Finite Automata and Regular Languages

Transcript

1. F I N I T E & R E G U L A R A U T O M A T A L A N G U A G E S @tomstuart / London JS / 2014-01-27
2. deterministic finite automata nondeterministic finite automata regular languages regular expressions
3. DETERMINISTIC FINITE AUTOMATA
4. a little imaginary machine with several possible states
5. 1 2 3
6. always has a current state
7. 1 2 3 1
8. changes state when it reads input
9. , 1 2 3 1 a b b a a b
10. 1 2 3 a b b , a a b a b bb aa 1
11. 2 3 a b b , a a b a b bb aa 2 1
12. 1 2 3 a b b , a a b a b bb aa 2
13. 1 3 a b b , a a b a b bb aa 3 2
14. 1 2 3 a b b , a a b a b bb aa 3
15. 1 2 3 a b b , a a b a b bb aa 3
16. 1 3 a b b , a a b a b bb aa 3 2
17. some states are "accept states"
18. , 1 2 3 1 a b b a a b
19. 1 2 3 a b b , a a b a b bb aa 1
20. 2 3 a b b , a a b a b bb aa 2 1
21. 1 2 3 a b b , a a b a b bb aa 2
22. 1 3 a b b , a a b a b bb aa 3 2
23. 1 2 3 a b b , a a b a b bb aa 3
24. 1 2 3 a b b , a a b a b bb aa 3
25. 1 3 a b b , a a b a b bb aa 3 2
26. "accepts" or "rejects" a string
27. 1 3 a b b , a a b a b bb aa 3 2 ✔
28. , 1 2 3 1 a b b a a b
29. a a 1 2 3 a b b , a a b b b bb 1
30. a a 1 2 3 1 a b , a a b b b bb b
31. a a 1 2 3 1 a b , a a b b b bb b
32. a a 1 2 3 1 a b , a a b b b bb b
33. a a 1 2 3 a b b , a a b b b bb 2
34. a a 1 2 3 a b b , a a b b b bb 2
35. a a 1 2 3 a b b , a a b b b bb 2
36. a a 1 2 3 a b b , a a b b b bb ✘ 2
37. bbbbaa baabba ✔ ✘ ab baba aaaab a baaaa b bbbbbb abbbb bbbbab bba ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘
38. 1 2 3 a b b , a a b 1
39. 1 2 3 a b b , a a b 2
40. 1 2 3 a b b , a a b 3
41. 1 2 3 a b b , a a b accepts strings containing "ab"
42. NONDETERMINISTIC FINITE AUTOMATA
43. accept strings whose third character is "b"
44. 1 3 4 5 2 a,b a,b a b a,b a,b
45. accept strings whose third-from-last character is "b"
46. 1 3 4 2 b a,b a,b a,b
47. accept strings whose length is a multiple of two or three
48. 1 2 a a
49. 1 2 a a 3 a
50. 1 2 a a 4 3 a a a
51. 4 5 6 a a a 1 2 3 a a
52. nondeterministic finite automata can choose between different rules and change state without reading input
53. but there's an algorithm to convert any nondeterministic finite automaton into a deterministic one
54. 1 3 4 2 b a,b a,b a,b
55. 1 1,2,4 1,3,4 1,3 b a a 1,2 1,4 1,2,3,4 1,2,3 b a b a b b a b a b a b a
56. REGULAR LANGUAGES
57. 1 2 3 a b b , a a b
58. bbbbaa baabba ✔ ✘ ab baba aaaab a baaaa b bbbbbb abbbb bbbbab bba ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘
59. baabba ab baba aaaab abbbb bbbbab this is part of the language that the machine recognises
60. some languages are finite, others are infinite
61. these machines can only recognise regular languages
62. some regular languages: • the strings "one", "two" and "three" • all strings that contain "ab" • all strings whose third character is "b" • all strings whose third-from-last character is "b" • all strings that start and end with "a"
63. some non-regular languages: • all strings that are palindromes • all strings that contain equal numbers of "a" and "b" characters • all strings of brackets that are balanced • all strings that are valid JSON • all strings that contain equal numbers of "a", "b" and "c" characters • all strings whose length is prime • all strings that contain a halting JavaScript program
64. ( () ✔ ✘ (()) (()(())) ()()()()() (( (() ) (((())))) ((((())))) (()((()))) (()))() ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘
65. 0 2 3 1 ( ) ( ) ( )
66. REGULAR EXPRESSIONS
67. /dog/ /cat/ /dogs?/ /dog|cat/ /do*g/
68. /a/
69. a
70. /ab/
71. a b followed by
72. a b
73. a b
74. a b b a
75. a b b a
76. a b b a
77. a b b a
78. a b b a
79. a b b a ✔
80. followed by
83. /abc/
84. c followed by a b
85. c a b
86. a b c
87. /a|b/
88. a b or
89. a b
90. a b a
91. a b a
92. a b a
93. a b a
94. a b a ✔
95. a b
96. a b b
97. a b b
98. a b b
99. a b b
100. a b b ✔
101. or
103. /a*/
104. a zero or more times
105. a
106. zero or more times
108. /(a(b|c))*/
109. /(a(b|c))*/
110. b c or
111. b c
112. /(a(b|c))*/
113. a followed by c b
114. a c b
115. /(a(b|c))*/
116. zero or more times a c b
117. a c b
118. b a c b a a ac b
119. b a c b a a ac b
120. b a c b a a ac b
121. b a c b a a ac b
122. b a c b a a ac b
123. b a c b a a ac b
124. b a c b a a ac b
125. b a c b a a ac b
126. b a c b a a ac b
127. b a c b a a ac b
128. b a c b a a ac b
129. b a c b a a ac b
130. b a c b a a ac b
131. b a c b a a ac b
132. b a c b a a ac b
133. b a c b a a ac b
134. b a c b a a ac b
135. b a c b a a ac b ✔
136. /a+/ = /aa*/ /a?/ = /a|/
137. every regular expression can be turned into an nondeterministic finite automaton
138. every nondeterministic finite automaton can be turned into a regular expression
139. regular expressions are a compact notation for nondeterministic finite automata
140. so regular expressions can only recognise regular languages
143. balanced = / \A (?<brackets> \( \g<brackets>* \) ) * \z /x
144. thank ! @tomstuart / tom@codon.com / computationbook.com