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Jiři Mockoř - One of the best experts on this subject based on the ideXlab platform.
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WILF - Any F-Transform Is Defined by a Powerset Theory.
Fuzzy Logic and Applications, 2019Co-Authors: Jiři MockořAbstract:Relationships between Powerset theories and F-transforms are investigated. Both these methods represent strong tools in fuzzy sets theory and applications. Although both methods deal with similar objects, both these methods use different tools and, so far, the relationship between the two methods has not been investigated. The aim of this paper is to show that there is a strong relationship between the two methods. Namely, arbitrary lower or upper F-transform of lattice-valued fuzzy sets can be derived from a special Powerset theory and, conversely, there exists a special class of Powerset theories, such that maps defined by these Powerset theories are lower or upper F-transforms. These results allow, among other things, to extend the range of methods and tools that are used in both theories.
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fuzzy type Powerset operators and f transforms
Modeling Decisions for Artificial Intelligence, 2018Co-Authors: Jiři MockořAbstract:We introduce two types of aggregation operators for lattice-valued fuzzy sets, called fuzzy type Powerset operators and fuzzy type F-transforms, which are derived from classical Powerset operators and F-transforms, respectively. We prove that, in contrast with classical Powerset operators, fuzzy type Powerset operators form a subclass of fuzzy type F-transforms. Some examples of fuzzy type Powerset operators are presented.
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MDAI - Fuzzy Type Powerset Operators and F-Transforms
Modeling Decisions for Artificial Intelligence, 2018Co-Authors: Jiři MockořAbstract:We introduce two types of aggregation operators for lattice-valued fuzzy sets, called fuzzy type Powerset operators and fuzzy type F-transforms, which are derived from classical Powerset operators and F-transforms, respectively. We prove that, in contrast with classical Powerset operators, fuzzy type Powerset operators form a subclass of fuzzy type F-transforms. Some examples of fuzzy type Powerset operators are presented.
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any f transform is defined by a Powerset theory
International Workshop on Fuzzy Logic and Applications, 2018Co-Authors: Jiři MockořAbstract:Relationships between Powerset theories and F-transforms are investigated. Both these methods represent strong tools in fuzzy sets theory and applications. Although both methods deal with similar objects, both these methods use different tools and, so far, the relationship between the two methods has not been investigated. The aim of this paper is to show that there is a strong relationship between the two methods. Namely, arbitrary lower or upper F-transform of lattice-valued fuzzy sets can be derived from a special Powerset theory and, conversely, there exists a special class of Powerset theories, such that maps defined by these Powerset theories are lower or upper F-transforms. These results allow, among other things, to extend the range of methods and tools that are used in both theories.
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some examples of relations between f transforms and Powerset theories
North American Fuzzy Information Processing Society, 2018Co-Authors: Jiři MockořAbstract:Six examples of Powerset theories based on lattice-valued fuzzy sets are presented and relations between these Powerset theories and F-transforms are investigated. It is proved that Powerset extensions corresponding to these Powerset theories are identical to, or restrictions of the F-transforms with respect to spaces with fuzzy partitions, consisting of objects of corresponding Powerset theories.
Enea Zaffanella - One of the best experts on this subject based on the ideXlab platform.
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widening operators for Powerset domains
Formal Methods, 2007Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. While most of the operations on the finite Powerset abstract domain are easily obtained by “lifting” the corresponding operations on the base-level domain, the problem of endowing finite Powersets with a provably correct widening operator is still open. In this paper we define three generic widening methodologies for the finite Powerset abstract domain. The widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators that allow all the flexibility required to tune the complexity/precision trade-off. As far as we know, this is the first time that the problem of deriving non-trivial, provably correct widening operators in a domain refinement is tackled successfully. We illustrate the proposed techniques by instantiating our widening methodologies on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
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VMCAI - Widening Operators for Powerset Domains
Lecture Notes in Computer Science, 2004Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. In this paper we define two generic widening operators for the finite Powerset abstract domain. Both widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators. We illustrate the proposed techniques by instantiating our widenings on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
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widening operators for Powerset domains
Lecture Notes in Computer Science, 2004Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. In this paper we define two generic widening operators for the finite Powerset abstract domain. Both widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators. We illustrate the proposed techniques by instantiating our widenings on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
Roberto Bagnara - One of the best experts on this subject based on the ideXlab platform.
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widening operators for Powerset domains
Formal Methods, 2007Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. While most of the operations on the finite Powerset abstract domain are easily obtained by “lifting” the corresponding operations on the base-level domain, the problem of endowing finite Powersets with a provably correct widening operator is still open. In this paper we define three generic widening methodologies for the finite Powerset abstract domain. The widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators that allow all the flexibility required to tune the complexity/precision trade-off. As far as we know, this is the first time that the problem of deriving non-trivial, provably correct widening operators in a domain refinement is tackled successfully. We illustrate the proposed techniques by instantiating our widening methodologies on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
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VMCAI - Widening Operators for Powerset Domains
Lecture Notes in Computer Science, 2004Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. In this paper we define two generic widening operators for the finite Powerset abstract domain. Both widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators. We illustrate the proposed techniques by instantiating our widenings on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
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widening operators for Powerset domains
Lecture Notes in Computer Science, 2004Co-Authors: Roberto Bagnara, Patricia M Hill, Enea ZaffanellaAbstract:The finite Powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. In this paper we define two generic widening operators for the finite Powerset abstract domain. Both widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators. We illustrate the proposed techniques by instantiating our widenings on Powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.
Stephen E. Rodabaugh - One of the best experts on this subject based on the ideXlab platform.
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Axiomatic Foundations For Uniform Operator Quasi-Uniformities
Topological and Algebraic Structures in Fuzzy Sets, 2020Co-Authors: Stephen E. RodabaughAbstract:Traditional uniformities have both the entourage approach of [33, 1], based on Powersets of the form 2 X ×X , as well as the uniform covering approach of [30, 10], based on double Powersets of the form \( {{2}^{{({{2}^{x}})}}} \).
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Relationship of algebraic theories to Powersets over objects in Set
2020Co-Authors: Stephen E. RodabaughAbstract:Thispaperdealswithaparticularquestion—WhendoPowersetsinlattice-valuedmathematicsformalgebraictheories(ormonads) incloneform?Ourapproachinthisandrelatedpapersistoconsider “Powersetsoverobjects”inthegroundcategoriesSetandSet×C 9 from the standpoint of algebraic theories in clone form (C is a particular subcategory of the dual of the category of semi-quantales). For both fixed-basis Powersets over objects of Set and variable-basis Powersets over objects of Set × C, necessary and sufficient 11 conditions are found under which the family of all such Powersets over a ground object forms an algebraic theory in clone form of standard construction. In such results a distinguished role emerges for unital quantales. 13
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Relationship of algebraic theories to Powersets over objects in Set and Set×C
Fuzzy Sets and Systems, 2010Co-Authors: Stephen E. RodabaughAbstract:This paper deals with a particular question-When do Powersets in lattice-valued mathematics form algebraic theories (or monads) in clone form? Our approach in this and related papers is to consider ''Powersets over objects'' in the ground categories Set and SetxC from the standpoint of algebraic theories in clone form (C is a particular subcategory of the dual of the category of semi-quantales). For both fixed-basis Powersets over objects of Set and variable-basis Powersets over objects of SetxC, necessary and sufficient conditions are found under which the family of all such Powersets over a ground object forms an algebraic theory in clone form of standard construction. In such results a distinguished role emerges for unital quantales.
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Point-set lattice-theoretic topology
Fuzzy Sets and Systems, 1991Co-Authors: Stephen E. RodabaughAbstract:This essay attempts to survey in a coherent way certain aspects of point-set lattice-theoretic or poslat topology, by which we mean (fuzzy) topology grounded in notions of sets, functions, Powersets, and Powerset operators - the latter two being lattice-theoretic in nature and examined from a lattice-theoretic point of view using methods of category theory. Connections with other related issues and developments are frequently given
Patrik Eklund - One of the best experts on this subject based on the ideXlab platform.
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EUSFLAT Conf. - A note on partially ordered generalized terms
2020Co-Authors: Patrik Eklund, M. A. Galán, Jesús Medina, Manuel Ojeda-aciego, Werner Gähler, Agustín ValverdeAbstract:In this paper we study the deflnition of generalized Powerset of terms in the context of partially ordered monads. We provide a deflnition of the Powerset of terms over acSLAT and study some possible directions for the deflnition of a partially ordered term monad.
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A framework for unication using Powersets of terms
2020Co-Authors: Patrik Eklund, Jesús Medina, Galán, M. Ojeda Aciego, Agustín ValverdeAbstract:Many-valued logic programming with generalised terms requires an extended notion of unification in order to handle Powersets of terms. In this paper we present substitutions and unifiers in a categorical framework based on Powersets of terms as monads. We build upon developments for monad compositions initiated in [4].
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Adding fuzziness to terms and powerobjects using a monadic approach
Fuzzy Sets and Systems, 2012Co-Authors: Patrik Eklund, Jari Kortelainen, Lawrence Neff StoutAbstract:Fuzzy mathematics often starts by taking a piece of classical mathematics and introducing fuzziness to existing mathematical concepts, and then proceeds to a more essential adoption of a fuzzy perspective. This paper explores this progression for two important monads: the Powerset monad and its generalizations to fuzzy powerobjects, and the term monad and its generalization to using fuzzy sets of operators. This brings together two lines of research previously discussed at the Linz seminars. The Powerset and its fuzzy analogs are important in the development of topology in a fuzzy world and the term monad and its fuzzy analogs are vital in understanding fuzzy computer science.
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Powersets of terms and composite monads
Fuzzy Sets and Systems, 2007Co-Authors: Patrik Eklund, M. A. Galán, Jesús Medina, Manuel Ojeda-aciego, Agustín ValverdeAbstract:Composing various Powerset functors with the term monad gives rise to the concept of generalized terms. This in turn provides a technique for handling many-valued sets of terms in a framework of variable substitutions, thus being the prerequisite for categorical unification in many-valued logic programming using an extended notion of terms. As constructions of monads involve complicated calculations with natural transformations, proofs are supported by a graphical approach that provides a useful tool for handling various conditions, such as those well known for distributive laws.
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ISMVL - The Rough Powerset Monad
37th International Symposium on Multiple-Valued Logic (ISMVL'07), 2007Co-Authors: Patrik Eklund, M. A. GalánAbstract:Rough sets provide a good environment to deal with vagueness and uncertainty situations. In this paper we show how monads can be used to generalize and interpret rough situations. In particular, the partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations.