Scala. Introduction to FP. Monads

Scala. Introduction to FP.Monads.

[email protected], 2015

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Functions as Objects

What about functions?

In fact function values are treated as objects in Scala.

The function type A => B is just an abbreviation for the class

scala.Function1[A, B], which is defined as follows.package scala

trait Function1[A, B] {

def apply(x: A): B

}

So functions are objects with apply methods.


An anonymous function such as

(x: Int) => x * x

is expanded to:

{

class AnonFun extends Function1[Int, Int] {

def apply(x: Int) = x * x

}

new AnonFun

}

or, shorter, using anonymous class syntax:

new Function1[Int, Int] {


}


A function call, such as f(a, b), where f is a value of some class

type, is expanded to

f.apply(a, b)

So the OO-translation of

val f = (x: Int) => x * x

f(7)

would be

val f = new Function1[Int, Int] {


}

f.apply(7)

Case Classes

A case class definition is similar to a normal class definition, except that it is preceded by the modifier case. For example:

trait Expr

case class Number(n: Int) extends Expr

case class Sum(e1: Expr, e2: Expr) extends Expr

Like before, this defines a trait Expr, and two concrete subclasses

Number and Sum

Case Classes

It also implicitly defines companion objects with apply methods.

object Number {

def apply(n: Int) = new Number(n)

}

object Sum {

def apply(e1: Expr, e2: Expr) = new Sum(e1, e2)

}

so you can write Number(1) instead of new Number(1).

Pattern matching

Pattern matching is a generalization of switch from C/Java to class hierarchies.

It’s expressed in Scala using the keyword match.

Example

eval(Sum(Number(1), Number(2)))

def eval(e: Expr): Int = e match {

case Number(n) => n

case Sum(e1, e2) => eval(e1) + eval(e2)

}

Lists

The list is a fundamental data stucture in functional programming.

A list having x1, ...,xn as elements is written List(x1, ...,xn)

Example

val fruit = List(«apples», «oranges», «pears»)

val nums = List(1, 2, 3, 4)

val diag3 = List(List(1, 0, 0), List(0, 1, 0), List(0, 0, 1))

val empty = List()

There are two important differences between lists and arrays.● List are imuttable - the elements of list cannot be changed● Lists are recursive, while arrays are flat

Constructors of Lists

All lists are constructed from:● The empty list Nil, and● The constructor operation :: (pronounces cons):

x :: xs gives a new list with the first element x, followed by the element of xs

Example

fruit = «apples» :: («oranges» :: («pears» :: Nil))

nums = 1 :: (2 :: (3 :: (4 :: Nil)))

empty = Nil

Map

A simple way to define map is as follows:

abstract class List[T] { ...

def map[U](f: T => U): List[U] = this match {

case Nil => Nil

case x :: xs => f(x) :: xs.map(f)

}

Example

def scaleList(xs: List[Double], factor: Double) =

xs map (x => x * factor)

flatMap

abstract class List[T] {

def flatMap[U](f: T => List[U]): List[U] = this match {case x :: xs => f(x) ++ xs.flatMap(f)

case Nil => Nil

}

}

Filter

This pattern is generalized by the method filter of the List class:

abstract class List[T] {

...

def filter(p: T => Boolean): List[T] = this match {

case Nil => Nil

case x :: xs => if (p(x)) x :: xs.filter(p) else xs.filter(p)

}

}

Example

def posElems(xs: List[Int]): List[Int] =

xs filter (x => x > 0)

For-Expression

Higher-order functions such as map, flatMap or filter provide powerful constructs for manipulating lists.

But sometimes the level of abstraction required by these function make the program difficult to understand.

In this case, Scala's for expression notation can help.

For-Expression Example

Let person be a list of elements of class Person, with fields name and age.

case class Person(name: String, age: Int)

To obtain the names of person over 20 years old, you can write:

for ( p <- persons if p.age > 20 ) yield p.name

Which is equivalent to:

persons filter (p => p.age > 20) map (p => p.name)

The for-expression is similar to loops in imperative languages, except that is builds a list of the result of all iterations.

Syntax of For

A for-expression is of the form:

for ( s ) yield e

where s is a sequence of generators and filters, and e is an expression whose value is returned by an iteration.

● A generator is of the form p <- e, where p is a pattern and e an expression whose value is a collection.

● A filter is of the form if f where f is a boolean expression.● The sequence must start with generator.● If there are several generators in the sequence, the last generators vary faster than the first.

Example

for {

i <- 1 until n

j <- 1 until i

if isPrime(i + j)

} yield (i, j)

Queries with forcase class Book(title: String, authors: List[String])

val books: List[Book] = List(

Book(title = ”Structure and Interpretation of Computer Programs”,

authors = List(”Abelson, Harald”, ”Sussman, Gerald J.”)),

Book(title = ”Introduction to Functional Programming”,

authors = List(”Bird, Richard”, ”Wadler, Phil”)),

Book(title = ”Effective Java”,

authors = List(”Bloch, Joshua”)),

Book(title = ”Java Puzzlers”,

authors = List(”Bloch, Joshua”, ”Gafter, Neal”)),

Book(title = ”Programming in Scala”,

authors = List(”Odersky, Martin”, ”Spoon, Lex”, ”Venners, Bill”)))

To find the names of all authors who have written at least two books present in the database:

for {

b1 <- books

b2 <- books

if b1.title < b2.title

a1 <- b1.authors

a2 <- b2.authors

if a1 == a2

} yield a1

For-Expressions and Higher-Order Functions

The syntax of for is closely related to the higher-order functions map, flatMap and filter.

First of all, these functions can all be defined in terms of for:

def map[T, U](xs: List[T], f: T => U): List[U] =

for (x <- xs) yield f(x)

def flatMap[T, U](xs: List[T], f: T => Iterable[U]): List[U] =

for (x <- xs; y <- f(x)) yield y

def filter[T](xs: List[T], p: T => Boolean): List[T] =

for (x <- xs if p(x)) yield x

And vice versa

Translation of For

In reality, the Scala compiler expresses for-expression in terms of map, flatMap and a lazy variant of filter.

1. A simple for expressionfor (x <- e1) yield e2

is translated to

e1.map(x => e2)

Translation of For

2: A for-expression

for (x <- e1 if f; s) yield e2

where f is a filter and s is a (potentially empty) sequence of generators and filters, is translated to

for (x <- e1.withFilter(x => f); s) yield e2

(and the translation continues with the new expression)

You can think of withFilter as a variant of filter that does not produce an intermediate list, but instead filters the following map or flatMap function application.

Translation of For

3: A for-expression

for (x <- e1; y <- e2; s) yield e3

where s is a (potentially empty) sequence of generators and filters,

is translated into

e1.flatMap(x => for (y <- e2; s) yield e3)

(and the translation continues with the new expression)

Example

Take the for-expression that computed pairs whose sum is prime:

for {

i <- 1 until n

j <- 1 until i

if isPrime(i + j)

} yield (i, j)

Applying the translation scheme to this expression gives:

(1 until n).flatMap(i =>

(1 until i).withFilter(j => isPrime(i+j)).map(j => (i, j)))

For and DatabasesFor example, books might not be a list, but a database stored on

some server.

As long as the client interface to the database defines the methods

map, flatMap and withFilter, we can use the for syntax for querying

the database.

This is the basis of the Scala data base connection frameworks

ScalaQuery and Slick.

Similar ideas underly Microsoft’s LINQ(for ( c < - coffees;

if c.sales > 999

) yield c.nam e).run

select "CO F_NAM E"from "CO FFEES"w here "SALES" > 999

Coffee break

As soon as you understand Monads, you will understand that this is a Monad, too.

{{alt: What if someone broke out of a hypothetical situation in your room right now?}}

Monads

Data structures with map and flatMap seem to be quite common.

In fact there’s a name that describes this class of a data structures together with some algebraic laws that they should have.

They are called monads.

What is a Monad?

A monad M is a parametric type M[T] with two operations, flatMap and unit, that have to satisfy some laws.

trait M[T] {

def flatMap[U](f: T => M[U]): M[U]

}

def unit[T](x: T): M[T]

In the literature, flatMap is more commonly called bind.

Examples of Monads

– List is a monad with unit(x) = List(x)

– Set is monad with unit(x) = Set(x)

– Option is a monad with unit(x) = Some(x)

– Generator is a monad with unit(x) = single(x)

– …......

flatMap is an operation on each of these types, whereas unit in Scala is di erent for ffeach monad.

Monads and map

map can be defined for every monad as a combination of flatMap and unit:

m map f

== m flatMap (x => unit(f(x)))

== m flatMap (f andThen unit)

Monad Laws

To qualify as a monad, a type has to satisfy three laws:

Associativity

m flatMap f flatMap g == m flatMap (x => f(x) flatMap g)

Left unit

unit(x) flatMap f == f(x)

Right unit

m flatMap unit == m

The Option monad

sealed trait Option[A] {def map[B](f: A => B): Option[B]

def flatMap[B](f: A => Option[B]): Option[B]

}

case class Some[A](a: A) extends Option[A]

case class None[A] extends Option[A]

The Option monad makes the possibility of missing data explicit in the type system, while hiding the boilerplate «if non-null» logic

Checking Monad Laws

Let’s check the monad laws for Option.

Here’s flatMap for Option:

abstract class Option[+T] {

def flatMap[U](f: T => Option[U]): Option[U] = this match {

case Some(x) => f(x)

case None => None

}

}

Try

Try resembles Option, but instead of Some/None there is a Success case with a value and a Failure case that contains an exception:

abstract class Try[+T]

case class Success[T](x: T) extends Try[T]

case class Failure(ex: Exception) extends Try[Nothing]

Try is used to pass results of computations that can fail with an exception between threads and computers

Creating a Try

You can wrap up an arbitrary computation in a Try.

Try(expr) // gives Success(someValue) or Failure(someException)

Here’s an implementation of Try:

object Try {

def apply[T](expr: => T): Try[T] =

try Success(expr)

catch {case NonFatal(ex) => Failure(ex)

}

}

Composing Try

Just like with Option, Try-valued computations can be composed in for

expresssions.

for {

x <- computeX

y <- computeY

} yield f(x, y)

If computeX and computeY succeed with results Success(x) and Success(y), this will return Success(f(x, y)).

If either computation fails with an exception ex, this will return

Failure(ex).

Definition of flatMap and map on Try

abstract class Try[T] {def flatMap[U](f: T => Try[U]): Try[U] = this match {

case Success(x) => try f(x) catch { case NonFatal(ex) => Failure(ex) }case fail: Failure => fail

}

def map[U](f: T => U): Try[U] = this match {

case Success(x) => Try(f(x))case fail: Failure => fail

}

}

The Try monad makes the possibility of errors explicit in the type system, while hiding the boilerplate «try/catch» logic

We have seen that for-expressions are useful not only for collections. Many other types also define map,flatMap, and withFilter operations and with them for-expressions.

Examples: Generator, Option, Try.

Many of the types defining flatMap are monads.

(If they also define withFilter, they are called “monads with zero”).

The three monad laws give useful guidance in the design of library APIs.

Reactive manifesto

[Merriam Webster] reactive: “readily responsive to a stimulus”.

▶ React to events (event-driven)

▶ React to load (scalable)

▶ React to failures (resilient)

▶ React to users (responsive)

Lets play a simple game:

trait Adventure {

def collectCoins(): List[Coin]

def buyTreasure(coins: List[Coin]): Treasure

}

val adventure = Adventure()

val coins = adventure.collectCoins()

val treasure = adventure.buyTreasure(coins)

Actions may fail

def collectCoins(): List[Coin] = {

if (eatenByMonster(this))

throw new GameOverException(“Ooops”)

List(Gold, Gold, Silver)

}

def buyTreasure(coins: List[Coin]): Treasure = {

if (coins.sumBy(_.value) < treasureCost)

throw new GameOverException(“Nice try!”)

Diamond

}




Sequential composition of actionsthat may fail



// block until coins are collected

// only continue if there is no exception


// block until treasure is bought


Expose possibility of failure in thetypes, honestly

T => S

T => Try[S]

Making failure evident intypes

import scala.util.{Try, Success, Failure}

abstract class Try[T]

case class Success[T](elem: T) extends Try[T]

case class Failure(t: Throwable) extends Try[Nothing]

object Try {

def apply[T](r: =>T): Try[T] = {

try { Success(r) }

catch { case t => Failure(t) }

}

trait Adventure {

def collectCoins(): Try[List[Coin]]

def buyTreasure(coins: List[Coin]): Try[Treasure]

}

Dealing with failure explicitly


val coins: Try[List[Coin]] = adventure.collectCoins()

val treasure: Try[Treasure] = coins match {

case Success(cs) adventure.buyTreasure(cs) ⇒

case failure @ Failure(t) failure ⇒

}

Noise reduction


val treasure: Try[Treasure] =

adventure.collectCoins().flatMap(coins { ⇒

adventure.buyTreasure(coins)

})

val treasure: Try[Treasure] = for {

coins <- adventure.collectCoins()

treasure <- buyTreasure(coins)

} yield treasure

Amonad that handles exceptions.

Try[T]

The Try monad makes the possibility of errors explicit in the type system, while hiding the boilerplate «try/catch» logic

trait Socket {

def readFromMemory(): Array[Byte]

def sendToEurope(packet: Array[Byte]): Array[Byte]

}

val socket = Socket()

val packet = socket.readFromMemory()

val confirmation = socket.sendToEurope(packet)

Timings for various operations ona typical PC



// block for 50,000 ns



// block for 150,000,000 ns


Lets translate this into human terms. 1 nanosecond → 1 second



// block for 3 days



// block for 5 years


Futures asynchronously notifyconsumers

Future[T] A monad that handles

exceptions and latency

import scala.concurrent._

import scala.concurrent.ExecutionContext.Implicits.global

trait Future[T] {

def onComplete(callback: Try[T] Unit) ⇒

(implicit executor: ExecutionContext): Unit

}

Futures asynchronously notifyconsumers

import scala.concurrent._

trait Future[T] {

def onComplete(callback: Try[T] Unit) ⇒ (implicit executor: ExecutionContext): Unit

}

trait Socket {

def readFromMemory(): Future[Array[Byte]]

def sendToEurope(packet: Array[Byte]): Future[Array[Byte]]

}

Sendpackets using futures


val packet: Future[Array[Byte]] = socket.readFromMemory()

packet onComplete {

case Success(p) { ⇒ val confirmation: Future[Array[Byte]] =

socket.sendToEurope(p)

}

case Failure(t) => …

}

Creating Futures

// Starts an asynchronous computation

// and returns a future object to which you

// can subscribe to be notified when the

// future completes

object Future {

def apply(body: T) ⇒

(implicit context: ExecutionContext): Future[T]

}

Creating Futures

import scala.concurrent.ExecutionContext.Implicits.global

import akka.serializer._

val memory = Queue[EMailMessage](

EMailMessage(from = “Erik”, to = “Roland”),

EMailMessage(from = “Martin”, to = “Erik”),

EMailMessage(from = “Roland”, to = “Martin”))

def readFromMemory(): Future[Array[Byte]] = Future {

val email = queue.dequeue()

val serializer = serialization.findSerializerFor(email)

serializer.toBinary(email)

}

Some Other Useful Monads

● The List monad makes nondeterminism explicit in the type system, while hiding the boilerplate of nested for-loops.

● The Promise monad makes asynchronous computation explicit in the type system, while hiding the boilerplate of threading logic

● The Transaction monad (non-standard) makes transactionality explicit in the type system, while hiding the boilerplate of invoking rollbacks

● … and more. Use monads wherever possible to keep your code clean!

Future[T] and Try[T] are dual

trait Future[T] {

def OnComplete[U](func: Try[T] U) ⇒

(implicit ex: ExecutionContext): Unit

}

Lets simplify:

(Try[T] Unit) Unit⇒ ⇒


(Try[T] Unit) Unit⇒ ⇒

Reverse:

Unit (Unit Try[T]) ⇒ ⇒

Simplify:

() (() Try[T])⇒ ⇒ ≈ Try[T]


Receive result of type Try[T] by passingcallback (Try[T] Unit)⇒ to method

def asynchronous(): Future[T] = { … }

Receive result of type Try[T] by blockinguntil method returns

def synchronous(): Try[T] = { … }

Iterable[T]

trait Iterable[T] {

def iterator(): Iterator[T]

}

trait Iterator[T] {

def hasNext: Boolean

def next(): T

}

() (() Try[Option[T]])⇒ ⇒

Iterable[T] vs Observables[T]

() (() Try[Option[T]]) ⇒ ⇒

Reverse:

(Try[Option[T]] Unit) Unit⇒ ⇒

Simplify:

( T Unit⇒

, Throwable Unit ⇒

, () Unit ⇒

) Unit⇒

( T Unit, ⇒, Throwable Unit ⇒, () Unit ⇒) Unit⇒

trait Observable[T] {

def Subscribe(observer: Observer[T]): Subscription

}

trait Observer[T] {

def onNext(value: T): Unit

def onError(error: Throwable): Unit

def onCompleted(): Unit

}

trait Subscription {

def unsubscribe(): Unit

}

Hello Observables

More INFO

● Functional Programming Principles in Scala https://www.coursera.org/course/progfun

● Principles of Reactive Programming https://www.coursera.org/course/reactive

● Programming Languages https://www.coursera.org/course/proglang● Paradigms of Computer Programming

https://www.edx.org/course/louvainx/louvainx-louv1-01x-paradigms-computer-1203

● Functional Reactive Programming & ClojureScript https://www.youtube.com/watch?v=R4sTvHXkToQ&list=PL6JL99ajzlXUbXUY0nwWr2imcrhJHT3Z7

● An Introduction to Functional Reactive Programming https://www.youtube.com/watch?v=ZOCCzDNsAtI

● The reactive manifesto http://www.reactivemanifesto.org/

https://www.coursera.org/course/progfun

https://www.coursera.org/course/reactive

https://www.coursera.org/course/proglang



https://www.youtube.com/watch?v=R4sTvHXkToQ&list=PL6JL99ajzlXUbXUY0nwWr2imcrhJHT3Z7

https://www.youtube.com/watch?v=R4sTvHXkToQ&list=PL6JL99ajzlXUbXUY0nwWr2imcrhJHT3Z7

https://www.youtube.com/watch?v=ZOCCzDNsAtI

http://www.reactivemanifesto.org/

Scala. Introduction to FP. Monads

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Transcript of Scala. Introduction to FP. Monads