Containers, monads and induction recursion

Abstract

Induction recursion offers the possibility of a clean, simple and yet powerful meta-language for the type system of a dependently typed programming language. At its crux, induction recursion allows us to define a universe, that is a set U of codes and a decoding function T : U → D which assigns to every code u : U, a value T, u of some type D, e.g. the large type Set of small types or sets. The name induction recursion refers to the build-up of codes in U using inductive clauses, simultaneously with the definition of the function T, by structural recursion on codes. Our contribution is to (i) bring out explicitly algebraic structure which is less visible in the original type-theoretic presentation – in particular showing how containers and monads play a pivotal role within induction recursion; and (ii) use these structures to present a clean and high level definition of induction recursion suitable for use in functional programming

Turing-Completeness Totally Free

“…Now, as Ghani and Hancock observe, IR I is a (relative) monad [13]. 5 Indeed, it is the free monad generated by σ and δ.…”

A Finite Axiomatisation of Inductive-Inductive Definitions

“…This is where we take most of our inspiration from. Recently, Ghani and Hancock [18] have shed new light on this construction. The idea of a universe of data types is also present in Epigram 2 [9], and has previously been used by Altenkirch, Ghani, Morris and McBride to study strictly positive types [23] and strictly positive families [22] (see also Morris' thesis [21]).…”