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   //explicit constructor for n-times

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)

}

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

}

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

  case Nil => r

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

}

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

//optional regular expression: one or zero times

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

// Test Cases

//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'))

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)

}

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

for (i <- 1 to 1201 by 100) {

  println(i + " " + "%.5f".format(time_needed(2, matches(EVIL1(i), "a" * i))))

}

for (i <- 1 to 1201 by 100) {

  println(i + " " + "%.5f".format(time_needed(2, matches(EVIL1(i), "a" * i))))

}

//test: (a*)* b

for (i <- 1 to 20) {

  println(i + " " + "%.5f".format(time_needed(2, matches(EVIL2, "a" * i))))

}

for (i <- 1 to 20) {

  println(i + " " + "%.5f".format(time_needed(2, matches(EVIL2, "a" * i))))

}

// size of a regular expressions - for testing purposes 

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(r) => 1 + size(r)

  case NTIMES(r, _) => 1 + size(r)

}

// EVIL1(n) has now a constant size, no matter

// what n is 

size(EVIL1(1))  // 7

size(EVIL1(3))  // 7

size(EVIL1(5))  // 7

size(EVIL1(7))  // 7

// but the size of the derivatives can still grow 

// quite dramatically

size(ders("".toList, EVIL2))     // 5

size(ders("a".toList, EVIL2))    // 12

size(ders("aa".toList, EVIL2))   // 28

size(ders("aaa".toList, EVIL2))  // 58

size(ders("aaaa".toList, EVIL2)) // 116