`> {-# LANGUAGE GeneralizedNewtypeDeriving, NoMonomorphismRestriction #-}`

# Applicative functors

CIS 194 Week 10 22 March 2012

Suggested reading:

- Applicative Functors from Learn You a Haskell
- The Typeclassopedia

## 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 `Name`

s and `String`

s to form `Employee`

s; or we could pair up the `Name`

s and `String`

s 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 is itself in a Functor context* to a value 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
```

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
```

**Lists**

How about an instance for lists? There are actually two possible instances: one that matches up the list of functions and list of arguments elementwise (that is, it “zips” them together), and one that combines functions and arguments in all possible ways.

First, let’s write the instance that does all possible combinations. (For reasons that will become clear next week, this is the default instance.) From this point of view, lists represent nondeterminism: that is, a value of type `[a]`

can be thought of as a single value with multiple possibilities. Then `(<*>)`

corresponds to nondeterministic function application—that is, the application of a nondeterministic function to a nondeterministic argument.

```
> instance Applicative [] where
> pure a = [a] -- a "deterministic" value
> [] <*> _ = []
> (f:fs) <*> as = map f as ++ fs <*> as
```

Here’s an example:

```
> names = ["Joe", "Sara", "Mae"]
> phones = ["555-5555", "123-456-7890", "555-4321"]
>
> employees1 = Employee <$> names <*> phones
```

Maybe this particular example doesn’t make that much sense, but it’s not hard to imagine situations where you want to combine things in all possible ways like this. For example, we can do nondeterministic arithmetic like so:

```
> (.+) = liftA2 (+) -- addition lifted to some Applicative context
> (.*) = liftA2 (*) -- same for multiplication
>
> -- nondeterministic arithmetic
> n = ([4,5] .* pure 2) .+ [6,1] -- (either 4 or 5) times 2, plus either 6 or 1
>
> -- and some possibly-failing arithmetic too, just for fun
> m1 = (Just 3 .+ Just 5) .* Just 8
> m2 = (Just 3 .+ Nothing) .* Just 8
```

Next, let’s write the instance that does elementwise combining. First, we must answer an important question: how should we handle lists of different lengths? Some thought reveals that the most sensible thing to do is to truncate the longer list to the length of the shorter, throwing away the extra elements. Of course there are other possible answers: we might, for instance, extend the shorter list by copying the last element (but then what do we do when one of the lists is empty); or extend the shorter list with a “neutral” element (but then we would have to require an instance of `Monoid`

, or an extra “default” argument for the application).

This decision in turn dictates how we must implement `pure`

, since we must obey the law

`pure f <*> xs === f <$> xs`

Notice that the right-hand side is a list with the same length as `xs`

, formed by applying `f`

to every element in `xs`

. The only way we can make the left-hand side turn out the same… is for `pure`

to create an infinite list of copies of `f`

, because we don’t know in advance how long `xs`

is going to be.

We implement the instance using a `newtype`

wrapper to distinguish it from the other list instance. The standard Prelude function `zipWith`

also comes in handy.

```
> newtype ZipList a = ZipList { getZipList :: [a] }
> deriving (Eq, Show, Functor)
>
> instance Applicative ZipList where
> pure = ZipList . repeat
> ZipList fs <*> ZipList xs = ZipList (zipWith ($) fs xs)
```

An example:

`> employees2 = getZipList $ Employee <$> ZipList names <*> ZipList phones`

**Reader/environment**

Let’s do one final example instance, for `(->) e`

. This is known as the *reader* or *environment* applicative, since it allows “reading” from the “environment” `e`

. Implementing the instance is not too hard, we just have to use our nose and follow the types:

```
> instance Functor ((->) e) where
> fmap = (.)
>
> instance Applicative ((->) e) where
> pure = const
> f <*> x = \e -> (f e) (x e)
```

An `Employee`

example:

```
> data BigRecord = BR { getName :: Name
> , getSSN :: String
> , getSalary :: Integer
> , getPhone :: String
> , getLicensePlate :: String
> , getNumSickDays :: Int
> }
>
> r = BR "Brent" "XXX-XX-XXX4" 600000000 "555-1234" "JGX-55T3" 2
>
> getEmp :: BigRecord -> Employee
> getEmp = Employee <$> getName <*> getPhone
>
> ex05 = getEmp r
```

`Generated 2012-03-29 23:16:35.955246`