[Caml-list] Encoding existential types without modules
Daniel Bünzli
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
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Daniel Bünzli
Besides, I would just like to point out that the way I solved the
problem of polymorphic recursion for the apply function in the example
of the list of composable function cannot be applied in general.
A general way to solve the problem of polymorphic recursion was given
by Brian Rogoff in a previous post [1]. Applying this solution gives
the following (much simpler and maybe more efficient) code.
Daniel
[1] http://caml.inria.fr/archives/200208/msg00334.html
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
(* List of composable functions.
The intended 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, 'z) list_scope = { bind_list : 'c. ('a -> 'c) * ('c, 'b) t
-> '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)
(* Type to handle the polymorphic recursion of the apply function *)
type poly_rec = { apply : 'a 'b. poly_rec -> ('a, 'b) t -> 'a -> 'b }
let apply' r l x = match l with
| Nil id -> id x
| Cons l ->
with_packed_list l { bind_list = function h,t -> r.apply r t (h x) }
let poly_rec = { apply = apply' }
let apply l x = apply' poly_rec l x
bro...@speakeasy.net
> Thanks for your feedback.
>
> Besides, I would just like to point out that the way I solved the
> problem of polymorphic recursion for the apply function in the example
> of the list of composable function cannot be applied in general.
>
> A general way to solve the problem of polymorphic recursion was given
> by Brian Rogoff in a previous post [1]. Applying this solution gives
> the following (much simpler and maybe more efficient) code.
I blame Jacques Garrigue for that trick! He is entitled to transfer the bla=
me
to whomever he chooses. ;-)
You may also want to consider using the recursive module feature introduced=
in
OCaml 3.07 for this too. Unfortunately, you can't (yet, Xavier says he's
working on it) apply it directly to Funlist on account of your nil. That w=
ould
be best, but by wrapping apply I think you can do it like so:
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
(* List of composable functions.
The intended 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, 'z) list_scope = { bind_list : 'c. ('a -> 'c) * ('c, 'b) =
t
-> '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)
module rec PolyRec : sig val apply : ('a, 'b) t -> 'a -> 'b end =
struct
let apply l x =
match l with
| Nil id -> id x
| Cons l ->
with_packed_list
l { bind_list = function h,t -> PolyRec.apply t (h x) }
end
let apply = PolyRec.apply
end
-- Brian
Jean-Christophe Filliatre
Daniel Bünzli writes:
>
> I think this may be usefull to others (e.g. to port some clever haskel=
l
> code). Below, I give two examples that show how to encode existential
l
> the encoding given by Pierce in [1]. It uses polymorphic record fields.
How tricky! Thanks for this post.
> 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.
--
Jean-Christophe
Daniel Bünzli writes:
> Hello,
>
> I think this may be usefull to others (e.g. to port some clever haskel=
l
> code). Below, I give two examples that show how to encode existential
> types in ocaml without using modules. This is done by adapting to ocam=
l
> 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_fl=
> and ('a, 'b, 'z) list_scope = { bind_list : 'c. ('a, 'b, 'c) list_da=
ta
> -> '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 polymorphi=
c
> 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.con=
s
> double Funlist.nil))
> let a, b = Funlist.apply list 2
>
>
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inria.fr
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