base-4.3.1.0: Basic libraries

Portability | portable
Stability | experimental
Maintainer | libraries@haskell.org

Data.Traversable

Description

Class of data structures that can be traversed from left to right, performing an action on each element.

See also

- Applicative Programming with Effects, by Conor McBride and Ross Paterson, online at http://www.soi.city.ac.uk/~ross/papers/Applicative.html.
- The Essence of the Iterator Pattern, by Jeremy Gibbons and Bruno Oliveira, in Mathematically-Structured Functional Programming, 2006, and online at http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator.

Note that the functions `mapM` and `sequence` generalize Prelude functions of the same names from lists to any `Traversable` functor. To avoid ambiguity, either import the Prelude hiding these names or qualify uses of these function names with an alias for this module.

Synopsis

- class (Functor t, Foldable t) => Traversable t where
  - traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
  - sequenceA :: Applicative f => t (f a) -> f (t a)
  - mapM :: Monad m => (a -> m b) -> t a -> m (t b)
  - sequence :: Monad m => t (m a) -> m (t a)
- for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b)
- forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b)
- 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)
- fmapDefault :: Traversable t => (a -> b) -> t a -> t b
- foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m

# Documentation

class (Functor t, Foldable t) => Traversable t where

Functors representing data structures that can be traversed from left to right.

Minimal complete definition: `traverse` or `sequenceA`.

Instances are similar to `Functor`, e.g. given a data type

     data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)

a suitable instance would be

     instance Traversable Tree where
        traverse f Empty = pure Empty
        traverse f (Leaf x) = Leaf <$> f x
        traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r

This is suitable even for abstract types, as the laws for `<*>` imply a form of associativity.

The superclass instances should satisfy the following:

- In the `Functor` instance, `fmap` should be equivalent to traversal with the identity applicative functor (`fmapDefault`).
- In the `Foldable` instance, `Data.Foldable.foldMap` should be equivalent to traversal with a constant applicative functor (`foldMapDefault`).

Methods

traverse :: Applicative f => (a -> f b) -> t a -> f (t b)

Map each element of a structure to an action, evaluate these actions from left to right, and collect the results.

sequenceA :: Applicative f => t (f a) -> f (t a)

Evaluate each action in the structure from left to right, and collect the results.

mapM :: Monad m => (a -> m b) -> t a -> m (t b)

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results.

sequence :: Monad m => t (m a) -> m (t a)

Evaluate each monadic action in the structure from left to right, and collect the results.

Instances

Traversable [] |
Traversable Maybe |
Ix i => Traversable (Array i) |

for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b)

`for` is `traverse` with its arguments flipped.

forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b)

`forM` is `mapM` with its arguments flipped.

mapAccumL :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)

The `mapAccumL` function behaves like a combination of `fmap` and `foldl`; it applies a function to each element of a structure, passing an accumulating parameter from left to right, and returning a final value of this accumulator together with the new structure.

mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)

The `mapAccumR` function behaves like a combination of `fmap` and `foldr`; it applies a function to each element of a structure, passing an accumulating parameter from right to left, and returning a final value of this accumulator together with the new structure.

fmapDefault :: Traversable t => (a -> b) -> t a -> t b

This function may be used as a value for `fmap` in a `Functor` instance.

foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m

This function may be used as a value for `Data.Foldable.foldMap` in a `Foldable` instance.