# 1 More Applications of the Pumping Lemma. 2 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we.

## Presentation on theme: "1 More Applications of the Pumping Lemma. 2 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we."— Presentation transcript:

1 More Applications of the Pumping Lemma

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 Regular languages Non-regular languages

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

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

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

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

8 From the Pumping Lemma: Thus:

9 From the Pumping Lemma: Thus:

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

12 Regular languages Non-regular languages

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

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

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

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

17 From the Pumping Lemma: Thus:

18 From the Pumping Lemma: Thus:

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

21 Regular languages Non-regular languages

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

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

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

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

26 From the Pumping Lemma: Thus:

27 From the Pumping Lemma: Thus:

28 Since: There must exist such that:

29 However:for for any