. 2. A role is called simple if it is neither transitive nor has any transitive sub-roles. This is crucial in order to get a decidable logic (=-=Horrocks, Sattler, & Tobies, 1999-=-). 278Reasoning with Very Expressive Fuzzy Description Logics An interpretation I satisfies a : C if aI ∈ CI , it satisfies 〈a, b〉 : R if 〈aI, bI 〉 ∈ RI , and it satisfies a ̸ . = b if aI ̸= bI . A S... |

162 | Description logic framework for information integration - Calvanese, Giacomo, et al. - 1998 |

155 | Reasoning within fuzzy description logics
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Citation Context ...that exist for Description Logics makes them ideal for encoding knowledge in many applications like the Semantic Web (Baader et al., 2002b; Pan, 2004), multimedia applications (Meghini, Sebastiani, & =-=Straccia, 2001-=-), medical applications (Rector & Horrocks, 1997), databases (Calvanese, De Giacomo, Lenzerini, Nardi, & Rosati, 1998) and many more. Interestingly, the current standard for Semantic Web ontology lang... |

150 | Terminological reasoning is inherently intractable
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Citation Context ...iom in T of which A appears on the left side. A knowledge base with an unfoldable TBox can be transformed into an equivalent one with an empty TBox by a transformation called unfolding, or expansion (=-=Nebel, 1990-=-): Concept inclusion introductions A ⊑ C are replaced by concept equivalence introductions A ≡ A ′ ⊓ C, where A ′ is a new concept name, which stands for the qualities that distinguish the elements of... |

140 | Complexity Results and Practical Algorithms for Logics in Knowledge Representation
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(Show Context)
Citation Context ...ide a reasoning algorithm for a less expressive, but more efficient fuzzy DL language, f-SI. The classical SI language is known to be Pspace-complete, in contrast to the Exptime-completeness of SHIN (=-=Tobies, 2001-=-), hence our algorithm for f-SI can be used for future research and for providing efficient and optimized implementations. Please note that fuzzy DLs (Straccia, 2001) are complementary to other approa... |

138 | Reasoning with Individuals for the Description Logic SHIQ
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(Show Context)
Citation Context ...pressive than SHOIN (OWL DL). In order to achieve our goal we again extend the techniques used for the classical SHIN language and which ensure correctness of the algorithm (Horrocks & Sattler, 1999; =-=Horrocks, Sattler, & Tobies, 2000-=-). Finally, we prove the decidability of the extended algorithm. There are many benefits on following such an approach. On the one hand we provide a gradual presentation to the very complex algorithm ... |

125 | Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Terminological Cycles. DFKI-Research Report RR-90-13, also published
- Baader
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Citation Context ...terpretation I satisfies C ⊑ D if C I ⊆ D I and it satisfies C ≡ D if C I = D I . Note that concept inclusion axioms of the above form are called general concept inclusions (Horrocks & Sattler, 1999; =-=Baader, 1990-=-). A SHIN RBox is a finite set of transitive role axioms (Trans(R)), and role inclusion axioms (R ⊑ S). An interpretation I satisfies Trans(R) if, for all x, y, z ∈ ∆ I , {〈x, y〉, 〈y, z〉} ⊆ R I → 〈x, ... |

124 | R: ‘Fuzzy Queries in Multimedia Database Systems
- Fagin
- 1998
(Show Context)
Citation Context ...ion. One such important theory is fuzzy set theory. Fuzzy Description Logics are very interesting logical formalisms as they can be used in numerous domains like multimedia and information retrieval (=-=Fagin, 1998-=-; Meghini et al., 2001) to provide ranking degrees, geospatial (Chen, Fellah, & Bishr, 2005) to cope with vague concepts like “near”, “far” and many more. 274Reasoning with Very Expressive Fuzzy Desc... |

123 | U.: A description logic with transitive and inverse roles and role hierarchies
- Horrocks, Sattler
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(Show Context)
Citation Context ...knowledge is inherently imprecise or vague. Concepts like that of a “near” destination (Berners-Lee et al., 2001), a “highQuality” audio system (McGuiness, 2003), “many” children, a “faulty” reactor (=-=Horrocks & Sattler, 1999-=-), “soon” and many more, require special modelling features. In the past many applications of various research areas, like decision making, image processing, robotics and medical diagnosis have adopte... |

121 | A Tableaux Decision Procedure for SHOIQ
- Horrocks, Sattler
- 2005
(Show Context)
Citation Context ...only happen after closely investigating the properties of the languages and after proving that these techniques also work in this new setting. In the current paper we have not addressed nominals (O) (=-=Horrocks & Sattler, 2005-=-). Note that in the fuzzy DL literature, there are proposals for crisp nominals (Stoilos et al., 2005c) and fuzzy nominals (Bobillo et al., 2006). Thus, the nominal constructor is not yet a mature not... |

112 |
is modal logic so robustly decidable
- Why
- 1997
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Citation Context ...f individuals. This propagation is crucial for reasoning algorithms in order to retain the tree-model property (Baader et al., 2002a), which is a property that leads to decidable decision procedures (=-=Vardi, 1997-=-). Our goal in the rest of the section is to investigate this property in the context of fuzzy Description Logics that allow for transitive role axioms. We have to determine if similar propagation occ... |

107 | P-CLASSIC: A tractable probabilistic description logic
- Koller, Levy, et al.
- 1997
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Citation Context ...r future research and for providing efficient and optimized implementations. Please note that fuzzy DLs (Straccia, 2001) are complementary to other approaches that extend DLs, like probabilistic DLs (=-=Koller, Levy, & Pfeffer, 1997-=-; Giugno & Lukasiewicz, 2002; Ding & Peng, 2004), or possibilistic DLs (Hollunder, 1994). More precisely, these theories are meant to be used for capturing different types of imperfect information and... |

107 |
Industrial Applications of Fuzzy Control
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Citation Context ...rt systems and decision making (Zimmermann, 1987), pattern recognition (Kandel, 1982), image analysis and computer vision (Krishnapuram & Keller, 1992), medicine (Oguntade & Beaumont, 1982), control (=-=Sugeno, 1985-=-), etc. 2.3 Fuzzy Set Theoretic Operations In this section, we will explain how to extend boolean operations and logical implications in the context of fuzzy sets and fuzzy logics. These operations ar... |

106 | A fuzzy description logic for the semantic web - Straccia |

99 | A Probabilistic Extension to Ontology Language OWL
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Citation Context ... implementations. Please note that fuzzy DLs (Straccia, 2001) are complementary to other approaches that extend DLs, like probabilistic DLs (Koller, Levy, & Pfeffer, 1997; Giugno & Lukasiewicz, 2002; =-=Ding & Peng, 2004-=-), or possibilistic DLs (Hollunder, 1994). More precisely, these theories are meant to be used for capturing different types of imperfect information and knowledge. Fuzziness is purposed for capturing... |

98 | DAML+OIL: a description logic for the semantic web
- Horrocks
- 2002
(Show Context)
Citation Context ...gent answers to complex user defined tasks. Examples of modern applications that have recently adopted knowledge representation languages are the World Wide Web (BernersLee, Hendler, & Lassila, 2001; =-=Baader, Horrocks, & Sattler, 2002-=-b), where knowledge is used to improve the abilities of agents and the interoperability between disparate systems, multimedia processing applications (Alejandro, Belle, & Smith, 2003; Benitez, Smith, ... |

87 | P-SHOQ(D): A probabilistic extension of SHOQ(D) for probabilistic ontologies in the semantic web
- Giugno, Lukasiewicz
- 2002
(Show Context)
Citation Context ...ding efficient and optimized implementations. Please note that fuzzy DLs (Straccia, 2001) are complementary to other approaches that extend DLs, like probabilistic DLs (Koller, Levy, & Pfeffer, 1997; =-=Giugno & Lukasiewicz, 2002-=-; Ding & Peng, 2004), or possibilistic DLs (Hollunder, 1994). More precisely, these theories are meant to be used for capturing different types of imperfect information and knowledge. Fuzziness is pur... |

83 | Subsumption Algorithms for Concept Description Languages
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- 1990
(Show Context)
Citation Context ...ch extension was presented by Stoilos et al. (2005b), but here we will revise that definition. Without loss of generality, we assume all concepts C occurring in A to be in negation normal form (NNF) (=-=Hollunder, Nutt, & Schmidt-Schaus, 1990-=-); i.e., negations occur in front of concept names only. A fKD-SI-concept can be transformed into an equivalent one in NNF by pushing negations inwards using a combination of the De Morgan laws (which... |

65 | A description logic for vague knowledge
- Tresp, Molitor
- 1998
(Show Context)
Citation Context ...zzy DLs the notation fALC is used (Straccia, 2004), but this notation is not so flexible to represent fuzzy DLs which use different norm operations, as we will see later on. In some other approaches (=-=Tresp & Molitor, 1998-=-; Hölldobler, Khang, & Störr, 2002) the naming ALCF is 276Reasoning with Very Expressive Fuzzy Description Logics participate in value and existential restrictions. In section 5 we give a detailed pr... |

55 |
Generalizing term subsumption languages to fuzzy logic
- Yen
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Citation Context ...hat all possible expansions result to a clash, thus the knowledge base is unsatisfiable. ♦ 7. Related Work There have been many efforts in the past to extend description logics with fuzzy set theory (=-=Yen, 1991-=-; Tresp & Molitor, 1998; Straccia, 2001; Hölldobler et al., 2002; Sanchez & Tettamanzi, 2006; Straccia, 2005b; Hajek, 2005; Li, Xu, Lu, & Kang, 2006b). The first effort was presented by Yen (1991). In... |

51 | A fuzzy description logic - Straccia - 1998 |

50 | Experience building a large, re-usable medical ontology using a description logic with transitivity and concept inclusions
- Rector, Horrocks
- 1997
(Show Context)
Citation Context ... them ideal for encoding knowledge in many applications like the Semantic Web (Baader et al., 2002b; Pan, 2004), multimedia applications (Meghini, Sebastiani, & Straccia, 2001), medical applications (=-=Rector & Horrocks, 1997-=-), databases (Calvanese, De Giacomo, Lenzerini, Nardi, & Rosati, 1998) and many more. Interestingly, the current standard for Semantic Web ontology languages, OWL (Bechhofer, van Harmelen, Hendler, Ho... |

49 | A concept language extended with different kinds of transitive roles
- Sattler
- 1996
(Show Context)
Citation Context ...lowing entities Arm ⊑ ∃isPartOf.Body Body ⊑ ∃isPartOf.Human where ⊑ is a subsumption relation and isPartOf is obviously a transitive relation. This knowledge can be captured with the aid of the S DL (=-=Sattler, 1996-=-). Moreover, one might want to capture the knowledge that the role hasPart is the inverse of the role isPartOf, writing hasPart := isPartOf− , thus being able to state that something that is a body an... |

47 | MediaNet: A Multimedia Information Network for
- Benitez, Smith, et al.
- 2000
(Show Context)
Citation Context ... & Sattler, 2002b), where knowledge is used to improve the abilities of agents and the interoperability between disparate systems, multimedia processing applications (Alejandro, Belle, & Smith, 2003; =-=Benitez, Smith, & Chang, 2000-=-), which use knowledge in order to bridge the “gap” between human percepc○2007 AI Access Foundation. All rights reserved.Stoilos, Stamou, Pan, Tzouvaras & Horrocks tion of the objects that exist with... |

46 | A model of multimedia information retrieval
- Meghini, Sebastiani, et al.
(Show Context)
Citation Context ...werful reasoning tools that exist for Description Logics makes them ideal for encoding knowledge in many applications like the Semantic Web (Baader et al., 2002b; Pan, 2004), multimedia applications (=-=Meghini, Sebastiani, & Straccia, 2001-=-), medical applications (Rector & Horrocks, 1997), databases (Calvanese, De Giacomo, Lenzerini, Nardi, & Rosati, 1998) and many more. Interestingly, the current standard for Semantic Web ontology lang... |

45 | Description logics with fuzzy concrete domains
- Straccia
- 2005
(Show Context)
Citation Context ...RI (a, b), CI (b))}. The complete set of semantics is depicted in Table 2. We have to note that there are many proposals for semantics of number restrictions in fuzzy DLs (Sánchez & Tettamanzi, 2004; =-=Straccia, 2005-=-b). We choose to follow the semantics proposed by Straccia (2005b) since they are based in the First-Order interpretation of number restrictions (Baader et al., 2002a). Moreover, as it is shown by Sto... |

42 | Transforming fuzzy description logics into classical description logics
- Straccia
- 2004
(Show Context)
Citation Context ... which we call f-SHIN . 1 language. Section 4 provides an investigation on the semantics of fuzzy DLs when fuzzy transitive relations 1. In a previous approach to fuzzy DLs the notation fALC is used (=-=Straccia, 2004-=-), but this notation is not so flexible to represent fuzzy DLs which use different norm operations, as we will see later on. In some other approaches (Tresp & Molitor, 1998; Hölldobler, Khang, & Störr... |

39 |
Making fuzzy description logic more general
- Hájek
- 2005
(Show Context)
Citation Context ... degrees. The infinite or finite property of interpretations comes from the constructs of the language, while the witnessed or non-witnessed property comes from the continuity of the fuzzy operators (=-=Hajek, 2005-=-). Consider for example a node x which contains some triple of the form 〈≤ 1R, ≥, 1〉. If a successor of x, say y, is blocked by some ancestor of x, say z, the dynamic blocking techniques would create ... |

39 | Fuzzy owl: Uncertainty and the semantic web
- Stoilos, Stamou, et al.
- 2005
(Show Context)
Citation Context ...niques also work in this new setting. In the current paper we have not addressed nominals (O) (Horrocks & Sattler, 2005). Note that in the fuzzy DL literature, there are proposals for crisp nominals (=-=Stoilos et al., 2005-=-c) and fuzzy nominals (Bobillo et al., 2006). Thus, the nominal constructor is not yet a mature notion in fuzzy DLs and more research is needed in order to find appropriate semantics for them, conside... |

36 | I.: The fuzzy description logic f-SHIN
- Stoilos, Stamou, et al.
(Show Context)
Citation Context ...niques also work in this new setting. In the current paper we have not addressed nominals (O) (Horrocks & Sattler, 2005). Note that in the fuzzy DL literature, there are proposals for crisp nominals (=-=Stoilos et al., 2005-=-c) and fuzzy nominals (Bobillo et al., 2006). Thus, the nominal constructor is not yet a mature notion in fuzzy DLs and more research is needed in order to find appropriate semantics for them, conside... |

32 |
On the structure of semigroups on a compact manifold with boundary
- Mostert, Shields
- 1957
(Show Context)
Citation Context ...t the same proofs can be modified to apply to the fuzzy logic defined by the fuzzy operators we are using in the current paper. That is because these operators are definable in the Lukasiewicz logic (=-=Mostert & Shields, 1957-=-). For the rest of the paper, without loss of generality, we are going to consider only witnessed models. In this paper, we will provide an algorithm to decide the fuzzy ABox consistency problem w.r.t... |

30 |
Fuzzy Techniques in Pattern recognition
- Kandel
- 1982
(Show Context)
Citation Context ... in the management of imprecise and vague information has been defined and used in a wide set of scientific areas including expert systems and decision making (Zimmermann, 1987), pattern recognition (=-=Kandel, 1982-=-), image analysis and computer vision (Krishnapuram & Keller, 1992), medicine (Oguntade & Beaumont, 1982), control (Sugeno, 1985), etc. 2.3 Fuzzy Set Theoretic Operations In this section, we will expl... |

30 |
Description Logics: Reasoning Support for the Semantic Web
- Pan
- 2004
(Show Context)
Citation Context ...the well-defined semantics and the powerful reasoning tools that exist for Description Logics makes them ideal for encoding knowledge in many applications like the Semantic Web (Baader et al., 2002b; =-=Pan, 2004-=-), multimedia applications (Meghini, Sebastiani, & Straccia, 2001), medical applications (Rector & Horrocks, 1997), databases (Calvanese, De Giacomo, Lenzerini, Nardi, & Rosati, 1998) and many more. I... |

27 | A crisp representation for fuzzy SHOIN with fuzzy nominals and general concept inclusions
- Bobillo, Delgado, et al.
- 2006
(Show Context)
Citation Context ...e current paper we have not addressed nominals (O) (Horrocks & Sattler, 2005). Note that in the fuzzy DL literature, there are proposals for crisp nominals (Stoilos et al., 2005c) and fuzzy nominals (=-=Bobillo et al., 2006-=-). Thus, the nominal constructor is not yet a mature notion in fuzzy DLs and more research is needed in order to find appropriate semantics for them, considering also the issue from the application po... |

24 | Generalizing quantification in fuzzy description logic
- Sanchez, Tettamanzi
- 2004
(Show Context)
Citation Context ...n in our language, we call it fKD-SHIN . This presentation follows the standard syntax and semantics of fuzzy DLs, that has been introduced in the literature (Straccia, 2001; Hölldobler et al., 2002; =-=Sánchez & Tettamanzi, 2004-=-). More precisely, fKD-SHIN was first presented by Straccia (2005b). For completeness reasons we will also present the language fKD-SHIN here. Please also note that our presentation differs from that ... |

23 | A fuzzy description logic with hedges as concept modifiers
- Hölldobler, Khang, et al.
- 2002
(Show Context)
Citation Context ...LC is used (Straccia, 2004), but this notation is not so flexible to represent fuzzy DLs which use different norm operations, as we will see later on. In some other approaches (Tresp & Molitor, 1998; =-=Hölldobler, Khang, & Störr, 2002-=-) the naming ALCF is 276Reasoning with Very Expressive Fuzzy Description Logics participate in value and existential restrictions. In section 5 we give a detailed presentation of the reasoning algori... |

22 | A fuzzy description logic for multimedia knowledge representation
- Stoilos, Stamou, et al.
- 2005
(Show Context)
Citation Context ...niques also work in this new setting. In the current paper we have not addressed nominals (O) (Horrocks & Sattler, 2005). Note that in the fuzzy DL literature, there are proposals for crisp nominals (=-=Stoilos et al., 2005-=-c) and fuzzy nominals (Bobillo et al., 2006). Thus, the nominal constructor is not yet a mature notion in fuzzy DLs and more research is needed in order to find appropriate semantics for them, conside... |

21 |
Semantic image segmentation and object labeling
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- 2007
(Show Context)
Citation Context ...e adjacent (since not all of their pixels are adjacent), one contained within the other, etc. Hence we can only decide about the membership of a region to a specific concept only to a certain degree (=-=Athanasiadis, Mylonas, Avrithis, & Kollias, 2007-=-). For example, in our case we could have that the object o1 isPartOf the object o2 to a degree of 0.8, that o2 isPartOf o3 to a degree of 0.9, that o1 is an Arm to a degree of 0.75 and that o2 is a B... |

21 | General concept inclusions in fuzzy description logics - Stoilos, Straccia, et al. - 2006 |

16 |
Fuzzy set theoretic approach to computer vision: an overview
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- 1992
(Show Context)
Citation Context ...n making, image processing, robotics and medical diagnosis have adopted special mathematical frameworks that are intended for modelling such types of concepts (Zimmermann, 1987; Larsen & Yager, 1993; =-=Krishnapuram & Keller, 1992-=-). One such a mathematical framework is fuzzy set theory (Zadeh, 1965). Though fuzzy extensions of various logical formalisms, like propositional, predicate or modal logics have been investigated in t... |

13 | Fuzzy Logics as the
- Běhounek, Cintula
(Show Context)
Citation Context ...x R and a TBox T iff there exists (does not exist) some model I of R and T for which there is some a ∈ ∆I such that CI (a) = n, and n ∈ (0, 1]. In this case, C is called n-satisfiable w.r.t. R and T (=-=Navara, 2000-=-). Let C and D be two fKD-SHIN -concepts. We say that C is subsumed by D w.r.t. R and T if for every model I of R and T it holds that, ∀x ∈ ∆I .CI (x) ≤ DI (x). Furthermore, an fKD-SHIN ABox A is cons... |

11 | Some complexity results on fuzzy description logics - Bonatti, Tettamanzi - 2004 |

10 |
The use of fuzzy relational thesauri for classificatory problem solving in information restrieval and exprert systems
- Larsen, Yager
- 1993
(Show Context)
Citation Context ...ch areas, like decision making, image processing, robotics and medical diagnosis have adopted special mathematical frameworks that are intended for modelling such types of concepts (Zimmermann, 1987; =-=Larsen & Yager, 1993-=-; Krishnapuram & Keller, 1992). One such a mathematical framework is fuzzy set theory (Zadeh, 1965). Though fuzzy extensions of various logical formalisms, like propositional, predicate or modal logic... |

9 |
Triangular norms. position paper i: basic analytical and algebraic properties
- Klement, Mesiar, et al.
(Show Context)
Citation Context ...y over the unit interval. These functions are usually called norm operations referred to as t-norms (t), in the case of fuzzy intersection, and t-conorms (or s-norms) (u), in the case of fuzzy union (=-=Klement, Mesiar, & Pap, 2004-=-). Again these operations should satisfy certain mathematical properties. More precisely, a t-norm (t-conorm) satisfies the boundary condition, t(a, 1) = a (u(a, 0) = a), is monotonic increasing, for ... |