1 // A simple matcher for basic regular expressions

     2 //

     3 // Call the test cases with X = {1,2,3}

     4 //

     5 //   amm re1.sc testX

     6 //

     7 // or 

     8 //

     9 //   amm re1.sc all

    10 

    11 // regular expressions

    12 abstract class Rexp

    13 case object ZERO extends Rexp                    // matches nothing

    14 case object ONE extends Rexp                     // matches an empty string

    15 case class CHAR(c: Char) extends Rexp            // matches a character c

    16 case class ALT(r1: Rexp, r2: Rexp) extends Rexp  // alternative

    17 case class SEQ(r1: Rexp, r2: Rexp) extends Rexp  // sequence

    18 case class STAR(r: Rexp) extends Rexp            // star

    19 

    20 // nullable function: tests whether a regular 

    21 // expression can recognise the empty string

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

    23   case ZERO => false

    24   case ONE => true

    25   case CHAR(_) => false

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

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

    28   case STAR(_) => true

    29 }

    30 

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

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

    33   case ZERO => ZERO

    34   case ONE => ZERO

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

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

    37   case SEQ(r1, r2) => 

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

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

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

    41 }

    42 

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

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

    45   case Nil => r

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

    47 }

    48 

    49 // the main matcher function

    50 def matcher(r: Rexp, s: String) : Boolean = 

    51   nullable(ders(s.toList, r))

    52 

    53 

    54 // some examples from the homework

    55 

    56 val r = STAR(ALT(SEQ(CHAR('a'), CHAR('b')), CHAR('b')))

    57 der('a', r)

    58 der('b', r)

    59 der('c', r)

    60 

    61 val r2 = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))

    62 der('x', r2)

    63 der('y', der('x', r2))

    64 der('z', der('y', der('x', r2)))

    65 

    66 

    67 // the optional regular expression (one or zero times)

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

    69 

    70 // the n-times regular expression (explicitly expanded)

    71 def NTIMES(r: Rexp, n: Int) : Rexp = n match {

    72   case 0 => ONE

    73   case 1 => r

    74   case n => SEQ(r, NTIMES(r, n - 1))

    75 }

    76 

    77 

    78 // Test Cases

    79 //============

    80 

    81 // the evil regular expression  a?{n} a{n}

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

    83 

    84 // the evil regular expression (a*)* b

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

    86 

    87 // for measuring time

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

    89   val start = System.nanoTime()

    90   for (j <- 1 to i) code

    91   val end = System.nanoTime()

    92   (end - start) / (i * 1.0e9)

    93 }

    94 

    95 

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

    97 @doc("Test (a?{n}) (a{n})")

    98 @main

    99 def test1() = {

   100   println("Test (a?{n}) (a{n})")

   101 

   102   for (i <- 0 to 20 by 2) {

   103     println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")

   104   }

   105 }

   106 

   107 // test: (a*)* b

   108 @doc("Test (a*)* b")

   109 @main

   110 def test2() = {

   111   println("Test (a*)* b")

   112 

   113   for (i <- 0 to 20 by 2) {

   114     println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")

   115   }

   116 }

   117 

   118 // the size of a regular expressions - for testing purposes 

   119 def size(r: Rexp) : Int = r match {

   120   case ZERO => 1

   121   case ONE => 1

   122   case CHAR(_) => 1

   123   case ALT(r1, r2) => 1 + size(r1) + size(r2)

   124   case SEQ(r1, r2) => 1 + size(r1) + size(r2)

   125   case STAR(r) => 1 + size(r)

   126 }

   127 

   128 // the expicit expansion in EVIL1(n) increases

   129 // drastically its size

   130 

   131 size(EVIL1(1))  // 5

   132 size(EVIL1(3))  // 17

   133 size(EVIL1(5))  // 29

   134 size(EVIL1(7))  // 41

   135 size(EVIL1(20)) // 119

   136 

   137 // given a regular expression and building successive

   138 // derivatives might result into bigger and bigger

   139 // regular expressions...here is an example for this:

   140 

   141 // (a+b)* o a o b o (a+b)*

   142 val BIG_aux = STAR(ALT(CHAR('a'), CHAR('b')))

   143 val BIG = SEQ(BIG_aux, SEQ(CHAR('a'),SEQ(CHAR('b'), BIG_aux)))

   144 

   145 size(ders("".toList, BIG))              // 13

   146 size(ders("ab".toList, BIG))            // 51

   147 size(ders("abab".toList, BIG))          // 112

   148 size(ders("ababab".toList, BIG))        // 191

   149 size(ders("abababab".toList, BIG))      // 288

   150 size(ders("ababababab".toList, BIG))    // 403

   151 size(ders("abababababab".toList, BIG))  // 536

   152 

   153 

   154 size(ders(("ab" * 200).toList, BIG))    // 366808

   155 

   156 @doc("Test (a + b)* o (a o b) o (a + b)*")

   157 @main

   158 def test3() = {

   159   println("Test (a + b)* o (a o b) o (a + b)*")

   160 

   161   for (i <- 0 to 200 by 10) {

   162     println(f"$i: ${time_needed(2, matcher(BIG, "ab" * i))}%.5f")

   163   }

   164 }

   165 

   166 

   167 

   168 

   169 @doc("All tests.")

   170 @main

   171 def all() = { test1(); test2() ; test3() }