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Fuzzy description logics under Gödel semantics

Profile image of Fernando BobilloFernando Bobillo

2009, International Journal of Approximate Reasoning

https://doi.org/10.1016/J.IJAR.2008.10.003
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Abstract

Classical ontologies are not suitable to represent vague pieces of information, which has lead to the birth of Fuzzy Description Logics as an appropriate formalism to represent this type of knowledge. Different families of fuzzy operators lead to Fuzzy Description Logics with different properties. This paper studies Fuzzy Description Logics under a semantics given by the Gödel family of fuzzy operators. We investigate some logical properties and show the decidability of a fuzzy extension of the logic SROIQ, theoretical basis of the language OWL 1.1, by providing a reasoning preserving procedure to obtain a crisp representation for it. Additionally, we show how to represent some types of concept and role modifiers.

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A Fuzzy Description Logic for the Semantic Web
Umberto Straccia

2006

In this paper we present a fuzzy version of SHOIN (D), the corresponding Description Logic of the ontology description language OWL DL. We show that the representation and reasoning capabilities of fuzzy SHOIN (D) go clearly beyond classical SHOIN (D). Interesting features are: (i) concept constructors are based on t-norm, t-conorm, negation and implication; (ii) concrete domains are fuzzy sets; (iii) fuzzy modifiers are allowed; and (iv) entailment and subsumption relationships may hold to some degree in the unit interval [0, 1]. mgb:Roadster (∃maker.{mg}) (∃speed.≤ 170km/h ) enzo:Car (∃maker.{ferrari}) (∃speed.> 350km/h ) tt:Car (∃maker.{audi}) (∃speed.= 243km/h ) Consider the knowledge base K = T , R, A . It is then easily verified that, e.g. K |= Roadster Car K |= mg:Maker K |= enzo:SportsCar K |= tt:¬SportsCar .

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A Fuzzy Description Logic
Umberto Straccia

1998

Description Logics (DLs, for short) allow reasoning about individuals and concepts, i.e. set of individuals with common properties. Typically, DLs are limited to dealing with crisp, well defined concepts. That is, concepts for which the problem whether an individual is an instance of it is a yes/no question. More often than not, the concepts encountered in the real world do not have a precisely defined criteria of membership: we may say that an individual is an instance of a concept only to a certain degree, depending on the individual's properties. Concepts of this kind are rather vague than precise. As fuzzy logic directly deals with the notion of vagueness and imprecision, it offers an appealing foundation for a generalisation of DLs to vague concepts. In this paper we present a general fuzzy DL, which combines fuzzy logic with DLs. We define its syntax, semantics and present constraint propagation calculi for reasoning in it.

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Optimizing the Crisp Representation of the Fuzzy Description Logic SROIQ
Fernando Bobillo

Iswc, 2007

Classical ontologies are not suitable to represent imprecise nor uncertain pieces of information. Fuzzy Description Logics were born to represent the former type of knowledge, but they require an appropriate fuzzy language to be agreed and an important number of available resources to be adapted. This paper faces these problems by presenting a reasoning preserving procedure to obtain a crisp representation for a fuzzy extension of the logic SROIQ which uses Gödel implication in the semantics of fuzzy concept and role subsumption. This reduction allows to reuse a crisp representation language as well as currently available reasoners. Our procedure is optimized with respect to the related work, reducing the size of the resulting knowledge base, and is implemented in DeLorean, the first reasoner supporting fuzzy OWL DL.

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Optimizing the Crisp Representation of the Fuzzy Description Logic $mathcalSROIQ$
Fernando Bobillo

2007

Classical ontologies are not suitable to represent imprecise nor uncertain pieces of information. Fuzzy Description Logics were born to represent the former type of knowledge, but they require an appropriate fuzzy language to be agreed and an important number of available resources to be adapted. This paper faces these problems by presenting a reasoning preserving procedure to obtain a crisp representation for a fuzzy extension of the logic SROIQ which uses Gödel implication in the semantics of fuzzy concept and role subsumption. This reduction allows to reuse a crisp representation language as well as currently available reasoners. Our procedure is optimized with respect to the related work, reducing the size of the resulting knowledge base, and is implemented in DeLorean, the first reasoner supporting fuzzy OWL DL.

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Transforming fuzzy description logic ALCFL into classical description logic ALCH
yining wu

icr.uni.lu

We consider ALC F L in this thesis. ALC F L is a fuzzy Description Logic with hedges as its fuzzy extension. Three example scenarios of ALC F L knowledge base are shown. The linear symmetric hedge algebra [4, is introduced in order to represent the truth domain of interpretations of the ALC F L knowledge base. The aim of this thesis is to present a satisfiability preserving transformation of ALC F L into the description logic ALCH, i.e., the extension of ALC by role hierarchies. The soundness and completeness of this transformation are proved. A contribution of this transformation is that we can do reasoning of ALC F L by using already existing Description Logic systems. The transformation is implemented in both the programming language JAVA and Prolog. Testing is done in JAVA using APIs of Jena, 1 KAON2 2 and Racer. 3 1 http://jena.sourceforge.net/ 2 http://kaon2.semanticweb.org/ 3 http://www.racer-systems.com/ iii iv Abstract Contents Acknowledgments i Abstract iii 1 Please visit http://www.w3.org/TR/owl-guide/ for more details

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Fuzzy quantification in fuzzy description logics
Daniel Sanchez, Andrea G B Tettamanzi

Capturing Intelligence, 2006

This chapter introduces reasoning procedures for ALCQ + F (D), a fuzzy description logic with extended qualified quantification [D. Sánchez, A.G.B. Tettamanzi, Generalizing quantification in fuzzy description logics, in: Proceedings 8th Dortmund Fuzzy Days, Dortmund, Germany, 2004]. The language allows for the definition of fuzzy quantifiers of the absolute and relative kind by means of piecewise linear functions on N and Q ∩ [0, 1] respectively. In order to reason about instances, the semantics of quantified expressions is defined by using method GD [M. Delgado, D. Sánchez, M. Vila, Fuzzy cardinality based evaluation of quantified sentences, Int. J. Approximate Reasoning 23 , 23-66], which is based on recently developed measures of the cardinality of fuzzy sets. The main contribution of this chapter is a procedure to calculate the fuzzy satisfiability of a fuzzy concept, which is a very important reasoning task. The procedure considers several different cases and provides direct solutions for the most frequent types of fuzzy concepts. In order to distinguish between these cases, a novel idea of concept independence is also introduced.

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Towards a Crisp Representation of Fuzzy Description Logics under Łukasiewicz Semantics
Fernando Bobillo

Lecture Notes in Computer Science, 2008

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Description Logics with Fuzzy Concrete Domains
Umberto Straccia

2005

We present a fuzzy version of description logics with concrete domains. Main features are: (i) concept constructors are based on t-norm, t-conorm, negation and implication; (ii) concrete domains are fuzzy sets; (iii) fuzzy modifiers are allowed; and (iv) the reasoning algorithm is based on a mixture of completion rules and bounded mixed integer programming.

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The fuzzy linguistic description logic ALCFL
Dzung Dinh Khac

Proceedings of the 11th …, 2006

Description Logics (DLs) have been studied and applied successfully in quite a lot of fields (see eg [1]). To deal with vague and imprecise information in real-world applications, fuzzy ALC [7] introduces fuzzy concepts. As a more general case of fuzzy ALC, a DL framework ...

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THE FUZZY DESCRIPTION LOGIC ALCFLH
Tran Khang

Description Logics, 2000

We present the fuzzy description logic ALC F LH . ALC F LH is based on ALC F H , but linear hedges are used instead of exponential ones. This allows to solve the entailment and the subsumption problem in a fuzzy description logic, where arbitrary concepts and roles may be modified.

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Joining Gödel and Zadeh Fuzzy Logics in Fuzzy Description Logics
Fernando Bobillo

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Ontologies have succeeded as a knowledge representation formalism in many domains of application. Nevertheless, they are not suitable to represent vague or imprecise information. To overcome this limitation, several extensions to classical ontologies based on fuzzy logic have been proposed. Even though different fuzzy logics lead to fuzzy ontologies with very different logical properties, the combined use of different fuzzy logics has received little attention to date. This paper proposes a fuzzy extension of the Description Logic SROIQ -the logic behind the ontology language OWL 2that joins Gödel and Zadeh fuzzy logics. We analyze the properties of the new fuzzy Description Logic in order to provide guidelines to ontology developers to exploit the best features of each fuzzy logic. The proposal also considers degrees of truth belonging to a finite set of linguistic terms rather than numerical values, thus being closer to real experts' reasonings. We prove the decidability of the combined logic by presenting a reasoning preserving procedure to obtain a crisp representation for it. This result is generalized to offer a similar reduction that can be applied when any other finite t-norms, t-conorms, negations or implications are considered in the logic.

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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 .

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