#### table of contents

I want to explore the Applicative functor, more than a functor and less than a monad. The reason to use it is that, by being less powerful, it can be applied in more situations.

This post is a companion to the talk I gave at HaskellMAD, you can find the slides of the talk here.

Applicatives or *Idioms*, as they were named back then, were introduced in
2008, in a Functional Pearl named *Idioms: applicative
programming with effects*.

As the begin of this writeup, let’s first introduce the typeclass:

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

Simple enough, this typeclass allows you to put elements inside an `Applicative`

context –with `pure`

–, and *smash* `Applicative`

values together through `<*>`

.

We can understand `<*>`

as `fmap`

from `Functor`

, but with the function lifted
to an `Applicative`

context.

But while you’re scratching your head trying to imagine a moment in your development career in which you have a function inside a container, let me show you an example:

```
Prelude> let square = \ x -> x * x
Prelude> let plusTwo = \ x -> x + 2
Prelude> let functions = [square, plusTwo]
Prelude> let numbers = [1,2,3]
Prelude> functions <*> numbers
[1,4,9,3,4,5]
```

As you can see, the `<*>`

implementation for `Applicative [a]`

, is the
application of each function in the first list to each element in the second
list.

But, you don’t need to tell me that functions inside a list is not the most
common thing in the world. Let’s explore a more useful way of using
`Applicative`

.

## Validation

Validation is a well known problem. You have some data that you need to
validate before it makes part of your domain, easy. For our validation
example we will use the `Either a b`

type. `Either`

is a *sum type*, meaning
that the number of possible values of `Either a b`

is the *sum* of possible values
of type `a`

, and possible values of type `b`

.

Basically, the `a`

s in our validations will be validation errors –represented as
strings– and the `b`

s, the final values.

```
data Person = Person {
name :: String,
lastName :: String
} deriving Show
data Err = String
validateName :: String -> Either Err String
validateName n = if n == "Pepe" then Right n
else Left "name is not Pepe"
validateLastName :: String -> Either Err String
validateLastName l = if l == "García" then Right l
else Left "last name is not García"
```

Our `validateName`

function takes a string as paramenter, and validates it as
a name, and the same does `validateLastName`

.

Now, let’b build up a `validatePerson`

function from the validation of each one
of its parts:

```
validatePersonM :: String -> String -> Either String Person
validatePersonM n l = do
vName <- validateName n
vLast <- validateLastName l
return $ Person vName vLast
```

Since `Either a`

is a monad, we can use all the monadic tricks in our hat to
compose it, for example this *do block*. But, the cool part of `Either`

,
is that it is an `Applicative`

, so we can use all the `Applicative`

machinery
to compose our program.

```
validatePersonA :: String -> String -> Either String Person
validatePersonA n l = Person <$> validateName n <*> validateLastName l
```

How can the last function possibly work? Well, in case you missed it, `<$>`

is
the infix version of `fmap`

. and the typing of the body of the function is the
following:

```
Prelude> :t Person
Person :: String -> String -> Person
Prelude> :t Person <$> validateName "pepe"
Person <$> validateName "pepe" :: Either String (String -> Person)
Prelude> :t Person <$> validateName "pepe" <*> validateLastName "Garcia"
Person <$> validateName "pepe" <*> validateLastName "Garcia" :: Either String Person
```

## Create your own Applicatives

You cand do this directly by providing an instance of the typeclass, like follows:

```
data Maybe a = Just a
| Nothing
instance Applicative Maybe where
pure = Just
Just f <*> m = fmap f m
Nothing <*> _ = Nothing
```

This will enable ourselves to use our new `Maybe`

data type as an `Applicative`

.
But there is another case I want to cover here. What can we do when what we
want is to provide an `Applicative`

instance for an *ADT* we have? We can use
*Free Applicatives*.

## Free Applicatives

*Free Applicatives* are an abstraction over Applicatives, and are basically
a lift from Applicative’s typeclass operations to constructors of an *Algebraic
Data Type*.

You can find more information about free applicatives here. Basically,
the definition of *Free Applicative* is:

```
data Ap f a where
Pure :: a -> Ap f a
Ap :: f a -> Ap f (a -> b) -> Ap f b
instance Functor (Ap f) where
fmap f (Pure a) = Pure (f a)
fmap f (Ap x y) = Ap x ((f .) <$> y)
instance Apply (Ap f) where
Pure f <.> y = fmap f y
Ap x y <.> z = Ap x (flip <$> y <.> z)
instance Applicative (Ap f) where
pure = Pure
Pure f <*> y = fmap f y
Ap x y <*> z = Ap x (flip <$> y <*> z)
```

Now, we can provide an instance of Applicatve for every data type with kind
`* -> *`

we have!

Let’s create a small *ADT* for all the operations in our blog. It could be
something like follows:

```
{-# LANGUAGE GADTs #-}
import Control.Applicative.Free
import Control.Applicative
data Author = Author {
name :: String,
lastName :: String
} deriving Show
data Post = Post {
id :: Int,
title :: String,
content :: String,
excerpt :: String
} deriving Show
data BlogF a where
GetPost :: Id -> BlogF Post
GetAuthor :: Id -> BlogF Author
type Blog a = Ap BlogF a
```

The most important part of this block is where we define our language, the `BlogF`

GADT, and where we lift it using a *Free Applicative*.

Also, we will need to add some operations to make our blog usable. Let’s create
*smart constructors* for `GetPost`

, and `GetAuthor`

operations, but lifted to
the `Blog`

type.

```
getPost :: Id -> Blog Post
getPost id = liftAp $ GetPost id
getAuthor :: Id -> Blog Author
getAuthor id = liftAp $ GetAuthor id
```

As you can imagine, `liftAp`

is part of the `Control.Applicative.Free`

library,
and it takes a value of the type `f a`

, and lifts it to the type `Ap f a`

.

Now let’s build a program for rendering a page of our blog!

```
data Page = Page {
post :: Post,
author :: Author
} deriving Show
getPage :: Id -> Id -> Blog Page
getPage postId authorId = Page <$> getPost postId
<*> getAuthor authorId
```

But, there is a part of our problem of *rendering a page* that we are lacking,
it is drawing the rest of the f*cking owl. Right now, we are not going to the
database to fetch our post and our author, we are just creating values of all
the operations in our program. We are building an *Abstract Syntax Tree*.

What we need right now is to interpret the *AST* to an actual value of `Page`

,
and that’s done via *Natural Transformations*. Natural Transformations are a
way of transforming one `Functor`

into another while mantaining the internal
structure. So, what we want to create is a function like this:

```
naturalTransformation :: BlogF a -> IO a
```

Why do we want to interpret to `IO`

? Because `IO`

is where side effects occur
in Haskell. Fetching elements from our database will return `IO a`

actions.
Now let’s do the actual implementation of the `interpret`

function:

```
interpret :: BlogF a -> IO a
interpret (GetPost id) = putStrLn ("getting post " ++ show id ++ " from DB") *> pure $ Post id "this is the post" "content of the post" "excerpt"
interpret (GetAuthor id) = putStrLn ("getting author " ++ show id ++ " from DB") *> pure $ Author "Pepe" "García"
```

And finally, let’s wire everything together:

```
main :: IO ()
main = do
page <- runAp interpret $ getPage 1 1
print page
-- Output:
-- getting post 1 from DB
-- getting author 1 from DB
-- Page {post = Post {id = 1, title = "this is the post", content = "content of the post", excerpt = "excerpt"}, author = Author {name = "Pepe", lastName = "Garc\237a"}}
```

*fully functioning*.

As you can imagine, `runAp`

is part of the free library, and it takes
first a *Natural Transformation* on any two functors, and then an `Ap f a`

, and
interprets the *AST*.

The newly introduced *AST* is just an *Abstract Syntax Trees*, and we mention it
because that’s what we are producing with the calls to `getPost`

and
`getAuthor`

. This means that what we were doing when calling `getPost`

, `getAuthor`

,
or `getPage`

we are just creating small programs, no executing code. The part of
the code execution is done by our interpreter.

## Static Analysis

So now, imagine that you want to limit the number of requests to the DB that
your `Blog`

programs have to 10. If you know that any expression `GetAuthor`

or
`GetPost`

is gonna be interpreted to a SQL sentence, you can just count the
number of these, and you’re done. How can we achieve this?

```
instance Monoid Int where
mempty = 0
mappend = (+)
countInstructions :: BlogF a -> Int
countInstructions _ = 1
main :: IO ()
main = do
putStrLn "NUMBER OF REQUESTS TO THE DB:"
print instructions
where instructions = runAp_ countInstructions page
page = getPage 1 1
-- Output:
--
-- NUMBER OF REQUESTS TO THE DB:
-- 2
```

`runAp_`

is a modified version of `runAp`

, and as an interpreter it takes
a function like:

```
fn :: Monoid b => f a -> b
```

And `mappend`

s all the elements produced by it together

## Conclusions

We need to identify when to use Applicatives. It’s a bit tricky in the beginning because we are not so used to it but as a few rules of thumb, I use the following:

Use Applicatives if your monadic expressions do not depends ones on the others,
or program does not depend on runtime values. This last sentence means that you
don’t need to evaluate functions with given arguments to keep evaluating your
*AST*.