abstract class Rexp +
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case object NULL extends Rexp+
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case object EMPTY extends Rexp+
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case class CHAR(c: Char) extends Rexp+
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp+
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp +
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case class STAR(r: Rexp) extends Rexp +
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case class NTIMES(r: Rexp, n: Int) extends Rexp +
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+
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def simp(r: Rexp): Rexp = r match {+
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  case ALT(r1, r2) => {+
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    val r1s = simp(r1)+
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    val r2s = simp(r2)+
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    (r1s, r2s) match {+
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      case (NULL, _) => r2s+
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      case (_, NULL) => r1s+
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      case _ => if (r1s == r2s) r1s else ALT(r1s, r2s)+
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    }+
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  }+
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  case SEQ(r1, r2) => {+
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    val r1s = simp(r1)+
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    val r2s = simp(r2)+
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    (r1s, r2s) match {+
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      case (NULL, _) => NULL+
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      case (_, NULL) => NULL+
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      case (EMPTY, _) => r2s+
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      case (_, EMPTY) => r1s+
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      case _ => SEQ(r1s, r2s)+
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    }+
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  }+
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  case NTIMES(r, n) => NTIMES(simp(r), n)    +
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  case r => r+
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}+
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+
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def nullable (r: Rexp) : Boolean = r match {+
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  case NULL => false+
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  case EMPTY => true+
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  case CHAR(_) => false+
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  case ALT(r1, r2) => nullable(r1) || nullable(r2)+
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  case SEQ(r1, r2) => nullable(r1) && nullable(r2)+
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  case STAR(_) => true+
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  case NTIMES(r, i) => if (i == 0) true else nullable(r)+
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}+
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+
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def der (c: Char, r: Rexp) : Rexp = r match {+
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  case NULL => NULL+
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  case EMPTY => NULL+
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  case CHAR(d) => if (c == d) EMPTY else NULL+
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  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))+
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  case SEQ(r1, r2) => +
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    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))+
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    else SEQ(der(c, r1), r2)+
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  case STAR(r) => SEQ(der(c, r), STAR(r))+
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  case NTIMES(r, i) => +
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    if (i == 0) NULL else SEQ(der(c, r), NTIMES(r, i - 1))+
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}+
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+
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def ders (s: List[Char], r: Rexp) : Rexp = s match {+
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  case Nil => r+
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  case c::s => ders(s, simp(der(c, r)))+
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}+
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+
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def matches(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))+
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+
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+
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//one or zero+
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def OPT(r: Rexp) = ALT(r, EMPTY)+
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+
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def EVIL(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))+
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+
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def time_needed[T](i: Int, code: => T) = {+
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  val start = System.nanoTime()+
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  for (j <- 1 to i) code+
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  val end = System.nanoTime()+
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  (end - start)/(i * 1.0e9)+
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}+
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+
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+
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for (i <- 1 to 12001 by 500) {+
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  println(i + " " + "%.5f".format(time_needed(1, matches(EVIL(i), "a" * i))))+
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}+
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+
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+
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