Fixed Points of Type Constructors and Primitive Recursion

摘要

For nested or heterogeneous datatypes, terminating recursion schemes considered so far have been instances of iteration, excluding efficient definitions of fixed-point unfolding. Two solutions of this problem are proposed: The first one is a system with equi-recursive non-strictly positive type constructors of arbitrary finite kinds, where fixed-point unfolding is computationally invisible due to its treatment on the level of type equality. Positivity is ensured by a polarized kinding system, and strong normalization is proven by a model construction based on saturated sets. The second solution is a formulation of primitive recursion for arbitrary type constructors of any rank. Although without positiv-ity restriction, the second system embeds—even operationally—into the first one.