Headline

A type for which a fold function can be defined

Illustration

Obviously, a fold function can be defined for lists. See also the concept of Maybe type for another simple example of a foldable type.

In Language:Haskell, there is a type class of foldable types:

class Foldable t
  where
    fold :: Monoid m => t m -> m
    foldMap :: Monoid m => (a -> m) -> t a -> m
    foldr :: (a -> b -> b) -> b -> t a -> b
    foldl :: (a -> b -> a) -> a -> t b -> a
    foldr1 :: (a -> a -> a) -> t a -> a
    foldl1 :: (a -> a -> a) -> t a -> a

The members foldr and foldl generalize the function signatures of the folklore fold functions for lists. It should be noted that a minimal complete definition requires either the definition of foldr or foldMap, as all other class members are then defined by appropriate defaults. Here is a particular attempt at such defaults:

class Foldable t
  where
    fold :: Monoid m => t m -> m
    foldMap :: Monoid m => (a -> m) -> t a -> m
    foldr :: (a -> b -> b) -> b -> t a -> b
    foldl :: (a -> b -> a) -> a -> t b -> a
    foldr1 :: (a -> a -> a) -> t a -> a
    foldl1 :: (a -> a -> a) -> t a -> a
    fold = foldr mappend mempty
    foldMap f = foldr (mappend . f) mempty
    foldr f z = foldr f z . toList
    foldl f z q = foldr (\x g a -> g (f a x)) id q z
    foldr1 f = foldr1 f . toList
    foldl1 f = foldl1 f . toList

In a number of places, we leverage a conversion function toList for going from a foldable type over an element type to the list type over the same element type. In this manner, we can reduce some operations on foldables to operations on lists. This conversion function is easily defined by a foldMap application:

toList :: Foldable t => t a -> [a]
toList = foldMap (\x->[x])

Looking at the defaults again and their use of toList, there is obviously an "unsound" circularity within the definitions, which however would be soundly broken, when either foldr or foldMap was defined for any given foldable type.

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