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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.
Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems
, 1998
"... The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representati ..."
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Cited by 2 (1 self)
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The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...
Possibility Measures And Possibility Integrals Defined On A Complete Lattice
"... . We consider the denition of possibility measures on complete lattices rather than on complete Boolean algebras of sets. We give a necessary and sucient condition for the extendability of any mapping to such a possibility measure. We also associate two types of integrals with these possibility meas ..."
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Cited by 1 (0 self)
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. We consider the denition of possibility measures on complete lattices rather than on complete Boolean algebras of sets. We give a necessary and sucient condition for the extendability of any mapping to such a possibility measure. We also associate two types of integrals with these possibility measures, and discuss some of their more important properties, amongst which a monotone convergence theorem. 1. Introduction Since possibility measures were introduced by Zadeh [18] in 1978, the measuretheoretic aspects of possibility measures and possibility integrals have been studied by various authors [2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20]. The most general denition of a possibility measure that can be distilled from this work is the following: a possibility measure is a mapping dened on a complete Boolean algebra [3] B and taking values in a complete lattice L, that is furthermore supremumpreserving , in that (sup j2J a j ) = sup j2J (a j ) for any family fa j j j ...
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.
Catalog of Axiom Systems
"... A catalog of axiom systems. Some introduction to the axiomatic method is presented. The axioms systems are provided as a general mathematical reference and as a source for theorem proving systems. DRAFT Contents DRAFT Part I Axioms and Proofs 1 Introduction Axioms are the backbone of all math ..."
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A catalog of axiom systems. Some introduction to the axiomatic method is presented. The axioms systems are provided as a general mathematical reference and as a source for theorem proving systems. DRAFT Contents DRAFT Part I Axioms and Proofs 1 Introduction Axioms are the backbone of all mathematics. They are the first principles, the starting points, the atoms. This is not to say that they are necessary. Historically, axioms began and stopped in ancient Greece, and there only in geometry, only to be started up again seriously in 19th century Europe. Axioms and the methods to derive theories from them lead to an important distinction in the practice of mathematics: intuitive, rigorous, and formal. Intuitive proofs will give you flashes of insight to great theorems only because of good habit with lesser principles. Rigorous proofs will prove theorems by consideration of all appropriate cases in a manner understandable, verifiable, and acceptable to others of similar ***. Formal...
On MApproximative Operators and MApproximative Systems
 IFSAEUSFLAT
, 2009
"... The concept of an Mapproximative system is introduced. Basic properties of the category of Mapproximative systems and in a natural way defined morphisms between them are studied. It is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as ..."
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The concept of an Mapproximative system is introduced. Basic properties of the category of Mapproximative systems and in a natural way defined morphisms between them are studied. It is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the
Classification Methods Based on Formal Concept Analysis
"... Abstract. Formal Concept Analysis (FCA) provides mathematical models for many domains of computer science, such as classification, categorization, text mining, knowledge management, software development, bioinformatics, etc. These models are based on the mathematical properties of concept lattices. ..."
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Abstract. Formal Concept Analysis (FCA) provides mathematical models for many domains of computer science, such as classification, categorization, text mining, knowledge management, software development, bioinformatics, etc. These models are based on the mathematical properties of concept lattices. The complexity of generating a concept lattice puts a constraint to the applicability of software systems. In this paper we report on some attempts to evaluate simple FCAbased classification algorithms. We present an experimental study of several benchmark datasets using FCAbased approaches. We discuss difficulties we encountered and make some suggestions concerning conceptbased classification algorithms.
A Note of the Discrete Monotonic Dynamical System
, 2004
"... Abstract. we give a upper bound of Lebesgue measure V (S(f,h,Ω)) of the set S(f,h,Ω) of points q ∈ Qd h for which the triple (h,q,Ω) is dynamically robust when f is monotonic and satisfies certain condition on some compact subset Ω ∈ Rd. 1. ..."
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Abstract. we give a upper bound of Lebesgue measure V (S(f,h,Ω)) of the set S(f,h,Ω) of points q ∈ Qd h for which the triple (h,q,Ω) is dynamically robust when f is monotonic and satisfies certain condition on some compact subset Ω ∈ Rd. 1.