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
   

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2. deterministic finite automata
   nondeterministic finite automata
   regular languages
   regular expressions
   

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3. DETERMINISTIC
   FINITE AUTOMATA
   

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4. a little imaginary machine
   with several possible states
   

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5. 1 2 3
   

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6. always has a current state
   

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7. 1 2 3
   1
   

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8. changes state when it reads input
   

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9. ,
   1 2 3
   1 a b
   b a a b
   

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10. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    1
    

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11. 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    2
    1
    

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12. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    2
    

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13. 1 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    2
    

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14. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    

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15. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    

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16. 1 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    2
    

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17. some states are "accept states"
    

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18. ,
    1 2 3
    1 a b
    b a a b
    

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19. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    1
    

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20. 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    2
    1
    

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21. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    2
    

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22. 1 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    2
    

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23. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    

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24. 1 2 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    

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25. 1 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    2
    

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26. "accepts" or "rejects" a string
    

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27. 1 3
    a b
    b ,
    a a b
    a
    b bb
    aa
    3
    2
    ✔
    

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28. ,
    1 2 3
    1 a b
    b a a b
    

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29. a
    a
    1 2 3
    a b
    b ,
    a a b
    b b
    bb
    1
    

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30. a
    a
    1 2 3
    1 a b
    ,
    a a b
    b b
    bb
    b
    

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31. a
    a
    1 2 3
    1 a b
    ,
    a a b
    b b
    bb
    b
    

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32. a
    a
    1 2 3
    1 a b
    ,
    a a b
    b b
    bb
    b
    

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33. a
    a
    1 2 3
    a b
    b ,
    a a b
    b b
    bb
    2
    

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34. a
    a
    1 2 3
    a b
    b ,
    a a b
    b b
    bb
    2
    

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35. a
    a
    1 2 3
    a b
    b ,
    a a b
    b b
    bb
    2
    

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36. a
    a
    1 2 3
    a b
    b ,
    a a b
    b b
    bb ✘
    2
    

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37. bbbbaa
    baabba ✔ ✘
    ab
    baba
    aaaab
    a
    baaaa
    b
    bbbbbb
    abbbb
    bbbbab bba
    ✔
    ✔
    ✔
    ✔
    ✔
    ✘
    ✘
    ✘
    ✘
    ✘
    

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38. 1 2 3
    a b
    b ,
    a a b
    1
    

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39. 1 2 3
    a b
    b ,
    a a b
    2
    

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40. 1 2 3
    a b
    b ,
    a a b
    3
    

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41. 1 2 3
    a b
    b ,
    a a b
    accepts strings containing "ab"
    

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42. NONDETERMINISTIC
    FINITE AUTOMATA
    

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43. accept strings whose third character is "b"
    

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44. 1 3
    4
    5
    2
    a,b a,b
    a
    b
    a,b
    a,b
    

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45. accept strings whose third-from-last
    character is "b"
    

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46. 1 3 4
    2
    b a,b a,b
    a,b
    

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47. accept strings whose length is
    a multiple of two or three
    

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48. 1 2
    a
    a
    

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49. 1 2
    a
    a
    3
    a
    

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50. 1
    2
    a
    a
    4
    3
    a
    a
    a
    

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51. 4
    5
    6
    a
    a
    a
    1
    2 3
    a
    a
    

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52. nondeterministic finite automata
    can choose between different rules
    and change state without reading input
    

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53. but there's an algorithm to convert
    any nondeterministic finite automaton
    into a deterministic one
    

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54. 1 3 4
    2
    b a,b a,b
    a,b
    

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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
    

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56. REGULAR
    LANGUAGES
    

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57. 1 2 3
    a b
    b ,
    a a b
    

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58. bbbbaa
    baabba ✔ ✘
    ab
    baba
    aaaab
    a
    baaaa
    b
    bbbbbb
    abbbb
    bbbbab bba
    ✔
    ✔
    ✔
    ✔
    ✔
    ✘
    ✘
    ✘
    ✘
    ✘
    

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59. baabba
    ab
    baba
    aaaab
    abbbb
    bbbbab
    this is part of the
    language that the
    machine recognises
    

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60. some languages are finite,
    others are infinite
    

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61. these machines
    can only recognise
    regular languages
    

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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"
    

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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
    

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64. (
    () ✔ ✘
    (())
    (()(()))
    ()()()()()
    ((
    (()
    )
    (((()))))
    ((((()))))
    (()((()))) (()))()
    ✔
    ✔
    ✔
    ✔
    ✔
    ✘
    ✘
    ✘
    ✘
    ✘
    

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65. 0 2 3
    1
    (
    )
    (
    )
    (
    )
    

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66. REGULAR
    EXPRESSIONS
    

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67. /dog/
    /cat/
    /dogs?/
    /dog|cat/
    /do*g/
    

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68. /a/
    

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69. a
    

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70. /ab/
    

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71. a
    b
    followed by
    

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72. a
    b
    

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73. a b
    

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74. a b
    b
    a
    

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75. a b
    b
    a
    

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76. a b
    b
    a
    

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77. a b
    b
    a
    

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78. a b
    b
    a
    

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79. a b
    b
    a ✔
    

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80. followed by
    

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83. /abc/
    

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84. c
    followed by
    a b
    

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85. c
    a b
    

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86. a b c
    

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87. /a|b/
    

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88. a
    b
    or
    

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89. a
    b
    

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90. a
    b
    a
    

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91. a
    b
    a
    

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92. a
    b
    a
    

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93. a
    b
    a
    

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94. a
    b
    a ✔
    

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95. a
    b
    

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96. a
    b
    b
    

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97. a
    b
    b
    

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98. a
    b
    b
    

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99. a
    b
    b
    

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100. a
     b
     b ✔
     

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101. or
     

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103. /a*/
     

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104. a
     zero or more times
     

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105. a
     

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106. zero or more times
     

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108. /(a(b|c))*/
     

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109. /(a(b|c))*/
     

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110. b
     c
     or
     

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111. b
     c
     

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112. /(a(b|c))*/
     

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113. a
     followed by
     c
     b
     

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114. a
     c
     b
     

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115. /(a(b|c))*/
     

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116. zero or more times
     a
     c
     b
     

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117. a
     c
     b
     

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118. b
     a
     c
     b
     a a ac
     b
     

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119. b
     a
     c
     b
     a a ac
     b
     

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120. b
     a
     c
     b
     a a ac
     b
     

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121. b
     a
     c
     b
     a a ac
     b
     

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122. b
     a
     c
     b
     a a ac
     b
     

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123. b
     a
     c
     b
     a a ac
     b
     

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124. b
     a
     c
     b
     a a ac
     b
     

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125. b
     a
     c
     b
     a a ac
     b
     

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126. b
     a
     c
     b
     a a ac
     b
     

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127. b
     a
     c
     b
     a a ac
     b
     

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128. b
     a
     c
     b
     a a ac
     b
     

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129. b
     a
     c
     b
     a a ac
     b
     

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130. b
     a
     c
     b
     a a ac
     b
     

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131. b
     a
     c
     b
     a a ac
     b
     

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132. b
     a
     c
     b
     a a ac
     b
     

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133. b
     a
     c
     b
     a a ac
     b
     

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134. b
     a
     c
     b
     a a ac
     b
     

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135. b
     a
     c
     b
     a a ac
     b ✔
     

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136. /a+/ = /aa*/
     /a?/ = /a|/
     

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137. every regular expression can be turned
     into an nondeterministic finite automaton
     

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138. every nondeterministic finite automaton
     can be turned into a regular expression
     

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139. regular expressions are a compact notation
     for nondeterministic finite automata
     

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140. so regular expressions can only
     recognise regular languages
     

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143. balanced =
     /
     \A
     (?
     \(
     \g*
     \)
     )
     *
     \z
     /x
     

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144. thank !
     @tomstuart / tom@codon.com / computationbook.com
     

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