Bounds for type parameter of FunctionK

Question

I\'m using cats FreeMonad. Here\'s a simplified version of the algebra:

sealed trait Op[A]

object Op {
    final case class Get[T](name: String         


        

Answer 1

(My original answer contained the same idea, but apparently it did not provide enough implementation details. This time, I wrote a more detailed step-by-step guide with a discussion of each intermediate step. Every section contains a separate compilable code snippet.)

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TL;DR

1. Implicits are required for each type T that occurs in get[T], therefore they must be inserted and stored when the DSL-program is constructed, not when it is executed. This solves the problem with the implicits.
2. There is a general strategy for gluing a natural transformation ~> from several restricted natural transformations trait RNT[R, F[_ <: R], G[_]]{ def apply[A <: R](x: F[A]): G[A] } using pattern matching. This solves the problem with the A <: Resource type bound. Details below.

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In your question, you have two separate problems:

1. implicit Format and Definition
2. <: Resource-type bound

I want to treat each of these two problems in isolation, and provide a reusable solution strategy for both. I will then apply both strategies to your problem.

My answer below is structured as follows:

1. First, I will summarize your question as I understand it.
2. Then I will explain what to do with the implicits, ignoring the type bound.
3. Then I will deal with the type bound, this time ignoring the implicits.
4. Finally, I apply both strategies to your particular problem.

Henceforth, I assume that you have scalaVersion 2.12.4, the dependencies

libraryDependencies += "org.typelevel" %% "cats-core" % "1.0.1"
libraryDependencies += "org.typelevel" %% "cats-free" % "1.0.1"

and that you insert

import scala.language.higherKinds

where appropriate. Note that the solution strategies are not specific to this particular scala version or the cats library.

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The setup

The goal of this section is to make sure that I'm solving the right problem, and also to provide very simple mock-up definitions of Resource, Format, Client etc., so that this answer is self-contained and compilable.

I assume that you want to build a little domain specific language using the Free monad. Ideally, you would like to have a DSL that looks approximately like this (I've used the names DslOp for the operations and Dsl for the generated free monad):

import cats.free.Free
import cats.free.Free.liftF

sealed trait DslOp[A]
case class Get[A](name: String) extends DslOp[A]

type Dsl[A] = Free[DslOp, A]
def get[A](name: String): Dsl[A] = liftF[DslOp, A](Get[A](name))

It defines a single command get that can get objects of type A given a string name.

Later, you want to interpret this DSL using a get method provided by some Client that you cannot modify:

import scala.concurrent.Future

trait Resource
trait Format[A <: Resource]
trait Definition[A <: Resource]

object Client {
  def get[A <: Resource](name: String)
    (implicit f: Format[A], d: Definition[A]): Future[A] = ???
}

Your problem is that the get method of the Client has a type bound, and that it requires additional implicits.

Dealing with implicits when defining interpreter for the Free monad

Let's first pretend that the get-method in client requires implicits, but ignore the type bound for now:

import scala.concurrent.Future

trait Format[A]
trait Definition[A]

object Client {
  def get[A](name: String)(implicit f: Format[A], d: Definition[A])
  : Future[A] = ???
}

Before we write down the solution, let's briefly discuss why you cannot supply all the necessary implicits when you are calling the apply method in ~>.

• When passed to foldMap, the apply of FunctionK is supposed to be able to cope with arbitrarily long programs of type Dsl[X] to produce Future[X].

• Arbitrarily long programs of type Dsl[X] can contain an unlimited number of get[T1], ..., get[Tn] commands for different types T1, ..., Tn.

• For each of those T1, ..., Tn, you have to get a Format[T_i] and Definition[T_i] somewhere.

• These implicit arguments must be supplied by the compiler.

• When you interpret the entire program of type Dsl[X], only the type X but not the types T1, ..., Tn are available, so the compiler cannot insert all the necessary Definitions and Formats at the call site.

• Therefore, all the Definitions and Formats must be supplied as implicit parameters to get[T_i] when you are constructing the Dsl-program, not when you are interpreting it.

The solution is to add Format[A] and Definition[A] as members to the Get[A] case class, and make the definition of get[A] with lift[DslOp, A] accept these two additional implicit parameters:

import cats.free.Free
import cats.free.Free.liftF
import cats.~>

sealed trait DslOp[A]
case class Get[A](name: String, f: Format[A], d: Definition[A]) 
  extends DslOp[A]

type Dsl[A] = Free[DslOp, A]
def get[A](name: String)(implicit f: Format[A], d: Definition[A])
: Dsl[A] = liftF[DslOp, A](Get[A](name, f, d))

Now, we can define the first approximation of the ~>-interpreter, which at least can cope with the implicits:

val clientInterpreter_1: (DslOp ~> Future) = new (DslOp ~> Future) {
  def apply[A](op: DslOp[A]): Future[A] = op match {
    case Get(name, f, d) => Client.get(name)(f, d)
  }
}

Type bounds in case classes defining the DSL-operations

Now, let's deal with the type bound in isolation. Suppose that your Client doesn't need any implicits, but imposes an additional bound on A:

import scala.concurrent.Future

trait Resource
object Client {
  def get[A <: Resource](name: String): Future[A] = ???
}

If you tried to write down the clientInterpreter in the same way as in the previous example, you would notice that the type A is too general, and that you therefore cannot work with the contents of Get[A] in Client.get. Instead, you have to find a scope where the additional type information A <: Resource is not lost. One way to achieve it is to define an accept method on Get itself. Instead of a completely general natural transformation ~>, this accept method will be able to work with natural transformations with restricted domain. Here is a trait to model that:

trait RestrictedNat[R, F[_ <: R], G[_]] {
  def apply[A <: R](fa: F[A]): G[A]
}

It looks almost like ~>, but with an additional A <: R restriction. Now we can define accept in Get:

import cats.free.Free
import cats.free.Free.liftF
import cats.~>

sealed trait DslOp[A]
case class Get[A <: Resource](name: String) extends DslOp[A] {
  def accept[G[_]](f: RestrictedNat[Resource, Get, G]): G[A] = f(this)
}

type Dsl[A] = Free[DslOp, A]
def get[A <: Resource](name: String): Dsl[A] = 
  liftF[DslOp, A](Get[A](name))

and write down the second approximation of our interpreter, without any nasty type-casts:

val clientInterpreter_2: (DslOp ~> Future) = new (DslOp ~> Future) {
  def apply[A](op: DslOp[A]): Future[A] = op match {
    case g @ Get(name) => {
      val f = new RestrictedNat[Resource, Get, Future] {
        def apply[X <: Resource](g: Get[X]): Future[X] = Client.get(g.name)
      }
      g.accept(f)
    }
  }
}

This idea can be generalized to an arbitrary number of type constructors Get_1, ..., Get_N, with type restrictions R1, ..., RN. The general idea corresponds to the construction of a piecewise defined natural transformation from smaller pieces that work only on certain subtypes.

Applying both solution strategies to your problem

Now we can combine the two general strategies into one solution for your concrete problem:

import scala.concurrent.Future
import cats.free.Free
import cats.free.Free.liftF
import cats.~>

// Client-definition with both obstacles: implicits + type bound
trait Resource
trait Format[A <: Resource]
trait Definition[A <: Resource]

object Client {
  def get[A <: Resource](name: String)
    (implicit fmt: Format[A], dfn: Definition[A])
  : Future[A] = ???
}


// Solution:
trait RestrictedNat[R, F[_ <: R], G[_]] {
  def apply[A <: R](fa: F[A]): G[A]
}

sealed trait DslOp[A]
case class Get[A <: Resource](
  name: String,
  fmt: Format[A],
  dfn: Definition[A]
) extends DslOp[A] {
  def accept[G[_]](f: RestrictedNat[Resource, Get, G]): G[A] = f(this)
}

type Dsl[A] = Free[DslOp, A]
def get[A <: Resource]
  (name: String)
  (implicit fmt: Format[A], dfn: Definition[A])
: Dsl[A] = liftF[DslOp, A](Get[A](name, fmt, dfn))


val clientInterpreter_3: (DslOp ~> Future) = new (DslOp ~> Future) {
  def apply[A](op: DslOp[A]): Future[A] = op match {
    case g: Get[A] => {
      val f = new RestrictedNat[Resource, Get, Future] {
        def apply[X <: Resource](g: Get[X]): Future[X] = 
          Client.get(g.name)(g.fmt, g.dfn)
      }
      g.accept(f)
    }
  }
}

Now, the clientInterpreter_3 can cope with both problems: the type-bound-problem is dealt with by defining a RestrictedNat for each case class that imposes an upper bound on its type arguments, and the implicits-problem is solved by adding an implicit parameter list to DSL's get-method.