Am Montag, 9. März 2009 17:46 schrieb 7stud:
This is an example that shows how foldl and foldr work (from RWH p.93-94):
foldl (+) 0 (1:2:3:[]) == foldl (+) (0 + 1) (2:3:[]) == foldl (+) ((0 + 1) + 2) (3:[]) == foldl (+) (((0 + 1) + 2) + 3) [] == (((0 + 1) + 2) + 3)
foldr (+) 0 (1:2:3:[]) == 1 + foldr (+) 0 (2:3:[]) == 1 + (2 + foldr (+) 0 (3:[]) == 1 + (2 + (3 + foldr (+) 0 [])) == 1 + (2 + (3 + 0))
The book says on p.94:
----- The difference between foldl and foldr should be clear from looking at where the parentheses and the empty list elements show up. With foldl, the empty list element is on the left, and all the parentheses group to the left. With foldr, the zero value is on the right, and the parentheses group to the right. ----
Huh? With foldl, the only empty list element I see is on the right.
What they meant was "the value that is the result in case the fold is applied to an empty list", in this case the 0, in the definition fold(l/r) f z xs = ... the 'z'.
Initially, it looked to me ike they did the same thing, and that the only difference was the way they called step. I think "step" is a horrible, non-descriptive name, so I'm going to use "accFunc" instead:
foldl calls: accFunc acc x
foldr calls: accFunc x acc
So it looks like you can define a function using either one and get the same result.
Note that in general the list elements and acc have different types, so only one of accFun acc x and accFun x acc typechecks. If the types are the same, in general accFun x acc /= accFun acc x, so foldr and foldl give different results, too.
Here is a test:
--I am going to use odd for pfunc and [1, 2, 3] for xs:
myFilter1 pfunc xs = foldl accFunc [] xs where accFunc acc x
| pfunc x = acc ++ [x] | otherwise = acc
myFilter2 pfunc xs = foldr accFunc [] xs where accFunc x acc
| pfunc x = acc ++ [x] | otherwise = acc
*Main> myFilter1 odd [1, 2, 3] [1,3] *Main> myFilter2 odd [1, 2, 3] [3,1]
Hmmm. So there is a difference. foldr appears to grab elements from the end of the list. Therefore, to get the same result from the function that uses foldr, I did this:
myFilter3 pfunc xs = foldr accFunc [] xs where accFunc x acc
| pfunc x = x : acc | otherwise = acc
*Main> myFilter3 odd [1, 2, 3] [1,3]
But then RWH explains that you would never use foldl in practice because it thunks the result, which for large lists can overwhelm the maximum memory alloted for a thunk. But it appears to me the same thunk problem would occur with foldr. So why is foldr used in practice but not foldl?
Since with foldr, the parentheses are grouped to the right: x0 `f` (x1 `f` (x2 `f` ... (xn `f` z) ... )), if f can start delivering the result without looking at its second argument, you can start consuming the result before the fold has traversed the whole list. Common examples are things like concat = foldr (++) [], so concat [l1,l2,l3,l4,l5] = l1 ++ (foldr (++) [] [l2,l3,l4,l5]) and the start (l1) can be used before further reducing the fold, and = foldr (&&) True and [True,False,..........] needs only inspect the list until it encounters the first False (if any, otherwise it must of course traverse the whole list) or = foldr (||) False foldr is useful if the combination function is lazy in its second argument. foldl on the other hand can't deliver anything before the whole list is consumed. So since foldl builds thunks (except in some easy cases where the optimiser sees it should be strict), which would have to be evaluated at the end when they've become rather large, foldl isn't as useful and one uses the strict left fold, foldl'.