1. 1 Closure Properties of Regular Languages L 1 and L 2 are regular. How about L 1  L 2, L 1  L 2, L 1 L 2, L 1, L 1 * ?

2. 2 Theorem 1 If L 1 and L 2 are regular, then so are L 1  L 2, L 1  L 2, L 1 L 2, L 1, L 1 *. (The family of regular languages is closed under intersection, union, concatenation, complement, and star-closure.)

3. 3 Proof L 1 = L(r 1 ) L 2 = L(r 2 ) L(r 1 + r 2 ) = L(r 1 )  L(r 2 ) L(r 1. r 2 ) = L(r 1 )L(r 2 ) L(r 1 * ) = (L(r 1 )) *

4. 4 Proof M = (Q, , , q 0, F) accepts L 1. M = (Q, , , q 0, Q – F) accepts L 1.

5. 5 Proof M 1 = (Q, ,  1, q 0, F 1 ) accepts L 1. M 2 = (P, ,  2, p 0, F 2 ) accepts L 2. q0q0 qfqf a1a1 anan p0p0 pfpf a1a1 anan  2 (p j, a) = p l  1 (q i, a) = q k  1 ((q i, p j ), a) = (q k, p l )

6. 6 Proof Example: L 1 = {ab n | n  0} L 2 = {a n b | n  0} L 1  L 2 = {ab}

7. 7 Theorem 2 The family of regular languages is closed under reversal: If L is regular, then so is L R.

8. 8 Proof ?

9. 9 Homomorphism Suppose  and  are alphabets. h:    * is called a homomorphism

10. 10 Homomorphism Extended definition: w = a 1 a 2... a n h(w) = h(a 1 )h(a 2 )... h(a n )

11. 11 Homomorphism If L is a language on , then its homomorphic image is defined as: h(L) = {h(w): w  L}

12. 12 Example  = {a, b}  = {a, b, c} h(a) = abh(b) = bbc h(aba) = abbbcab L = {aa, aba}  h(L) = {abab, abbbcab}

13. 13 Homomorphism If L is the language on  of a regular expression r, then the regular expression for h(L) is obtained by applying the homomorphism to each  symbol of r.

14. 14 Example  = {a, b}  = {b, c, d} h(a) = dbcch(b) = bdc r = (a + b * )(aa) * L = L(r) h(r) = (dbcc + (bdc) * )(dbccdbcc) *

15. 15 Theorem 3 The family of regular languages is closed under homomorphism: If L is regular, then so is h(L).

16. 16 Proof Let L(r) = L for some regular expression r. Prove: h(L(r)) = L(h(r)).

17. 17 Homework Exercises: 2, 4, 6, 8, 9, 11, 18, 22 of Section 4.1 - Linz’s book. Presentation: Section 4.3 - Linz’s book (pigeonhole principle & pumping lemma and examples).

18. 18 Right Quotient Let L 1 and L 2 be languages on the same alphabet. Then the right quotient of L 1 with L 2 is defined as: L 1 /L 2 = {x | xy  L 1 and y  L 2 }

19. 19 Example L 1 = {a n b m | n  1, m  0}  {ba} L 2 = {b m | m  1} L 1 /L 2 = ?

20. 20 Example L 1 = {a n b m | n  1, m  0}  {ba} L 2 = {b m | m  1} L 1 /L 2 = {a n b m | n  1, m  0}

21. 21 Example q0q0 q1q1 q2q2 b b b a q3q3 a a, b a q4q4 q5q5 a L 1 = {a n b m | n  1, m  0}  {ba}

22. 22 Example q0q0 q1q1 q2q2 b b b a q3q3 a a, b a q4q4 q5q5 a L 1 = {a n b m | n  1, m  0}  {ba}L 2 = {b m | m  1}  * (q 0, x) = q i  * (q i, y)  F and y  L 2

23. 23 Example q0q0 q1q1 q2q2 b b b a q3q3 a a, b a q4q4 q5q5 a L 1 = {a n b m | n  1, m  0}  {ba}L 2 = {b m | m  1}  * (q 0, x) = q i  * (q i, y)  F and y  L 2

24. 24 Theorem The family of regular languages is closed under right quotient: If L 1 and L 2 are regular, then so is L 1 /L 2.

25. 25 Proof M = (Q, , , q 0, F) accepts L 1. M ^ = (P, , , q 0, F ^ ) accepts L 1 /L 2. If y  L 2 and  * (q i, y)  F  add q i to F ^

26. 26 Proof M = (Q, , , q 0, F) accepts L 1. M ^ = (P, , , q 0, F ^ ) accepts L 1 /L 2. If y  L 2 and  * (q i, y)  F  add q i to F ^ M i = (P, , , q i, F) and L(M i )  L 2  .

27. 27 Example L 1 = L(a * baa * ) L 2 = L(ab * ) L 1 /L 2 = ?

28. 28 Example q0q0 q1q1 q2q2 a a b b b a, b q3q3 a L 1 = L(a * baa * ) L(M 0 )  L 2 =  L(M 1 )  L 2 = {a} L(M 2 )  L 2 = {a} L(M 3 )  L 2 =  L 1 = L(ab * )

29. 29 Example q0q0 q1q1 q2q2 a a b b b a, b q3q3 a L 1 = L(a * baa * ) L(M 0 )  L 2 =  L(M 1 )  L 2 = {a} L(M 2 )  L 2 = {a} L(M 3 )  L 2 =  L 1 = L(ab * )

30. 30 Questions about RLs 1.Given a regular language L on  and any w   *, is there an algorithm to determine whether or not w  L?.

31. 31 Questions about RLs 1.Given a regular language L on  and any w   *, is there an algorithm to determine whether or not w  L?. Yes.

32. 32 Questions about RLs 1.Is there an algorithm to determine whether or not a regular language is empty, finite, or infinite?.

33. 33 Questions about RLs 1.Is there an algorithm to determine whether or not a regular language is empty, finite, or infinite?. Yes.

34. 34 Questions about RLs 1.Given two regular languages L 1 and L 2, is there an algorithm to determine whether or not L 1 = L 2 ?.

35. 35 Questions about RLs 1.Given two regular languages L 1 and L 2, is there an algorithm to determine whether or not L 1 = L 2 ?. Yes.

36. 36 Homework Exercises: 1, 2, 3, 5, 9 of Section 4.2 - Linz’s book. Exercises: 3, 4, 5, 6, 8, 10, 12 of Section 4.3 - Linz’s book.