1. 1 The Pumping Lemma for Context-Free Languages

2. 2 Take an infinite context-free language Example: Generates an infinite number of different strings

3. 3 A derivation:

4. 4 Derivation tree

5. 5 Derivation tree repeated

6. 6

7. 7 Repeated part

8. 8 A possible derivation

9. 9

10. 10

11. 11

12. 12 Therefore, the string is generated by the grammar

13. 13 We know This string is also generated:

14. 14 We know This string is also generated:

15. 15 Therefore, knowing that is generated, we also know that is generated

16. 16 In general: We are given an infinite context-free grammar We take the derivation of a long enough string

17. 17 Some variable must be repeated in the derivation Take the length ofBigger than = Productions * (largest production)

18. 18 repeated

19. 19 repeated Possible derivations:

20. 20 We know: This string is also generated:

21. 21 We know: This string is also generated: (the original )

22. 22 We know: This string is also generated:

23. 23 We know: This string is also generated:

24. 24 We know: This string is also generated:

25. 25 Therefore, any string of the form Is generated by the grammar

26. 26 knowing that we also know that Therefore,

27. 27

28. 28

29. 29 The pumping lemma: there exists an integer for any string we can write For context-free language with Such that: