03a nondeterministic finite accepters
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Non Deterministic Automata
2
{a}• Alphabet =
Nondeterministic Finite Accepter (NFA)
0q
1q
3q
2qa
a
a
3
{a}• Alphabet =
Nondeterministic Finite Accepter (NFA)
0q
1q
3q
2qa
a
a
Two choices
4
{a}• Alphabet =
Nondeterministic Finite Accepter (NFA)
0q
1q
3q
2qa
a
a
Two choices
No transition
No transition
5
Nondeterministic Finite Accepter (NFA)
0q
1q
3q
2qa
a
a
Input Stringa a
6
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a
7
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a
8
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a
All input is consumed
“accept"
9
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a
10
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a
11
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a
No further transition:the automaton hangs
12
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a
The input cannot be consumed
“reject"
13
NFA: Acceptance• 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 a final state.
14
Example
0q
1q
3q
2qa
a
a
is accepted by the NFA
“accept"
aa
0q
1q
3q
2qa
a
a“reject"
because this computation accepts
15
Rejection Example
0q
1q
3q
2qa
a
a
Input Stringa
16
First Choice
0q
1q
3q
2qa
a
a
Input Stringa
“reject"
17
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa
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Second Choice
0q
1q
3q
2qa
a
a
Input Stringa
“reject"
19
NFA: RejectionAn 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
20
Example
0q
1q
3q
2qa
a
a
is rejected by the NFA
“reject"
a
0q
1q
3q
2qa
a
a“reject"
All possible computations leads to rejection
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Rejection Example
0q
1q
3q
2qa
a
a
Input Stringa a a
22
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a a
23
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a a
No transition; NFA hangs
24
First Choice
0q
1q
3q
2qa
a
a
Input Stringa a a
Input cannot be consumed
“reject"
25
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a a
No transition; NFA hangs
26
Second Choice
0q
1q
3q
2qa
a
a
Input Stringa a a
Input cannot be consumed
“reject"
27
Example
0q
1q
3q
2qa
a
a
is rejected by the NFA
“reject"
aaa
0q
1q
3q
2qa
a
a“reject"
All possible computations leads to rejection
28
Language
0q
1q
3q
2qa
a
a
• Language accepted by the NFA: L {aa}
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Lambda Transitions
0q 1q 2q 3qa a
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Lambda Transitions
0q 1q 2q 3qa a
Input Stringa a
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Lambda Transitions
0q 1q 2q 3qa a
Input Stringa a
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Lambda Transitions
0q 1q 2q 3qa a
Input Stringa a
Read head does not move
33
Lambda Transitions
0q 1q 2q 3qa a
Input Stringa a
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Lambda Transitions
0q 1q 2q 3qa a
Input Stringa a
String is acceptedaa
“accept”
35
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
36
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
37
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
Read head does not move
38
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
39
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
No transition; NFA hangs
40
Rejection Example
0q 1q 2q 3qa a
Input Stringa a a
String is rejectedaaa
“reject”
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Language
0q 1q 2q 3qa a
Language accepted: L {aa}
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Another NFA Example
0q 1q 2q 3qa b
Input Stringa b
43
Another NFA Example
0q 1q 2q 3qa b
Input Stringa b
44
Another NFA Example
0q 1q 2q 3qa b
Input Stringa b
45
Another NFA Example
0q 1q 2q 3qa b
Input Stringa b
“accept”
46
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
47
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
48
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
49
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
50
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
51
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
52
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
53
Another NFA Example
0q 1q 2q 3qa b
Another Input Stringa b a b
“accept”
54
Language Accepted
0q 1q 2q 3qa b
L ab, abab, ababab, ... ab
55
Another NFA Example
0q 1q 2q10,1
0
L(M) , 10, 1010, 101010, ... 10 *
Language accepted
56
Remarks
0q1M
1L(M )
• The symbol never appears on the input tape
• Extreme automata:
0q2M
2L(M )
57
Remarks
0q
1M
1L(M ) a
0q
2M
2L(M ) a
a0q 0q
a
• NFAs are interesting because we can express languages easier than DFAs.
aNFA DFA
58
Formal Definition of NFAs
0M Q, , , q , F
Q :
:
0q :
F :
Set of states, i.e. 0 1 2q , q , q
: Input aplhabet, i.e. a, b
Transition function
Initial state
Final states
59
0 1q ,1 q
Transition Function
0q 1q 2q10,1
0
60
1 0 2q , 0 q ,q
Transition Function
0q 1q 2q10,1
0
61
1 0 2q , q ,q
Transition Function
0q 1q 2q10,1
0
62
2q ,1
Transition Function
0q 1q 2q10,1
0
63
Extended Transition Function
0 1q ,a q
0q 1q 2q 3qa b
4q 5q
a a
64
Extended Transition Function
0 4 5q ,aa q ,q
0q 1q 2q 3qa b
4q 5q
a a
65
Extended Transition Function
0 0 2 3q ,ab q ,q ,q
0q 1q 2q 3qa b
4q 5q
a a
66
Formally
j iq q ,w
It holds
if and only if
there is a walk from towith label
iq jqw
67
The Language of an NFA
0 4 5q ,aa q ,q
M
aa L(M)
0 5F q ,q
0q 1q 2q 3qa b
4q 5q
a a
68
The Language of an NFA
0 2 3 0q ,ab q ,q ,q
M
ab L(M)
0 5F q ,q
0q 1q 2q 3qa b
4q 5q
a a
69
The Language of an NFA
0 4 5q ,abaa q ,q
M
abaa L(M)
0 5F q ,q
0q 1q 2q 3qa b
4q 5q
a a
70
The Language of an NFA
0 1q ,aba q
M
aba L(M)
0 5F q ,q
0q 1q 2q 3qa b
4q 5q
a a
71
The Language of an NFA
L(M) {aa} {ab} {ab} aa
M
0q 1q 2q 3qa b
4q 5q
a a
72
FormallyThe language accepted by NFA is:
where
and there is some (final state)
M
1 2 3L M w ,w ,w ,...
0 m i j(q ,w ) {q ,q ,...}
kq F
73
0q kq
w
w
w
0(q ,w) w L M
kq F
iq
jq
Formally
74
1. Design an NFA for the language
2. Design an NFA for the language
3. Design an NFA for the language
4. Design an NFA no more than 5 states for the language
Exercises
75
5. Construct an NFA that accepts the language ∗
6. Construct an NFA with 3 states that accepts the language
∗
7. Find an NFA that accepts the language
Exercises
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