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1 // A version with simplification of derivatives; |
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2 // this keeps the regular expressions small, which |
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3 // is good for the run-time |
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4 |
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5 |
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6 abstract class Rexp |
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7 case object ZERO extends Rexp |
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8 case object ONE extends Rexp |
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9 case class CHAR(c: Char) extends Rexp |
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10 case class ALT(r1: Rexp, r2: Rexp) extends Rexp |
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11 case class SEQ(r1: Rexp, r2: Rexp) extends Rexp |
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12 case class STAR(r: Rexp) extends Rexp |
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13 case class NTIMES(r: Rexp, n: Int) extends Rexp |
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14 |
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15 |
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16 |
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17 // the nullable function: tests whether the regular |
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18 // expression can recognise the empty string |
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19 def nullable (r: Rexp) : Boolean = r match { |
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20 case ZERO => false |
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21 case ONE => true |
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22 case CHAR(_) => false |
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23 case ALT(r1, r2) => nullable(r1) || nullable(r2) |
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24 case SEQ(r1, r2) => nullable(r1) && nullable(r2) |
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25 case STAR(_) => true |
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26 case NTIMES(r, i) => if (i == 0) true else nullable(r) |
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27 } |
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28 |
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29 // the derivative of a regular expression w.r.t. a character |
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30 def der (c: Char, r: Rexp) : Rexp = r match { |
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31 case ZERO => ZERO |
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32 case ONE => ZERO |
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33 case CHAR(d) => if (c == d) ONE else ZERO |
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34 case ALT(r1, r2) => ALT(der(c, r1), der(c, r2)) |
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35 case SEQ(r1, r2) => |
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36 if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) |
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37 else SEQ(der(c, r1), r2) |
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38 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
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39 case NTIMES(r, i) => |
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40 if (i == 0) ZERO else SEQ(der(c, r), NTIMES(r, i - 1)) |
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41 } |
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42 |
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43 def simp(r: Rexp) : Rexp = r match { |
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44 case ALT(r1, r2) => (simp(r1), simp(r2)) match { |
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45 case (ZERO, r2s) => r2s |
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46 case (r1s, ZERO) => r1s |
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47 case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s) |
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48 } |
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49 case SEQ(r1, r2) => (simp(r1), simp(r2)) match { |
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50 case (ZERO, _) => ZERO |
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51 case (_, ZERO) => ZERO |
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52 case (ONE, r2s) => r2s |
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53 case (r1s, ONE) => r1s |
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54 case (r1s, r2s) => SEQ(r1s, r2s) |
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55 } |
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56 case r => r |
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57 } |
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58 |
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59 |
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60 // the derivative w.r.t. a string (iterates der) |
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61 def ders(s: List[Char], r: Rexp) : Rexp = s match { |
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62 case Nil => r |
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63 case c::s => ders(s, simp(der(c, r))) |
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64 } |
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65 |
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66 |
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67 // the main matcher function |
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68 def matcher(r: Rexp, s: String) : Boolean = |
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69 nullable(ders(s.toList, r)) |
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70 |
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71 |
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72 // one or zero |
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73 def OPT(r: Rexp) = ALT(r, ONE) |
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74 |
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75 |
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76 // Test Cases |
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77 |
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78 // evil regular expressions: (a?){n} a{n} and (a*)* b |
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79 def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n)) |
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80 val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
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81 |
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82 |
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83 def time_needed[T](i: Int, code: => T) = { |
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84 val start = System.nanoTime() |
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85 for (j <- 1 to i) code |
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86 val end = System.nanoTime() |
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87 (end - start)/(i * 1.0e9) |
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88 } |
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89 |
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90 |
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91 //test: (a?{n}) (a{n}) |
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92 for (i <- 0 to 8000 by 1000) { |
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93 println(f"$i: ${time_needed(3, matcher(EVIL1(i), "a" * i))}%.5f") |
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94 } |
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95 |
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96 //test: (a*)* b |
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97 for (i <- 0 to 6000000 by 500000) { |
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98 println(f"$i: ${time_needed(3, matcher(EVIL2, "a" * i))}%.5f") |
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99 } |
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100 |
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101 |
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102 // size of a regular expressions - for testing purposes |
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103 def size(r: Rexp) : Int = r match { |
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104 case ZERO => 1 |
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105 case ONE => 1 |
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106 case CHAR(_) => 1 |
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107 case ALT(r1, r2) => 1 + size(r1) + size(r2) |
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108 case SEQ(r1, r2) => 1 + size(r1) + size(r2) |
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109 case STAR(r) => 1 + size(r) |
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110 case NTIMES(r, _) => 1 + size(r) |
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111 } |
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112 |
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113 |
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114 // now the size of the derivatives grows |
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115 // much, much slower |
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116 |
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117 size(ders("".toList, EVIL2)) // 5 |
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118 size(ders("a".toList, EVIL2)) // 8 |
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119 size(ders("aa".toList, EVIL2)) // 8 |
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120 size(ders("aaa".toList, EVIL2)) // 8 |
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121 size(ders("aaaa".toList, EVIL2)) // 8 |
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122 size(ders("aaaaa".toList, EVIL2)) // 8 |
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123 |
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124 |
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125 |
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126 |
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127 |