中文摘要：

粒计算的形式化研究一直没有被仔细讨论．文中在集合论框架下，对粒计算做了系统研究，给出了粒度空间的三层模型（论域，基，粒结构）．借用逻辑语言L判定粒的可定义性，将经典粗糙集通过此模型重新解释．根据模型中从基到粒结构不同的构造规则，引出并可约和交可约粒度空间的定义，分别讨论了不同粒度空间下覆盖、基和粒结构的关系，从而给出从覆盖求基的方法；进一步，利用子系统表示方法对扩展粗糙集以及一般的交可约与并可约空间的上下近似进行了研究，分析了现有的4种基于覆盖的粗糙集模型的合理性；研究了形式概念分析以及知识空间的粒度空间模型，给出这两种理论中上下近似的概念．

英文摘要：

Granular computing has been widely studied in recent years. However, it still lacks of a formal framework which can cover all the affiliate models of granular computing, such as rough sets, formal concept analysis and knowledge spaces, In this paper, a three levels＇ model of gran ular spaces （the universe, the basis and the granular structure） in set-theoretic formulation is proposed. By using the definability defined by the logic language L, a three levels＇ model of granular spaces in Pawlak rough sets is established. In terms of the different construction rules from the basis to the granular structure, the definitions of the union reducible granular spaces and the intersection reducible granular spaces are given. We discuss the relationship among the cover ing, the basis, and the corresponding granular structure, and give the methods for seeking the basis from the covering in the two types of granular spaces respectively. Then we focus on set- theoretic setting and generalized rough set aPproximations with subsystem-based definition, The generalized approximations are also investigated in the union reducible and the intersection reduci- ble granular spaces. Furthermore, the rationality of the four types of covering-based rough setmodels is examined. Finally, the three levels＇ models of granular spaces in formal concept analy-sis and knowledge spaces are expressed. The generalized approximations are also explored in the two theories. It is shown that the set-framework of granular computing can help us to study the branch models of granular computing in a more formal view.