// A version with simplification of derivatives;+
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// this keeps the regular expressions small, which+
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// is good for the run-time+
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+
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abstract class Rexp+
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case object ZERO extends Rexp+
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case object ONE 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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+
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+
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// the nullable function: tests whether the regular +
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// expression can recognise the empty string+
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def nullable (r: Rexp) : Boolean = r match {+
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  case ZERO => false+
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  case ONE => 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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// the derivative of a regular expression w.r.t. a character+
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def der (c: Char, r: Rexp) : Rexp = r match {+
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  case ZERO => ZERO+
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  case ONE => ZERO+
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  case CHAR(d) => if (c == d) ONE else ZERO+
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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(r1) => SEQ(der(c, r1), STAR(r1))+
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  case NTIMES(r, i) => +
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    if (i == 0) ZERO else SEQ(der(c, r), NTIMES(r, i - 1))+
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}+
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+
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def simp(r: Rexp) : Rexp = r match {+
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  case ALT(r1, r2) => (simp(r1), simp(r2)) match {+
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    case (ZERO, r2s) => r2s+
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    case (r1s, ZERO) => r1s+
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    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)+
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  }+
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  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {+
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    case (ZERO, _) => ZERO+
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    case (_, ZERO) => ZERO+
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    case (ONE, r2s) => r2s+
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    case (r1s, ONE) => r1s+
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    case (r1s, r2s) => SEQ(r1s, r2s)+
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  }+
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  case r => r+
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}+
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+
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+
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// the derivative w.r.t. a string (iterates der)+
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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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+
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// the main matcher function+
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def matcher(r: Rexp, s: String) : Boolean = +
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  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, ONE)+
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+
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+
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// Test Cases+
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+
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// evil regular expressions: (a?){n} a{n}  and (a*)* b+
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def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))+
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val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))+
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+
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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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//test: (a?{n}) (a{n})+
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for (i <- 0 to 8000 by 1000) {+
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  println(f"$i: ${time_needed(3, matcher(EVIL1(i), "a" * i))}%.5f")+
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}+
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+
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//test: (a*)* b+
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for (i <- 0 to 6000000 by 500000) {+
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  println(f"$i: ${time_needed(3, matcher(EVIL2, "a" * i))}%.5f")+
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}+
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+
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+
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// size of a regular expressions - for testing purposes +
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def size(r: Rexp) : Int = r match {+
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  case ZERO => 1+
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  case ONE => 1+
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  case CHAR(_) => 1+
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  case ALT(r1, r2) => 1 + size(r1) + size(r2)+
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  case SEQ(r1, r2) => 1 + size(r1) + size(r2)+
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  case STAR(r) => 1 + size(r)+
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  case NTIMES(r, _) => 1 + size(r)+
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}+
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+
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+
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// now the size of the derivatives grows +
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// much, much slower+
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+
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size(ders("".toList, EVIL2))      // 5+
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size(ders("a".toList, EVIL2))     // 8+
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size(ders("aa".toList, EVIL2))    // 8+
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size(ders("aaa".toList, EVIL2))   // 8+
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size(ders("aaaa".toList, EVIL2))  // 8+
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size(ders("aaaaa".toList, EVIL2)) // 8+
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