Applicative functor

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

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 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.

4 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.

5 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).

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.

7 See also

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

Feb	OCT	Dec
	30
2017	2018	2019

Personal tools

Views

Applicative functor

From HaskellWiki

Contents

1 Example

2 Usage

3 Some advantages of applicative functors

4 Applicative transformers

5 How to switch from monads

6 Alternative terms

7 See also

Navigation

Tools