The Experts below are selected from a list of 264 Experts worldwide ranked by ideXlab platform
Ling Zhang - One of the best experts on this subject based on the ideXlab platform.
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GrC - An Update and Summary of Multi-granular Computing and Quotient Structure
2010 IEEE International Conference on Granular Computing, 2010Co-Authors: Ling Zhang, Bo ZhangAbstract:Human always conceptualizes the world at different granularities and deals with it hierarchically. This is one of human’s characteristics that underlie his/her power. The Quotient space model that we presented is a mathematical model of human multi-granular problem solving. In the paper, we introduce the model and discuss how to use the model to deal with the granularity relation, hierarchical problem solving. We also present the future research directions.
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An Update and Summary of Multi-granular Computing and Quotient Structure
2010 IEEE International Conference on Granular Computing, 2010Co-Authors: Ling Zhang, Bo ZhangAbstract:Human always conceptualizes the world at different granularities and deals with it hierarchically. This is one of human's characteristics that underlie his/her power. The Quotient space model that we presented is a mathematical model of human multi-granular problem solving. In the paper, we introduce the model and discuss how to use the model to deal with the granularity relation, hierarchical problem solving. We also present the future research directions.
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Multi-granular representation-the key to machine intelligence
2008 3rd International Conference on Intelligent System and Knowledge Engineering, 2008Co-Authors: Bo Zhang, Ling ZhangAbstract:One of the basic characteristics in human problem solving is the ability to conceptualize the world at different granularities and translate from one abstraction level to the others easily. But so far computers can only deal with one abstraction level in problem solving generally. It seems important to develop new techniques which will in some way enable the computers to represent the world at different granularities. So the multi-granular representation is the key to machine intelligence. In the talk, we first introduce the Quotient space based problem solving theory. In the theory, a problem is represented by a triplet (X,F,T), where X - the universe with the finest grain-size, F -the attribute of X, and T- the Structure of X. When we view the same problem at a coarser grain size, we have a coarse-grained universe denoted by [X]. Then we have a new representation ([X],[F],[T]) of the problem. The coarse universe [X] is defined by an equivalence relation R on X. Then, representation ([X],[F],[T]) is called a Quotient space of(X,F,T), where [X] -the Quotient set of X, [F] -the Quotient attribute of F, and [T] -the Quotient Structure of T. Obviously, the set of representations of a problem at different granularities composes a complete semi-order lattice. That is, in the theory the concept, Quotient space, in algebra is used as a mathematical model to represent the relationship between representations with different grain-sizes. Multi-granular representation methodology can be used both in problem solving and machine learning. In multi-granular problem solving, a problem is solved from the coarse grain-size to the fine one hierarchically. The aim of hierarchical problem solving is intended to reduce the computational complexity. Multi-granular machine learning is intended to learn the knowledge from representations with different grain-size, i.e., the so-called multi-information fusion. Generally speaking, the fine representation has more details but less robustness. Conversely, the coarse representation has more robustness but less expressiveness. They are complement so multi-granular learning can benefit from them. We also present some examples in hierarchical problem solving and machine learning to show the advantages of using multi-granular representation.
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Granular computing based on fuzzy and tolerance relations
2008 IEEE International Conference on Granular Computing, 2008Co-Authors: Bo Zhang, Ling ZhangAbstract:Granular computing (GC) as a new field has grown rapidly since the term was proposed. There are two basic concepts underlying GC, i.e., fuzziness/fuzzification and granularity/granulation. Fuzzy set theory is an effective tool to deal with the fuzzification problem. There has been a large number of works on this aspect such as computing with words. The difficulty is in the concept of granulation. There was only an informal definition proposed by Zadeh for the concept. So far only a few works addressed the problem such as rough set and Quotient space theories. In these works, the granulation is mainly defined by equivalence relations, i.e., a partition model. For example, in Quotient space theory, a problem is represented by a triplet (X,F,T), where X -the universe with the finest grain-size, F -the attribute of X, and T- the Structure of X. When we view the same problem at a coarser grain size, we have a coarse-grained universe denoted by [X]. Then we have a new representation ([X],[F],[T]) of the problem. The coarse universe [X] is defined by an equivalence relation R on X. Then, representation ([X],[F],[T]) is called a Quotient space of(X,F,T), where [X] -the Quotient set of X, [F] -the Quotient attribute of F, and [T] -the Quotient Structure of T. Obviously, the set of representations of a problem at different granularities composes a complete semi-order lattice. But in many real applications, we must deal with non-partition models such as tolerance relations (or similarity relations, neighboring relations). In the talk, we will discuss the granular computing based on fuzzy and tolerance relations from the Quotient space theory point of view. We focus on the connections among different grain-size worlds, especially, the two basic properties between Quotient spaces, i.e., falsity and truth preserving properties. We show that these two basic properties still hold in fuzzy and tolerance worlds. Using these properties, the computational complexity can be reduced either in problem solving or machine learning. We will extend the Quotient space theory based multi-granular computing from crispy world to fuzzy and tolerance worlds. The optimal path finding in complex networks is given as an example to show the application of the theoretical results.
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KSEM - Quotient space based multi-granular analysis
Knowledge Science Engineering and Management, 2007Co-Authors: Ling Zhang, Bo ZhangAbstract:We presented a Quotient space model that can represent a problem at different granularities; each model has three components: the universe X, property f and Structure T. So a multi-granular analysis can be implemented based on the model. The basic properties among different Quotient spaces such as the falsity preserving, the truth preserving properties are discussed. There are three Quotient-space model construction approaches, i.e., the construction based on universe, based on property and based on Structure. Four examples are given to show how a Quotient space model can be constructed from a real problem and how benefit we can get from the multi-granular analysis. First, by adding statistical inference method to heuristic search, a statistical heuristic search approach is presented. Due to the hierarchical and multi-granular problem solving strategy, the computational complexity of the new search algorithm is reduced greatly. Second, in the collision-free paths planning in robotics, the topological model is constructed from geometrical one. By using the truth preserving property between these two models, the paths planning can be implemented in the coarser and simpler topological space so that the computational cost is saved. Third, we discuss the Quotient space approximation and the multi-resolution signal analysis. And the second-generation wavelet analysis can be obtained from Quotientspace based function approximation. It shows the equivalence relation between the Quotient space model based analysis and wavelet transform. Fourth, in the automatic assembly sequence planning of mechanical product, we mainly show how a Quotient Structure can be constructed from the original one. By using the simpler Quotient Structure, the assembly sequence planning can be simplified greatly. In conclusion, the Quotientspace model enables us to implement a multi-granular analysis. And we can get great benefit from the analysis.
Bo Zhang - One of the best experts on this subject based on the ideXlab platform.
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GrC - An Update and Summary of Multi-granular Computing and Quotient Structure
2010 IEEE International Conference on Granular Computing, 2010Co-Authors: Ling Zhang, Bo ZhangAbstract:Human always conceptualizes the world at different granularities and deals with it hierarchically. This is one of human’s characteristics that underlie his/her power. The Quotient space model that we presented is a mathematical model of human multi-granular problem solving. In the paper, we introduce the model and discuss how to use the model to deal with the granularity relation, hierarchical problem solving. We also present the future research directions.
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An Update and Summary of Multi-granular Computing and Quotient Structure
2010 IEEE International Conference on Granular Computing, 2010Co-Authors: Ling Zhang, Bo ZhangAbstract:Human always conceptualizes the world at different granularities and deals with it hierarchically. This is one of human's characteristics that underlie his/her power. The Quotient space model that we presented is a mathematical model of human multi-granular problem solving. In the paper, we introduce the model and discuss how to use the model to deal with the granularity relation, hierarchical problem solving. We also present the future research directions.
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Multi-granular representation-the key to machine intelligence
2008 3rd International Conference on Intelligent System and Knowledge Engineering, 2008Co-Authors: Bo Zhang, Ling ZhangAbstract:One of the basic characteristics in human problem solving is the ability to conceptualize the world at different granularities and translate from one abstraction level to the others easily. But so far computers can only deal with one abstraction level in problem solving generally. It seems important to develop new techniques which will in some way enable the computers to represent the world at different granularities. So the multi-granular representation is the key to machine intelligence. In the talk, we first introduce the Quotient space based problem solving theory. In the theory, a problem is represented by a triplet (X,F,T), where X - the universe with the finest grain-size, F -the attribute of X, and T- the Structure of X. When we view the same problem at a coarser grain size, we have a coarse-grained universe denoted by [X]. Then we have a new representation ([X],[F],[T]) of the problem. The coarse universe [X] is defined by an equivalence relation R on X. Then, representation ([X],[F],[T]) is called a Quotient space of(X,F,T), where [X] -the Quotient set of X, [F] -the Quotient attribute of F, and [T] -the Quotient Structure of T. Obviously, the set of representations of a problem at different granularities composes a complete semi-order lattice. That is, in the theory the concept, Quotient space, in algebra is used as a mathematical model to represent the relationship between representations with different grain-sizes. Multi-granular representation methodology can be used both in problem solving and machine learning. In multi-granular problem solving, a problem is solved from the coarse grain-size to the fine one hierarchically. The aim of hierarchical problem solving is intended to reduce the computational complexity. Multi-granular machine learning is intended to learn the knowledge from representations with different grain-size, i.e., the so-called multi-information fusion. Generally speaking, the fine representation has more details but less robustness. Conversely, the coarse representation has more robustness but less expressiveness. They are complement so multi-granular learning can benefit from them. We also present some examples in hierarchical problem solving and machine learning to show the advantages of using multi-granular representation.
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Granular computing based on fuzzy and tolerance relations
2008 IEEE International Conference on Granular Computing, 2008Co-Authors: Bo Zhang, Ling ZhangAbstract:Granular computing (GC) as a new field has grown rapidly since the term was proposed. There are two basic concepts underlying GC, i.e., fuzziness/fuzzification and granularity/granulation. Fuzzy set theory is an effective tool to deal with the fuzzification problem. There has been a large number of works on this aspect such as computing with words. The difficulty is in the concept of granulation. There was only an informal definition proposed by Zadeh for the concept. So far only a few works addressed the problem such as rough set and Quotient space theories. In these works, the granulation is mainly defined by equivalence relations, i.e., a partition model. For example, in Quotient space theory, a problem is represented by a triplet (X,F,T), where X -the universe with the finest grain-size, F -the attribute of X, and T- the Structure of X. When we view the same problem at a coarser grain size, we have a coarse-grained universe denoted by [X]. Then we have a new representation ([X],[F],[T]) of the problem. The coarse universe [X] is defined by an equivalence relation R on X. Then, representation ([X],[F],[T]) is called a Quotient space of(X,F,T), where [X] -the Quotient set of X, [F] -the Quotient attribute of F, and [T] -the Quotient Structure of T. Obviously, the set of representations of a problem at different granularities composes a complete semi-order lattice. But in many real applications, we must deal with non-partition models such as tolerance relations (or similarity relations, neighboring relations). In the talk, we will discuss the granular computing based on fuzzy and tolerance relations from the Quotient space theory point of view. We focus on the connections among different grain-size worlds, especially, the two basic properties between Quotient spaces, i.e., falsity and truth preserving properties. We show that these two basic properties still hold in fuzzy and tolerance worlds. Using these properties, the computational complexity can be reduced either in problem solving or machine learning. We will extend the Quotient space theory based multi-granular computing from crispy world to fuzzy and tolerance worlds. The optimal path finding in complex networks is given as an example to show the application of the theoretical results.
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KSEM - Quotient space based multi-granular analysis
Knowledge Science Engineering and Management, 2007Co-Authors: Ling Zhang, Bo ZhangAbstract:We presented a Quotient space model that can represent a problem at different granularities; each model has three components: the universe X, property f and Structure T. So a multi-granular analysis can be implemented based on the model. The basic properties among different Quotient spaces such as the falsity preserving, the truth preserving properties are discussed. There are three Quotient-space model construction approaches, i.e., the construction based on universe, based on property and based on Structure. Four examples are given to show how a Quotient space model can be constructed from a real problem and how benefit we can get from the multi-granular analysis. First, by adding statistical inference method to heuristic search, a statistical heuristic search approach is presented. Due to the hierarchical and multi-granular problem solving strategy, the computational complexity of the new search algorithm is reduced greatly. Second, in the collision-free paths planning in robotics, the topological model is constructed from geometrical one. By using the truth preserving property between these two models, the paths planning can be implemented in the coarser and simpler topological space so that the computational cost is saved. Third, we discuss the Quotient space approximation and the multi-resolution signal analysis. And the second-generation wavelet analysis can be obtained from Quotientspace based function approximation. It shows the equivalence relation between the Quotient space model based analysis and wavelet transform. Fourth, in the automatic assembly sequence planning of mechanical product, we mainly show how a Quotient Structure can be constructed from the original one. By using the simpler Quotient Structure, the assembly sequence planning can be simplified greatly. In conclusion, the Quotientspace model enables us to implement a multi-granular analysis. And we can get great benefit from the analysis.
Yue Yang - One of the best experts on this subject based on the ideXlab platform.
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TAMC - On the Quotient Structure of computably enumerable degrees modulo the noncuppable ideal
Lecture Notes in Computer Science, 2006Co-Authors: Angsheng Li, Guohua Wu, Yue YangAbstract:We show that minimal pairs exist in the Quotient Structure of $\mathcal{R}$ modulo the ideal of noncuppable degrees.
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on the Quotient Structure of computably enumerable degrees modulo the noncuppable ideal
Lecture Notes in Computer Science, 2006Co-Authors: Angsheng Li, Guohua Wu, Yue YangAbstract:We show that minimal pairs exist in the Quotient Structure of R modulo the ideal of noncuppable degrees.
Guohua Wu - One of the best experts on this subject based on the ideXlab platform.
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a minimal pair in the Quotient Structure m ncup
Conference on Computability in Europe, 2007Co-Authors: Guohua WuAbstract:In this paper, we prove the existence of a minimal pair of c.e.degrees a and b such that both of them are cuppable, and noincomplete c.e. degree can cup both of them to 0'. As aconsequence, [a] and [b] form a minimal pair inM/NCup, the Quotient Structure of the cappabledegrees modulo noncuppable degrees. We also prove that the dual ofLempp's conjecture is true.
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CiE - A Minimal Pair in the Quotient Structure M/NCup
Lecture Notes in Computer Science, 2007Co-Authors: Guohua WuAbstract:In this paper, we prove the existence of a minimal pair of c.e.degrees a and b such that both of them are cuppable, and noincomplete c.e. degree can cup both of them to 0'. As aconsequence, [a] and [b] form a minimal pair inM/NCup, the Quotient Structure of the cappabledegrees modulo noncuppable degrees. We also prove that the dual ofLempp's conjecture is true.
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TAMC - On the Quotient Structure of computably enumerable degrees modulo the noncuppable ideal
Lecture Notes in Computer Science, 2006Co-Authors: Angsheng Li, Guohua Wu, Yue YangAbstract:We show that minimal pairs exist in the Quotient Structure of $\mathcal{R}$ modulo the ideal of noncuppable degrees.
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on the Quotient Structure of computably enumerable degrees modulo the noncuppable ideal
Lecture Notes in Computer Science, 2006Co-Authors: Angsheng Li, Guohua Wu, Yue YangAbstract:We show that minimal pairs exist in the Quotient Structure of R modulo the ideal of noncuppable degrees.
J M E Hyland - One of the best experts on this subject based on the ideXlab platform.
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modified realizability toposes and strong normalization proofs
International Conference on Typed Lambda Calculi and Applications, 1993Co-Authors: J M E HylandAbstract:This paper is motivated by the discovery that an appropriate Quotient SN of the strongly normalising untyped λ*-terms (where * is just a formal constant) forms a partial applicative Structure with the inherent application operation. The Quotient Structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative Structures conditionally partial combinalory algebras (c-pca). Remarkably, an arbitrary rightabsorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizabilily, as opposed to the standard Kleene-style realizability. Starting from an arbitrary right-absorptive C-PCA U, the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm(U) of non-standard sets equipped with an equality predicate. Church's Thesis is internally valid in TOP m (K1) (where the pca k1 is “Kleene's first model” of natural numbers) but not Markov's Principle. There is a topos inclusion of SET-the “classical” topos of sets-into TOPm(U); the image of the inclusion is just sheaves for the ⌝⌝-topology. Separated objects of the ⌝⌝-topology are characterized. We identify the appropriate notion of PER's (partial equivalence relations) in the modified realizability setting and state its completeness properties. The topos TOP m (U) has enough completeness property to provide a category-theoretic semantics for a family of higher type theories which include Girard's System F and the Calculus of Constructions due to Coquand and Huet. As an important application, by interpreting type theories in the topos TOP m (SN.), a clean semantic explanation of the Tait-Girard style strong normalization argument is obtained. We illustrate how a strong normalization proof for an impredicative and dependent type theory may be assembled from two general “stripping arguments” in the framework of the topos TOP m (SN.). This opens up the possibility of a “generic” strong normalization argument for an interesting class of type theories.
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TLCA - Modified Realizability Toposes and Strong Normalization Proofs
Lecture Notes in Computer Science, 1993Co-Authors: J M E HylandAbstract:This paper is motivated by the discovery that an appropriate Quotient SN of the strongly normalising untyped λ*-terms (where * is just a formal constant) forms a partial applicative Structure with the inherent application operation. The Quotient Structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative Structures conditionally partial combinalory algebras (c-pca). Remarkably, an arbitrary rightabsorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizabilily, as opposed to the standard Kleene-style realizability. Starting from an arbitrary right-absorptive C-PCA U, the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm(U) of non-standard sets equipped with an equality predicate. Church's Thesis is internally valid in TOP m (K1) (where the pca k1 is “Kleene's first model” of natural numbers) but not Markov's Principle. There is a topos inclusion of SET-the “classical” topos of sets-into TOPm(U); the image of the inclusion is just sheaves for the ⌝⌝-topology. Separated objects of the ⌝⌝-topology are characterized. We identify the appropriate notion of PER's (partial equivalence relations) in the modified realizability setting and state its completeness properties. The topos TOP m (U) has enough completeness property to provide a category-theoretic semantics for a family of higher type theories which include Girard's System F and the Calculus of Constructions due to Coquand and Huet. As an important application, by interpreting type theories in the topos TOP m (SN.), a clean semantic explanation of the Tait-Girard style strong normalization argument is obtained. We illustrate how a strong normalization proof for an impredicative and dependent type theory may be assembled from two general “stripping arguments” in the framework of the topos TOP m (SN.). This opens up the possibility of a “generic” strong normalization argument for an interesting class of type theories.
