// Core Part about Regular Expression Matching

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

object CW9c {

// 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)

}

// (5) 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(c) => false

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

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

		case STAR(r) => true

	}

}

// (6) 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.

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

	r match {

		case ZERO => ZERO

		case ONE => ZERO

		case CHAR(d) => if(d == c) 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(r) => SEQ(der(c, r), STAR(r))

	}

}

// (7) 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) => STAR(r) // does not process r star

		case SEQ(r1, r2) => {

			val x = (simp(r1), simp(r2))

			if(x._1 == ZERO) ZERO else

			if(x._2 == ZERO) ZERO else

			if(x._1 == ONE) simp(x._2) else 

			if(x._2 == ONE) simp(x._1) else

			if(x._1 == x._2) simp(x._2) else

			SEQ(simp(x._1), simp(x._2))

		}

		case ALT(r1, r2) => {

			val x = (simp(r1), simp(r2))

			if(x._1 == ZERO) simp(x._2) else

			if(x._2 == ZERO) simp(x._1) else

			if(x._1 == x._2) simp(x._2) else

			ALT(simp(x._1), simp(x._2))

		}

		case r => r // if single regex, return it

	}

}

// (8) 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 = {

	val listOfCharacters = s.toList

	val result = ders(listOfCharacters, r)

	nullable(result)

}

// (9) 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(c) => 1

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

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

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

	}

}

// 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 

simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(50).next) == ONE

// the Iterator produces the rexp

//

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

//

//    where SEQ is nested 50 times.

*/

}