Athasit Surarerks
2110711
THEORY OF COMPUTATION
08 KLEENE’S THEOREM
IMPORTANT
2
All three methods of defining languages regular expression, acceptance by finite automata and acceptance by transition graph are equivalent
KLEENE’S THEOREM
3
Ø
Language accepted by a deterministic finite automaton can be defined by a transition graph.
Ø
Language accepted by a transition graph can be defined by a regular expression.
Ø
Language generated by a regular expression can be defined by a deterministic finite automaton.
KLEENE’S THEOREM
4
ü
Language accepted by a deterministic finite automaton can be defined by a transition graph.
Ø
Language accepted by a transition graph can be defined by a regular expression.
Ø
Language generated by a regular expression can be defined
by a deterministic finite automaton.
KLEENE’S THEOREM
5
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 graph
¤Guarantee to finish its job in a finite time.
KLEENE’S THEOREM
6
STATEGY
Only one initial state.
A
C
D
B
E
…
0
01 110
KLEENE’S THEOREM
7
STATEGY
Only one initial state.
A C
D
B
E
…
0
01
S 110
Λ Λ
Λ
KLEENE’S THEOREM
8
STATEGY
Only one final state.
A
C
D
B
E
…
0
01 110
010
KLEENE’S THEOREM
9
STATEGY
Only one final state.
A C
D
B
E
…
0
01 110
010
F
Λ Λ
KLEENE’S THEOREM
10
STATEGY
Eliminate 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.
KLEENE’S THEOREM
11
STATEGY
Eliminate 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.
KLEENE’S THEOREM
12
STATEGY
Eliminate all internal states.
Let A,B be states in a transition graph.
r, s, t and u be any substrings.
r
… A
t
B C …
S, T be regular expressions.
s
A B
…
r ST
…
KLEENE’S THEOREM
13
STATEGY
Eliminate all internal states.
Let A,B be states in a transition graph.
r, s, t and u be any substrings.
r
… A
t
B C …
S, T, U be regular expressions.
s
A B
…
r SU*T
…
u
KLEENE’S THEOREM
14
EXAMPLE
A
C
D B
E
…
0
01 010 110
00
KLEENE’S THEOREM
15
EXAMPLE
A
C
D B
E
…
0
01 010 110
0(010)*00
KLEENE’S THEOREM
16
EXAMPLE
A
C
D B
E
…
0
0(010)*01 010 110
0(010)*00
KLEENE’S THEOREM
17
EXAMPLE
A
C
D B
E
…
0
0(010)*01 0(010)*110 010
0(010)*00
KLEENE’S THEOREM
18
EXAMPLE
A
C
D
E
0(010)*01
…
0(010)*110
0(010)*00
KLEENE’S THEOREM
19
Eliminate B, by create AC = r1r2*r3 create AD = r1r2*r4 delete AB
erase B when B has no input.
A
C
D
B …
r1
r4 r3 r2
KLEENE’S THEOREM
20
Determine the regular expression corresponding to this transition graph.
A B C
r1
r2
r3
r4
r5 r6
r7 r8
r9
KLEENE’S THEOREM
21
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 Λ)
¤Close under +, concatenation and Kleene’s star.
CONCLUSION
22
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*.
