Induction, Coinduction, and Fixed Points in PL Type Theory

Abstract: Recently we presented a concise survey of the formulation of the induction and coinduction principles, and some concepts related to them, in programming languages type theory and four other mathematical disciplines. The presentation in type theory involved the separate formulation of these concepts, first, in the theory of types of functional programming languages and, next, in the theory of types of object-oriented programming languages. In this article we show that separating these two formulations helps demonstrate some of the fundamental differences between structural subtyping, predominant in functional programming languages, and nominal subtyping, predominant in object-oriented programming languages---including differences concerning type negation and concerning the existence of inductive types, of coinductive types, and of approximations thereof. In the article we also motivate mutual coinduction and mutual coinductive types, and their approximations, and we discuss in brief the potential relevance of these concepts to object-oriented programming (OOP) type theory.