1. Non-Deterministic Finite Automata
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2. Nondeterministic Finite Automaton (NFA)
Alphabet =
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3. Alphabet =
Two choices
No transition
No transition
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4. First Choice
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5. First Choice
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6. First Choice
All input is consumed
“accept”
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7. Second Choice
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8. Second Choice Input cannot be consumed Automaton Halts “reject”
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9. if there is a computation path of the NFA that accepts the string
A NFA accepts a string:
if there is a computation path of the NFA
that accepts the string
all the symbols of input string are processed
and
the last is an accepting state
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10. is accepted by the NFA: “accept” “reject” because this computation
accepts
this computation
is ignored
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11. Rejection example
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12. First Choice
“reject”
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13. Second Choice
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14. Second Choice
“reject”
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15. Another Rejection example
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16. First Choice
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17. First Choice Input cannot be consumed “reject” Automaton halts
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18. Second Choice
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19. Second Choice Input cannot be consumed Automaton halts “reject”
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20. OR A NFA rejects a string: if there is no computation of the NFA
that accepts the string.
For each possible computation path:
All the input is consumed and the
automaton is in a non accepting state OR
The input cannot be consumed
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21. All possible computations lead to rejection
is rejected by the NFA:
“reject”
“reject”
All possible computations lead to rejection
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22. All possible computations lead to rejection
is rejected by the NFA:
“reject”
“reject”
All possible computations lead to rejection
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23. Language accepted:
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24. Epsilon Transitions
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27. input tape head does not move
Automaton changes state
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28. all input is consumed
“accept”
String is accepted
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29. Rejection Example
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31. (read head doesn’t move)
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32. Input cannot be consumed
Automaton halts
“reject”
String is rejected
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33. Language accepted:
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34. Another NFA Example
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37. “accept”
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38. Another String
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44. “accept”
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45. Language accepted
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46. Another NFA Example
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47. Language accepted
(redundant
state)
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48. Simple Automata
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49. NFAs are interesting because we can express languages easier than DFAs
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50. Formal Definition of NFAs
Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Accepting states
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51. Transition Function resulting states reached
by following one transition
with input symbol
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52. States reachable from scanning
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53. States reachable from scanning
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54. States reachable from with one transition scanning no input symbol
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55. States reachable from scanning
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56. Extended Transition Function
Similar with but applied on strings
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57. States reachable from scanning
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58. States reachable from scanning
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59. Special case:
for any state
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60. In general
: there is a walk from to
with label
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61. The Language of an NFA The language accepted by is: Where for each
and there is some
(accepting state)
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68. NFAs accept the Regular Languages
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69. Equivalence of Machines
Definition:
Machine is equivalent to machine if
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70. Example of equivalent machines
NFA
DFA
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71. Theorem: Languages accepted Regular by NFAs Languages Languages
by DFAs
NFAs and DFAs have the same computation power,
namely, they accept the same set of languages
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72. Proof: we need to show Languages accepted Regular by NFAs Languages
AND
Languages
accepted
by NFAs
Regular
Languages
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73. Every DFA is trivially a NFA
Proof-Step 1
Languages
accepted
by NFAs
Regular
Languages
Every DFA is trivially a NFA
Any language accepted by a DFA
is also accepted by a NFA
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74. Any NFA can be converted to an equivalent DFA
Proof-Step 2
Languages
accepted
by NFAs
Regular
Languages
Any NFA can be converted to an
equivalent DFA
Any language accepted by a NFA
is also accepted by a DFA
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75. Conversion of NFA to DFA
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76. NFA
DFA
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77. empty set
NFA
DFA
trap state
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78. NFA
union
DFA
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79. NFA
union
DFA
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80. NFA
DFA
trap state
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81. END OF CONSTRUCTION
NFA
DFA
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82. General Conversion Procedure
Input: an NFA
Output: an equivalent DFA
with
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83. The DFA has states from the power set
The NFA has states
The DFA has states from the power set
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84. 1. Initial state of NFA: Conversion Procedure Steps step
Initial state of DFA:
step
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85. Example
NFA
DFA
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86. step 2. For every DFA’s state compute in the NFA Union
add transition to DFA
Union
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87. Example
NFA
DFA
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88. 3. Repeat Step 2 for every state in DFA and symbols in alphabet until no more states can be added in the DFA
step
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89. Example
NFA
DFA
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90. if some is accepting state in NFA
step
4. For any DFA state
if some is accepting state in NFA
Then,
is accepting state in DFA
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91. Example
NFA
DFA
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92. then the two automata are equivalent:
Lemma:
If we convert NFA to DFA
then the two automata are equivalent:
Proof:
We need to show:
AND
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93. First we show:
We only need to prove:
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94. NFA
Consider
symbols
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95. denotes a possible sub-path like
symbol
denotes a possible sub-path like
symbol …… ……
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96. We will show that if NFA then DFA state state label label
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97. More generally, we will show that if in :
(arbitrary prefix)
NFA
then
DFA
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98. is true by construction of
Proof by induction on
Induction Basis:
NFA
DFA
is true by construction of
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99. Induction hypothesis:
Suppose that the following hold
NFA
DFA
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100. Then this is true by construction of
Induction Step:
Then this is true by construction of
NFA
DFA
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101. Therefore if
NFA
then
DFA
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102. We have shown: With a similar proof we can show: Therefore:
END OF PROOF
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