Automating Combinatorics

Doron Zeilberger建议继续开发利用计算机代数的巨大潜力在组合学及相关领域进行研究的方法论，并为(严格的)计算机辅助和计算机生成研究设计实验。他特别希望开发一种新的算法理论，命名为符号矩演算，它将产生计算有趣的组合数量的(统计)矩的算法。他还计划发展递归方程的一般理论，其中包括计算生物学中普遍存在的具有最大值的动态规划递归，以及在组合博弈理论中发生的Mex运算。他还建议继续他在人工数学中的努力并开发了发现和严格证明计数置换类的计数方案以及自动推导母函数的算法。另一项研究涉及自动行列式计算，它在应用数学和科学中有潜在的应用。这项研究应该是共生的，因为预计具体结果和基本方法都将帮助计算机代数开发人员改进和增强他们的系统。还希望这项研究将有助于实验数学的萌芽领域，因为它将有助于开发一种研究方法，用于进行严格(和有趣！)的计算机实验。数学定理(和证明)，而不仅仅是验证和表述猜想。这项研究是在组合学领域，其对科学和技术的用处是众所周知的。特别是，计算机科学主要以组合学为基础，电子通信和万维网也是如此。

Doron Zeilberger proposes to continue to developmethodologies for harnessing the great potential ofComputer Algebra to do research in Combinatorics andrelated areas, and design experiments for (rigorous)computer-assisted and computer-generated research.In particular he hopes to develop a new algorithmic theoryto be named `Symbolic Moment Calculus', that would producealgorithms for computing, symbolically and automatically, (statistical) moments of interesting combinatorial quantities.He also plans to develop a general theory of recurrence equationsthat would include both the `Dynamic Programming' recurrencesfeaturing the maximum, ubiquitous in Computational Biology,and the mex operation, that occurs in the theory ofCombinatorial Games.He also proposes to continue hisefforts in `Artificial Combinatorics', and develop algorithmsfor the discovery and rigorous proof of enumeration schemes for counting permutation classes, and for automatically deducing generating functions. Anotherline of research concerns automatic determinant evaluationsthat has potential applications in applied mathematics andscience.This research should be symbiotic, as it isexpected that both the concrete results and the underlying methodologies, would help computer algebra developersto improve and enhance their systems. It is also hoped thatthis research will contribute to the budding field of ExperimentalMathematics, in that it will help develop a research methodology forconducting computer experiments that output rigorous (and interesting!) mathematical theorems(and proofs), rather than just verifying and formulating conjectures.This research is in the field of Combinatorics, whose usefulnessto science and technology is well-known. In particular, computerscience is largely based on combinatorics, as is electroniccommunication and the World Wide Web.