# # Traversable

The `Traversable`

class generalises the function formerly known as `mapM :: Monad m => (a -> m b) -> [a] -> m [b]`

to work with `Applicative`

effects over structures other than lists.

## # Traversing a structure in reverse

A traversal can be run in the opposite direction with the help of the `Backwards`

applicative functor, which flips an existing applicative so that composed effects take place in reversed order.

```
newtype Backwards f a = Backwards { forwards :: f a }
instance Applicative f => Applicative (Backwards f) where
pure = Backwards . pure
Backwards ff <*> Backwards fx = Backwards ((\x f -> f x) <$> fx <*> ff)
```

`Backwards`

can be put to use in a "reversed `traverse`

". When the underlying applicative of a `traverse`

call is flipped with `Backwards`

, the resulting effect happens in reverse order.

```
newtype Reverse t a = Reverse { getReverse :: t a }
instance Traversable t => Traversable (Reverse t) where
traverse f = fmap Reverse . forwards . traverse (Backwards . f) . getReverse
ghci> traverse print (Reverse "abc")
'c'
'b'
'a'
```

The `Reverse`

newtype is found under Data.Functor.Reverse.

## # Definition of Traversable

```
class (Functor t, Foldable t) => Traversable t where
{-# MINIMAL traverse | sequenceA #-}
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
traverse f = sequenceA . fmap f
sequenceA :: Applicative f => t (f a) -> f (t a)
sequenceA = traverse id
mapM :: Monad m => (a -> m b) -> t a -> m (t b)
mapM = traverse
sequence :: Monad m => t (m a) -> m (t a)
sequence = sequenceA
```

`Traversable`

structures `t`

are finitary containers of elements `a`

which can be operated on with an effectful "visitor" operation. The visitor function `f :: a -> f b`

performs a side-effect on each element of the structure and `traverse`

composes those side-effects using `Applicative`

. Another way of looking at it is that `sequenceA`

says `Traversable`

structures commute with `Applicative`

s.

## # An instance of Traversable for a binary tree

Implementations of `traverse`

usually look like an implementation of `fmap`

lifted into an `Applicative`

context.

```
data Tree a = Leaf
| Node (Tree a) a (Tree a)
instance Traversable Tree where
traverse f Leaf = pure Leaf
traverse f (Node l x r) = Node <$> traverse f l <*> f x <*> traverse f r
```

This implementation performs an in-order traversal of the tree.

```
ghci> let myTree = Node (Node Leaf 'a' Leaf) 'b' (Node Leaf 'c' Leaf)
-- +--'b'--+
-- | |
-- +-'a'-+ +-'c'-+
-- | | | |
-- * * * *
ghci> traverse print myTree
'a'
'b'
'c'
```

The `DeriveTraversable`

extension allows GHC to generate `Traversable`

instances based on the structure of the type. We can vary the order of the machine-written traversal by adjusting the layout of the `Node`

constructor.

```
data Inorder a = ILeaf
| INode (Inorder a) a (Inorder a) -- as before
deriving (Functor, Foldable, Traversable) -- also using DeriveFunctor and DeriveFoldable
data Preorder a = PrLeaf
| PrNode a (Preorder a) (Preorder a)
deriving (Functor, Foldable, Traversable)
data Postorder a = PoLeaf
| PoNode (Postorder a) (Postorder a) a
deriving (Functor, Foldable, Traversable)
-- injections from the earlier Tree type
inorder :: Tree a -> Inorder a
inorder Leaf = ILeaf
inorder (Node l x r) = INode (inorder l) x (inorder r)
preorder :: Tree a -> Preorder a
preorder Leaf = PrLeaf
preorder (Node l x r) = PrNode x (preorder l) (preorder r)
postorder :: Tree a -> Postorder a
postorder Leaf = PoLeaf
postorder (Node l x r) = PoNode (postorder l) (postorder r) x
ghci> traverse print (inorder myTree)
'a'
'b'
'c'
ghci> traverse print (preorder myTree)
'b'
'a'
'c'
ghci> traverse print (postorder myTree)
'a'
'c'
'b'
```

## # Instantiating Functor and Foldable for a Traversable structure

```
import Data.Traversable as Traversable
data MyType a = -- ...
instance Traversable MyType where
traverse = -- ...
```

Every `Traversable`

structure can be made a `Foldable`

`Functor`

using the `fmapDefault`

and `foldMapDefault`

functions found in `Data.Traversable`

.

```
instance Functor MyType where
fmap = Traversable.fmapDefault
instance Foldable MyType where
foldMap = Traversable.foldMapDefault
```

`fmapDefault`

is defined by running `traverse`

in the `Identity`

applicative functor.

```
newtype Identity a = Identity { runIdentity :: a }
instance Applicative Identity where
pure = Identity
Identity f <*> Identity x = Identity (f x)
fmapDefault :: Traversable t => (a -> b) -> t a -> t b
fmapDefault f = runIdentity . traverse (Identity . f)
```

`foldMapDefault`

is defined using the `Const`

applicative functor, which ignores its parameter while accumulating a monoidal value.

```
newtype Const c a = Const { getConst :: c }
instance Monoid m => Applicative (Const m) where
pure _ = Const mempty
Const x <*> Const y = Const (x `mappend` y)
foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m
foldMapDefault f = getConst . traverse (Const . f)
```

## # Traversable structures as shapes with contents

If a type `t`

is `Traversable`

then values of `t a`

can be split into two pieces: their "shape" and their "contents":

```
data Traversed t a = Traversed { shape :: t (), contents :: [a] }
```

where the "contents" are the same as what you'd "visit" using a `Foldable`

instance.

Going one direction, from `t a`

to `Traversed t a`

doesn't require anything but `Functor`

and `Foldable`

```
break :: (Functor t, Foldable t) => t a -> Traversed t a
break ta = Traversed (fmap (const ()) ta) (toList ta)
```

but going back uses the `traverse`

function crucially

```
import Control.Monad.State
-- invariant: state is non-empty
pop :: State [a] a
pop = state $ \(a:as) -> (a, as)
recombine :: Traversable t => Traversed t a -> t a
recombine (Traversed s c) = evalState (traverse (const pop) s) c
```

The `Traversable`

laws require that `break . recombine`

and `recombine . break`

are both identity. Notably, this means that there are exactly the right number elements in `contents`

to fill `shape`

completely with no left-overs.

`Traversed t`

is `Traversable`

itself. The implementation of `traverse`

works by visiting the elements using the list's instance of `Traversable`

and then reattaching the inert shape to the result.

```
instance Traversable (Traversed t) where
traverse f (Traversed s c) = fmap (Traversed s) (traverse f c)
```

## # Transforming a Traversable structure with the aid of an accumulating parameter

The two `mapAccum`

functions combine the operations of folding and mapping.

```
-- A Traversable structure
-- |
-- A seed value |
-- | |
-- |-| |---|
mapAccumL, mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)
-- |---|---|---|---|---|---|---|---|---|---------------| |---|---|---|---|---|---|---|---|---|-----|
-- | |
-- A folding function which produces a new mapped |
-- element 'c' and a new accumulator value 'a' |
-- |
-- Final accumulator value
-- and mapped structure
```

These functions generalise `fmap`

in that they allow the mapped values to depend on what has happened earlier in the fold. They generalise `foldl`

/`foldr`

in that they map the structure in place as well as reducing it to a value.

For example, `tails`

can be implemented using `mapAccumR`

and its sister `inits`

can be implemented using `mapAccumL`

.

```
tails, inits :: [a] -> [[a]]
tails = uncurry (:) . mapAccumR (\xs x -> (x:xs, xs)) []
inits = uncurry snoc . mapAccumL (\xs x -> (x `snoc` xs, xs)) []
where snoc x xs = xs ++ [x]
ghci> tails "abc"
["abc", "bc", "c", ""]
ghci> inits "abc"
["", "a", "ab", "abc"]
```

`mapAccumL`

is implemented by traversing in the `State`

applicative functor.

```
{-# LANGUAGE DeriveFunctor #-}
newtype State s a = State { runState :: s -> (s, a) } deriving Functor
instance Applicative (State s) where
pure x = State $ \s -> (s, x)
State ff <*> State fx = State $ \s -> let (t, f) = ff s
(u, x) = fx t
in (u, f x)
mapAccumL f z t = runState (traverse (State . flip f) t) z
```

`mapAccumR`

works by running `mapAccumL`

in reverse.

```
mapAccumR f z = fmap getReverse . mapAccumL f z . Reverse
```

## # Transposing a list of lists

Noting that `zip`

transposes a tuple of lists into a list of tuples,

```
ghci> uncurry zip ([1,2],[3,4])
[(1,3), (2,4)]
```

and the similarity between the types of `transpose`

and `sequenceA`

,

```
-- transpose exchanges the inner list with the outer list
-- +---+-->--+-+
-- | | | |
transpose :: [[a]] -> [[a]]
-- | | | |
-- +-+-->--+---+
-- sequenceA exchanges the inner Applicative with the outer Traversable
-- +------>------+
-- | |
sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a)
-- | |
-- +--->---+
```

the idea is to use `[]`

's `Traversable`

and `Applicative`

structure to deploy `sequenceA`

as a sort of **n-ary zip**, zipping together all the inner lists together pointwise.

`[]`

's default "prioritised choice" `Applicative`

instance is not appropriate for our use - we need a "zippy" `Applicative`

. For this we use the `ZipList`

newtype, found in `Control.Applicative`

.

```
newtype ZipList a = ZipList { getZipList :: [a] }
instance Applicative ZipList where
pure x = ZipList (repeat x)
ZipList fs <*> ZipList xs = ZipList (zipWith ($) fs xs)
```

Now we get `transpose`

for free, by traversing in the `ZipList`

`Applicative`

.

```
transpose :: [[a]] -> [[a]]
transpose = getZipList . traverse ZipList
ghci> let myMatrix = [[1,2,3],[4,5,6],[7,8,9]]
ghci> transpose myMatrix
[[1,4,7],[2,5,8],[3,6,9]]
```