Recursion schemes are functions that automate the traversal of nested data structures. With them, recursive code can be written in a way that is much simpler and easier to understand. The best introduction in the world to recursion schemes is written by Patrick Thomson on Sum Type Of Way, and the documentation for the Haskell recursion-schemes library is a nice complement. (But in case you don’t like Haskell, I’ll briefly go over them as I implement them.)
Recursion schemes are popular in Haskell, but I think they actually make even more sense for OCaml. That’s because they abstract away the actual process of recursing through a data structure. That means they can be implemented in such a way that all operations are tail-recursive and therefore stack-safe, which is otherwise a pain to do by hand.
So, let’s get into it.
Opening Up Recursive Types
The basic operation we want to do is this transformation:
type List('a) = Nil | Cons('a, List('a)) ↓ type List('a, 'f) = Nil | Cons('a, 'f)
This turns a recursive type to a type that supports “open recursion”. If you have rectypes enabled, you can actually recover the original type using
List('a, 'f) as 'f.
(In case that’s confusing, here’s a hint: it’s kind of like saying
'f = List('a, 'f). Substituting
'f on the right hand side, you get
'f = List('a, List('a, 'f)). Repeat and you get
'f = List('a, List('a, List('a, List('a, ...)))) to infinity.)