36 def % = STAR(s) |
37 def % = STAR(s) |
37 def ~ (r: Rexp) = SEQ(s, r) |
38 def ~ (r: Rexp) = SEQ(s, r) |
38 def ~ (r: String) = SEQ(s, r) |
39 def ~ (r: String) = SEQ(s, r) |
39 } |
40 } |
40 |
41 |
41 // (1a) Complete the function nullable according to |
42 // (1) Complete the function nullable according to |
42 // the definition given in the coursework; this |
43 // the definition given in the coursework; this |
43 // function checks whether a regular expression |
44 // function checks whether a regular expression |
44 // can match the empty string |
45 // can match the empty string and Returns a boolean |
|
46 // accordingly. |
45 |
47 |
46 def nullable (r: Rexp) : Boolean = r match { |
48 def nullable (r: Rexp) : Boolean = r match { |
47 case ZERO => false |
49 case ZERO => false |
48 case ONE => true |
50 case ONE => true |
49 case CHAR(_) => false |
51 case CHAR(_) => false |
50 case ALT(r1, r2) => nullable(r1) || nullable(r2) |
52 case ALT(r1, r2) => nullable(r1) || nullable(r2) |
51 case SEQ(r1, r2) => nullable(r1) && nullable(r2) |
53 case SEQ(r1, r2) => nullable(r1) && nullable(r2) |
52 case STAR(_) => true |
54 case STAR(_) => true |
53 } |
55 } |
54 |
56 |
55 // (1b) Complete the function der according to |
57 // (2) Complete the function der according to |
56 // the definition given in the coursework; this |
58 // the definition given in the coursework; this |
57 // function calculates the derivative of a |
59 // function calculates the derivative of a |
58 // regular expression w.r.t. a character |
60 // regular expression w.r.t. a character. |
59 |
61 |
60 def der (c: Char, r: Rexp) : Rexp = r match { |
62 def der (c: Char, r: Rexp) : Rexp = r match { |
61 case ZERO => ZERO |
63 case ZERO => ZERO |
62 case ONE => ZERO |
64 case ONE => ZERO |
63 case CHAR(d) => if (c == d) ONE else ZERO |
65 case CHAR(d) => if (c == d) ONE else ZERO |
66 if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) |
68 if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) |
67 else SEQ(der(c, r1), r2) |
69 else SEQ(der(c, r1), r2) |
68 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
70 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
69 } |
71 } |
70 |
72 |
71 // (1c) Complete the function der according to |
73 // (3) Complete the simp function according to |
72 // the specification given in the coursework; this |
74 // the specification given in the coursework; this |
73 // function simplifies a regular expression; |
75 // function simplifies a regular expression from |
74 // however it does not simplify inside STAR-regular |
76 // the inside out, like you would simplify arithmetic |
75 // expressions |
77 // expressions; however it does not simplify inside |
|
78 // STAR-regular expressions. |
76 |
79 |
77 def simp(r: Rexp) : Rexp = r match { |
80 def simp(r: Rexp) : Rexp = r match { |
78 case ALT(r1, r2) => (simp(r1), simp(r2)) match { |
81 case ALT(r1, r2) => (simp(r1), simp(r2)) match { |
79 case (ZERO, r2s) => r2s |
82 case (ZERO, r2s) => r2s |
80 case (r1s, ZERO) => r1s |
83 case (r1s, ZERO) => r1s |
88 case (r1s, r2s) => SEQ(r1s, r2s) |
91 case (r1s, r2s) => SEQ(r1s, r2s) |
89 } |
92 } |
90 case r => r |
93 case r => r |
91 } |
94 } |
92 |
95 |
93 // (1d) Complete the two functions below; the first |
96 |
|
97 // (4) Complete the two functions below; the first |
94 // calculates the derivative w.r.t. a string; the second |
98 // calculates the derivative w.r.t. a string; the second |
95 // is the regular expression matcher taking a regular |
99 // is the regular expression matcher taking a regular |
96 // expression and a string and checks whether the |
100 // expression and a string and checks whether the |
97 // string matches the regular expression |
101 // string matches the regular expression. |
98 |
102 |
99 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
103 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
100 case Nil => r |
104 case Nil => r |
101 case c::s => ders(s, simp(der(c, r))) |
105 case c::s => ders(s, simp(der(c, r))) |
102 } |
106 } |
103 |
107 |
104 // main matcher function |
108 // main matcher function |
105 def matcher(r: Rexp, s: String): Boolean = nullable(ders(s.toList, r)) |
109 def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r)) |
106 |
110 |
107 // (1e) Complete the size function for regular |
111 // (5) Complete the size function for regular |
108 // expressions according to the specification |
112 // expressions according to the specification |
109 // given in the coursework. |
113 // given in the coursework. |
|
114 |
110 |
115 |
111 def size(r: Rexp): Int = r match { |
116 def size(r: Rexp): Int = r match { |
112 case ZERO => 1 |
117 case ZERO => 1 |
113 case ONE => 1 |
118 case ONE => 1 |
114 case CHAR(_) => 1 |
119 case CHAR(_) => 1 |
118 } |
123 } |
119 |
124 |
120 |
125 |
121 |
126 |
122 // some testing data |
127 // some testing data |
123 /* |
128 |
124 matcher(("a" ~ "b") ~ "c", "abc") // => true |
129 //matcher(("a" ~ "b") ~ "c", "abc") // => true |
125 matcher(("a" ~ "b") ~ "c", "ab") // => false |
130 //matcher(("a" ~ "b") ~ "c", "ab") // => false |
126 |
131 |
127 // the supposedly 'evil' regular expression (a*)* b |
132 // the supposedly 'evil' regular expression (a*)* b |
128 val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
133 val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
129 |
134 |
130 matcher(EVIL, "a" * 1000 ++ "b") // => true |
135 //matcher(EVIL, "a" * 1000 ++ "b") // => true |
131 matcher(EVIL, "a" * 1000) // => false |
136 //matcher(EVIL, "a" * 1000) // => false |
132 |
137 |
133 // size without simplifications |
138 // size without simplifications |
134 size(der('a', der('a', EVIL))) // => 28 |
139 //size(der('a', der('a', EVIL))) // => 28 |
135 size(der('a', der('a', der('a', EVIL)))) // => 58 |
140 //size(der('a', der('a', der('a', EVIL)))) // => 58 |
136 |
141 |
137 // size with simplification |
142 // size with simplification |
138 size(simp(der('a', der('a', EVIL)))) // => 8 |
143 //size(simp(der('a', der('a', EVIL)))) // => 8 |
139 size(simp(der('a', der('a', der('a', EVIL))))) // => 8 |
144 //size(simp(der('a', der('a', der('a', EVIL))))) // => 8 |
140 |
145 |
141 // Java needs around 30 seconds for matching 28 a's with EVIL. |
146 // 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 |
|
148 // around 30 seconds. |
142 // |
149 // |
143 // Lets see how long it takes to match strings with |
150 // Lets see how long it takes to match strings with |
144 // 0.5 Million a's...it should be in the range of some |
151 // 5 Million a's...it should be in the range of a |
145 // seconds. |
152 // couple of seconds. |
146 |
153 |
147 def time_needed[T](i: Int, code: => T) = { |
154 def time_needed[T](i: Int, code: => T) = { |
148 val start = System.nanoTime() |
155 val start = System.nanoTime() |
149 for (j <- 1 to i) code |
156 for (j <- 1 to i) code |
150 val end = System.nanoTime() |
157 val end = System.nanoTime() |
151 (end - start)/(i * 1.0e9) |
158 (end - start)/(i * 1.0e9) |
152 } |
159 } |
153 |
160 |
154 for (i <- 0 to 5000000 by 500000) { |
161 //for (i <- 0 to 5000000 by 500000) { |
155 println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i)))) |
162 // println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i))) + " secs.") |
156 } |
163 //} |
157 */ |
|
158 |
164 |
159 } |
165 // another "power" test case |
|
166 //simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next) == ONE |
|
167 |
|
168 // the Iterator produces the rexp |
|
169 // |
|
170 // SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE) |
|
171 // |
|
172 // where SEQ is nested 100 times. |
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173 |
|
174 |
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175 |
|
176 //} |