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  1. More Applications ofthe Pumping Lemma Prof. Busch - LSU

  2. The Pumping Lemma: • Given a infinite regular language • there exists an integer (critical length) • for any string with length • we can write • with and • such that: Prof. Busch - LSU

  3. Non-regular languages Regular languages Prof. Busch - LSU

  4. Theorem: The language is not regular Proof: Use the Pumping Lemma Prof. Busch - LSU

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

  6. Let be the critical length for Pick a string such that: and length We pick Prof. Busch - LSU

  7. From the Pumping Lemma: we can write: with lengths: Thus: Prof. Busch - LSU

  8. From the Pumping Lemma: Thus: Prof. Busch - LSU

  9. From the Pumping Lemma: Thus: Prof. Busch - LSU

  10. BUT: CONTRADICTION!!! Prof. Busch - LSU

  11. Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language END OF PROOF Prof. Busch - LSU

  12. Non-regular languages Regular languages Prof. Busch - LSU

  13. Theorem: The language is not regular Proof: Use the Pumping Lemma Prof. Busch - LSU

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

  15. Let be the critical length of Pick a string such that: and length We pick Prof. Busch - LSU

  16. From the Pumping Lemma: We can write With lengths Thus: Prof. Busch - LSU

  17. From the Pumping Lemma: Thus: Prof. Busch - LSU

  18. From the Pumping Lemma: Thus: Prof. Busch - LSU

  19. BUT: CONTRADICTION!!! Prof. Busch - LSU

  20. Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language END OF PROOF Prof. Busch - LSU

  21. Non-regular languages Regular languages Prof. Busch - LSU

  22. Theorem: The language is not regular Proof: Use the Pumping Lemma Prof. Busch - LSU

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

  24. Let be the critical length of Pick a string such that: length We pick Prof. Busch - LSU

  25. From the Pumping Lemma: We can write With lengths Thus: Prof. Busch - LSU

  26. From the Pumping Lemma: Thus: Prof. Busch - LSU

  27. From the Pumping Lemma: Thus: Prof. Busch - LSU

  28. Since: There must exist such that: Prof. Busch - LSU

  29. However: for for any Prof. Busch - LSU

  30. BUT: CONTRADICTION!!! Prof. Busch - LSU

  31. Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language END OF PROOF Prof. Busch - LSU