In this paper we present an axiomatic theory for a class of algorithms, called problem reduction generators, that includes dynamic programming, general branch-and-bound, and game tree search as special cases. This problem reduction theory is used as the basis for a mechanizable design tactic that transforms formal specifications into problem reduction generators. The theory and tactic are illustrated by application to the problem of enumerating optimal binary search trees. Contents 1. Introduction 3 2. Basic Concepts And Notation 3 2.1. Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Signatures and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Problem Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Enumerating Feasible Solutions 6 3.1. Problem Reduction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2. Design Tactic -- Enumerating Feasible Solutions . . . . . . . . . . . . . ....

The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other non-relational calculi. The graph

Dynamic programming has long been used as an algorithm design technique, with various mathematical theories proposed to model it. Here we take a different perspective, using a relational calculus to model the problems and solutions using dynamic programming. This approach serves to shed new light on the different styles of dynamic programming, representing them by different search strategies of the tree-like space of partial solutions. 1 INTRODUCTION AND HISTORY Dynamic programming is an algorithm design technique for solving many different types of optimization problem, applicable to such diverse fields as operations research (Ecker and Kupferschmid, 1988) and neutron transport theory (Bellman, Kagiwada and Kalaba, 1967). The mathematical theory of the subject dates back to 1957, when Richard Bellman (Bellman, 1957) first popularized the idea, producing a mathematical theory to model multi-stage decision processes and to solve related optimization problems. He was also the first to i...

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies. 1 Introduction Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynami...