All geek questions in one place

How to implement "efficient generalized fold" in F #?

In a paper by Martin et al., I read about efficient generalized folds for nestet data types. The document talks about Haskell, and I want to try it in F #.

So far, I have been able to follow the example Nest, including the implementation gfold.

type Pair<'a> = 'a * 'a
type Nest<'a> = Nil | Cons of 'a * Nest<Pair<'a>>

let example =
    Cons(1,
        Cons((2, 3),
            Cons(((4, 5), (6, 7)),
                Nil
            )
        )
    )

let pair (f:'a -> 'b) ((a, b):Pair<'a>) : Pair<'b> = f a, f b

let rec nest<'a, 'r> (f:'a -> 'r) : Nest<'a> -> Nest<'r> = function
    | Nil -> Nil
    | Cons(x, xs) -> Cons(f x, nest (pair f) xs)

//val gfold : e:'r -> f:('a * 'r -> 'r) -> g:(Pair<'a> -> 'a) -> _arg1:Nest<'a> -> 'r
let rec gfold e f g : Nest<'a> -> 'r = function
    | Nil -> e
    | Cons(x, xs) ->
        f(x, gfold e f g (nest g xs))

let uncurry f (a, b) = f a b

let up = uncurry (+)

let sum = example |> gfold 0 up up

Unfortunately, gfoldit seems to have quadratic complexity and therefore the authors came up with efold. As you probably guessed, I could not work. After messing around with many type annotations, I came up with this version, which has only a slight deviation left:

let rec efold<'a, 'b, 'r> (e:'r) (f:'a * 'r -> 'r) (g:(Pair<'a> -> Pair<'a>) -> 'a -> 'a) (h:_) (nest:Nest<'a>) : 'r =
    match nest with
    | Nil -> e
    | Cons(x, xs) -> f(h x, efold e f g ((g << pair) h) xs)
                                                        ^^

The only remaining unspecified type is h. The compiler reports val h : ('a -> 'a), but I think there should be different types.

.      < 'a >
    Nest < Pair < ' →
'' a '' Pair < 'a > '

h . , Haskell F #.

. .

: , kvb:

So h , , . g , f . , e .

h , . g, , , h .

f , , , , h . , . , .. .

?

+4
1

efold Haskell - - :

efold :: forall n m b.
    (forall a. n a)->
    (forall a.(m a, n (Pair a)) -> n a)->
    (forall a.Pair (m a) -> m (Pair a))->
    (forall a.(a -> m b) -> Nest a -> n b) 
efold e f g h Nil = e 
efold e f g h (Cons (x,xs)) = f (h x, efold e f g (g . pair h) xs

F # , n m " " - , , F # ( .NET).

, . , , , - , , . - :

efold e f g h example ≡
    f (h 1, f ((g << pair h) (2, 3), f ((g << pair (g << pair h)) ((4,5), (6,7)), e)))

, h , f   . g h ( h a -> m b Pair a -> m (Pair b) Pair (Pair a) -> m (Pair (Pair b)) ..) f , h f. , e , f.

, , . f, , . g, , (, , , g, fun (a,b) -> b,a).

- efold n, m, . , , Nest, n _ m _ int. , n _ m _ :

let rec efold<'n,'m,'a> (e:'n) (f:'m*'n->'n) (g:Pair<'m> -> 'm) (h:'a->'m) : Nest<'a> -> 'n = function
| Nil -> e
| Cons(x,xs) -> f (h x, efold e f g (g << (pair h)) xs)

let total = efold 0 up up id example

, n m , (, , , F # - ). , , n 'a= list<'a> m 'b= 'b. e , forall 'a.list<'a> [], :

type ListIdF =
    abstract Apply : 'a * list<Pair<'a>> -> list<'a>

type ListIdG =
    abstract Apply : Pair<'a> -> Pair<'a>

let rec efold<'a,'b> (f:ListIdF) (g:ListIdG) (h:'a -> 'b) : Nest<'a> -> list<'b> = function
| Nil -> []
| Cons(x,xs) -> f.Apply(h x, efold f g (pair h >> g.Apply) xs)

let toList n = efold { new ListIdF with member __.Apply(a,l) = a::(List.collect (fun (x,y) -> [x;y]) l) } { new ListIdG with member __.Apply(p) = p } id n

F # , , . , Higher. .

App<'T,'a>, - T<'a>, , App<_,_>:

type App<'F, 'T>(token : 'F, value : obj) = 
    do
        if obj.ReferenceEquals(token, Unchecked.defaultof<'F>) then
            raise <| new System.InvalidOperationException("Invalid token")

    // Apply the secret token to have access to the encapsulated value
    member self.Apply(token' : 'F) : obj =
        if not (obj.ReferenceEquals(token, token')) then
            raise <| new System.InvalidOperationException("Invalid token")
        value

, ( ):

// App<Const<'a>, 'b> represents a value of type 'a (that is, ignores 'b)
type Const<'a> private () =
    static let token = Const ()
    static member Inj (value : 'a) =
        App<Const<'a>, 'b>(token, value)
    static member Prj (app : App<Const<'a>, 'b>) : 'a =
        app.Apply(token) :?> _

// App<List, 'a> represents list<'a>
type List private () = 
    static let token = List()
    static member Inj (value : 'a list) =
        App<List, 'a>(token, value)
    static member Prj (app : App<List, 'a>) : 'a list =
        app.Apply(token) :?> _

// App<Id, 'a> represents just a plain 'a
type Id private () =
    static let token = Id()
    static member Inj (value : 'a) =
        App<Id, 'a>(token, value)
    static member Prj (app : App<Id, 'a>) : 'a =
        app.Apply(token) :?> _

// App<Nest, 'a> represents a Nest<'a>
type Nest private () =
    static let token = Nest()
    static member Inj (value : Nest<'a>) =
        App<Nest, 'a>(token, value)
    static member Prj (app : App<Nest, 'a>) : Nest<'a> =
        app.Apply(token) :?> _

:

// forall a. n a
type E<'N> =
    abstract Apply<'a> : unit -> App<'N,'a>

// forall a.(m a, n (Pair a)) -> n a)
type F<'M,'N> =
    abstract Apply<'a> : App<'M,'a> * App<'N,'a*'a> -> App<'N,'a>

// forall a.Pair (m a) -> m (Pair a))
type G<'M> =
    abstract Apply<'a> : App<'M,'a> * App<'M,'a> -> App<'M,'a*'a>

:

let rec efold<'N,'M,'a,'b> (e:E<'N>) (f:F<'M,'N>) (g:G<'M>) (h:'a -> App<'M,'b>) : Nest<'a> -> App<'N,'b> = function
| Nil -> e.Apply()
| Cons(x,xs) -> f.Apply(h x, efold e f g (g.Apply << pair h) xs)

efold Inj Prj, , :

let toList n = 
    efold { new E<_> with member __.Apply() = List.Inj [] } 
          { new F<_,_> with member __.Apply(m,n) = Id.Prj m :: (n |> List.Prj |> List.collect (fun (x,y) -> [x;y])) |> List.Inj }
          { new G<_> with member __.Apply(m1,m2) = (Id.Prj m1, Id.Prj m2) |> Id.Inj }
          Id.Inj
          n
    |> List.Prj

let sumElements n =
    efold { new E<_> with member __.Apply() = Const.Inj 0 }
          { new F<_,_> with member __.Apply(m,n) = Const.Prj m + Const.Prj n |> Const.Inj }
          { new G<_> with member __.Apply(m1,m2) = Const.Prj m1 + Const.Prj m2 |> Const.Inj }
          Const.Inj
          n
    |> Const.Prj

let reverse n = 
    efold { new E<_> with member __.Apply() = Nest.Inj Nil }
          { new F<_,_> with member __.Apply(m,n) = Cons(Id.Prj m, Nest.Prj n) |> Nest.Inj }
          { new G<_> with member __.Apply(m1,m2) = (Id.Prj 2, Id.Prj m1) |> Id.Inj }
          Id.Inj
          n
    |> Nest.Prj

, , , , App<_,_>. inline ( ):

let inline (|Prj|) (app:App< ^T, 'a>) = (^T : (static member Prj : App< ^T, 'a> -> 'b) app)
let inline prj (Prj x) = x
let inline inj x = (^T : (static member Inj : 'b -> App< ^T, 'a>) x)

let toList n = 
    efold { new E<List> with member __.Apply() = inj [] } 
          { new F<Id,_> with member __.Apply(Prj m, Prj n) = m :: (n |> List.collect (fun (x,y) -> [x;y])) |> inj }
          { new G<_> with member __.Apply(Prj m1,Prj m2) = (m1, m2) |> inj }
          inj
          n
    |> prj

let sumElements n =
    efold { new E<Const<_>> with member __.Apply() = inj 0 }
          { new F<Const<_>,_> with member __.Apply(Prj m, Prj n) = m + n |> inj }
          { new G<_> with member __.Apply(Prj m1,Prj m2) = m1 + m2 |> inj }
          inj
          n
    |> prj

let reverse n = 
    efold { new E<_> with member __.Apply() = Nest.Inj Nil }
          { new F<Id,_> with member __.Apply(Prj m,Prj n) = Cons(m, n) |> inj }
          { new G<_> with member __.Apply(Prj m1,Prj m2) = (m2, m1) |> inj }
          inj
          n
    |> prj
+6

Source: https://habr.com/ru/post/1661080/

More articles:


All Articles