Symbolic Computation and Combinatorics

Doron Zeilberger建议继续开发方法，以利用符号计算的巨大潜力在组合学和相关领域进行研究。特别是，他希望引入新的计算和概念框架，将所谓的威尔夫-泽尔伯格证明理论扩展到更广泛的恒等式和定理类别。他还建议继续他在人工组合学方面的努力，并开发发现和证明组合数学中的定理的算法，这些定理的复杂性使其不适用于人类证明。这项研究应该是共生的，因为预计具体的结果和基本的方法论都将有助于计算机代数开发人员改进和提高他们的系统。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。这项研究也是在符号计算的一般领域，它试图教计算机执行以前需要大量人力资源的研究。这一领域的进展有望对科学和技术产生重要影响。

Doron Zeilberger proposes to continue to develop methodologies for harnessing the great potential of Symbolic Computation to do research in Combinatorics and related areas. In particular he hopes to introduce new computational and conceptual frameworks that would extend the so-called Wilf-Zeilberger proof theory to much wider classes of identities and theorems. He also proposes to continue his efforts in `Artificial Combinatorics', and develop algorithms for the discovery and {\it rigorous} proof of theorems in combinatorics whose complexity make them unfeasible for human proofs. This research should be symbiotic, as it is expected that both the concrete results and the underlying methodologies, would help computer algebra developers to improve and enhance their systems.This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. This research is also in the general area of Symbolic Computation, that attempts to teach computers to perform research that previously required extensive human resources. Progress in this area promises to have important ramifications to science and technology.