// A version with simplification of derivatives;// this keeps the regular expressions small, which// is good for the run-timeabstract class Rexpcase object ZERO extends Rexpcase object ONE extends Rexpcase class CHAR(c: Char) extends Rexpcase 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 stringdef 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 characterdef 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 functiondef matcher(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))// one or zerodef OPT(r: Rexp) = ALT(r, ONE)// Test Cases// evil regular expressions: (a?){n} a{n} and (a*)* bdef 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: ${time_needed(3, matcher(EVIL1(i), "a" * i))}%.5f")}//test: (a*)* bfor (i <- 0 to 6000000 by 500000) { println(f"$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 slowersize(ders("".toList, EVIL2)) // 5size(ders("a".toList, EVIL2)) // 8size(ders("aa".toList, EVIL2)) // 8size(ders("aaa".toList, EVIL2)) // 8size(ders("aaaa".toList, EVIL2)) // 8size(ders("aaaaa".toList, EVIL2)) // 8