44 // function checks whether a regular expression |
42 // function checks whether a regular expression |
45 // can match the empty string and Returns a boolean |
43 // can match the empty string and Returns a boolean |
46 // accordingly. |
44 // accordingly. |
47 |
45 |
48 def nullable (r: Rexp) : Boolean = r match { |
46 def nullable (r: Rexp) : Boolean = r match { |
49 case ZERO => false |
47 case ZERO => false |
50 case ONE => true |
48 case ONE => true |
51 case CHAR(_) => false |
49 case CHAR(_) => false |
52 case ALT(r1, r2) => nullable(r1) || nullable(r2) |
50 case ALT(r1, r2) => nullable(r1) | nullable(r2) |
53 case SEQ(r1, r2) => nullable(r1) && nullable(r2) |
51 case SEQ(r1, r2) => nullable(r1) & nullable(r2) |
54 case STAR(_) => true |
52 case STAR(_) => true |
55 } |
53 } |
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54 |
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55 |
56 |
56 |
57 // (2) Complete the function der according to |
57 // (2) Complete the function der according to |
58 // the definition given in the coursework; this |
58 // the definition given in the coursework; this |
59 // function calculates the derivative of a |
59 // function calculates the derivative of a |
60 // regular expression w.r.t. a character. |
60 // regular expression w.r.t. a character. |
61 |
61 |
|
62 //TODO: debug |
|
63 //TODO: understand this more. |
|
64 // first test runs |
|
65 // test 2 fails |
|
66 // test 3 runs |
|
67 // test 4 runs |
62 def der (c: Char, r: Rexp) : Rexp = r match { |
68 def der (c: Char, r: Rexp) : Rexp = r match { |
63 case ZERO => ZERO |
69 //TODO: debug |
64 case ONE => ZERO |
70 case ZERO => ZERO |
65 case CHAR(d) => if (c == d) ONE else ZERO |
71 case ONE => ZERO |
66 case ALT(r1, r2) => ALT(der(c, r1), der(c, r2)) |
72 case CHAR(r1) => if (c == r1) ONE else ZERO |
67 case SEQ(r1, r2) => |
73 case ALT(r1, r2) => ALT(der(c, r1), der(c, r2)) |
68 if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) |
74 case SEQ(r1, r2) => if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) else SEQ(der(c, r1), r2) |
69 else SEQ(der(c, r1), r2) |
75 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
70 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
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71 } |
76 } |
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77 |
72 |
78 |
73 // (3) Complete the simp function according to |
79 // (3) Complete the simp function according to |
74 // the specification given in the coursework; this |
80 // the specification given in the coursework; this |
75 // function simplifies a regular expression from |
81 // function simplifies a regular expression from |
76 // the inside out, like you would simplify arithmetic |
82 // the inside out, like you would simplify arithmetic |
77 // expressions; however it does not simplify inside |
83 // expressions; however it does not simplify inside |
78 // STAR-regular expressions. |
84 // STAR-regular expressions. |
79 |
85 |
80 def simp(r: Rexp) : Rexp = r match { |
86 def simp(r: Rexp) : Rexp = r match { |
81 case ALT(r1, r2) => (simp(r1), simp(r2)) match { |
87 case STAR(_) => r |
82 case (ZERO, r2s) => r2s |
88 case SEQ(r1, r2) => (simp(r1), simp(r2)) match { // potential failure |
83 case (r1s, ZERO) => r1s |
89 case (_, ZERO) => ZERO |
84 case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s) |
90 case (ZERO, _) => ZERO |
85 } |
91 case (r1, ONE) => simp(r1) |
86 case SEQ(r1, r2) => (simp(r1), simp(r2)) match { |
92 case (ONE, r2) => simp(r2) |
87 case (ZERO, _) => ZERO |
93 case (r1, r2) => SEQ(r1, r2) |
88 case (_, ZERO) => ZERO |
94 } |
89 case (ONE, r2s) => r2s |
95 case ALT(r1, r2) => (simp(r1), simp(r2)) match { |
90 case (r1s, ONE) => r1s |
96 case (r1, ZERO) => simp(r1) |
91 case (r1s, r2s) => SEQ(r1s, r2s) |
97 case (ZERO, r1) => simp(r1) |
92 } |
98 case (r1, r2) if r1 == r2 => simp(r1) |
93 case r => r |
99 case (r1, r2) => ALT(r1, r2) |
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100 } |
|
101 case r => r |
94 } |
102 } |
95 |
103 |
96 |
104 |
97 // (4) Complete the two functions below; the first |
105 // (4) Complete the two functions below; the first |
98 // calculates the derivative w.r.t. a string; the second |
106 // calculates the derivative w.r.t. a string; the second |
99 // is the regular expression matcher taking a regular |
107 // is the regular expression matcher taking a regular |
100 // expression and a string and checks whether the |
108 // expression and a string and checks whether the |
101 // string matches the regular expression. |
109 // string matches the regular expression |
102 |
110 |
103 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
111 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
104 case Nil => r |
112 case Nil => r |
105 case c::s => ders(s, simp(der(c, r))) |
113 case c::cs => ders(cs, simp(der(c, r))) |
106 } |
114 } |
107 |
115 |
108 // main matcher function |
116 def matcher(r: Rexp, s: String): Boolean = { |
109 def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r)) |
117 nullable(ders(s.toList, r)) |
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118 } |
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119 |
110 |
120 |
111 // (5) Complete the size function for regular |
121 // (5) Complete the size function for regular |
112 // expressions according to the specification |
122 // expressions according to the specification |
113 // given in the coursework. |
123 // given in the coursework. |
114 |
124 |
115 |
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116 def size(r: Rexp): Int = r match { |
125 def size(r: Rexp): Int = r match { |
117 case ZERO => 1 |
126 case ZERO => 1 |
118 case ONE => 1 |
127 case ONE => 1 |
119 case CHAR(_) => 1 |
128 case CHAR(_) => 1 |
120 case ALT(r1, r2) => 1 + size(r1) + size (r2) |
129 case SEQ(r1, r2) => 1 + size(r1) + size(r2) |
121 case SEQ(r1, r2) => 1 + size(r1) + size (r2) |
130 case ALT(r1, r2) => 1 + size(r1) + size(r2) |
122 case STAR(r1) => 1 + size(r1) |
131 case STAR(r1) => 1 + size(r1) |
123 } |
132 } |
124 |
133 |
125 |
134 |
|
135 // some testing data |
126 |
136 |
127 // some testing data |
137 //matcher(("a" ~ "b") ~ "c", "abc") // => true |
128 /* |
138 //matcher(("a" ~ "b") ~ "c", "ab") // => false |
129 matcher(("a" ~ "b") ~ "c", "abc") // => true |
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130 matcher(("a" ~ "b") ~ "c", "ab") // => false |
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131 |
139 |
132 // the supposedly 'evil' regular expression (a*)* b |
140 // the supposedly 'evil' regular expression (a*)* b |
133 val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
141 val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
134 |
142 |
135 matcher(EVIL, "a" * 1000 ++ "b") // => true |
143 //matcher(EVIL, "a" * 1000 ++ "b") // => true |
136 matcher(EVIL, "a" * 1000) // => false |
144 //matcher(EVIL, "a" * 1000) // => false |
137 |
145 |
138 // size without simplifications |
146 // size without simplifications |
139 size(der('a', der('a', EVIL))) // => 28 |
147 //size(der('a', der('a', EVIL))) // => 28 |
140 size(der('a', der('a', der('a', EVIL)))) // => 58 |
148 //size(der('a', der('a', der('a', EVIL)))) // => 58 |
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149 |
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150 |
141 |
151 |
142 // size with simplification |
152 // size with simplification |
143 size(simp(der('a', der('a', EVIL)))) // => 8 |
153 //size(simp(der('a', der('a', EVIL)))) // => 8 |
144 size(simp(der('a', der('a', der('a', EVIL))))) // => 8 |
154 //size(simp(der('a', der('a', der('a', EVIL))))) // => 8 |
145 |
155 |
146 // Python needs around 30 seconds for matching 28 a's with EVIL. |
156 // Python needs around 30 seconds for matching 28 a's with EVIL. |
147 // Java 9 and later increase this to an "astonishing" 40000 a's in |
157 // Java 9 and later increase this to an "astonishing" 40000 a's in |
148 // around 30 seconds. |
158 // 30 seconds. |
149 // |
159 // |
150 // Lets see how long it takes to match strings with |
160 // Lets see how long it really takes to match strings with |
151 // 5 Million a's...it should be in the range of a |
161 // 5 Million a's...it should be in the range of a couple |
152 // couple of seconds. |
162 // of seconds. |
153 |
163 |
154 def time_needed[T](i: Int, code: => T) = { |
164 def time_needed[T](i: Int, code: => T) = { |
155 val start = System.nanoTime() |
165 val start = System.nanoTime() |
156 for (j <- 1 to i) code |
166 for (j <- 1 to i) code |
157 val end = System.nanoTime() |
167 val end = System.nanoTime() |
158 (end - start)/(i * 1.0e9) |
168 (end - start)/(i * 1.0e9) |
159 } |
169 } |
160 |
170 |
161 for (i <- 0 to 5000000 by 500000) { |
171 //for (i <- 0 to 5000000 by 500000) { |
162 println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i)))) |
172 // println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i)))) |
163 } |
173 //} |
164 |
174 |
165 // another "power" test case |
175 // another "power" test case |
166 simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next) == ONE |
176 println(simp(Iterator.iterate(ONE:Rexp)(r => ALT(r, r)).drop(40).next)) |
167 |
177 |
168 // the Iterator produces the rexp |
178 // the Iterator produces the rexp |
169 // |
179 // |
170 // SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE) |
180 // SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE) |
171 // |
181 // |
172 // where SEQ is nested 100 times. |
182 // where SEQ is nested 50 times. |
173 |
|
174 */ |
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175 |
183 |
176 //} |
184 |