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on 12/11/2009 9:05 AM

A noticeable omission in F# standard library is Seq.foldBack, or the famous Haskell foldr. The semantics of foldr is very simple to remember: it replaces the native cons and nil of a list with arbitrary computations:

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    foldr cons nil []     = nil
    foldr cons nil (x:xs) = cons x (foldr cons nil xs)

In particular, replacing the native cons and nil with themselves is always equivalent to the original list, e.g. forall x: foldr (:) [] x == x

Surprisingly, the above equation holds for infinite lists as well. This is something important to remember when porting these ideas to F#.

A naive F# translation would use this type:

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 foldBack : ('T1 -> 'T2 -> 'T2) -> 'T1 -> seq<'T1> -> 'T2

However, by being strict in the second argument, cons will now prematurely force the evaluation of infinite sequences.

Here is a more faithful translation using LazyList from the FSharp.PowerPack.dll:

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    /// Implements the lazy right-to-left fold.
    let foldBack (f: 'T1 -> Lazy<'T2> -> 'T2) (z: 'T2) (xs: seq<'T1>) : 'T2 =
        let rec foldr = function
            | LazyList.Nil         -> z
            | LazyList.Cons(x, xs) -> f x (lazy (foldr xs))
        foldr (LazyList.ofSeq xs)

Now let us test the code to make sure we have been faithful to Haskell in our translation:

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    Seq.FoldBack 
        (fun x xs -> LazyList.consDelayed x (fun () -> Lazy.force xs)) 
        (LazyList.empty ())
        (Seq.initInfinite (fun x -> x))
    |> LazyList.toSeq
    |> Seq.take 10
    |> Seq.toArray
    |> printfn "%A"
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