// Part 1 about Regular Expression Matching

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

// Regular Expressions

abstract class Rexp

case object ZERO extends Rexp

case object ONE extends Rexp

case class CHAR(c: Char) extends Rexp

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

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

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

// some convenience for typing 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)

}

// (1) Complete the function nullable according to

// the definition given in the coursework; this 

// function checks whether a regular expression

// can match the empty string and Returns a boolean

// accordingly.

def nullable (r: Rexp) : Boolean = r match {

    case ZERO => false

    case ONE => true

    case CHAR(_) => false

    case ALT(r1, r2) => nullable(r1) | nullable(r2)

    case SEQ(r1, r2) => nullable(r1) & nullable(r2)

    case STAR(_) => true

}

// (2) Complete the function der according to

// the definition given in the coursework; this

// function calculates the derivative of a 

// regular expression w.r.t. a character.

//TODO: debug

//TODO: understand this more. 

// first test runs

// test 2 fails

// test 3 runs

// test 4 runs

def der (c: Char, r: Rexp) : Rexp = r match {

    //TODO: debug

    case ZERO => ZERO

    case ONE => ZERO

    case CHAR(r1) => if (c == r1) ONE else ZERO

    case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))

    case SEQ(r1, r2) => if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) else SEQ(der(c, r1), r2)

    case STAR(r1) => SEQ(der(c, r1), STAR(r1))

}

// (3) Complete the simp function according to

// the specification given in the coursework; this

// function simplifies a regular expression from

// the inside out, like you would simplify arithmetic 

// expressions; however it does not simplify inside 

// STAR-regular expressions.

def simp(r: Rexp) : Rexp = r match {

    case STAR(_) => r

    case SEQ(r1, r2) => (simp(r1), simp(r2)) match { // potential failure

        case (_, ZERO) => ZERO

        case (ZERO, _) => ZERO

        case (r1, ONE) => simp(r1)

        case (ONE, r2) => simp(r2)

        case (r1, r2) => SEQ(r1, r2)

    }

    case ALT(r1, r2) => (simp(r1), simp(r2)) match {

        case (r1, ZERO) => simp(r1)

        case (ZERO, r1) => simp(r1)

        case (r1, r2) if r1 == r2 => simp(r1)

        case (r1, r2) => ALT(r1, r2)

    }

    case r => r

}

// (4) Complete the two functions below; the first 

// calculates the derivative w.r.t. a string; the second

// is the regular expression matcher taking a regular

// expression and a string and checks whether the

// string matches the regular expression

def ders (s: List[Char], r: Rexp) : Rexp = s match {

    case Nil => r

    case c::cs => ders(cs, simp(der(c, r)))

}

def matcher(r: Rexp, s: String): Boolean = {

    nullable(ders(s.toList, r))

}

// (5) Complete the size function for regular

// expressions according to the specification 

// given in the coursework.

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

    case ZERO => 1

    case ONE => 1

    case CHAR(_) => 1

    case SEQ(r1, r2) => 1 + size(r1) + size(r2)

    case ALT(r1, r2) => 1 + size(r1) + size(r2)

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

}

// some testing data

//matcher(("a" ~ "b") ~ "c", "abc")  // => true

//matcher(("a" ~ "b") ~ "c", "ab")   // => false

// the supposedly 'evil' regular expression (a*)* b

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

//matcher(EVIL, "a" * 1000 ++ "b")   // => true

//matcher(EVIL, "a" * 1000)          // => false

// size without simplifications

//size(der('a', der('a', EVIL)))             // => 28

//size(der('a', der('a', der('a', EVIL))))   // => 58

// size with simplification

//size(simp(der('a', der('a', EVIL))))           // => 8

//size(simp(der('a', der('a', der('a', EVIL))))) // => 8

// Python needs around 30 seconds for matching 28 a's with EVIL. 

// Java 9 and later increase this to an "astonishing" 40000 a's in

// 30 seconds.

//

// Lets see how long it really takes to match strings with 

// 5 Million a's...it should be in the range of a couple

// of seconds.

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

  val start = System.nanoTime()

  for (j <- 1 to i) code

  val end = System.nanoTime()

  (end - start)/(i * 1.0e9)

}

//for (i <- 0 to 5000000 by 500000) {

//  println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i))))

//}

// another "power" test case 

println(simp(Iterator.iterate(ONE:Rexp)(r => ALT(r, r)).drop(40).next))

// the Iterator produces the rexp

//

//      SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)

//

//    where SEQ is nested 50 times.