How to Take the Inverse of a Type

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 this 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. Our adventure begins with an exploration of the properties and applications of this construction, visiting various topics from the literature including program calculation, Laurent polynomials, and derivatives of data types. Examples are given throughout using Haskell’s linear types extension to demonstrate the ideas. We then step through the looking glass to discover what might be possible in richer settings; the functional language Granule offers linear functions that incorporate local side effects, which allow us to demonstrate further algebraic structure. Lastly, we discuss whether dualities in linear logic might permit the related notion of an additive inverse.