1 The Pumping Lemma for Context-Free Languages

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

3 A derivation:

4 Derivation tree

5 Derivation tree repeated

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7 Repeated part

8 A possible derivation

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12 Therefore, the string is generated by the grammar

13 We know This string is also generated:

14 We know This string is also generated:

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

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

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

18 repeated

19 repeated Possible derivations:

20 We know: This string is also generated:

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

22 We know: This string is also generated:

23 We know: This string is also generated:

24 We know: This string is also generated:

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

26 knowing that we also know that Therefore,

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