Concept:
Foldable type
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. See the concept of rose tree for a more powerful illustration of a foldables.
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.
Ralf Lämmel
edited this article at
Fri, 12 Jun 2020 15:20:51 +0200
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