Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
* o...@okmij.org [2013-02-19 06:27:10-] > > As others have pointed out, _in principle_, foldr is not at all > deficient. We can, for example, express foldl via foldr. Moreover, we > can express head, tail, take, drop and even zipWith through > foldr. That is, the entire list processing library can be written in > terms of foldr: > > http://okmij.org/ftp/Algorithms.html#zip-folds > > That said, to express foldl via foldr, we need a higher-order > fold. There are various problems with higher-order folds, related to > the cost of building closures. The problems are especially severe > in strict languages or strict contexts. Indeed, > > foldl_via_foldr f z l = foldr (\e a z -> a (f z e)) id l z > > first constructs the closure and then applies it to z. The closure has > the same structure as the list -- it is isomorphic to the > list. However, the closure representation of a list takes typically > quite more space than the list. So, in strict languages, expressing > foldl via foldr is a really bad idea. It won't work for big lists. If we unroll foldr once (assuming l is not empty), we'll get \z -> foldr (\e a z -> a (f z e)) id (tail l) (f z (head l)) which is a (shallow) closure. In order to observe what you describe (a closure isomorphic to the list) we'd need a language which does reductions inside closures. Am I wrong? Roman ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
As others have pointed out, _in principle_, foldr is not at all deficient. We can, for example, express foldl via foldr. Moreover, we can express head, tail, take, drop and even zipWith through foldr. That is, the entire list processing library can be written in terms of foldr: http://okmij.org/ftp/Algorithms.html#zip-folds That said, to express foldl via foldr, we need a higher-order fold. There are various problems with higher-order folds, related to the cost of building closures. The problems are especially severe in strict languages or strict contexts. Indeed, foldl_via_foldr f z l = foldr (\e a z -> a (f z e)) id l z first constructs the closure and then applies it to z. The closure has the same structure as the list -- it is isomorphic to the list. However, the closure representation of a list takes typically quite more space than the list. So, in strict languages, expressing foldl via foldr is a really bad idea. It won't work for big lists. BTW, this is why foldM is _left_ fold. The arguments against higher-order folds as a `big hammer' were made back in 1998 by Gibbons and Jones http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.1735 So, the left-fold with the early termination has a good justification. In fact, this is how Iteratees were first presented, at the DEFUN08 tutorial (part of the ICFP2008 conference). The idea of left fold with early termination is much older though. For example, Takusen (a database access framework) has been using it since 2003 or so. For a bit of history, see http://okmij.org/ftp/Streams.html#fold-stream ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
On 18/02/13 16:10, Petr Pudlák wrote: > - `foldr` is unsuitable because it counts the elements from the end, > while `!!` needs counting from the start (and it's not tail recursive). It is common misconception that foldr processes the list "from the right". foldr "brackets" from the right, but this has nothing to do with processing direction; all [a] are processed left to right, since this is the only way to structurally deconstruct them. This is the reason why it is possible to write foldr (:) [] [1..] If foldr processed the list from the right, it would on infinite lists - and it does. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
Thanks Roman and Andres for the tip. I knew the trick with accumulating a function, but I had never imagined it could work so efficiently. I thought the problem with using foldr would be that the function would be neither tail recursive nor efficient and so I hadn't even tried. Apparently that was wrong. After your suggestion I checked its performance and how it compiles to core and to my surprise GHC optimizes the whole thing into a most-efficient tail recursive function! Best regards, Petr 2013/2/18 Roman Cheplyaka > * Petr Pudlįk [2013-02-18 17:10:26+0100] > > Dear Haskellers, > > > > while playing with folds and trying to implement `!!` by folding, I came > to > > the conclusion that: > > > > - `foldr` is unsuitable because it counts the elements from the end, > while > > `!!` needs counting from the start (and it's not tail recursive). > > - `foldl` is also unsuitable, because it always traverses the whole list. > > Every structurally-recursive function is definable through foldr, > because foldr *is* the structural recursion (aka catamorphism) operation > for lists. > > Here the trick is to make the accumulator a function. This way you can > pass a value from left to right. > > Something like > > foldr (\x rest n -> ...) id list 0 > > I'll leave filling in the dots as an exercise. > > Roman > ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
* Roman Cheplyaka [2013-02-18 18:28:47+0200] > * Petr Pudlák [2013-02-18 17:10:26+0100] > > Dear Haskellers, > > > > while playing with folds and trying to implement `!!` by folding, I came to > > the conclusion that: > > > > - `foldr` is unsuitable because it counts the elements from the end, while > > `!!` needs counting from the start (and it's not tail recursive). > > - `foldl` is also unsuitable, because it always traverses the whole list. > > Every structurally-recursive function is definable through foldr, > because foldr *is* the structural recursion (aka catamorphism) operation > for lists. > > Here the trick is to make the accumulator a function. This way you can > pass a value from left to right. > > Something like > > foldr (\x rest n -> ...) id list 0 > > I'll leave filling in the dots as an exercise. Er, my template is not quite right — I had 'length' in mind while writing it. Anyway, Andres showed the correct definition. Roman ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
* Petr Pudlák [2013-02-18 17:10:26+0100] > Dear Haskellers, > > while playing with folds and trying to implement `!!` by folding, I came to > the conclusion that: > > - `foldr` is unsuitable because it counts the elements from the end, while > `!!` needs counting from the start (and it's not tail recursive). > - `foldl` is also unsuitable, because it always traverses the whole list. Every structurally-recursive function is definable through foldr, because foldr *is* the structural recursion (aka catamorphism) operation for lists. Here the trick is to make the accumulator a function. This way you can pass a value from left to right. Something like foldr (\x rest n -> ...) id list 0 I'll leave filling in the dots as an exercise. Roman ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
Hi. > while playing with folds and trying to implement `!!` by folding, I came to > the conclusion that: > > - `foldr` is unsuitable because it counts the elements from the end, while > `!!` needs counting from the start (and it's not tail recursive). What is the problem with the following definition using foldr? > index :: Int -> [a] -> a > index n xs = > foldr > (\ x r n -> if n == 0 then x else r (n - 1)) > (const (error $ "No such index")) > xs > n Cheers, Andres ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
Dear Haskellers, while playing with folds and trying to implement `!!` by folding, I came to the conclusion that: - `foldr` is unsuitable because it counts the elements from the end, while `!!` needs counting from the start (and it's not tail recursive). - `foldl` is also unsuitable, because it always traverses the whole list. I came up with the following tail-recursive generalization of `foldl` that allows exiting the computation prematurely: foldlE :: (a -> c) -> (a -> b -> Either c a) -> Either c a -> [b] -> c foldlE f g = fld where fld (Left c) _ = c fld (Right a) []= f a fld (Right a) (x:xs)= fld (g a x) xs `foldl` can be defined from it as foldl'' :: (a -> b -> a) -> a -> [b] -> a foldl'' f z = foldlE id ((Right .) . f) (Right z) and `!!` as: -- Checks for a negative index omitted for brevity. index :: Int -> [a] -> a index i = foldlE (error $ "No such index") f (Right i) where f 0 x = Left x f n _ = Right (n - 1) Is something like that already available somewhere? Best regards, Petr Pudlak ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] generalized, tail-recursive left fold that can finish tne computation prematurely
Dear Haskellers, while playing with folds and trying to implement `!!` by folding, I came to the conclusion that: - `foldr` is unsuitable because it counts the elements from the end, while `!!` needs counting from the start (and it's not tail recursive). - `foldl` is also unsuitable, because it always traverses the whole list. I came up with the following tail-recursive generalization of `foldl` that allows exiting the computation prematurely: foldlE :: (a -> c) -> (a -> b -> Either c a) -> Either c a -> [b] -> c foldlE f g = fld where fld (Left c) _ = c fld (Right a) []= f a fld (Right a) (x:xs)= fld (g a x) xs `foldl` can be defined from it as foldl'' :: (a -> b -> a) -> a -> [b] -> a foldl'' f z = foldlE id ((Right .) . f) (Right z) and `!!` as: -- Checks for a negative index omitted for brevity. index :: Int -> [a] -> a index i = foldlE (error $ "No such index") f (Right i) where f 0 x = Left x f n _ = Right (n - 1) Is something like that already available somewhere? Best regards, Petr Pudlak ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
