Class of data structures that can be folded to a summary value.

The Foldabla class generalises some common Data.List functions to structures that can be reduced to a summary value one element at a time.

Left and right folds

The contribution of each element to the final result is combined with an accumulator via a suitable operator. The operator may be explicitly provided by the caller as with foldr or may be implicit as in length. In the case of foldMap, the caller provides a function mapping each element into a suitable Monoid, which makes it possible to merge the per-element contributions via that monoid's mappend function.

A key distinction is between left-associative and right-associative folds:

• In left-associative folds the accumulator is a partial fold over the elements that precede the current element, and is passed to the operator as its first (left) argument. The outermost application of the operator merges the contribution of the last element of the structure with the contributions of all its predecessors.
• In right-associative folds the accumulator is a partial fold over the elements that follow the current element, and is passed to the operator as its second (right) argument. The outermost application of the operator merges the contribution of the first element of the structure with the contributions of all its successors.

These two types of folds are typified by the left-associative strict foldl' and the right-associative lazy foldr.

foldl' :: Foldable t => (b -> a -> b) -> b -> t a -> b
foldr  :: Foldable t => (a -> b -> b) -> b -> t a -> b

Example usage:

>>> foldl' (+) 0 [1..100]
5050
>>> foldr (&&) True (repeat False)
False

The first argument of both is an explicit operator that merges the contribution of an element of the structure with a partial fold over, respectively, either the preceding or following elements of the structure.

The second argument of both is an initial accumulator value z of type b. This is the result of the fold when the structure is empty. When the structure is non-empty, this is the accumulator value merged with the first element in left-associative folds, or with the last element in right-associative folds.

The third and final argument is a Foldable structure containing elements (a, b, c, …).

• foldl' takes an operator argument of the form:

  f :: b -- accumulated fold of the initial elements
    -> a -- current element
    -> b -- updated fold, inclusive of current element
  

  If the structure's last element is y, the result of the fold is:

  g y . … . g c . g b . g a $ z
    where g element !acc = f acc element
  

  Since foldl' is strict in the accumulator, this is always a strict reduction with no opportunity for early return or intermediate results. The structure must be finite, since no result is returned until the last element is processed. The advantage of strictness is space efficiency: the final result can be computed without storing a potentially deep stack of lazy intermediate results.

• foldr takes an operator argument of the form:

  f :: a -- current element
    -> b -- accumulated fold of the remaining elements
    -> b -- updated fold, inclusive of current element
  

  the result of the fold is:

  f a . f b . f c . … $ z

  If each call of f on the current element e, (referenced as (f e) below) returns a structure in which its second argument is captured in a lazily-evaluated component, then the fold of the remaining elements is available to the caller of foldr as a pending computation (thunk) that is computed only when that component is evaluated.

  Alternatively, if any of the (f e) ignore their second argument, the fold stops there, with the remaining elements unused. As a result, foldr is well suited to define both corecursive and short-circuit reductions.

  When the operator is always strict in its second argument, foldl' is generally a better choice than foldr. When foldr is called with a strict operator, evaluation cannot begin until the last element is reached, by which point a deep stack of pending function applications may have been built up in memory.

Foldable structures are generally expected to be efficiently iterable from left to right. Right-to-left iteration may be substantially more costly, or even impossible (as with, for example, infinite lists). The text in the sections that follow that suggests performance differences between left-associative and right-associative folds assumes left-handed structures in which left-to-right iteration is cheaper than right-to-left iteration.

In finite structures for which right-to-left sequencing no less efficient than left-to-right sequencing, there is no inherent performance distinction between left-associative and right-associative folds. If the structure's Foldable instance takes advantage of this symmetry to also make strict right folds space-efficient and lazy left folds corecursive, one need only take care to choose either a strict or lazy method for the task at hand.

Foldable instances for symmetric structures should strive to provide equally performant left-associative and right-associative interfaces. The main limitations are:

• The lazy fold, foldMap and toList methods have no right-associative counterparts.
• The strict foldMap' method has no left-associative counterpart.

Thus, for some foldable structures foldr' is just as efficient as foldl' for strict reduction, and foldl may be just as appropriate for corecursive folds as foldr.

Finally, in some less common structures (e.g. snoc lists) right to left iterations are cheaper than left to right. Such structures are poor candidates for a Foldable instance, and are perhaps best handled via their type-specific interfaces. If nevertheless a Foldable instance is provided, the material in the sections that follow applies to these also, by replacing each method with one with the opposite associativity (when available) and switching the order of arguments in the fold's operator.

You may need to pay careful attention to strictness of the fold's operator when its strictness is different between its first and second argument. For example, while (+) is expected to be commutative and strict in both arguments, the list concatenation operator (++) is not commutative and is only strict in the initial constructor of its first argument. The fold:

myconcat xs = foldr (\a b -> a ++ b) [] xs

is subtantially cheaper (linear in the length of the consumed portion of the final list, thus e.g. constant time/space for just the first element) than:

revconcat xs = foldr (\a b -> b ++ a) [] xs

In which the total cost scales up with both the number of lists combined and the number of elements ultimately consumed. A more efficient way to combine lists in reverse order, is to use:

revconcat = foldr (++) [] . reverse

As observed in the above description of left and right folds, there are three general ways in which a structure can be reduced to a summary value:

• Recursive reduction, which is strict in all the elements of the structure. This produces a single final result only after processing the entire input structure, and so the input must be finite.
• Corecursion, which yields intermediate results as it encounters additional input elements. Lazy processing of the remaining elements makes the intermediate results available even before the rest of the input is processed. The input may be unbounded, and the caller can stop processing intermediate results early.
• Short-circuit reduction, which examines some initial sequence of the input elements, but stops once a termination condition is met, returning a final result based only on the elements considered up to that point. The remaining elements are not considered. The input should generally be finite, because the termination condition might otherwise never be met.

Whether a fold is recursive, corecursive or short-circuiting can depend on both the method chosen to perform the fold and on the operator passed to that method (which may be implicit, as with the mappend method of a monoid instance).

There are also hybrid cases, where the method and/or operator are not well suited to the task at hand, resulting in a fold that fails to yield incremental results until the entire input is processed, or fails to strictly evaluate results as it goes, deferring all the work to the evaluation of a large final thunk. Such cases should be avoided, either by selecting a more appropriate Foldable method, or by tailoring the operator to the chosen method.

The distinction between these types of folds is critical, both in deciding which Foldable method to use to perform the reduction efficiently, and in writing Foldable instances for new structures. Below is a more detailed overview of each type.

Common examples of strict recursive reduction are the various aggregate functions, like sum, product, length, as well as more complex summaries such as frequency counts. These functions return only a single value after processing the entire input structure. In such cases, lazy processing of the tail of the input structure is generally not only unnecessary, but also inefficient. Thus, these and similar folds should be implemented in terms of strict left-associative Foldable methods (typically foldl') to perform an efficient reduction in constant space.

Conversely, an implementation of Foldable for a new structure should ensure that foldl' actually performs a strict left-associative reduction.

The foldMap' method is a special case of foldl', in which the initial accumulator is mempty and the operator is mappend . f, where f maps each input element into the Monoid in question. Therefore, foldMap' is an appropriate choice under essentially the same conditions as foldl', and its implementation for a given Foldable structure should also be a strict left-associative reduction.

While the examples below are not necessarily the most optimal definitions of the intended functions, they are all cases in which foldMap' is far more appropriate (as well as more efficient) than the lazy foldMap.

length  = getSum     . foldMap' (const (Sum 1))
sum     = getSum     . foldMap' Sum
product = getProduct . foldMap' Product

[ The actual default definitions employ coercions to optimise out getSum and getProduct. ]

The full list of strict recursive functions in this module is:

• Provided the operator is strict in its left argument:

  foldl' :: Foldable t => (b -> a -> b) -> b -> t a -> b

• Provided mappend is strict in its left argument:

  foldMap' :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

• Provided the instance is correctly defined:

  length    :: Foldable t => t a -> Int
  sum       :: (Foldable t, Num a) => t a -> a
  product   :: (Foldable t, Num a) => t a -> a
  maximum   :: (Foldable t, Ord a) => t a -> a
  minimum   :: (Foldable t, Ord a) => t a -> a
  maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
  minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a
  

Common examples of lazy corecursive reduction are functions that map and flatten a structure to a lazy stream of result values, i.e. an iterator over the transformed input elements. In such cases, it is important to choose a Foldable method that is lazy in the tail of the structure, such as foldr (or foldMap, if the result Monoid has a lazy mappend as with e.g. ByteString Builders).

Conversely, an implementation of foldr for a structure that can accommodate a large (and possibly unbounded) number of elements is expected to be lazy in the tail of the input, allowing operators that are lazy in the accumulator to yield intermediate results incrementally. Such folds are right-associative, with the tail of the stream returned as a lazily evaluated component of the result (an element of a tuple or some other non-strict constructor, e.g. the (:) constructor for lists).

The toList function below lazily transforms a Foldable structure to a List. Note that this transformation may be lossy, e.g. for a keyed container (Map, HashMap, …) the output stream holds only the values, not the keys. Lossless transformations to/from lists of (key, value) pairs are typically available in the modules for the specific container types.

toList = foldr (:) []

A more complex example is concatenation of a list of lists expressed as a nested right fold (bypassing (++)). We can check that the definition is indeed lazy by folding an infinite list of lists, and taking an initial segment.

>>> myconcat = foldr (\x z -> foldr (:) z x) []
>>> take 15 $ myconcat $ map (\i -> [0..i]) [0..]
[0,0,1,0,1,2,0,1,2,3,0,1,2,3,4]

Of course in this case another way to achieve the same result is via a list comprehension:

myconcat xss = [x | xs <- xss, x <- xs]

The full list of lazy corecursive functions in this module is:

• Provided the reduction function is lazy in its second argument, (otherwise best to use a strict recursive reduction):

  foldr  :: Foldable t => (a -> b -> b) -> b -> t a -> b
  foldr1 :: Foldable t => (a -> a -> a) -> t a -> a
  

• Provided the Monoid mappend is lazy in its second argument (otherwise best to use a strict recursive reduction):

  fold    :: Foldable t => Monoid m => t m -> m
  foldMap :: Foldable t => Monoid m => (a -> m) -> t a -> m
  

• Provided the instance is correctly defined:

  toList    :: Foldable t => t a -> [a]
  concat    :: Foldable t => t [a] -> [a]
  concatMap :: Foldable t => (a -> [b]) -> t a -> [b]
  

Examples of short-cicuit reduction include various boolean predicates that test whether some or all the elements of a structure satisfy a given condition. Because these don't necessarily consume the entire list, they typically employ foldr with an operator that is conditionally strict in its second argument. Once the termination condition is met the second argument (tail of the input structure) is ignored. No result is returned until that happens.

The key distinguishing feature of these folds is conditional strictness in the second argument, it is sometimes evaluated and sometimes not.

The simplest (degenerate case) of these is null, which determines whether a structure is empty or not. This only needs to look at the first element, and only to the extent of whether it exists or not, and not its value. In this case termination is guaranteed, and infinite input structures are fine. Its default definition is of course in terms of the lazy foldr:

null = foldr (\_ _ -> False) True

A more general example is any, which applies a predicate to each input element in turn until it finds the first one for which the predicate is true, at which point it returns success. If, in an infinite input stream the predicate is false for all the elements, any will not terminate, but since it runs in constant space, it typically won't run out of memory, it'll just loop forever.

The full list of short-circuit folds in this module is:

• Boolean predicate folds. These functions examine elements strictly until a condition is met, but then return a result ignoring the rest (lazy in the tail). These may loop forever given an unbounded input where no elements satisy the termination condition.

  null    :: Foldable t => t a -> Bool
  elem    :: Foldable t => Eq a => a -> t a -> Bool
  notElem :: (Foldable t, Eq a) => a -> t a -> Bool
  and     :: Foldable t => t Bool -> Bool
  or      :: Foldable t => t Bool -> Bool
  find    :: Foldable t => (a -> Bool) -> t a -> Maybe a
  any     :: Foldable t => (a -> Bool) -> t a -> Bool
  all     :: Foldable t => (a -> Bool) -> t a -> Bool
  

• Many instances of (<|>) (e.g. the Maybe instance) are conditionally lazy, and use or don't use their second argument depending on the value of the first. These are used with the folds below, which terminate as early as possible, but otherwise generally keep going. Some instances (e.g. for List) are always strict, but the result is lazy in the tail of the output, so that asum for a list of lists is in fact corecursive. These folds are defined in terms of foldr.

  asum :: (Foldable t, Alternative f) => t (f a) -> f a
  msum :: (Foldable t, MonadPlus m) => t (m a) -> m a
  

• Likewise, the (*>) operator in some Applicative functors, and (>>) in some monads are conditionally lazy and can short-circuit a chain of computations. The below folds will terminate as early as possible, but even infinite loops can be productive here, when evaluated solely for their stream of IO side-effects. See Data.Traversable for some additional discussion.

  traverse_  :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f ()
  for_       :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f ()
  sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f ()
  mapM_      :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
  forM_      :: (Foldable t, Monad m) => t a -> (a -> m b) -> m ()
  sequence_  :: (Foldable t, Monad m) => t (m a) -> m ()
  

• Finally, there's one more special case, foldlM:

  foldlM :: (Foldable t, Monad m) => (b -> a -> m b) -> b -> t a -> m b

  The sequencing of monadic effects proceeds from left to right. If at some step the bind operator (>>=) short-circuits (as with, e.g., mzero with a MonadPlus, or an exception with a MonadThrow, etc.), then the evaluated effects will be from an initial portion of the element sequence.

  >>> :set -XBangPatterns
  >>> import Control.Monad
  >>> import Control.Monad.Trans.Class
  >>> import Control.Monad.Trans.Maybe
  >>> import Data.Foldable
  >>> let f !_ e = when (e > 3) mzero >> lift (print e)
  >>> runMaybeT $ foldlM f () [0..]
  0
  1
  2
  3
  Nothing
  

  Contrast this with foldrM, which sequences monadic effects from right to left, and therefore diverges when folding an unbounded input structure without ever having the opportunity to short-circuit.

  >>> let f e _ = when (e > 3) mzero >> lift (print e)
  >>> runMaybeT $ foldrM f () [0..]
  ...hangs...
  

  When the structure is finite foldrM performs the monadic effects from right to left, possibly short-circuiting after processing a tail portion of the element sequence.

  >>> let f e _ = when (e < 3) mzero >> lift (print e)
  >>> runMaybeT $ foldrM f () [0..5]
  5
  4
  3
  Nothing
  

The below folds, are neither strict reductions that produce a final answer in constant space, nor lazy corecursions, and so have limited applicability. They do have specialised uses, but are best avoided when in doubt.

foldr' :: Foldable t => (a -> b -> b) -> b -> t a -> b
foldl  :: Foldable t => (b -> a -> b) -> b -> t a -> b
foldl1 :: Foldable t => (a -> a -> a) -> t a -> a
foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b

The lazy left-folds (used corecursively) and foldrM (used to sequence actions right-to-left) can be performant in structures whose Foldable instances take advantage of efficient right-to-left iteration to compute lazy left folds outside-in from the rightmost element.

The strict foldr' is the least likely to be useful, structures that support efficient sequencing only right-to-left are not common.

So far, we have not discussed generative recursion. Unlike recursive reduction or corecursion, instead of processing a sequence of elements already in memory, generative recursion involves producing a possibly unbounded sequence of values from an initial seed value. The canonical example of this is unfoldr for Lists, with variants available for Vectors and various other structures.

A key issue with lists, when used generatively as iterators, rather than as poor-man's containers (see [1]), is that such iterators tend to consume memory when used more than once. A single traversal of a list-as-iterator will run in constant space, but as soon as the list is retained for reuse, its entire element sequence is stored in memory, and the second traversal reads the copy, rather than regenerates the elements. It is sometimes better to recompute the elements rather than memoise the list.

Memoisation happens because the built-in Haskell list [] is represented as data, either empty or a cons-cell holding the first element and the tail of the list. The Foldable class enables a variant representation of iterators as functions, which take an operator and a starting accumulator and output a summary result.

The fmlist package takes this approach, by representing a list via its foldMap action.

Below we implement an analogous data structure using a representation based on foldr. This is an example of Church encoding (named after Alonzo Church, inventor of the lambda calculus).

{-# LANGUAGE RankNTypes #-}
newtype FRList a = FR { unFR :: forall b. (a -> b -> b) -> b -> b }

The unFR field of this type is essentially its foldr method with the list as its first rather than last argument. Thus we immediately get a Foldable instance (and a toList function mapping an FRList to a regular list).

instance Foldable FRList where
    foldr f z l = unFR l f z
    -- With older versions of @base@, also define sum, product, ...
    -- to ensure use of the strict 'foldl''.
    -- sum = foldl' (+) 0
    -- ...

We can convert a regular list to an FRList with:

fromList :: [a] -> FRList a
fromList as = FRList $ \ f z -> foldr f z as

However, reuse of an FRList obtained in this way will typically memoise the underlying element sequence. Instead, we can define FRList terms directly:

-- | Immediately return the initial accumulator
nil :: FRList a
nil = FRList $ \ _ z -> z
{-# INLINE nil #-}

-- | Fold the tail to use as an accumulator with the new initial element
cons :: a -> FRList a -> FRList a
cons a l = FRList $ \ f z -> f a (unFR l f z)
{-# INLINE cons #-}

More crucially, we can also directly define the key building block for generative recursion:

-- | Generative recursion, dual to `foldr`.
unfoldr :: (s -> Maybe (a, s)) -> s -> FRList a
unfoldr g s0 = FR generate
  where generate f z = loop s0
          where loop s | Just (a, t) <- g s = f a (loop t)
                       | otherwise = z
{-# INLINE unfoldr #-}

Which can, for example, be specialised to number ranges:

-- | Generate a range of consecutive integral values.
range :: (Ord a, Integral a) => a -> a -> FRList a
range lo hi =
    unfoldr (\s -> if s > hi then Nothing else Just (s, s+1)) lo
{-# INLINE range #-}

The program below, when compiled with optimisation:

main :: IO ()
main = do
    let r :: FRList Int
        r = range 1 10000000
     in print (sum r, length r)

produces the expected output with no noticeable garbage-collection, despite reuse of the FRList term r.

(50000005000000,10000000)
    52,120 bytes allocated in the heap
     3,320 bytes copied during GC
    44,376 bytes maximum residency (1 sample(s))
    25,256 bytes maximum slop
         3 MiB total memory in use (0 MB lost due to fragmentation)

The Weak Head Normal Form of an FRList is a lambda abstraction not a data value, and reuse does not lead to memoisation. Reuse of the iterator above is somewhat contrived, when computing multiple folds over a common list, you should generally traverse a list only once. The goal is to demonstrate that the separate computations of the sum and length run efficiently in constant space, despite reuse. This would not be the case with the list [1..10000000].

This is, however, an artificially simple reduction. More typically, there are likely to be some allocations in the inner loop, but the temporary storage used will be garbage-collected as needed, and overall memory utilisation will remain modest and will not scale with the size of the list.

If we go back to built-in lists (i.e. []), but avoid reuse by performing reduction in a single pass, as below:

data PairS a b = P !a !b -- We define a strict pair datatype

main :: IO ()
main = do
    let l :: [Int]
        l = [1..10000000]
     in print $ average l
  where
    sumlen :: PairS Int Int -> Int -> PairS Int Int
    sumlen (P s l) a = P (s + a) (l + 1)

    average is =
        let (P s l) = foldl' sumlen (P 0 0) is
         in (fromIntegral s :: Double) / fromIntegral l

the result is again obtained in constant space:

5000000.5
         102,176 bytes allocated in the heap
           3,320 bytes copied during GC
          44,376 bytes maximum residency (1 sample(s))
          25,256 bytes maximum slop
               3 MiB total memory in use (0 MB lost due to fragmentation)

(and, in fact, faster than with FRList by a small factor).

The [] list structure works as an efficient iterator when used just once. When space-leaks via list reuse are not a concern, and/or memoisation is actually desirable, the regular list implementation is likely to be faster. This is not a suggestion to replace all your uses of [] with a generative alternative.

The FRList type could be further extended with instances of Functor, Applicative, Monad, Alternative, etc., and could then provide a fully-featured list type, optimised for reuse without space-leaks. If, however, all that's required is space-efficient, re-use friendly iteration, less is perhaps more, and just Foldable may be sufficient.

In applications where you want to compute a composite function of a structure, which requires more than one aggregate as an input, it is generally best to compute all the aggregates in a single pass, rather than to traverse the same structure repeatedly.

The foldl package implements a robust general framework for dealing with this situation. If you choose to to do it yourself, with a bit of care, the simplest cases are not difficult to handle directly. You just need to accumulate the individual aggregates as strict components of a single data type, and then apply a final transformation to it to extract the composite result. For example, computing an average requires computing both the sum and the length of a (non-empty) structure and dividing the sum by the length:

import Data.Foldable (foldl')

data PairS a b = P !a !b -- We define a strict pair datatype

-- | Compute sum and length in a single pass, then reduce to the average.
average :: (Foldable f, Fractional a) => f a -> a
average xs =
    let sumlen (P s l) a = P (s + a) (l + 1 :: Int)
        (P s l) = foldl' sumlen (P 0 0) xs
     in s / fromIntegral l

The above example is somewhat contrived, some structures keep track of their length internally, and can return it in \(\mathcal{O}(1)\) time, so this particular recipe for averages is not always the most efficient. In general, composite aggregate functions of large structures benefit from single-pass reduction. This is especially the case when reuse of a list and memoisation of its elements is thereby avoided,

For many structures reasonably efficient Foldable instances can be derived automatically, by enabling the DeriveFoldable GHC extension. When this works, it is generally not necessary to define a custom instance by hand. Though in some cases one may be able to get slightly faster hand-tuned code, care is required to avoid producing slower code, or code that is not sufficiently lazy, strict or lawful.

The hand-crafted instances can get away with only defining one of foldr or foldMap. All the other methods have default definitions in terms of one of these. The default definitions have the expected strictness and the expected asymptotic runtime and space costs, modulo small constant factors. If you choose to hand-tune, benchmarking is advised to see whether you're doing better than the default derived implementations, plus careful tests to ensure that the custom methods are correct.

Below we construct a Foldable instance for a data type representing a (finite) binary tree with depth-first traversal.

data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)

a suitable instance would be:

instance Foldable Tree where
   foldr f z Empty = z
   foldr f z (Leaf x) = f x z
   foldr f z (Node l k r) = foldr f (f k (foldr f z r)) l

The Node case is a right fold of the left subtree whose initial value is a right fold of the rest of the tree.

For example, when f is (:), all three cases return an immediate value, respectively z or a cons cell holding x or l, with the remainder the structure, if any, encapsulated in a lazy thunk. This meets the expected efficient corecursive behaviour of foldr.

Alternatively, one could define foldMap:

instance Foldable Tree where
   foldMap f Empty = mempty
   foldMap f (Leaf x) = f x
   foldMap f (Node l k r) = foldMap f l <> f k <> foldMap f r

And indeed some efficiency may be gained by directly defining both, avoiding some indirection in the default definitions that express one in terms of the other. If you implement just one, likely foldr is the better choice.

A binary tree typically (when balanced, or randomly biased) provides equally efficient access to its left and right subtrees. This makes it possible to define a foldl optimised for corecursive folds with operators that are lazy in their first (left) argument.

instance Foldable Tree where
   foldr f z Empty = z
   foldr f z (Leaf x) = f x z
   foldr f z (Node l k r) = foldr f (f k (foldr f z r)) l
   --
   foldMap f Empty = mempty
   foldMap f (Leaf x) = f x
   foldMap f (Node l k r) = foldMap f l <> f k <> foldMap f r
   --
   foldl f z Empty = z
   foldl f z (Leaf x) = f z x
   foldl f z (Node l k r) = foldl f (f (foldl f z l) k) r

Now left-to-right and right-to-left iteration over the structure elements are equally efficient (note the mirror-order output when using foldl):

>>> foldr (\e acc -> e : acc) [] (Node (Leaf 1) 2 (Leaf 3))
[1,2,3]
>>> foldl (\acc e -> e : acc) [] (Node (Leaf 1) 2 (Leaf 3))
[3,2,1]

We can carry this further, and define more non-default methods...

The structure definition actually admits trees that are unbounded on either or both sides. The only fold that can plausibly terminate for a tree unbounded on both left and right is null, when defined as shown below. The default definition in terms of foldr diverges if the tree is unbounded on the left. Here we define a variant that avoids travelling down the tree to find the leftmost element and just examines the root node.

   null Empty = True
   null _     = False

This is a sound choice also for finite trees.

In practice, unbounded trees are quite uncommon, and can barely be said to be Foldable. They would typically employ breadth first traversal, and would support only corecursive and short-circuit folds (diverge under strict reduction).

Returning to simpler instances, defined just in terms of foldr, it is somewhat surprising that a fairly efficient default implementation of the strict foldl' is defined in terms of lazy foldr when only the latter is explicitly provided by the instance. It may be instructive to take a look at how this works.

Sometimes, it is useful for the result of applying foldr to be a function. This is done by mapping the structure elements to functions with the same argument and result types. The per-element functions are then composed to give the final result.

For example, we can flip the strict left fold foldl' by writing:

foldl' f z xs = flippedFoldl' f xs z

with the function flippedFoldl' defined as below, with seq used to ensure the strictness in the accumulator:

flippedFoldl' f [] z = z
flippedFoldl' f (x : xs) z = z `seq` flippedFoldl' f xs (f z x)

Rewriting to use lambdas, this is:

flippedFoldl' f [] = \ b -> b
flippedFoldl' f (x : xs) = \ b -> b `seq` r (f b x)
    where r = flippedFoldl' f xs

The above has the form of a right fold, enabling a rewrite to:

flippedFoldl' f = \ xs -> foldr f' id xs
    where f' x r = \ b -> b `seq` r (f b x)

We can now unflip this to get foldl':

foldl' f z = \ xs -> foldr f' id xs z
          -- \ xs -> flippedFoldl' f xs z
  where f' x r = \ b -> b `seq` r (f b x)

The function foldr f' id xs applied to z is built corecursively, and its terms are applied to an eagerly evaluated accumulator before further terms are applied to the result. As required, this runs in constant space, and can be optimised to an efficient loop.

(The actual definition of foldl' labels the lambdas in the definition of f' above as oneShot, which enables further optimisations).

The type constructor Endo from Data.Monoid, associates with each type b the newtype-encapulated type of functions mapping b to itself. Functions from a type to itself are called endomorphisms, hence the name Endo. The type Endo b is a Monoid under function composition:

newtype Endo b = Endo { appEndo :: b -> b }
instance Semigroup Endo b where
    Endo f <> Endo g = Endo (f . g)
instance Monoid Endo b where
    mempty = Endo id

For every Monoid m, we also have a Dual monoid Dual m which combines elements in the opposite order:

newtype Dual m = Dual { getDual :: m }
instance Semigroup m => Semigroup Dual m where
    Dual a <> Dual b = Dual (b <> a)
instance Monoid m => Monoid Dual m where
    mempty = Dual mempty

With the above preliminaries out of the way, Foldable instances are expected to satisfy the following laws:

The foldr method must be equivalent in value and strictness to replacing each element a of a Foldable structure with Endo (f a), composing these via foldMap and applying the result to the base case z:

foldr f z t = appEndo (foldMap (Endo . f) t ) z

Likewise, the foldl method must be equivalent in value and strictness to composing the functions flip f a in reverse order and applying the result to the base case:

foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z

When the elements of the structure are taken from a Monoid, the defintion of fold must agree with foldMap id:

fold = foldMap id

The length method must agree with a foldMap mapping each element to Sum 1 (The Sum type abstracts numbers as a monoid under addition).

length = getSum . foldMap (Sum . const 1)

sum, product, maximum, and minimum should all be essentially equivalent to foldMap forms, such as

sum     = getSum     . foldMap' Sum
product = getProduct . foldMap' Product

but are generally more efficient when defined more directly as:

sum = foldl' (+) 0
product = foldl' (*) 1

If the Foldable structure has a Functor instance, then for every function f mapping the elements into a Monoid, it should satisfy:

foldMap f = fold . fmap f

which implies that

foldMap f . fmap g = foldMap (f . g)

Since Foldable does not have Functor as a superclass, it is possible to define Foldable instances for structures that constrain their element types. Therefore, Set can be Foldable, even though sets keep their elements in ascending order. This requires the elements to be comparable, which precludes defining a Functor instance for Set.

The Foldable class makes it possible to use idioms familiar from the List type with container structures that are better suited to the task at hand. This supports use of more appropriate Foldable data types, such as Seq, Set, NonEmpty, etc., without requiring new idioms (see [1] for when not to use lists).

The more general methods of the Foldable class are now exported by the Prelude in place of the original List-specific methods (see the FTP Proposal). The List-specific variants are still available in Data.List.

Surprises can arise from the Foldable instance of the 2-tuple (a,) which now behaves as a 1-element Foldable container in its second slot. In contexts where a specific monomorphic type is expected, and you want to be able to rely on type errors to guide refactoring, it may make sense to define and use less-polymorphic variants of some of the Foldable methods.

Below are two examples showing a definition of a reusable less-polymorphic sum and a one-off in-line specialisation of length:

{-# LANGUAGE TypeApplications #-}

mySum :: Num a => [a] -> a
mySum = sum

type SlowVector a = [a]
slowLength :: SlowVector -> Int
slowLength v = length @[] v

In both cases, if the data type to which the function is applied changes to something other than a list, the call-site will no longer compile until appropriate changes are made.

It is perhaps worth noting that since the elem function in the Foldable class carries only an Eq constraint on the element type, search for the presence or absence of an element in the structure generally takes \(\mathcal{O}(n)\) time, even for ordered structures like Set that are potentially capable of performing the search faster. (The member function of the Set module carries an Ord constraint, and can perform the search in \(\mathcal{O}(log\ n)\) time).

An alternative to Foldable's elem method is required in order to abstract potentially faster than linear search over general container structures. This can be achieved by defining an additional type class (e.g. HasMember below). Instances of such a type class (that are also Foldable) can employ the elem linear search as a last resort, when faster search is not supported.

{-# LANGUAGE FlexibleInstances, MultiParamTypeClasses #-}

import qualified Data.Set as Set

class Eq a => HasMember t a where
    member :: a -> t a -> Bool

instance Eq a => HasMember [] a where
    member = elem
[...]
instance Ord a => HasMember Set.Set a where
    member = Set.member

The above suggests that elem may be a misfit in the Foldable class. Alternative design ideas are solicited on GHC's bug tracker via issue #20421.

Note that some structure-specific optimisations may of course be possible directly in the corresponding Foldable instance, e.g. with Set the size of the set is known in advance, without iterating to count the elements, and its length instance takes advantage of this to return the size directly.

• [1] "When You Should Use Lists in Haskell (Mostly, You Should Not)", by Johannes Waldmann, in arxiv.org, Programming Languages (cs.PL), at https://arxiv.org/abs/1808.08329.
• [2] "The Essence of the Iterator Pattern", by Jeremy Gibbons and Bruno Oliveira, in Mathematically-Structured Functional Programming, 2006, online at http://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/#iterator.
• [3] "A tutorial on the universality and expressiveness of fold", by Graham Hutton, J. Functional Programming 9 (4): 355–372, July 1999, online at http://www.cs.nott.ac.uk/~pszgmh/fold.pdf.