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Foldable and Traversable

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(fmap and toList must commute [fmap can't change the structure and toList's order can't depend on the values])
(Foldable: Set and StorableVector are Foldable although they are not Functors)
 
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operations.
 
operations.
   
When you've been using it for a little while, there seem to be some
+
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 "<hask>map</hask>" and "<hask>toList</hask>".
baffling omissions from the API. The first couple you are likely to
+
The answer to these lies in the long list of instances which Sequence has:
notice are the absence of "<hask>map</hask>" and "<hask>toList</hask>".
+
* The Sequence version of map is "<hask>fmap</hask>", which comes from the Functor class.
+
* The Sequence version of <hask>toList</hask> is in the <hask>Foldable</hask> [[class]].
The answer to these lies in the long list of instances which Sequence
 
has. The Sequence version of map is "<hask>fmap</hask>", which comes from the
 
Functor class. The Sequence version of <hask>toList</hask> is in the <hask>Foldable</hask> [[class]].
 
   
 
When working with <hask>Sequence</hask> you also want to refer to the documentation
 
When working with <hask>Sequence</hask> you also want to refer to the documentation
for at least <hask>Foldable</hask> and <hask>Traversable</hask>. <hask>Functor</hask> only has the single
+
for at least <hask>Foldable</hask> and <hask>Traversable</hask>. <hask>Functor</hask> only has the single [[method]], so we've already covered that.
[[method]], so we've already covered that.
 
   
 
==What do these classes all mean? A brief tour:==
 
==What do these classes all mean? A brief tour:==
Line 22: Line 22:
 
===<hask>Functor</hask>===
 
===<hask>Functor</hask>===
   
A [[functor]] is simply a [[container]]. Given a container, and a [[function]]
+
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 "<hask>map</hask>" does exactly this.
which works on the elements, we can apply that function to each
 
element. For lists, the familiar "<hask>map</hask>" does exactly this.
 
   
 
Note that the function can produce elements of a different [[type]], so we
 
Note that the function can produce elements of a different [[type]], so we
Line 38: Line 38:
 
===Foldable===
 
===Foldable===
   
A <hask>Foldable</hask> [[type]] is also a [[container]] (although the [[class]] does not
+
A <hask>Foldable</hask> [[type]] is also a [[container]].
technically require <hask>Functor</hask>, interesting <hask>Foldable</hask>s are all <hask>Functor</hask>s). It is a container with the added property that its items
+
The [[class]] does not require <hask>Functor</hask> superclass
can be 'folded' to a summary value. In other words, it is a type which
+
in order to allow containers like <hask>Set</hask> or <hask>StorableVector</hask>
supports "<hask>foldr</hask>".
+
that have additional constraints on the element type.
  +
But many interesting <hask>Foldable</hask>s are also <hask>Functor</hask>s.
  +
A foldable container 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 "<hask>foldr</hask>".
   
Once you support <hask>foldr</hask>, of course, you can be turned into a list, by
+
Once you support <hask>foldr</hask>, of course, it can be turned into a list, by using <hask>toList = foldr (:) []</hask>. This means that all <hask>Foldable</hask>s have a representation as a list, but the order of the items may or may not have any particular significance. However, if a <hask>Foldable</hask> is also a <hask>Functor</hask>, [[parametricity]] and the [[Functor law]] guarantee that <hask>toList</hask> and <hask>fmap</hask> commute. Further, in the case of <hask>Data.Sequence</hask>, there '''is''' a well defined order and it is exposed as expected by <hask>toList</hask>.
using <hask>toList = foldr (:) []</hask>. This means that all <hask>Foldable</hask>s have a
 
representation as a list, but the order of the items may or may
 
not have any particular significance. However, if a <hask>Foldable</hask> is
 
also a <hask>Functor</hask>, [[parametricity]] and the [[Functor law]] guarantee that <hask>toList</hask> and <hask>fmap</hask> commute. Further, in the case of <hask>Data.Sequence</hask>, there '''is''' a well defined order and it is exposed as expected by <hask>toList</hask>.
 
   
A particular kind of fold well-used by Haskell programmers is
+
A particular kind of fold well-used by Haskell programmers is <hask>mapM_</hask>, which is a kind of fold over <hask>(>>)</hask>, and <hask>Foldable</hask> provides this along with the related <hask>sequence_</hask>.
<hask>mapM_</hask>, which is a kind of fold over
 
<hask>(>>)</hask>, and <hask>Foldable</hask> provides this along with the
 
related <hask>sequence_</hask>.
 
   
 
===Traversable===
 
===Traversable===
   
A <hask>Traversable</hask> [[type]] is a kind of upgraded <hask>Foldable</hask>. Where <hask>Foldable</hask>
+
A <hask>Traversable</hask> [[type]] is a kind of upgraded <hask>Foldable</hask>. Where <hask>Foldable</hask> gives you the ability to go through the structure processing the elements (<hask>foldr</hask>) but throwing away the shape, <hask>Traversable</hask> allows you to do that whilst preserving the shape and, e.g., putting new values in.
gives you the ability to go through the structure processing the
 
elements (<hask>foldr</hask>) but throwing away the shape, <hask>Traversable</hask> allows you
 
to do that whilst preserving the shape and, e.g., putting new values
 
in.
 
   
<hask>Traversable</hask> is what we need for <hask>mapM</hask> and
+
<hask>Traversable</hask> is what we need for <hask>mapM</hask> and <hask>sequence</hask> : note the apparently surprising fact that the "_" versions are in a different [[typeclass]].
<hask>sequence</hask> : note the apparently surprising fact that the
 
"_" versions are in a different [[typeclass]].
 
   
 
== Some trickier functions: concatMap and filter ==
 
== Some trickier functions: concatMap and filter ==
   
Neither <hask>Traversable</hask> nor <hask>Foldable</hask> contain elements for <hask>concatMap</hask> and <hask>filter</hask>. That is because <hask>Foldable</hask> is about tearing down the structure
+
Neither <hask>Traversable</hask> nor <hask>Foldable</hask> contain elements for <hask>concatMap</hask> and <hask>filter</hask>. That is because <hask>Foldable</hask> is about tearing down the structure completely, while <hask>Traversable</hask> is about preserving the structure exactly as-is. On the other hand <hask>concatMap</hask> tries to 'squeeze more elements in' at a place and <hask>filter</hask> tries to cut them out.
completely, while <hask>Traversable</hask> is about preserving the structure
 
exactly as-is. On the other hand <hask>concatMap</hask> tries to
 
'squeeze more elements in' at a place and <hask>filter</hask> tries to
 
cut them out.
 
   
 
You can write <hask>concatMap</hask> for <hask>Sequence</hask> as follows:
 
You can write <hask>concatMap</hask> for <hask>Sequence</hask> as follows:
Line 64: Line 64:
 
</haskell>
 
</haskell>
   
But why does it work? It works because sequence is an instance of<hask>Monoid</hask>, where the [[monoid]]al operation is "appending". The same
+
But why does it work? It works because sequence is an instance of <hask>Monoid</hask>, where the [[monoid]]al operation is "appending". The same definition works for lists, and we can write it more generally as:
definition works for lists, and we can write it more generally as:
 
   
 
<haskell>
 
<haskell>
Line 71: Line 71:
 
</haskell>
 
</haskell>
   
And that works with lists and sequences both. Does it work with any
+
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 <hask>toList (f `mappend` g) = toList f ++ toList g</hask> 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.
Monoid which is Foldable? Only if the Monoid 'means the right
 
thing'. If you have <hask>toList (f `mappend` g) = toList f ++ toList g</hask> 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.
 
   
<hask>filter</hask> turns out to be slightly harder still. You need
+
<hask>filter</hask> turns out to be slightly harder still. You need something like 'singleton' (from <hask>Sequence</hask>), or <hask>\a -> [a]</hask> for lists. We can use <hask>pure</hask> from <hask>Applicative</hask>, although it's not really right to bring <hask>Applicative</hask> in for this, and get:
something like 'singleton' (from <hask>Sequence</hask>), or <hask>\a -> [a]</hask>
 
for lists. We can use <hask>pure</hask> from <hask>Applicative</hask>, although
 
it's not really right to bring <hask>Applicative</hask> in for this, and get:
 
   
 
<haskell>
 
<haskell>
Line 81: Line 81:
 
</haskell>
 
</haskell>
   
It's interesting to note that, under these conditions, we have a candidate
+
It's interesting to note that, under these conditions, we have a candidate to help us turn the <hask>Foldable</hask> into a <hask>Monad</hask>, since <hask>concatMap</hask> is a good definition for <hask>>>=</hask>, and we can use <hask>pure</hask> for <hask>return</hask>.
to help us turn the <hask>Foldable</hask> into a <hask>Monad</hask>, since <hask>concatMap</hask> is a good
 
definition for <hask>>>=</hask>, and we can use <hask>pure</hask> for <hask>return</hask>.
 
   
 
== Generalising zipWith ==
 
== Generalising zipWith ==
   
Another really useful list [[combinator]] that doesn't appear in the
+
Another really useful list [[combinator]] that doesn't appear in the interfaces for <hask>Sequence</hask>, <hask>Foldable</hask> or <hask>Traversable</hask> is <hask>zipWith</hask>. The most general kind of <hask>zipWith</hask> over <hask>Traversable</hask>s will keep the exact shape of the <hask>Traversable</hask> on the left, whilst zipping against the values on the right. It turns out you can get away with a <hask>Foldable</hask> on the right, but you need to use a <hask>Monad</hask> (or an <hask>Applicative</hask>, actually) to thread the values through:
interfaces for <hask>Sequence</hask>, <hask>Foldable</hask> or <hask>Traversable</hask> is <hask>zipWith</hask>. The most general kind of <hask>zipWith</hask> over <hask>Traversable</hask>s will keep the exact shape of
 
the <hask>Traversable</hask> on the left, whilst zipping against the values on the right. It turns out you can get away with a <hask>Foldable</hask> on the right, but you need to use a <hask>Monad</hask> (or an <hask>Applicative</hask>, actually) to thread the
 
values through:
 
   
 
<haskell>
 
<haskell>
Line 201: Line 201:
 
where map_one (x:xs) y = (xs, g y x)
 
where map_one (x:xs) y = (xs, g y x)
 
</haskell>
 
</haskell>
Replace <hask>mapAccumL</hask> with <hask>mapAccumR</hask> and the elements of the Foldable are zipped in reverse order.
+
Replace <hask>mapAccumL</hask> with <hask>mapAccumR</hask> and the elements of the Foldable are zipped in reverse order. Similarly, we can define a generalization of <hask>reverse</hask> on Traversables, which preserves the shape but reverses the left-to-right position of the elements:
Similarly, we can define a generalization of <hask>reverse</hask> on Traversables, which preserves the shape but reverses the left-to-right position of the elements:
 
 
<haskell>
 
<haskell>
 
reverseT :: (Traversable t) => t a -> t a
 
reverseT :: (Traversable t) => t a -> t a

Latest revision as of 10:16, 12 January 2013


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

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

FunctorHierarchy.svg

[edit] 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"]

[edit] 1.2 Foldable

A
Foldable
type is also a container. The class does not require
Functor
superclass in order to allow containers like
Set
or
StorableVector

that have additional constraints on the element type.

But many interesting
Foldable
s are also
Functor
s.

A foldable container 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, it 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_
.

[edit] 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.

[edit] 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
.

[edit] 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)