Polytypic Functions Over Nested Datatypes

The theory and practice of polytypic programming is intimately connected with the initial algebra semantics of datatypes. This is both a blessing and a curse. It is a blessing because the underlying theory is beautiful and well developed. It is a curse because the initial algebra semantics is restricted to so-called regular datatypes. Recent work by R. Bird and L. Meertens (1998) on the semantics of non-regular or nested datatypes suggests that an extension to general datatypes is not entirely straightforward. Here we propose an alternative which extends polytypism to arbitrary datatypes, including nested datatypes and mutually recursive datatypes. The central idea is to use rational trees over a suitable set of functor symbols as index sets for polytypic functions. Besides covering a wider range of types the approach is also simpler and technically less involving than previous ones. The prize one has to pay is a certain loss in expressiveness: general recursion schemes such as fold an..