1. 1 More Applications of the Pumping Lemma

2. 2 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we can write with and such that:

3. 3 Regular languages Non-regular languages

4. 4 Theorem: The language is not regular Proof: Use the Pumping Lemma

5. 5 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

6. 6 We pick Let be the integer in the Pumping Lemma Pick a string such that: length and

7. 7 Write it must be that length From the Pumping Lemma Thus:

8. 8 From the Pumping Lemma: Thus:

9. 9 From the Pumping Lemma: Thus:

10. 10 BUT: CONTRADICTION!!!

11. 11 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

12. 12 Regular languages Non-regular languages

13. 13 Theorem: The language is not regular Proof: Use the Pumping Lemma

14. 14 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

15. 15 We pick Let be the integer in the Pumping Lemma Pick a string such that: length and

16. 16 Write it must be that length From the Pumping Lemma Thus:

17. 17 From the Pumping Lemma: Thus:

18. 18 From the Pumping Lemma: Thus:

19. 19 BUT: CONTRADICTION!!!

20. 20 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

21. 21 Regular languages Non-regular languages

22. 22 Theorem: The language is not regular Proof: Use the Pumping Lemma

23. 23 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

24. 24 We pick Let be the integer in the Pumping Lemma Pick a string such that: length

25. 25 Write it must be that length From the Pumping Lemma Thus:

26. 26 From the Pumping Lemma: Thus:

27. 27 From the Pumping Lemma: Thus:

28. 28 Since: There must exist such that:

29. 29 However:for for any

30. 30 BUT: CONTRADICTION!!!

31. 31 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: