FIx data type in OCaml

Question

How can the following data type from Haskell be expressed in OCaml or SML?

newtype Fix f = In (f (Fix f))

Answer 1

I already answered this question on the mailing-list (and I must say I am mildly displeased that you ask the question in two different places without a good couple of day of delay, because of the duplication of efforts it could provoke), but let's reproduce it here.

There is a difficulty here because OCaml doesn't support higher-ranked type variables. In this declaration, f is not a "type", but a "type operator" (kind * -> *). To do the same in OCaml, you can use a functor (not a Haskell functor; in OCaml the word "functor" denotes a higher-order module that may depend on other modules/functors); functors are higher-kinded.

module type ParamType = sig
  type ('a, 'b) t
end

module Fix (M : ParamType) = struct
  type 'b fix = In of ('b fix, 'b) M.t
end

module List = struct
  module Param = struct
    type ('a, 'b) t = Nil | Cons of 'b * 'a
  end
  include Fix(Param)
end

open List.Param
open List

let rec to_usual_list =
  function
  | In Nil -> []
  | In (Cons (x, xs)) -> x :: to_usual_list xs

The good news is that OCaml also supports equi-recursive rather than iso-recursive types, which allows you to remove the "In" wrapper at each recursion layer. For that you must compile the incumbing module (and all the modules that also see this equirecursion through an interface) with the "-rectypes" option. Then you can write:

module type ParamType = sig
  type ('a, 'b) t
end

module EqFix (M : ParamType) = struct
  type 'b fix = ('b fix, 'b) M.t
end

module EqList = struct
  module Param = struct
    type ('a, 'b) t = Nil | Cons of 'b * 'a
  end
  include EqFix(Param)
end

open EqList.Param

let rec to_usual_list =
  function
  | Nil -> []
  | (Cons (x, xs)) -> x :: to_usual_list xs

The syntax of modules is quite heavy and could appear frightening. If you insist you can use first-class modules to move some of these uses from functors to simple functions. I choose to begin with the "simple" way to do it first.

Higher-kinded variables envy is probably the most severe illness about OCaml type worshippers (or Haskellers that for some (good!) reason come to wander in these parts of Functional County). In practice we do without it with not too much problems, but heavy use of monad transformers would be complicated indeed by this functor step, which is one of the reason it's not a very popular style around here. You may also distract yourself by thinking about the imperfections of higher-kinded variables in the languages that do support them; the limitation to constructor polymorphism rather than arbitrary type-level functions make them less expressive than you would like. The day we work out the details of the absolutely perfect higher-order type abstraction, maybe OCaml will jump to it?

Answer 2

I don't think OCaml lets you abstract over type constructors. For particular applications of Fix you can get a similar effect using -rectypes, I think.

$ ghci
GHCi, version 7.4.2: http://www.haskell.org/ghc/  :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Prelude> newtype Fix f = In (f (Fix f))
Prelude> type L = Fix []
Prelude> let w = In [] :: L
Prelude> let x = In [x] :: L

$ ocaml -rectypes
        OCaml version 4.00.0

# type l = l list;;
type l = l list
# ([] : l);;
- : l = []
# let rec x = [x];;
val x : 'a list as 'a = [[[...]]]
# (x : l);;
- : l = [[[...]]]

I am not a module type expert. Probably there's a way to use modules to get closer than this. Everything seems possible using the module system.