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Computational Complexity pp 1929–1938Cite as

Multi-Granular Computing and Quotient Structure

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ArticleOutline

Glossary

Definition of the Subject

Introduction

The Granularity Relation

Hierarchy

Combination

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Granularity:

Granularity is the relative size, scale, level of detail, or depth of penetration that characterizes an object or system.

Multi‐granular computing:

Humans are good at viewing and solving a problem at different grain‐sizes (abstraction levels) and translating from one abstraction level to the others easily. This is one of basic characteristics of human intelligence. The aim of multi‐granular computing is intended to investigate the granulation problem in human cognition and endow computers with the same capability to make them more efficient in problem solving.

Quotient set:

Given a universe X and an equivalence relation R on X, define a set \( [X]=\{[x]|x\in X\},[x]=\{y|y\sim x,y\in X\} \). [X] is called a quotient set with respect to R, or simply a quotient set.

Quotient space:

Given a topologic space \( { (X,T) } \), T is a topology on X and R is an equivalence relation on X. Define a quotient structure on [X] as \( [T]=\{u|p^{-1}(u)\in T,u\subset [X]\} \), where \( { p\colon X\to [X] } \) is a natural projection from X to [X]. Construct a topologic space \( ([X], [T]) \). Space \( ([X], [T]) \) is a quotient space corresponding to R. There are authors who keep the neighborhood system structured but remove the axioms of topology [13,14].

Quotient space model:

The quotient space model is a mathematical model to represent a problem at different grain‐sizes by using the concept of quotient space in algebra. In the model a problem (or a system) is described by a triple \( { (X,T,f) } \), with universe (domain) X, structure T and attribute f. If X represents the universe composed of the objects with the finest grain‐size, when we view the same universe X at a coarser grain‐size, we have a coarse‐grained universe denoted by [X]. Then we have a new problem space \( { ([X],[T],[f]) } \), where [X] is the quotient universe of X, [T] the corresponding quotient structure and [f] the quotient attribute. The coarse space \( { ([X],[T],[f]) } \) is called a quotient space of space \( { (X,T,f) } \). Therefore, a problem with different grain‐sizes can be represented by a family of quotient spaces.

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Acknowledgments

The work issupported by the National Natural Science Foundation of Chinaunder Grant. No. 60621062, and the National Key FoundationR&D Project under Grant. No.2003CB317007,2007CB311003.

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Authors and Affiliations

  1. Artificial Intelligence Institute, Anhui University, Hefei, Anhui, China

    Ling Zhang

  2. Department of Computer Science, State Key Lab of Intelligent Technology and Systems, Tsinghua University, Beijing, China

    Bo Zhang

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  1. Ling Zhang
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2012 Springer-Verlag

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Zhang, L., Zhang, B. (2012). Multi-Granular Computing and Quotient Structure. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_125

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