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Undecidability of fuzzy description logics
 In Proc. of the 13th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2012
, 2012
"... Abstract. Fuzzy Description Logics (DLs) with tnorm semantics have been studied as a means for representing and reasoning with vague knowledge. Recent work has shown that even fairly inexpressive fuzzy DLs become undecidable for a wide variety of tnorms. We complement those results by providing ..."
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Cited by 32 (17 self)
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Abstract. Fuzzy Description Logics (DLs) with tnorm semantics have been studied as a means for representing and reasoning with vague knowledge. Recent work has shown that even fairly inexpressive fuzzy DLs become undecidable for a wide variety of tnorms. We complement those results by providing a class of tnorms and an expressive fuzzy DL for which ontology consistency is linearly reducible to crisp reasoning, and thus has its same complexity. Surprisingly, in these same logics crisp models are insufficient for deciding fuzzy subsumption. 1
R.: On the undecidability of fuzzy description logics with GCIs and product tnorm
 Proc. of the 8th Int. Symp. on Frontiers of Combining Systems (FroCoS’11
, 2011
"... Abstract. The combination of Fuzzy Logics and Description Logics (DLs) has been investigated for at least two decades because such fuzzy DLs can be used to formalize imprecise concepts. In particular, tableau algorithms for crisp Description Logics have been extended to reason also with their fuzzy ..."
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Cited by 19 (6 self)
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Abstract. The combination of Fuzzy Logics and Description Logics (DLs) has been investigated for at least two decades because such fuzzy DLs can be used to formalize imprecise concepts. In particular, tableau algorithms for crisp Description Logics have been extended to reason also with their fuzzy counterparts. Recently, it has been shown that, in the presence of general concept inclusion axioms (GCIs), some of these fuzzy DLs actually do not have the finite model property, thus throwing doubt on the correctness of tableau algorithm for which it was claimed that they can handle fuzzy DLs with GCIs. In a previous paper, we have shown that these doubts are indeed justified, by proving that a certain fuzzy DL with product tnorm and involutive negation is undecidable. In the present paper, we show that undecidability also holds if we consider a tnormbased fuzzy DL where disjunction and involutive negation are replaced by the constructor implication, which is interpreted as the residuum. The only condition on the tnorm is that it is a continuous tnorm “starting ” with the product tnorm, which covers an uncountable family of tnorms. 1
Description logics over lattices with multivalued ontologies
 In Proc. IJCAI’11
, 2011
"... Uncertainty is unavoidable when modeling most application domains. In medicine, for example, symptoms (such as pain, dizziness, or nausea) are always subjective, and hence imprecise and incomparable. Additionally, concepts and their relationships may be inexpressible in a crisp, clearcut manner. ..."
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Cited by 9 (8 self)
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Uncertainty is unavoidable when modeling most application domains. In medicine, for example, symptoms (such as pain, dizziness, or nausea) are always subjective, and hence imprecise and incomparable. Additionally, concepts and their relationships may be inexpressible in a crisp, clearcut manner. We extend the description logicALC with multivalued semantics based on lattices that can handle uncertainty on concepts as well as on the axioms of the ontology. We introduce reasoning methods for this logic w.r.t. general concept inclusions and show that the complexity of reasoning is not increased by this new semantics. 1
On the undecidability of fuzzy description logics with GCIs with Łukasiewicz tnorm
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GCIs Make Reasoning in Fuzzy DL with the Product Tnorm Undecidable
"... Fuzzy variants of Description Logics (DLs) were introduced in order to deal with applications where not all concepts can be defined in a precise way. A great variety of fuzzy DLs have been investigated in the literature [12,8]. In fact, compared to crisp DLs, fuzzy DLs offer an additional degree of ..."
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Cited by 7 (0 self)
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Fuzzy variants of Description Logics (DLs) were introduced in order to deal with applications where not all concepts can be defined in a precise way. A great variety of fuzzy DLs have been investigated in the literature [12,8]. In fact, compared to crisp DLs, fuzzy DLs offer an additional degree of freedom when
R.: Fuzzy ontologies over lattices with tnorms
 In: Proc. DL’11
, 2011
"... In some knowledge domains, a correct handling of vagueness and imprecision is fundamental for adequate knowledge representation and reasoning. For example, when trying to diagnose a disease, medical experts need to confront symptoms described by the patient, which are by definition subjective, and h ..."
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Cited by 7 (5 self)
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In some knowledge domains, a correct handling of vagueness and imprecision is fundamental for adequate knowledge representation and reasoning. For example, when trying to diagnose a disease, medical experts need to confront symptoms described by the patient, which are by definition subjective, and hence vague.
Gödel Negation Makes Unwitnessed Consistency Crisp ⋆
"... Abstract. Ontology consistency has been shown to be undecidable for a wide variety of fairly inexpressive fuzzy Description Logics (DLs). In particular, for any tnorm “starting with ” the Lukasiewicz tnorm, consistency of crisp ontologies (w.r.t. witnessed models) is undecidable in any fuzzy DL wi ..."
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Cited by 4 (4 self)
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Abstract. Ontology consistency has been shown to be undecidable for a wide variety of fairly inexpressive fuzzy Description Logics (DLs). In particular, for any tnorm “starting with ” the Lukasiewicz tnorm, consistency of crisp ontologies (w.r.t. witnessed models) is undecidable in any fuzzy DL with conjunction, existential restrictions, and (residual) negation. In this paper we show that for any tnorm with Gödel negation, that is, any tnorm not starting with Lukasiewicz, ontology consistency for a variant of fuzzy SHOI is linearly reducible to crisp reasoning, and hence decidable in exponential time. Our results hold even if reasoning is not restricted to the class of witnessed models only. 1
R.: NonGödel negation makes unwitnessed consistency undecidable
 In: Proc. of the 25th Int. Workshop on Description Logics (DL 2012). CEUR Workshop Proceedings
, 2012
"... Abstract. Recent results show that ontology consistency is undecidable for a wide variety of fuzzy Description Logics (DLs). Most notably, undecidability arises for a family of inexpressive fuzzy DLs using only conjunction, existential restrictions, and residual negation, even if the ontology itself ..."
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Cited by 4 (4 self)
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Abstract. Recent results show that ontology consistency is undecidable for a wide variety of fuzzy Description Logics (DLs). Most notably, undecidability arises for a family of inexpressive fuzzy DLs using only conjunction, existential restrictions, and residual negation, even if the ontology itself is crisp. All those results depend on restricting reasoning to witnessed models. In this paper, we show that ontology consistency for inexpressive fuzzy DLs using any tnorm starting with the Łukasiewicz tnorm is also undecidable w.r.t. general models. 1
Fuzzy ontologies and fuzzy integrals
 11th International Conference on Intelligent Systems Design and Applications, IEEE
, 2011
"... AbstractFuzzy ontologies extend classical ontologies to allow the representation of imprecise and vague knowledge. Although a relatively important amount of work has been carried out in the last years and they have been successfully used in several applications, several notions from fuzzy logic, s ..."
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Cited by 2 (1 self)
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AbstractFuzzy ontologies extend classical ontologies to allow the representation of imprecise and vague knowledge. Although a relatively important amount of work has been carried out in the last years and they have been successfully used in several applications, several notions from fuzzy logic, such as fuzzy integrals, have not been considered yet in fuzzy ontologies. In this work, we show how to support fuzzy integrals in fuzzy ontologies. As a theoretical formalism, we provide the syntax and semantics of a fuzzy Description Logic with fuzzy integrals. We also provide a reasoning algorithm for a family of fuzzy integrals and show how to encode them into the language Fuzzy OWL 2.
A Framework for Reasoning with Expressive Continuous Fuzzy Description Logics
"... Abstract. In the current paper we study the reasoning problem for fuzzy SI (fSI) under arbitrary continuous fuzzy operators. Our work can be seen as an extension of previous works that studied reasoning algorithms for fSI, but focused on specific fuzzy operators, e.g. fKDSI and of reasoning algor ..."
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Abstract. In the current paper we study the reasoning problem for fuzzy SI (fSI) under arbitrary continuous fuzzy operators. Our work can be seen as an extension of previous works that studied reasoning algorithms for fSI, but focused on specific fuzzy operators, e.g. fKDSI and of reasoning algorithms for less expressive fuzzy DLs, like fLALC and fPALC (fuzzy ALC under the Lukasiewicz and product fuzzy operators, respectively). We show how transitivity can be handled for all the range of continuous fuzzy DLs and discuss about blocking and correctness in this setting. Based on these analysis, we present a unifying framework for reasoning over the class of continuous fuzzy DLs. Finally use the results to prove decidability of several fuzzy SI DLs. 1