Primitive (Co)Recursion and Course-of-Value (Co)Iteration, Categorically

Abstract

In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic func- tion definition schemes of iteration and coiteration are modelled by constructions known as cata- morphisms and anamorphisms. Primitive recursion has been captured by a construction called paramorphisms. We draw attention to the dual construction of apomorphisms, and show on ex- amples that primitive corecursion is a useful function definition scheme. We also put forward and study two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of course-of-value iteration and its dual, respectively, and argue that even these are helpful.