Applications of Regular Closure

1. Applications of Regular Closure

2. The intersection of a context-free language and a regular language is a context-free language Regular Closure context free regular context-free

3. An Application of Regular Closure Prove that: is context-free

4. We know: is context-free

5. We also know: is regular is regular

6. context-free regular (regular closure) is context-free context-free

7. Another Application of Regular Closure Prove that: is not context-free

8. Therefore, is not context free is context-free If (regular closure) Then context-free regular context-free Impossible!!!

9. Decidable Propertiesof Context-Free Languages

10. Parsers Membership Algorithms: • Exhaustive search parser • CYK parsing algorithm Membership Question: for context-free grammar find if string

11. Algorithm: • Remove useless variables • Check if start variable is useless Empty Language Question: for context-free grammar find if

12. Infinite Language Question: Algorithm: 1. Remove useless variables 2. Remove unit and productions 3. Create dependency graph for variables 4. If there is a loop in the dependency graph then the language is infinite for context-free grammar find if is infinite

13. Example: Infinite language Dependency graph

15. The Pumping LemmaforContext-Free Languages

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

17. A derivation:

18. Derivation tree string

19. Derivation tree string repeated

21. Repeated Part

22. Another possible derivation

26. Therefore, the string is also generated by the grammar

27. We know: We also know this string is generated:

28. We know: Therefore, this string is also generated:

29. We know: Therefore, this string is also generated:

30. We know: Therefore, this string is also generated:

31. Therefore, knowing that is generated by grammar , we also know that is generated by

32. In general: We are given an infinite context-free grammar Assume has no unit-productions no -productions

33. Take a string with length bigger than > (Number of productions) x (Largest right side of a production) Consequence: Some variable must be repeated in the derivation of

34. String Last repeated variable repeated

35. Possible derivations:

36. We know: This string is also generated:

37. We know: This string is also generated: The original

38. We know: This string is also generated:

39. We know: This string is also generated:

40. We know: This string is also generated:

41. Therefore, any string of the form is generated by the grammar

42. Therefore, knowing that we also know that

43. Observation: Since is the last repeated variable

44. Observation: Since there are no unit or productions

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

46. Applicationsof The Pumping Lemma

47. Non-context free languages Context-free languages

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

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

50. Pumping Lemma gives a magic number such that: Pick any string with length We pick: