// Part 1 about Regular Expression Matching
//==========================================

object CW8a {

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 
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 
case class STAR(r: Rexp) extends Rexp 

// 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)
}

// (1a) Complete the function nullable according to
// the definition given in the coursework; this 
// function checks whether a regular expression
// can match the empty string

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
}

// (1b) 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 (c == d) 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))
}

// (1c) Complete the function der according to
// the specification given in the coursework; this
// function simplifies a regular expression;
// however it does not simplify inside STAR-regular
// expressions

def simp(r: Rexp) : Rexp = r match {
  case ALT(r1, r2) => (simp(r1), simp(r2)) match {
    case (ZERO, r2s) => r2s
    case (r1s, ZERO) => r1s
    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)
  }
  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {
    case (ZERO, _) => ZERO
    case (_, ZERO) => ZERO
    case (ONE, r2s) => r2s
    case (r1s, ONE) => r1s
    case (r1s, r2s) => SEQ(r1s, r2s)
  }
  case r => r
}

// (1d) 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::s => ders(s, simp(der(c, r)))
}

// main matcher function
def matcher(r: Rexp, s: String): Boolean = nullable(ders(s.toList, r))

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

// Java needs around 30 seconds for matching 28 a's with EVIL. 
//
// Lets see how long it takes to match strings with 
// 0.5 Million a's...it should be in the range of some
// 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))))
}
*/

}
