Quotient Space Based Multi-granular Analysis

Abstract

We presented a quotient space model that can represent a problem at different granularities; each model has three components: the universe X, property f and structure T. So a multi-granular analysis can be implemented based on the model. The basic properties among different quotient spaces such as the falsity preserving, the truth preserving properties are discussed. There are three quotient-space model construction approaches, i.e., the construction based on universe, based on property and based on structure. Four examples are given to show how a quotient space model can be constructed from a real problem and how benefit we can get from the multi-granular analysis. First, by adding statistical inference method to heuristic search, a statistical heuristic search approach is presented. Due to the hierarchical and multi-granular problem solving strategy, the computational complexity of the new search algorithm is reduced greatly. Second, in the collision-free paths planning in robotics, the topological model is constructed from geometrical one. By using the truth preserving property between these two models, the paths planning can be implemented in the coarser and simpler topological space so that the computational cost is saved. Third, we discuss the quotient space approximation and the multi-resolution signal analysis. And the second-generation wavelet analysis can be obtained from quotient-space based function approximation. It shows the equivalence relation between the quotient space model based analysis and wavelet transform. Fourth, in the automatic assembly sequence planning of mechanical product, we mainly show how a quotient structure can be constructed from the original one. By using the simpler quotient structure, the assembly sequence planning can be simplified greatly. In conclusion, the quotient-space model enables us to implement a multi-granular analysis. And we can get great benefit from the analysis.