// A Version with an explicit n-times regular expression;// this keeps the size of the regular expression in the// EVIL1 test-case quite smallabstract class Rexp case 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 //explicit constructor for n-timesdef 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)}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(r1, i) => if (i == 0) ZERO else SEQ(der(c, r1), NTIMES(r1, i - 1))}def ders (s: List[Char], r: Rexp) : Rexp = s match { case Nil => r case c::s => ders(s, der(c, r))}def matcher(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))// the optional regular expression: one or zero times// this regular expression is still defined in terms of ALTdef OPT(r: Rexp) = ALT(r, ONE)// Test Cases// evil regular expressionsdef 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 1000 by 100) { println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")}// test: (a*)* bfor (i <- 0 to 20) { println(f"$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)}// EVIL1(n) has now a constant size, no matter// what n is; also the derivative only grows // moderately size(EVIL1(1)) // 7size(EVIL1(3)) // 7size(EVIL1(5)) // 7size(EVIL1(7)) // 7size(EVIL1(20)) // 7size(ders("".toList, EVIL1(5))) // 7size(ders("a".toList, EVIL1(5))) // 16size(ders("aa".toList, EVIL1(5))) // 35size(ders("aaa".toList, EVIL1(5))) // 59size(ders("aaaa".toList, EVIL1(5))) // 88size(ders("aaaaa".toList, EVIL1(5))) // 122size(ders("aaaaaa".toList, EVIL1(5))) // 151// but the size of the derivatives can still grow // quite dramatically in case of EVIL2size(ders("".toList, EVIL2)) // 5size(ders("a".toList, EVIL2)) // 12size(ders("aa".toList, EVIL2)) // 28size(ders("aaa".toList, EVIL2)) // 58size(ders("aaaa".toList, EVIL2)) // 116size(ders("aaaaa".toList, EVIL2)) // 230size(ders("aaaaaa".toList, EVIL2)) // 456size(ders(("a" * 20).toList, EVIL2)) // 7340068