Download presentation

Presentation is loading. Please wait.

1
Fall 2006Costas Busch - RPI1 More Applications of the Pumping Lemma

2
Fall 2006Costas Busch - RPI2 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:

3
Fall 2006Costas Busch - RPI3 Regular languages Non-regular languages

4
Fall 2006Costas Busch - RPI4 Theorem: The language is not regular Proof: Use the Pumping Lemma

5
Fall 2006Costas Busch - RPI5 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

6
Fall 2006Costas Busch - RPI6 We pick Let be the critical length for Pick a string such that: lengthand

7
Fall 2006Costas Busch - RPI7 we can write: with lengths: From the Pumping Lemma: Thus:

8
Fall 2006Costas Busch - RPI8 From the Pumping Lemma: Thus:

9
Fall 2006Costas Busch - RPI9 From the Pumping Lemma: Thus:

10
Fall 2006Costas Busch - RPI10 BUT: CONTRADICTION!!!

11
Fall 2006Costas Busch - RPI11 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF

12
Fall 2006Costas Busch - RPI12 Regular languages Non-regular languages

13
Fall 2006Costas Busch - RPI13 Theorem: The language is not regular Proof: Use the Pumping Lemma

14
Fall 2006Costas Busch - RPI14 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

15
Fall 2006Costas Busch - RPI15 We pick Let be the critical length of Pick a string such that: length and

16
Fall 2006Costas Busch - RPI16 We can write With lengths From the Pumping Lemma: Thus:

17
Fall 2006Costas Busch - RPI17 From the Pumping Lemma: Thus:

18
Fall 2006Costas Busch - RPI18 From the Pumping Lemma: Thus:

19
Fall 2006Costas Busch - RPI19 BUT: CONTRADICTION!!!

20
Fall 2006Costas Busch - RPI20 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF

21
Fall 2006Costas Busch - RPI21 Regular languages Non-regular languages

22
Fall 2006Costas Busch - RPI22 Theorem: The language is not regular Proof: Use the Pumping Lemma

23
Fall 2006Costas Busch - RPI23 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

24
Fall 2006Costas Busch - RPI24 We pick Let be the critical length of Pick a string such that: length

25
Fall 2006Costas Busch - RPI25 We can write With lengths From the Pumping Lemma: Thus:

26
Fall 2006Costas Busch - RPI26 From the Pumping Lemma: Thus:

27
Fall 2006Costas Busch - RPI27 From the Pumping Lemma: Thus:

28
Fall 2006Costas Busch - RPI28 Since: There must exist such that:

29
Fall 2006Costas Busch - RPI29 However:for for any

30
Fall 2006Costas Busch - RPI30 BUT: CONTRADICTION!!!

31
Fall 2006Costas Busch - RPI31 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google