Type-I Topological Logic C1I and Approximate Reasoning

We introduce the consistent topological structure and neighborhood structure into the logical framework for providing the logical foundation and logical normalization for the approximate reasoning. We present the concept of the formulae mass, the knowledge mass and the approximating knowledge closure of the knowledge library by means of topological closure. We obtain the fundamental framework of type-I topological logics. In this framework, we present the type-I topological algorithm of the simple approximate reasoning and multi-approximate reasoning. In the frameworks of type-I strong topological logics, we present the type-I topological algorithm of multidimensional approximate reasoning and multiple multidimensional approximate reasoning. We study the type-I completeness and type-I perfection of the knowledge library in the framework of topological logical frameworks. We construct the type-I knowledge universe and prove that the second class knowledge universe of type-I is coincident with the first class knowledge universe of type-I, therefore the type-I knowledge universe is stable. We construct a self-extensive type-I knowledge library and the type-I expert system. In this expert system, the new approximate knowledge acquired by the self-extensive type-I knowledge library KI will not beyond the type-I approximate knowledge closure, (K0)−−, of the initial knowledge library K0. Therefore, the precision of all new acquired approximate knowledge of this automatic reasoning system will be controlled well by the type-I approximate knowledge closure (K0)−− of the initial knowledge library K0.