Active Coevolutionary Learning of Deterministic Finite Automata
Non-Deterministic Automata Regular Expressions
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1 Non-Deterministic Automata Regular Expressions
description
Non-Deterministic Automata Regular Expressions. NFA Example. NFA Example. Input aplhabet, i.e. Formal Definition of NFAs. Set of states, i.e. Transition function. Initial state. Final states. Transition Function. Extended Transition Function. Formally. It holds. if and only if. - PowerPoint PPT Presentation
Transcript of Non-Deterministic Automata Regular Expressions
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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
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,,,*
,,*
,*
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
