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96
Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty
 Applied Intelligence
, 1996
"... In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preferenc ..."
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Cited by 74 (13 self)
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In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instanciations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instanciations are provided owing to the relaxation of ...
Computations with Imprecise Parameters in Engineering Design: Background and Theory
 ASME JOURNAL OF MECHANISMS, TRANSMISSIONS, AND AUTOMATION IN DESIGN
, 1989
"... A technique to perform design calculations on imprecise representations of parameters has been developed and is presented. The level of imprecision in the description of design elements is typically high in the preliminary phase of engineering design. This imprecision is represented using the fuzzy ..."
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Cited by 51 (18 self)
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A technique to perform design calculations on imprecise representations of parameters has been developed and is presented. The level of imprecision in the description of design elements is typically high in the preliminary phase of engineering design. This imprecision is represented using the fuzzy calculus. Calculations can be performed using this method, to produce (imprecise) performance parameters from imprecise (input) design parameters. The Fuzzy Weighted Average technique is used to perform these calculations. A new metric, called the γlevel measure, is introduced to determine the relative coupling between imprecise inputs and outputs. The background and theory supporting this approach are presented, along with one example.
MultiValued Symbolic ModelChecking
 ACM TRANSACTIONS ON SOFTWARE ENGINEERING AND METHODOLOGY
, 2003
"... This paper introduces the concept and the general theory of multivalued model checking, and describes a multivalued symbolic modelchecker \Chi Chek. Multivalued ..."
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Cited by 50 (16 self)
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This paper introduces the concept and the general theory of multivalued model checking, and describes a multivalued symbolic modelchecker \Chi Chek. Multivalued
The Maslov Dequantization, Idempotent and Tropical Mathematics: a Very Brief Introduction
, 2005
"... ..."
Order Norms On Bounded Partially Ordered Sets
 THE JOURNAL OF FUZZY MATHEMATICS
, 1994
"... In this paper, we extend the domains of affirmation and negation operators, and more important, of triangular (semi)norms and (semi)conorms from the unit interval to bounded partially ordered sets. The fundamental properties of the original operators are proven to be conserved under this extension. ..."
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Cited by 24 (14 self)
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In this paper, we extend the domains of affirmation and negation operators, and more important, of triangular (semi)norms and (semi)conorms from the unit interval to bounded partially ordered sets. The fundamental properties of the original operators are proven to be conserved under this extension. This clearly shows that they are essentially based upon ordertheoretic notions. Consequently, a rather general ordertheoretic invariance study of these operators is undertaken. Also, in a brief algebraic excursion, the notion of weak invertibility of these operators is introduced, and the relation with the ordertheoretic concept of residuals is studied. The importance of these results for fuzzy set theory and possibility theory is briefly discussed.
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.
Possibility theory. I. The measure and integraltheoretic groundwork
 Internat. J. Gen. Systems
, 1997
"... In this paper, I provide the basis for a measure and integraltheoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno’s fuzzy integral – the seminormed fuzzy integral, or possibility integral –, a unified and ..."
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Cited by 21 (16 self)
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In this paper, I provide the basis for a measure and integraltheoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno’s fuzzy integral – the seminormed fuzzy integral, or possibility integral –, a unified and consistent account can be given of many of the possibilistic results extant in the literature. The striking formal analogy between this treatment of possibility theory, using possibility integrals, and Kolmogorov’s measuretheoretic formulation of probability theory, using Lebesgue integrals, is explored and exploited. I introduce and study possibilistic and fuzzy variables as possibilistic counterparts of stochastic and real stochastic variables respectively, and develop the notion of a possibility distribution for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a RadonNikodymlike theorem in possibility theory. Following the example set by the classical theory of integration, product possibility measures and multiple possibility integrals are introduced, and a Fubinilike theorem is proven. In this way, the groundwork is laid for a unifying measure and integraltheoretic treatment of conditional possibility and possibilistic independence, discussed in more detail in Parts II and III of this series of three papers. INDEX TERMS: Possibility measure, seminormed fuzzy integral, possibilistic variable, fuzzy
Possibility and necessity integrals
 Fuzzy Sets and Systems
, 1996
"... Abstract: In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction ..."
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Cited by 15 (9 self)
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Abstract: In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Legesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno’s fuzzy integrals. A large number of important general properties of these integrals is studied. Furthermore, and most importantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied. It is shown that these combinations are very natural, and have properties which are analogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure and integraltheoretic account of possibility and necessity theory, in very much the same way as the theory of Lebesgue integration provides a proper framework for a unifying and formal account of probability theory.
Fuzzy Lattice Neurocomputing (FLN) Models
, 2000
"... In this work it is shown how fuzzy lattice neurocomputing (FLN) emerges as a connectionist paradigm in the framework of fuzzy lattices (FLframework) whose advantages include the capacity to deal rigorously with: disparate types of data such as numeric and linguistic data, intervals of values, "miss ..."
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Cited by 15 (7 self)
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In this work it is shown how fuzzy lattice neurocomputing (FLN) emerges as a connectionist paradigm in the framework of fuzzy lattices (FLframework) whose advantages include the capacity to deal rigorously with: disparate types of data such as numeric and linguistic data, intervals of values, "missing" and "don't care" data. A novel notation for the FLframework is introduced here in order to simplify mathematical expressions without losing content. Two concrete FLN models are presented, namely " FLN" for competitive clustering, and "FLN with tightest fits (FLNtf)" for supervised clustering. Learning by the FLN, is rapid as it requires a single pass through the data, whereas learning by the FLNtf , is incremental, data order independent, polynomial O(n³), and it guarantees maximization of the degree of inclusion of an input in a learned class as explained in the text. Convenient geometric interpretations are provided. The FLN is presented here as fuzzyART 's extension in the FLfr...