Free Applicative

FreeApplicatives are similar to Free (monads) in that they provide a nice way to represent computations as data and are useful for building embedded DSLs (EDSLs). However, they differ from Free in that the kinds of operations they support are limited, much like the distinction between Applicative and Monad.


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


Consider building an EDSL for validating strings - to keep things simple we’ll just have a way to check a string is at least a certain size and to ensure the string contains numbers.

sealed abstract class ValidationOp[A]
case class Size(size: Int) extends ValidationOp[Boolean]
case object HasNumber extends ValidationOp[Boolean]

Much like the Free monad tutorial, we use smart constructors to lift our algebra into the FreeApplicative.


type Validation[A] = FreeApplicative[ValidationOp, A]

def size(size: Int): Validation[Boolean] = lift(Size(size))

val hasNumber: Validation[Boolean] = lift(HasNumber)

Because a FreeApplicative only supports the operations of Applicative, we do not get the nicety of a for-comprehension. We can however still use Applicative syntax provided by Cats.

import cats.implicits._

val prog: Validation[Boolean] = (size(5) |@| hasNumber).map { case (l, r) => l && r}

As it stands, our program is just an instance of a data structure - nothing has happened at this point. To make our program useful we need to interpret it.

import cats.Id
import cats.arrow.FunctionK
import cats.implicits._

// a function that takes a string as input
type FromString[A] = String => A

val compiler =
   λ[FunctionK[ValidationOp, FromString]] { fa =>
      str =>
        fa match {
          case Size(size) => str.size >= size
          case HasNumber  => str.exists(c => "0123456789".contains(c))
val validator = prog.foldMap[FromString](compiler)
// validator: FromString[Boolean] = scala.Function1$$Lambda$2476/1848486024@48d61010

// res7: Boolean = false

// res8: Boolean = true

Differences from Free

So far everything we’ve been doing has been not much different from Free - we’ve built an algebra and interpreted it. However, there are some things FreeApplicative can do that Free cannot.

Recall a key distinction between the type classes Applicative and Monad - Applicative captures the idea of independent computations, whereas Monad captures that of dependent computations. Put differently Applicatives cannot branch based on the value of an existing/prior computation. Therefore when using Applicatives, we must hand in all our data in one go.

In the context of FreeApplicatives, we can leverage this static knowledge in our interpreter.


Because we have everything we need up front and know there can be no branching, we can easily write a validator that validates in parallel.

import cats.implicits._
import scala.concurrent.Future

// recall Kleisli[Future, String, A] is the same as String => Future[A]
type ParValidator[A] = Kleisli[Future, String, A]

val parCompiler =
  λ[FunctionK[ValidationOp, ParValidator]] { fa =>
    Kleisli { str =>
      fa match {
        case Size(size) => Future { str.size >= size }
        case HasNumber  => Future { str.exists(c => "0123456789".contains(c)) }

val parValidation = prog.foldMap[ParValidator](parCompiler)


We can also write an interpreter that simply creates a list of strings indicating the filters that have been used - this could be useful for logging purposes. Note that we need not actually evaluate the rules against a string for this, we simply need to map each rule to some identifier. Therefore we can completely ignore the return type of the operation and return just a List[String] - the Const data type is useful for this.

import cats.implicits._

type Log[A] = Const[List[String], A]

val logCompiler =
  λ[FunctionK[ValidationOp, Log]] {
    case Size(size) => Const(List(s"size >= $size"))
    case HasNumber  => Const(List("has number"))

def logValidation[A](validation: Validation[A]): List[String] =
// res16: List[String] = List(size >= 5, has number)

logValidation(size(5) *> hasNumber *> size(10))
// res17: List[String] = List(size >= 5, has number, size >= 10)

logValidation((hasNumber |@| size(3)).map(_ || _))
// res18: List[String] = List(has number, size >= 3)

Why not both?

It is perhaps more plausible and useful to have both the actual validation function and the logging strings. While we could easily compile our program twice, once for each interpreter as we have above, we could also do it in one go - this would avoid multiple traversals of the same structure.

Another useful property Applicatives have over Monads is that given two Applicatives F[_] and G[_], their product type FG[A] = (F[A], G[A]) is also an Applicative. This is not true in the general case for monads.

Therefore, we can write an interpreter that uses the product of the ParValidator and Log Applicatives to interpret our program in one go. We can create this interpreter easily by using FunctionK#and.


type ValidateAndLog[A] = Prod[ParValidator, Log, A]

val prodCompiler: FunctionK[ValidationOp, ValidateAndLog] = parCompiler and logCompiler

val prodValidation = prog.foldMap[ValidateAndLog](prodCompiler)


Deeper explanations can be found in this paper Free Applicative Functors by Paolo Capriotti