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Applicative functor

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An applicative functor has more structure than a functor but less than a monad. See the Haddock docs for <div class="inline-code">
Control.Applicative
</div>
.

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)
The (fictive) functions
openDialog
and
openWindow
could not only open dialogs and windows but could also register some cleanup routine in the
CleanIO
.
 runAndCleanup
would first run the opening actions and afterwards the required cleanup actions.

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.

Consider the example above where the choice between
openDialog
and
openWindow
depends on the outcome of
 getLine
. You cannot run initialization code for either
openDialog
or
openWindow
, because you do not know which one will be called before executing
 getLine
.

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")
where
 writeToWindow
registers an initialization routine which opens the window.

2 Usage

If you have the variables

f :: a -> b -> c
a :: f a
b :: f b
you can combine them in the following ways with the same result of type
f c
:
pure f <*> a <*> b
 
liftA2 f a b
But how to cope with
let
and sharing in the presence of effects?

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
runs the effect of
fy
twice. E.g. if
fy
writes something to the terminal then
fh <*> fy <*> fy
writes twice.

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

Actually, using the
liftA
commands we can pull results of applicative functors

into 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
The order of effects is entirely determined by the order of arguments to
liftA3
.

3 Some advantages of applicative functors

  • Code that uses only on the
    Applicative
    interface are more general than ones uses the
    Monad
    interface, because there are more applicative functors than monads. The
    ZipList
    is an applicative functor on lists, where
    liftA2
    is implemented by
    zipWith
    . It is a typical example of an applicative functor that is not a monad.
  • Programming with
    Applicative
    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.

4 How to switch from monads

  • Start using
    liftM
    ,
    liftM2
    , etc or
    ap
    where you can, in place of
    do
    /
    (>>=)
    . You will often encounter code like
do x <- fx
   y <- fy
   return (g x y)
It can be rewritten to
liftM2 g fx fy
. 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 with
liftM
.
  • When you notice you're only using those monad methods, then import
    Control.Applicative
    and replace
    return
    with
    pure
    ,
    liftM
    with
    (<$>)
    (or
    fmap
    or
    liftA
    ),
    liftM2
    with
    liftA2
    , etc, and
    ap
    with
    (<*>)
    . If your function signature was
    Monad m => ...
    , change to
    Applicative m => ...
    (and maybe rename
    m
    to
    f
    or whatever).

5 See also