Meta-Rules: Reasoning About Control

How can we insure that knowledge embedded in a program is applied effectively? Traditionally the answer to this question has been sought in different problem solving paradigms and in different approaches to encoding and indexing knowledge. Each of these is useful with a certain variety of problem, but they all share a common problem: they become ineffective in the face of a sufficiently large knowledge base. How then can we make it possible for a system to continue to function in the face of a very large number of plausibly useful chunks of knowledge? In response to this question we propose a framework for viewing issues of knowledge indexing and retrieval, a framework that includes what appears to be a useful perspective on the concept of a strategy. We view strategies as a means of controlling invocation in situations where traditional selection mechanisms become ineffective. We examine ways to effect such control, and describe meta-rules, a means of specifying strategies which offers a number of advantages. We consider at some length how and when it is useful to reason about control, and explore the advantages meta-rules offer for doing this.

Basis Reduction Algorithms and Subset Sum Problems

This thesis investigates a new approach to lattice basis reduction suggested by M. Seysen. Seysen's algorithm attempts to globally reduce a lattice basis, whereas the Lenstra, Lenstra, Lovasz (LLL) family of reduction algorithms concentrates on local reductions. We show that Seysen's algorithm is well suited for reducing certain classes of lattice bases, and often requires much less time in practice than the LLL algorithm. We also demonstrate how Seysen's algorithm for basis reduction may be applied to subset sum problems. Seysen's technique, used in combination with the LLL algorithm, and other heuristics, enables us to solve a much larger class of subset sum problems than was previously possible.