When Do Datatypes Commute?

Polytypic programs are programs that are parameterised by type constructors (like List), unlike polymorphic programs which are parameterised by types (like Int). In this paper we formulate precisely the polytypic programming problem of \commut-ing" two datatypes. The precise formulation involves a novel notion of higher order polymorphism. We demonstrate via a number of examples the relevance and interest of the problem, and we show that all \regular datatypes" (the sort of datatypes that one can deene in a functional programming language) do indeed commute according to our speciication. The framework we use is the theory of allegories, a combination of category theory with the point-free relation calculus. 1 Polytypism The ability to abstract is vital to success in computer programming. At the macro level of requirements engineering the successful designer is the one able to abstract from the particular wishes of a few clients a general purpose product that can capture a large market 31]. At the micro level of programming the ability to write so-called \generic" code capturing commonly occurring patterns is vital to reusability and thus to programmer productivity. One of the most signiicant contributions to generic programming has been the notion of parametric polymorphism | rst introduced by Strachey 32] and later incorporated in the language ML by Milner 25, 26]. The use of parametric polymorphism eliminates the compulsion in languages like Pascal to provide irrelevant type information. For example, it is irrelevant to the computation of the length of a list whether the elements of the list 1