Haskell: Applicative

And a bit of Traversable

updated 2021-05-25

What are they?


  • An Applicative is short for a strong lax monoidal functor or some people might call it an Applicative Functor.

  • In Haskell - it is a Functor that comes with an operation <*> :: f (a -> b) -> f a -> f b that tells it how to Apply a higher level Functor to another Functor.

  • If you remember from the Functor article, we already defined what a Functor is and what fmap (aka. <$>) does - let’s compare them side by side:

<?> ::   (a -> b) -> f a -> f b
<*> :: f (a -> b) -> f a -> f b
  • In essence the difference is that with <$> the function is flat - i.e. outside the realms of the Functor and it is injected inside.

  • In <*> ( a.k.a. “splat” - as I have just found out during an interview earlier this week) the first argument is a lifted function - i.e. the Functor contains a higher order type.

  • for Haskell to believe you a type is an instance of an Applicative it is sufficient to prove it has the functions:
    • pure of type Applicative f => a -> f a, and
    • either:
      • <*> of type Applicative f => f (a -> b) -> f a -> f b, or
      • liftA2 of type Applicative f => (a -> b -> c) -> f a -> f b -> f c.
  • Other laws that it must satisfy:
    • Identity pure id <*> v = v
    • Composition pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
    • Homomorphism pure f <*> pure x = pure (f x)
    • Interchange u <*> pure y = pure ($ y) <*> u
      • For the whole list with further conversation it would be good to consult hackage.


Lets say we want to have a type Doer that stores both a type and its meta sign- which is a property we came up with.

The meta sign has the following behaviour: when we do operations with these type, the signs interact and the resulting sign follow the normal sign rule i.e. :

  • + & + = +,
  • + & - = -,
  • - & + = -,
  • - & - = +.

I would also like to introduce the use of GADTs in this case, as it makes working with the type nicer. Note that this could have been done without it, but I thought it would be clearer what we are doing by using them.


  • As mentioned before lets enable GADTs, and also let’s import the Applicative module
import Control.Applicative

  • And continue by defining our types. Let us define the type Op which is our meta sign. We also implement a Show so we can display it nicely.
data Op = Add | Sub deriving (Eq)

instance Show Op where
  show o = if o == Add then "+" else "-"

  • Now we can define our sign rule function that combines the two Ops.
signRule :: Op -> Op -> Op
signRule x y
  = case x of
      Add -> case y of
                Add -> Add
                Sub -> Sub
      Sub -> case y of
                Add -> Sub
                Sub -> Add
  • But if we look, we can see a pattern - Op is a monoid where Add is the neutral element. So let’s make use of this and also enrich our type with another of its inherent - discovered properties.
  • Note: algebraic properties are discovered and not designed - they are inherent to the type itself. You can design your types to make use of types that have the properties or not - but you cannot design the property into a type - if it doesn’t have it.
instance Semigroup Op where
  (<>) = signRule

instance Monoid Op where
  mempty = Add 

  • Next let’s define our type Doer. In this implementation, our type constructor ON takes a meta sign Op and any type a and it creates an instance of Doer a - think of it like an embellished type that contains our meta-sign.
data Doer a where
  ON :: Op -> a -> Doer a
  deriving (Show, Eq)

  • And let’s look at a few examples that would make it clear
ex0 =  ON Sub 3            -- ON - 3
ex1 =  ON Add "Hello"      -- ON + "Hello"
ex2 = [ON Sub 3, ON Add 2] -- [ON - 3,ON + 2]

We can see that Doer a is a Functor. Let’s prove it.

instance Functor Doer where
  fmap f (ON o x) = ON o $ f x

Now, let’s prove it is an Applicative.

instance Applicative Doer where
  pure                      = ON (mempty::Op)
  (<*>) (ON o f) (ON o' n') = ON (o <> o') (f $ n')

Note: we made use in (o <> o') of our previously discovered Monoid property. The reader (yes, you) can check that the rest of the laws of the Applicative Functor stand.


  • OK I’m glad to say that we’ve covered most of the info regarding Applicative, some examples to see it in action would be good.

Example 1:

Let’s use our splat to apply the lifted function (+1) to a functor:

ex01 = ON Sub (+1) <*> ON Sub 1 -- ON + 2
--   = ON - (+1) <*> ON - 1     -- this is what happens
--   = ON ( - <> - ) (1 $ (+1))
--   = ON + 2

So it works as intended, we have the result of the operation and the result of the interaction between their meta signs.

Example 2:

We want to compose two functions and apply them to a Functor

ex02 = pure ((+1). (+2)) <*> ON Sub 3 -- ON - 6

Is there another way to write this? Of course, and it might give some insight into what actually happens:

ex03 = pure (.) <*> pure (+1) <*> pure (+2) <*> ON Sub 3 -- ON - 6

Explanation: we lift the composition, we partially apply it to the lifted (+1) we then apply it to the lifted (+2) giving us the lifted composition of (+1) and (+2). Then in the end we apply this to lifted 3 and we get lifted 6. As the functions are pure our meta sign is not altered.

Example 3:

We want to do the previous while changing our meta sign accordingly.

ex04 = pure (.) <*> ON Sub (+1) <*> ON Add (+2) <*> ON Sub 3 -- ON + 6

Isn’t that fantastic:

  • we already know that 6 is the result of the function application
  • - + - = + - which gets through to us in the type.

Example 4:

Say we have a list of Doers and a higher typed Doer and we want to fmap it.

ex2  = [ON Sub 3, ON Add 2]
ex05 = (pure (+1) <*>) <$> ex2    -- [ON - 4,ON + 3]

This is interesting - we can observe here what is going on - if we refactor again with something that we’ve seen before:

ex05' = (fmap (+1)) <$>  ex2    -- [ON - 4,ON + 3]
ex05''= fmap (fmap (+1)) ex2    -- [ON - 4,ON + 3]

This behaviour is linked to the definition of pure. But the added benefit of being able to interact through <*> is that it easily allows us to change the meta-sign.

ex06 = ((ON Sub (+1)) <*> ) <$> ex2  -- [ON + 4,ON - 3]

Observe how in ex06 the meta signs have flipped.

Traversable Time

What is it ?

  • It is a typeclass that can be traversed from left to right - but its uses are a bit more subtle than that - and we’ll talk a bit more about it when time comes.

  • I’m not going to go very in depth with regards to Traversable, because the post would fork too much (I will make a dedicated post at some point) - but it’s important to note that Traversable, Applicatives, Monads are very tightly connected. (The wiki article is a good read)

  • Note: in the examples the Traversable, Foldable t is the [] type.


  • Example: say we take a list of Doers and we want to nest it into a Doer type itself, with the correct meta-sign (the sign of folding all the meta signs inside the list with the <> we defined) while in essence unwrapping the elements. The type of this function would look something like:
squishSign :: [Doer a] -> Doer [a]

So here are the challenges we are facing:

  • if we foldMap :: Monoid m => (a -> m) -> t a -> m we need to do quite a bit of work to to get our Doer to match that type - not ideal - and we are lazy.
  • Wouldn’t it be nice if we had some sort of function that would traverse our list, do something with the type and then at the end gives us that something? Don’t worry Haskell has our back:
sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a)
  • observe how that t and f swap positions - that is what we want to do. We know Doer is Applicative, and we know that [] is Traversable so what are we waiting for:
squishSign :: [Doer a] -> Doer [a]
squishSign  = sequenceA 

I wrote it as a section as it reads a bit better - but let’s see it in action:

ex11 = squishSign [ON Sub 3, ON Add 2]           -- ON - [3,2]
ex12 = squishSign [ON Sub 3, ON Add 2, ON Sub 3] -- ON + [3,2,3]

Now we can apply some more functy (funky + functor) things we’ve seen earlier.

ex13 = pure sum <*> squishSign [ON Sub 3, ON Add 2, ON Sub 3] -- ON + [8]

This has been a well long post, and we’ve had a lot of functy time together. What other cool Applicative and Traversable design patterns do you use or have discovered? Let me know!

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