Foldl as foldr

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When you wonder whether to choose foldl or foldr you may remember,

that both

foldl

and

foldl'

can be expressed as

foldr

. (

foldr

may lean so far right it came back left again.)

It holds

foldl :: (a -> b -> a) -> a -> [b] -> a
foldl f a bs =
   foldr (\b g x -> g (f x b)) id bs a

(The converse is not true, since

foldr

may work on infinite lists, which

foldl

variants never can do. However, for finite lists,

foldr

can also be written in terms of

foldl

(although losing laziness in the process), in a similar way like this:

foldr :: (b -> a -> a) -> a -> [b] -> a
foldr f a bs =
   foldl (\g b x -> g (f b x)) id bs a

)

Now the question are:

How can someone find a convolved expression like this?
How can we benefit from this rewrite?

1 Folding by concatenating updates
2 foldl which may terminate early
3 Practical example: Parsing numbers using a bound
4 See also

1 Folding by concatenating updates

Instead of thinking in terms of

foldr

and a function

g

as argument to the accumulator function,

I find it easier to imagine a fold as a sequence of updates. An update is a function mapping from an old value to an updated new value.

newtype Update a = Update {evalUpdate :: a -> a}

We need a way to assemble several updates.

To this end we define a

Monoid

instance.

instance Monoid (Update a) where
   mempty = Update id
   mappend (Update x) (Update y) = Update (y.x)

Now left-folding is straight-forward.

foldlMonoid :: (a -> b -> a) -> a -> [b] -> a
foldlMonoid f a bs =
   flip evalUpdate a $
   mconcat $
   map (Update . flip f) bs

Now, where is the

foldr

? It is hidden in

mconcat

mconcat :: Monoid a => [a] -> a
mconcat = foldr mappend mempty

Since

mappend

must be associative (and is actually associative for our

Update

monoid),

mconcat

could also be written as

foldl

, but this is avoided, precisely

foldl

fails on infinite lists.

By the way:

Update a

is just

Dual (Endo a)

. If you use a

State

monad instead of a monoid, you obtain an alternative implementation of

mapAccumL

2 foldl which may terminate early

The answer to the second question is:

Using the

foldr

expression we can write variants of

foldl

that behave slightly different from the original one.

E.g. we can write a

foldl

that can stop before reaching the end of the input list

and thus may also terminate on infinite input.

The function

foldlMaybe

terminates with

Nothing

as result when it encounters a

Nothing

as interim accumulator result.

foldlMaybe :: (a -> b -> Maybe a) -> a -> [b] -> Maybe a
foldlMaybe f a bs =
   foldr (\b g x -> f x b >>= g) Just bs a

Maybe the monoidic version is easier to understand. The implementation of the fold is actually the same, we do only use a different monoid.

import Control.Monad ((>=>), )
 
newtype UpdateMaybe a = UpdateMaybe {evalUpdateMaybe :: a -> Maybe a}
 
instance Monoid (UpdateMaybe a) where
   mempty = UpdateMaybe Just
   mappend (UpdateMaybe x) (UpdateMaybe y) = UpdateMaybe (x>=>y)
 
foldlMaybeMonoid :: (a -> b -> Maybe a) -> a -> [b] -> Maybe a
foldlMaybeMonoid f a bs =
   flip evalUpdateMaybe a $
   mconcat $
   map (UpdateMaybe . flip f) bs

3 Practical example: Parsing numbers using a bound

As a practical example consider a function that converts an integer string to an integer, but that aborts when the number exceeds a given bound.

With this bound it is possible to call

readBounded 1234 $ repeat '1'

which will terminate with

Nothing

readBounded :: Integer -> String -> Maybe Integer
readBounded bound str =
   case str of
      ""  -> Nothing
      "0" -> Just 0
      _ -> foldr
         (\digit addLeastSig mostSig ->
            let n = mostSig*10 + toInteger (Char.digitToInt digit)
            in  guard (Char.isDigit digit) >>
                guard (not (mostSig==0 && digit=='0')) >>
                guard (n <= bound) >>
                addLeastSig n)
         Just str 0
 
readBoundedMonoid :: Integer -> String -> Maybe Integer
readBoundedMonoid bound str =
   case str of
      ""  -> Nothing
      "0" -> Just 0
      _ ->
         let m digit =
               UpdateMaybe $ \mostSig ->
                  let n = mostSig*10 + toInteger (Char.digitToInt digit)
                  in  guard (Char.isDigit digit) >>
                      guard (not (mostSig==0 && digit=='0')) >>
                      guard (n <= bound) >>
                      Just n
         in  evalUpdateMaybe (mconcat $ map m str) 0

4 See also

Graham Hutton: A tutorial on the universality and expressiveness of fold
Fold
Foldr Foldl Foldl'

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