Post on 21-Jan-2016
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transcript
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Non-Deterministic Finite Automata
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1q 2q
3q
a
a
a
0q
}{aAlphabet =
Nondeterministic Finite Automaton (NFA)
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1q 2q
3q
a
a
a
0q
Two choices
}{aAlphabet =
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No transition
1q 2q
3q
a
a
a
0q
Two choices No transition
}{aAlphabet =
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a a
0q
1q 2q
3q
a
a
First Choice
a
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a a
0q
1q 2q
3q
a
a
a
First Choice
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a a
0q
1q 2q
3q
a
a
First Choice
a
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a a
0q
1q 2q
3q
a
a
a “accept”
First Choice
All input is consumed
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a a
0q
1q 2q
3q
a
a
Second Choice
a
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a a
0q
1q 2qa
a
Second Choice
a
3q
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a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:the automaton hangs
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a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
Input cannot be consumed
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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:
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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
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a
0q
1q 2q
3q
a
a
Rejection example
a
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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
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Second Choice
a
0q
1q 2qa
a
a
3q
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Second Choice
a
0q
1q 2qa
a
a
3q “reject”
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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:
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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
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a a
0q
1q 2q
3q
a
a
First Choice
a
a
No transition:the automaton hangs
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a a
0q
1q 2q
3q
a
a
a “reject”
First Choice
a
Input cannot be consumed
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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
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a a
0q
1q 2qa
a
a
3q
Second Choice
No transition:the automaton hangs
a
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a a
0q
1q 2qa
a
a
3q
Second Choice
“reject”
a
Input cannot be consumed
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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
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1q 2q
3q
a
a
a
0q
Language accepted: }{aaL
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Lambda Transitions
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
(read head does not move)
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a a
1q 3qa0q
2q a
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a a
1q 3qa0q
2q a
“accept”
String is acceptedaa
all input is consumed
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a a
1q 3qa0q
2q a
Rejection Example
a
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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
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a a
1q 3qa0q
2q a
“reject”
String is rejectedaaa
a
Input cannot be consumed
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Language accepted: }{aaL
1q 3qa0q
2q a
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Another NFA Example
0q 1q 2qa b
3q
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a b
0q 1q 2qa b
3q
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0q 2qa b
3q
a b
1q
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a b
0q 1qa b
3q2q
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a b
0q 1qa b
3q2q
“accept”
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0qa b
a b
Another String
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
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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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a b a b
0qa b
1q 2q 3q
“accept”
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ab
ababababababL ...,,,
Language accepted
0q 1q 2qa b
3q
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Another NFA Example
0q 1q 2q0
11,0
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{ }{ }*10=
...,101010,1010,10,λ=)(ML
0q 1q 2q0
11,0
Language accepted
(redundant state)
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Remarks:
•The symbol never appears on the input tape
0q2M
0q1M
{}=)M(L 1 }λ{=)M(L 2
•Simple automata:
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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
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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
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10 1, qq
0
11,0
Transition Function
0q 1q 2q
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0q
0
11,0
},{)0,( 201 qqq
1q 2q
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0q
0
11,0
1q 2q
},{),( 200 qqq
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0q
0
11,0
1q 2q
)1,( 2q
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Extended Transition Function
*
0q
5q4q
3q2q1qa
aa
b
10 ,* qaq
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540 ,,* qqaaq
0q
5q4q
3q2q1qa
aa
b
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0320 ,,,* qqqabq
0q
5q4q
3q2q1qa
aa
b
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Formally
wqq ij ,* : there is a walk from to with label
iq jqw
iq jq
w
kw 211 2 k
iq jq
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The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
540 ,,* qqaaq
M
)(MLaa
50 ,qqF
F
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0q
5q4q
3q2q1qa
aa
b
0320 ,,,* qqqabq MLab
50 ,qqF
F
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0q
5q4q
3q2q1qa
aa
b
50 ,qqF
540 ,,* qqabaaq )(MLaabaF
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0q
5q4q
3q2q1qa
aa
b
50 ,qqF
10 ,* qabaq MLabaF
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0q
5q4q
3q2q1qa
aa
b
}{* aaabML
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FormallyThe language accepted by NFA is:
where
and there is some
M
,...,, 321 wwwML
},,...,,{),(* 0 kjim qqqwq
Fqk (accepting state)
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0q kq
w
w
w
),(* 0 wq MLw
Fqk
iq
jq