// Scala Lecture 4

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

// Polymorphic Types

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

// You do not want to write functions like contains, first, 

// length and so on for every type of lists.

def length_string_list(lst: List[String]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length_string_list(xs)

}

def length_int_list(lst: List[Int]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length_int_list(xs)

}

length_string_list(List("1", "2", "3", "4"))

length_int_list(List(1, 2, 3, 4))

//-----

def length[A](lst: List[A]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length(xs)

}

length(List("1", "2", "3", "4"))

length(List(1, 2, 3, 4))

def map[A, B](lst: List[A], f: A => B): List[B] = lst match {

  case Nil => Nil

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

}

map(List(1, 2, 3, 4), (x: Int) => x.toString)

// Remember?

def first[A, B](xs: List[A], f: A => Option[B]) : Option[B] = ...

// distinct / distinctBy

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

ls.distinct

ls.minBy(_._2)

ls.sortBy(_._1)

def distinctBy[B, C](xs: List[B], 

                     f: B => C, 

                     acc: List[C] = Nil): List[B] = xs match {

  case Nil => Nil

  case x::xs => {

    val res = f(x)

    if (acc.contains(res)) distinctBy(xs, f, acc)  

    else x::distinctBy(xs, f, res::acc)

  }

} 

// distinctBy  with the identity function is 

// just distinct

distinctBy(ls, (x: Int) => x)

val cs = List('A', 'b', 'a', 'c', 'B', 'D', 'd')

distinctBy(cs, (c:Char) => c.toUpper)

// Type inference is local in Scala

def id[T](x: T) : T = x

val x = id(322)          // Int

val y = id("hey")        // String

val z = id(Set[Int](1,2,3,4)) // Set[Int]

// The type variable concept in Scala can get really complicated.

//

// - variance (OO)

// - bounds (subtyping)

// - quantification

// Java has issues with this too: Java allows

// to write the following incorrect code, and

// only recovers by raising an exception

// at runtime.

// Object[] arr = new Integer[10];

// arr[0] = "Hello World";

// Scala gives you a compile-time error, which

// is much better.

var arr = Array[Int]()

arr(0) = "Hello World"

//

// 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

Bird("Sparrow")

println(Bird("Sparrow"))

println(Bird("Sparrow").toString)

// you can override methods

case class Bird(name: String) extends Animal {

  override def toString = name

}

// There is a very convenient short-hand notation

// for constructors:

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

  def numer = x

  def denom = y

}

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

val half = Fraction(1, 2)

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)

// 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 not 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.numer, denom + other.denom)

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

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

}

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

val half = Fraction(1, 2)

val third = Fraction (1, 3)

half + third

half / third

// not sure if one can get this to work

// properly, since Scala just cannot find out

// if / is for ints or for Fractions 

(1 / 3) + half

(1 / 2) + third

// either you have to force the Fraction-type by

// using a method that is not defined for ints

(1 /% 3) + half

(1 /% 2) + third

// ...or explicitly give the type in order to allow

// Scala to do the conversion to Fractions 

((1:Fraction) / 3) + half

(1 / (3: Fraction)) + 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

// another DFA with a Sink state

abstract class S

case object S0 extends S

case object S1 extends S

case object S2 extends S

case object Sink extends S

// transition function with a sink state

val sigma : (S, Char) => S = 

  { case (S0, 'a') => S1

    case (S1, 'a') => S2

    case _ => Sink

  }

val dfa2 = DFA(S0, sigma, Set[S](S2))

dfa2.accepts("aa".toList)        // true

dfa2.accepts("".toList)          // false

dfa2.accepts("ab".toList)        // false

//  we could also have a dfa for numbers

val sigmai : (S, Int) => S = 

  { case (S0, 1) => S1

    case (S1, 1) => S2

    case _ => Sink

  }

val dfa3 = DFA(S0, sigmai, Set[S](S2))

dfa3.accepts(List(1, 1))        // true

dfa3.accepts(Nil)               // false

dfa3.accepts(List(1, 2))        // 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(nfa1).accepts("aa".toList)             // false

subset(nfa1).accepts("aaaaa".toList)          // false

subset(nfa1).accepts("aaaaab".toList)         // true

subset(nfa1).accepts("aaaaabbb".toList)       // true

subset(nfa1).accepts("aaaaabbbaaa".toList)    // false

subset(nfa1).accepts("ac".toList)             // false

// Cool Stuff in Scala

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

// Implicits or How to Pimp my Library

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

//

// For example adding your own methods to Strings:

// Imagine you want to increment strings, like

//

//     "HAL".increment

//

// you can avoid ugly fudges, like a MyString, by

// using implicit conversions.

implicit class MyString(s: String) {

  def increment = for (c <- s) yield (c + 1).toChar 

}

"HAL".increment

// Regular expressions - the power of DSLs in Scala

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

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*

// writing (ab)* in the format above is 

// tedious

val r0 = STAR(SEQ(CHAR('a'), CHAR('b')))

// 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)

val r1 = STAR("ab")

val r2 = STAR(ALT("ab", "baa baa black sheep"))

val r3 = STAR(SEQ("ab", ALT("a", "b")))

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.% 

// Lazy Evaluation

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

//

// Do not evaluate arguments just yet:

// this uses the => in front of the type