Solutions to "Scala with Cats": Chapter 4

April 4, 2023

These are my solutions to the exercises of chapter 4 of Scala with Cats.

Exercise 4.1.2: Getting Func-y
Exercise 4.3.1: Monadic Secret Identities
Exercise 4.4.5: What is Best?
Exercise 4.5.4: Abstracting
Exercise 4.6.5: Safer Folding using Eval
Exercise 4.7.3: Show Your Working
Exercise 4.8.3: Hacking on Readers
Exercise 4.9.3: Post-Order Calculator
Exercise 4.10.1: Branching out Further with Monads

Exercise 4.1.2: Getting Func-y

We have pure and flatMap to define map. We want to start from an F[A] and get to an F[B] from a function A => B. As such, we want to call flatMap over the value. We can’t use func directly, though. However, we can produce a function that would lift our value to an F using pure (a => pure(func(a))):

trait Monad[F[_]] {
  def pure[A](a: A): F[A]

  def flatMap[A, B](value: F[A])(func: A => F[B]): F[B]

  def map[A, B](value: F[A])(func: A => B): F[B] =
    flatMap(value)(func.andThen(pure))
}

Exercise 4.3.1: Monadic Secret Identities

pure, map and flatMap for Id can be implemented as follows:

def pure[A](a: A): Id[A] =
  a

def flatMap[A, B](value: Id[A])(func: A => Id[B]): Id[B] =
  func(value)

def map[A, B](value: Id[A])(func: A => B): Id[B] =
  func(value)

Since Id[A] is just a type alias for A, we can notice that we avoid all boxing in the implementations and, due to that fact, flatMap and map are identical.

Exercise 4.4.5: What is Best?

The answer depends on what we are looking for in specific instances, but some things that the previous examples for error handling don’t cover are:

We can’t accumulate errors. The proposed examples all fail fast.
We can’t tell exactly where the error was raised.
It’s not easy to do error recovery.

Exercise 4.5.4: Abstracting

A possible implementation for validateAdult is the following:

import cats.{Applicative, MonadError}

def validateAdult[F[_]](age: Int)(implicit me: MonadError[F, Throwable]): F[Int] =
  if (age >= 18) Applicative[F].pure(age)
  else me.raiseError(new IllegalArgumentException("Age must be greater than or equal to 18"))
}

If age is greater than or equal to 18, we summon an Applicative for F (which we must have in scope due to MonadError) and lift the age to F. If age is less than 18, we use the MonadError instance we have in scope to lift an IllegalArgumentException to F.

Exercise 4.6.5: Safer Folding using Eval

One way to make the naive implementation of foldRight stack safe using Eval is the following:

import cats.Eval

def foldRightEval[A, B](as: List[A], acc: B)(fn: (A, B) => B): B = {
  def aux(as: List[A], acc: B): Eval[B] =
    as match {
      case head :: tail =>
        Eval.defer(aux(tail, acc)).map(fn(head, _))
      case Nil =>
        Eval.now(acc)
    }

  aux(as, acc).value
}

We defer the call to the recursive step and then map over it to apply fn, all within the context of Eval.

Exercise 4.7.3: Show Your Working

A possible rewrite of factorial so that it captures the log messages in a Writer is the following:

import cats.data.Writer

def factorial(n: Int): Writer[Vector[String], Int] = {
  slowly {
    if (n == 0)
      Writer.apply(Vector("fact 0 1"), 1)
    else
      factorial(n - 1).mapBoth { (log, res) =>
        val ans = res * n
        (log :+ s"fact $n $ans", ans)
      }
  }
}

We can show that this allows us to reliably separate the logs for concurrent computations because we have the logs for each instance captured in each Writer instance:

Await.result(Future.sequence(Vector(
  Future(factorial(5)),
  Future(factorial(5))
)).map(_.map(_.written)), 5.seconds)
// Returns Vector(
//   Vector("fact 0 1", "fact 1 1", "fact 2 2", "fact 3 6", "fact 4 24", "fact 5 120"),
//   Vector("fact 0 1", "fact 1 1", "fact 2 2", "fact 3 6", "fact 4 24", "fact 5 120")
// )

Exercise 4.8.3: Hacking on Readers

To create a type alias for a Reader that consumes a Db we want to fix the first type parameter of Reader to Db, while still leaving the result type as a type parameter:

import cats.data.Reader

type DbReader[A] = Reader[Db, A]

The findUsername and checkPassword functions can be implemented as follows:

def findUsername(userId: Int): DbReader[Option[String]] =
  Reader.apply(db => db.usernames.get(userId))

def checkPassword(username: String, password: String): DbReader[Boolean] =
  Reader.apply(db => db.passwords.get(username).contains(password))

The checkLogin method can be implemented as follows:

def checkLogin(userId: Int, password: String): DbReader[Boolean] =
  for {
    usernameOpt <- findUsername(userId)
    validLogin <- usernameOpt.map(checkPassword(_, password)).getOrElse(Reader.apply((_: Db) => false))
  } yield validLogin

We are making use of the findUsername and checkPassword methods. There are two scenarios in which checkLogin can return a false for a given Db: when the username doesn’t exist and when the password doesn’t match.

Exercise 4.9.3: Post-Order Calculator

A possible implementation of evalOne with no proper error handling is the following:

def evalOne(sym: String): CalcState[Int] = {
  def op(f: (Int, Int) => Int): CalcState[Int] = State {
    case y :: x :: rest =>
      val ans = f(x, y)
      (ans :: rest, ans)
    case _ =>
      throw new IllegalArgumentException("Insufficient stack size")
  }

  def num(value: String): CalcState[Int] = State { s =>
    val ans = value.toInt
    (ans :: s, ans)
  }

  sym match {
    case "+"   => op(_ + _)
    case "*"   => op(_ * _)
    case "-"   => op(_ - _)
    case "/"   => op(_ / _)
    case other => num(other)
  }
}

We’re not told which operands to support, so I assumed at least +, *, - and /.

For the evalAll implementation, we’re not told what to do in case the input is empty. I assumed it would be OK to just have an exception thrown (since that was the case before), and relied on reduce over the evalOne calls:

def evalAll(input: List[String]): CalcState[Int] =
  input.map(evalOne).reduce((e1, e2) => e1.flatMap(_ => e2))

The evalInput method can rely on a call to evalAll after splitting the input by whitespaces:

def evalInput(input: String): Int =
  evalAll(input.split("\\s+").toList).runA(Nil).value

Exercise 4.10.1: Branching out Further with Monads

One implementation of Monad for Tree is the following:

implicit val treeMonad: Monad[Tree] = new Monad[Tree] {
  def pure[A](a: A): Tree[A] =
    Leaf(a)

  def flatMap[A, B](fa: Tree[A])(f: A => Tree[B]): Tree[B] =
    fa match {
      case Branch(left, right) =>
        Branch(flatMap(left)(f), flatMap(right)(f))

      case Leaf(value) =>
        f(value)
    }

  def tailRecM[A, B](a: A)(f: A => Tree[Either[A, B]]): Tree[B] =
    flatMap(f(a)) {
      case Left(value) =>
        tailRecM(value)(f)

      case Right(value) =>
        Leaf(value)
    }
}

However, tailRecM isn’t tail-recursive. We can make it tail-recursive by making the recursion explicit in the heap. In this case, we’re using two mutable stacks: one of open nodes to visit and one of already visited nodes. On that stack, we use None to signal a non-leaf node and a Some to signal a leaf node.

implicit val treeMonad: Monad[Tree] = new Monad[Tree] {
  def pure[A](a: A): Tree[A] =
    Leaf(a)

  def flatMap[A, B](fa: Tree[A])(f: A => Tree[B]): Tree[B] =
    fa match {
      case Branch(left, right) =>
        Branch(flatMap(left)(f), flatMap(right)(f))

      case Leaf(value) =>
        f(value)
    }

  def tailRecM[A, B](a: A)(f: A => Tree[Either[A, B]]): Tree[B] = {
    import scala.collection.mutable

    val open = mutable.Stack.empty[Tree[Either[A, B]]]
    val closed = mutable.Stack.empty[Option[Tree[B]]]

    open.push(f(a))

    while (open.nonEmpty) {
      open.pop() match {
        case Branch(l, r) =>
          open.push(r)
          open.push(l)
          closed.push(None)

        case Leaf(Left(value)) =>
          open.push(f(value))

        case Leaf(Right(value)) =>
          closed.push(Some(pure(value)))
      }
    }

    val ans = mutable.Stack.empty[Tree[B]]

    while (closed.nonEmpty) {
      closed.pop() match {
        case None    => ans.push(Tree.branch(ans.pop(), ans.pop()))
        case Some(v) => ans.push(v)
      }
    }

    ans.pop()
  }
}

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