And why they are so useful Monads And why they are so useful

flatMap revisited The flatMap method is like Map, but removes one level of nesting from a sequence scala> List(2, 3, 4, 5) map (x => List(x, x * x, x * x * x)) res21: List[List[Int]] = List(List(2, 4, 8), List(3, 9, 27), List(4, 16, 64), List(5, 25, 125)) scala> List(2, 3, 4, 5) flatMap (x => List(x, x * x, x * x * x)) res22: List[Int] = List(2, 4, 8, 3, 9, 27, 4, 16, 64, 5, 25, 125) Used with a sequence of Option, flatMap effectively reduces Some(x) to x, and entirely deletes None scala> List(1, -1, 2, 4, -5, 9) flatMap (root(_)) res17: List[Double] = List(1.0, 1.4142135623730951, 2.0, 3.0)

map and for…yield val nums = List(1, 2, 3) List(1, 2, 3) nums map (_ * 2) List(2, 4, 6) for (n <- nums) yield (n * 2) List(2, 4, 6) nums map (n => List(n, 2 * n, 3 * n)) List(List(1, 2, 3), List(2, 4, 6), List(3, 6, 9)) nums map (n => List(n, 2 * n, 3 * n)) flatten List(1, 2, 3, 2, 4, 6, 3, 6, 9) nums flatMap (n => List(n, 2 * n, 3 * n)) List(1, 2, 3, 2, 4, 6, 3, 6, 9)

More map and for…yield for (x <- List(1, 2, 3); y <- List(10, 100)) yield x * y List(10, 100, 20, 200, 30, 300) List(1, 2, 3) flatMap { x => List(10, 100) map (y => x * y) } List(10, 100, 20, 200, 30, 300) for (x <- List(1, 2, 3, 4) if x != 3) yield 2 * x List(2, 4, 8) List(1, 2, 3, 4) filter (_ != 3) map (2 * _) List(2, 4, 8) for…yield is not a loop—it is syntactic sugar for a sequence of map, flatmap, and filter operations Of course, those operations may be implemented using loops for without a yield is equivalent to a foreach

Functors In Scala terms, a functor is anything that has a map operation Scala’s collections (lists, arrays, sets, maps, strings, etc.) have a map operation It is implemented differently for different collection types trait F[T] { def map(f: T => S): F[S] }

The Option type Scala has null because it interoperates with Java; it shouldn’t be used any other time Instead, use an Option type, with values Some(value) and None def max(list: List[Int]) = { if (list.length > 0) { val big = list reduce {(a, b) => if (a > b) a else b} Some(big) } else { None } max(myList) match { case Some(x) => println("The largest number is " + x) case None => println("There are no numbers here!!!") }

Try scala> import scala.util.{Try, Success, Failure} import scala.util.{Try, Success, Failure} scala> def root2(x: Double): Try[Double] = | if (x >= 0) Success(math.sqrt(x)) else | Failure(new Exception("Imaginary root")) root2: (x: Double)scala.util.Try[Double] scala> root2(10) res28: scala.util.Try[Double] = Success(3.1622776601683795) scala> root2(-10) res29: scala.util.Try[Double] = Failure(java.lang.Exception: Imaginary root)

Containers I The map operation is also useful for various types of containers Some(5) map (x => 2 * x) Some(10) val nope: Option[Int] = None None import scala.util.{Try, Success, Failure} Success(5) map (x => 2 * x) Success(10) val fail: Try[Int] = Failure(new Exception) fail map (x => 2 * x) Failure(java.lang.Exception)

Containers II import scala.concurrent._ import scala.concurrent.duration._ import scala.concurrent.ExecutionContext.Implicits.global val foo = Future{42} map (x => 2 * x) List() Await.result(foo, 1 nano) 84 Therefore, Option (Some, None), Try (Success, Failure), and Future are also functors

Monad as a trait In Scala, we could define a monad trait as follows: trait M[A] { def flatMap[B](f: A => M[B]): M[B] } def unit[A](x: A): M[A] You can think of unit as a constructor, or better yet, as a factory method There isn’t actually any unit method in Scala, but there is apply Notice that monads have a type parameter (A) If a Scala object has a constructor with a type parameter and a flatMap operation, it’s a monad! Scala is full of monads Examples: List, Set, Map, Array, String, Option, Try, Future, to name a few

Comparing map and flatMap val listA: List[A] = … final def map[B](f: (A) ⇒ B): List[B] final def flatMap[B](f: (A) ⇒ M[B]): List[B] where M = GenTraversableOnce val nums = List(1, 10, 100) x map (v => List(v - 1, v, v + 1)) res0: List[List[Int]] = List(List(0, 1, 2), List(9, 10, 11), List(99, 100, 101)) x flatMap (v => List(v - 1, v, v + 1)) res1: List[Int] = List(0, 1, 2, 9, 10, 11, 99, 100, 101)

Monad in Haskell and Scala Remember Haskell? A monad consists of three things: A type constructor M A bind operation, (>>=) :: (Monad m) => m a -> (a -> m b) -> m b A return operation, return :: (Monad m) => a -> m a Compare: A Haskell type constructor with a Scala type declaration A Haskell bind operation with Scala’s flatMap Haskell: m a -> (a -> m b) -> m b Scala: M(A).(f: A => M[B]): M[B] A Haskell return operation with a Scala constructor (or unit)

andThen does sequencing scala> def double(x: Int) = 2 * x double: (x: Int)Int scala> def triple(x: Int) = 3 * x triple: (x: Int)Int scala> ((double _) andThen (triple _))(5) res4: Int = 30 scala> def upper(s: String) = s.toUpperCase upper: (s: String)String scala> def addXs(s: String) = "x" + s + "x" addXs: (s: String)String scala> ((upper _) andThen (addXs _))("Hello") res10: String = xHELLOx

foo1 Method to print a string, then return its length: scala> def foo1(bar: String) = { | println(bar) | bar.size | } foo1: (bar: String)Int scala> foo1("Hello") Hello res0: Int = 5

foo2 Here’s the same method, but replacing each expression with an anonymous function: scala> def foo1(bar: String) = { | (() => println(bar))() | (() => bar.length)() | } foo1: (bar: String)Int scala> foo2("Hello") Hello res1: Int = 5

andThen applied to foo Consider this form: The above almost works… def foo(bar: String) = { ({ () => println(bar) } andThen { () => bar.length })() } The above almost works… andThen is not defined for 0-argument functions Basically, what this achieves is sequencing in a purely functional manner In pure functions, there is no concept of sequencing

Thing Compare: def foo(i: Int) = i + 1 val a = 1 val b = foo(a) With: case class Thing[+A](value: A) val a = Thing(1) val b = Thing(2) def foo(i: Int) = Thing(i + 1) val a = Thing(1) val b = foo(a.value) The difference is that in the second, the value is “wrapped” in a Thing container

Monads as wrappers Is Thing a monad? A monad consists of three things: A type constructor M A bind operation, (>>=) :: (Monad m) => m a -> (a -> m b) -> m b A return operation, return :: (Monad m) => a -> m a Is Thing a monad? It has a type constructor, Thing It has a return operation, Thing(i) Let’s give it a bind operation: case class Thing[+A](value: A) { def bind[B](f: A => Thing[B]) = f(value) }

The Thing monad Here’s what we had before: Here’s what we have now: scala> val a = Thing(1) a: Thing[Int] = Thing(1) scala> val b = foo(a.value) b: Thing[Int] = Thing(2) Here’s what we have now: scala> val a = Thing(1) a: Thing[Int] = Thing(1) scala> val b = a bind foo b: Thing[Int] = Thing(2) We have additional syntax, but really, nothing’s changed

The monad pattern Any time you start with something which you pull apart and use to compute a new something of that same type, you have a monad. val a = Thing(1) The first thing is that I can wrap up a value inside of a new Thing. Object-oriented developers might call this a “constructor”. Monads call it “the unit function”. Haskell calls it “return” (maybe we shouldn’t try to figure out that one). a bind { i => Thing(i + 1) } We also have this fancy bind function, which digs inside our Thing and allows a function which we supply to use that value to create a new Thing. Scala calls this function “flatMap”. Haskell calls it “>>=”. …What’s interesting here is the fact that bind is how you combine two things together in sequence. Directly quoted from www.codecommit.com/blog/

bind == flatMap Scala’s for expression is translated into map, flatMap, and withFilter operations Multiple generators lead to a flatMap for (x <- expr1; y <- expr2; seq) yield expr3 gets translated to expr1.flatMap(x => for (y <- expr2; seq) yield expr3) Repeated use of flatMap will change List[List[List[items]]] into just List[items]

Using flatMap scala> for (v <- List(1, 2, 3, -1, 4)) { | val Some(rootOfV) = root(v) | println(rootOfV) | } 1.0 1.4142135623730951 1.7320508075688772 scala.MatchError: None (of class scala.None$) scala> for (v <- List(1, 2, 3, -1, 4) flatMap (root(_))) println(v) 1.0 1.4142135623730951 1.7320508075688772 2.0

Option A value of type Option[T] can be either Some[value] or None, where value is of type T scala> def root(x: Double): Option[Double] = | if (x >= 0) Some(math.sqrt(x)) else None root: (x: Double)Option[Double] scala> root(10.0) res14: Option[Double] = Some(3.1622776601683795) scala> root(-5.0) res15: Option[Double] = None

Repeated flatMap Let’s say we want to load a user from the database and if he exists we want to see if he has a grandchild. We need to invoke these three functions: String → Option[User] // load from db User → Option[User] // get child User → Option[User] // get child’s child Here’s the code. val result = UserService.loadUser("mike") .flatMap(getChild) .flatMap(getChild) Or we can do this: val result = for { user <- UserService.loadUser("mike) usersChild <- user.child usersGrandChild <- usersChild.child } yield usersGrandChild Example from: https://medium.com/@sinisalouc/demystifying-the-monad-in-scala-cc716bb6f534#.jf1x02xm9

bind for Option sealed trait Option[+A] { def bind[B](f: A => Option[B]): Option[B] } case class Some[+A](value: A) extends Option[A] { def bind[B](f: A => Option[B]) = f(value) } case object None extends Option[Nothing] { def bind[B](f: Nothing => Option[B]) = None }

Maintaining state A purely functional language has no notion of “state” (or time, or change…) Everything relevant to a function is in its parameters Therefore, a function that “changes state” must be called recursively with different parameters Consider an adventure game State includes the location of each object and the location of the player—this is easily done with a Map The state usually includes other information (is the dragon alive?)—we can put this in a tuple along with the Map Player’s actions can be implemented with a function that takes a State and computes a new State—that is, a monad

Life, the Universe, and Everything Passing around the entire “state of the universe” in parameters seems excessive, but… Typically a very large proportion of the information is immutable, and need not be part of the state You have to depend on the quality of the implementation of persistent data structures Scala has a specific State monad I haven’t explored this, but I’ve read that it’s complicated

A “functional” println def foo(bar: String, stdout: Vector[String]) = { val stdout2 = println(bar, stdout) (bar.length, stdout2) } def println(str: String, stdout: Vector[String]) = stdout + str Now we can use andThen Functional input is trickier—we won’t go there Haskell does this for everything!

And then there’s the IO monad… Haskell’s IO monad is like our earlier “functional” println, only richer and with a better syntax Like all monads, it pulls apart some kind of thing, and creates a new thing from it The weird part is, I/O happens along the way Output doesn’t affect the result Input does affect the result The IO monad (1) achieves sequencing, and (2) isolates the I/O side effects from the rest of the program In Scala we could use a functional println to implement (part of) an IO monad—but why bother?

The End Substantial portions of this talk taken from: http://www.codecommit.com/blog/