abstract class Rexp

case object NULL extends Rexp

case object EMPTY 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 

// nullable function: tests whether the regular 

// expression can recognise the empty string

def nullable (r: Rexp) : Boolean = r match {

  case NULL => false

  case EMPTY => true

  case CHAR(_) => false

  case ALT(r1, r2) => nullable(r1) || nullable(r2)

  case SEQ(r1, r2) => nullable(r1) && nullable(r2)

  case STAR(_) => true

}

// derivative of a regular expression w.r.t. a character

def der (c: Char, r: Rexp) : Rexp = r match {

  case NULL => NULL

  case EMPTY => NULL

  case CHAR(d) => if (c == d) EMPTY else NULL

  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))

}

// 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, der(c, r))

}

// main matcher function

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

//example from the homework

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

der('a', r)

der('b', r)

der('c', r)

//optional: one or zero times

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

//n-times

def NTIMES(r: Rexp, n: Int) : Rexp = n match {

  case 0 => EMPTY

  case 1 => r

  case n => SEQ(r, NTIMES(r, n - 1))

}

// the evil regular expression  a?{n} a{n}

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

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

//for measuring time

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 <- 1 to 20) {

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

}

for (i <- 1 to 6502 by 500) {

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

}