# Applicative functor

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## Revision as of 05:53, 17 December 2011

An applicative functor has more structure than a functor but less than a monad. See the Haddock docs for Control.Applicative.

## Contents |

## 1 Example

It has turned out that many applications do not require monad functionality but only those of applicative functors. Monads allow you to run actions depending on the outcomes of earlier actions.

do text <- getLine if null text then putStrLn "You refuse to enter something?" else putStrLn ("You entered " ++ text)

This is obviously necessary in some cases, but in other cases it is disadvantageous.

Consider an extended IO monad which handles automated closing of allocated resources. This is possible with a monad.

openDialog, openWindow :: String -> CleanIO () liftToCleanup :: IO a -> CleanIO a runAndCleanup :: CleanIO a -> IO a runAndCleanup $ do text <- liftToCleanup getLine if null text then openDialog "You refuse to enter something?" else openWindow ("You entered " ++ text)

I.e. if the dialog was opened, the dialog must be closed, but not the window. That is, the cleanup procedure depends on the outcomes of earlier actions.

Now consider the slightly different task, where functions shall register *initialization* routines
that shall be run before the actual action takes place.
(See the original discussion started by Michael T. Richter in Haskell-Cafe:
Practical Haskell Question)
This is impossible in the monadic framework.

If you eliminate this dependency, you end up in an applicative functor and there you can do the initialization trick. You could write

initializeAndRun $ liftA2 (liftToInit getLine) (writeToWindow "You requested to open a window")

## 2 Usage

If you have the variables

f :: a -> b -> c a :: f a b :: f b

pure f <*> a <*> b liftA2 f a b

Consider the non-functorial expression:

x :: x g :: x -> y h :: y -> y -> z let y = g x in h y y

Very simple. Now we like to generalize this to

fx :: f x fg :: f (x -> y) fh :: f (y -> y -> z)

However, we note that

let fy = fg <*> fx in fh <*> fy <*> fy

This could be intended, but how can we achieve, that the effect is run only once and the result is used twice?

Actually, using theinto a scope where we can talk exclusively about functor results and not about effects. Note that functor results can also be functions. This scope is simply a function, which contains the code that we used in the non-functorial setting.

liftA3 (\x g h -> let y = g x in h y y) fx fg fh

## 3 Some advantages of applicative functors

- Code that uses only on the interface are more general than ones uses theApplicativeinterface, because there are more applicative functors than monads. TheMonadis an applicative functor on lists, whereZipListis implemented byliftA2. It is a typical example of an applicative functor that is not a monad.zipWith
- Programming with has a more applicative/functional feel. Especially for newbies, it may encourage functional style even when programming with effects. Monad programming with do notation encourages a more sequential & imperative style.Applicative

## 4 Applicative transfomers

From the Monad Transformer Library we are used to have two flavours of every monad:

A base monad likeIn the transformers package we even have only monad transformers

except theSo where are applicative transformers? The answer is, that we do not need special transformers for applicative functors since they can be combined in a generic way.

h :: f (g (a -> b)) a :: f (g a) liftA2 (<*>) h a :: f (g b)

The first one needs a lot of type extensions, whereas the second one is entirely Haskell 98.

It can be useful to use the applicative composition even when you have a monad transformer at hand.

In the example abovethat is used for counting the number of involved applicative actions. Since in an applicative functor the number of run actions is independent from interim results, the writer can count the actions at compile time.

## 5 How to switch from monads

- Start using ,liftM, etc orliftM2where you can, in place ofap/do. You will often encounter code like(>>=)

do x <- fx y <- fy return (g x y)

- It can be rewritten to . In general, whenever the choice or construction of monadic actions does not depend on the outcomes of previous monadic actions, then it should be possible to rewrite everything withliftM2 g fx fy.liftM

- When you notice you're
*only*using those monad methods, then importand replaceControl.Applicativewithreturn,purewithliftM(or(<$>)orfmap),liftAwithliftM2, etc, andliftA2withap. If your function signature was(<*>), change toMonad m => ...(and maybe renameApplicative m => ...tomor whatever).f

## 6 Alternative terms

Applicative functors were introduced by several people under different names:

- Ross Paterson called them Sequence
- Conor McBride called them Idiom
- The same kind of structure is used in the UU Parsing-Combinators.