# Foldable and Traversable

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− | + | [[Category:Code]] [[Category:Idioms]] | |

+ | |||

+ | <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 | + | 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 10: | ||

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 41: | Line 42: | ||

===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 | + | 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 |

− | + | ||

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 | + | 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 | + | 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 | + | 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 90: | Line 89: | ||

</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 | + | |

definition works for lists, and we can write it more generally as: | definition works for lists, and we can write it more generally as: | ||

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<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 117: | Line 115: | ||

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 | + | 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 171: | Line 166: | ||

</haskell> | </haskell> | ||

− | The code above fails with a pattern match error when the | + | 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 "The answer to these lies in the long list of instances which Sequence

has. The Sequence version of map is "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 "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

Acan be 'folded' to a summary value. In other words, it is a type which

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

not have any particular significance. In particular if aalso a <hask>Functor

*after*the

*before*the

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

A particular kind of fold well-used by Haskell programmers is

### 1.3 Traversable

Agives you the ability to go through the structure processing the

elements (to do that whilst preserving the shape and, e.g., putting new values in.

"_" versions are in a different typeclass.

## 2 Some trickier functions: concatMap and filter

Neithercut them out.

You can writeconcatMap :: (a -> Seq b) -> Seq a -> Seq b concatMap = foldMap

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 havecondition is stronger than needed; it would be good enough if they were permutations of each other.

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## 3 Generalising zipWith

Another really useful list combinator that doesn't appear in the

interfaces forvalues 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)

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)