On lists
The folding of the list [1,2,3,4,5] with the addition operator would result in 15, the sum of the elements of the list [1,2,3,4,5]. To a rough approximation, one can think of this fold as replacing the commas in the list with the + operation, giving 1 + 2 + 3 + 4 + 5.[1]
In the example above, + is an associative operation, so the final result will be the same regardless of parenthesization, although the specific way in which it is calculated will be different. In the general case of non-associative binary functions, the order in which the elements are combined may influence the final result's value. On lists, there are two obvious ways to carry this out: either by combining the first element with the result of recursively combining the rest (called a right fold), or by combining the result of recursively combining all elements but the last one, with the last element (called a left fold). This corresponds to a binary operator being either right-associative or left-associative, in Haskell's or Prolog's terminology. With a right fold, the sum would be parenthesized as 1 + (2 + (3 + (4 + 5))), whereas with a left fold it would be parenthesized as (((1 + 2) + 3) + 4) + 5.
In practice, it is convenient and natural to have an initial value which in the case of a right fold is used when one reaches the end of the list, and in the case of a left fold is what is initially combined with the first element of the list. In the example above, the value 0 (the additive identity) would be chosen as an initial value, giving 1 + (2 + (3 + (4 + (5 + 0)))) for the right fold, and ((((0 + 1) + 2) + 3) + 4) + 5 for the left fold. For multiplication, an initial choice of 0 wouldn't work: 0 * 1 * 2 * 3 * 4 * 5 = 0. The identity element for multiplication is 1. This would give us the outcome 1 * 1 * 2 * 3 * 4 * 5 = 120 = 5!.
Linear vs. tree-like folds
The use of an initial value is necessary when the combining function f is asymmetrical in its types (e.g. a → b → b), i.e. when the type of its result is different from the type of the list's elements. Then an initial value must be used, with the same type as that of f 's result, for a linear chain of applications to be possible. Whether it will be left- or right-oriented will be determined by the types expected of its arguments by the combining function. If it is the second argument that must be of the same type as the result, then f could be seen as a binary operation that associates on the right, and vice versa.
When the function is a magma, i.e. symmetrical in its types (a → a → a), and the result type is the same as the list elements' type, the parentheses may be placed in arbitrary fashion thus creating a binary tree of nested sub-expressions, e.g., ((1 + 2) + (3 + 4)) + 5. If the binary operation f is associative this value will be well-defined, i.e., same for any parenthesization, although the operational details of how it is calculated will be different. This can have significant impact on efficiency if f is non-strict.
Whereas linear folds are node-oriented and operate in a consistent manner for each node of a list, tree-like folds are whole-list oriented and operate in a consistent manner across groups of nodes.
Special folds for non-empty lists
One often wants to choose the identity element of the operation f as the initial value z. When no initial value seems appropriate, for example, when one wants to fold the function which computes the maximum of its two parameters over a non-empty list to get the maximum element of the list, there are variants of foldr and foldl which use the last and first element of the list respectively as the initial value. In Haskell and several other languages, these are called foldr1 and foldl1, the 1 making reference to the automatic provision of an initial element, and the fact that the lists they are applied to must have at least one element.
These folds use type-symmetrical binary operation: the types of both its arguments, and its result, must be the same. Richard Bird in his 2010 book proposes[2] "a general fold function on non-empty lists" foldrn which transforms its last element, by applying an additional argument function to it, into a value of the result type before starting the folding itself, and is thus able to use type-asymmetrical binary operation like the regular foldr to produce a result of type different from the list's elements type.
Implementation
Linear folds
Using Haskell as an example, foldl and foldr can be formulated in a few equations.
foldl :: (b -> a -> b) -> b -> [a] -> b
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
If the list is empty, the result is the initial value. If not, fold the tail of the list using as new initial value the result of applying f to the old initial value and the first element.
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f z [] = z
foldr f z (x:xs) = f x (foldr f z xs)
If the list is empty, the result is the initial value z. If not, apply f to the first element and the result of folding the rest.
Tree-like folds
Lists can be folded over in a tree-like fashion, both for finite and for indefinitely defined lists:
foldt f z [] = z
foldt f z [x] = f x z
foldt f z xs = foldt f z (pairs f xs)
foldi f z [] = z
foldi f z (x:xs) = f x (foldi f z (pairs f xs))
pairs f (x:y:t) = f x y : pairs f t
pairs _ t = t
In the case of foldi function, to avoid its runaway evaluation on indefinitely defined lists the function f must not always demand its second argument's value, at least not all of it, or not immediately (see example below).
Folds for non-empty lists
foldl1 f [x] = x
foldl1 f (x:y:xs) = foldl1 f (f x y : xs)
foldr1 f [x] = x
foldr1 f (x:xs) = f x (foldr1 f xs)
foldt1 f [x] = x
foldt1 f (x:y:xs) = foldt1 f (f x y : pairs f xs)
foldi1 f [x] = x
foldi1 f (x:xs) = f x (foldi1 f (pairs f xs))
Evaluation order considerations
In the presence of lazy, or non-strict evaluation, foldr will immediately return the application of f to the head of the list and the recursive case of folding over the rest of the list. Thus, if f is able to produce some part of its result without reference to the recursive case on its "right" i.e., in its second argument, and the rest of the result is never demanded, then the recursion will stop (e.g., head == foldr (\a b->a) (error "empty list")). This allows right folds to operate on infinite lists. By contrast, foldl will immediately call itself with new parameters until it reaches the end of the list. This tail recursion can be efficiently compiled as a loop, but can't deal with infinite lists at all — it will recurse forever in an infinite loop.
Having reached the end of the list, an expression is in effect built by foldl of nested left-deepening f-applications, which is then presented to the caller to be evaluated. Were the function f to refer to its second argument first here, and be able to produce some part of its result without reference to the recursive case (here, on its left i.e., in its first argument), then the recursion would stop. This means that while foldr recurses on the right, it allows for a lazy combining function to inspect list's elements from the left; and conversely, while foldl recurses on the left, it allows for a lazy combining function to inspect list's elements from the right, if it so chooses (e.g., last == foldl (\a b->b) (error "empty list")).
Reversing a list is also tail-recursive (it can be implemented using rev = foldl (\ys x -> x : ys) []). On finite lists, that means that left-fold and reverse can be composed to perform a right fold in a tail-recursive way (cf. 1+>(2+>(3+>0)) == ((0<+3)<+2)<+1), with a modification to the function f so it reverses the order of its arguments (i.e., foldr f z == foldl (flip f) z . foldl (flip (:)) []), tail-recursively building a representation of expression that right-fold would build. The extraneous intermediate list structure can be eliminated with the continuation-passing style technique, foldr f z xs == foldl (\k x-> k . f x) id xs z; similarly, foldl f z xs == foldr (\x k-> k . flip f x) id xs z ( flip is only needed in languages like Haskell with its flipped order of arguments to the combining function of foldl unlike e.g., in Scheme where the same order of arguments is used for combining functions to both foldl and foldr).
Another technical point is that, in the case of left folds using lazy evaluation, the new initial parameter is not being evaluated before the recursive call is made. This can lead to stack overflows when one reaches the end of the list and tries to evaluate the resulting potentially gigantic expression. For this reason, such languages often provide a stricter variant of left folding which forces the evaluation of the initial parameter before making the recursive call. In Haskell this is the foldl' (note the apostrophe, pronounced 'prime') function in the Data.List library (one needs to be aware of the fact though that forcing a value built with a lazy data constructor won't force its constituents automatically by itself). Combined with tail recursion, such folds approach the efficiency of loops, ensuring constant space operation, when lazy evaluation of the final result is impossible or undesirable.
Examples
Using a Haskell interpreter, the structural transformations which fold functions perform can be illustrated by constructing a string:
λ> foldr (\x y -> concat ["(",x,"+",y,")"]) "0" (map show [1..13])
"(1+(2+(3+(4+(5+(6+(7+(8+(9+(10+(11+(12+(13+0)))))))))))))"
λ> foldl (\x y -> concat ["(",x,"+",y,")"]) "0" (map show [1..13])
"(((((((((((((0+1)+2)+3)+4)+5)+6)+7)+8)+9)+10)+11)+12)+13)"
λ> foldt (\x y -> concat ["(",x,"+",y,")"]) "0" (map show [1..13])
"(((((1+2)+(3+4))+((5+6)+(7+8)))+(((9+10)+(11+12))+13))+0)"
λ> foldi (\x y -> concat ["(",x,"+",y,")"]) "0" (map show [1..13])
"(1+((2+3)+(((4+5)+(6+7))+((((8+9)+(10+11))+(12+13))+0))))"
Infinite tree-like folding is demonstrated e.g., in recursive primes production by unbounded sieve of Eratosthenes in Haskell:
primes = 2 : _Y ((3 :) . minus [5,7..] . foldi (\(x:xs) ys -> x : union xs ys) []
. map (\p-> [p*p, p*p+2*p..]))
_Y g = g (_Y g) -- = g . g . g . g . ...
where the function union operates on ordered lists in a local manner to efficiently produce their set union, and minus their set difference.
A finite prefix of primes is concisely defined as a folding of set difference operation over the lists of enumerated multiples of integers, as
primesTo n = foldl1 minus [[2*x,3*x..n] | x <- [1..n]]
For finite lists, e.g., merge sort (and its duplicates-removing variety, nubsort) could be easily defined using tree-like folding as
mergesort xs = foldt merge [] [[x] | x <- xs]
nubsort xs = foldt union [] [[x] | x <- xs]
with the function merge a duplicates-preserving variant of union.
Functions head and last could have been defined through folding as
head = foldr (\x r -> x) (error "head: Empty list")
last = foldl (\a x -> x) (error "last: Empty list")
