Research On Reduction Theory And Approach Of Set-Valued Formal Contexts

Rough set and formal concept analysis are proposed respectively in the early1980s. For now, research on the two theories has developed rapidly and many applications based on these theories have been successful in many fields. A set-valued information system is an information system that can reflect the uncertainty and the variety of the value to be taken. In the present research work on this system, there is a limitation that many studies are mainly based on rough set theory while few based on formal concept analysis. In this thesis, by redefining the set-valued information system as a set-valued formal context which is the generalization of a classical formal context, the set-valued information system is studied by means of the approach of formal concept analysis. The research methods about classical formal contexts are extended, and the reduction theory and approach of set-valued formal contexts are studied in this thesis, and the research results provide the theoretical basis for acquiring more knowledge from a set-valued information system.The main work and conclusions obtained in this thesis are summarized as follows.Firstly, many basic concepts about set-valued formal contexts are defined, and one pair of operators which satisfy Galois connection is constructed. The set-valued concept lattice is also proved to be a complete lattice. And the classical formal context is the special case of the set-valued formal context.Secondly, in the sense that two set-valued concept lattices are isomorphism, the concepts of the set-valued consistent set, the set-valued reduction, the set of set-valued consistent attributes, and the set of set-valued reduction attributes are proposed. A series of necessary and sufficient conditions about judging the set-valued consistent set and the set-valued reduction are obtained. The characteristics of set-valued primitive elements which are the basic elements of a set-valued consistent set are studied, and meanwhile, an approach which can be used to judge three types of set-valued primitive elements is presented. Based on the cardinality of a set-valued extension, two definitions are proposed, which are the measurement of the structure of extensions of a set-valued concept lattice and the significance of a set-valued primitive element. Taking into account the two definitions respectively, corresponding judgment theorems for set-valued consistent sets and set-valued reduction are obtained.Finally, the concept of a discernibility matrix is presented, and an algorithm for reducing set-valued primitive elements based on this matrix is also proposed. For improving the efficiency of obtaining the set-valued reduction, on the basis of the cardinality of a set-valued extension, a heuristic algorithm for reducing set-valued primitive elements is proposed. The numerical examples demonstrate the validity of the two algorithms.