Monadologie —
professional help for
type anxiety

Christopher League
12 July 2010

Monadologie —
github.com/league/scala-type-anxiety

@chrisleague
league@contrapunctus.net

Principles of
Programming
Languages

January 1997 § Paris

“Java might be the vehicle Principles of
that will help us carry Programming
modern programming
language innovations Languages
into industrial practice.”
– me
January 1997 § Paris

Monadologie
continuations
monads
existentials
variance
effects

Monadologie
continuations – control ﬂow
monads – data ﬂow

def plus [A] (x: Int, y: Int, k: Int=>A): A =
k(x+y)

def times [A] (x: Int, y: Int, k: Int=>A): A =
k(x*y)

def less [A] (x: Int, y: Int,
kt: => A, kf: => A): A =
if(x < y) kt else kf

def factorial [A] (n: Int, k: Int => A): A =
less(n, 2, k(1),
plus(n, -1, (m:Int) =>
factorial(m, (f:Int) =>
times(n, f, k))))

scala> factorial(5, println)
120
scala> factorial(3, factorial(_, println))
720
scala> val n = factorial(3, r => r)
n: Int = 6

CPS
makes some programs simpler

class Regex
case class Literal(s: String) extends Regex
case class Concat(r1: Regex, r2: Regex)
extends Regex
case class Choice(r1: Regex, r2: Regex)
extends Regex
case class Star(r: Regex) extends Regex

Concat(Star(Literal("ab")), Literal("abc"))
// (ab*)abc matches abababc

Choice(Star(Literal("ab")),
Concat(Literal("a"), Star(Literal("ba"))))
// (ab)*|a(ba)* matches abababa

def accept (regex: Regex, chars: Seq[Char],
k: Seq[Char] => Boolean): Boolean =
regex match {
case Literal(expect) =>
if(chars.take(expect.length).sameElements(expect))
k(chars.drop(expect.length))
else false
case Concat(regex1, regex2) =>
accept(regex1, chars, remaining =>
accept(regex2, remaining, k))
case Choice(regex1, regex2) =>
accept(regex1, chars, k) || accept(regex2, chars, k)
case Star(repeatable) =>
k(chars) ||
accept(repeatable, chars, remaining =>
accept(regex, remaining, k))
}

def accept (regex: Regex, chars: Seq[Char],
k: Seq[Char] => Boolean): Boolean =
...

def complete (remaining: Seq[Char]): Boolean =
remaining.length == 0

accept(regex1, "abababc", complete) // true
accept(regex2, "abababa", complete) // true
accept(regex1, "abababcd", complete) // false

Delimited
Continuations
via compiler plugin (2.8)

Delimited
Continuations
shift & reset

def doSomething0 = reset {
println("Ready?")
val result = 1 + special * 3
println(result)
}

Punch a hole in your code.
What type of value is expected in the hole?
What happens after the hole?
What is result type of the reset block?

println("Ready?")
println(result)
}

Think of the rest of the computation as
a function with the hole as its parameter.
Call it the continuation.

println("Ready?")
println(result)
}

def special = shift {
k: (Int => Unit) =>
println(99)
"Gotcha!" shift captures the continuation and
} then determines its own future.

println("Ready?")
println(result)
}

def special = shift {
k: (Int => Unit) =>
println(99)
"Gotcha!" shift doSomething1
scala> captures the continuation and
} Ready?determines its own future.
then
99
res0: java.lang.String = Gotcha!

println("Ready?")
val result = 1 + wacky * 3
println(result)
}

def wacky = shift {
k: (Int => Unit) =>
k(2)
println("Yo!")
k(3)
}

println("Ready?")
val result = 1 + wacky * 3
println(result)
}

def wacky = shift {
k: (Int => Unit) =>
k(2)
println("Yo!") scala> doSomething2
k(3) Ready?
} 7
Yo!
10

Continuation
low-level control-ﬂow primitive that
can implement:
exceptions
concurrency (actors)
suspensions (lazy)
…

def multi = reset {
println("This function returns tentatively")
println("but you can always ask for more.")
produce(42)
println("Didn't like that? Back again.")
produce(99)
println("Still not OK? I'm out of ideas!")
None
}

def multi = reset {
produce(42)
scala> multi
produce(99) function returns tentatively
This
println("Still can alwaysI'm out more.
but you not OK? ask for of ideas!")
None res0: Option[ReturnThunk[Int]] = Some(...)
} scala> println(res0.get.value)
42
scala> res0.get.proceed
Didn’t like that? Back again.
res1: Option[ReturnThunk[Int]] = Some(...)
scala> println(res1.get.value)
99
scala> res1.get.proceed
Still not OK? I’m out of ideas!
res2: Option[ReturnThunk[Int]] = None

case class ReturnThunk[A]
(value: A,
proceed: Unit => Option[ReturnThunk[A]])

def produce [A] (value: A):
Unit @cps[Option[ReturnThunk[A]]] =
shift {
k: (Unit => Option[ReturnThunk[A]]) =>
Some(ReturnThunk(value, k))
}
def multi = reset {
produce(42)
produce(99)
println("Still not OK? I'm out of ideas!")
None
}

def interact = reset {
val first = ask("Please give me a number")
val second = ask("Enter another number")
printf("The sum of your numbers is: %dn",
first + second)
}

first + second)
}
scala> interact
Please give me a number
respond with: submit(0x28d092b7, ...)
scala> submit(0x28d092b7, 14)
Enter another number
respond with: submit(0x1ddb017b, ...)
scala> submit(0x1ddb017b, 28)
The sum of your numbers is: 42

type UUID = Int
def uuidGen: UUID = Random.nextInt
type Data = Int
val sessions = new HashMap[UUID, Data=>Unit]

def ask(prompt: String): Data @cps[Unit] = shift {
k: (Data => Unit) => {
val id = uuidGen
printf("%snrespond with: submit(0x%x, ...)n",
prompt, id)
sessions += id -> k
}
}

def submit(id: UUID, data: Data) = sessions(id)(data)

first + second)
}

def harmful = reset {
var i = 0
println("Hello world!")
label("loop")
println(i)
i = i + 1
if(i < 20) goto("loop")
println("Done.")
}

var i = 0
println("Hello world!") Hello world!
label("loop") 0
println(i) 1
i = i + 1 2
.
if(i < 20) goto("loop") .
println("Done.") .
} 18
19
Done.

val labelMap = new HashMap[String, Unit=>Unit]

def label(name:String) =
shift { k:(Unit=>Unit) =>
labelMap += name -> k
k()
}
def goto(name:String) =
shift { k:(Unit=>Unit) => labelMap(name)() }
var i = 0
println("Hello world!")
label("loop")
println(i)
i = i + 1
if(i < 20) goto("loop")
println("Done.")
}

Monads
the leading design pattern
for functional programming

A type constructor M is a monad
if it supports these operations:
def unit[A] (x: A): M[A]

def flatMap[A,B] (m: M[A]) (f: A => M[B]): M[B]

def map[A,B] (m: M[A]) (f: A => B): M[B] =
flatMap(m){ x => unit(f(x)) }

def andThen[A,B] (ma: M[A]) (mb: M[B]): M[B] =
flatMap(ma){ x => mb }

Option is a monad.
def unit[A] (x: A): Option[A] = Some(x)

def flatMap[A,B](m:Option[A])(f:A =>Option[B]):
Option[B] =
m match {
case None => None
case Some(x) => f(x)
}

List is a monad.
def unit[A] (x: A): List[A] = List(x)

def flatMap[A,B](m:List[A])(f:A =>List[B]): List[B] =
m match {
case Nil => Nil
case x::xs => f(x) ::: flatMap(xs)(f)
}

For comprehension: convenient syntax for monadic structures.
for(i <- 1 to 4;
j <- 1 to i;
k <- 1 to j)
yield { i*j*k }

Compiler translates it to:
(1 to 4).flatMap { i =>
(1 to i).flatMap { j =>
(1 to j).map { k =>
i*j*k }}}

Example: a series of operations, where each may fail.
lookupVenue: String => Option[Venue]
getLoggedInUser: SessionID => Option[User]
reserveTable: (Venue, User) => Option[ConfNo]


val user = getLoggedInUser(session)
val confirm =
if(!user.isDefined) { None }
else lookupVenue(name) match {
case None => None
case Some(venue) => {
val confno = reserveTable(venue, user.get)
if(confno.isDefined)
mailTo(confno.get, user.get)
confno
}
}


val confirm =
for(venue <- lookupVenue(name);
user <- getLoggedInUser(session);
confno <- reserveTable(venue, user))
yield {
mailTo(confno, user)
confno
}

Example from Lift form validation:
def addUser(): Box[UserInfo] =
for {
firstname <- S.param("firstname") ?~
"firstname parameter missing" ~> 400
lastname <- S.param("lastname") ?~
"lastname parameter missing"
email <- S.param("email") ?~
"email parameter missing"
} yield {
val u = User.create.firstName(firstname).
lastName(lastname).email(email)

S.param("password") foreach u.password.set

u.saveMe
}

Example: use monad to pass around state behind the scenes

class Tree[A]
case class Leaf[A](elem: A) extends Tree[A]
case class Branch[A](left: Tree[A], right: Tree[A])
extends Tree[A]

inject(‘Q’, ) =>

f a Q c
d b f g

e c h g d e a h

def inject[A] (root: Tree[A], cur: A): (Tree[A], A) =
root match {
case Leaf(old) => (Leaf(cur), old)
case Branch(left, right) =>
val (t1, last1) = inject(left, cur)
val (t2, last2) = inject(right, last1)
(Branch(t1,t2), last2)
}

def inject[A] (root: Tree[A], cur: A): (Tree[A], A) =
root match {
case Leaf(old) => (Leaf(cur), old)
val (t1, last1) = inject(left, cur)
val (t2, last2) = inject(right, last1)
(Branch(t1,t2), last2)
}
def injectST[A] (root: Tree[A]): ST[A, Tree[A]] =
root match {
case Leaf(old) =>
for(cur <- init[A];
u <- update[A](_ => old))
yield Leaf(cur)
for(t1 <- injectST(left);
t2 <- injectST(right))
yield Branch(t1,t2)
}

case class ST[S,A](exec: S => (S,A)) {
def flatMap[B] (f: A => ST[S,B]): ST[S,B] =
ST { s0 =>
val (s1, a) = exec(s0)
f(a).exec(s1)
}

def map[B] (f: A => B)
(implicit unit: B => ST[S,B]) : ST[S,B] =
flatMap { x => unit(f(x)) }
}

implicit def unitST[S,A] (x: A): ST[S,A] =
ST { s0 => (s0, x) }

def init[S]: ST[S,S] =
ST { s0 => (s0, s0) }

def update[S] (g: S => S): ST[S,Unit] =
ST { s0 => (g(s0), ()) }

case class ST[S,A](exec: S => (S,A)) {
def flatMap[B] (f: scala> ST[S,B]): ST[S,B] =
A => drawTree(t1,"")
ST { s0 => ______f
| _____d
val (s1, a) = exec(s0)
| _____e
f(a).exec(s1) | __c
} _____a
________h
| __g
def map[B] (f: A => B) __b
(implicit unit: B => ST[S,B]) : ST[S,B] =
flatMap { x => unit(f(x)) m }= injectST(t1)
scala> val
m: ST[Char,Tree[Char]] = ST(<function1>)
}
scala> val (_,t2) = m.exec('Q')
t2: Tree[Char] = ...
implicit def unitST[S,A] (x: A): ST[S,A] =
ST { s0 => (s0, x) scala> drawTree(t2,"")
}
______Q
def init[S]: ST[S,S] | _____f
=
| _____d
ST { s0 => (s0, s0)| } __e
_____c
________a
def update[S] (g: S => S): ST[S,Unit]
| __h
=
ST { s0 => (g(s0), ()) } __g

Monadologie
github.com/league/scala-type-anxiety

@chrisleague
league@contrapunctus.net

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