Free Monad

API Documentation: Free

What is it?

A free monad is a construction which allows you to build a monad from any Functor. Like other monads, it is a pure way to represent and manipulate computations.

In particular, free monads provide a practical way to:

(In Cats, the type representing a free monad is abbreviated as Free[_].)

Using Free Monads

If you'd like to use Cats' free monad, you'll need to add a library dependency for the cats-free module.

A good way to get a sense for how free monads work is to see them in action. The next section uses Free[_] to create an embedded DSL (Domain Specific Language).

If you're interested in the theory behind free monads, the What is Free in theory? section discusses free monads in terms of category theory.

Study your topic

Let's imagine that we want to create a DSL for a key-value store. We want to be able to do three things with keys:

The idea is to write a sequence of these operations in the embedded DSL as a "program", compile the "program", and finally execute the "program" to interact with the actual key-value store.

For example:

put("toto", 3)
get("toto") // returns 3
delete("toto")

But we want:

Study your grammar

We have 3 commands to interact with our KeyValue store:

Create an ADT representing your grammar

ADT stands for Algebraic Data Type. In this context, it refers to a closed set of types which can be combined to build up complex, recursive values.

We need to create an ADT to represent our key-value operations:

sealed trait KVStoreA[A]
case class Put[T](key: String, value: T) extends KVStoreA[Unit]
case class Get[T](key: String) extends KVStoreA[Option[T]]
case class Delete(key: String) extends KVStoreA[Unit]

Free your ADT

There are five basic steps to "freeing" the ADT:

  1. Create a type based on Free[_] and KVStoreA[_].
  2. Create smart constructors for KVStore[_] using liftF.
  3. Build a program out of key-value DSL operations.
  4. Build a compiler for programs of DSL operations.
  5. Execute our compiled program.

1. Create a Free type based on your ADT

import cats.free.Free

type KVStore[A] = Free[KVStoreA, A]

2. Create smart constructors using liftF

These methods will make working with our DSL a lot nicer, and will lift KVStoreA[_] values into our KVStore[_] monad (note the missing "A" in the second type).

import cats.free.Free.liftF

// Put returns nothing (i.e. Unit).
def put[T](key: String, value: T): KVStore[Unit] =
  liftF[KVStoreA, Unit](Put[T](key, value))

// Get returns a T value.
def get[T](key: String): KVStore[Option[T]] =
  liftF[KVStoreA, Option[T]](Get[T](key))

// Delete returns nothing (i.e. Unit).
def delete(key: String): KVStore[Unit] =
  liftF(Delete(key))

// Update composes get and set, and returns nothing.
def update[T](key: String, f: T => T): KVStore[Unit] =
  for {
    vMaybe <- get[T](key)
    _ <- vMaybe.map(v => put[T](key, f(v))).getOrElse(Free.pure(()))
  } yield ()

3. Build a program

Now that we can construct KVStore[_] values we can use our DSL to write "programs" using a for-comprehension:

def program: KVStore[Option[Int]] =
  for {
    _ <- put("wild-cats", 2)
    _ <- update[Int]("wild-cats", (_ + 12))
    _ <- put("tame-cats", 5)
    n <- get[Int]("wild-cats")
    _ <- delete("tame-cats")
  } yield n

This looks like a monadic flow. However, it just builds a recursive data structure representing the sequence of operations.

4. Write a compiler for your program

As you may have understood now, Free[_] is used to create an embedded DSL. By itself, this DSL only represents a sequence of operations (defined by a recursive data structure); it doesn't produce anything.

Free[_] is a programming language inside your programming language!

So, like any other programming language, we need to compile our abstract language into an effective language and then run it.

To do this, we will use a natural transformation between type containers. Natural transformations go between types like F[_] and G[_] (this particular transformation would be written as FunctionK[F,G] or as done here using the symbolic alternative as F ~> G).

In our case, we will use a simple mutable map to represent our key value store:

import cats.arrow.FunctionK
import cats.{Id, ~>}
import scala.collection.mutable

// the program will crash if a type is incorrectly specified.
def impureCompiler: KVStoreA ~> Id  =
  new (KVStoreA ~> Id) {

    // a very simple (and imprecise) key-value store
    val kvs = mutable.Map.empty[String, Any]

    def apply[A](fa: KVStoreA[A]): Id[A] =
      fa match {
        case Put(key, value) =>
          println(s"put($key, $value)")
          kvs(key) = value
          ()
        case Get(key) =>
          println(s"get($key)")
          kvs.get(key).asInstanceOf[A]
        case Delete(key) =>
          println(s"delete($key)")
          kvs.remove(key)
          ()
      }
  }

Please note this impureCompiler is impure -- it mutates kvs and also produces logging output using println. The whole purpose of functional programming isn't to prevent side-effects, it is just to push side-effects to the boundaries of your system in a well-known and controlled way.

Id[_] represents the simplest type container: the type itself. Thus, Id[Int] is just Int. This means that our program will execute immediately, and block until the final value can be returned.

However, we could easily use other type containers for different behavior, such as:

5. Run your program

The final step is naturally running your program after compiling it.

Free[_] is just a recursive structure that can be seen as sequence of operations producing other operations. In this way it is similar to List[_]. We often use folds (e.g. foldRight) to obtain a single value from a list; this recurses over the structure, combining its contents.

The idea behind running a Free[_] is exactly the same. We fold the recursive structure by:

This operation is called Free.foldMap:

final def foldMap[M[_]](f: FunctionK[S,M])(M: Monad[M]): M[A] = ...

M must be a Monad to be flattenable (the famous monoid aspect under Monad). As Id is a Monad, we can use foldMap.

To run your Free with previous impureCompiler:

val result: Option[Int] = program.foldMap(impureCompiler)
// put(wild-cats, 2)
// get(wild-cats)
// put(wild-cats, 14)
// put(tame-cats, 5)
// get(wild-cats)
// delete(tame-cats)
// result: Option[Int] = Some(value = 14)

An important aspect of foldMap is its stack-safety. It evaluates each step of computation on the stack then unstack and restart. This process is known as trampolining.

As long as your natural transformation is stack-safe, foldMap will never overflow your stack. Trampolining is heap-intensive but stack-safety provides the reliability required to use Free[_] for data-intensive tasks, as well as infinite processes such as streams.

6. Use a pure compiler (optional)

The previous examples used an effectful natural transformation. This works, but you might prefer folding your Free in a "purer" way. The State data structure can be used to keep track of the program state in an immutable map, avoiding mutation altogether.

import cats.data.State

type KVStoreState[A] = State[Map[String, Any], A]
val pureCompiler: KVStoreA ~> KVStoreState = new (KVStoreA ~> KVStoreState) {
  def apply[A](fa: KVStoreA[A]): KVStoreState[A] =
    fa match {
      case Put(key, value) => State.modify(_.updated(key, value))
      case Get(key) =>
        State.inspect(_.get(key).asInstanceOf[A])
      case Delete(key) => State.modify(_ - key)
    }
}

(You can see that we are again running into some places where Scala's support for pattern matching is limited by the JVM's type erasure, but it's not too hard to get around.)

val result: (Map[String, Any], Option[Int]) = program.foldMap(pureCompiler).run(Map.empty).value
// result: (Map[String, Any], Option[Int]) = (
//   Map("wild-cats" -> 14),
//   Some(value = 14)
// )

Composing Free monads ADTs.

Real world applications often time combine different algebras. The injection type class described by Swierstra in Data types à la carte lets us compose different algebras in the context of Free.

Let's see a trivial example of unrelated ADT's getting composed as a EitherK that can form a more complex program.

import cats.data.EitherK
import cats.free.Free
import cats.{Id, InjectK, ~>}
import scala.collection.mutable.ListBuffer
/* Handles user interaction */
sealed trait Interact[A]
case class Ask(prompt: String) extends Interact[String]
case class Tell(msg: String) extends Interact[Unit]

/* Represents persistence operations */
sealed trait DataOp[A]
case class AddCat(a: String) extends DataOp[Unit]
case class GetAllCats() extends DataOp[List[String]]

Once the ADTs are defined we can formally state that a Free program is the EitherK of its Algebras.

type CatsApp[A] = EitherK[DataOp, Interact, A]

In order to take advantage of monadic composition we use smart constructors to lift our Algebra to the Free context.

class Interacts[F[_]](implicit I: InjectK[Interact, F]) {
  def tell(msg: String): Free[F, Unit] = Free.liftInject[F](Tell(msg))
  def ask(prompt: String): Free[F, String] = Free.liftInject[F](Ask(prompt))
}

object Interacts {
  implicit def interacts[F[_]](implicit I: InjectK[Interact, F]): Interacts[F] = new Interacts[F]
}

class DataSource[F[_]](implicit I: InjectK[DataOp, F]) {
  def addCat(a: String): Free[F, Unit] = Free.liftInject[F](AddCat(a))
  def getAllCats: Free[F, List[String]] = Free.liftInject[F](GetAllCats())
}

object DataSource {
  implicit def dataSource[F[_]](implicit I: InjectK[DataOp, F]): DataSource[F] = new DataSource[F]
}

ADTs are now easily composed and trivially intertwined inside monadic contexts.

def program(implicit I : Interacts[CatsApp], D : DataSource[CatsApp]): Free[CatsApp, Unit] = {

  import I._, D._

  for {
    cat <- ask("What's the kitty's name?")
    _ <- addCat(cat)
    cats <- getAllCats
    _ <- tell(cats.toString)
  } yield ()
}

Finally we write one interpreter per ADT and combine them with a FunctionK to EitherK so they can be compiled and applied to our Free program.

object ConsoleCatsInterpreter extends (Interact ~> Id) {
  def apply[A](i: Interact[A]) = i match {
    case Ask(prompt) =>
      println(prompt)
      readLine()
    case Tell(msg) =>
      println(msg)
  }
}

object InMemoryDatasourceInterpreter extends (DataOp ~> Id) {

  private[this] val memDataSet = new ListBuffer[String]

  def apply[A](fa: DataOp[A]) = fa match {
    case AddCat(a) => memDataSet.append(a); ()
    case GetAllCats() => memDataSet.toList
  }
}

val interpreter: CatsApp ~> Id = InMemoryDatasourceInterpreter or ConsoleCatsInterpreter

Now if we run our program and type in "snuggles" when prompted, we see something like this:

import DataSource._, Interacts._
val evaled: Unit = program.foldMap(interpreter)
// What's the kitty's name?
// List(snuggles)

For the curious ones: what is Free in theory?

Mathematically-speaking, a free monad (at least in the programming language context) is a construction that is left adjoint to a forgetful functor whose domain is the category of Monads and whose co-domain is the category of Endofunctors. Huh?

Concretely, it is just a clever construction that allows us to build a very simple Monad from any functor.

The above forgetful functor takes a Monad and:

By reversing all arrows to build the left-adjoint, we deduce that the free monad is basically a construction that:

In terms of implementation, to build a monad from a functor we use the following classic inductive definition:

sealed abstract class Free[F[_], A]
case class Pure[F[_], A](a: A) extends Free[F, A]
case class Suspend[F[_], A](a: F[Free[F, A]]) extends Free[F, A]

(This generalizes the concept of fixed point functor.)

In this representation:

So a typical Free structure might look like:

Suspend(F(Suspend(F(Suspend(F(....(Pure(a))))))))

Free is a recursive structure. It uses A in F[A] as the recursion "carrier" with a terminal element Pure.

From a computational point of view, Free recursive structure can be seen as a sequence of operations.

Please note this Free construction has the interesting quality of encoding the recursion on the heap instead of the stack as classic function calls would. This provides the stack-safety we heard about earlier, allowing very large Free structures to be evaluated safely.

For the very curious ones

If you look at implementation in cats, you will see another member of the Free[_] ADT:

case class FlatMapped[S[_], B, C](c: Free[S, C], f: C => Free[S, B]) extends Free[S, B]

FlatMapped represents a call to a subroutine c and when c is finished, it continues the computation by calling the function f with the result of c.

It is actually an optimization of Free structure allowing to solve a problem of quadratic complexity implied by very deep recursive Free computations.

It is exactly the same problem as repeatedly appending to a List[_]. As the sequence of operations becomes longer, the slower a flatMap "through" the structure will be. With FlatMapped, Free becomes a right-associated structure not subject to quadratic complexity.

FreeT

Often times we want to interleave the syntax tree when building a Free monad with some other effect not declared as part of the ADT. FreeT solves this problem by allowing us to mix building steps of the AST with calling action in other base monad.

In the following example a basic console application is shown. When the user inputs some text we use a separate State monad to track what the user typed.

As we can observe in this case FreeT offers us the alternative to delegate denotations to State monad with stronger equational guarantees than if we were emulating the State ops in our own ADT.

import cats.free._
import cats._
import cats.data._

/* A base ADT for the user interaction without state semantics */
sealed abstract class Teletype[A] extends Product with Serializable
final case class WriteLine(line : String) extends Teletype[Unit]
final case class ReadLine(prompt : String) extends Teletype[String]

type TeletypeT[M[_], A] = FreeT[Teletype, M, A]
type Log = List[String]

type TeletypeState[A] = State[List[String], A]

/** Teletype smart constructors */
object TeletypeOps {
  def writeLine(line : String) : TeletypeT[TeletypeState, Unit] =
	FreeT.liftF[Teletype, TeletypeState, Unit](WriteLine(line))
  def readLine(prompt : String) : TeletypeT[TeletypeState, String] =
	FreeT.liftF[Teletype, TeletypeState, String](ReadLine(prompt))
  def log(s : String) : TeletypeT[TeletypeState, Unit] =
	FreeT.liftT[Teletype, TeletypeState, Unit](State.modify(s :: _))
}

def program : TeletypeT[TeletypeState, Unit] = {
  for {
	userSaid <- TeletypeOps.readLine("what's up?!")
	_ <- TeletypeOps.log(s"user said : $userSaid")
	_ <- TeletypeOps.writeLine("thanks, see you soon!")
  } yield ()
}

def interpreter = new (Teletype ~> TeletypeState) {
  def apply[A](fa : Teletype[A]) : TeletypeState[A] = {
	fa match {
	  case ReadLine(prompt) =>
		println(prompt)
		val userInput = "hanging in here" //scala.io.StdIn.readLine()
		StateT.pure[Eval, List[String], A](userInput)
	  case WriteLine(line) =>
		StateT.pure[Eval, List[String], A](println(line))
	}
  }
}

import TeletypeOps._

val state = program.foldMap(interpreter)
// state: TeletypeState[Unit] = cats.data.IndexedStateT@6d77dc25
val initialState = Nil
// initialState: Nil.type = List()
val (stored, _) = state.run(initialState).value
// what's up?!
// thanks, see you soon!
// stored: List[String] = List("user said : hanging in here")

Another example is more basic usage of FreeT with some context Ctx for which we provide Try interpreter, combined with OptionT for reducing boilerplate.

import cats.free._
import cats._
import cats.data._
import cats.syntax.all._
import scala.util.Try

sealed trait Ctx[A]

case class Action(value: Int) extends Ctx[Int]

def op1: FreeT[Ctx, Option, Int] =
  FreeT.liftF[Ctx, Option, Int](Action(7))

def op2: FreeT[Ctx, Option, Int] =
  FreeT.liftT[Ctx, Option, Int](Some(4))

def op3: FreeT[Ctx, Option, Int] =
  FreeT.pure[Ctx, Option, Int](1)

val opComplete: FreeT[Ctx, Option, Int] =
  for {
    a <- op1
    b <- op2
    c <- op3
  } yield a + b + c
// opComplete: FreeT[Ctx, Option, Int] = FlatMapped(
//   a0 = Suspend(a = Some(value = Left(value = Action(value = 7)))),
//   f0 = <function1>
// )


/*  Our interpreters  */

type OptTry[A] = OptionT[Try, A]

def tryInterpreter: Ctx ~> OptTry = new (Ctx ~> OptTry) {
  def apply[A](fa: Ctx[A]): OptTry[A] = {
    fa match {
      case Action(value) =>
        OptionT.liftF(Try(value))
    }
  }
}

def optTryLift: Option ~> OptTry = new (Option ~> OptTry) {
  def apply[A](fa: Option[A]): OptTry[A] = {
    fa match {
      case Some(value) =>
        OptionT(Try(Option(value)))
      case None =>
        OptionT.none
    }
  }
}

val hoisted = opComplete.hoist(optTryLift)
// hoisted: FreeT[Ctx, [A]OptTry[A], Int] = FlatMapped(
//   a0 = Suspend(
//     a = OptionT(
//       value = Success(value = Some(value = Left(value = Action(value = 7))))
//     )
//   ),
//   f0 = scala.Function1$$Lambda$11875/0x00007f0fb9fa0bb8@3708fcdd
// )
val evaluated = hoisted.foldMap(tryInterpreter)
// evaluated: OptTry[Int] = OptionT(value = Success(value = Some(value = 12)))
val result = evaluated.value
// result: Try[Option[Int]] = Success(value = Some(value = 12))

Future Work (TODO)

There are many remarkable uses of Free[_]. In the future, we will include some here, such as:

We will also discuss the Coyoneda Trick.

Credits

This article was written by Pascal Voitot and edited by other members of the Cats community.