The Property of Different Granule and Granular Methods Based on Quotient Space

Abstract

Nowadays, we have entered the era of big data, and we have to deal with complex systems and massive data frequently. Facing complicated objects, how to describe or present objects is the base to solve questions frequently. So we suppose that a problem solving space, or a problem space for short, is described by a triplet (X, f, Γ), and assume that X is a domain, R is an equivalence relation on X, Г is a topology of X, [X] is a quotient set under R. Regarding [X] as a new domain, we have a new world to analyse and to research this object, consequently we describe or present a question into different granule worlds, these granular worlds are called the quotient space. Further we are able to predigest and solve a question, i.e. we apply quotient space and granulate to represent an object. Comparing rough set and decision-making tree, the quotient space has the stronger representation. Not only it can represent vectors of the problem domain, different structures between vectors, but also it can define different attribute functions and operations etc. In this paper, we discuss the method how to represent and to partition an object in granular worlds, and educe the relationship of different granular worlds and confirm the degree of granule. We will prove three important theorems of different granules, i.e. to preserve false property theorem and to preserve true property theorem. To solve a problem in different granular worlds, the process procedure of quotient approximate will be applied. We also supply an example of solving problem by different granule worlds—the shortest path of a complex network. The example indicates that to describe or present a complicated object is equal to construct quotient space. In quotient set [X], the complexity to solve a problem is lower than X. We have a new solution method to analysis a big data based on the quotient space theory.