Applicative functor

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

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.

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.

Some advantages of applicative functors

• Code that uses only the Applicative interface is more general than code that 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.

Applicative transformers

From the Monad Transformer Library we are used to have two flavours of every monad: a base monad like State and a transformer variant StateT. In the transformers package we even have only monad transformers except the Identity monad. So where are applicative transformers? The answer is, that in most situations, 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)

That is, liftA2 (<*>) is essentially the definition for <*> for the composition of the functors f and g. This is implemented in the TypeCompose library as type constructor O and in transformers library in module Data.Functor.Compose. 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 above f might be Writer (Sum Int) that 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.

It is not true that transformers are unnecessary for applicatives, though. Consider State s (IO a) == s -> (s, IO a), which behaves like an applicative with the above trick, but it is different from StateT s IO a == s -> IO (s, a). The latter is more useful in some situations, and it is not a composition of any two applicatives.

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 replacereturn 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).

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.

See also

• Applicative Programming with Effects
• The blog article The basics of applicative functors, put to practical work