Building Lenses

Implementing basic Haskell lenses in twenty exercises

What are Lenses?

Lenses are purely functional references to pieces of data.

They let us focus on a specific part of a structure, to see it more closely.

Difficulty

The inner workings of lenses have a reputation for being difficult to learn. This isn’t entirely fair—for example, I think monads are harder to gain a good intuition for. However, there are many different things that go into learning lenses, so the difficulty of putting them all together adds up.

Some of the perceived difficulty probably comes from seeing code like this:

a & b %~ (c ^. _1)

Admittedly, this is near-impossible for a beginner to understand. Even worse, its components are tough to search for when attempting to piece it apart! However, with enough background for how lenses work, and some helpful mnemonics, even this code will eventually become readable.

Besides strange unreadable operators, lenses often have confusing type signatures, can be autogenerated by Template Haskell (for ease of use, but not ease of understanding), and are called in where your normal tools fall apart—whenever you need to operate on deeply nested, complicated structures.

Focus

In this post I’d like to mainly focus not on how to use lenses, but on how to write them yourself. I think this tends to (eventually) lead to deeper levels of understanding.

Many fields don’t have the luxury of being able to pry open the things they’re working with at almost any level of abstraction. Physicists can’t construct alternate universes to play with (yet). Poets who want to invent a new language just for their poetry will have a hard time finding an audience.

Luckily, in programming we can build our own tools as we go along. It’s worth quite a bit of effort to do so, and the rewards in terms of deeper understanding can be impressive.

For these reasons, we’ll be focusing on how lenses work on the inside. You’ll be building your own little optics library, including Getter, Setter and of course Lens.

There are 20 exercises (the bits with the blue backgrounds) strewn throughout. I’ve provided answers for all but one of them, since that one is more open-ended and meant to help you start your own exploration. When answers appear, they’re presented inline, but inside spoiler blocks.

I highly recommend actually following along and writing the code, but I won’t tell anybody if you just want to breeze through.

Code accompanying this post can be found in this github repo

Getting Into It

Let’s talk about a function! Functional programmers like those, right? The specific one I have in mind is

get :: s -> a

which takes some “source” structure of type s and gives us back our desired value of type a.

We can make a fun newtype from this. Since it’s fairly simple and it’s used for getting values, we’ll call it a SimpleGetter.

newtype SimpleGetter s a =
  SimpleGetter (s -> a)

Here’s the first exercise!

fst :: (a, b) -> a

If you wrote SimpleGetter fst then congratulations! Let’s take a closer look at it, especially at its type signature

SimpleGetter (a, b) a

This means our type s is (a, b) and our type a is…well, a. This SimpleGetter thing is really simple! If you think about it though, that’s not very surprising. A SimpleGetter is just a function from s to a. With that in mind, some other things we can write SimpleGetters to get are

• The first element of a list
• A specific piece of a deeply-nested structure
• Every third element of a list
• The successor of a number
• An identical thing back

View and To

Unfortunately, even though we can do all these things, SimpleGetter can’t do them with just a type. We need a function to actually run! This function is traditionally called view.

view :: SimpleGetter s a -> s -> a

view (SimpleGetter fst) (1, 2)

Hopefully this exercise isn’t too bad. We’ve already got all the components, so we just have to unwrap the newtype to produce view, our first function that works with a Haskell optic!

view (SimpleGetter get) s = get s

It’s also time for us to define the first of many operators (^.), and the first of many mnemonics for remembering operators. There are a couple mnemonics to choose from here. An upside-down V (from “view”) looks kind of like a ^. The caret also looks kind of like something being extracted from the ground (so we can view it). Anyways, we define ^. like

infixl 8 ^.

(^.) :: s -> SimpleGetter s a -> a
(^.) = flip view

The arguments are flipped in such a way that they kind of match the order you may be used to from imperative languages. For example, a.getB() would be something like a ^. b. We’ll see this ordering thing come up again in a moment.

Before that though, there’s one more thing we should do with our SimpleGetter, so long as it’s sitting around.

Alright, so that one was pretty easy too. But it’s good to be able to write these kinds of functions, since we’ll be writing similar ones as we deal with more complicated optics. A simple solution looks like

toSimpleGetter :: (s -> a) -> SimpleGetter s a
toSimpleGetter f = SimpleGetter f

This also gets us a fun new ability! Or rather, a new way to do something we were already quite capable of doing. Since ^. gets an a from a SimpleGetter, and toSimpleGetter gets a SimpleGetter from an s -> a, combining the two as x ^. toSimpleGetter f is just function application—f x.

Continuing on with Getters

Let’s get back to the comment about the order of operations feeling like object-oriented programming. Haskell lens libraries—Control.Lens, Lens.Micro, and others—often feature lens composition (.) that feels “backwards” to functional programmers, but the right way around to people from a more object-oriented background.

To understand why this is, we’ll be digging into continuation-passing style a little bit, and then talking about contravariant functors. The “backwardness” of contravariance will then help us explain why optic composition can feel like it goes the wrong way.

Continuation passing converts a function into a function that takes a callback. That is, if we have a function

f :: s -> a

we will also pass in a callback, something to do once we have an a, which changes our function’s type to

f :: (a -> r) -> s -> r

length :: String -> Int

lengthCont :: (Int -> r) -> String -> r

So, for this exercise, we have to define length, but with a callback function as its first parameter, and then we have to call that callback once we get the string’s length.

lengthCont callback str =
  callback $ length str

We can now test that

lengthCont (*2) "hello"

evaluates to 10.

Let’s also do the same kind of transformation to our SimpleGetter to write a Getter (it’s no longer very simple). Instead of it being a newtype wrapper around a plain function, we’ll change it to use continuation-passing style, so that it’ll better compose with other optics we add later.

The type signature might be a little confusing, so bear with me. All we’re really going to do is replace r with a contravariant functor over s and a.

Contravariant is like a normal functor, but the order feels “backwards”. Check out its typeclass

class Contravariant f where
  contramap :: (a -> b) -> f b -> f a

Comparing this to the type signature of fmap, we can pretty quickly see where the backwardness comes from. The last two arguments are flipped.

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

It amuses me that the usual operator for contramap is >$<, as opposed to fmap’s <$>.

newtype MakeString a =
  MakeString { mkString :: a -> String }

instance Contravariant MakeString where
  contramap = _

While you’re playing around with this, note that usual functors with fmap are what is called “covariant”. In contravariance, lifted functions go the opposite direction (what we’ve been calling “backwards”) as plain functions. Covariance just means functions at both levels go in the same direction.

contramap for MakeString might look a little something like this.

  contramap f (MakeString mkStr) =
    MakeString (mkStr . f)

Our First Real Optic

All of this machinery has been leading up to defining our first real Getter. It combines ideas from continuation-passing style, contravariance, Const functors, and of course just plain function composition, so don’t worry if it’s a little tricky to figure out what’s going on at first. Here’s the type signature of Getter, where f is our contravariant functor since we need it to work in both directions. You’ll also need RankNTypes enabled for most of the code from here on out.

{-# LANGUAGE RankNTypes #-}

type Getter s a =
  forall f. (Contravariant f, Functor f)
  => (a -> f a) -> s -> f s

newtype Const a b = Const { getConst :: a }

instance Functor (Const m)
  where fmap _ (Const a) = Const a

instance Contravariant (Const m)
  where contramap _ (Const b) = Const b

Give this exercise a try, and if you’re still lost, then feel free to keep reading. One good thing to remember is that if you put _ instead of a real value, GHC will find the “type hole” and give you some information about what could possibly go in it. Good luck!

Just for fun, we’ll implement the operator version first here.

(^.) :: s -> Getter s a -> a
s ^. g  = getConst (g Const s)

view :: Getter s a -> s -> a
view g s = s ^. g

What is going on here? Let’s take it piece by piece. First, we name our Getter “g” and our structure of type s “s”. g Const s is applying two arguments to our Getter type. We can think of those two arguments as Const :: a -> f a and s :: s. Finally, of course, our return value will be something of type f s. This value gets passed to getConst, which extracts the final value out.

Congratulations! You’ve written your first Haskell optic.

A Getter Workout

The Getter type above is the same as in Control.Lens, which means it has the full general power of those Getters. To simplify our understanding a bit, we can specialize f to Const r and use this Getter type:

type Getter s a =
  forall r. (a -> Const r a) -> s -> Const r s

I recommend trying out this Const r version of Getter for the next exercise, unless you happen to know/find the generalization here early!

Hopefully this exercise is more tedious than difficult, once you figure out how to write one of the Getters

headGetter :: Getter [a] a
headGetter f = Const . getConst . f . head

nestedGetter :: Getter (a, ((b, (c, d)), e)) d
nestedGetter f = Const . getConst . f . snd . snd . fst . snd

everyThirdGetter :: Getter [a] [a]
everyThirdGetter f = Const . getConst . f . everyThird
  where everyThird (_:_:x:xs) = x : everyThird xs
        everyThird _ = []

succGetter :: Getter Integer Integer
succGetter f = Const . getConst . f . succ

idGetter :: Getter a a
idGetter = id -- bit of a special case

Some of those Getters are deeply useful in practice, while others don’t come up that much, but should be good training for writing Getters.

You’ve probably noticed a pattern here (if not, then I applaud your clever solution) of using Const . getConst which has type Const r a -> Const r s. There’s actually a more-general way to do this operation, called phantom. Just like our general Getter, phantom operates on contravariant functors, and is useful to go from an f a to a f b. It uses <$, which is just fmap . const, but also the likely-unfamiliar $<, which is a flipped version of contramap . const.

phantom :: (Contravariant f, Functor f) => f a -> f b
phantom x = () <$ x $< ()

If you did the exercise above, you should be able to spot the generalization available here

toGetter :: (s -> a) -> Getter s a
toGetter get f =
  phantom . f . get

Setting the Stage

Just like how a Getter is analogous to a plain function, a Setter is analogous to a plain Functor. Using a Setter is like fmapping over a structure.

Let’s get right into the thick of things. Similar to how our Getter implementation used the Const functor, Setter is going to use Identity. The definition for Identity looks like

newtype Identity a =
  Identity { runIdentity :: a }

instance Functor Identity
  where fmap = coerce

By the way, if you’d like more information on how to use Identity or coercions (especially the Coercible composition operator (#.)) I might recommend reading Composing Coercions before moving on. Be warned that it gives away the implementation for set (which is an exercise below), but if you get stuck when implementing set it could be a good first resource to look at.

Anyways, just like with Getter, let’s start with a SimpleSetter. Don’t worry, we’ll be getting rid of this one rather quickly. Full-blown setters are way more powerful and fun to play with.

Immediately, here’s another exercise, but using the answer to the previous exercise

type SimpleSetter s a =
  (a -> Identity a) -> s -> Identity s

Here’s what you should have now:

type LessSimpleSetter s t a b =
  (a -> Identity b) -> s -> Identity t

Once again, we’ll quickly jump into another exercise. It’ll help to know that the definition of Settable is

class ( Applicative f
      , Distributive f
      , Traversable f
      ) => Settable f

What we have now is the same Setter type as defined in Control.Lens.

type Setter s t a b =
  forall f. Settable f =>
  (a -> f b) -> s -> f t

Set for Success

Like view for Getter, Setter’s main action—changing values inside structures—is called set, which works like this

set mySetter newValue structure

We’re “setting” a pure functional value using a setter! Of course, what we’re really doing is more like “replacing”. The operator equivalent of set is .~, so if we see

mySetter .~ newValue myStructure

We can read it as “set newValue in myStructure using mySetter”.

The type signature for set looks like this. Note that we never have to pass in a value of type a—since it’s being overwritten with a new value of type b, it doesn’t really matter what was there before.

set :: Setter s t a b -> b -> s -> t

One implementation of set looks like

set s b = runIdentity #. s (\_ -> Identity b)

setter :: Setter (Int, String) (String, String) Int String

This we can figure out by following the types. It also feels quite a bit like a function call in some imperative language.

setter .~ "Hello" (1, "world")

One easy generalization for set is to make it take a function instead of a constant value. The usual name for such a function is over. We can always recover set from over with a simple set setter x = over setter (const x).

over :: Setter s t a b -> (a -> b) -> s -> t

over hits a sweet spot. It’s general enough to be really powerful, but not so general that figuring out how to write it is trivial.

over setter f =
  runIdentity #. setter (Identity #. f)

Something surprising and magical has just happened. We’ve unlocked the front gate to the operator zoo for Haskell lenses. Using over, or its operator version (%~), we can get all kinds of helpful setters by passing useful functions to over.

For example, in programming languages with mutable data, you often see a += operator to add to a number. The Setter version of this looks like +~, as is implemented by…well, you tell me.

Hopefully that one wasn’t too bad, at least compared to implementing over!

(+~) :: Num a => Setter s t a a -> a -> s -> t
setter +~ a = over setter (+a)

Tons of useful operators have Setter forms. Along with +~, there are operators like -~, *~, /~, &&~, <>~, etc. The key thing to remember here is that the squiggle means Setter.

There are also Setters specifically for handling Maybe and State values, which come in handy.

(.=) :: MonadState s m => Setter s s a b -> b -> m ()

This exercise shows some of the similarities between operating within a stateful monad, and using a setter. Both are ways to imagine doing something like imperative programming, but in a purely functional way.

(.=) :: MonadState s m => Setter s s a b -> b -> m ()
setter .= b = modify $ set setter b

(%=) :: MonadState s m => Setter s s a b -> (a -> b) -> m ()
setter %= f = modify $ over setter f

We’re getting really close to imperative programming now, what with all this setting and manipulation of state. Let’s take advantage of that for a simple version of a stateful environment: a game.

Lenses as Purely Functional References

Lenses are among the most useful, and certainly among the most-talked-about kinds of optics. You can think of a Lens as combining a Setter and a Getter to provide a purely functional reference to some piece of data.

They’re especially useful when working with deeply nested data structures, since they save the programmer from having to unpack and repack the structure every time they want to manipulate one of its inner elements.

The type of a Lens is

type Lens s t a b =
  forall f. Functor f =>
  (a -> f b) -> s -> f t

Just like we had SimpleSetter and SimpleGetter, we can also have a SimpleLens by reducing generalization and making the source and target types the same.

This exercise isn’t too bad. We just make the source and target types the same for both the structure and the value within it.

type SimpleLens s a = Lens s s a a

Lenses really are just the combination of a getting function and a setting function. Here, why don’t you prove that?

makeLens :: (s -> a) -> (s -> b -> t) -> Lens s t a b

This exercise is a bit tricky, so I’ve broken it down into a few parts and tried to name them appropriately. get :: s -> a and set :: s -> b -> t should both be pretty straightforward to understand. transform :: a -> f b transforms our a into a b and also wraps it in a Functor called f. structure :: s should also be clear.

Reading from right to left, we first get a value from our structure. This value has type a. We then transform that value, to get an f b. We then fmap a set structure on this f b, to get out an f t (set structure has type b -> t, and so the lifted version is f b -> f t).

makeLens get set transform structure =
  set structure <$> transform (get structure)

This actually works in a fairly intuitive way. To do something to a value, first we have to get it from our structure, then we can transform it in some way, and then we have to set that newly-transformed value in the structure.

Here’s another crazy operator. We write %%~ to indicate that the left side has a lens, and the right side has a transform function and some structure of data that our lens can operate on.

(%%~) ::
  Functor f => 
  Lens s t a b -> (a -> f b) -> s -> f t

Alright, so that last exercise is a little cheesy. The answer is

(%%~) = id

To double-check this, look at the types of Lens and %%~ and groan.

Wrapping Up

Congratulations! If you’ve completed all twenty exercises, you’ve probably got a decent understanding of how basic lenses work under the hood, but there’s definitely plenty more left to learn about the wild and wonderful world of Haskell optics. Like with anything else, the best way to solidify your knowledge is with practice, which is why I tried to focus so heavily on the exercises here. I hope this post was at least somewhat helpful.

Haskell isn’t the only language with lenses, of course. I’ve also seen them in PureScript, and have even used these concepts in Elm (though it was called a Focus). You could also write them in your favorite non-purely-functional language, though those languages usually already have idiomatic mutable ways to affect some piece of a large data structure. (Unfortunately those languages don’t often provide strong guarantees like the type safety you maintain by using lenses.)

One more fantastic thing about Lenses is that they compose! Programmers love things that compose nicely. This composition is especially nice for lenses because it lets us combine them over very complicated data structures in straightforward ways, maintaining pure functional style the whole way.

Again, code accompanying this post can be found in this github repo