Introduction

Abstract

A mathematical theory of algorithms is introduced here, which has been developed with an aim toward finding new algorithms, rather than analyzing known ones. One important approach to the study of algorithms is based on the idea that each one can be expressed by a program on a Turing Machine or some equivalent model of computation. However, it is not feasible to try to find a desired algorithm by exploring the set of all Turing Machine programs. What is proposed here, instead, is a theory in which an algorithm is expressed as a combination of major computational steps, each of which can be carried out in a manner which is assumed to be known. In other words, an algorithm is subdivided into a relatively small number of parts, so that its global structure may be readily expressible in terms of the incidence relations among them. Such a theory will be developed in chapter 1, and applications of it will be discussed in subsequent chapters.