Embed Size (px)
DESCRIPTION
Non-Deterministic Finite Automata. Nondeterministic Finite Automaton (NFA). Alphabet =. Alphabet =. Two choices. Alphabet =. Two choices. No transition. No transition. First Choice. First Choice. First Choice. First Choice. All input is consumed. “accept”. Second Choice. - PowerPoint PPT Presentation
Citation preview
1
Non-Deterministic Finite Automata
2
1q 2q
3q
a
a
a
0q
}{aAlphabet =
Nondeterministic Finite Automaton (NFA)
3
1q 2q
3q
a
a
a
0q
Two choices
}{aAlphabet =
4
No transition
1q 2q
3q
a
a
a
0q
Two choices No transition
}{aAlphabet =
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 automatonis in an accepting state
There is a computation:
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
17
a
0q
1q 2q
3q
a
a
a
First Choice
“reject”
18
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
OR
For each computation:
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
23
Rejection example
a a
0q
1q 2q
3q
a
a
a
a
24
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
28
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
34
a a
1q 3qa0q
2q a
35
a a
1q 3qa0q
2q a
36
a a
1q 3qa0q
2q a
(read head does not move)
37
a a
1q 3qa0q
2q a
38
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
41
a a
1q 3qa0q
2q a
(read head doesn’t move)
a
42
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
53
0qa b
a b a b
1q 2q 3q
54
0qa b
a b a b
1q 2q 3q
55
0qa b
a b a b
1q 2q 3q
56
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
(redundant state)
61
Remarks:
•The symbol never appears on the input tape
0q2M
0q1M
{}=)M(L 1 }λ{=)M(L 2
•Simple automata:
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
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 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 ,* : there is a walk from to with label
iq jqw
iq jq
w
kw 211 2 k
iq jq
72
The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
540 ,,* qqaaq
M
)(MLaa
50 ,qqF
F
73
0q
5q4q
3q2q1qa
aa
b
0320 ,,,* qqqabq MLab
50 ,qqF
F
74
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
540 ,,* qqabaaq )(MLaabaF
75
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
10 ,* qabaq MLabaF
76
0q
5q4q
3q2q1qa
aa
b
}{* aaabML
77
FormallyThe language accepted by NFA is:
where
and there is some
M
,...,, 321 wwwML
},,...,,{),(* 0 kjim qqqwq
Fqk (accepting state)
78
0q kq
w
w
w
),(* 0 wq MLw
Fqk
iq
jq
![Reduksi DFA [Deterministic Finite Automata] · PDF file[Deterministic Finite Automata] ... FSA dengan jumlah state yang lebih sedikit merupakan FSA yang paling efisien](https://img.dokumen.tips/doc/110x75/5a9650b37f8b9a30358cf790/reduksi-dfa-deterministic-finite-automata-deterministic-finite-automata-fsa.jpg)
