85
1 Non Deterministic Automata
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1
Non Deterministic Automata
2
1q 2q
3q
a
a
a
0q
}{aAlphabet =
Nondeterministic Finite Accepter (NFA)
3
1q 2q
3q
a
a
a
0q
Two choices
}{aAlphabet =
Nondeterministic Finite Accepter (NFA)
4
No transition
1q 2q
3q
a
a
a
0q
Two choices No transition
}{aAlphabet =
Nondeterministic Finite Accepter (NFA)
5
a a
0q
1q 2q
3q
a
a
First Choice
a
6
a a
0q
1q 2q
3q
a
a
a
First Choice
7
a a
0q
1q 2q
3q
a
a
First Choice
a
8
a a
0q
1q 2q
3q
a
a
a “accept”
First Choice
All input is consumed
9
a a
0q
1q 2q
3q
a
a
Second Choice
a
10
a a
0q
1q 2qa
a
Second Choice
a
3q
11
a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:the automaton hangs
12
a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
Input cannot be consumed
13
An NFA accepts a string:when there is a computation of the NFAthat accepts the string
•All the input is consumed and the automaton is in a final state
14
Example
aa is accepted by the NFA:
0q
1q 2q
3q
a
a
a
“accept”
0q
1q 2qa
a
a
3q “reject”
because this computationaccepts aa
15
a
0q
1q 2q
3q
a
a
Rejection example
a
16
a
0q
1q 2q
3q
a
a
a
First Choice
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a
0q
1q 2q
3q
a
a
a
First Choice
“reject”
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Second Choice
a
0q
1q 2q
3q
a
a
a
19
Second Choice
a
0q
1q 2qa
a
a
3q
20
Second Choice
a
0q
1q 2qa
a
a
3q “reject”
21
An NFA rejects a string:when there is no computation of the NFAthat accepts the string
• All the input is consumed and the automaton is in a non final state
• The input cannot be consumed
22
Example
a is rejected by the NFA:
0q
1q 2qa
a
a
3q “reject”
0q
1q 2qa
a
a
3q
“reject”
All possible computations lead to rejection
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Rejection example
a a
0q
1q 2q
3q
a
a
a
a
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a a
0q
1q 2q
3q
a
a
a
First Choice
a
25
a a
0q
1q 2q
3q
a
a
First Choice
a
a
No transition:the automaton hangs
26
a a
0q
1q 2q
3q
a
a
a “reject”
First Choice
a
Input cannot be consumed
27
a a
0q
1q 2q
3q
a
a
Second Choice
a
a
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a a
0q
1q 2qa
a
Second Choice
a
3q
a
29
a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:the automaton hangs
a
30
a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
a
Input cannot be consumed
31
aaais rejected by the NFA:
0q
1q 2q
3q
a
a
a
“reject”
0q
1q 2qa
a
a
3q “reject”
All possible computations lead to rejection
32
1q 2q
3q
a
a
a
0q
Language accepted: }{aaL
33
Lambda Transitions
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
35
a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
(read head doesn’t move)
37
a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
“accept”
String is acceptedaa
all input is consumed
39
a a
1q 3qa0q
2q a
Rejection Example
a
40
a a
1q 3qa0q
2q a
a
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a a
1q 3qa0q
2q a
(read head doesn’t move)
a
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a a
1q 3qa0q
2q a
a
No transition:the automaton hangs
43
a a
1q 3qa0q
2q a
“reject”
String is rejectedaaa
a
Input cannot be consumed
44
Language accepted: }{aaL
1q 3qa0q
2q a
45
Another NFA Example
0q 1q 2qa b
3q
46
a b
0q 1q 2qa b
3q
47
0q 2qa b
3q
a b
1q
48
a b
0q 1qa b
3q2q
49
a b
0q 1qa b
3q2q
“accept”
50
0qa b
a b
Another String
a b
1q 2q 3q
51
0qa b
a b a b
1q 2q 3q
52
0qa b
a b a b
1q 2q 3q
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0qa b
a b a b
1q 2q 3q
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0qa b
a b a b
1q 2q 3q
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0qa b
a b a b
1q 2q 3q
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0qa b
a b a b
1q 2q 3q
57
a b a b
0qa b
1q 2q 3q
“accept”
58
ab
ababababababL ...,,,
Language accepted
0q 1q 2qa b
3q
59
Another NFA Example
0q 1q 2q0
11,0
60
{ }{ }*10=
...,101010,1010,10,λ=)(ML
0q 1q 2q0
11,0
Language accepted
61
Remarks:
•The symbol never appears on the input tape
0q2M
0q1M
{}=)M(L 1 }λ{=)M(L 2
•Extreme automata:
62
0q
2q
1qa
a a
a
0q 1qa
}{=)( 1 aML
2M1M
}{=)( 2 aML
NFA DFA
•NFAs are interesting because we can express languages easier than DFAs
63
Formal Definition of NFAs FqQM ,,,, 0
:Q
::0q
:F
Set of states, i.e. 210 ,, qqq
: Input aplhabet, i.e. ba,Transition function
Initial state
Final states
64
10 1, qq
0
11,0
Transition Function
0q 1q 2q
65
0q
0
11,0
},{)0,( 201 qqq
1q 2q
66
0q
0
11,0
1q 2q
},{),( 200 qqq
67
0q
0
11,0
1q 2q
)1,( 2q
68
Extended Transition Function
*
0q
5q4q
3q2q1qa
aa
b
10 ,* qaq
69
540 ,,* qqaaq
0q
5q4q
3q2q1qa
aa
b
70
0320 ,,,* qqqabq
0q
5q4q
3q2q1qa
aa
b
71
Formally
wqq ij ,*It holds
if and only if
there is a walk from towith label
iq jqw
72
The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
540 ,,* qqaaq
M
)(MLaa
50 ,qqF
73
0q
5q4q
3q2q1qa
aa
b
0320 ,,,* qqqabq MLab
50 ,qqF
74
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
540 ,,* qqabaaq )(MLaaba
75
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
10 ,* qabaq MLaba
76
0q
5q4q
3q2q1qa
aa
b
aaababaaML *
77
FormallyThe language accepted by NFA is:
where
and there is some
M
,...,, 321 wwwML
,...},{),(* 0 jim qqwq
Fqk (final state)
78
0q kq
w
w
w
),(* 0 wq MLw
Fqk
iq
jq
79
Equivalence of NFAs and DFAs
80
Equivalence of Machines
For DFAs or NFAs:
Machine is equivalent to machine
if
1M 2M
21 MLML
81
Example
0q 1q 2q
0
11,0
0q 1q 2q
0
11
0
1,0
NFA
DFA
*}10{1 ML
*}10{2 ML
1M
2M
82
Since
machines and are equivalent
*1021 MLML
1M 2M
0q 1q 2q
0
11,0
0q 1q 2q
0
11
0
1,0
DFA
NFA 1M
2M
83
Equivalence of NFAs and DFAs
Question: NFAs = DFAs ?
Same power?Accept the same languages?
84
Equivalence of NFAs and DFAs
Question: NFAs = DFAs ?
Same power?Accept the same languages?
YES!
85
Languages acceptedby NFAs
Languages acceptedby DFAs
Regular LanguagesRegular Languages
Thus, NFAs accept the regular languages
