Explain Free Monad Like I am Five (Part 1)

Photo by Meagan Carsience

In Cats documentation:

A free monad is a construction that allows you to build a monad from any Functor.

Another definition says it is an idea to switch from effect functions (that might be impure) to plain data structures representing our domain logic.

Simply put, it is like a wrapper that wraps any ADT into a monadic structure DSL that you able to create a program out of it. It separates your program DSL with its interpreter so that you can pick and choose various interpreters.

This post will start with a simple Todo application (no concurrency and all that), from the imperative standpoint, and slowly transform it into a Free monad style.

Creating Algebra

case class Todo(id: Long, description: String, isFinished: Boolean)

sealed trait Action[T] extends Product with Serializable
case class Create(description: String) extends Action[Unit]
case class Find(description: String) extends Action[Unit]
case class Mark(id: Long) extends Action[Unit]
case class Delete(id: Long) extends Action[Unit]
case object Read extends Action[Unit]

The above, we created the Todo case class, with a simple CRUD.

The initial way to embed DSL to other programming languages was by using ADT to encode every sentence in your programming languages as a form of the ADT.

Imperative Program

We will want to do a series of operations such as Create a Todo. Then, we can Find the id belongs to the Todo List. Read the Todo List, to show all the results of the Todo list that we had, and Mark the Todo that we have finished.

The imperative way of doing such things can be to construct a List of these ADT like this:

val program = List(
    Create("Do Laundry"),
    Read,
    Find("Do Laundry"),
    Mark(0L)
  )

Therefore, the program above is just a description. To execute the plan, we need some executing interpreter to run it.

Let’s create an “execute” interpreter:

object Interpreter {
  import Dsl._
  var map = scala.collection.mutable.Map[Long, Todo]()
  var id = 0L

  def execute[T](action: Action[T]): T = {
    action match {
      case Read =>
        println(map.values.toList)
        ().asInstanceOf[T]
      case Find(id) =>
        println(map.get(id))
        map.get(id).asInstanceOf[T]
      case Mark(id) =>
        println(
          map
            .get(id)
            .flatMap(t => map.put(id, t.copy(isFinished = !t.isFinished)))
        )
        map
          .get(id)
          .flatMap(t => map.put(id, t.copy(isFinished = !t.isFinished)))
          .asInstanceOf[T]
      case Delete(id) =>
        println(s"removing ${map.get(id)}")
        map.remove(id).asInstanceOf[T]
      case Create(description) =>
        println(s"creating todo list ${description} in id ${id}")
        (map += (id -> Todo(id, description, false)))
        id += 1
        ().asInstanceOf[T]
    }
  }

}

We use pattern matching to do all kinds of action when we run the program’s description above. This is where we all the side-effect and all the mutation happens. Therefore, we will loop through the program and run execute on each of the descriptions:

program.foreach(t => Interpreter.execute(t))

So far, so good. However, the ADT that we describe above is not very useful. We cannot, for instance, Find an id, and based on that id marked the Todo list. Ideally, we want to do a sequential operation, such as creating a Todo List, Reading all the list of todos, and marking the finished ones.

We need to find a way to get the previous operation’s value and do some other sequential process based on the prior operation’s evaluated value. Sounds like a Monad, right?

Ideally, we want to do something like this:

val program = for {
  todo <- Create("Do Laundry")
  listTodos <- Read
  idZero <- Find(0L)
  _ <- Mark(todo.id)
} yield ()

The problem now is that since the program is not a List anymore, how do we create the interpreter?

Since we are creating a general data structure on the program, we want to have some sort of “wrapper” to wrap these data structures with a Monadic bind to construct a monadic type of program.

Big Rewrite

We need to return our algebra to “return” some value to capture it in the monadic bind:



case class Todo(id: Long, description: String, isFinished: Boolean)

sealed trait Action[T] extends Product with Serializable
                                                  // The return type
                                                    // vv
case class Create(description: String) extends Action[Todo]
case class Find(description: String) extends Action[Option[Todo]]
case class Mark(id: Long) extends Action[Unit]
case class Delete(id: Long) extends Action[Option[Todo]]
case object Read extends Action[List[Todo]]

Let’s write a wrapper of our program so that it can have that monadic bind function. The monadic operation can be translated to something like this:

FlatMap(Create("Do Laundry"), todo => 
  FLatMap(Create("Clean Bedroom"), cleanBedroom => 
    FlatMap(Read, (listTodos) => 
      FlatMap(Find(0L), idZero => 
        Pure(Mark(idZero))
      )
    )
  )
)

We introduce FlatMap and Pure to bind our original algebraic type to Monad:

sealed trait Free[F[_], A]

case class FlatMap[F[_], A, B](fa: F[A], f: A => F[B]) extends Free[F,B]

case class Pure[F[_],A](fa: F[A]) extends Free[F, A]

Imagine that F[_] being like Todo, but it can be any type of F[_]. It doesn’t have any constraint, and FlatMap and Pure are analogous to flatMap and pure in Monad, where it binds the context into a sequential operation.

Free is a recursive data structure where each subsequent computation can access the previous calculation. This is all we need to build the programs using a straightforward data structure that is free to its interpretation.

How can we make the above Free ADT works with the for-comprehension above?

Ideally, we want to come into this conclusion:

val program =
for {
  todo <- Create("Do Laundry")
  _ <- Mark(todo.id)
  listOfTodo <- Read
  } yield {
    println(listOfTodo)
  }

To do so:

  1. Free has to be a monad. It needs to have some flatMap and map so that scala can detect and do “for-comprehension”.
  2. We want the program to do flatMap on Free, not the “Action” ADT that we defined. That leaves the action just a data structure that we can wire to our interpreter later on.

Creating Free as a Monad

Let’s construct Free function, which it needs to have a map and a flatMap method.

sealed trait Free[F[_], A] {
  def flatMap[B](func: A => Free[F,B]): Free[F,B] = this match {
    case FlatMap(fa, f) => FlatMap(fa, f andThen (a => a.flatMap(func))) // a is a Free[F[_], A] here
    case Pure(a) => func(a)
  }
  
  def map[B](func : A => B): Free[F,B] = flatMap(a => Pure(func(a)))
}

The code flatMap above recursively doing flatMap until it hits Pure, and apply the func to a. f andThen (a => a.flatMap(func)) means it compose the function by applying the input to f first. The return value of f(a) is a Free[F[_], B]. The return value, then, is applied to the consequent function a => flatMap(func). If you still don’t get it, slap the above code into IntelliJ, and try solving the type yourself. The more you look at the input type and return type, the more you know how to create the function above.

Once we have the map and flatMap function ready, we can start constructing our program.

However, how do we construct Free?

Lifting Free stuff

We want to lift the Action to a Free[F[_],A].

Let’s create a lift function that will do that:

def lift[A](fa: Action[A]): Free[Action,A] = FlatMap(fa, Pure.apply)

Then, we can create our program like this:

val program = for {
  todo <- lift(Create("Do Laundry"))
  cleanBedroom  <- lift(Create("Clean Bedroom"))
  listTodos <- lift(Read)
  idZero <- lift(Find(0L))
  _ <- lift(Mark(idZero))
} yield ()

We can also just create a dsl to make it more readable way by creating an implicit conversion:

def lift[A](fa: Action[A]): Free[Action,A] = FlatMap(fa, Pure.apply)

val program =
  for {
    todo <- Create("Do Laundry")

    _ <- Mark(todo.id)
    listOfTodo <- Read

  } yield {
    println(listOfTodo)
  }

Interpreter

Now that we have created our program, we also need to bind the program with an interpreter somehow.

Let’s look back at the first interpreter that we defined:

 def execute[T](action: Action[T]): T = {
    action match {
      case Read =>
        println(map.values.toList)
        map.values.toList.asInstanceOf[T]
      case Find(description) =>
        println(map.values.find(t => t.description == description))
        map.values.find(t => t.description == description).asInstanceOf[T]
      case Mark(id) =>
        println(
          map
            .get(id)
            .flatMap(t => map.put(id, t.copy(isFinished = !t.isFinished)))
        )
        ().asInstanceOf[T]
      case Delete(id) =>
        println(s"removing ${map.get(id)}")
        map.remove(id).asInstanceOf[T]
      case Create(description) =>
        println(s"creating todo list ${description} in id ${id}")
        val todo = Todo(id, description, false)
        (map += (id -> Todo(id, description, false)))
        id += 1
        todo.asInstanceOf[T]
    }
  }

The above code looks good, but how do we connect the interpreter above with our Free Monad?

We want to do something like this - given the following Free[F,A] data structure, we want to traverse the Free structure, evaluating each step and thread the result to the next subsequent computation. We want to fold the List of the program description.

Ultimately, we will do the same for Free, by creating a pattern matching for FlatMap and Return:

def runProgram[A](free: Free[Action,A]): A = free match {
  case Pure(a) => a
  case FlatMap(fa, fn) => ???
}

I put a ??? above because we don’t have any access to the regular interpreter that we created. If only if we also supply the interpreter as another argument to evaluate the Action:


 def execute[T](action: Action[T]): T = {
    action match {
      case Read =>
        println(map.values.toList)
        map.values.toList.asInstanceOf[T]
      case Find(description) =>
        println(map.values.find(t => t.description == description))
        map.values.find(t => t.description == description).asInstanceOf[T]
      case Mark(id) =>
        println(
          map
            .get(id)
            .flatMap(t => map.put(id, t.copy(isFinished = !t.isFinished)))
        )
        ().asInstanceOf[T]
      case Delete(id) =>
        println(s"removing ${map.get(id)}")
        map.remove(id).asInstanceOf[T]
      case Create(description) =>
        println(s"creating todo list ${description} in id ${id}")
        val todo = Todo(id, description, false)
        (map += (id -> Todo(id, description, false)))
        id += 1
        todo.asInstanceOf[T]
    }
}

def runProgram[A](program:Free[Action,A]): A = program match {
  case Pure(a) => a
  case FlatMap(fa, fn) =>
    // execute the Action here
    val res = interpreter(fa)
    // thread the function into a new Free
    val newFree = fn(res)
    // execute the next function
    runProgram(newFree, interpreter)
}

We can run our program with the existing interpreter like this:

runProgram(program)

Conclusion

We have come a long way by first introducing the traditional imperative way of creating a program description. We realized that we have no access to the previously computed value within a regular’ List’ of description and cannot make a sequentially like computation that we want to.

Then, we created a Free structure that enabled us to wrap our existing ADT into something more monadic. We introduce Free ADT by having FlatMap and Pure, which is analogous to monadic bind for flatMap and pure. Besides, we also created a way to lift the Action type to a Free variety to comprehend them.

Lastly, we touch on using the existing interpreter that we created,execute, and execute the Free program.

In part 2 of this series, I want to dive further into how we can generalize the Free ADT and create a free structure to create a DSL in any program ultimately. Stay tuned!

Full source code is here.

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