* Re: GADT examples: composable functions list (Was: Re: [Caml-list] Wanted: GADT examples: string length, counting module x)
2012-03-22 9:46 ` Daniel Bünzli
2012-03-22 10:54 ` Roberto Di Cosmo
@ 2012-03-22 14:31 ` Markus Mottl
1 sibling, 0 replies; 4+ messages in thread
From: Markus Mottl @ 2012-03-22 14:31 UTC (permalink / raw)
To: Daniel Bünzli; +Cc: Roberto Di Cosmo, caml-list
On Thu, Mar 22, 2012 at 05:46, Daniel Bünzli
<daniel.buenzli@erratique.ch> wrote:
> You don't need to cheat the type system with Obj without GADT.
>
> http://caml.inria.fr/pub/ml-archives/caml-list/2004/01/52732867110697f55650778d883ae5e9.en.html
>
> Not to say that it's not involved, but it's possible.
Indeed, this encoding is not the easiest to reason about. The pipe
operator mentioned earlier is essentially arrow composition. I once
played around with this encoding and arrows and though it's kind of
fun, I'm not sure I'd want to write significant amounts of code that
way.
The code is below if anybody is curious. The "SimpleDataContArrow"
demonstrates how it can be done. It also uses continuation passing
style for "chasing arrows" (i.e. running computations) to prevent
stack overflows.
--------- arrow.ml
let id x = x
(* Simple arrows with application *)
module type SIMPLE_ARROW = sig
type ('a, 'b) t (* Type of arrows *)
val arr : ('a -> 'b) -> ('a, 'b) t
(* [arr f] projects an OCaml-function to a morphism (arrow) in the category
of computations. *)
val (>>>) : ('a, 'b) t -> ('b, 'c) t -> ('a, 'c) t
(* [af >>> ag] composes the two computations [af] and [ag]. *)
val app : unit -> (('a, 'b) t * 'a, 'b) t
(* [app ()] @return an arrow that represents a computation which takes
another arrow and a value as argument and returns the result of applying
the latter to the former. *)
val run : ('a, 'b) t -> 'a -> 'b
(* [run af x] runs the computation represented by arrow [af] on input [x]. *)
end
(* Implementation of simple arrows using plain functions *)
module SimpleArrow : SIMPLE_ARROW = struct
type ('a, 'b) t = 'a -> 'b
let arr f = f
let (>>>) f g x = g (f x)
let app () (f, x) = f x
let run = arr
end
(* Implementation of simple arrows using continuations.
Does not blow stack with deeply nested arrows! *)
module SimpleContArrow : SIMPLE_ARROW = struct
type ('a, 'b) t = { f : 'z. 'a -> ('b -> 'z) -> 'z }
let arr f = { f = fun x cont -> cont (f x) }
let (>>>) af ag = { f = fun x cont -> af.f x (fun yf -> ag.f yf cont) }
let app () = { f = fun (af, x) -> af.f x }
let run af x = af.f x id
end
(* Helper signature required for recursive module below *)
module type DATA_ARROW = sig
include SIMPLE_ARROW
val run_cont : ('a, 'b) t -> 'a -> cont : ('b -> 'c) -> 'c
end
(* Implementation of simple arrows using continuations and representing
them as variants *)
module rec SimpleDataContArrow : DATA_ARROW = struct
type ('a, 'b) t =
| Arr of ('a -> 'b)
| Comp of ('a, 'b) comp
| App of ('a, 'b) app
and ('a, 'b) comp =
{
comp_open : 'z. ('a, 'b, 'z) comp_scope -> 'z
}
and ('a, 'b, 'z) comp_scope =
{
comp_bind : 'c. ('a, 'c) t -> ('c, 'b) t -> 'z
}
and ('a, 'b) app =
{
app_open : 'z. ('a, 'b, 'z) app_scope -> 'z
}
and ('a, 'b, 'z) app_scope =
{
app_bind :
'c. ('a -> ('c, 'b) t * 'c) -> (('c, 'b) t * 'c, 'b) t -> 'z
}
let arr f = Arr f
let (>>>) af ag = Comp { comp_open = fun scope -> scope.comp_bind af ag }
let rec run_cont a x ~cont =
match a with
| Arr f -> cont (f x)
| Comp comp ->
comp.comp_open
{
comp_bind = fun af ag ->
SimpleDataContArrow.run_cont af x
~cont:(SimpleDataContArrow.run_cont ag ~cont)
}
| App app ->
app.app_open
{
app_bind = fun unpack af ->
SimpleDataContArrow.run_cont af (unpack x) ~cont
}
let app () =
App
{
app_open = fun scope ->
let f (af, x) = SimpleDataContArrow.run_cont af x ~cont:id in
scope.app_bind id (Arr f)
}
let run a x = run_cont a x ~cont:id
end
(* Fully-featured arrows with many more operators *)
module type ARROW = sig
include SIMPLE_ARROW
val first : ('a, 'b) t -> ('a * 'c, 'b * 'c) t
(* [first af] takes a computation [af] accepting argument [a].
@return a computation, which takes a pair [(a, c)], and returns the pair
[(b, c)], where [b] is the result of running computation [ag] on [a],
and [c] is a passed-through variable. *)
val second : ('a, 'b) t -> ('c * 'a, 'c * 'b) t
(* [second af] is a dual of [first], and passes the constant variable
as first argument. *)
val ( *** ) : ('a, 'b) t -> ('c, 'd) t -> ('a * 'c, 'b * 'd) t
(* [af *** ag] @return computation that performs computation [af] and
[ag] on the first and respectively second argument of the input pair,
returning the two results as a pair. *)
val (&&&) : ('a, 'b) t -> ('a, 'c) t -> ('a, 'b * 'c) t
(* [af &&& ag] @return computation that passes its input to two
computations [af] and [ag] and returns the pair of the results. *)
val liftA2 : ('a -> 'b -> 'c) -> ('d, 'a) t -> ('d, 'b) t -> ('d, 'c) t
(* [liftA2 f af ag] @return computation that applies the function [f]
to the results of [af] and [ag], which both receive the input. *)
type ('a, 'b) either = Left of 'a | Right of 'b
val left : ('a, 'b) t -> (('a, 'c) either, ('b, 'c) either) t
(* [left af] @return computation that applies computation [af] to
[l] if the input is [Left l], returning [Left result] and otherwise
passes [Right r] through unchanged. *)
val right : ('a, 'b) t -> (('c, 'a) either, ('c, 'b) either) t
(* [right af] is the dual of [left]. *)
val (<+>) : ('a, 'c) t -> ('b, 'd) t -> (('a, 'b) either, ('c, 'd) either) t
(* [af <+> ag] @return a computation that either performs [af] or
[ag] depending on its input, returning either [Left res_af] or
[Right res_ag] respectively. *)
val (|||) : ('a, 'c) t -> ('b, 'c) t -> (('a, 'b) either, 'c) t
(* [af ||| ag] @return a computation that either performs [af] or [ag]
depending on input. *)
val test : ('a, bool) t -> ('a, ('a, 'a) either) t
(* [test acond] @return a computation that tests its input with [acond]
and returns either [Left res] if the predicate is true or [Right res]
otherwise. *)
end
(* Functor from simple arrows with "apply" to fully-featured arrows *)
module MkArrow (SA : SIMPLE_ARROW)
: ARROW with type ('a, 'b) t = ('a, 'b) SA.t =
struct
include SA
let swap (x, y) = y, x
let first af = arr (fun (a, c) -> af >>> arr (fun b -> b, c), a) >>> app ()
let second af = arr swap >>> first af >>> arr swap
let ( *** ) af ag = first af >>> second ag
let ( &&& ) af ag = arr (fun x -> x, x) >>> af *** ag
let liftA2 f af ag = af &&& ag >>> arr (fun (b, c) -> f b c)
type ('a, 'b) either = Left of 'a | Right of 'b
let left af =
arr (function
| Left l -> arr (fun () -> l) >>> af >>> arr (fun x -> Left x), ()
| Right _ as right -> arr (fun () -> right), ())
>>> app ()
let mirror = function Left x -> Right x | Right x -> Left x
let right af = arr mirror >>> left af >>> arr mirror
let (<+>) af ag = left af >>> right ag
let (|||) af ag = af <+> ag >>> arr (function Left x | Right x -> x)
let test acond =
acond &&& arr id >>> arr (fun (b, x) -> if b then Left x else Right x)
end
(* Instantiations of fully-featured arrows *)
module Arrow = MkArrow (SimpleArrow)
module ContArrow = MkArrow (SimpleContArrow)
module DataContArrow = MkArrow (SimpleDataContArrow)
(* Some silly fun with Kleisli categories *)
(* Monad specification *)
module type MONAD = sig
type 'a t
val return : 'a -> 'a t
val (>>=) : 'a t -> ('a -> 'b t) -> 'b t
val run : 'a t -> 'a
end
(* Functor from arrow with apply operator to monad *)
module MkArrowMonad (SA : SIMPLE_ARROW)
: MONAD with type 'a t = (unit, 'a) SA.t =
struct
open SA
type 'a t = (unit, 'a) SA.t
let return x = arr (fun () -> x)
let (>>=) af g = af >>> arr (fun x -> g x, ()) >>> app ()
let run af = SA.run af ()
end
(* Functor from monads to their Kleisli category *)
module MkKleisli (M : MONAD) : ARROW with type ('a, 'b) t = 'a -> 'b M.t =
MkArrow (struct
open M
type ('a, 'b) t = 'a -> 'b M.t
let arr f x = return (f x)
let (>>>) f g x = f x >>= g
let app () (f, x) = f x
let run f x = M.run (f x)
end)
(* Example instantiation of a monad made from DataContArrows *)
module MonadFromArrow = MkArrowMonad (DataContArrow)
(* And now going back from monads to arrows *)
module ArrowFromMonad = MkKleisli (MonadFromArrow)
(* Some test code *)
open DataContArrow
let bump = arr succ
let bump_n_times n =
let rec loop n arrow =
if n <= 0 then arrow
else loop (n - 1) (arrow >>> bump)
in
loop n (arr id)
let () =
let n = 424242 in
let arrow = bump_n_times n in
let result = run arrow 0 in
Printf.printf "%d\n" result
---------
Regards,
Markus
--
Markus Mottl http://www.ocaml.info markus.mottl@gmail.com
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