# Foldl as foldr

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(→See also: + Foldr Foldl Foldl', Fold) |
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that both <hask>foldl</hask> and <hask>foldl'</hask> can be expressed as <hask>foldr</hask>. | that both <hask>foldl</hask> and <hask>foldl'</hask> can be expressed as <hask>foldr</hask>. | ||

(<hask>foldr</hask> may [http://www.willamette.edu/~fruehr/haskell/evolution.html lean so far right] it came back left again.) | (<hask>foldr</hask> may [http://www.willamette.edu/~fruehr/haskell/evolution.html lean so far right] it came back left again.) | ||

− | |||

− | |||

It holds | It holds | ||

<haskell> | <haskell> | ||

Line 10: | Line 8: | ||

foldr (\b g x -> g (f x b)) id bs a | foldr (\b g x -> g (f x b)) id bs a | ||

</haskell> | </haskell> | ||

+ | |||

+ | |||

+ | (The converse is not true, since <hask>foldr</hask> may work on infinite lists, | ||

+ | which <hask>foldl</hask> variants never can do. However, for ''finite'' lists, <hask>foldr</hask> ''can'' also be written in terms of <hask>foldl</hask> (although losing laziness in the process), in a similar way like this: | ||

+ | <haskell> | ||

+ | foldr :: (b -> a -> a) -> a -> [b] -> a | ||

+ | foldr f a bs = | ||

+ | foldl (\g b x -> g (f b x)) id bs a | ||

+ | </haskell> | ||

+ | ) | ||

Now the question are: | Now the question are: | ||

* How can someone find a convolved expression like this? | * How can someone find a convolved expression like this? | ||

* How can we benefit from this rewrite? | * How can we benefit from this rewrite? | ||

+ | |||

+ | |||

+ | == Folding by concatenating updates == | ||

+ | |||

+ | Instead of thinking in terms of <hask>foldr</hask> and a function <hask>g</hask> 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. | ||

+ | <haskell> | ||

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

+ | </haskell> | ||

+ | We need a way to assemble several updates. | ||

+ | To this end we define a <hask>Monoid</hask> instance. | ||

+ | <haskell> | ||

+ | instance Monoid (Update a) where | ||

+ | mempty = Update id | ||

+ | mappend (Update x) (Update y) = Update (y.x) | ||

+ | </haskell> | ||

+ | Now left-folding is straight-forward. | ||

+ | <haskell> | ||

+ | foldlMonoid :: (a -> b -> a) -> a -> [b] -> a | ||

+ | foldlMonoid f a bs = | ||

+ | flip evalUpdate a $ | ||

+ | mconcat $ | ||

+ | map (Update . flip f) bs | ||

+ | </haskell> | ||

+ | Now, where is the <hask>foldr</hask>? | ||

+ | It is hidden in <hask>mconcat</hask>. | ||

+ | <haskell> | ||

+ | mconcat :: Monoid a => [a] -> a | ||

+ | mconcat = foldr mappend mempty | ||

+ | </haskell> | ||

+ | Since <hask>mappend</hask> must be associative | ||

+ | (and is actually associative for our <hask>Update</hask> monoid), | ||

+ | <hask>mconcat</hask> could also be written as <hask>foldl</hask>, | ||

+ | but this is avoided, precisely <hask>foldl</hask> fails on infinite lists. | ||

+ | |||

+ | By the way: | ||

+ | <hask>Update a</hask> is just <hask>Dual (Endo a)</hask>. | ||

+ | If you use a <hask>State</hask> monad instead of a monoid, | ||

+ | you obtain an alternative implementation of <hask>mapAccumL</hask>. | ||

+ | |||

+ | |||

+ | == foldl which may terminate early == | ||

The answer to the second question is: | The answer to the second question is: | ||

− | + | Using the <hask>foldr</hask> expression we can write variants of <hask>foldl</hask> | |

+ | that behave slightly different from the original one. | ||

+ | E.g. we can write a <hask>foldl</hask> that can stop before reaching the end of the input list | ||

and thus may also terminate on infinite input. | and thus may also terminate on infinite input. | ||

+ | The function <hask>foldlMaybe</hask> terminates with <hask>Nothing</hask> as result | ||

+ | when it encounters a <hask>Nothing</hask> as interim accumulator result. | ||

+ | <haskell> | ||

+ | foldlMaybe :: (a -> b -> Maybe a) -> a -> [b] -> Maybe a | ||

+ | foldlMaybe f a bs = | ||

+ | foldr (\b g x -> f x b >>= g) Just bs a | ||

+ | </haskell> | ||

+ | |||

+ | Maybe the monoidic version is easier to understand. | ||

+ | The implementation of the fold is actually the same, we do only use a different monoid. | ||

+ | <haskell> | ||

+ | 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 | ||

+ | </haskell> | ||

+ | |||

+ | |||

+ | == 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 <hask>readBounded 1234 $ repeat '1'</hask> | ||

+ | which will terminate with <hask>Nothing</hask>. | ||

+ | <haskell> | ||

+ | 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 | ||

+ | </haskell> | ||

+ | |||

+ | == See also == | ||

+ | |||

+ | * Graham Hutton: [http://www.cs.nott.ac.uk/~gmh/fold.pdf A tutorial on the universality and expressiveness of fold] | ||

+ | * [[Fold]] | ||

+ | * [[Foldr Foldl Foldl']] | ||

[[Category:Idioms]] | [[Category:Idioms]] |

## Revision as of 11:01, 21 November 2011

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

that bothIt holds

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

*finite*lists,

*can*also be written in terms of

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 |

## 1 Folding by concatenating updates

Instead of thinking in terms ofI 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 ainstance 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

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

By the way:

## 2 foldl which may terminate early

The answer to the second question is:

Using thethat behave slightly different from the original one.

E.g. we can write aand thus may also terminate on infinite input.

The functionfoldlMaybe :: (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 callreadBounded :: 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