Monads

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 2

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 3

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 4

andThen applied to foo Consider this form: 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 5

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 6

Monads as wrappers 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) } 7

The Thing monad Here’s what we had before: 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 8

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 just yet). 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. 9 Directly quoted from

bind == flatMap Scala’s for expression is translated into map, flatMap, and withFilter operations Multiple generators lead to a flatMap for (x for (y <- expr 2 ; seq ) yield expr 3 ) Repeated use of flatMap will change List[List[List[ items ]]] into just List[ items ] 10

flatMap 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, , 2.0, 3.0) 11

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

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( ) scala> root(-5.0) res15: Option[Double] = None 13

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 } 14

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 Functional input is trickier—we won’t go there Now let’s do this for everything! 15

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 16

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 17

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 a 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 18

Formal definition of 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 And the operations must obey some simple rules: return x >>= f = f x return just sends its result to the next function m >>= return = m Returning the result of an action is equivalent to just doing the action do {x <- m1; y <- m2; m3} = do {y <- do {x <- m1; m2} m3} >>= is associative

The End Substantial portions of this talk taken from: 20

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( ) scala> root2(-10) res29: scala.util.Try[Double] = Failure(java.lang.Exception: Imaginary root) 21