# Foldl as foldr

Note: there is an alternative explanation of some of the basics from a more elementary perspective.

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?

## Contents

## 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`

.

## 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
```

## 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
```