广义粗集的拓扑结构、方法及应用研究

Rough set theory, proposed by Pawlak in 1982, is a powerful tool to describe the dependences among attributes,evaluate the significance of attributes, and derive decision rules. It has attracted the interesting of researchers and practitioners in various fields of science and technology. The concept of topological structures and their generalizations are one of the most powerful notion in system analysis. It is also one of the most important structure of rough sets. Studies of topological structure not only can provide more insight into and a full understanding of rough set theory but also can find new applications.. In this project, the generalized rough sets include rough sets induced by general binary relations, fuzzy relations, and coverings. The aims of the project is to study topological structures of generalized rough sets and their applications by means of matrix approaches. First, we study three types of topological structures induced by binary relations, fuzzy relations and coverings, respectively. In general, for a binary relation R, the upper approximation induced by R is a closure, however, the upper approximation may be not a topological closure. We can extened the closure to topological closure. We consider the relationship between the closure and topological closure. We also study similar problems in the context of fuzzy sets. As for covering rough sets, we know that not each pair of approximations can induce a topology, we investigate the relationship between coverings and binary relations and consider the condition which approximations can induce a topological structure. We also discuss the neighborhood systems of Frechet topological space induced by generalized rough sets.Second, many problems related to decision making are: given a decision table, we need to find all minimal decision algorithms associated with the table. we use topological approaches, with the help of matrices, to find the minimal decision algorithms. We will proposed the concept of invariable matrices to reflect the invariable information of the decision table and consider the reduction of decision table. Finally, we consider two new applications: There are many applications of fuzzy dynamic systems to information science, natural language processing and artificial intelligence. We use topological theory of fuzzy rough sets as a tool to study fuzzy dynamic systems; Fuzzy multiplecriteria decision making was introduced as a promising and important field of study in the early 1970s, as another application, we use fuzzy rough set theory as a tool to study fuzzy multiplecriteria decision making.

粗集是智能信息处理的重要工具，在知识发现等领域的作用日益显著。拓扑结构是粗集最重要、最基础的结构之一，是知识提取与知识表达的重要基础，本项目拟利用拓扑理论描述粗空间的结构。所指的广义粗集包括经典粗集的各种推广与发展，但主要指由一般二元关系诱导的粗集、模糊粗集和覆盖粗集。在理论方面，用布尔矩阵或模糊矩阵的方法，深入研究一般二元关系诱导的粗集的拓扑结构、模糊粗集确定的模糊拓扑结构及在一定条件限制下的覆盖粗集的拓扑结构；研究对应的闭包扩展成拓扑闭包的变化规律，考虑具有附带条件的覆盖粗集的拓扑结构与二元关系粗集的拓扑结构的相互转化问题；通过对广义粗集拓扑结构的研究，以拓扑为方法，以矩阵为工具，考虑复杂信息系统（或决策表）的知识约简问题，给出简单的约简算法。在应用方面主要考虑（1）应用模糊粗集诱导的拓扑的闭集的性质研究模糊离散动力系统中的平衡态问题。（2）应用模糊粗集的方法研究模糊综合评判问题。

粗集作为一种处理不确定性问题的理论，自上世纪九十年代以来就引起人们的重视，广义粗集作为经典粗集的发展，受到了越来越多科研人员的关注，广义粗集的拓扑结构是粗集结构中较为困难的部分，本项目紧紧抓住粗集的线性及拓扑结构开展工作，按照申请书的思路进行研究，达到了预期的目的，拓宽了研究领域。取得了丰富的研究成果。有关属性约简的结果有较强的应用前景，到目前为止共发表学术论文11篇，其中国际SCI收录的论文9篇，EI收录的国际会议论文1篇，一般国际杂志论文1篇。. 在决策表的属性约简方面，我们提练出不变矩阵的概念，分别对于绝对约简、正区域约简及分布约简给出不变矩阵的具体形式，基于不变矩阵给出这三类约简的统一算法。我们还把属性约简的算法推广到一般的关系决策系统，经过一步步的探索，对于正区域约简，终于给出一个十分简洁的约简算法，该算法不要求决策属性为等价关系，从而大幅扩大了应用范围，且统一了过去正区域约简的算法。.针对覆盖，定义了最大、最小关系，通过布尔矩阵的方法，把覆盖最大、最小约简问题转化成矩阵或关系的约简问题，给出了保持覆盖最大、最小关系不变的两类覆盖约简算法。. 覆盖粗集是广义粗集的一种，研究了由覆盖粗集诱导的拓扑结构，通过关系来替代领域的思想，用关系描述了覆盖粗集诱导的拓扑结构，研究了这些拓扑的可分离性，得到了简洁的结构关系，我们知道，在拟离散拓扑中，闭包算子扩充到传递闭包算子的过程对应于二元关系扩充到该关系的传递关系，我们把这个结果推广到模糊闭包算子的情况。. 针对人们提出了许多不同类型的广义粗集近似对，我们研究了一些不同的广义粗集（含覆盖粗集）之间的关系，用数学公式给出它们之间精准的关系，有了这些关系后就知道哪些近似对是不需要研究的，从而简化了人们对近似对的研究。. 我们还研究了一类特殊的覆盖并由此导出偏序关系决定的粗集的公理化。此外，我们还研究了近似算子成为自同态算子的条件。.