Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.
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Transcript of Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.
![Page 1: Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.](https://reader036.fdocuments.in/reader036/viewer/2022062802/56649e9e5503460f94ba0460/html5/thumbnails/1.jpg)
Athasit Surarerks
2110711THEORY OF COMPUTATION
08 KLEENE’S THEOREM
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IMPORTANT2
All three methods of defining languages regular expression, acceptance by finite automata and acceptance by transition
graph are equivalent
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KLEENE’S THEOREM3
Language accepted by a deterministic finite automaton can be defined by a transition graph.
Language accepted by a transition graph can be defined by a regular expression.
Language generated by a regular expression can be defined by a deterministic finite automaton.
![Page 4: Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.](https://reader036.fdocuments.in/reader036/viewer/2022062802/56649e9e5503460f94ba0460/html5/thumbnails/4.jpg)
KLEENE’S THEOREM4
Language accepted by a deterministic finite automaton can be defined by a transition graph.
Language accepted by a transition graph can be defined by a regular expression.
Language generated by a regular expression can be defined by a deterministic finite automaton.
![Page 5: Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.](https://reader036.fdocuments.in/reader036/viewer/2022062802/56649e9e5503460f94ba0460/html5/thumbnails/5.jpg)
KLEENE’S THEOREM5
Proof:Language accepted by a transition graph can be defined by a regular expression.
Given a transition graph TG.Find a regular expression that defines the same language.Construct an algorithm that satisfies two criteria.Work for every conceivable transition graphGuarantee to finish its job in a finite time.
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KLEENE’S THEOREM6
STATEGYOnly one initial state.
A
C
D
B
E
…
0
01
110
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KLEENE’S THEOREM7
STATEGYOnly one initial state.
A
C
D
B
E
…
0
01
110S
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KLEENE’S THEOREM8
STATEGYOnly one final state.
A
C
D
B
E
…
0
01
110
010
![Page 9: Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.](https://reader036.fdocuments.in/reader036/viewer/2022062802/56649e9e5503460f94ba0460/html5/thumbnails/9.jpg)
KLEENE’S THEOREM9
STATEGYOnly one final state.
A
C
D
B
E
…
0
01
110
010
F
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KLEENE’S THEOREM1
0
STATEGYEliminate all internal states.
Let A be any state in a transition graph.r, s and t be any substrings.
A
t
r s
… … A
R+S+T
… …
R, S, T be regular expressions.
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KLEENE’S THEOREM1
1
STATEGYEliminate all internal states.
Let A,B be states in a transition graph.r, s, t and u be any substrings.
B
t
r
s
……
A
u
B
r
S+T……A
u
S, T be regular expressions.
![Page 12: Athasit Surarerks 2110711 THEORY OF COMPUTATION 08 KLEENE’S THEOREM.](https://reader036.fdocuments.in/reader036/viewer/2022062802/56649e9e5503460f94ba0460/html5/thumbnails/12.jpg)
KLEENE’S THEOREM1
2
STATEGYEliminate all internal states.
Let A,B be states in a transition graph.r, s, t and u be any substrings.
r
… AB
t
…C
S, T be regular expressions.
s
A B…r
…ST
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KLEENE’S THEOREM1
3
STATEGYEliminate all internal states.
Let A,B be states in a transition graph.r, s, t and u be any substrings.
r
… AB
t
…C
S, T, U be regular expressions.
s
A B…r
…SU*T
u
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KLEENE’S THEOREM1
4
EXAMPLE
A
C
DB
E
…0
01
110010
00
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KLEENE’S THEOREM1
5
EXAMPLE
A
C
DB
E
…0
01
110010
0(010)*00
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KLEENE’S THEOREM1
6
EXAMPLE
A
C
DB
E
…0
0(010)*01
110010
0(010)*00
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KLEENE’S THEOREM1
7
EXAMPLE
A
C
DB
E
…0
0(010)*01
0(010)*110010
0(010)*00
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KLEENE’S THEOREM1
8
EXAMPLE
A
C
D
E
…0(010)*01
0(010)*110
0(010)*00
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KLEENE’S THEOREM1
9
Eliminate B, by create AC = r1r2*r3
create AD = r1r2*r4
delete ABerase B when B has no input.
A
C
DB
…r1
r4
r3
r2
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KLEENE’S THEOREM2
0
Determine the regular expression corresponding to this transition graph.
A CB
r1
r2
r3
r4
r5
r6
r7r8
r9
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KLEENE’S THEOREM2
1
Proof:Language generated by a regular expression can be defined by a deterministic finite automaton.
Given a regular expression.Find an automaton that defines the same language.Construct an algorithm that satisfies two criteria.Accepts any particular letter of the alphabet. (or )Close under +, concatenation and Kleene’s star.
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CONCLUSION2
2
Let FA1 accepts the language defined by regular expression r1,
FA2 accepts the language defined by regular expression r2.
Then there is a FA3 accepts the language (r1+r2).
Then there is a FA4 accepts the language r1r2.
Then there is a FA5 accepts the language r*.