Applicative functors in Haskell

Functor

According to Haskell, a functor is simply something that can be mapped over. In other words, it is an abstraction for a context with the ability to apply a function to all the things inside the context.
The context can be defined as container, computation etc.
E.g.; a list or sequence is a container of homogenous elements. You can apply a function to each of the elements, which produces a new sequence of elements transformed by the function.

Let’s create a new functor, implementing myfmap such that Data.Map is an instance of the new functor typeclass. myfmap applies a function f to the value(s) inside functor’s context while preserving the context.

import Data.Map as DataMap
import Data.List as DataList

class MyFunctor f where
  myfmap :: (a -> b) -> f a -> f b

instance (Ord k) => MyFunctor (DataMap.Map k) where
  myfmap f x = DataMap.fromList $ 
                     DataList.map (\(p,q) ->(p,f q)) $ 
                     DataMap.toList x

Applicative Functor

In computer science, applicative is an abstraction for a context, and it has the ability to apply functions in the same type of context to all elements in the context. For example, a sequence A which has homogenous elements, and sequence B which consists of functions that can be applied to sequence A , which produce a new sequence of elements transformed by all the functions. (Super confusing? Sorry 😕)

The Applicative typeclass in Haskell is located in Control.Applicative module and defines pure and (<*>) .

class (Functor f) => Applicative f where
    pure :: a -> f a
    (<*>) :: f (a -> b) -> f a -> f b

• An Applicative has to be Functor. This is a class constraint.

• pure takes any context and returns an Applicative the value of context inside it.

• (<*>) is a representation of fmap , where <*> takes a functor that has a function in it and another functor and run the function from first functor and maps it over the second functor. Whereas fmap which takes a function and functor and applies the function inside the functor.

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List an an Applicative!

List also instantiates an applicative typeclass, with implementation as :

instance Applicative [] where
    pure x = [x]
    fs <*> xs = [f x | f <- fs, x <- xs]

Here, the implementation of (<*>) is basically a list comprehension, where every function is applied to every value. For example :

ghci> (*) <$> [2, 3] <*> [4, 5]
[8, 10, 12, 15]

If you want to apply each function in first list to the respective value in second list, the ZipList typeclass is very handy.

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Curious case of Either

The base Monad instance for Either is defined as follows.

instance Monad (Either e) where
  return = Right
  Left e  >>= _ = Left e
  Right a >>= f = f a

This instance has inherent short-circuiting, where we go from left to right and once a single computation fails into the Left , then all do as well. But in case you would like to collect error messages which occur anywhere in the above computation, it goes against (>>=) and lazy-evaluation/short-circuiting .

Lazy evaluation is when we proceed from left to right, when a single computation “fails” into the Left then all the rest do as well.

(>>=) takes a function, maps it over an instance of a monad and then flattens the result.
(>>=) :: m a -> (a -> m b) -> m b

(>>=) produces m b from m a so long as it can run (a -> m b) . This demands that the value of a should ideally exists during the time of computation, and this is impossible for Either .

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So let’s try to solve the above problem by defining a Functor instance of Either .

instance Functor (Either a) where
    fmap f (Left x) = Left x
    fmap f (Right y) = Right (f y)

Things we can understand from the above definition :

• We know the definition of fmap :: (c -> d) -> f c -> f d .

• If we replace f with Either a , we get 
  fmap :: (c -> d) -> Either a c -> Either a d

The problem with this implementation of Either is that we cannot map over Left .
(😱Why?) To understand that, let Either a b computation, which may succeed and return b or fail with error a , similar to monad instance. So the functor instance does not map over Left values since you would want to map over the computation, if it fails, there is nothing to manipulate.

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Let’s see if we can implement an Applicative instance instead, which cannot have a corresponding Monad , to solve the problem.

As we saw the definition of Applicative, we will define pure and (<*>) .

Defining pure is rather simple, as we want it return the Right element. Implementation of (<*>) is rather tricky . The following cases need to be considered for defining (<*>)

instance Applicative (Either e) where
    pure                 =  Right
    Right f  <*> Right a  = Right (f a)    
    Left  e  <*> Right _ = Left e          
    Right _  <*> Left  e = Left e          
    Left e1  <*> Left e2 = Left (e1 <> e2)

• The first statement is the pure statement.

• (<*>) allows evaluation in parallel instead of necessarily needing results from previous computation to compute present values.

• Thus, we can use our purely Applicative Either to collect errors, ignoring Right if any Left exist in the sequence.

• As soon as it hits a Left, it aborts and returns that Left.
  

ghci> (++) <$> Left "Hello" <*> undefined 
Left "Hello"                              -- not undefined

ghci> (++) <$> Right "Hello" <*> undefined 
*** Exception: Prelude.undefined          -- undefined

ghci> (++)  <$> Right "Hello" <*> Left " World"
Left " World"

ghci> (++)  <$> Right "Hello" <*> Right " World"
Right "Hello World"

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Limitations

There’s some limitations to using purely applicative functor. As we saw in the definition of (>>=) :: m a -> (a -> m b) -> m b ; which means that without (>>=) you can’t pick “what to do next based on what came before”.
Also, if you take a generally pure function and feed the applicative arguments to it, like

> f :: a -> b -> c 
> f <$> getLine <*> getProcessID "chrome" <*> getFreeMemory

All the arguments will get evaluated, no matter what. That is, you cannot express — “if the second argument exits, abort the rest of the computation”. Due to this, anything recursive, as well as most interactive programs do not use Applicatives. Monads, on the other hand are a good choice in that case.

Original Article - Applicative functors in Haskell

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More Resources :

http://learnyouahaskell.com/functors-applicative-functors-and-monoids
http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html
https://hackage.haskell.org/package/base-4.8.1.0/docs/Control-Applicative.html
https://stackoverflow.com/questions/17409260/what-advantage-does-monad-give-us-over-an-applicative

Comments

[sam]

I don't think your Either example will work as written--since you're using <>, you need something like

Monoid e => instance Applicative (Either e) where
    ...
    Left e1 <*> Left e2 = Left (e1 <> e2)