Research On Knowledge Acquisition And Analysis Of Mathematical Concepts

Mathematical knowledge representation and acquisition are major tasks in many mathematical applications, including knowledge-based automated theorem proving, integration of different mathematical software systems, mathematical semantic Web, and high-level mathematical instruction. Content-oriented mathematical markup languages, such as MathML, OpenMath and OMDoc, have obtained more and more attention, and a number of projects for building concrete mathematical knowledge bases or libraries (e.g. MBase, HELM, and MOWGLI) have been initiated. Formally launched in early 2000 in the Institute of Computing Technology, Chinese Academy of Sciences, the project of National Knowledge Infrastructure (NKI) aims to build a multi-domain knowledge base to be shared by various knowledge-intensive sophisticated applications, such as natural language understanding, speech understanding, planning and diagnosis, through a standard knowledge application programming interface (KAPI). In 2001, a subproject called NKIMath was initiated to build a multi-branch mathematical knowledge base as a component of NKI. So far, thousands of mathematical concepts and assertions have been acquired, covering algebra, set theory, number theory, mathematical analysis, graph theory, general topology, etc.The dissertation addresses the knowledge representation, acquisition, verification, organization and management of mathematical concepts. The main contributions of this dissertation are as follows:(1) Building a multi-purpose knowledge representation framework for mathematical concepts. The framework has three levels: predicate logic level, knowledge statement level, and conceptual relation level. The representation system has several features, such as knowledge inheritance between concepts, multi-granularity representation for concepts, knowledge sharing and reuse, transformation between knowledge formats.(2) Developing a knowledge formalization method for mathematical concepts based conceptual associative space. The conceptual associations have been defined, and the set of concepts has been proved to be a partially ordered space under the association relations. Four measurement units - relative abstraction degree, association degree, out-degree and descriptive complexity degree - are introduced to describe the abstraction, importance, fundamentality and complexity of concepts. Guided by the conceptual associative space, a knowledge formalization...