// 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.

*/

}