// A version with simplification of derivatives;

// this keeps the regular expressions small, which

// is good for the run-time

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 

// 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)

}

// 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(r, i) => 

    if (i == 0) ZERO else SEQ(der(c, r), NTIMES(r, 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 matcher(r: Rexp, s: String) : Boolean = 

  nullable(ders(s.toList, r))

// one or zero

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

// Test Cases

// evil regular expressions: (a?){n} a{n}  and (a*)* b

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 <- 0 to 8000 by 1000) {

  println(f"$i: ${time_needed(3, matcher(EVIL1(i), "a" * i))}%.5f")

}

//test: (a*)* b

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

  println(f"$i: ${time_needed(3, matcher(EVIL2, "a" * i))}%.5f")

}

// 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)

}

// now the size of the derivatives grows 

// much, much slower

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

size(ders("a".toList, EVIL2))     // 8

size(ders("aa".toList, EVIL2))    // 8

size(ders("aaa".toList, EVIL2))   // 8

size(ders("aaaa".toList, EVIL2))  // 8

size(ders("aaaaa".toList, EVIL2)) // 8