Re: GADT examples: composable functions list (Was: Re: [Caml-list] Wanted: GADT examples: string length, counting module x)
On Thu, Mar 22, 2012 at 05:46, Daniel Bünzli
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. *)
Re: GADT examples: composable functions list (Was: Re: [Caml-list] Wanted: GADT examples: string length, counting module x)
Hi Daniel, sure, never said you need, just that you can get away cheating the type system. But I need to show my students elegant and simple solutions: I want to get more OCaml users around, not scare them off :-) --Roberto On Thu, Mar 22, 2012 at 10:46:33AM +0100, Daniel Bünzli wrote: > > > Le jeudi, 22 mars 2012 à 10:28, Roberto Di Cosmo a écrit : > > > Without GADT > > > > > > One can get away cheating the type system and declaring the type > 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. > > Best, > > Daniel > > > -- --Roberto Di Cosmo -- Professeur En delegation a l'INRIA PPS E-mail: robe...@dicosmo.org Universite Paris Diderot WWW : http://www.dicosmo.org Case 7014Tel : ++33-(0)1-57 27 92 20 5, Rue Thomas Mann F-75205 Paris Cedex 13 Identica: http://identi.ca/rdicosmo FRANCE. Twitter: http://twitter.com/rdicosmo -- Attachments: MIME accepted, Word deprecated http://www.gnu.org/philosophy/no-word-attachments.html -- Office location: Bureau 6C08 (6th floor) 175, rue du Chevaleret, XIII Metro Chevaleret, ligne 6 -- -- Caml-list mailing list. Subscription management and archives: https://sympa-roc.inria.fr/wws/info/caml-list Beginner's list: http://groups.yahoo.com/group/ocaml_beginners Bug reports: http://caml.inria.fr/bin/caml-bugs
Re: GADT examples: composable functions list (Was: Re: [Caml-list] Wanted: GADT examples: string length, counting module x)
Le jeudi, 22 mars 2012 à 10:28, Roberto Di Cosmo a écrit : > Without GADT > > > One can get away cheating the type system and declaring the type 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. Best, Daniel -- Caml-list mailing list. Subscription management and archives: https://sympa-roc.inria.fr/wws/info/caml-list Beginner's list: http://groups.yahoo.com/group/ocaml_beginners Bug reports: http://caml.inria.fr/bin/caml-bugs
GADT examples: composable functions list (Was: Re: [Caml-list] Wanted: GADT examples: string length, counting module x)
GADT come in really handy is when you have data structures that need existential
type variables.
A nice example is the case of lists of composable functions: say you want to
build a list containing functions f_i : A_i -> A_{i+1}
Without GADT
One can get away cheating the type system and declaring the type
type ('a,'b) cfl = ('a -> 'b) list;;
which is really incorrect: 'a is the first input type, 'b is the last output
type, and that's ok, but it is really not true that the list will contain
functions of type 'a -> 'b ...
This shows up as soon as one tries to do something useful with this list, like
adding one element at the bebinning: to keep the type checker happy, we call
Obj.magic in for help
let add (f: 'a -> 'b) (fl : ('b,'c) cfl) : ('a,'c) cfl =
(Obj.magic f):: (Obj.magic fl);;
And you will need Obj.magic's help in writing map, fold, compute, whatever...
You may argue that if all the hectic primitives are well hidden behind a module
signature, and the module programmer is very smart, all will be well, but
that's ugly, isn't it?
Here is the elegant way of doing it using GADT
--
Declare the type cfl of a composable function list as follows
type ('a,'b) cfl =
Nilf: ('a,'a) cfl
|Consf: ('a -> 'b) * ('b,'c) cfl -> ('a,'c) cfl;;
Now you can write useful functions which are well typed
let rec compute : type a b. a -> (a,b) cfl -> b = fun x ->
function
| Nilf -> x (* here 'a = 'b *)
| Consf (f,rl) -> compute (f x) rl;;
Try it... it works!
let cl = Consf ((fun x -> Printf.sprintf "%d" x), Nilf);;
let cl' = Consf ((fun x -> truncate x), cl);;
compute 3.5 cl';;
Notice that the type of Consf contains a variable 'b which is
not used in the result type: one can check that
('a -> 'b) * ('b,'c) cfl -> ('a,'c) cfl
can be seen as
\forall 'a 'c. (\exists 'b.('a -> 'b) * ('b,'c) cfl) -> ('a,'c) cfl
so, when deconstructing a cfl, one gets of course a function and the
rest of the list, but now we know that their type is
\exists 'b.('a -> 'b) * ('b,'c) cfl
Well, isn't this a contrived example?
-
Actually, not at all... back in 1999, when developing a parallel
programming library named ocamlp3l, we implemented high-level
parallelism combinators that allowed to write expressions like this
(hey, isn't this map/reduce? well, yes... indeed that was an ooold idea)
(seq(intervals 10) ||| mapvector(seq(seq_integr f),5) |||
reducevector(seq(sum),2))
These combinators could be interpreted sequentially or graphically quite
easily, but turning them into a distributed program required a lot of
work, and the first step was to build an AST from these expressions:
here is a snippet of the actual type declaration from the old code in parp3l.ml
(* the type of the p3l cap *)
type ('a,'b) p3ltree = Farm of (('a,'b) p3ltree * int)
| Pipe of ('a,'b) p3ltree list
| Map of (('a,'b) p3ltree * int)
| Reduce of (('a,'b) p3ltree * int)
| Seq of ('a -> 'b)
;;
And here is one of the simplification steps we had to perform on the AST
let (|||) (t1 : ('a,'b) p3ltree) (t2 : ('b,'c) p3ltree) =
match ((Obj.magic t1 : ('a,'c) p3ltree), (Obj.magic t2 : ('a,'c) p3ltree))
with
(Pipe l1, Pipe l2) -> Pipe(l1 @ l2)
| (s1, Pipe l2) -> Pipe(s1 :: l2)
| (Pipe l1, s2) -> Pipe(l1 @ [s2])
| (s1, s2) -> Pipe [s1; s2];;
I am sure you see the analogy with the composable function list: a series
of functions in a paralle pipeline have exactly the same type structure.
With GADTs, onw can can finally write this 1999 code in a clean way in OCaml,
so many thanks to the OCaml team, and keep up the good work!
--Roberto
--
Professeur En delegation a l'INRIA
PPS E-mail: robe...@dicosmo.org
Universite Paris Diderot WWW : http://www.dicosmo.org
Case 7014Tel : ++33-(0)1-57 27 92 20
5, Rue Thomas Mann
F-75205 Paris Cedex 13 Identica: http://identi.ca/rdicosmo
FRANCE. Twitter: http://twitter.com/rdicosmo
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