Building Algebra Systems by Overloading Lisp: Automatic Differentiation

Some of the earliest computer algebra systems (CAS) looked like overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula Algol, and others, each took advantage of a pre-existing language base and expanded the notion of a numeric value to include mathematical expressions. Much more recently, perhaps encouraged by the growth in popularity of C++, we have seen a renewal of the use of overloading to implement a CAS. This paper makes three points. 1. It is easy to do overloading in Common Lisp, and show how to do it in detail. 2. Overloading per se provides an easy solution to some simple programming problems. We show how it can be used for a “demonstration ” CAS. Other simple and plausible overloadings interact nicely with this basic system. 3. Not all goes so smoothly: we can view overloading as a case study and perhaps an object lesson since it fails to solve a number of fairly-well articulated and difficult design issues in CAS for which other approaches are preferable.