On modeling similarity and three-way decision under incomplete information in rough set theory

Although incomplete information is a well studied topic in rough set theory, there still does not exist a general agreement on the semantics of various types of incomplete information. This has led to some confusions and many definitions of similarity or tolerance relations on a set of objects, without a sound of semantical justification. The main objective of this paper is to address semantics issues related to incomplete information. We present a four-step model of Pawlak rough set analysis, in order to gain insights on how an indiscernibility relation (i.e., an equivalence relation) is defined and used under complete information. The results enable us to propose a conceptual framework for studying the similarity of objects under incomplete information. The framework is based on a classification of four types of incomplete information (i.e., “do-not-care value”, “partially-known value”, “class-specific value”, and “non-applicable value”) and two groups of methods (i.e., relation-based and granule-based methods) for modeling similarity. We examine existing studies on similarity and their relationships. In spite of their semantics differences, all four types of incomplete information can be uniformly represented in a set-valued table. We are therefore able to have a common conceptual possible-world semantics. Finally, to demonstrate the value of the proposed framework, we examine three-way decisions under incomplete information.