Reductions and Saturation Reductions of (Abstract) Knowledge Bases

Rough set theory is a useful tool for dealing with fuzzyness and uncertainty of knowledge. In rough set theory, knowledge reductions and generatings are important research topics and critical steps of knowledge acquisition. This paper generalize knowledge bases to abstract knowledge bases and study (abstract) knowledge bases on infinite universe by considering the problem of existence of finite reductions of infinite knowledge bases. For abstract knowledge bases, the concept of saturations and saturation reductions are introduced. Global properties of saturations and saturation reductions of abstract knowledge bases are investigated. It is proved that for a given abstract knowledge base which is closed w.r.t. arbitrary unions on a finite universe U, its saturation augmented the unverse U forms a topology, whereas a counterexample is constructed to show that this may not be true if U is infinite. Making use of the saturation of an abstract knowledge base, some sufficient and/or necessary conditions for existence of finite reductions of an infinite abstract knowledge base are given. It is proved that for an abstract knowledge base on finite universe, there is one and only one saturation reduction. Some examples are constructed to reveal various cases of existence of knowledge reductions. Simple applications of saturation reductions are also given.