数学知识表示是知识表示中的一个重要方面,是数学知识检索、自动定理机器证明、智能教学系统等的基础.根据在设计NKI(national knowledge infrastructure)的数学知识表示语言中遇到的问题,并在讨论了数学对象的本体论假设的基础上提出了两种数学知识的表示方法:一种是以一个逻辑语言上的公式为属性值域的描述逻辑;另一种是以描述逻辑描述的本体为逻辑语言的一部分的一阶逻辑.在前者的表示中,如果对公式不作任何限制,那么得到的知识库中的推理不是可算法化的;在后者的表示中,以描述逻辑描述的本体中的推理是可算法化的,而以本体为逻辑语言的一部分的一阶逻辑所表示的数学知识中的推理一般是不可算法化的.因此,在表示数学知识时,需要区分概念性的知识(本体中的知识)和非概念性的知识(用本体作为语言表示的知识).框架或者描述逻辑可以表示和有效地推理概念性知识,但如果将非概念性知识加入到框架或知识库中,就可能使得原来可以有效推理的框架所表示的知识库不存在有效的推理算法,甚至不存在推理算法.为此,建议在表示数学知识时,用框架或描述逻辑来表示概念性知识;然后,用这样表示的知识库作为逻辑语言的一部分,以表示非概念性知识.

The representation of mathematical knowledge is an important aspect of knowledge representation. It is the foundation for knowledge-based automated theorem proving, mathematical knowledge retrieval and intelligent tutoring systems, etc. According to the problems that are encountered in designing the mathematical knowledge representation language in NKI (national knowledge infrastructure) and after the discussion of ontological assumptions for mathematical objects, two kinds of formalisms for the representation of mathematical knowledge are provided. One is a description logic in which the range of an attribute can be a formula in some logical language; and another is a first order logic in which an ontology represented by the description logic is a part of the logical language. In the former representation, if no restrictions are imposed on formulas, then there is no algorithm to realize the reasoning in the resulted knowledge base; in the latter representation, the reasoning in the ontology represented by the description logic is decidable, while in general, for mathematical knowledge described by the first order logic which contains the ontology represented by the description logic, there is no algorithm to realize its reasoning. Hence, in the representation of mathematical knowledge, it is necessary to distinguish conceptual knowledge (knowledge in an ontology) and non-conceptual knowledge (knowledge represented by a language containing the ontology). Frames and description logics can represent and reason effectively about conceptual knowledge, but the addition of non-conceptual knowledge to frames or knowledge bases may make the reasoning in the resulted knowledge bases not decidable and there is even no algorithm to reason about the knowledge bases. Therefore, it is suggested that in representing mathematical knowledge, frames or description logics are used todescribe conceptual knowledge, and the logical languages containing the knowledge base represented by the frames or description logics are used to represent non-conceptual knowledge.

Supported by the National Natural Science Foundation of China under Grant Nos.60273019, 60373042, 60496326, 60573063, 60573064 (国家自然科学基金); the National Grand Fundamental Research 973 Program of China under Grant Nos.2003CB317008, G1999032701 (国家重点基础研究发展计划(973))