Mathematical Sciences: Toward a Theory of Well-Founded Orderings for Use in Automated Deduction

建立良好的序在自动演绎中无处不在，是基于重写的演绎系统中计算的终止证明发展的基础，也是旨在限制饱和过程中搜索空间的演绎策略的基础。然而，确定适合于特定目的的有充分根据的命令通常需要大量的技术专门知识，并且由于重写的终止通常是不可确定的，因此可以期望的最佳结果是对足够多的命令有充分的了解，以便能够应对实践中出现的各种终止和效率问题。这项研究计划拨款旨在发展基于逻辑和数值不变量的有充分根据的有序的综合理论。进一步研究的两个具体方向是将已知的结果扩展到更大的项类，以及为更一般的成立良好的排序（如指数排序）获得类似的结果。作为进一步的，相关的调查线，一个理论的非完全有理有据的排序将探讨。由于顺序类型的概念对于非全排序没有意义，因此需要确定适当的逻辑不变量。＊＊＊

9510164 Johann Well-founded orderings are ubiquitous in automated deduction, being fundamental to the development of termination proofs for computations in rewrite-based deduction systems, as well as of deduction strategies aiming to restrict search spaces in saturation processes. The determination of well-founded orderings suitable for a particular purpose, however, typically requires a good deal of technical expertise, and because termination of rewriting is undecidable in general the best that can be hoped for is to have a sufficient understanding of a sufficient variety of orderings to be able to cope with the many kinds of termination and efficiency problems which occur in practice. This research planning grant undertakes to develop a comprehensive theory of well-founded orderings based on their logical and numerical invariants. Two specific directions for further research are extending known results to larger classes of terms, and obtaining similar results for more general well- founded orderings, such as exponential orderings. As a further, related line of inquiry, a theory of nontotal well-founded orderings will be explored. Since the notion of order type makes no sense for nontotal orderings, appropriate logical invariants will need to be determined. ***