On the Design and Construction of Algebraic Manipulation Systems

Abstract

We compare and contrast several techniques for the implementation of components of an algebraic manipulation system. On one hand is the mathematical-algebraic approach which characterizes (for example) IBM's Axiom. On the other hand is the more ad hoc approach which characterizes many other popular systems (for example, Macsyma, Reduce, Maple, and Mathematica). While the algebraic approach has generally positive results, careful examination suggests that there are significant remaining problems, especially in the representation and manipulation of analytical, as opposed to algebraic, mathematics. We describe some of these problems and some general approaches for solutions. INTRODUCTION Symbolic algebraic mathematics programs have been in use for several decades now. Some programs implement stand-alone algorithms which accomplish a particular task: say, factoring a univariate polynomial. Other programs appear as part of a monolithic system such as Macsyma, Mathlab '68, Axiom (formerly S..