What's a usage pattern for existential type variables?

Online, (Scala/Java) and Haskell-specific tutorials on existential types seem to be dominant. Because I wouldn’t get stuck too much on notation, I went with Haskell tutorials, and they all seem to imply that existentials that are unconstrained are useless.

The usual example for existentials is information (but not trait) hiding, in which for example you can have a definition like so (with some syntax parallels for those unfamiliar with haskell):

{-
ocaml: `type gadt = Constr : typexpr`
haskell: `data Gadt where Constr :: typexpr`
-}

data Hidden where
  Hide :: Show a => a -> Hidden

{-
ocaml: iter (fun (Hide x) -> ...) [Hide 1; Hide ...]
-}

main :: IO ()
main =
  mapM_ (\(Hide x) -> print x) [Hide 1, Hide "a", Hide 4.2]

The problem with such implication is that I can’t find a usage for an existential introduced by an OCaml GADT because I don’t know how I’d express the Show a => a constraint.

type hidden = Hide : 'a -> hidden

Would simply completely cloak 'a, and any attempt at interacting with it would produce a type scope escaping error.

That considered, since Show a => a is best emulated now with passing a first-class module, that was the vector I was walking in with my existentials experiments, but I have a feeling there’s better, more valid, usage of those type variables that isn’t reliant on constraining.

So my question is, have you ever come across a pattern where an existential type variable is the solution?

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Your showable example can be written as:

type showable = Show: 'a * ('a -> string) -> showable

And there is fact many similar way to constraint existential types, either using type constructors:
if you have an 'a. 'a t, you may still be able to write useful functions, or coupling various producers or consumers of the existential types.

The showable example is a little artificial here but that are many examples where existential types are useful for:

  • representing internal types.
  • hiding some type information at an abstraction barrier.

I can offer one examples that combines the two to implement a zipper through mutually recursive trees.
Starting from a simplified data structure

type a_tree =
  | Leaf of int
  | Node_A of b_tree * b_tree
and b_tree =
  | Leaf of float
  | Node_B of a_tree * a_tree

Then zipper over a binary tree would be simply:

type dir = Left | Right
type zipper = { value: tree; path: (dir * tree) list }

but here our problem is that we have alternatively a_tree and b_tree. We can use existential types to keep a similar representation as the simpler zipper described above. The first step is to describe a step in the data structure as

type ('a,'b) step =
  | A: dir * b_tree -> (b_tree,a_tree) step
  | B: dir * a_tree -> (a_tree,b_tree) step

Then we can connect the step using an existential types to describe that if we have a path from
nodes of type top to a node of type bottom, we don’t necessarily know the intermediary types

type ('bottom,'top) path =
  | []: ('top,'top) path
  | (::):
  ('bottom , 'previous_bottom) step * ('previous_bottom,'top) path ->
    ('bottom,'top) path

Then a zipper is simply:

type ('bottom,'top) zipper = { value: 'bottom; path:('bottom,'top) path }

However, sometime we don’t care if we don’t know where we are in the tree, thus we may need to erase the bottom information from the zipper:

type 'top any_zipper = Any: ('bottom, 'top) zipper -> 'top any_zipper

and we can still use this any_zipper type to write an up function for instance:

let up (type top) (Any z: top any_zipper) =
  match z.path with
  | [] -> Any z
  | A (Left,r) :: rest ->
    Any { path=rest; value = Node_A(z.value,r) }
  | B (Left,r) :: rest ->
    Any { path=rest; value = Node_B(z.value,r) }
  | A (Right,l) :: rest ->
    Any { path=rest; value = Node_A(l, z.value) }
  | B (Right,l) :: rest ->
    Any { path=rest; value = Node_B(l, z.value) }
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thanks for the detailed example :sparkles:
gonna take my time to process it and wire the types together

similar ways as in we put the type of the consumer right in the constructor, like you did with showable?

like in a record?

Any time you have useful functions of type 'a. 'a t -> .., you can quantify existentially that type parameter and still retain some functionality. For instance, I can compute a length for an any_list:

type any_list = Any_list : 'a list -> any_list
let len (Any_list l) = List.length l

then there is also the possibility to use type witnesses to recover the quantified type at a later date:

 type 'a num_ty =
| Int: int num
| Float: float num
type num = Num: 'a num_ty * 'a -> num
type num_array = Narray: 'a num_ty * 'a array -> num_array
let sum (Narray (witness, array)) = match witness with
| Int -> Num(Int, Array.fold_left (+) 0 array)
| Float -> Num(Float, Array.fold_left (+.) 0. array)
1 Like

@octachron’s examples are very good. For variety, here is a small example of a slightly different flavor that we use in ocamlformat to wrap Cmdliner: (from here)

include Cmdliner

(** existential package of a Term and a setter for a ref to receive the
    parsed value *)
type arg = Arg : 'a Term.t * ('a -> unit) -> arg

(** convert a list of arg packages to a term for the tuple of all the arg
    terms, and apply it to a function that sets all the receiver refs *)
let tuple args =
  let pair (Arg (trm_x, set_x)) (Arg (trm_y, set_y)) =
    let trm_xy = Term.(const (fun a b -> (a, b)) $ trm_x $ trm_y) in
    let set_xy (a, b) = set_x a ; set_y b in
    Arg (trm_xy, set_xy)
  in
  let init = Arg (Term.const (), fun () -> ()) in
  let (Arg (trm, set)) = Base.List.fold_right ~f:pair args ~init in
  Term.app (Term.const set) trm

let args : arg list ref = ref []

let mk ~default arg =
  let var = ref default in
  let set x = var := x in
  args := Arg (arg, set) :: !args ;
  var

let parse info validate =
  Term.eval (Term.(ret (const validate $ tuple !args)), info)

The idea here is similar to an “any list”, where an existential around a pair of an 'a Cmdliner.Term.t representing a command-line argument yielding an 'a value, and a function of type 'a -> unit to set a ref to the parsed value. Then a list of such packages can be built using mk and then converted with tuple to a single Cmdliner.Term for the list as a tuple.

I don’t think this example needs anything more from the type system, but is an almost minimal example of packaging a value with the interesting operations over it into an existentially-typed value.

3 Likes

I have made my own futures library and one of the features is futures are connected to each other creating a dependency tree. This is done so that you can do Future.abort fut and it will abort that future and everything connected to it. I accomplish this by each future storing a list of those futures that depend on it. But those futures do not necessarily have the same type. So I hide the type in an existential so I can still perform actions on the future that do not depend on the type.

Note, I stole this from @dbuenzli 's futures library which I used to start mine.

Here is the code from my library:

  (* The concrete type of a future.  Future's are mutable, but once they are
     determined they become immutable.  A future has a state which starts as
     undetermined, can become determined or an alias.  An alias is a future that
     needs to exist because some unknown computation will eventually become its
     value, and once that computation is found out, we set the future to an
     alias to that future. *)
  type 'a u = { mutable state : 'a state }

  and 'a state =
    [ 'a Abb_intf.Future.Set.t
    | `Undet of 'a undet
    | `Alias of 'a u
    ]

  (* An undetermined has an optional function, which is some work to be
     executed, watchers are executed when this undetermined future becomes
     determined, deps are futures that are not required to be executed before
     this future is determined but in some meaningful way connected to it, the
     abort function is what to do if this future is aborted, and finally num_ops
     is how many operations have been performed on this future.  The definition
     of an "operation" is kind of vague but basically it corresponds to mutating
     this undetermined future in some way. *)
  and 'a undet = {
    mutable f : (State.t -> State.t) option;
    mutable watchers : 'a Watcher.t list;
    mutable deps : dep list;
    abort : abort;
    mutable num_ops : int;
  }

  (* A dependency can be any future and it will not have the same type as this
     future, so we have to hide the actual type in an existential so we can
     reference any future as a dependency. *)
  and dep = Dep : 'a u -> dep
1 Like