1 

     2 abstract class Rexp

     3 case object ZERO extends Rexp

     4 case object ONE extends Rexp

     5 case class CHAR(c: Char) extends Rexp

     6 case class ALT(r1: Rexp, r2: Rexp) extends Rexp 

     7 case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 

     8 case class STAR(r: Rexp) extends Rexp 

     9 case class NTIMES(r: Rexp, n: Int) extends Rexp 

    10 case class UPNTIMES(r: Rexp, n: Int) extends Rexp 

    11 

    12 // the nullable function: tests whether the regular 

    13 // expression can recognise the empty string

    14 def nullable (r: Rexp) : Boolean = r match {

    15   case ZERO => false

    16   case ONE => true

    17   case CHAR(_) => false

    18   case ALT(r1, r2) => nullable(r1) || nullable(r2)

    19   case SEQ(r1, r2) => nullable(r1) && nullable(r2)

    20   case STAR(_) => true

    21   case NTIMES(r, i) => if (i == 0) true else nullable(r)

    22   case UPNTIMES(r: Rexp, n: Int) => true

    23 }

    24 

    25 // the derivative of a regular expression w.r.t. a character

    26 def der (c: Char, r: Rexp) : Rexp = r match {

    27   case ZERO => ZERO

    28   case ONE => ZERO

    29   case CHAR(d) => if (c == d) ONE else ZERO

    30   case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))

    31   case SEQ(r1, r2) => 

    32     if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))

    33     else SEQ(der(c, r1), r2)

    34   case STAR(r1) => SEQ(der(c, r1), STAR(r1))

    35   case NTIMES(r1, i) => 

    36     if (i == 0) ZERO else

    37     if (nullable(r1)) SEQ(der(c, r1), UPNTIMES(r1, i - 1))

    38     else SEQ(der(c, r1), NTIMES(r1, i - 1))

    39   case UPNTIMES(r1, i) => 

    40     if (i == 0) ZERO

    41     else SEQ(der(c, r1), UPNTIMES(r1, i - 1)) 

    42 }

    43 

    44 def simp(r: Rexp) : Rexp = r match {

    45   case ALT(r1, r2) => (simp(r1), simp(r2)) match {

    46     case (ZERO, r2s) => r2s

    47     case (r1s, ZERO) => r1s

    48     case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)

    49   }

    50   case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {

    51     case (ZERO, _) => ZERO

    52     case (_, ZERO) => ZERO

    53     case (ONE, r2s) => r2s

    54     case (r1s, ONE) => r1s

    55     case (r1s, r2s) => SEQ(r1s, r2s)

    56   }

    57   case r => r

    58 }

    59 

    60 

    61 // the derivative w.r.t. a string (iterates der)

    62 def ders (s: List[Char], r: Rexp) : Rexp = s match {

    63   case Nil => r

    64   case c::s => ders(s, simp(der(c, r)))

    65 }

    66 

    67 

    68 // the main matcher function

    69 def matches(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))

    70 

    71 // one or zero

    72 def OPT(r: Rexp) = ALT(r, ONE)

    73 

    74 // evil regular expressions

    75 def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

    76 val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

    77 val EVIL3 = SEQ(STAR(ALT(CHAR('a'), SEQ(CHAR('a'),CHAR('a')))), CHAR('b'))

    78 

    79 def time_needed[T](i: Int, code: => T) = {

    80   val start = System.nanoTime()

    81   for (j <- 1 to i) code

    82   val end = System.nanoTime()

    83   "%.5f".format((end - start) / (i * 1.0e9))

    84 }

    85 

    86 

    87 // test: (a?{n}) (a{n})

    88 for (i <- 0 to 8000 by 1000) {

    89   println(s"$i: ${time_needed(2, matches(EVIL1(i), "a" * i))}")

    90 }

    91 

    92 // test: (a*)* b

    93 for (i <- 0 to 6000000 by 500000) {

    94   println(s"$i: ${time_needed(2, matches(EVIL2, "a" * i))}")

    95 }

    96 

    97 

    98 val r0 = simp(der('a', EVIL3))

    99 val r1 = simp(der('a', r0))

   100 val r2 = simp(der('a', r1))

   101 val r3 = simp(der('a', r2))

   102 val r4 = simp(der('a', r3))

   103 val r5 = simp(der('a', r4))

   104 val r6 = simp(der('a', r5))

   105 

   106 //test: (a|aa)* b

   107 /*

   108 for (i <- 0 to 100 by 10) {

   109   println(s"$i: ${time_needed(2, matches(EVIL3, "a" * i ++ "c"))}")

   110 }

   111  */

   112