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A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 52 (3 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
A systematic approach to the assessment of fuzzy association rules. Data Mining and Knowledge Discovery
, 2006
"... In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such ru ..."
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Cited by 30 (6 self)
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In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such rules by means of appropriate quality measures is less obvious. Particularly, it assumes an understanding of the semantic meaning of a fuzzy rule. This aspect has been ignored by most existing proposals, which must therefore be considered as adhoc to some extent. In this paper, we develop a systematic approach to the assessment of fuzzy association rules. To this end, we proceed from the idea of partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data. Evaluation measures are then derived from the cardinalities of the corresponding subsets. The problem of finding a proper partition has a rather obvious solution for standard association rules but becomes less trivial in the fuzzy case. Our results not only provide a sound justification for commonly used measures but also suggest a means for constructing meaningful alternatives. 1.
On the extension of pseudoboolean functions for the aggregation of interacting criteria
 European Journal of Operational Research
, 2003
"... The paper presents an analysis on the use of integrals defined for nonadditive measures (or capacities) as the Choquet and the ˇ Sipoˇs integral, and the multilinear model, all seen as extensions of pseudoBoolean functions, and used as a means to model interaction between criteria in a multicriter ..."
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Cited by 26 (3 self)
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The paper presents an analysis on the use of integrals defined for nonadditive measures (or capacities) as the Choquet and the ˇ Sipoˇs integral, and the multilinear model, all seen as extensions of pseudoBoolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a “neutral level”, allowing to introduce the absolute notion of attractive or repulsive act. It is shown
Structure of uninorms
 Internat. J. Uncertain. Fuzziness KnowledgeBased Systems
, 1997
"... Abstract: In this paper we characterize those uninorms which are rational functions (i.e., quotients of two polynomials). These are closely related to the wellknown parametric families of tnorms and tconorms studied and characterized by Hamacher [8]. ..."
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Cited by 24 (3 self)
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Abstract: In this paper we characterize those uninorms which are rational functions (i.e., quotients of two polynomials). These are closely related to the wellknown parametric families of tnorms and tconorms studied and characterized by Hamacher [8].
On the representation of intuitionistic fuzzy tnorms and tconorms
 IEEE Transactions on Fuzzy Systems
, 2004
"... Abstract—Intuitionistic fuzzy sets form an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a nonmembership degree. The only constraint on those two degrees is that their sum must be smaller than ..."
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Cited by 22 (11 self)
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Abstract—Intuitionistic fuzzy sets form an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a nonmembership degree. The only constraint on those two degrees is that their sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms and conorms is the class of continuous Archimedean nilpotent triangular norms and conorms. It has been shown that for suchnorms there exists a permutation of [0,1] such that is thetransform of the Łukasiewicznorm. In this paper we introduce the notion of intuitionistic fuzzynorm andconorm, and investigate under which conditions a similar representation theorem can be obtained. Index Terms—Archimedean property, intuitionistic fuzzy set, intuitionistic fuzzy triangular norm and conorm, nilpotency, representation theorem,transform. I.
Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence
 ARTIFICIAL INTELLIGENCE
, 2007
"... Dempster’s rule plays a central role in the theory of belief functions. However, it assumes the combined bodies of evidence to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commuta ..."
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Cited by 17 (9 self)
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Dempster’s rule plays a central role in the theory of belief functions. However, it assumes the combined bodies of evidence to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine belief functions induced by reliable, but possibly overlapping bodies of evidence. A dual operator, the bold disjunctive rule, is also introduced. This operator is also commutative, associative and idempotent, and can be used to combine belief functions issues from possibly overlapping and unreliable sources. Finally, the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms.
Characterization of Measures Based on Strict Triangular Norms
 J. Math. Anal. Appl
"... As a natural generalization of a measure space, Butnariu and Klement introduced T tribes of fuzzy sets with T measures. They gave a complete characterization of T measures for a Frank triangular norm T . Here we characterize Tmeasures with respect to nonFrank strict triangular norms. We show th ..."
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Cited by 16 (7 self)
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As a natural generalization of a measure space, Butnariu and Klement introduced T tribes of fuzzy sets with T measures. They gave a complete characterization of T measures for a Frank triangular norm T . Here we characterize Tmeasures with respect to nonFrank strict triangular norms. We show the specific roles of Frank triangular norms and a newly introduced family, nearly Frank triangular norms. 1 The notion of Tmeasure We start with the basic definitions from [4]. Let X be a set and B a oealgebra of subsets of X . The Bgenerated tribe is the collection T of all functions A: X ! [0; 1] (fuzzy subsets of X) which are Bmeasurable. In order to define measures on T , we fix a tnorm T (fuzzy conjunction), i.e., a binary operation T : [0; 1] 2 ! [0; 1] which is commutative, associative, nondecreasing, and satisfies the boundary condition T (a; 1) = a for all a 2 [0; 1] (see [15]). For the other necessary fuzzy logical operations, we take the standard fuzzy negation 0 : [0; 1...
Flexible neurofuzzy systems
 IEEE TRANS. NEURAL NETW
, 2003
"... In this paper, we derive new neurofuzzy structures called flexible neurofuzzy inference systems or FLEXNFIS. Based on the input–output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to f ..."
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Cited by 15 (3 self)
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In this paper, we derive new neurofuzzy structures called flexible neurofuzzy inference systems or FLEXNFIS. Based on the input–output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to fuzzy implication operators, to aggregation of rules and to connectives of antecedents; 2) certainty weights to aggregation of rules and to connectives of antecedents; and 3) parameterized families of Tnorms and Snorms to fuzzy implication operators, to aggregation of rules and to connectives of antecedents. Our approach introduces more flexibility to the structure and design of neurofuzzy systems. Through computer simulations, we show that Mamdanitype systems are more suitable to approximation problems, whereas logicaltype systems may be preferred for classification problems.
Automorphisms of the algebra of fuzzy truth values II
 INT J. OF UNCERTAINTY, FUZZINESS AND KNOWLEDGEBASED SYSTEMS
, 2008
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A Survey on Different Triangular NormBased Fuzzy Logics
, 1999
"... Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizati ..."
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Cited by 13 (1 self)
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Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, manyvalued logic, Frank tnorm 1 Introduction A manyvalued propositional logic with a continuum of truth values modelled by the unit interval [0; 1] is quite often called a fuzzy logic. In such a logic, the conjunction is usually interpreted by a triangular norm. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...