Interval Arithmetic, Extended Numbers and Computer Algebra Systems

Many ambitious computer algebra systems were initially designed in a flush of enthusiasm, with the goal of automating any symbolic mathematical manipulation “correctly.” Historically, this approach resulted in programs that implicitly used certain identities to simplify expressions. These identities, which very likely seemed universally true to the programmers in the heat of writing the CAS, (and often were true in well-known abstract algebraic domains) later needed re-examination when such systems were extended for dealing with kinds of objects unanticipated in the original design. These new objects are generally introduced to the CAS by extending “generically” the arithmetic or other operations. For example, approximate floats do not have the mathematical properties of exact integers or rationals. Complex numbers may strain a system designed for real-valued variables. In the situation examined here, we consider two categories of “extended” numbers: ∞ or undefined, and real intervals. We comment on issues raised by these two troublesome notions, how their introduction into a computer algebra system may require a (sometimes painful) reconsideration and redesign of parts of the program, and how they are related. An alternative (followed most notably by the Axiom system is to essentially envision a “meta” CAS defined in terms of categories and inheritance with only the most fundamental built-in concepts; from these one can build many variants of specific CAS features. This approach is appealing but can fails to accommodate extensions that violate some mathematical tenets in the cause of practicality.