Stack overflow

Folds

First, read Performance/Accumulating parameter. If you are not writing your code tail-recursively, then that is why you are getting stack overflows. However, making code tail-recursive in a lazy language is not quite the same as in a eager language. This page is more geared to the latter case using foldr/l as the prime culprit/example. As such Fold may be helpful, but isn't too critical. Also knowing what seq and ($!) do, as covered in Making Haskell programs faster and smaller and in the Haskell Report is necessary.

The definitions of the three folds we'll be looking at are as follows:

foldr  f z []     =  z
foldr  f z (x:xs) =  f x (foldr f z xs)

foldl  f z []     =  z
foldl  f z (x:xs) =  foldl f (f z x) xs

foldl' f z []     =  z
foldl' f z (x:xs) =  (foldl' f $! f z x) xs

foldl' (found in e.g. Data.List) is just a stricter version of foldl.

The one-line summary for folds: if the binary operation is strict use foldl', otherwise use foldr.

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Common newbie stack overflowing code:

mysum :: [Integer] -> Integer
mysum = foldr (+) 0

main = print (mysum [1..1000000])

If you've read Performance/Accumulating parameter, you should immediately see the problem from the definition of foldr above. Quite simply, foldr isn't tail-recursive! But,

concat xss = foldr (++) [] xss

This is from the Haskell Report. Surely they know what they are doing! And sure enough,

main = print (length (concat [[x] | x <- [1..1000000]]))

works fine.

Less common newbie stack overflowing code:

mysum :: [Integer] -> Integer
mysum = foldl (+) 0 -- foldl instead of foldr this time

main = print (mysum [1..1000000])

So what's going on here? Looking at the code for foldl, it looks tail-recursive. Well, much like you can see the problem with a non-tail-recursive factorial by unfolding a few iterations, let's do the same for our foldl definition of sum, but making sure to use a call-by-name/need evaluation order. Here is the unfolding,

mysum [1..10] ->
foldl (+) 0 (1:[2..10]) ->
foldl (+) (0+1) (2:[3..10]) ->
foldl (+) (0+1+2) (3:[4..10]) ->
foldl (+) (0+1+2+3) (4:[5..10]) -> <nowiki>...</nowiki>

I think you get the idea. The problem is that we are building up a chain of thunks that will evaluate the sum instead of just maintaining a running sum. What we need to do is to force the addition before recursing. This is exactly what foldl' does.

Just to check,

mysum :: [Integer] -> Integer
mysum = foldl' (+) 0

main = print (mysum [1..1000000])

works fine.

Now let's go back to the foldr sum and concat. What's the difference between sum and concat that makes the sum definition wrong, but the concat definition right. Again, let's evaluate each by hand.

mysum (+) 0 [1..10] ->
foldr (+) 0 (1:[2..10]) ->
1+foldr (+) 0 (2:[3..10]) ->
1+(2+foldr (+) 0 (3:[4..10])) -> <nowiki>...</nowiki>

Okay, no surprise there.

concat [[1],[2],[3],<nowiki>...</nowiki>] ->
foldr (++) [[]] ([1]:[[2],[3],<nowiki>...</nowiki>]) ->
(1:[])++foldr (++) [[]] [[2],[3],<nowiki>...</nowiki>] ->
1:([]++foldr (++) [[]] [[2],[3],<nowiki>...</nowiki>])

Notice that there is no '-> ...' at the end. That was the complete evaluation. There is no reason to do anything more, unless we look at the result. We may well GC the 1 before we look at the tail, and GC the first cons cell before we look at the second. So, concat runs in a constant amount of stack and further can handle infinite lists (as a note, it's immediately obvious foldl(') can never work on infinite lists because we'll always be in the (:) case and that always immediately recurses). The differentiator between mysum and concat is that (++) is not strict* in its second argument; we don't have to evaluate the rest of the foldr to know the beginning of concat. In mysum, since (+) is strict in its second argument, we need the results of the whole foldr before we can compute the final result.

So, we arrive at the one-line summary: A function strict* in its second argument will always require linear stack space with foldr, so foldl' should be used instead in that case. If the function is lazy/non-strict in its second argument we should use foldr to 1) support infinite lists and 2) to allow a streaming use of the input list where only part of it needs to be in memory at a time.

Okay, both here and in the one-line summary, there is no mention of foldl. When should foldl be used? The pragmatic answer is: by and far, it shouldn't be used. A case where it makes a difference is if the function is conditionally strict in its first argument depending on its second, where I use conditionally strict to mean a function that is strict or not in one argument depending on another argument(s). For an example, consider a definition of (*) that builds up ASTs of arithmetic expressions and incorporates a simplification (a*0 = 0 and then 0*a = 0); then if product is defined by foldl (*) 1, product [⊥,0] will terminate with 0 while a definition in terms of foldl' wouldn't. However, I can't think of a really convincing example. In most cases, foldl' is what you want.

* A strict function is a function f, such that f⊥=⊥. Typically, we think of a function "being strict" in an argument as a function that "forces" its argument, but the above definition of strict should immediately suggest another function that is strict and doesn't "force" it's argument in the intuitive sense, namely id. ([]++) = id and therefore is a strict function. Sure enough, if you were to evaluate (concat (repeat [])) it would not terminate. As such, (++) is a conditionally strict function. This also makes the "always" slightly imprecise, a function that is strict because it just returns it's argument, will not use up stack space (but is, as mentioned, still an issue for infinitely long lists).

Weak Head Normal Form

Common newbie stack overflowing code:

myAverage = uncurry (/) . foldl' (\(acc, len) x -> (acc+x, len+1)) (0,0)

People who understand seq and weak head normal form (whnf) can immediately understand what goes wrong here. (acc+x, len+1) is already in whnf, so seq, which reduces a value to whnf, does nothing to this. This code will build up thunks just like the original foldl example, they'll just be inside a tuple. The solution is just to force the components of the tuple, e.g.

myAverage = uncurry (/) . foldl' (\(acc, len) x -> acc `seq` len `seq` (acc+x, len+1)) (0,0)

or more clearly and concisely using a recent GHC extension

myAverage = uncurry (/) . foldl' (\(!acc, !len) x -> (acc+x, len+1)) (0,0)

Scans

A subtle stack-overflow surprise comes when

print (scanl (+) 0 [1..1000000])

completes successfully but

print (last (scanl (+) 0 [1..1000000]))

causes a stack overflow.

The latter stack overflow is explained exactly as before, namely,

last (scanl (+) 0 [1..5]) ->
<nowiki>... several steps ...</nowiki> ->
((((0+1)+2)+3)+4)+5

This is exactly like foldl, building a deep thunk, then evaluating, needing much stack.

Most puzzling is why the former succeeds without a stack overflow. This is caused by a combination of two factors:

• thunks in the list produced by scanl enjoy sharing: late thunks build upon early thunks
• printing a list of numbers evaluates early thunks and then late thunks

To exemplify, here is an abridged progression. I use this pseudo format to depict sharing of thunks

expr where var=expr, var=expr

although in reality it is more like a pointer graph.

print (scanl (+) 0 [1..1000000]) ->
print (a : case [1..1000000] of <nowiki>...</nowiki> x:xs -> scanl (+) (a+x) xs) where a=0 ->
<nowiki>... evaluate a to 0 for printing, I/O, some more steps ...</nowiki> ->

print (scanl (+) (a+1) [2..1000000]) where a=0 ->
print (b : case [2..1000000] of <nowiki>...</nowiki> x:xs -> scanl (+) (b+x) xs) where a=0, b=a+1 ->
<nowiki>... evaluate b to 1 for printing, I/O, some more steps ...</nowiki> ->

print (scanl (+) (b+2) [3..1000000]) where b=1 ->
print (c : case [3..1000000] of <nowiki>...</nowiki> x:xs -> scanl (+) (c+x) xs) where b=1, c=b+2 ->
<nowiki>... evaluate c to 3 for printing, I/O, some more steps ...</nowiki> ->

print (scanl (+) (c+3) [4..1000000]) where c=3 ->
print (d : case [4..1000000] of <nowiki>...</nowiki> x:xs -> scanl (+) (d+x) xs) where c=3, d=c+3 ->
<nowiki>... evaluate d to 6 for printing, I/O, some more steps ...</nowiki> ->

print (scanl (+) (d+4) [5..1000000]) where d=6 -> etc.

The important thing to watch is the life cycle of intermediate thunks, e.g., c is created at some point as a 1-level deep addition, then almost immediately reduced to a number out of necessity, before a later thunk d builds upon it. Therefore there is no growth and no stack overflow.

In contrast, again, last (scanl (+) 0 [1..1000000]) skips over to the last thunk right away. Since early items are not reduced yet, the last item remains a huge chain and causes overflow.

As an addendum, there are three ways of handling this problem and similar ones:

1. You can have your traversal functions (in this case, last) force the list as it goes along.
2. You can use (perhaps custom) versions of the list (or data structure, in general) producing functions (in this case, scanl) that force the elements as it builds the data structure.
3. You can use a data structure that's strict in its elements, in this case it would be a head strict list.