A blog exploring functional programming and Swift.

Case Study: Algebraic Data Types

Monday Apr 23, 2018


In our episode on algebraic data types we showed how there is algebra lurking in the Swift type system, and then used that knowledge to refactor our data types so that the invalid states are unrepresentable by the compiler. You simply were not allowed to construct invalid values, and it was proven to us by the compiler!

In today’s Point-Free Pointer we are going to apply this to a real world problem. In fact, we are going to analyze a data type that I made for a feature on this very site. I did it completely wrong the first time, and it held lots of values that were just completely nonsensical. I convinced myself that it wasn’t going to be a problem, and just rolled with it for awhile. But, I kept finding myself adding lil if lets here and lil guard lets there, until finally I said enough is enough, it’s time to refactor. The amazing part is that I literally used algebra to do this refactoring, and so today I want to walk you through exactly how I dissected this problem.

Episode Credits

But first, let me describe the feature I was working on.

While most of our episodes are for subscribers only, we wanted to give people the opportunity to see a video of their choosing for free. All they had to do was sign up for our newsletter, and they would get a credit that could be used on any subscriber-only episode.

Now, when a user is on the episode page, we have this lil module to callout that users can subscriber to our series or sign up to get a credit. The messaging in that box depends on quite a few states.



It can change depending on whether or not you are logged in, or if you are a subscriber already or not, or if you’ve already used your credit for this episode, and finally if this episode is free to the public or subscriber only. Naively this would be 2^4 = 16 states, many of which don’t make any sense. Like, you can’t be logged out and a subscriber. We want to omit those states, so let’s use algebra!

First Attempt

Here is what I first started with:

struct EpisodePermission {
  let hasUsedCredit: Bool
  let isLoggedIn: Bool
  let isSubscriber: Bool
  let isSubscriberOnly: Bool
}

It’s just a simple struct with four fields of Bool values for each of the conditions that can affect the messaging in the box.

This worked great for awhile. It got me making progress on the feature quickly and it was easy to understand. But soon I was having to guard against states that I knew were not possible. Let’s break this down algebraically and see if we can whittle away the invalid states.

Using Algebra

Here is our type algebraically:

// (Is/Not)LoggedIn
//   * (Is/Not)Subscriber
//   * (Has/Not)UsedCredit
//   * (Is/Not)SubscriberOnly (16)

Our type is a struct, which is a product type, so we have used * to denote that we are multplying the fields together. Written this way it is clear there are 16 states.

Let’s take this one step at a time. I first want to consider what states make sense when the user is not logged in:

// NotLoggedIn
//   * (Is/Not)Subscriber
//   * (Has/Not)UsedCredit
//   * (Is/Not)SubscriberOnly (8)

There are 8 states because 2 * 2 * 2 = 8. Which of these are reasonable? Well, you can’t be logged out and a subscriber. You also can’t be logged out and have used a credit. It is possible to be logged out and for an episode to be subscriber only, those things are completely independent.

So, we’ve just eliminated 6 states from the full set of 16:

// Valid states:
// NotLoggedIn * NotSubscriber * NotUsedCredit * (Is/Not)SubscriberOnly (2)

// Invalid states:
// NotLoggedIn * IsSubscriber  * HasUsedCredit * IsSubscriberOnly  (1)
// NotLoggedIn * NotSubscriber * HasUsedCredit * IsSubscriberOnly  (1)
// NotLoggedIn * IsSubscriber  * NotUsedCredit * IsSubscriberOnly  (1)
// NotLoggedIn * IsSubscriber  * HasUsedCredit * NotSubscriberOnly (1)
// NotLoggedIn * NotSubscriber * HasUsedCredit * NotSubscriberOnly (1)
// NotLoggedIn * IsSubscriber  * NotUsedCredit * NotSubscriberOnly (1)

That’s pretty nice, with just a lil bit of work we’ve whittled the 16 states down to 10. But, we haven’t even considered the logged in case yet. Let’s look at that:

// LoggedIn
//   * (Is/Not)Subscriber
//   * (Has/Not)UsedCredit
//   * (Is/Not)SubscriberOnly (8)

Now, all of these states are technically possible, but some are redundant when it comes to what we want to message to the user. For example, If you are a subscriber, it doesn’t really matter if you had previously used a credit on this episode (which you may have done before you became a subscriber), and it doesn’t matter if the episode is subscriber only or not. Let’s list them all our so that we can chip away at them one-by-one:

// LoggedIn * IsSubscriber  * HasUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * NotSubscriber * HasUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * IsSubscriber  * NotUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * IsSubscriber  * HasUsedCredit * NotSubscriberOnly (1)
// LoggedIn * NotSubscriber * HasUsedCredit * NotSubscriberOnly (1)
// LoggedIn * IsSubscriber  * NotUsedCredit * NotSubscriberOnly (1)
// LoggedIn * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)

So, as we just explained, when you are a subscriber all of the other states don’t matter. So all states that have IsSubscriber should really just constitute one state:

// LoggedIn
//   * IsSubscriber
//   * (Has/Not)UsedCredit
//   * (Is/Not)SubscriberOnly (4)
// ⬇️
// LoggedIn
//   * IsSubscriber
//   * Void
//   * Void                   (1)

So we’ve now gotten rid of 3 invalid states, which means we have just 7 from the original 16.

We still have 4 more states to consider, the case where you are logged in but not a subscriber.

// LoggedIn * NotSubscriber * HasUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * NotSubscriber * HasUsedCredit * NotSubscriberOnly (1)
// LoggedIn * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)

There are two states here that are kind of redundant. For example, if you have used a credit to see this particular episode, then it does not matter if the episode was originally subscriber only or not. In particular these two states represent just one that we are actually interested in:

// LoggedIn * NotSubscriber * HasUsedCredit * IsSubscriberOnly  (1)
// LoggedIn * NotSubscriber * HasUsedCredit * NotSubscriberOnly (1)
// ⬇️
// LoggedIn * NotSubscriber * HasUsedCredit * Void              (1)

So we have reduced those two states to one, which brings our original 16 down to just 6! A more than 60% reduction in states! Let’s gather all the states we are actually interested in:

NotLoggedIn * NotSubscriber * NotUsedCredit * (Is/Not)SubscriberOnly (2)
LoggedIn    * IsSubscriber  * Void          * Void                   (1)
LoggedIn    * NotSubscriber * NotUsedCredit * IsSubscriberOnly       (1)
LoggedIn    * NotSubscriber * NotUsedCredit * NotSubscriberOnly      (1)
LoggedIn    * NotSubscriber * HasUsedCredit * Void                   (1)

Translate into Swift

Ok, this has been fun, but we’ve been entirely working in comments and pseudocode. It’s now our job to translate this to a Swift data type. Well, we want the sum of all these states, so I’m thinking at the root level we want an enum. We can see that it splits first at the question of logged in or not logged in. So let’s start there!

enum EpisodePermission {
  // LoggedIn    * IsSubscriber    * Void             * Void         (1)
  // LoggedIn    * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
  // LoggedIn    * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)
  // LoggedIn    * NotSubscriber * HasUsedCredit    * Void           (1)
  case loggedIn

  // NotLoggedIn * NotSubscriber * NotUsedCredit * (Is/Not)SubscriberOnly (2)
  case loggedOut
}

The loggedOut case seems to be the simplest because we don’t know anything about subscriber state or credit state, we only care about whether or not the episode is for subscribers only. So we can fill that in with a boolean:

enum EpisodePermission {
  // LoggedIn    * IsSubscriber    * Void             * Void         (1)
  // LoggedIn    * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
  // LoggedIn    * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)
  // LoggedIn    * NotSubscriber * HasUsedCredit    * Void           (1)
  case loggedIn

  case loggedOut(isEpisodeSubscriberOnly: Bool)
}

In the loggedIn state we can see that we next split on the question of whether or not the user is a subscriber. Sounds like we can introduce a nested enum for that:

enum EpisodePermission {
  case loggedIn(subscriberPermission: SubscriberPermission)

  enum SubscriberPermission {
    // LoggedIn * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
    // LoggedIn * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)
    // LoggedIn * NotSubscriber * HasUsedCredit  * Void             (1)
    case isNotSubscriber

    // LoggedIn * IsSubscriber * Void * Void (1)
    case isSubscriber
  }

  case loggedOut(isEpisodeSubscriberOnly: Bool)
}

Looks like the isSubscriber case is already done, no other data to put there. The isNotSubscriber state, however, has a few more choices in it. Looks like we split on the question of whether or not the user has used a credit for this episode. Sounds like a job for yet another nested enum!

enum EpisodePermission {
  case loggedIn(subscriberPermission: SubscriberPermission)

  enum SubscriberPermission {
    case isNotSubscriber(creditPermission: CreditPermission)

    enum CreditPermission {
      // LoggedIn * NotSubscriber * NotUsedCredit * NotSubscriberOnly (1)
      // LoggedIn * NotSubscriber * NotUsedCredit * IsSubscriberOnly  (1)
      case hasNotUsedCredit

      // LoggedIn * NotSubscriber * HasUsedCredit * Void              (1)
      case hasUsedCredit
    }

    case isSubscriber
  }

  case loggedOut(isEpisodeSubscriberOnly: Bool)
}

Ok we’re so close! The hasUsedCredit is finished, no extra data is needed, but the hasNotUsedCredit needs to further know if the episode is for subscribers only. A simple boolean will solve that:

enum EpisodePermission {
  case loggedIn(subscriberPermission: SubscriberPermission)

  enum SubscriberPermission {
    case isNotSubscriber(creditPermission: CreditPermission)

    enum CreditPermission {
      case hasNotUsedCredit(isEpisodeSubscriberOnly: Bool)
      case hasUsedCredit
    }

    case isSubscriber
  }

  case loggedOut(isEpisodeSubscriberOnly: Bool)
}

And we are done! This is a bit messy, so let’s clean up real quick by grouping the cases together and putting the nested enums last:

enum EpisodePermission {
  case loggedIn(subscriberPermission: SubscriberPermission)
  case loggedOut(isEpisodeSubscriberOnly: Bool)

  enum SubscriberPermission {
    case isNotSubscriber(creditPermission: CreditPermission)
    case isSubscriber

    enum CreditPermission {
      case hasNotUsedCredit(isEpisodeSubscriberOnly: Bool)
      case hasUsedCredit
    }
  }
}

That is so simple! And this is precisely the data type we use in the code on this site! It cleaned up the code that dealt with this permissions type a lot. I was able to delete let’s of guarding and if leting and instead just focus on the states I knew were valid.

So, that’s it for this Point-Free Pointer. I hope you can see how understanding algebra in the Swift type system can greatly simplify the types we work with. Also, the code for this entire website, including everything we discussed today, is fully open sourced on GitHub. If you are curious about this technique, and any of the other things we do on Point-Free, feel free to poke around and ask us questions on Twitter!


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