Structures for structural recursion

Abstract

Our goal is to develop co-induction from our understanding of induction, putting them on level ground as equal partners for reasoning about programs. We investigate several structures which represent well-founded forms of recursion in programs. These simple structures encapsulate reasoning by primitive and noetherian induction principles, and can be composed together to form complex recursion schemes for programs operating over a wide class of data and co-data types. At its heart, this study is guided by duality: each structure for recursion has a dual form, giving perfectly symmetric pairs of equal and opposite data and co-data types for representing recursion in programs. Duality is brought out through a framework presented in sequent style, which inherently includes control effects that are interpreted logically as classical reasoning principles. To accommodate the presence of effects, we give a calculus parameterized by a notion of strategy, which is strongly normalizing for a wide range of strategies. We also present a more traditional calculus for representing effect-free functional programs, but at the cost of losing some of the founding dualities.