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Non-Deterministic Finite Automata
EC303, Lecture - 14
http://users.comlab.ox.ac.uk/luke.ong/teaching/moc/nfa2up.pdf
2
1q 2q
3q
a
a
a
0q
}{aAlphabet =
NondeterministicFinite Automaton (NFA)
EC303, Lecture - 14
3
1q 2q
3q
a
a
a
0q
Two choices
}{aAlphabet =
EC303, Lecture - 14
4
No transition
1q 2q
3q
a
a
a
0q
Two choicesNo transition
}{aAlphabet =
EC303, Lecture - 14
5
a a
0q
1q 2q
3q
a
a
First Choice
a
EC303, Lecture - 14
6
a a
0q
1q 2q
3q
a
a
a
First Choice
EC303, Lecture - 14
7
a a
0q
1q 2q
3q
a
a
First Choice
a
EC303, Lecture - 14
8
a a
0q
1q 2q
3q
a
a
a “accept”
First Choice
All input is consumed
EC303, Lecture - 14
9
a a
0q
1q 2q
3q
a
a
Second Choice
a
EC303, Lecture - 14
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a a
0q
1q 2qa
a
Second Choice
a
3q
EC303, Lecture - 14
11
a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:
the automaton hangs
EC303, Lecture - 14
12
a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
Input cannot be consumed
EC303, Lecture - 14
13
An NFA accepts a string:
when there is a computation of the NFA
that accepts the string
all the input is consumed and the automaton
is in an accepting state
There is a computation:
EC303, Lecture - 14
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
computation
accepts aaEC303, Lecture - 14
15
a
0q
1q 2q
3q
a
a
Rejection example
a
EC303, Lecture - 14
16
a
0q
1q 2q
3q
a
a
a
First Choice
EC303, Lecture - 14
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a
0q
1q 2q
3q
a
a
a
First Choice
“reject”
EC303, Lecture - 14
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Second Choice
a
0q
1q 2q
3q
a
a
a
EC303, Lecture - 14
19
Second Choice
a
0q
1q 2qa
a
a
3q
EC303, Lecture - 14
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Second Choice
a
0q
1q 2qa
a
a
3q “reject”
EC303, Lecture - 14
21
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
• The input cannot be consumed
OR
For each computation:
EC303, Lecture - 14
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 rejectionEC303, Lecture - 14
23
Rejection example
a a
0q
1q 2q
3q
a
a
a
a
EC303, Lecture - 14
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a a
0q
1q 2q
3q
a
a
a
First Choice
a
EC303, Lecture - 14
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a a
0q
1q 2q
3q
a
a
First Choice
a
a
No transition:
the automaton hangs
EC303, Lecture - 14
26
a a
0q
1q 2q
3q
a
a
a “reject”
First Choice
a
Input cannot be consumed
EC303, Lecture - 14
27
a a
0q
1q 2q
3q
a
a
Second Choice
a
a
EC303, Lecture - 14
28
a a
0q
1q 2qa
a
Second Choice
a
3q
a
EC303, Lecture - 14
29
a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:
the automaton hangs
a
EC303, Lecture - 14
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a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
a
Input cannot be consumed
EC303, Lecture - 14
31
aaa is rejected by the NFA:
0q
1q 2q
3q
a
a
a
“reject”
0q
1q 2qa
a
a
3q “reject”
All possible computations lead to rejectionEC303, Lecture - 14
32
1q 2q
3q
a
a
a
0q
Language accepted: }{aaL
EC303, Lecture - 14
33
Lambda Transitions
1q 3qa0q 2q a
EC303, Lecture - 14
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a a
1q 3qa0q 2q a
EC303, Lecture - 14
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a a
1q 3qa0q 2q a
EC303, Lecture - 14
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a a
1q 3qa0q 2q a
(read head does not move)
EC303, Lecture - 14
37
a a
1q 3qa0q 2q a
EC303, Lecture - 14
38
a a
1q 3qa0q 2q a
“accept”
String is acceptedaa
all input is consumed
EC303, Lecture - 14
39
a a
1q 3qa0q 2q a
Rejection Example
a
EC303, Lecture - 14
40
a a
1q 3qa0q 2q a
a
EC303, Lecture - 14
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a a
1q 3qa0q 2q a
(read head doesn’t move)
a
EC303, Lecture - 14
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a a
1q 3qa0q 2q a
a
No transition:
the automaton hangsEC303, Lecture - 14
43
a a
1q 3qa0q 2q a
“reject”
String is rejectedaaa
a
Input cannot be consumed
EC303, Lecture - 14
44
Language accepted: }{aaL
1q 3qa0q 2q a
EC303, Lecture - 14
45
Another NFA Example
0q 1q 2qa b3q
EC303, Lecture - 14
46
a b
0q 1q 2qa b3q
EC303, Lecture - 14
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0q 2qa b3q
a b
1q
EC303, Lecture - 14
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a b
0q 1qa b3q2q
EC303, Lecture - 14
49
a b
0q 1qa b3q2q
“accept”
EC303, Lecture - 14
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0q a b
a b
Another String
a b
1q 2q 3q
EC303, Lecture - 14
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0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
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0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
53
0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
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0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
55
0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
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0q a b
a b a b
1q 2q 3q
EC303, Lecture - 14
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a b a b
0q a b1q 2q 3q
“accept”
EC303, Lecture - 14
58
ab
ababababababL ...,,,
Language accepted
0q 1q 2qa b3q
EC303, Lecture - 14
59
Another NFA Example
0q 1q 2q
0
11,0
EC303, Lecture - 14
60
{ }
{ }*10=
...,101010,1010,10,λ=)(ML
0q 1q 2q
0
11,0
Language accepted
(redundant
state)
EC303, Lecture - 14
61
Remarks:
•The symbol never appears on the
input tape
0q
2M
0q
1M
{}=)M(L 1 }λ{=)M(L 2
•Simple automata:
EC303, Lecture - 14
62
0q
2q
1qa
a
a
0q 1qa
}{=)( 1 aML
2M1M
}{=)( 2 aML
NFA FA
•NFAs are interesting because we can
express languages easier than FAs
EC303, Lecture - 14
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
Accepting statesEC303, Lecture - 14
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10 1, qq
0
11,0
Transition Function
0q 1q 2q
EC303, Lecture - 14
65
0q
0
11,0
},{)0,( 201 qqq
1q 2q
EC303, Lecture - 14
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0q
0
11,0
1q 2q
},{),( 200 qqq
EC303, Lecture - 14
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0q
0
11,0
1q 2q
)1,( 2q
EC303, Lecture - 14
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Extended Transition Function *
0q
5q4q
3q2q1qa
aa
b
10 ,* qaq
EC303, Lecture - 14
69
540 ,,* qqaaq
0q
5q4q
3q2q1qa
aa
b
EC303, Lecture - 14
70
0320 ,,,* qqqabq
0q
5q4q
3q2q1qa
aa
b
EC303, Lecture - 14
71
Formally
wqq ij ,* : there is a walk from to
with label iq jq
w
iq jq
w
kw 21
1 2 k
iq jq
EC303, Lecture - 14
72
The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
540 ,,* qqaaq
M
)(MLaa
50,qqF
FEC303, Lecture - 14
73
0q
5q4q
3q2q1qa
aa
b
0320 ,,,* qqqabq MLab
50,qqF
FEC303, Lecture - 14
74
0q
5q4q
3q2q1qa
aa
b
50,qqF
540 ,,* qqabaaq )(MLaaba
FEC303, Lecture - 14
75
0q
5q4q
3q2q1qa
aa
b
50,qqF
10,* qabaq MLaba
FEC303, Lecture - 14
76
0q
5q4q
3q2q1qa
aa
b
}{* aaabML
EC303, Lecture - 14
77
Formally
The language accepted by NFA is:
where
and there is some
M
,...,, 321 wwwML
},,...,,{),(* 0 kjim qqqwq
Fqk (accepting state)
EC303, Lecture - 14
78
0q kq
w
w
w
),(* 0 wqMLw
Fqk
iq
jq
EC303, Lecture - 14
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