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Foldable and Traversable

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=Notes on Foldable, Traversable and other useful classes=
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[[Category:Code]] [[Category:Idioms]]
  +
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<center>'''Notes on Foldable, Traversable and other useful classes'''</center>
 
<center>'' or "Where is Data.Sequence.toList?"''</center>
 
<center>'' or "Where is Data.Sequence.toList?"''</center>
   
Data.Sequence is recommended as an efficient alternative to lists,
+
Data.Sequence is recommended as an efficient alternative to [list]s,
 
with a more symmetric feel and better complexity on various
 
with a more symmetric feel and better complexity on various
 
operations.
 
operations.
Line 8: Line 8:
 
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
 
baffling omissions from the API. The first couple you are likely to
notice are the absence of "map" and "toList".
+
notice are the absence of "<hask>map</hask>" and "<hask>toList</hask>".
   
 
The answer to these lies in the long list of instances which Sequence
 
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
+
has. The Sequence version of map is "<hask>fmap</hask>", which comes from the
Functor class. The Sequence version of toList is in the Foldable
+
Functor class. The Sequence version of <hask>toList</hask> is in the <hask>Foldable</hask> [[class]].
class.
 
   
When working with Sequence 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 Foldable and Traversable. Functor 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:==
   
===Functor===
+
===<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
 
which works on the elements, we can apply that function to each
element. For lists, the familiar "map" does exactly this.
+
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
 
may have a different type at the end.
 
may have a different type at the end.
   
Line 40: Line 40:
 
===Foldable===
 
===Foldable===
   
A Foldable type is also a container (although the class does not
+
A <hask>Foldable</hask> [[type]] is also a [[container]] (although the [[class]] does not
technically require Functor, interesting Foldables are all
+
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
Functors). 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
 
can be 'folded' to a summary value. In other words, it is a type which
supports "foldr".
+
supports "<hask>foldr</hask>".
   
Once you support foldr, of course, you can be turned into a list, by
+
Once you support <hask>foldr</hask>, of course, you can be turned into a list, by
using <hask>foldr (:) []</hask>. This means that all Foldables have a
+
using <hask>foldr (:) []</hask>. This means that all <hask>Foldable</hask>s have a
 
representation as a list; however the order of the items may or may
 
representation as a list; however the order of the items may or may
not have any particular significance. In particular if a Foldable is
+
not have any particular significance. In particular if a <hask>Foldable<hask> is
also a Functor, toList and fmap need not perfectly commute; the list
+
also a <hask>Functor</hask>, <hask>toList</hask> and <hask>fmap</hask> need not perfectly commute; the list
given ''after'' the fmap may be in a different order to the list
+
given ''after'' the <hask>fmap</hask> may be in a different order to the list
''before'' the fmap. In the particular case of Data.Sequence, though,
+
''before'' the <hask>fmap</hask>. In the particular case of <hask>Data.Sequence</hask>, though,
there *is* a well defined order and it is preserved as expected by
+
there '''is''' a well defined order and it is preserved as expected by
fmap and exposed by toList.
+
<hask>fmap</hask> and exposed 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>mapM_</hask>, which is a kind of fold over
<hask>(>>)</hask>, and Foldable provides this along with the
+
<hask>(>>)</hask>, and <hask>Foldable</hask> provides this along with the
 
related <hask>sequence_</hask>.
 
related <hask>sequence_</hask>.
   
 
===Traversable===
 
===Traversable===
   
A Traversable type is a kind of upgraded Foldable. Where Foldable
+
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
 
gives you the ability to go through the structure processing the
elements (foldr) but throwing away the shape, Traversable allows you
+
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
 
to do that whilst preserving the shape and, e.g., putting new values
 
in.
 
in.
   
Traversable 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
 
<hask>sequence</hask> : note the apparently surprising fact that the
"_" versions are in a different typeclass.
+
"_" versions are in a different [[typeclass]].
   
 
== Some trickier functions: concatMap and filter ==
 
== Some trickier functions: concatMap and filter ==
   
Neither Traversable nor Foldable contain elements for concatMap and
+
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
filter. That is because Foldable is about tearing down the structure
+
completely, while <hask>Traversable</hask> is about preserving the structure
completely, while Traversable is about preserving the structure
 
 
exactly as-is. On the other hand <hask>concatMap</hask> tries to
 
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
 
'squeeze more elements in' at a place and <hask>filter</hask> tries to
 
cut them out.
 
cut them out.
   
You can write concatMap for Sequence as follows:
+
You can write <hask>concatMap</hask> for <hask>Sequence</hask> as follows:
   
 
<haskell>
 
<haskell>
Line 87: Line 87:
 
</haskell>
 
</haskell>
   
But why does it work? It works because sequence is an instance of
+
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
Monoid, where the monoidal 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:
   
Line 102: Line 102:
   
 
<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 Sequence), or <hask>\a -> [a]</hask>
+
something like 'singleton' (from <hask>Sequence</hask>), or <hask>\a -> [a]</hask>
for lists. We can use <hask>pure</hask> from Applicative, although
+
for lists. We can use <hask>pure</hask> from <hask>Applicative</hask>, although
it's not really right to bring Applicative in for this, and get:
+
it's not really right to bring <hask>Applicative</hask> in for this, and get:
   
 
<haskell>
 
<haskell>
Line 113: Line 113:
   
 
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 Foldable into a Monad, since concatMap is a good
+
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 pure for return.
+
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 Sequence, Foldable or Traversable is zipWith. The most
+
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
general kind of zipWith over Traversables 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
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:
 
values through:
   
Line 164: Line 164:
 
</haskell>
 
</haskell>
   
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.
+
The code above fails with a [[pattern match]] error when the <hask>Foldable</hask> container doesn't have enough input. Here is an alternative version which provides friendlier error reports and makes use of <hask>State</hask> instead of the self defined Supply [[monad]].
   
 
<haskell>
 
<haskell>

Revision as of 16:00, 20 April 2008


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:

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
foldr (:) []
. This means that all
Foldable
s have a

representation as a list; however the order of the items may or may

not have any particular significance. In particular if a
Foldable<hask> is
also a <hask>Functor
,
toList
and
fmap
need not perfectly commute; the list given after the
fmap
may be in a different order to the list before the
fmap
. In the particular case of
Data.Sequence
, though,

there is a well defined order and it is preserved as expected by

fmap
and exposed 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)