// Main Part 3 about Regular Expression Matching
//=============================================

object M3 {

// Regular Expressions
abstract class Rexp
case object ZERO extends Rexp
case object ONE extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALTs(rs: List[Rexp]) extends Rexp      // alternatives 
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp   // sequence
case class STAR(r: Rexp) extends Rexp             // star


// some convenience for typing regular expressions

//the usual binary choice can be defined in terms of ALTs
def ALT(r1: Rexp, r2: Rexp) = ALTs(List(r1, r2))


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(c) => false
  case ALTs(rs) => {
    if (rs.size == 0) false
    else if (nullable(rs.head)) true
    else nullable(ALTs(rs.tail))
  }
  case SEQ(c, s) => nullable(c) && nullable(s)
  case STAR(r) => true
  case _ => false
}


// (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.

def der (c: Char, r: Rexp) : Rexp = r match {
  case ZERO => ZERO
  case ONE => ZERO
  case CHAR(x) => {
    if (x==c) ONE
    else ZERO
  }
  case ALTs(rs) => ALTs(for (i <- rs) yield der(c, i))
  case SEQ(x, y) => {
    if (nullable(x)) ALTs(List(SEQ(der(c, x), y), der(c, y)))
    else SEQ(der(c, x), y)
  }
  case STAR(x) => SEQ(der(c, x), STAR(x))
}


// (3) Implement the flatten function flts. It
// deletes 0s from a list of regular expressions
// and also 'spills out', or flattens, nested 
// ALTernativeS.

def flts(rs: List[Rexp]) : List[Rexp] = rs match {
  case Nil => Nil
  case ZERO::rest => flts(rest)
  case ALTs(rs_other)::rest => rs_other ::: flts(rest)
  case r::rest => r::flts(rest)
}



// (4) Complete the simp function according to
// the specification given in the coursework description; 
// 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. Use the _.distinct and 
// flts functions.

def simp(r: Rexp) : Rexp = r match {
  case SEQ(x, ZERO) => ZERO
  case SEQ(ZERO, x) => ZERO
  case SEQ(x, ONE) => x
  case SEQ(ONE, x) => x
  case SEQ(x, y) => SEQ(simp(x), simp(y))
  case ALTs(rs) => {
    val list = flts(for (x <- rs) yield simp(x)).distinct
    if (list.size == 0) ZERO
    else if (list.size == 1) list.head
    else ALTs(list)
  }
  case x => x
}


// (5) 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::rest => {
    val deriv = simp(der(c,r))
    ders(rest, deriv)
  }
}

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


// (6) Complete the size function for regular
// expressions according to the specification 
// given in the coursework.

def size(r: Rexp): Int = r match {
  case Nil => 0
  case ZERO => 1
  case ONE => 1
  case CHAR(x) => 1
  case ALTs(rs) => 1 + (for (x <- rs) yield size(x)).sum
  case SEQ(x, y) => 1 + size(x) + size(y)
  case STAR(x) => 1 + size(x)
}


// 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()
//   "%.5f".format((end - start)/(i * 1.0e9))
// }

// for (i <- 0 to 5000000 by 500000) {
//   println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.") 
// }

// 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.

// This a dummy comment. Hopefully it works!

}