1. THEORY OF COMPUTATION
08 KLEENE’S THEOREM

2. IMPORTANT
All three methods of defining languages regular expression, acceptance by finite automata and acceptance by transition graph are equivalent

3. KLEENE’S THEOREM
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.

4. KLEENE’S THEOREM
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.

5. KLEENE’S THEOREM 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.

6. KLEENE’S THEOREM
STATEGY Only one initial state. A B … C
110 E 01 D

7. KLEENE’S THEOREM STATEGY Only one initial state. … A B S C E D   110A B  … S  C
110 E  01 D

8. KLEENE’S THEOREM
STATEGY Only one final state.
010 A B … C
110 E 01 D

9. KLEENE’S THEOREM STATEGY Only one final state. … A B F C E D 010  110A B  … F C
110 E  01 D

10. KLEENE’S THEOREM R, S, T be regular expressions.
STATEGY Eliminate all internal states. Let A be any state in a transition graph. r, s and t be any substrings. r s
R+S+T A … A … … … t
R, S, T be regular expressions.

11. KLEENE’S THEOREM
STATEGY Eliminate all internal states. Let A,B be states in a transition graph. r, s, t and u be any substrings. r u r u s … A B
S+T … … B A … t
S, T be regular expressions.

12. KLEENE’S THEOREM C B … … A … … B A S, T be regular expressions.
STATEGY Eliminate all internal states. Let A,B be states in a transition graph. r, s, t and u be any substrings. r t C B … … A s ST … … B A r
S, T be regular expressions.

13. KLEENE’S THEOREM C B … … A … … B A S, T, U be regular expressions.
STATEGY Eliminate all internal states. Let A,B be states in a transition graph. r, s, t and u be any substrings. r t u C B … … A s
SU*T … … B A r
S, T, U be regular expressions.

14. KLEENE’S THEOREM
EXAMPLE C
010
110 A B … 01 D 00 E

15. KLEENE’S THEOREM
EXAMPLE C
010
110 A B … 01 D
0(010)*00 E

16. KLEENE’S THEOREM
EXAMPLE C
010
110 A B … D
0(010)*01
0(010)*00 E

17. KLEENE’S THEOREM … C A B D E EXAMPLE 010 0(010)*110 0(010)*01A B … D
0(010)*01
0(010)*00 E

18. KLEENE’S THEOREM
EXAMPLE C
0(010)*110 A … D
0(010)*01
0(010)*00 E

19. KLEENE’S THEOREM … C A B D
Eliminate B, by create AC = r1r2*r3 create AD = r1r2*r4 delete AB erase B when B has no input. r2 C r3 r1 A B r4 … D

20. KLEENE’S THEOREM
Determine the regular expression corresponding to this transition graph. r6 r3 r1 r5 r2 r4 A B C r8 r7 r9

21. KLEENE’S THEOREM 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.

22. CONCLUSION
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*.