# Foldr Foldl Foldl'

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<haskell>> import Prelude hiding (foldr, foldl)</haskell> | <haskell>> import Prelude hiding (foldr, foldl)</haskell> | ||

+ | |||

+ | ==Foldr== | ||

Say we want to calculate the sum of a very big list: | Say we want to calculate the sum of a very big list: | ||

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

− | + | For a nice interactive animation of the above behavior see: http://foldr.com | |

− | + | ||

− | + | ||

− | + | The problem is that (+) is strict in both of its arguments. This means that both arguments must be fully evaluated before (+) can return a result. So to evaluate: | |

+ | |||

+ | <haskell>1 + (2 + (3 + (4 + (...))))</haskell> | ||

+ | |||

+ | <tt>1</tt> is pushed on the stack. Then: | ||

+ | |||

+ | <haskell>2 + (3 + (4 + (...)))</haskell> | ||

+ | |||

+ | is evaluated. So <tt>2</tt> is pushed on the stack. Then: | ||

+ | |||

+ | <haskell>3 + (4 + (...))</haskell> | ||

+ | |||

+ | is evaluated. So <tt>3</tt> is pushed on the stack. Then: | ||

+ | |||

+ | <haskell>4 + (...)</haskell> | ||

+ | |||

+ | is evaluated. So <tt>4</tt> is pushed on the stack. Then: ... | ||

+ | |||

+ | ... your limited stack will eventually run full when you evaluate a large enough chain of (+)s. This then triggers the stack overflow exception. | ||

Lets think about how to solve it... | Lets think about how to solve it... | ||

+ | |||

+ | ==Foldl== | ||

One problem with the chain of (+)'s is that we can't make it | One problem with the chain of (+)'s is that we can't make it | ||

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> foldl f z [] = z | > foldl f z [] = z | ||

> foldl f z (x:xs) = let z' = z `f` x | > foldl f z (x:xs) = let z' = z `f` x | ||

− | + | > in foldl f z' xs | |

> sum2 = foldl (+) 0 | > sum2 = foldl (+) 0 | ||

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in z1000000 --> | in z1000000 --> | ||

− | -- Now a large chain of +'s | + | -- Now a large chain of +'s will be created: |

let z1 = 0 + 1 | let z1 = 0 + 1 | ||

Line 230: | Line 250: | ||

(((((((((0 + 1) + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> | (((((((((0 + 1) + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> | ||

+ | |||

+ | -- Now we can actually start reducing: | ||

((((((((1 + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> | ((((((((1 + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> | ||

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

− | For a nice interactive animation of the above | + | For a nice interactive animation of the above behavior see: http://foldl.com (actually this animation is not quite the same :-( ) |

− | Well, you clearly see that the redexen < | + | Well, you clearly see that the redexen are created. But instead of being directly reduced, they are allocated on the heap: |

− | + | ||

+ | <haskell> | ||

+ | let z1 = 0 + 1 | ||

+ | z2 = z1 + 2 | ||

+ | z3 = z2 + 3 | ||

+ | z4 = z3 + 4 | ||

+ | ... | ||

+ | z999997 = z999996 + 999997 | ||

+ | z999998 = z999997 + 999998 | ||

+ | z999999 = z999998 + 999999 | ||

+ | z100000 = z999999 + 1000000 | ||

+ | in z1000000 | ||

+ | </haskell> | ||

+ | |||

+ | Note that your heap is only limited by the amount of memory in your system (RAM and swap). So the only thing this does is filling up a large part of your memory. | ||

+ | |||

+ | The problem starts when we finally evaluate z1000000: | ||

+ | |||

+ | Note that <tt>z1000000 = z999999 + 1000000</tt>. | ||

+ | So <tt>1000000</tt> is pushed on the stack. | ||

+ | Then <tt>z999999</tt> is evaluated. | ||

+ | |||

+ | Note that <tt>z999999 = z999998 + 999999</tt>. | ||

+ | So <tt>999999</tt> is pushed on the stack. | ||

+ | Then <tt>z999998</tt> is evaluated: | ||

+ | |||

+ | Note that <tt>z999998 = z999997 + 999998</tt>. | ||

+ | So <tt>999998</tt> is pushed on the stack. | ||

+ | Then <tt>z999997</tt> is evaluated: | ||

+ | So ... | ||

+ | |||

+ | ...your limited stack will eventually run full when you evaluate a large enough chain of (+)s. This then triggers the stack overflow exception. | ||

+ | |||

+ | But this is exactly the problem we had in the foldr case! Only now the chain of (+) is going to the left instead of going to the right. | ||

+ | |||

+ | So why doesn't the chain reduce sooner than | ||

before? | before? | ||

Line 265: | Line 322: | ||

first. In this case it are the outer <tt>foldl (+) ... [1..10000]</tt> | first. In this case it are the outer <tt>foldl (+) ... [1..10000]</tt> | ||

redexen which are repeatedly reduced. | redexen which are repeatedly reduced. | ||

− | So the inner <tt> | + | So the inner <tt>z1, z2, z3, ...</tt> redexen only get reduced when |

the foldl is completely gone. | the foldl is completely gone. | ||

+ | |||

+ | ==Foldl'== | ||

We somehow have to tell the system that the inner redex should be | We somehow have to tell the system that the inner redex should be |

## Revision as of 16:05, 26 March 2009

To *foldr*, *foldl* or *foldl'* that's the question! This article demonstrates the differences between these different folds by a simple example.

If you want you can copy/paste this article into your favorite editor and run it.

We are going to define our own folds so we hide the ones from the Prelude:

> import Prelude hiding (foldr, foldl)

## 1 Foldr

Say we want to calculate the sum of a very big list:

> veryBigList = [1..1000000]

Lets start with the following:

> foldr f z [] = z > foldr f z (x:xs) = x `f` foldr f z xs > sum1 = foldr (+) 0 > try1 = sum1 veryBigList

If we evaluate *try1* we get:

`*** Exception: stack overflow`

Too bad... So what happened:

try1 --> sum1 veryBigList --> foldr (+) 0 veryBigList --> foldr (+) 0 [1..1000000] --> 1 + (foldr (+) 0 [2..1000000]) --> 1 + (2 + (foldr (+) 0 [3..1000000])) --> 1 + (2 + (3 + (foldr (+) 0 [4..1000000]))) --> 1 + (2 + (3 + (4 + (foldr (+) 0 [5..1000000])))) --> -- ... -- ... My stack overflows when there's a chain of around 500000 (+)'s !!! -- ... But the following would happen if you got a large enough stack: -- ... 1 + (2 + (3 + (4 + (... + (999999 + (foldr (+) 0 [1000000]))...)))) --> 1 + (2 + (3 + (4 + (... + (999999 + (1000000 + ((foldr (+) 0 []))))...)))) --> 1 + (2 + (3 + (4 + (... + (999999 + (1000000 + 0))...)))) --> 1 + (2 + (3 + (4 + (... + (999999 + 1000000)...)))) --> 1 + (2 + (3 + (4 + (... + 1999999 ...)))) --> 1 + (2 + (3 + (4 + 500000499990))) --> 1 + (2 + (3 + 500000499994)) --> 1 + (2 + 500000499997) --> 1 + 500000499999 --> 500000500000

For a nice interactive animation of the above behavior see: http://foldr.com

The problem is that (+) is strict in both of its arguments. This means that both arguments must be fully evaluated before (+) can return a result. So to evaluate:

1 + (2 + (3 + (4 + (...))))

`1` is pushed on the stack. Then:

2 + (3 + (4 + (...)))

is evaluated. So `2` is pushed on the stack. Then:

3 + (4 + (...))

is evaluated. So `3` is pushed on the stack. Then:

4 + (...)

is evaluated. So `4` is pushed on the stack. Then: ...

... your limited stack will eventually run full when you evaluate a large enough chain of (+)s. This then triggers the stack overflow exception.

Lets think about how to solve it...

## 2 Foldl

One problem with the chain of (+)'s is that we can't make it smaller (reduce it) until at the very last moment when it's already too late.

The reason we can't reduce it, is that the chain doesn't contain an
expression which can be reduced (a so called "redex" for **red**ucible
**ex**pression.) If it did we could reduce that expression before going
to the next element.

Well, we can introduce a redex by forming the chain in another way. If
instead of the chain `1 + (2 + (3 + (...)))` we could form the chain
`(((0 + 1) + 2) + 3) + ...` then there would always be a redex.

We can form the latter chain by using a function called *foldl*:

> foldl f z [] = z > foldl f z (x:xs) = let z' = z `f` x > in foldl f z' xs > sum2 = foldl (+) 0 > try2 = sum2 veryBigList

Lets evaluate *try2*:

`*** Exception: stack overflow`

Good Lord! Again a stack overflow! Lets see what happens:

try2 --> sum2 veryBigList --> foldl (+) 0 veryBigList --> foldl (+) 0 [1..1000000] --> let z1 = 0 + 1 in foldl (+) z1 [2..1000000] --> let z1 = 0 + 1 z2 = z1 + 2 in foldl (+) z2 [3..1000000] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 in foldl (+) z3 [4..1000000] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 in foldl (+) z4 [5..1000000] --> -- ... after many foldl steps ... let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 in foldl (+) z999997 [999998..1000000] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 in foldl (+) z999998 [999999..1000000] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 z999999 = z999998 + 999999 in foldl (+) z999999 [1000000] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 z999999 = z999998 + 999999 z100000 = z999999 + 1000000 in foldl (+) z1000000 [] --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 z999999 = z999998 + 999999 z100000 = z999999 + 1000000 in z1000000 --> -- Now a large chain of +'s will be created: let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 z999999 = z999998 + 999999 in z999999 + 1000000 --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 in (z999998 + 999999) + 1000000 --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 in ((z999997 + 999998) + 999999) + 1000000 --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... in (((z999996 + 999997) + 999998) + 999999) + 1000000 --> -- ... -- ... My stack overflows when there's a chain of around 500000 (+)'s !!! -- ... But the following would happen if you got a large enough stack: -- ... let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 in (((((z4 + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 in ((((((z3 + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> let z1 = 0 + 1 z2 = z1 + 2 in (((((((z2 + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> let z1 = 0 + 1 in ((((((((z1 + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> (((((((((0 + 1) + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> -- Now we can actually start reducing: ((((((((1 + 2) + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> (((((((3 + 3) + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> ((((((6 + 4) + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> (((((10 + 5) + ...) + 999997) + 999998) + 999999) + 1000000 --> ((((15 + ...) + 999997) + 999998) + 999999) + 1000000 --> (((499996500006 + 999997) + 999998) + 999999) + 1000000 --> ((499997500003 + 999998) + 999999) + 1000000 --> (499998500001 + 999999) + 1000000 --> 499999500000 + 1000000 --> 500000500000 -->

For a nice interactive animation of the above behavior see: http://foldl.com (actually this animation is not quite the same :-( )

Well, you clearly see that the redexen are created. But instead of being directly reduced, they are allocated on the heap:

let z1 = 0 + 1 z2 = z1 + 2 z3 = z2 + 3 z4 = z3 + 4 ... z999997 = z999996 + 999997 z999998 = z999997 + 999998 z999999 = z999998 + 999999 z100000 = z999999 + 1000000 in z1000000

Note that your heap is only limited by the amount of memory in your system (RAM and swap). So the only thing this does is filling up a large part of your memory.

The problem starts when we finally evaluate z1000000:

Note that `z1000000 = z999999 + 1000000`.
So `1000000` is pushed on the stack.
Then `z999999` is evaluated.

Note that `z999999 = z999998 + 999999`.
So `999999` is pushed on the stack.
Then `z999998` is evaluated:

Note that `z999998 = z999997 + 999998`.
So `999998` is pushed on the stack.
Then `z999997` is evaluated:
So ...

...your limited stack will eventually run full when you evaluate a large enough chain of (+)s. This then triggers the stack overflow exception.

But this is exactly the problem we had in the foldr case! Only now the chain of (+) is going to the left instead of going to the right.

So why doesn't the chain reduce sooner than before?

The answer is that GHC uses a lazy reduction strategy. This means that GHC only reduces an expression when its value is actually needed.

The reduction strategy works by reducing the outer-left-most redex
first. In this case it are the outer `foldl (+) ... [1..10000]`
redexen which are repeatedly reduced.
So the inner `z1, z2, z3, ...` redexen only get reduced when
the foldl is completely gone.

## 3 Foldl'

We somehow have to tell the system that the inner redex should be
reduced before the outer. Fortunately this is possible with the
*seq* function:

seq :: a -> b -> b

*seq* is a primitive system function that when applied to *x* and
*y* will first reduce *x*, then reduce *y* and return the result of
the latter. The idea is that *y* references *x* so that when *y* is
reduced *x* will not be a big unreduced chain anymore.

Now lets fill in the pieces:

> foldl' f z [] = z > foldl' f z (x:xs) = let z' = z `f` x > in seq z' $ foldl' f z' xs > sum3 = foldl' (+) 0 > try3 = sum3 veryBigList

If we now evaluate *try3* we get the correct answer and we get it very quickly:

`500000500000`

Lets see what happens:

try3 --> foldl' (+) 0 [1..1000000] --> foldl' (+) 1 [2..1000000] --> foldl' (+) 3 [3..1000000] --> foldl' (+) 6 [4..1000000] --> foldl' (+) 10 [5..1000000] --> -- ... -- ... You see that the stack doesn't overflow -- ... foldl' (+) 499999500000 [1000000] --> foldl' (+) 500000500000 [] --> 500000500000

You can clearly see that the inner redex is repeatedly reduced first.

*first*argument,

> (?) :: Int -> Int -> Int > _ ? 0 = 0 > x ? y = x*y > > list :: [Int] > list = [2,3,undefined,5,0] > > okey = foldl (?) 1 list > > boom = foldl' (?) 1 list

Let's see what happens:

okey --> foldl (?) 1 [2,3,undefined,5,0] --> foldl (?) (1 ? 2) [3,undefined,5,0] --> foldl (?) ((1 ? 2) ? 3) [undefined,5,0] --> foldl (?) (((1 ? 2) ? 3) ? undefined) [5,0] --> foldl (?) ((((1 ? 2) ? 3) ? undefined) ? 5) [0] --> foldl (?) (((((1 ? 2) ? 3) ? undefined) ? 5) ? 0) [] --> ((((1 ? 2) ? 3) ? undefined) ? 5) ? 0 --> 0 boom --> foldl' (?) 1 [2,3,undefined,5,0] --> 1 ? 2 --> 2 foldl' (?) 2 [3,undefined,5,0] --> 2 ? 3 --> 6 foldl' (?) 6 [undefined,5,0] --> 6 ? undefined --> <nowiki>*** Exception: Prelude.undefined</nowiki>

For another explanation about folds see the Fold article.