Research on Formula Manipulation Expert System Based on Database of Mathematical Formulas

本研究基本建立了数学公式数据库的构建方法，并阐明了数学公式数据库的有效性。本研究的主要内容如下：1）找到了一种简单有效的数学公式索引方法; 2）为用户访问数据库确定了一个简洁实用的语句; 3）实现了一种高效的公式检索方法，并对多种公式进行了测试; 4）开发了一个能接受多种形式公式的解析器; 5）实现了一个模式匹配器，用于数学表达式的一般化简; 6）给出了数学公式数据库的构造方法和有效性，特别是，我们的数据库检索方法使得不仅通过包括在每个公式中的函数名而且通过公式的模式和属性来检索数学公式，但由于研究术语的限定性，在以下几个方面的研究尚未完成：1）按人名、公式或学科、领域检索数据库; 2）公式教材中公式的重新编排; 3）将数据库与计算机代数系统GAL连接起来。这些工作正在进行中。由于公式数据库本身并不那么值钱，但当它被应用于计算积分、解微分方程、简化数学表达式等时，它就变得非常有用。因此，在我们的研究基础上，公式数据库的应用将在不久的将来成为公式操作的一个重要主题。

We have almost established the construction method and clarified the effectiveness of database of mathematical formulas in this research. The items performed in this research are as follows:1) We found a simple and very effective indexing method for mathematical formulas; 2) We decided a concise and useful sentence for the user access to the databese; 3) We implemented an efficient retrieval method and tested it with many formulas; 4) We developed a parser which accepts quite many formulas in various forms; 5) We implemented a pattern matcher for general simplification of mathematical expressions; 6) We showed a construction method and effectiveness of database of mathematical formulas.In particular, our database retrieval method is such that the mathematical formulas are retrieved by not only the function names included in each formula but also the pattern and property of the formula, and that the retrieval is done in a similar way as the hand retrieval from formula textbooks.However, the research has not finished in the following points because of shortness of the research term approved:1) Retrieval of database by names of persons, formulas, or scientific subjects and fields; 2) Rearrangement of formulas in formula textbook; 3) Connecting the database with the computer algebra system GAL.These works are being carried out now.Since the formula database is not so worthy by itself but it becomes very useful when it is applied to calculating integrals, solving differential equations, simplifying mathematical expressions, and so on. Therefore, on the basis of our research, the application of formula database will become an important theme of formula manipulation in a very near future.