88
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
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”
13
Observation
An NFA accepts a stringifthere is a computation of the NFAthat accepts the string
14
Example
aa is accepted by the NFA:
0q
1q 2q
3q
a
a
a
15
Lambda Transitions
1q 3qa0q
1q a
16
a a
1q 3qa0q
2q a
17
a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
(read head doesn’t move)
19
a a
1q 3qa0q
2q a
20
a a
1q 3qa0q
2q a
“accept”
String is acceptedaa
21
Language accepted: }{aaL
1q 3qa0q
2q a
22
Another NFA Example
0q 1q 2qa b
3q
23
a b
0q 1q 2qa b
3q
24
0q 2qa b
3q
a b
1q
25
a b
0q 1qa b
3q2q
26
a b
0q 1qa b
3q2q
“accept”
27
0qa b
a b
Another String
a b
1q 2q 3q
28
0qa b
a b a b
1q 2q 3q
29
0qa b
a b a b
1q 2q 3q
30
0qa b
a b a b
1q 2q 3q
31
0qa b
a b a b
1q 2q 3q
32
0qa b
a b a b
1q 2q 3q
33
0qa b
a b a b
1q 2q 3q
34
a b a b
0qa b
1q 2q 3q
“accept”
35
ab
ababababababL ...,,,
Language accepted
0q 1q 2qa b
3q
36
Another NFA Example
0q 1q 2q0
11,0
37
*10
...,101010,1010,10,
L
0q 1q 2q0
11,0
Language accepted
38
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
39
10 1, qq
0
11,0
Transition Function
0q 1q 2q
40
0q
0
11,0
},{)0,( 201 qqq
1q 2q
41
0q
0
11,0
1q 2q
},{),( 200 qqq
42
0q
0
11,0
1q 2q
)1,( 2q
43
Extended Transition Function
*
0q
5q4q
3q2q1qa
aa
b
10 ,* qaq
44
540 ,,* qqaaq
0q
5q4q
3q2q1qa
aa
b
45
0320 ,,,* qqqabq
0q
5q4q
3q2q1qa
aa
b
46
Formally
wqq ij ,*It holds
if and only if
there is a walk from towith label
iq jqw
47
The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
540 ,,* qqaaq
M
)(MLaa
50 ,qqF
48
0q
5q4q
3q2q1qa
aa
b
0320 ,,,* qqqabq MLab
50 ,qqF
49
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
540 ,,* qqabaaq )(MLaaba
50
0q
5q4q
3q2q1qa
aa
b
50 ,qqF
10 ,* qabaq MLaba
51
0q
5q4q
3q2q1qa
aa
b
aaababaaML *
52
FormallyThe language accepted by NFA is:
where
and there is some
M
,...,, 321 wwwML
,...},{),(* 0 jim qqwq
Fqk (final state)
53
0q kq
w
w
w
),(* 0 wq MLw
Fqk
iq
jq
54
Equivalence of NFAs and DFAs
55
Equivalence of Machines
For DFAs or NFAs:
Machine is equivalent to machine
if
1M 2M
21 MLML
56
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
57
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
58
Equivalence of NFAs and DFAs
Question: NFAs = DFAs ?
Same power?Accept the same languages?
59
Equivalence of NFAs and DFAs
Question: NFAs = DFAs ?
Same power?Accept the same languages?
YES!
60
We will prove:
Languages acceptedby NFAs
Languages acceptedby DFAs
61
We will prove:
Languages acceptedby NFAs
Languages acceptedby DFAs
NFAs and DFAs have the same computation power
62
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 1
63
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 1
Proof: Every DFA is also an NFA
64
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 1
Proof: Every DFA is also an NFA
A language accepted by a DFAis also accepted by an NFA
65
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 2
66
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 2
Proof: Any NFA can be converted to anequivalent DFA
67
Languages acceptedby NFAs
Languages acceptedby DFAs
Step 2
Proof: Any NFA can be converted to anequivalent DFA
A language accepted by an NFAis also accepted by a DFA
68
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q
69
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
70
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
71
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
a
72
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
73
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
74
NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
75
NFA to DFA: Remarks
We are given an NFA
We want to convert it to an equivalent DFA
With
M
M
)(MLML
76
If the NFA has states
the DFA has states in the powerset
,...,, 210 qqq
,....,,,,,,, 7432110 qqqqqqq
77
Procedure NFA to DFA
1. Initial state of NFA:
Initial state of DFA:
0q
0q
78
Example
a
b
a
0q 1q 2q
NFA
DFA
0q
79
Procedure NFA to DFA
2. For every DFA’s state
Compute in the NFA
Add transition
},...,,{ mji qqq
...
,,*
,,*
aq
aq
j
i
},...,,{ mji qqq
},...,,{},,...,,{ mjimji qqqaqqq
80
Exampe
a
b
a
0q 1q 2q
NFA
0q 21,qqa
DFA
},{),(* 210 qqaq
210 ,, qqaq
81
Procedure NFA to DFA
Repeat Step 2 for all letters in alphabet,
untilno more transitions can be added.
82
Example
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
83
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
},...,,{ mji qqq
jq
},...,,{ mji qqq
84
Example
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
Fq 1
Fqq 21,
85
Theorem Take NFA M
Apply procedure to obtain DFA M
Then and are equivalent :M M
MLML
86
Finally
We have proven
Languages acceptedby NFAs
Languages acceptedby DFAs
87
Languages acceptedby NFAs
Languages acceptedby DFAs
We have proven
Regular Languages
88
Languages acceptedby NFAs
Languages acceptedby DFAs
We have proven
Regular LanguagesRegular Languages
