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Foldable and Traversable

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Notes on Foldable, Traversable and other useful classes
or "Where is Data.Sequence.toList?"

Data.Sequence is recommended as an efficient alternative to [list]s, with a more symmetric feel and better complexity on various operations.

When you've been using it for a little while, there seem to be some baffling omissions from the API. The first couple you are likely to notice are the absence of "
map
" and "
toList
".

The answer to these lies in the long list of instances which Sequence has:

  • The Sequence version of map is "
    fmap
    ", which comes from the Functor class.
  • The Sequence version of
    toList
    is in the
    Foldable
    class.
When working with
Sequence
you also want to refer to the documentation for at least
Foldable
and
Traversable
.
Functor
only has the single method, so we've already covered that.

Contents

1 What do these classes all mean? A brief tour:

Image:FunctorHierarchy.svg

1.1
Functor

A functor is simply a container. Given a container, and a function which works on the elements, we can apply that function to each element. For lists, the familiar "
map
" does exactly this.

Note that the function can produce elements of a different type, so we may have a different type at the end.

Examples: 

Prelude Data.Sequence> map (\n -> replicate n 'a') [1,3,5]
["a","aaa","aaaaa"]
Prelude Data.Sequence> fmap (\n -> replicate n 'a') (1 <| 3 <| 5 <| empty)
fromList ["a","aaa","aaaaa"]

1.2 Foldable

A
Foldable
type is also a container (although the class does not technically require
Functor
, interesting
Foldable
s are all
Functor
s). It is a container with the added property that its items can be 'folded' to a summary value. In other words, it is a type which supports "
foldr
". Once you support
foldr
, of course, you can be turned into a list, by using
toList = foldr (:) []
. This means that all
Foldable
s have a representation as a list, but the order of the items may or may not have any particular significance. However, if a
Foldable
is also a
Functor
, parametricity and the Functor law guarantee that
toList
and
fmap
commute. Further, in the case of
Data.Sequence
, there is a well defined order and it is exposed as expected by
toList
. A particular kind of fold well-used by Haskell programmers is
mapM_
, which is a kind of fold over
(>>)
, and
Foldable
provides this along with the related
sequence_
.

1.3 Traversable

A
Traversable
type is a kind of upgraded
Foldable
. Where
Foldable
gives you the ability to go through the structure processing the elements (
foldr
) but throwing away the shape,
Traversable
allows you to do that whilst preserving the shape and, e.g., putting new values in.
Traversable
is what we need for
mapM
and
sequence
 : note the apparently surprising fact that the "_" versions are in a different typeclass.

2 Some trickier functions: concatMap and filter

Neither
Traversable
nor
Foldable
contain elements for
concatMap
and
filter
. That is because
Foldable
is about tearing down the structure completely, while
Traversable
is about preserving the structure exactly as-is. On the other hand
concatMap
tries to 'squeeze more elements in' at a place and
filter
tries to cut them out. You can write
concatMap
for
Sequence
as follows: 
concatMap :: (a -> Seq b) -> Seq a -> Seq b
concatMap = foldMap
But why does it work? It works because sequence is an instance of
Monoid
, where the monoidal operation is "appending". The same definition works for lists, and we can write it more generally as: 
concatMap :: (Foldable f, Monoid (f b)) => (a -> f b) -> f a -> f b
concatMap = foldMap
And that works with lists and sequences both. Does it work with any Monoid which is Foldable? Only if the Monoid 'means the right thing'. If you have
toList (f `mappend` g) = toList f ++ toList g
then it definitely makes sense. In fact this easy to write condition is stronger than needed; it would be good enough if they were permutations of each other.
filter
turns out to be slightly harder still. You need something like 'singleton' (from
Sequence
), or
\a -> [a]
for lists. We can use
pure
from
Applicative
, although it's not really right to bring
Applicative
in for this, and get: 
filter :: (Applicative f, Foldable f, Monoid (f a)) => 
          (a -> Bool) -> f a -> f a
filter p = foldMap (\a -> if p a then pure a else mempty)
It's interesting to note that, under these conditions, we have a candidate to help us turn the
Foldable
into a
Monad
, since
concatMap
is a good definition for
>>=
, and we can use
pure
for
return
.

3 Generalising zipWith

Another really useful list combinator that doesn't appear in the interfaces for
Sequence
,
Foldable
or
Traversable
is
zipWith
. The most general kind of
zipWith
over
Traversable
s will keep the exact shape of the
Traversable
on the left, whilst zipping against the values on the right. It turns out you can get away with a
Foldable
on the right, but you need to use a
Monad
(or an
Applicative
, actually) to thread the

values through: 

import Prelude hiding (sequence)
 
import Data.Sequence
import Data.Foldable
import Data.Traversable
import Control.Applicative
 
 
data Supply s v = Supply { unSupply :: [s] -> ([s],v) }
 
instance Functor (Supply s) where 
  fmap f av = Supply (\l -> let (l',v) = unSupply av l in (l',f v))
 
instance Applicative (Supply s) where
  pure v    = Supply (\l -> (l,v))
  af <*> av = Supply (\l -> let (l',f)  = unSupply af l
                                (l'',v) = unSupply av l'
                            in (l'',f v))
 
runSupply :: (Supply s v) -> [s] -> v
runSupply av l = snd $ unSupply av l
 
supply :: Supply s s
supply = Supply (\(x:xs) -> (xs,x))
 
zipTF :: (Traversable t, Foldable f) => t a -> f b -> t (a,b)
zipTF t f = runSupply (traverse (\a -> (,) a <$> supply) t) (toList f)
 
zipWithTF :: (Traversable t,Foldable f) => (a -> b -> c) -> t a -> f b -> t c
zipWithTF g t f = runSupply  (traverse (\a -> g a <$> supply) t) (toList f)
 
zipWithTFM :: (Traversable t,Foldable f,Monad m) => 
              (a -> b -> m c) -> t a -> f b -> m (t c)
zipWithTFM g t f = sequence (zipWithTF g t f)
 
zipWithTFA :: (Traversable t,Foldable f,Applicative m) => 
              (a -> b -> m c) -> t a -> f b -> m (t c)
zipWithTFA g t f = sequenceA (zipWithTF g t f)
The code above fails with a pattern match error when the
Foldable
container doesn't have enough input. Here is an alternative version which provides friendlier error reports and makes use of
State
instead of the self defined Supply monad. 
module GenericZip 
 (zipWithTF,
  zipTF,
  zipWithTFA,
  zipWithTFM) where
 
 
import Data.Foldable
import Data.Traversable
import qualified Data.Traversable as T
import Control.Applicative
import Control.Monad.State 
 
-- | The state contains the list of values obtained form the foldable container
--   and a String indicating the name of the function currectly being executed
data ZipState a = ZipState {fName :: String,
                            list  :: [a]}
 
-- | State monad containing ZipState
type ZipM l a = State (ZipState l) a
 
-- | pops the first element of the list inside the state
pop :: ZipM l l
pop = do 
 st <- get 
 let xs = list st
     n = fName st
 case xs of
   (a:as) -> do put st{list=as}
                return a
   [] -> error $ n ++ ": insufficient input"
 
-- | pop a value form the state and supply it to the second 
--   argument of a binary function 
supplySecond :: (a -> b -> c) -> a -> ZipM b c
supplySecond f a = do b <- pop  
                      return $ f a b
 
zipWithTFError :: (Traversable t,Foldable f) => 
                  String -> (a -> b -> c) -> t a -> f b -> t c  
zipWithTFError str g t f = evalState (T.mapM (supplySecond g) t) 
                                     (ZipState str (toList f))
 
 
zipWithTF :: (Traversable t,Foldable f) => (a -> b -> c) -> t a -> f b -> t c
zipWithTF = zipWithTFError "GenericZip.zipWithTF"
 
zipTF :: (Traversable t, Foldable f) => t a -> f b -> t (a,b)
zipTF = zipWithTFError "GenericZip.zipTF"  (,) 
 
 
zipWithTFM :: (Traversable t,Foldable f,Monad m) => 
              (a -> b -> m c) -> t a -> f b -> m (t c)
zipWithTFM g t f = T.sequence (zipWithTFError "GenericZip.zipWithTFM"  g t f)
 
zipWithTFA :: (Traversable t,Foldable f,Applicative m) => 
              (a -> b -> m c) -> t a -> f b -> m (t c)
zipWithTFA g t f = sequenceA (zipWithTFError "GenericZip.zipWithTFA" g t f)
Recent versions of
Data.Traversable
include generalizations of
mapAccumL
and
mapAccumR
from lists to Traversables (encapsulating the state monad used above): 
mapAccumL :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)
mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)

Using these, the first version above can be written as 

zipWithTF :: (Traversable t, Foldable f) => (a -> b -> c) -> t a -> f b -> t c
zipWithTF g t f = snd (mapAccumL map_one (toList f) t)
  where map_one (x:xs) y = (xs, g y x)
Replace
mapAccumL
with
mapAccumR
and the elements of the Foldable are zipped in reverse order. Similarly, we can define a generalization of
reverse
on Traversables, which preserves the shape but reverses the left-to-right position of the elements: 
reverseT :: (Traversable t) => t a -> t a
reverseT t = snd (mapAccumR (\ (x:xs) _ -> (xs, x)) (toList t) t)
Retrieved from "http://www.haskell.org/haskellwiki/Foldable_and_Traversable"

Categories: Code | Idioms