Applicative functors, Part I

CIS 194 Week 10
25 March 2012

Motivation

Consider the following Employee type:

type Name = String

data Employee = Employee { name    :: Name
                         , phone   :: String }
                deriving Show

Of course, the Employee constructor has type

Employee :: Name -> String -> Employee

That is, if we have a Name and a String, we can apply the Employee constructor to build an Employee object.

Suppose, however, that we don’t have a Name and a String; what we actually have is a Maybe Name and a Maybe String. Perhaps they came from parsing some file full of errors, or from a form where some of the fields might have been left blank, or something of that sort. We can’t necessarily make an Employee. But surely we can make a Maybe Employee. That is, we’d like to take our (Name -> String -> Employee) function and turn it into a (Maybe Name -> Maybe String -> Maybe Employee) function. Can we write something with this type?

(Name -> String -> Employee) ->
(Maybe Name -> Maybe String -> Maybe Employee)

Sure we can, and I am fully confident that you could write it in your sleep by now. We can imagine how it would work: if either the name or string is Nothing, we get Nothing out; if both are Just, we get out an Employee built using the Employee constructor (wrapped in Just). But let’s keep going…

Consider this: now instead of a Name and a String we have a [Name] and a [String]. Maybe we can get an [Employee] out of this? Now we want

(Name -> String -> Employee) ->
([Name] -> [String] -> [Employee])

We can imagine two different ways for this to work: we could match up corresponding Names and Strings to form Employees; or we could pair up the Names and Strings in all possible ways.

Or how about this: we have an (e -> Name) and (e -> String) for some type e. For example, perhaps e is some huge data structure, and we have functions telling us how to extract a Name and a String from it. Can we make it into an (e -> Employee), that is, a recipe for extracting an Employee from the same structure?

(Name -> String -> Employee) ->
((e -> Name) -> (e -> String) -> (e -> Employee))

No problem, and this time there’s really only one way to write this function.

Generalizing

Now that we’ve seen the usefulness of this sort of pattern, let’s generalize a bit. The type of the function we want really looks something like this:

(a -> b -> c) -> (f a -> f b -> f c)

Hmm, this looks familiar… it’s quite similar to the type of fmap!

fmap :: (a -> b) -> (f a -> f b)

The only difference is an extra argument; we might call our desired function fmap2, since it takes a function of two arguments. Perhaps we can write fmap2 in terms of fmap, so we just need a Functor constraint on f:

fmap2 :: Functor f => (a -> b -> c) -> (f a -> f b -> f c)
fmap2 h fa fb = undefined

Try hard as we might, however, Functor does not quite give us enough to implement fmap2. What goes wrong? We have

h  :: a -> b -> c
fa :: f a
fb :: f b

Note that we can also write the type of h as a -> (b -> c). So, we have a function that takes an a, and we have a value of type f a… the only thing we can do is use fmap to lift the function over the f, giving us a result of type:

h         :: a -> (b -> c)
fmap h    :: f a -> f (b -> c)
fmap h fa :: f (b -> c)

OK, so now we have something of type f (b -> c) and something of type f b… and here’s where we are stuck! fmap does not help any more. It gives us a way to apply functions to values inside a Functor context, but what we need now is to apply a functions which are themselves in a Functor context to values in a Functor context.

Applicative

Functors for which this sort of “contextual application” is possible are called applicative, and the Applicative class (defined in Control.Applicative) captures this pattern.

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

The (<*>) operator (often pronounced “ap”, short for “apply”) encapsulates exactly this principle of “contextual application”. Note also that the Applicative class requires its instances to be instances of Functor as well, so we can always use fmap with instances of Applicative. Finally, note that Applicative also has another method, pure, which lets us inject a value of type a into a container. For now, it is interesting to note that fmap0 would be another reasonable name for pure:

pure  :: a             -> f a
fmap  :: (a -> b)      -> f a -> f b
fmap2 :: (a -> b -> c) -> f a -> f b -> f c

Now that we have (<*>), we can implement fmap2, which in the standard library is actually called liftA2:

liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c
liftA2 h fa fb = (h `fmap` fa) <*> fb

In fact, this pattern is so common that Control.Applicative defines (<$>) as a synonym for fmap,

(<$>) :: Functor f => (a -> b) -> f a -> f b
(<$>) = fmap

so that we can write

liftA2 h fa fb = h <$> fa <*> fb

What about liftA3?

liftA3 :: Applicative f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d
liftA3 h fa fb fc = ((h <$> fa) <*> fb) <*> fc

(Note that the precedence and associativity of (<$>) and (<*>) are actually defined in such a way that all the parentheses above are unnecessary.)

Nifty! Unlike the jump from fmap to liftA2 (which required generalizing from Functor to Applicative), going from liftA2 to liftA3 (and from there to liftA4, …) requires no extra power—Applicative is enough.

Actually, when we have all the arguments like this we usually don’t bother calling liftA2, liftA3, and so on, but just use the f <$> x <*> y <*> z <*> ... pattern directly. (liftA2 and friends do come in handly for partial application, however.)

But what about pure? pure is for situations where we want to apply some function to arguments in the context of some functor f, but one or more of the arguments is not in f—those arguments are “pure”, so to speak. We can use pure to lift them up into f first before applying. Like so:

liftX :: Applicative f => (a -> b -> c -> d) -> f a -> b -> f c -> f d
liftX h fa b fc = h <$> fa <*> pure b <*> fc

Applicative laws

There is only one really “interesting” law for Applicative:

f `fmap` x === pure f <*> x

Mapping a function f over a container x ought to give the same results as first injecting the function into the container, and then applying it to x with (<*>).

There are other laws, but they are not as instructive; you can read about them on your own if you really want.

Applicative examples

Maybe

Let’s try writing some instances of Applicative, starting with Maybe. pure works by injecting a value into a Just wrapper; (<*>) is function application with possible failure. The result is Nothing if either the function or its argument are.

instance Applicative Maybe where
  pure              = Just
  Nothing <*> _     = Nothing
  _ <*> Nothing     = Nothing
  Just f <*> Just x = Just (f x)

Let’s see an example:

m_name1, m_name2 :: Maybe Name
m_name1 = Nothing
m_name2 = Just "Brent"

m_phone1, m_phone2 :: Maybe String
m_phone1 = Nothing
m_phone2 = Just "555-1234"

ex01 = Employee <$> m_name1 <*> m_phone1
ex02 = Employee <$> m_name1 <*> m_phone2
ex03 = Employee <$> m_name2 <*> m_phone1
ex04 = Employee <$> m_name2 <*> m_phone2

Generated 2013-04-04 15:15:39.515041