Research On Models Of Granular Computing Based On Quotient Space Theory And Rough Set Theory

Perception, understanding and representation, as well as analysis, synthesis and reasoning, of real world problem together with its solution at different levels of granularity, is an obvious feature in the process of human problem solving, and it also embodies the outstanding ability of human problem solving. In a sense, it is precisely the intelligence in the process of human problem solving. Considering such ability of human, researchers of artificial intelligence have made some further investigation and presented many formal models. As an emerging research sub-field of artificial intelligence, granular computing, whose philosophy is to implement the problem solving at different levels of granularity, aims to establish much more general model reflecting the process of human problem solving. Granule, a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality, is the primitive notion of granular computing. In a narrow sense, granular computing can be understood as the computing and reasoning with granules at different levels of granularity. In a broader sense, granular computing is a unified notion of theories, methodologies, techniques and tools that utilize granules in the process of problem solving. Three basic components of a granular computing model are granule, granular level derived by a given granulation criteria and hierarchy composed of all granular levels. Granulation and computing or reasoning with granules, either of them can be studied from both the semantic and algorithmic perspectives, are two related issues of granular computing.The thesis mainly studies two crisp models of granular computing in the framework of set theory, namely, quotient space theory of problem solving and rough set theory. More specifically, the research includes:1, Quotient space theory of problem solving is generalized from two directions, namely the structure of the universe of discourse and granulation criteria. In the light of granular computing, quotient space theory of problem solving exploits topology to describe the structure of the universe of discourse, and utilizes equivalence relation to implement granulation, and relies on the natural mapping to realize the translations among different levels of granularity. Closure operation in the sense of Cech is more general mathematical language that can be used to describe the structure of the universe of discourse. When generalizing quotient space theory of problem sohing. the thesis exploits closure operation of Cech sense to describe structure. It turns out thatconclusions of classical quotient space theory keep being true in such a general framework. The axiom of transitivity is a tough condition for many applications, which may confine the applicable fields of quotient space theory of problem solving, so a weakened version of quotient space theory with topology describing the structure, compatibility relation that is reflexive and symmetric binary relation implementing granulation, set-valued function playing the role of natural mapping, are presented. The discussion implies that there are corresponding conclusions of classical quotient space theory. To some degree, these two generalizations enrich the intension of quotient space theory of problem solving, and extend its applicable fields.2 > The thesis presents the topological interpretation of the upper and lower approximate operators of generalized rough set model based on reflexive binary relation, as well as the order-theoretic interpretation of the upper and lower approximate operators of generalized rough set model based on covering. Classical rough set theory lays its foundation on the indiscemibility relation, which is represented as equivalence relation in mathematics, between objects. Generalized rough set models based on binary relation and covering respectively are two important studies. The thesis proves that the upper approximate operator of generalized based rough set models based on binary relation is precisely quasi-discrete closure operation, in the sense of Cech. derived by the given binary relation. Dually, the lower approximate operator is preciseh- the quasi-discrete interior operation. Then the topological understandings of generalized approximate rough operators are proposed. As far as covering based rough approximate operators are concerned, it is proved, by constructing Galois connection between the power set of object set and the power set of the covering, that the upper approximate operator is preciseh' the closure operator, which satisfies special axioms, in the sense of order theory. Also, the dual result about the lower approximate operator is given. The studies of the thesis provide, to some extent, the foundations for the further application of generalized rough set theory.3-, A generalized multi-valued information system and its corresponding formal language, a -decision logic language, are proposed. Simply, information system, also called information table, is a 2-dimension table used as the carrier of information, which includes the finite set of objects, die finite set of attribute (name), the finite domain for each attribute and the information function for each attribute that maps objects to the domain of this attribute. It is the information function that distinguishes different ft-pe information systems. Based on the analysis of single-valuedinformation system, multi-valued one, relational attribute one and fuzzified single-valued one, a fuzzified multi-valued information system, which generalizes all of the mentioned ones, is presented in which the information function for each attribute maps a object to a fuzzy set of corresponding domain. Consequently, a -decision logic language that generalizes decision logic language of single information system is proposed. By the selection of parameter a one can describes, analyzes and solves of problem at different level of granularity.Undoubtedly, considering the philosophy of granular computing, the thesis discusses the problem of granular computing at rather coarse granularity. Namely, the granular computing is studied at a very abstract level. Therefore, how to combine discussed theories and methods of granular computing with concrete problems is an important issue of further research.