粗糙集的拓扑结构研究
摘要
粗糙集理论是一种新的处理不确定性知识的数学工具。它能有效地分析不完备、不相容的信息并发现其中所隐含的知识,从而揭示出事物中潜在的规律。经过三十余年的发展,已经在理论及应用方面获得了大量重要成果。拓扑学是基础数学中的一个重要分支,在数学的诸多领域中具有广泛应用。本文研究粗糙集的拓扑结构、代数结构等相关问题,主要有如下四个方面的研究成果：
一、对于近似空间中下近似集构成的拓扑空间,讨论了基于一般二元关系的粗糙集模型以及基于自反、对称二元关系的粗糙集模型中近似算子的拓扑性质,通过相应的下近似集构造了广义粗糙拓扑空间；对于一般论域,证明了U上的所有拓扑构成的集合与u上基于自反、传递关系的下近似集构成的拓扑集合之间存在一一对应关系。提出了广义近似空间连续映射的概念,得到了连续映射的一些基本性质。
二、研究了二元组形式的粗糙集构成的粗糙拓扑空间M.基于Pawlak近似算子,给出了M中的内部算子与闭包算子的解析表达式,并给出了它的拓扑基；基于自反、传递关系下的粗糙近似算子,讨论了当论域为无限集时粗糙拓扑空间M的结构与性质,刻画了它的内部算子及闭包算子；基于粗糙模糊近似算子,构造了粗糙模糊集构成的粗糙模糊拓扑空间,讨论了它的结构与性质。
三、研究了粗糙集代数的非经典逻辑代数结构。给出了剩余格中乘积运算存在伴随对的充分必要条件并刻画了伴随蕴涵的结构；在所有粗糙集构成的集合上定义了并、交、补、蕴涵、乘积运算,证明了相应运算下的粗糙集代数构成一个格,分别构造了粗糙集构成BL代数及Mv代数。讨论了粗糙集代数中滤子的性质,刻画了滤子的结构。
四、给出了集值信息系统基于对称限制相容关系的属性约简的判定定理,通过区分矩阵与区分函数给出了约简计算方法。讨论了集值决策表基于对称限制相容关系的约简理论与方法,给出了集值决策表的分配约简和正域约简的判定定理,借助区分矩阵与区分函数给出了约简计算方法。作为集值决策表约简理论的应用,提出了一种道路交通事故成因分析方法。
Rough sets theory is a new mathematical tool for dealing with uncertain knowledge. It can be used effectively to analyze incomplete and inconsistent information and to discover the knowledge hidden in information systems. There are many important results in the theory and application of rough set theory after thirty years of development. Topology is an important branch of pure mathematics. It has been applied in many fields of mathematics. The thesis is devoted to the discussion of the topological structure, algebraic structures and the related problems of rough sets. The main achievements are as follows.
1. The topological space induced by lower approximations in an approximation space is investigated. The topological properties of approximation operators with respect to rough sets model based on general binary relation and reflexive and symmetric relation are discussed. The related rough topological spaces induced by lower approximations are constructed. For general universe, it is proved that there exists a one-to-one correspondence between the set of all topologies induced by lower approximations based on reflexive and transitive relation on U and the set of all topologies on U. The notion of continuous mapping of generalized approximation space is proposed and some basic properties of continuous mapping are obtained.
2. The rough topological space M induced by two-tuples rough sets is studied. The analytic expression of interior and closure operators of M with respect to Pawlak approximation operators and the topological basis of M are presented. Based on rough approximation operators induced by reflexive and transitive relation, when the universe is an infinite set the structure and properties of rough topological space M are analyzed. The interior and closure operators are constructed. The rough fuzzy topological space consisting of rough fuzzy sets is constructed based on rough fuzzy approximation operators.
3. The non-classic logical algebraic structure of rough set algebra is studied. In residuated lattice, the necessary and sufficient condition for the existing of adjoint pair with respect to a multiplication operation is given. The adjoint implication is constructed. The union, the intersection, the complement, the implication and the multiplication operations on the set of all rough sets are defined. It is proved that the rough sets algebra is a lattice with respect to these operations. The BL algebra and MV algebra of rough sets are constructed respectively. The filter of rough algebra is discussed and the structure of the filter is portrayed.
4. Based on symmetry restriction tolerance relation, the judgment theorem of attribute reduction of set-valued information system is given. The reduction method is proposed with the assistance of discernibility matrix and discernibility function. The reduction theory and methods of set-valued decision table based on symmetry restriction tolerance relation are discussed. The judgment theorems of distribution reduction and positive region reduction of set-valued decision table are presented. The reduction methods are proposed with the assistance of discernibility matrix and discernibility function. In the end, as an application of the reduction theory of set-valued decision table, we propose an approach to analyze the cause road traffic accidents.
一、对于近似空间中下近似集构成的拓扑空间,讨论了基于一般二元关系的粗糙集模型以及基于自反、对称二元关系的粗糙集模型中近似算子的拓扑性质,通过相应的下近似集构造了广义粗糙拓扑空间；对于一般论域,证明了U上的所有拓扑构成的集合与u上基于自反、传递关系的下近似集构成的拓扑集合之间存在一一对应关系。提出了广义近似空间连续映射的概念,得到了连续映射的一些基本性质。
二、研究了二元组形式的粗糙集构成的粗糙拓扑空间M.基于Pawlak近似算子,给出了M中的内部算子与闭包算子的解析表达式,并给出了它的拓扑基；基于自反、传递关系下的粗糙近似算子,讨论了当论域为无限集时粗糙拓扑空间M的结构与性质,刻画了它的内部算子及闭包算子；基于粗糙模糊近似算子,构造了粗糙模糊集构成的粗糙模糊拓扑空间,讨论了它的结构与性质。
三、研究了粗糙集代数的非经典逻辑代数结构。给出了剩余格中乘积运算存在伴随对的充分必要条件并刻画了伴随蕴涵的结构；在所有粗糙集构成的集合上定义了并、交、补、蕴涵、乘积运算,证明了相应运算下的粗糙集代数构成一个格,分别构造了粗糙集构成BL代数及Mv代数。讨论了粗糙集代数中滤子的性质,刻画了滤子的结构。
四、给出了集值信息系统基于对称限制相容关系的属性约简的判定定理,通过区分矩阵与区分函数给出了约简计算方法。讨论了集值决策表基于对称限制相容关系的约简理论与方法,给出了集值决策表的分配约简和正域约简的判定定理,借助区分矩阵与区分函数给出了约简计算方法。作为集值决策表约简理论的应用,提出了一种道路交通事故成因分析方法。
Rough sets theory is a new mathematical tool for dealing with uncertain knowledge. It can be used effectively to analyze incomplete and inconsistent information and to discover the knowledge hidden in information systems. There are many important results in the theory and application of rough set theory after thirty years of development. Topology is an important branch of pure mathematics. It has been applied in many fields of mathematics. The thesis is devoted to the discussion of the topological structure, algebraic structures and the related problems of rough sets. The main achievements are as follows.
1. The topological space induced by lower approximations in an approximation space is investigated. The topological properties of approximation operators with respect to rough sets model based on general binary relation and reflexive and symmetric relation are discussed. The related rough topological spaces induced by lower approximations are constructed. For general universe, it is proved that there exists a one-to-one correspondence between the set of all topologies induced by lower approximations based on reflexive and transitive relation on U and the set of all topologies on U. The notion of continuous mapping of generalized approximation space is proposed and some basic properties of continuous mapping are obtained.
2. The rough topological space M induced by two-tuples rough sets is studied. The analytic expression of interior and closure operators of M with respect to Pawlak approximation operators and the topological basis of M are presented. Based on rough approximation operators induced by reflexive and transitive relation, when the universe is an infinite set the structure and properties of rough topological space M are analyzed. The interior and closure operators are constructed. The rough fuzzy topological space consisting of rough fuzzy sets is constructed based on rough fuzzy approximation operators.
3. The non-classic logical algebraic structure of rough set algebra is studied. In residuated lattice, the necessary and sufficient condition for the existing of adjoint pair with respect to a multiplication operation is given. The adjoint implication is constructed. The union, the intersection, the complement, the implication and the multiplication operations on the set of all rough sets are defined. It is proved that the rough sets algebra is a lattice with respect to these operations. The BL algebra and MV algebra of rough sets are constructed respectively. The filter of rough algebra is discussed and the structure of the filter is portrayed.
4. Based on symmetry restriction tolerance relation, the judgment theorem of attribute reduction of set-valued information system is given. The reduction method is proposed with the assistance of discernibility matrix and discernibility function. The reduction theory and methods of set-valued decision table based on symmetry restriction tolerance relation are discussed. The judgment theorems of distribution reduction and positive region reduction of set-valued decision table are presented. The reduction methods are proposed with the assistance of discernibility matrix and discernibility function. In the end, as an application of the reduction theory of set-valued decision table, we propose an approach to analyze the cause road traffic accidents.
引文
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