// A version which attempts to move whole strings, not// just characters, under derivatives whenever possibleabstract 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(r1, i) => if (i == 0) ZERO else SEQ(der(c, r1), NTIMES(r1, 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}// an exampleval r = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))der('x', r)der('y', der('x', r))der('z', der('y', der('x', r)))simp(der('z', der('y', der('x', r))))// *new*// the derivative w.r.t. a string (iterates der)def ders2(s: List[Char], r: Rexp) : Rexp = (s, r) match { case (Nil, r) => r case (s, ZERO) => ZERO case (s, ONE) => if (s == Nil) ONE else ZERO case (s, CHAR(c)) => if (s == List(c)) ONE else if (s == Nil) CHAR(c) else ZERO case (s, ALT(r1, r2)) => ALT(ders2(s, r1), ders2(s, r2)) case (c::s, r) => ders2(s, simp(der(c, r)))}// the main matcher functiondef matcher(r: Rexp, s: String) : Boolean = nullable(ders2(s.toList, r))// one or zerodef OPT(r: Rexp) = ALT(r, ONE)// Test Casesdef EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))// for measuring timedef 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 11000 by 1000) { println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")}// test: (a*)* bfor (i <- 0 to 7000000 by 500000) { println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")}// the size of a regular expressions - for testing purposes def size(r: Rexp) : Int = r match { case ZERO => 1 case ONE => 1 case CHAR(_) => 1 case ALT(r1, r2) => 1 + size(r1) + size(r2) case SEQ(r1, r2) => 1 + size(r1) + size(r2) case STAR(r) => 1 + size(r) case NTIMES(r, _) => 1 + size(r)}// string of a regular expressions - for testing purposes def string(r: Rexp) : String = r match { case ZERO => "0" case ONE => "1" case CHAR(c) => c.toString case ALT(r1, r2) => s"(${string(r1)} + ${string(r2)})" case SEQ(CHAR(c), CHAR(d)) => s"${c}${d}" case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})" case STAR(r) => s"(${string(r)})*" case NTIMES(r, n) => s"(${string(r)}){${n}}"}// test: ("a" | "aa")*val EVIL3 = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))// test: ("" | "a" | "aa")*val EVIL4 = STAR(ALT(ONE, ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a')))))val t1 = ders2("a".toList, EVIL3)val t1s = simp(t1) val t2 = ders2("aa".toList, t1s)val t2s = simp(t2)val t3 = ders2("aaa".toList, t2s)val t3s = simp(t3)val t4 = ders2("aaaa".toList, t3s)val t4s = simp(t4)println(string(t1) + " " + size(t1))println("s " + string(t1s) + " " + size(t1s))println(string(t2) + " " + size(t2))println("s " + string(t2s) + " " + size(t2s))println(string(t3) + " " + size(t3))println("s " + string(t3s) + " " + size(t3s))println(string(t4) + " " + size(t4))println("s " + string(t4s) + " " + size(t4s))