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 

case class NTIMES(r: Rexp, n: Int) extends Rexp 

case class UPNTIMES(r: Rexp, n: Int) extends Rexp 

// the nullable function: tests whether the regular 

// expression can recognise 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

  case NTIMES(r, i) => if (i == 0) true else nullable(r)

  case UPNTIMES(r: Rexp, n: Int) => true

}

// 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))

  case NTIMES(r1, i) => 

    if (i == 0) ZERO else

    if (nullable(r1)) SEQ(der(c, r1), UPNTIMES(r1, i - 1))

    else SEQ(der(c, r1), NTIMES(r1, i - 1))

  case UPNTIMES(r1, i) => 

    if (i == 0) ZERO

    else SEQ(der(c, r1), UPNTIMES(r1, i - 1)) 

}

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

}

// the derivative w.r.t. a string (iterates der)

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

  case Nil => r

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

}

// the main matcher function

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

// one or zero

def OPT(r: Rexp) = ALT(r, ONE)

// evil regular expressions

def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

val EVIL3 = SEQ(STAR(ALT(CHAR('a'), SEQ(CHAR('a'),CHAR('a')))), CHAR('b'))

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))

}

// test: (a?{n}) (a{n})

for (i <- 0 to 8000 by 1000) {

  println(s"$i: ${time_needed(2, matches(EVIL1(i), "a" * i))}")

}

// test: (a*)* b

for (i <- 0 to 6000000 by 500000) {

  println(s"$i: ${time_needed(2, matches(EVIL2, "a" * i))}")

}

val r0 = simp(der('a', EVIL3))

val r1 = simp(der('a', r0))

val r2 = simp(der('a', r1))

val r3 = simp(der('a', r2))

val r4 = simp(der('a', r3))

val r5 = simp(der('a', r4))

val r6 = simp(der('a', r5))

//test: (a|aa)* b

/*

for (i <- 0 to 100 by 10) {

  println(s"$i: ${time_needed(2, matches(EVIL3, "a" * i ++ "c"))}")

}

 */