How to Take the Inverse of a Type (Artifact)

Abstract

In functional programming, regular types are a subset of algebraic data types formed from products and sums with their respective units. One can view regular types as forming a commutative semiring but where the usual axioms are isomorphisms rather than equalities. In the pearl, we show that regular types in a linear setting permit a useful notion of multiplicative inverse, allowing us to ‘divide’ one type by another. We begin with an exploration of the properties and applications of this construction, and this artifact includes examples in Haskell demonstrating its relevance to various topics from the literature including program calculation, Laurent polynomials, and derivatives of data types. Some examples from the paper require a richer setting than Haskell can offer; the artifact replays the first set of examples in Granule, while also presenting additional examples demonstrating further algebraic structure through linear functions that incorporate local side effects.