Encoding existential types without modules

I think this may be usefull to others (e.g. to port some clever haskell
code). Below, I give two examples that show how to encode existential
types in ocaml without using modules. This is done by adapting to ocaml
the encoding given by Pierce in [1]. It uses polymorphic record fields.

The examples are a little bit silly but their aim is to show the
concept of the encoding.
The first example is a counter abstract datatype. The second one is a
datatype that can hold a list of composable function, that is a type
that expresses something like

 type ('a, 'b) funlist = Nil of ('a ->'b) | Cons of exists 'c. ('a ->
'c) * ('c,'b) funlist

I usually need three types to encode an existential type. Does anybody
see a simpler way of doing that ?

Daniel

[1] Benjamin C. Pierce, Types and Programming Languages, section 24.3


--- Abstract counter datatype

(* The type expressed by the three types below is :
   type packed_counter =
      exists 'x. { create : 'x; get : ('x -> int); inc : ('x -> 'x)}
*)
type 'x counter = { create : 'x;  get : ('x -> int); inc : ('x -> 'x) }
type 't counter_scope = { bind_counter : 'x. 'x counter -> 't }
type packed_counter = { open_counter : 't. 't counter_scope -> 't }

(* Creating a package from a counter implementation *)
let pack_counter impl = { open_counter = fun scope ->
scope.bind_counter impl }

(* Using a package with a scoped expression *)
let with_packed_counter p e = p.open_counter e

(* Two different implementations of the counter *)
let int_impl = { create = 1 ; get = (function i -> i) ; inc = (fun i ->
i+1) }
let float_impl = { create = 1.0; get = (function i -> (int_of_float i))
;
                   inc = (fun i -> i +. 1.0) }

let counter = pack_counter int_impl
let counter' = pack_counter float_impl

(* An expression using an abstract counter *)
let expr =
  { bind_counter = fun counter -> (* counter is bound to the <<
interface >> *)
    (counter.get (counter.inc counter.create)) }

let result = with_packed_counter counter expr
let result' = with_packed_counter counter' expr

(*
   This doesn't type, the counter type is abstract !
   let expr =
   { bind_counter = fun counter ->
     (counter.get (counter.inc (counter.get counter.create))) }
*)

(*
   This doesn't type, the abstract type tries to escape its scope !
   let expr = { bind_counter = fun counter -> (counter.create) }
*)


--- Lists of composable functions.

module Funlist : sig

(* The funlist datatype *)
type ('a, 'b) t

(* Constructors *)
val nil : ('a, 'a) t
val cons : ('a -> 'b) -> ('b, 'c) t -> ('a, 'c) t

(* Applying a value to a composition *)
val apply : ('a, 'b) t -> 'a -> 'b

end = struct

(* The type expressed by the four types below is :
   type ('a, 'b) t = Nil of ('a -> 'b)
                   | Cons of exists 'c. ('a -> 'c) * ('c, 'b) t  *)

type ('a, 'b) t = Nil of ('a -> 'b) | Cons of ('a, 'b) packed_list
and ('a, 'b, 'c) list_data = ('a -> 'c) * ('c, 'b) t
and ('a, 'b, 'z) list_scope = { bind_list : 'c. ('a, 'b, 'c) list_data
-> 'z}
and ('a, 'b) packed_list = { open_list : 'z. ('a, 'b, 'z) list_scope ->
'z }

(* Packing and unpacking lists *)
let pack_list h t = { open_list = fun scope -> scope.bind_list (h,t) }
let with_packed_list p e = p.open_list e

(* Constructors *)
let nil = Nil(fun x -> x)
let cons h t = Cons(pack_list h t)


(* The following type is introduced to avoid the problem of polymorphic
recursion that comes while attempting a naive implementation of the
apply
funtion. The idea was taken from Laufer, Odersky, Polymorphic Type
Inference
and Abstract Data Types, 1994. *)

(* The type expressed by the three types below is :
    type 'b packed_apply = exists 'a. ('a, 'b) t * 'a
*)
type ('a, 'b) apply_data = ('a, 'b) t * 'a
type ('b, 'z) apply_scope = { bind_apply : 'a. ('a, 'b) apply_data ->
'z}
type 'b packed_apply = { open_apply : 'z. ('b, 'z) apply_scope -> 'z}

(* Packing and unpacking applications *)
let pack_apply l x = { open_apply = fun scope -> scope.bind_apply (l,x)
}
let with_packed_apply p e = p.open_apply e

let rec apply' a =
  with_packed_apply
    a { bind_apply = function
      | Nil id, x -> id x
      | Cons l, x ->
          with_packed_list
            l { bind_list = function h,t -> apply' (pack_apply t (h x))}}

let apply l x = apply' (pack_apply l x)

end

(* Example of use *)
let twice x = 2*x
let double x = (x, x)
let list = Funlist.cons twice (Funlist.cons (( = ) 4) (Funlist.cons
double Funlist.nil))
let a, b = Funlist.apply list 2
    

> I usually need three types to encode an existential type. Does anybody
> see a simpler way of doing that ?

I guess  you really need  these three types  (at least before  we have
rank-2 polymorphism  in ocaml). As a  attempt to justify  this, let us
try to make a more generic, functorized, code from your first example.
It takes  the first of your  three types as argument  ('a counter) and
returns the third one (packed_counter) as a result (so obviously these
two types you need).

The functor looks like

        module Make(X : sig type 'a t end) : sig
          type t (* the existential type exists 'a. 'a t *)
          val pack : 'a X.t -> t
          ...
        end = struct
          ...
        end

To be  able to use  the abstract  type t, we  would like to  provide a
function similar to your with_packed_counter, of type

        val use : ('a. 'a X.t -> 'b) -> t -> 'b

Since we do not have  rank-2 polymorphism, your solution introduces an
intermediate record type, as the type of the first argument.

Finally we get this:

        module Make(X : sig type 'a t end) : sig
          type t
          val pack : 'a X.t -> t 
          type 'a user = { f : 'b. 'b X.t -> 'a }
          val use : 'a user -> t -> 'a
        end = struct
          type 'a user = { f : 'b. 'b X.t -> 'a }
          type t = { pack : 'a. 'a user -> 'a }
          let pack impl = { pack = fun user -> user.f impl }
          let use f p = p.pack f
        end

and it seems  that we can't avoid the three  types solution. Note that
for other examples requiring the returned type to be polymorphic (e.g. 
the compositional list example) you need to write another functor.
    

Adding an ocaml interpreter to my C program