1. Non Deterministic Automata

2. Nondeterministic Finite Accepter (NFA)
Alphabet =

3. Nondeterministic Finite Accepter (NFA)
Alphabet =
Two choices

4. Nondeterministic Finite Accepter (NFA)
Alphabet =
Two choices
No transition
No transition

5. First Choice

6. First Choice

7. First Choice

8. First Choice
All input is consumed
“accept”

9. Second Choice

10. Second Choice

11. Second Choice
No transition:
the automaton hangs

12. Second Choice
Input cannot be consumed
“reject”

13. AND An NFA accepts a string: when there is a computation of the NFA
that accepts the string
AND
all the input is consumed and the automaton
is in a final state

14. Example is accepted by the NFA: “accept” “reject” because this
computation
accepts

15. Rejection example

16. First Choice

17. First Choice
“reject”

18. Second Choice

19. Second Choice

20. Second Choice
“reject”

21. OR An NFA rejects a string: when there is no computation of the NFA
that accepts the string:
All the input is consumed and the
automaton is in a non final state OR
The input cannot be consumed

22. Example is rejected by the NFA: “reject” “reject”
All possible computations lead to rejection

23. Rejection example

24. First Choice

25. First Choice
No transition:
the automaton hangs

26. First Choice
Input cannot be consumed
“reject”

27. Second Choice

28. Second Choice

29. Second Choice
No transition:
the automaton hangs

30. Second Choice
Input cannot be consumed
“reject”

31. is rejected by the NFA:
“reject”
“reject”
All possible computations lead to rejection

32. Language accepted:

33. Lambda Transitions

36. (read head does not move)

38. all input is consumed
“accept”
String is accepted

39. Rejection Example

41. (read head doesn’t move)

42. No transition:
the automaton hangs

43. Input cannot be consumed
“reject”
String is rejected

44. Language accepted:

45. Another NFA Example

49. “accept”

50. Another String

57. “accept”

58. Language accepted

59. Another NFA Example

60. Language accepted
(redundant
state)

61. Remarks:
The symbol never appears on the
input tape
Simple automata:

62. NFAs are interesting because we can
express languages easier than DFAs
NFA
DFA

63. Formal Definition of NFAs
Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Final states

64. Transition Function

68. Extended Transition Function

71. Formally
: there is a walk from to
with label

72. The Language of an NFA

77. Formally The language accepted by NFA is: where and there is some
(final state)

79. NFAs accept the Regular Languages

80. Equivalence of Machines
Definition for Automata:
Machine is equivalent to machine if

81. Example of equivalent machines
NFA
DFA

82. same computation power
We will prove:
Languages
accepted
by NFAs
Regular
Languages
Languages
accepted
by DFAs
NFAs and DFAs have the
same computation power

83. Step 1
Languages
accepted
by NFAs
Regular
Languages
Proof:
Every DFA is trivially an NFA
Any language accepted by a DFA
is also accepted by an NFA

84. Step 2
Languages
accepted
by NFAs
Regular
Languages
Proof:
Any NFA can be converted to an
equivalent DFA
Any language accepted by an NFA
is also accepted by a DFA

85. Convert NFA to DFA
NFA
DFA

86. Convert NFA to DFA
NFA
DFA

87. Convert NFA to DFA
NFA
DFA

88. Convert NFA to DFA
NFA
DFA

89. Convert NFA to DFA
NFA
DFA

90. Convert NFA to DFA
NFA
DFA

91. Convert NFA to DFA
NFA
DFA

92. NFA to DFA: Remarks We are given an NFA We want to convert it
to an equivalent DFA
With

93. If the NFA has states
the DFA has states in the powerset

94. Procedure NFA to DFA
1. Initial state of NFA:
Initial state of DFA:

95. Example
NFA
DFA

96. Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA
Add transition to DFA

97. Exampe
NFA
DFA

98. Procedure NFA to DFA Repeat Step 2 for all letters in alphabet, until
no more transitions can be added.

99. Example
NFA
DFA

100. Procedure NFA to DFA 3. For any DFA state
If some is a final state in the NFA
Then,
is a final state in the DFA

101. Example
NFA
DFA

102. Theorem Take NFA Apply procedure to obtain DFA
Then and are equivalent :

103. Proof
AND

104. First we show:
Take arbitrary:
We will prove:

106. We will show that if

107. More generally, we will show that if in :
(arbitrary string)

108. Proof by induction on
Induction Basis:

109. Induction hypothesis:

110. Induction Step:

111. Induction Step:

112. Therefore if

113. We have shown:
We also need to show:
(proof is similar)