| Rough set theory as a new mathematical tool is fit for data mining. Rough set theory has received a lot of attention on areas in both of real-life applications and the theory itself. Meanwhile, the real-life applications promote rough set theory research by means of extending structures of rough set. Pawlak rough set is discussed in approximation space, which is constituted by universe, equivalence relation, and approximation space. The promoted research of rough set is extension of universe, equivalence relation and approximation space. At the aspect of approximation space, many research communities have done much in rough set theory and application research on extension of equivalence relation and approximation space. However, the study on rough set on universe extension has just started internally. Definitions about rough set on universe extension are lack of unity. Properties of rough set on universe extension are not perfect. Applications from rough set on universe extension are few. On the other hand, practical problems are mostly related to two universes. For example, characteristics of customer and attributes of product in marketing, problems and solutions in the diagnosis of enterprise business, symptoms of disease and diagnosis in fault diagnosis and so on are problems with two different universes. Rough set on universe extension is especially applicable to solve these issues, which can help us make decisions. Thus, rough set on universe extension is significant in theory and applications.In this paper, rough set on universe extension is discussed by relax tolerance relation to general relation, which is named Rough Set over Dual-universe (RSDU for short). Approximation operators, basic concepts and uncertainty measures of RSDU are discussed. RSDU in fuzzy information system and probability space are also studied in this dissertation. Applications based on RSDU, FRSDU and VPRSDU are illustrated in expert system. Specifically, the main contributions can be formulated as follows:(1) This paper has made positive contributions to the enrichment and development of RSDU theory. In accordance with the limitation of research on traditional rough set on universe extension, we proposed RSDU by general relation in place of tolerance relation. Attribute significance, attribute reduction, information entropy and information granular of RSDU are presented. The relations between Pawlak rough set and RSDU are studied and uncertainty measures of RSDU are discussed.(2) For vague description of problems often comes up in our real life. RSDU are researched in fuzzy information system, which is named Fuzzy Rough Set over Dual-universes(FRSDU for short). FRSDU is introduced by discussing fuzzy relation and cut set. Approximation operators are derived. Attribute significance and attribute reduction of FRSDU are presented. Properties of FRSDU are discussed from the angles of rough set and fuzzy set and examples are advanced to illustrate FRSDU.(3) For dependency degree formula is accurately described in RSDU and noise data exerts a considerable influence when we deal with problems. Variable Precision Rough Set over Dual-universes (VPRSDU) is introduced by discussing inclusion degree. Approximation operators of VPRSDU are derived by using general relation and inclusion degree. Properties of VPRSDU are discussed from the angles of inclusion degree and threshold. Attribute significance, attribute reduction and relative attribute reduction of VPRSDU are presented. Then, two parametric variable precision rough set over Dual-universes is presented and examples are advanced to illustrate VPRSDU.(4) For research objects and expert system may be not perfectly matched, RSDU, FRSDU and VPRSDU are utilized to construct expert system. Algorithms for lower and upper approximation of RSDU, FRSDU and VPRSDU are advanced. Positive region, negative region, boundary region and possibility region are obtained. Expert knowledge and research objects draw certainty rules and possible rules.In this paper, constructive method is utilized to research rough set on universe extension:RSDU. New concepts of RSDU are presented by comparing with traditional concepts. RSDU in fuzzy and probability information system are researched for uncertainty in real life. Applications of rough set on universe extension are discussed in expert system.
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