## Granular Computing: basic issues and possible solutions (2000)

Venue: | Proceedings of the 5th Joint Conference on Information Sciences |

Citations: | 29 - 17 self |

### BibTeX

@INPROCEEDINGS{Yao00granularcomputing:,

author = {Y. Y. Yao},

title = {Granular Computing: basic issues and possible solutions},

booktitle = {Proceedings of the 5th Joint Conference on Information Sciences},

year = {2000},

pages = {186--189}

}

### Years of Citing Articles

### OpenURL

### Abstract

Granular computing (GrC) may be regarded as a label of theories, methodologies, techniques, and tools that make use of granules, i.e., groups, classes, or clusters of a universe, in the process of problem solving. The main objective of this paper is to discuss basic issues of GrC, with emphasis on the construction of granules and computation with granules. After a brief review of existing studies, a set-theoretic model of GrC is proposed based on the notion of power algebras.

### Citations

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Citation Context ...{x | a ≤ x ≤ a}. (6) Degenerate intervals of the form [a, a] are equivalent to real numbers. One can perform arithmetic operations on interval numbers by lifting arithmetic operations on real numbers =-=[4]-=-. Let A = [a, a] and B = [b, b] be two interval numbers, we have: A + B = {x + y | x ∈ A, y ∈ B} = [a + b, a + b],sA − B = {x − y | x ∈ A, y ∈ B} = [a − b, a − b], A · B = {x · y | x ∈ A, y ∈ B} = [mi... |

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Citation Context ..., such as interval fuzzy reasoning [11], interval probabilistic reasoning [6, 9], and reasoning with granular probabilities [2]. It can be easily extended to study fuzzy arithmetic with fuzzy numbers =-=[3]-=-. 3.3 Interval set algebra Given two subsets A 1 , A 2 # 2 U with A 1 # A 2 , the subset of 2 U , A = [A 1 , A 2 ] = {X # 2 U | A 1 # X # A 2 }, (8) is called a closed interval set [8]. The set A 1 is... |

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Citation Context ...ept of generalized constraints. Relationships between granules are represented in terms of fuzzy graphs or fuzzy if-then rules. The associated computation method is known as computing with words (CW) =-=[7, 15]-=-. Let X be a variable taking values in a universe U . A generalized constraint on the values of X can be expressed as X isr R, where R is a constraining relation, isr is a variable copula and r is a d... |

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Citation Context ...lated fields, such as interval analysis, quantization, rough set theory, Dempster-Shafer theory of belief functions, divide and conquer, cluster analysis, machine learning, databases, and many others =-=[16]-=-. The topic of fuzzy information granulation was first proposed and discussed by Zadeh [14] in 1979. There is a fast growing and renewed interest in this topic [13]. Granular computing is likely to pl... |

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Citation Context ... intervals of real numbers [11]. Interval number algebra may serve as a basis for interval reasoning with numeric truth values, such as interval fuzzy reasoning [11], interval probabilistic reasoning =-=[6, 9]-=-, and reasoning with granular probabilities [2]. It can be easily extended to study fuzzy arithmetic with fuzzy numbers [3]. 3.3 Interval set algebra Given two subsets A 1 , A 2 # 2 U with A 1 # A 2 ,... |

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Citation Context ...problem involves incomplete, uncertain, or vague information, it may be di#cult to di#erentiate distinct elements and one is forced to consider granules. A typical example is the theory of rough sets =-=[5]-=-. The lack of information may only allow us to define granules rather than individuals. In some situations, although detailed information may be available, it may be su#cient to use granules in order ... |

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Citation Context ...eory of belief functions, divide and conquer, cluster analysis, machine learning, databases, and many others [16]. The topic of fuzzy information granulation was first proposed and discussed by Zadeh =-=[14]-=- in 1979. There is a fast growing and renewed interest in this topic [13]. Granular computing is likely to play an important role in the evolution of fuzzy logic and its applications. 1.1 What is GrC?... |

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Citation Context ...h fuzzy numbers [3]. 3.3 Interval set algebra Given two subsets A 1 , A 2 # 2 U with A 1 # A 2 , the subset of 2 U , A = [A 1 , A 2 ] = {X # 2 U | A 1 # X # A 2 }, (8) is called a closed interval set =-=[8]-=-. The set A 1 is called the lower bound of the interval set and A 2 the upper bound. Degenerate interval sets of the form [A, A] are equivalent to ordinary sets. By lifting standard set-theoretic oper... |

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Citation Context ...s on fuzzy granules can therefore be defined by operations on #-cuts. 3.1 Power algebras Let # be a binary operation on a universe U . One can define a binary operation # + on subsets of U as follows =-=[1]-=-: X # + Y = {x # y | x # X, y # Y }, (4) for any X,Y # U . In general, one may lift any operationsf on elements of U to an operation f + on subsets of U , called the power operation of f . Supposesf :... |

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Citation Context ...nd causation. "Granulation involves decomposition of whole into parts, organization involves integration of parts into whole, and causation involves association of causes and e#ects." Yager =-=and Filev [7] pointed out that &q-=-uot;human beings have been developed a granular view of the world", and ". . . objects with which mankind perceives, measures, conceptualizes and reasons are granular". From a more prac... |

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Citation Context ...ment, one can lift operations in a Boolean algebra or a lattice [11]. Such interval algebras may be used for reasoning with interval extension of classical logic [10], and interval incidence calculus =-=[12]-=-. 4 Conclusion Granular computing may have a great impact on the design and implementation of intelligent information systems, and on real world problem solving. The results from existing studies show... |

8 |
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Citation Context ...earning, databases, and many others [16]. The topic of fuzzy information granulation was first proposed and discussed by Zadeh [14] in 1979. There is a fast growing and renewed interest in this topic =-=[13]-=-. Granular computing is likely to play an important role in the evolution of fuzzy logic and its applications. 1.1 What is GrC? The following quotations from Zadeh may help us in understanding the sco... |

6 |
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Citation Context ...oolean algebra. By using the same argument, one can lift operations in a Boolean algebra or a lattice [11]. Such interval algebras may be used for reasoning with interval extension of classical logic =-=[10]-=-, and interval incidence calculus [12]. 4 Conclusion Granular computing may have a great impact on the design and implementation of intelligent information systems, and on real world problem solving. ... |

4 |
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Citation Context ...lgebra may serve as a basis for interval reasoning with numeric truth values, such as interval fuzzy reasoning [11], interval probabilistic reasoning [6, 9], and reasoning with granular probabilities =-=[2]-=-. It can be easily extended to study fuzzy arithmetic with fuzzy numbers [3]. 3.3 Interval set algebra Given two subsets A 1 , A 2 # 2 U with A 1 # A 2 , the subset of 2 U , A = [A 1 , A 2 ] = {X # 2 ... |

4 |
Interval Analysis. Englewood Cli s
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(Show Context)
Citation Context ...{x | a # x # a}. (6) Degenerate intervals of the form [a, a] are equivalent to real numbers. One can perform arithmetic operations on interval numbers by lifting arithmetic operations on real numbers =-=[4]-=-. Let A = [a, a] and B = [b, b] be two interval numbers, we have: A +B = {x + y | x # A, y # B} = [a + b, a + b], A -B = {x - y | x # A, y # B} = [a - b, a - b], A B = {xsy | x # A, y # B} = [min(a b,... |

4 | A comparison of two interval-valued probabilistic reasoning methods
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(Show Context)
Citation Context ... intervals of real numbers [11]. Interval number algebra may serve as a basis for interval reasoning with numeric truth values, such as interval fuzzy reasoning [11], interval probabilistic reasoning =-=[6, 9]-=-, and reasoning with granular probabilities [2]. It can be easily extended to study fuzzy arithmetic with fuzzy numbers [3]. 3.3 Interval set algebra Given two subsets A 1 , A 2 # 2 U with A 1 # A 2 ,... |

2 | Interval based uncertain reasoning using fuzzy and rough sets
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(Show Context)
Citation Context ...ber operations are again closed and bounded intervals. When 0 # B, A/B is undefined. One may lift any operations on real numbers, such as min and max, to power operations on intervals of real numbers =-=[11]-=-. Interval number algebra may serve as a basis for interval reasoning with numeric truth values, such as interval fuzzy reasoning [11], interval probabilistic reasoning [6, 9], and reasoning with gran... |

2 | Announcement of GrC - Zadeh - 1997 |