// Scala Lecture 5

//=================

// Laziness with style

//=====================

// The concept of lazy evaluation doesn’t really 

// exist in non-functional languages. C-like languages

// are (sort of) strict. To see the difference, consider

def square(x: Int) = x * x

square(42 + 8)

// This is called "strict evaluation".

// On the contrary, say we have a pretty expensive operation:

def peop(n: BigInt): Boolean = peop(n + 1) 

val a = "foo"

val b = "foo"

if (a == b || peop(0)) println("true") else println("false")

// This is called "lazy evaluation":

// you delay compuation until it is really 

// needed. Once calculated though, the result

// does not need to be re-calculated.

// A useful example is

def time_needed[T](i: Int, code: => T) = {

  val start = System.nanoTime()

  for (j <- 1 to i) code

  val end = System.nanoTime()

  f"${(end - start) / (i * 1.0e9)}%.6f secs"

}

// A slightly less obvious example: Prime Numbers.

// (I do not care how many) primes: 2, 3, 5, 7, 9, 11, 13 ....

def generatePrimes (s: LazyList[Int]): LazyList[Int] =

  s.head #:: generatePrimes(s.tail.filter(_ % s.head != 0))

val primes = generatePrimes(LazyList.from(2))

// the first 10 primes

primes.take(100).toList

time_needed(1, primes.filter(_ > 100).take(3000).toList)

time_needed(1, primes.filter(_ > 100).take(3000).toList)

// A Stream (LazyList) of successive numbers:

LazyList.from(2).take(10)

LazyList.from(2).take(10).force

// An Iterative version of the Fibonacci numbers

def fibIter(a: BigInt, b: BigInt): LazyList[BigInt] =

  a #:: fibIter(b, a + b)

fibIter(1, 1).take(10).force

fibIter(8, 13).take(10).force

fibIter(1, 1).drop(10000).take(1)

fibIter(1, 1).drop(10000).take(1).force

// LazyLists are good for testing

// Regular expressions - the power of DSLs in Scala

//                                     and Laziness

//==================================================

abstract class Rexp

case object ZERO extends Rexp                     // nothing

case object ONE extends Rexp                      // the empty string

case class CHAR(c: Char) extends Rexp             // a character c

case class ALT(r1: Rexp, r2: Rexp) extends Rexp   // alternative  r1 + r2

case class SEQ(r1: Rexp, r2: Rexp) extends Rexp   // sequence     r1 . r2  

case class STAR(r: Rexp) extends Rexp             // star         r*

// some convenience for typing in regular expressions

import scala.language.implicitConversions    

import scala.language.reflectiveCalls 

def charlist2rexp(s: List[Char]): Rexp = s match {

  case Nil => ONE

  case c::Nil => CHAR(c)

  case c::s => SEQ(CHAR(c), charlist2rexp(s))

}

implicit def string2rexp(s: String): Rexp = 

  charlist2rexp(s.toList)

implicit def RexpOps (r: Rexp) = new {

  def | (s: Rexp) = ALT(r, s)

  def % = STAR(r)

  def ~ (s: Rexp) = SEQ(r, s)

}

implicit def stringOps (s: String) = new {

  def | (r: Rexp) = ALT(s, r)

  def | (r: String) = ALT(s, r)

  def % = STAR(s)

  def ~ (r: Rexp) = SEQ(s, r)

  def ~ (r: String) = SEQ(s, r)

}

//example regular expressions

val digit = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

val sign = "+" | "-" | ""

val number = sign ~ digit ~ digit.% 

// Task: enumerate exhaustively regular expressions

// starting from small ones towards bigger ones.

// 1st idea: enumerate them all in a Set

// up to a level

def enuml(l: Int, s: String) : Set[Rexp] = l match {

  case 0 => Set(ZERO, ONE) ++ s.map(CHAR).toSet

  case n =>  

    val rs = enuml(n - 1, s)

    rs ++

    (for (r1 <- rs; r2 <- rs) yield ALT(r1, r2)) ++

    (for (r1 <- rs; r2 <- rs) yield SEQ(r1, r2)) ++

    (for (r1 <- rs) yield STAR(r1))

}

enuml(1, "a")

enuml(1, "a").size

enuml(2, "a").size

enuml(3, "a").size // out of heap space

def enum(rs: LazyList[Rexp]) : LazyList[Rexp] = 

  rs #::: enum( (for (r1 <- rs; r2 <- rs) yield ALT(r1, r2)) #:::

                (for (r1 <- rs; r2 <- rs) yield SEQ(r1, r2)) #:::

                (for (r1 <- rs) yield STAR(r1)) )

enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b'))).take(200).force

enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b'))).take(5_000_000).force

def depth(r: Rexp) : Int = r match {

  case ZERO => 0

  case ONE => 0

  case CHAR(_) => 0

  case ALT(r1, r2) => Math.max(depth(r1), depth(r2)) + 1

  case SEQ(r1, r2) => Math.max(depth(r1), depth(r2)) + 1 

  case STAR(r1) => depth(r1) + 1

}

val is = 

  (enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b')))

    .dropWhile(depth(_) < 3)

    .take(10).foreach(println))

// (Immutable)

// Object Oriented Programming in Scala

// =====================================

abstract class Animal 

case class Bird(name: String) extends Animal {

   override def toString = name

}

case class Mammal(name: String) extends Animal

case class Reptile(name: String) extends Animal

Mammal("Zebra")

println(Mammal("Zebra"))

println(Mammal("Zebra").toString)

Bird("Sparrow")

println(Bird("Sparrow"))

println(Bird("Sparrow").toString)

Bird("Sparrow").copy(name = "House Sparrow")

def group(a : Animal) = a match {

  case Bird(_) => "It's a bird"

  case Mammal(_) => "It's a mammal"

}

// There is a very convenient short-hand notation

// for constructors:

class Fraction(x: Int, y: Int) {

  def numer = x

  def denom = y

}

val half = new Fraction(1, 2)

half.numer

case class Fraction(numer: Int, denom: Int)

val half = Fraction(1, 2)

half.numer

half.denom

// In mandelbrot.scala I used complex (imaginary) numbers 

// and implemented the usual arithmetic operations for complex 

// numbers.

case class Complex(re: Double, im: Double) { 

  // represents the complex number re + im * i

  def +(that: Complex) = Complex(this.re + that.re, this.im + that.im)

  def -(that: Complex) = Complex(this.re - that.re, this.im - that.im)

  def *(that: Complex) = Complex(this.re * that.re - this.im * that.im,

                                 this.re * that.im + that.re * this.im)

  def *(that: Double) = Complex(this.re * that, this.im * that)

  def abs = Math.sqrt(this.re * this.re + this.im * this.im)

}

val test = Complex(1, 2) + Complex (3, 4)

import scala.language.postfixOps

List(5,4,3,2,1).sorted.reverse

// this could have equally been written as

val test = Complex(1, 2).+(Complex (3, 4))

// this applies to all methods, but requires

import scala.language.postfixOps

List(5, 2, 3, 4).sorted

List(5, 2, 3, 4) sorted

// ...to allow the notation n + m * i

import scala.language.implicitConversions   

val i = Complex(0, 1)

implicit def double2complex(re: Double) = Complex(re, 0)

val inum1 = -2.0 + -1.5 * i

val inum2 =  1.0 +  1.5 * i

// All is public by default....so no public is needed.

// You can have the usual restrictions about private 

// values and methods, if you are MUTABLE !!!

case class BankAccount(init: Int) {

  private var balance = init

  def deposit(amount: Int): Unit = {

    if (amount > 0) balance = balance + amount

  }

  def withdraw(amount: Int): Int =

    if (0 < amount && amount <= balance) {

      balance = balance - amount

      balance

    } else throw new Error("insufficient funds")

}

// BUT since we are completely IMMUTABLE, this is 

// virtually of no concern to us.

// another example about Fractions

import scala.language.implicitConversions

import scala.language.reflectiveCalls

case class Fraction(numer: Int, denom: Int) {

  override def toString = numer.toString + "/" + denom.toString

  def +(other: Fraction) = 

    Fraction(numer * other.denom + other.numer * denom, 

             denom * other.denom)

  def *(other: Fraction) = Fraction(numer * other.numer, denom * other.denom)

 }

implicit def Int2Fraction(x: Int) = Fraction(x, 1)

val half = Fraction(1, 2)

val third = Fraction (1, 3)

half + third

half * third

1 + half

// DFAs in Scala  

//===============

import scala.util.Try

// A is the state type

// C is the input (usually characters)

case class DFA[A, C](start: A,              // starting state

                     delta: (A, C) => A,    // transition function

                     fins:  A => Boolean) { // final states (Set)

  def deltas(q: A, s: List[C]) : A = s match {

    case Nil => q

    case c::cs => deltas(delta(q, c), cs)

  }

  def accepts(s: List[C]) : Boolean = 

    Try(fins(deltas(start, s))).getOrElse(false)

}

// the example shown in the handout 

abstract class State

case object Q0 extends State

case object Q1 extends State

case object Q2 extends State

case object Q3 extends State

case object Q4 extends State

val delta : (State, Char) => State = 

  { case (Q0, 'a') => Q1

    case (Q0, 'b') => Q2

    case (Q1, 'a') => Q4

    case (Q1, 'b') => Q2

    case (Q2, 'a') => Q3

    case (Q2, 'b') => Q2

    case (Q3, 'a') => Q4

    case (Q3, 'b') => Q0

    case (Q4, 'a') => Q4

    case (Q4, 'b') => Q4 

    case _ => throw new Exception("Undefined") }

val dfa = DFA(Q0, delta, Set[State](Q4))

dfa.accepts("abaaa".toList)     // true

dfa.accepts("bbabaab".toList)   // true

dfa.accepts("baba".toList)      // false

dfa.accepts("abc".toList)       // false

// NFAs (Nondeterministic Finite Automata)

case class NFA[A, C](starts: Set[A],          // starting states

                     delta: (A, C) => Set[A], // transition function

                     fins:  A => Boolean) {   // final states 

  // given a state and a character, what is the set of 

  // next states? if there is none => empty set

  def next(q: A, c: C) : Set[A] = 

    Try(delta(q, c)).getOrElse(Set[A]()) 

  def nexts(qs: Set[A], c: C) : Set[A] =

    qs.flatMap(next(_, c))

  // depth-first version of accepts

  def search(q: A, s: List[C]) : Boolean = s match {

    case Nil => fins(q)

    case c::cs => next(q, c).exists(search(_, cs))

  }

  def accepts(s: List[C]) : Boolean =

    starts.exists(search(_, s))

}

// NFA examples

val nfa_trans1 : (State, Char) => Set[State] = 

  { case (Q0, 'a') => Set(Q0, Q1) 

    case (Q0, 'b') => Set(Q2) 

    case (Q1, 'a') => Set(Q1) 

    case (Q2, 'b') => Set(Q2) }

val nfa = NFA(Set[State](Q0), nfa_trans1, Set[State](Q2))

nfa.accepts("aa".toList)             // false

nfa.accepts("aaaaa".toList)          // false

nfa.accepts("aaaaab".toList)         // true

nfa.accepts("aaaaabbb".toList)       // true

nfa.accepts("aaaaabbbaaa".toList)    // false

nfa.accepts("ac".toList)             // false

// Q: Why the kerfuffle about the polymorphic types in DFAs/NFAs?

// A: Subset construction. Here the state type for the DFA is

//    sets of states.

def subset[A, C](nfa: NFA[A, C]) : DFA[Set[A], C] = {

  DFA(nfa.starts, 

      { case (qs, c) => nfa.nexts(qs, c) }, 

      _.exists(nfa.fins))

}

subset(nfa).accepts("aa".toList)             // false

subset(nfa).accepts("aaaaa".toList)          // false

subset(nfa).accepts("aaaaab".toList)         // true

subset(nfa).accepts("aaaaabbb".toList)       // true

subset(nfa).accepts("aaaaabbbaaa".toList)    // false

subset(nfa).accepts("ac".toList)             // false

import scala.math.pow

// The End ... Almost Christmas

//===============================

// I hope you had fun!

// A function should do one thing, and only one thing.

// Make your variables immutable, unless there's a good