39 def % = STAR(s) |
41 def % = STAR(s) |
40 def ~ (r: Rexp) = SEQ(s, r) |
42 def ~ (r: Rexp) = SEQ(s, r) |
41 def ~ (r: String) = SEQ(s, r) |
43 def ~ (r: String) = SEQ(s, r) |
42 } |
44 } |
43 |
45 |
44 // (1) Complete the function nullable according to |
46 // (1) |
45 // the definition given in the coursework; this |
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46 // function checks whether a regular expression |
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47 // can match the empty string and Returns a boolean |
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48 // accordingly. |
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49 |
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50 def nullable (r: Rexp) : Boolean = r match { |
47 def nullable (r: Rexp) : Boolean = r match { |
51 case ZERO => false |
48 case ZERO => false |
52 case ONE => true |
49 case ONE => true |
53 case CHAR(_) => false |
50 case CHAR(_) => false |
54 case ALTs(rs) => rs.exists(nullable) |
51 case ALTs(rs) => rs.exists(nullable) |
55 case SEQ(r1, r2) => nullable(r1) && nullable(r2) |
52 case SEQs(rs) => rs.forall(nullable) |
56 case STAR(_) => true |
53 case STAR(_) => true |
57 } |
54 } |
58 |
55 |
59 // (2) Complete the function der according to |
56 // (2) |
60 // the definition given in the coursework; this |
57 def der(c: Char, r: Rexp) : Rexp = r match { |
61 // function calculates the derivative of a |
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62 // regular expression w.r.t. a character. |
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63 |
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64 def der (c: Char, r: Rexp) : Rexp = r match { |
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65 case ZERO => ZERO |
58 case ZERO => ZERO |
66 case ONE => ZERO |
59 case ONE => ZERO |
67 case CHAR(d) => if (c == d) ONE else ZERO |
60 case CHAR(d) => if (c == d) ONE else ZERO |
68 case ALTs(rs) => ALTs(rs.map(der(c, _))) |
61 case ALTs(rs) => ALTs(rs.map(der(c, _))) |
69 case SEQ(r1, r2) => |
62 case SEQs(Nil) => ZERO |
70 if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2)) |
63 case SEQs(r1::rs) => |
71 else SEQ(der(c, r1), r2) |
64 if (nullable(r1)) ALT(SEQs(der(c, r1)::rs), der(c, SEQs(rs))) |
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65 else SEQs(der(c, r1):: rs) |
72 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
66 case STAR(r1) => SEQ(der(c, r1), STAR(r1)) |
73 } |
67 } |
74 |
68 |
75 |
69 |
76 // (3) Implement the flatten function flts. It |
70 // (3) |
77 // deletes 0s from a list of regular expressions |
71 def denest(rs: List[Rexp]) : List[Rexp] = rs match { |
78 // and also 'spills out', or flattens, nested |
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79 // ALTernativeS. |
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80 |
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81 def flts(rs: List[Rexp]) : List[Rexp] = rs match { |
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82 case Nil => Nil |
72 case Nil => Nil |
83 case ZERO::tl => flts(tl) |
73 case ZERO::tl => denest(tl) |
84 case ALTs(rs1)::rs2 => rs1 ::: flts(rs2) |
74 case ALTs(rs1)::rs2 => rs1 ::: denest(rs2) |
85 case r::rs => r :: flts(rs) |
75 case r::rs => r :: denest(rs) |
86 } |
76 } |
87 |
77 |
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78 // (4) |
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79 def flts(rs: List[Rexp], acc: List[Rexp] = Nil) : List[Rexp] = rs match { |
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80 case Nil => acc |
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81 case ZERO::rs => ZERO::Nil |
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82 case ONE::rs => flts(rs, acc) |
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83 case SEQs(rs1)::rs => flts(rs, acc ::: rs1) |
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84 case r::rs => flts(rs, acc :+ r) |
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85 } |
88 |
86 |
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87 // (5) |
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88 def ALTs_smart(rs: List[Rexp]) : Rexp = rs match { |
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89 case Nil => ZERO |
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90 case r::Nil => r |
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91 case rs => ALTs(rs) |
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92 } |
89 |
93 |
90 // (4) Complete the simp function according to |
94 def SEQs_smart(rs: List[Rexp]) : Rexp = rs match { |
91 // the specification given in the coursework; this |
95 case Nil => ONE |
92 // function simplifies a regular expression from |
96 case ZERO::nil => ZERO |
93 // the inside out, like you would simplify arithmetic |
97 case r::Nil => r |
94 // expressions; however it does not simplify inside |
98 case rs => SEQs(rs) |
95 // STAR-regular expressions. |
99 } |
96 |
100 |
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101 // (6) |
97 |
102 |
98 def simp(r: Rexp) : Rexp = r match { |
103 def simp(r: Rexp) : Rexp = r match { |
99 case ALTs(rs) => (flts(rs.map(simp)).distinct) match { |
104 case ALTs(rs) => |
100 case Nil => ZERO |
105 ALTs_smart(denest(rs.map(simp)).distinct) |
101 case r::Nil => r |
106 case SEQs(rs) => |
102 case rs => ALTs(rs) |
107 SEQs_smart(flts(rs.map(simp))) |
103 } |
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104 case SEQ(r1, r2) => (simp(r1), simp(r2)) match { |
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105 case (ZERO, _) => ZERO |
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106 case (_, ZERO) => ZERO |
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107 case (ONE, r2s) => r2s |
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108 case (r1s, ONE) => r1s |
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109 case (r1s, r2s) => SEQ(r1s, r2s) |
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110 } |
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111 case r => r |
108 case r => r |
112 } |
109 } |
113 |
110 |
114 simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE)) |
111 //println("Simp tests") |
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112 //println(simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE))) |
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113 //println(simp(((CHAR('a') | ZERO) ~ ONE) | |
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114 // (((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO)))) |
115 |
115 |
116 // (5) Complete the two functions below; the first |
116 |
117 // calculates the derivative w.r.t. a string; the second |
117 // (7) |
118 // is the regular expression matcher taking a regular |
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119 // expression and a string and checks whether the |
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120 // string matches the regular expression. |
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121 |
118 |
122 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
119 def ders (s: List[Char], r: Rexp) : Rexp = s match { |
123 case Nil => r |
120 case Nil => r |
124 case c::s => ders(s, simp(der(c, r))) |
121 case c::s => ders(s, simp(der(c, r))) |
125 } |
122 } |
126 |
123 |
127 // main matcher function |
124 // main matcher function |
128 def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r)) |
125 def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r)) |
129 |
126 |
130 // (6) Complete the size function for regular |
127 // (8) |
131 // expressions according to the specification |
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132 // given in the coursework. |
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133 |
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134 |
128 |
135 def size(r: Rexp): Int = r match { |
129 def size(r: Rexp): Int = r match { |
136 case ZERO => 1 |
130 case ZERO => 1 |
137 case ONE => 1 |
131 case ONE => 1 |
138 case CHAR(_) => 1 |
132 case CHAR(_) => 1 |
139 case ALTs(rs) => 1 + rs.map(size).sum |
133 case ALTs(rs) => 1 + rs.map(size).sum |
140 case SEQ(r1, r2) => 1 + size(r1) + size (r2) |
134 case SEQs(rs) => 1 + rs.map(size).sum |
141 case STAR(r1) => 1 + size(r1) |
135 case STAR(r1) => 1 + size(r1) |
142 } |
136 } |
143 |
137 |
144 |
138 |
145 |
139 |
146 // some testing data |
140 // some testing data |
147 |
141 /* |
148 //matcher(("a" ~ "b") ~ "c", "abc") // => true |
142 println(matcher(("a" ~ "b") ~ "c", "abc")) // => true |
149 //matcher(("a" ~ "b") ~ "c", "ab") // => false |
143 println(matcher(("a" ~ "b") ~ "c", "ab")) // => false |
150 |
144 |
151 // the supposedly 'evil' regular expression (a*)* b |
145 // the supposedly 'evil' regular expression (a*)* b |
152 // val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
146 val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b')) |
153 |
147 |
154 //println(matcher(EVIL, "a" * 1000 ++ "b")) // => true |
148 println(matcher(EVIL, "a" * 1000 ++ "b")) // => true |
155 //println(matcher(EVIL, "a" * 1000)) // => false |
149 println(matcher(EVIL, "a" * 1000)) // => false |
156 |
150 |
157 // size without simplifications |
151 // size without simplifications |
158 //println(size(der('a', der('a', EVIL)))) // => 28 |
152 println(size(der('a', der('a', EVIL)))) // => 36 |
159 //println(size(der('a', der('a', der('a', EVIL))))) // => 58 |
153 println(size(der('a', der('a', der('a', EVIL))))) // => 83 |
160 |
154 |
161 // size with simplification |
155 // size with simplification |
162 //println(simp(der('a', der('a', EVIL)))) |
156 println(simp(der('a', der('a', EVIL)))) |
163 //println(simp(der('a', der('a', der('a', EVIL))))) |
157 println(simp(der('a', der('a', der('a', EVIL))))) |
164 |
158 |
165 //println(size(simp(der('a', der('a', EVIL))))) // => 8 |
159 println(size(simp(der('a', der('a', EVIL))))) // => 7 |
166 //println(size(simp(der('a', der('a', der('a', EVIL)))))) // => 8 |
160 println(size(simp(der('a', der('a', der('a', EVIL)))))) // => 7 |
167 |
161 |
168 // Python needs around 30 seconds for matching 28 a's with EVIL. |
162 // Python needs around 30 seconds for matching 28 a's with EVIL. |
169 // Java 9 and later increase this to an "astonishing" 40000 a's in |
163 // Java 9 and later increase this to an "astonishing" 40000 a's in |
170 // around 30 seconds. |
164 // around 30 seconds. |
171 // |
165 // |
172 // Lets see how long it takes to match strings with |
166 // Lets see how long it takes to match strings with |
173 // 5 Million a's...it should be in the range of a |
167 // 5 Million a's...it should be in the range of a |
174 // couple of seconds. |
168 // few seconds. |
175 |
169 |
176 def time_needed[T](i: Int, code: => T) = { |
170 def time_needed[T](i: Int, code: => T) = { |
177 val start = System.nanoTime() |
171 val start = System.nanoTime() |
178 for (j <- 1 to i) code |
172 for (j <- 1 to i) code |
179 val end = System.nanoTime() |
173 val end = System.nanoTime() |
180 "%.5f".format((end - start)/(i * 1.0e9)) |
174 "%.5f".format((end - start)/(i * 1.0e9)) |
181 } |
175 } |
182 |
176 |
183 //for (i <- 0 to 5000000 by 500000) { |
177 for (i <- 0 to 5000000 by 500000) { |
184 // println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.") |
178 println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.") |
185 //} |
179 } |
186 |
180 |
187 // another "power" test case |
181 // another "power" test case |
188 //simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next) == ONE |
182 println(simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next()) == ONE) |
189 |
183 |
190 // the Iterator produces the rexp |
184 // the Iterator produces the rexp |
191 // |
185 // |
192 // SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE) |
186 // SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE) |
193 // |
187 // |
194 // where SEQ is nested 50 times. |
188 // where SEQ is nested 100 times. |
195 |
189 */ |
196 |
190 |
197 |
191 |
198 } |
192 } |
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193 |