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Complicated Haskell Words - Isomorphism

November 29, 20208 min read

As someone who has been learning Haskell on and off for the last year, I know the struggles of just how complicated Haskell can be for beginners. You are instantly thrown off in a new world full of new terms, each one more obscure than the other. Being familiar with a statically typed language like Elm certainly helped, but I felt I struggled the most with the seemingly endless list of new words I would bump into every day… Isomorphism, functor, monoid, covariant, and so on and so forth.

So my goal here is to explain these terms using the simplest analogies I’ve come across. Some of them might not be the most complete or the most faithful, but my hope is that they will be useful lifeboats when you find yourself lost at sea. I will try to keep this series updated over time by adding new terms and simpler explanations. One word of warning: I won’t be talking about monads since there are plenty of tutorials and burritos out there.

Isomorphism

Let’s start with something that’s used all the time. If two things are isomorphic, it means that for all intents and purposes they are the same. Anytime you read isomorphic, just read more or less the same for what we care about.

This should be enough information for you to get on with your day. You can close the tab now. Thanks for reading!

The rest of this post is about building an intuitive understanding of why isomorphisms are just so damn cool.

Let’s say we want to describe if we are happy or sad.

data Mood = Happy | Sad

Now we could have a function that tells us if we are happy:

areWeHappy :: Mood -> Bool
areWeHappy Happy = True
areWeHappy Sad = False

We could come up with a way to derive a Mood from a Bool:

moodFromBool :: Bool -> Mood
moodFromBool True = Happy
moodFromBool False = Sad

If we take a look at this pair of functions we can see that we can chain them together.

mysteriousFunction =
  -- first run moodFromBool, then areWeHappy
  areWeHappy . moodFromBool

Let’s try plugging some values to this mysterious function:

  • if we pass True, it gets converted to Happy, then back to True
  • if we pass False, it gets converted to Sad, then back to False

So we basically implemented an identity function:

id :: a -> a
id x = x

This doesn’t sound too interesting, I know. What’s also not too interesting is that there is another way to combine those two functions:

otherMysteriousFunction =
  -- first run areWeHappy, then moodFromBool
  moodFromBool . areWeHappy
  • if we pass Happy, it gets converted to True, then back to Happy
  • if we pass Sad, it gets converted to False, then back to Sad

Another day, another identity function. What’s new?

Intermezzo, or why is zero cool

The ancient Greeks were thoroughly confused by the number zero and used to wonder:

How can nothing be something?

Adding zero to an existing number is a no-op, it doesn’t do anything at all. It’s a complete waste of effort!

The great thing about zero is that it helps us to define a relationship between numbers. The number 3 has now this beautiful property with itself:

3 - 3 = 0

Not only that, without zero we wouldn’t even have negative numbers! Zero stands as the legendary gatekeeper between positive and negative numbers. Now each positive number a has a friend in the nether world a', which relation is defined as:

a + a' = 0

Back to identity functions

That’s the same reason why identity functions are cool. They are a way to establish interesting relationships between other functions.

Let’s do a quick refresher on the functions we had:

areWeHappy :: Mood -> Bool
areWeHappy Happy = True
areWeHappy Sad = False

moodFromBool :: Bool -> Mood
moodFromBool True = Happy
moodFromBool False = Sad

mysteriousFunction :: Bool -> Bool
mysteriousFunction =
  areWeHappy . moodFromBool

If our mysterious function is an identity function, it means we have successfully undone what moodFromBool did in the first place. We were able to implement Control-Z for a function! We went from Bool to Mood and then back to Mood without losing any information.

otherMysteriousFunction :: Mood -> Mood
otherMysteriousFunction =
  moodFromBool . areWeHappy

In the other mysterious function, we were able to go from Mood to Bool and then back to Mood without losing information.

The definition™

Two types A and B are isomorphic if two conversion functions exist such that:

f :: A -> B
g :: B -> A

And these two properties are valid:

f . g = id
g . f = id

Having this pair of functions means that we can losslessly convert between values of type A and values of type B.

So this is what we meant at the beginning when we said that for all intents and purposes A and B are the same!

Many isomorphisms

Note that the multiple isomorphisms can exist between two types. For example we could have this:

areWeSad :: Mood -> Bool
areWeSad Sad = True
areWeSad Happy = False

moodFromBoolSad :: Bool -> Mood
moodFromBoolSad True = Sad
moodFromBoolSad False = Happy

What’s the lesson here? All isomorphisms are born equal, and no isomorphism is more equal than others.

Some examples of isomorphisms

We can easily prove that (a, b) and (b, a) are isomorphic by importing the Data.Tuple module:

f :: (a, b) -> (b, a)
f = swap

g :: (b, a) -> (a, b)
g = swap

In the same module, there’s another couple of functions:

curry :: ((a, b) -> c) -> a -> b -> c
uncurry :: (a -> b -> c) -> (a, b) -> c

Those type signatures look quite funky, right? curry lets us apply a function, originally intended to be used on a tuple, on two separate arguments:

Prelude> :t fst
fst :: (a, b) -> a

Prelude> fst (1, 2)
1

Prelude> curry fst 1 2
1

While uncurry lets us apply a function, originally intended to be used on two arguments, on a tuple:

Prelude> (+) 1 2
3

Prelude> uncurry (+) (1, 2)
3

Let’s prove that these functions create an isomorphism:

Prelude> uncurry (curry fst) (1, 2)
1

Prelude> curry (uncurry (+)) 1 2
3

After looking at the type signatures again we can say that these following functions are isomorphic:

((a, b) -> c) -> a -> b -> c
(a -> b -> c) -> (a, b) -> c

Pretty neat!

Unit

In Haskell you would sometimes see this type:

foo :: ()

It’s an empty tuple, which is referred to as the unit. It can contain a single value, which can only be:

()

It turns out to be quite useful when you need a type that contains a single value.

For example, we could prove that Bool and Either () () are isomorphic:

f :: Bool -> Either () ()
f True = Right ()
f False = Left ()

g :: Either () () -> Bool
g (Right ()) = True
g (Left ()) = False

Or that Maybe a is isomorphic to Either () a

f :: Maybe a -> Either () a
f (Just x) = Right a
f Nothing = Left ()

g :: Either () a -> Maybe a
g (Right a) = Just a
g (Left ()) = Nothing

One more thing…

Peano numbers are a simple way of representing natural numbers using a zero value and a successor function:

data PeanoNum
  = Zero
  | Succ PeanoNum
  deriving (Show)

We can construct an isomorphism between this representation and Int (let’s just pretend for a moment that negative numbers aren’t a thing):

toInt :: PeanoNum -> Int
toInt Zero = 0
toInt (Succ x) = toInt x + 1

fromInt :: Int -> PeanoNum
fromInt 0 = Zero
fromInt x = Succ (fromInt (x - 1))

Let’s take it for a spin:

Prelude> toInt Zero
0

Prelude> toInt (Succ (Succ Zero))
2

Prelude> fromInt 3
Succ (Succ (Succ Zero))

Prelude> toInt (fromInt 3)
3

🎉

Thanks for reading, I hope you have emerged from this with a newfound sense of awe for isomorphisms and identity functions!


Ju Liu

Personal blog by Ju Liu.

I try to write code that doesn't suck.