A language for computational algebra

This paper reports ongoing research at the IBM Research Center on the development of a language with extensible parameterized types and generic operators for computational algebra. The language provides an abstract data type mechanism for defining algorithms which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categories designate collections of domains having common operations with stated mathematical properties. Domains and categories are computed objects which may be dynamically assigned to variables, passed as arguments, and returned by functions. Although the language has been carefully tailored for the application of algebraic computation, it actually provides a very general abstract data type mechanism. Our notion of a category to group domains with common properties appears novel among programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithms missing from other work on data types known to the authors.