1 non-deterministic automata regular expressions
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1
Non-Deterministic Automata
Regular Expressions
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NFA Example
0q 1q 2q
0
11,0
*10
...,101010,1010,10,
L
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NFA Example
0q 1q 2qa b
3q
ab
abababababL ...,,,
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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
Final states
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Transition Function
0q 1q 2q
0
11,0
200
0
10
,,
0,
1,
qqq
q
1,
,0,
2
201
q
qqq
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Extended Transition Function
*
0q
5q4q
3q2q1qa
aa
b
0320
540
10
,,,*
,,*
,*
qqqabq
qqaaq
qaq
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Formally
wqq ij ,*
It holds
if and only if
there is a walk from towith label
iq jqw
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The Language of an NFA
0q
5q4q
3q2q1qa
aa
b
0320
540
,,,*
,,*
qqqabq
qqaaq
M
MLab
MLaa
)(
50 ,qqF
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0q
5q4q
3q2q1qa
aa
b
50 ,qqF
10
540
,*
,,*
qabaq
qqabaaq
MLaba
MLaaba
)(
10
0q
5q4q
3q2q1qa
aa
b
aaababaaML *
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FormallyThe languageaccepted by NFA is:
where
and there is some
)(MLM
,...,, 321 wwwML
,...,),(* 0 ji qqwq
Fqk (final state)
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0q kq
w
w
w
),(* 0 wq MLw
Fqk
iq
jq
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Equivalence of NFAs and DFAs
For DFAs or NFAs:
Machine is equivalent to
if
1M 2M
21 MLML
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Example
0q 1q 2q
0
11,0
0q 1q 2q
0
11
0
1,0
NFA
DFA
*}10{1 ML
*}10{2 ML
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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
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Equivalence of NFAs and DFAs
Every DFA is also an NFA
A language accepted by a DFA will be accepted by an NFA
An NFA is as least as powerful as a DFA
Is an DFA as powerful as an NFA?
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Is a DFA as Powerful as an NFA?
Answer: YES!
A language accepted by an NFA will be accepted by some DFA
For every NFA there is an equivalent DFAthat accepts the same language
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NFAs Accept the Regular Languages
For every NFA there is an equivalent DFA
The language accepted by a DFA is regular
The language accepted by an NFA is regular
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
a
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
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NFA to DFA
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
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NFA to DFA Observations
We are given an NFA
We want to convert it to an equivalent DFA
With
M
M
)(MLML
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If the NFA has states
The DFA has states in the powerset
,...,, 210 qqq
,....,,,,,,, 7432110 qqqqqqq
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Procedure NFA to DFA
1. Initial state of NFA:
Initial state of DFA:
0q
0q
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Example
a
b
a
0q 1q 2q
NFA
DFA
0q
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Procedure NFA to DFA 2. For every DFA’s state
Compute in the NFA
Giving union
Add a transition
mji qqq ,...,,
...,,*
,,*
aq
aq
j
i
mji qqq ,...,,
mjimji qqqaqqq ,...,,,,...,,
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Exampe
a
b
a
0q 1q 2q
NFA
0q 21,qqa
DFA
},{),(* 210 qqaq
210 ,, qqaq
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Procedure NFA to DFA
Repeat Step 2 for all letters, untilno more transitions can be added.
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Example
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
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Procedure NFA to DFA 3. For any DFA state
If some is a final state for the NFA
Then, is a final state for the DFA
mji qqq ,...,,
jq
mji qqq ,...,,
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Example
a
b
a
0q 1q 2q
NFA
DFA
0q 21,qqa
b
ab
ba,
Fq 1
Fqq 21,
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Theorem Take NFA M
Apply procedure to obtain DFA M
Then and are equivalent :M M
MLML