Athasit Surarerks

2110711THEORY OF COMPUTATION

08 KLEENE’S THEOREM

IMPORTANT2

All three methods of defining languages regular expression, acceptance by finite automata and acceptance by transition

graph are equivalent

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.

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.

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.

KLEENE’S THEOREM6

STATEGYOnly one initial state.

A

C

D

B

E

…

0

01

110

KLEENE’S THEOREM7

STATEGYOnly one initial state.

A

C

D

B

E

…

0

01

110S

KLEENE’S THEOREM8

STATEGYOnly one final state.

A

C

D

B

E

…

0

01

110

010

KLEENE’S THEOREM9

STATEGYOnly one final state.

A

C

D

B

E

…

0

01

110

010

F

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.

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.

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

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

KLEENE’S THEOREM1

4

EXAMPLE

A

C

DB

E

…0

01

110010

00

KLEENE’S THEOREM1

5

EXAMPLE

A

C

DB

E

…0

01

110010

0(010)*00

KLEENE’S THEOREM1

6

EXAMPLE

A

C

DB

E

…0

0(010)*01

110010

0(010)*00

KLEENE’S THEOREM1

7

EXAMPLE

A

C

DB

E

…0

0(010)*01

0(010)*110010

0(010)*00

KLEENE’S THEOREM1

8

EXAMPLE

A

C

D

E

…0(010)*01

0(010)*110

0(010)*00

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

KLEENE’S THEOREM2

0

Determine the regular expression corresponding to this transition graph.

A CB

r1

r2

r3

r4

r5

r6

r7r8

r9

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.

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*.