粗糙集的数学基础研究与两个广义粗糙集模型的探讨
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摘要
本文主要进行了粗糙集的数学基础研究，与程度粗糙集和变精度粗糙集两个广义粗糙集模型的探讨。
第一章，主要研究了粗糙集与拓扑的关系。首先修正粗糙集近似算子，提出近似集的新型并、交、补运算和近似幂集空间，从算子论和集合论两者的角度丰富了粗糙集理论；再结合拓扑，定义和研究了粗糙拓扑、近似拓扑。用知识分化论域的观点，提出和研究了知识论域、知识拓扑。在论域、知识分类、拓扑三者组成的系统中，提出了集合邻域、近似邻域等更深刻的概念，研究了粗糙集与拓扑的复合关系，得出了两者间的深刻联系。最后，用覆盖空间统一了粗糙集与拓扑。
第二章，主要研究了粗糙集与模糊集、测度、积分、格、群的关系和结合。第1节，研究了粗糙集与模糊集的关系，定义了关联函数，统一了两者。第2节，以知识分类为基础，构造开区间，定义和研究了一种具体的勒贝格测度——知识测度。第3节，在粗糙集理论的知识库中，加强知识为分划，定义和研究了知识积分和与知识积分，证明了知识熵是一种知识积分和的本质，并以知识积分和与知识积分为手段研究了知识熵。第4节，在知识、等价关系和分类三个层面上定义了知识集、知识等价关系集、知识分类集，研究了其上的并交格和精细格的性质和相互关系，得出了粗糙集与格的复合关系。第5节，重新定义了粗糙群理论的一系列概念，并在传统的和新定义的两种粗糙群理论体系中，研究了基于子群的群的粗糙的性质。
第三章，研究了程度粗糙集和变精度粗糙集两种广义粗糙集模型的性质、
关系和统一。在近似空间中定义了更一般的程度近似和变精度近似，研究了两
者各自的性质和相互关系，得到了程度近似与精度近似相互转化的重要公式。
定义了程度与精度并交补的近似和积的近似，研究了其结构与幂性等基本性质。
用程度与精度并交的近似和积的近似统一了程度近似和变精度近似，从而用程
度与精度并交的近似和积的近似的性质，推导了程度近似与变精度近似更深刻
的性质。
第四章，基于粗糙集隶属度，提出了一个规则提取算法，并用实例演示了
该算法可以得出满意的结果，展现了粗糙集理论的一个实际应用。
最后，对粗糙集理论的前景进行了展望，给出了结论，提出了一些值得思
考的问题。
In this paper, the mathematical foundation study of rough sets and the study of two general rough sets models are mainly researched.
In chapter 1, the connection of rough sets and topology is researched. At first the approximate operators of rough sets are modified, so the new operators of union, intersection and complement and approximate power sets are presented. This has enriched rough sets theory from the operator-oriented and set-oriented views. Then by combining topology, rough topology and approximate topologies are defined and researched. Knowledge universe and knowledge topology are presented and researched by parting universe of knowledge. In the system of universe, knowledge and topology, set neighborhood and approximate neighborhood, the deep notions, are presented. The combination of rough sets and topology is researched, and the deep connection of them is achieved. At last, rough sets and topology are unified by covering space.
In chapter 2, the connection and combination of rough sets and other mathematical subjects, such as fuzzy sets, measure, integral, lattice, group, are researched. In section 1, the connection of rough sets and fuzzy sets is researched, and the relationship function is defined to unify them. In section 2, based on knowledge class open intervals are constructed, then a kind of specific Lebesgue measures, called knowledge measure, is defined and researched. In section 3, in
knowledge base of rough sets, knowledge is reinforced to class of measure, and knowledge integral sum and knowledge integral are defined and researched. That the entropy of knowledge is a kind of knowledge integral sums in nature is proved, and by knowledge integral sum and knowledge integral the entropy of knowledge is researched. In section 4, knowledge sets, and knowledge equivalent relation sets and knowledge class sets are defined respectively on knowledge, equivalent relation and class parts. The connection of union-intersection lattice and fine lattice of them is researched, and the combination of rough sets and lattice is achieved. In section 5, a system of notions are redefined in rough groups theory, and in the traditional and the redefined rough group theory systems, the properties of roughness of group based on subgroup are researched.
In chapter 3, the relationship, combination and unification of graded rough sets and variable rough sets are researched. The more general graded approximations and variable precision approximations are defined in approximate space. The relationship between them is researched, and the important formula of conversion is achieved. The union, intersection, complement and product approximations of grade and precision are defined. Their constructions and some basic properties, such as the power actions of approximation operators, are researched. Graded rough set model and variable precision rough set model are unified by the combination models, so their old and new properties are achieved from the new model, the combination and unification model.
In chapter 4, based on rough membership value a vital algorithm of getting rules is presented, and the practical example shows that this algorithm has a content result. This has showed an aspect of the practice of the rough sets theory.
At last, the future of rough sets theory is prospected. The conclusion is reached and some question with thinking value is presented.
第一章，主要研究了粗糙集与拓扑的关系。首先修正粗糙集近似算子，提出近似集的新型并、交、补运算和近似幂集空间，从算子论和集合论两者的角度丰富了粗糙集理论；再结合拓扑，定义和研究了粗糙拓扑、近似拓扑。用知识分化论域的观点，提出和研究了知识论域、知识拓扑。在论域、知识分类、拓扑三者组成的系统中，提出了集合邻域、近似邻域等更深刻的概念，研究了粗糙集与拓扑的复合关系，得出了两者间的深刻联系。最后，用覆盖空间统一了粗糙集与拓扑。
第二章，主要研究了粗糙集与模糊集、测度、积分、格、群的关系和结合。第1节，研究了粗糙集与模糊集的关系，定义了关联函数，统一了两者。第2节，以知识分类为基础，构造开区间，定义和研究了一种具体的勒贝格测度——知识测度。第3节，在粗糙集理论的知识库中，加强知识为分划，定义和研究了知识积分和与知识积分，证明了知识熵是一种知识积分和的本质，并以知识积分和与知识积分为手段研究了知识熵。第4节，在知识、等价关系和分类三个层面上定义了知识集、知识等价关系集、知识分类集，研究了其上的并交格和精细格的性质和相互关系，得出了粗糙集与格的复合关系。第5节，重新定义了粗糙群理论的一系列概念，并在传统的和新定义的两种粗糙群理论体系中，研究了基于子群的群的粗糙的性质。
第三章，研究了程度粗糙集和变精度粗糙集两种广义粗糙集模型的性质、
关系和统一。在近似空间中定义了更一般的程度近似和变精度近似，研究了两
者各自的性质和相互关系，得到了程度近似与精度近似相互转化的重要公式。
定义了程度与精度并交补的近似和积的近似，研究了其结构与幂性等基本性质。
用程度与精度并交的近似和积的近似统一了程度近似和变精度近似，从而用程
度与精度并交的近似和积的近似的性质，推导了程度近似与变精度近似更深刻
的性质。
第四章，基于粗糙集隶属度，提出了一个规则提取算法，并用实例演示了
该算法可以得出满意的结果，展现了粗糙集理论的一个实际应用。
最后，对粗糙集理论的前景进行了展望，给出了结论，提出了一些值得思
考的问题。
In this paper, the mathematical foundation study of rough sets and the study of two general rough sets models are mainly researched.
In chapter 1, the connection of rough sets and topology is researched. At first the approximate operators of rough sets are modified, so the new operators of union, intersection and complement and approximate power sets are presented. This has enriched rough sets theory from the operator-oriented and set-oriented views. Then by combining topology, rough topology and approximate topologies are defined and researched. Knowledge universe and knowledge topology are presented and researched by parting universe of knowledge. In the system of universe, knowledge and topology, set neighborhood and approximate neighborhood, the deep notions, are presented. The combination of rough sets and topology is researched, and the deep connection of them is achieved. At last, rough sets and topology are unified by covering space.
In chapter 2, the connection and combination of rough sets and other mathematical subjects, such as fuzzy sets, measure, integral, lattice, group, are researched. In section 1, the connection of rough sets and fuzzy sets is researched, and the relationship function is defined to unify them. In section 2, based on knowledge class open intervals are constructed, then a kind of specific Lebesgue measures, called knowledge measure, is defined and researched. In section 3, in
knowledge base of rough sets, knowledge is reinforced to class of measure, and knowledge integral sum and knowledge integral are defined and researched. That the entropy of knowledge is a kind of knowledge integral sums in nature is proved, and by knowledge integral sum and knowledge integral the entropy of knowledge is researched. In section 4, knowledge sets, and knowledge equivalent relation sets and knowledge class sets are defined respectively on knowledge, equivalent relation and class parts. The connection of union-intersection lattice and fine lattice of them is researched, and the combination of rough sets and lattice is achieved. In section 5, a system of notions are redefined in rough groups theory, and in the traditional and the redefined rough group theory systems, the properties of roughness of group based on subgroup are researched.
In chapter 3, the relationship, combination and unification of graded rough sets and variable rough sets are researched. The more general graded approximations and variable precision approximations are defined in approximate space. The relationship between them is researched, and the important formula of conversion is achieved. The union, intersection, complement and product approximations of grade and precision are defined. Their constructions and some basic properties, such as the power actions of approximation operators, are researched. Graded rough set model and variable precision rough set model are unified by the combination models, so their old and new properties are achieved from the new model, the combination and unification model.
In chapter 4, based on rough membership value a vital algorithm of getting rules is presented, and the practical example shows that this algorithm has a content result. This has showed an aspect of the practice of the rough sets theory.
At last, the future of rough sets theory is prospected. The conclusion is reached and some question with thinking value is presented.
引文
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[2]Y.Y. Yao. Two Views of the Theory of Rough Sets in Finite Universes. International of Approximate Reasoning, 1996, 15: 291-317
[3]Y.Y. Yao. A Comparative Study of Fuzzy Sets and Rough Sets. Information Sciences, 1998, 109:227-242
[4]Y.Y. Yao. Constructive and Algebraic Methods of the Theory of Rough Sets. Information Sciences, 1998, 109:21-47
[5]S. Comer. An Algebraic Approach to the Approximation of Information. Fundamenta Informaticae, 1991, 14:492-502
[6]T.Y. Lin, Q. Lin. Rough Approximate Operators: Axiomatic Rough Set Theory. In: W. P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery. Berlin: Springer, 1994: 256-260
[7]U. Wybraniec-Skardowska. On a Generalization of Approximation Space. Bulletin of the Polish Academy of Sciences, Mathematics, 1989, 37:51-61
[8]Y. Y. Yao, T. Y. Lin. Generalization of Rough Sets Using Modal Logic. Intelligent Automation and Soft Computing, an International Journal, 1996, 2:103-120
[9]M. Gehrke, E.Walker. On the Structure of Rough Sets. Bulletin of the Polish Academy of Sciences Mathematics, 1994, 40:235-245
[10]A. Wasilewska. Topological Rough Algebras. In: T. Y. Lin, N. Cercone (Ed.), Rough Sets and Data Mining: Analysis for Imprecise Data. Dordrecht: Kluwer Academic Publishers, 1997:411-425
[11]Z. Pawlak. Rough Sets and Fuzzy Sets. Fuzzy Sets and Systems, 1985, 17:99-102
[12]Duntsch I. Rough Relation Algebras. Fundamenta Informaticae, 1994, 21:321-331
[13]Z. Pawlak. Rough Logic. Bull. Polish Acad. Aci. Tech. 35/5-6, 1987:253-258
[14]E. Orlowska. A Logic of Indiscernibility Relation. In: A. Skowron (Ed.), Computation
Theory, Lecture Notes in Computer Science 208, 1985:177-186
[15]Q. Lin. The OI-Resolution of Operator Rough Logic. LNAI 1424, Springer 1998, 6: 432-435
[16]Z. Pawlak. Rough Sets, Rough Relations and Rough Function. Fundamenta Informaticae, 1996, 27 (8): 2-3
[17]Z. Pawlak. Rough Functions. Bull. Polish Acad. Sci. Tech. 35/5-6, 1987:249-251
[18]刘真．实函数Rough离散化．南京大学学报，2000，36(Computer Issue)：188-191
[19]Z. Pawlak and A. Skowron. Rough Membership Functions: A Tool for Reasoning with Uncertainty. In: C. Rauszer (Ed.), Algebra Methods in Logic and Computer Science. Warsaw: Banach Center Publications 28, Polish/Academy of Science, 1993:135-150
[20]刘旺金，温华永．拓扑学基础．武汉：武汉大学出版社，1992
[21]Benjamin T Sims. Fundamentals of Topology. New York: Macmillan Publishing Co., Inc, 1978
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