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Pumping Lemma for Context-free Languages

Pumping Lemma for Context-free Languages. Take an infinite context-free language. Generates an infinite number of different strings. Example:. In a derivation of a “long” enough string, variables are repeated. A possible derivation:. Derivation Tree. Repeated variable. Derivation Tree.

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Pumping Lemma for Context-free Languages

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  1. Pumping LemmaforContext-free Languages Prof. Busch - LSU

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

  3. In a derivation of a “long” enough string, variables are repeated A possible derivation: Prof. Busch - LSU

  4. Derivation Tree Repeated variable Prof. Busch - LSU

  5. Derivation Tree Repeated variable Prof. Busch - LSU

  6. Prof. Busch - LSU

  7. Prof. Busch - LSU

  8. Prof. Busch - LSU

  9. Putting all together Prof. Busch - LSU

  10. We can remove the middle part Prof. Busch - LSU

  11. Prof. Busch - LSU

  12. We can repeated middle part two times 1 2 Prof. Busch - LSU

  13. Prof. Busch - LSU

  14. We can repeat middle part three times 1 2 3 Prof. Busch - LSU

  15. Prof. Busch - LSU

  16. Repeat middle part times 1 i Prof. Busch - LSU

  17. For any Prof. Busch - LSU

  18. From Grammar and given string We inferred that a family of strings is in for any Prof. Busch - LSU

  19. Arbitrary Grammars Consider now an arbitrary infinite context-free language Let be the grammar of Take so that it has no unit-productions and no -productions (remove them) Prof. Busch - LSU

  20. Let be the number of variables Let be the maximum right-hand size of any production Example: Prof. Busch - LSU

  21. Claim: Take string with . Then in the derivation tree of there is a path from the root to a leaf where a variable of is repeated Proof: Proof by contradiction Prof. Busch - LSU

  22. Derivation tree of We will show: some variable is repeated Prof. Busch - LSU

  23. First we show that the tree of has at least levels of nodes Suppose the opposite: At most Levels Prof. Busch - LSU

  24. Maximum number of nodes per level Level 0: nodes Level 1: nodes nodes The maximum right-hand side of any production Prof. Busch - LSU

  25. Maximum number of nodes per level Level 0: nodes Level 1: nodes Level 2: nodes nodes nodes nodes Prof. Busch - LSU

  26. Maximum number of nodes per level Level 0: nodes At most Level : nodes Levels Level : nodes Maximum possible string length = max nodes at level = Prof. Busch - LSU

  27. Therefore, maximum length of string : However we took, Contradiction!!! Therefore, the tree must have at least levels Prof. Busch - LSU

  28. Thus, there is a path from the root to a leaf with at least nodes (root) At least Variables Levels symbol (leaf) Prof. Busch - LSU

  29. Since there are at most different variables, some variable is repeated Pigeonhole principle END OF CLAIM PROOF Prof. Busch - LSU

  30. Take now a string with From claim: some variable is repeated subtree of Take to be the deepest, so that only is repeated in subtree Prof. Busch - LSU

  31. We can write yield yield yield yield yield Strings of terminals Prof. Busch - LSU

  32. Example: Prof. Busch - LSU

  33. Possible derivations Prof. Busch - LSU

  34. Example: Prof. Busch - LSU

  35. Remove Middle Part Yield: Prof. Busch - LSU

  36. Repeat Middle part two times 1 2 Yield: Prof. Busch - LSU

  37. Prof. Busch - LSU

  38. Repeat Middle part times 1 i Yield: Prof. Busch - LSU

  39. Prof. Busch - LSU

  40. Therefore, If we know that: then we also know: For all since Prof. Busch - LSU

  41. Observation 1: Since has no unit and -productions At least one of or is not Prof. Busch - LSU

  42. Observation 2: since in subtree only variable is repeated subtree of Explanation follows…. Prof. Busch - LSU

  43. subtree of Various yields since no variable is repeated in Maximum right-hand side of any production Prof. Busch - LSU

  44. Thus, if we choose critical length then, we obtain the pumping lemma for context-free languages Prof. Busch - LSU

  45. The Pumping Lemma: For any infinite context-free language there exists an integer such that for any string we can write with lengths and it must be that: Prof. Busch - LSU

  46. Applicationsof The Pumping Lemma Prof. Busch - LSU

  47. Non-context free languages Context-free languages Prof. Busch - LSU

  48. Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages Prof. Busch - LSU

  49. Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma Prof. Busch - LSU

  50. Let be the critical length of the pumping lemma Pick any string with length We pick: Prof. Busch - LSU

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