Abstract

this paper, we discuss generic recursion combinators, i.e., combinator codings of recursive function definition schemes working uniformly on any inductive source type of functions. Such combinators are polymorphic in a type constructor inducing an inductive type. Generic recursion schemes are numerous. The most fundamental and easy to code by a combinator are (simple) iteration and primitive-recursion. Some other schemes that are more general in terms of direct expression power, e.g., the course-of-value strengthenings of iteration and primitive-recursion, are equally useful in programming and do not lead to considerably higher sophistication in the formulation [1]. Too particular generalizations, however, get too esoteric and sophisticated. To make a library primitive out of every powerful imaginable recursion scheme is therefore not a good idea. It might be the case that a better approach is to add another dimension of genericity. In this paper, we suggest a "many-in-one" recursion combinator that neatly captures a whole family of recursion schemes that prima facie present themselves as diverging generalizations of simple iteration. The combinator is doubly generic: in addition to being polymorphic in a signature for an inductive type, it is polymorphic in a comonad and a natural transformation, which together determine a particular recursion scheme. The family includes simple primitive-recursion, course-of-value versions of iteration and primitive-recursion, and much more. Below is a brief mathematical summary. Let C be a category and F : C # C be a functor. An F -algebra is a morphism # : F C #C, where C (its carrier) is an object of C

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