Knowledge structures in a tolerance knowledge base and their uncertainty measures

•Dependence between knowledge structures in a tolerance knowledge base is proposed.•Group, mapping and lattice characterizations of knowledge structures are obtained.•Measuring uncertainty of knowledge structures is investigated.•Effectiveness analysis of the proposed measures is conducted from the angle of statistics. Rough set theory is a useful tool for dealing with imprecise knowledge. Its important notion is a knowledge base. In a knowledge base, one can approximately describe the target notion in terms of existing knowledge structures. A tolerance knowledge base is the generalization of knowledge bases. This paper investigates knowledge structures in a tolerance knowledge base and their uncertainty measures. Knowledge structures in a tolerance knowledge base are first depicted by means of set vectors. Then, dependence and independence between knowledge structures are described by using inclusion degree. Next, mapping and lattice characterizations of knowledge structures are given. Finally, measuring uncertainty of knowledge structures in a tolerance knowledge base is studied, two numerical experiments on the congressional voting records data set that comes from UCI Repository of machine learning databases are conducted, and based on these numerical experiments, effectiveness analysis from the angle of statistics is given to evaluate the performance of the proposed measures. These results will be helpful for establishing a framework of granular computing in knowledge bases.