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
is context-free

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

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

9. Decidable Properties of Context-Free Languages

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

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

12. Infinite Language Question:
for context-free grammar
find if is infinite
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

13. Example:
Infinite language
Dependency graph

15. The Pumping Lemma for Context-Free Languages

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

17. Variables are repeated
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. Applications of The Pumping Lemma

47. Non-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:

51. We can write:
with lengths and

52. Pumping Lemma says:
for all

53. We examine all the possible locations
of string in

54. Case 1:
is within

55. Case 1:
and consist from only

56. Case 1:
Repeating and

57. Case 1:
From Pumping Lemma:

58. Case 1:
From Pumping Lemma:
However:
Contradiction!!!

59. Case 2:
is within

60. Case 2:
Similar analysis with case 1

61. Case 3:
is within

62. Case 3:
Similar analysis with case 1

63. Case 4:
overlaps and

64. Case 4:
Possibility 1:
contains only
contains only

65. Case 4:
Possibility 1:
contains only
contains only

66. Case 4:
From Pumping Lemma:

67. Case 4:
From Pumping Lemma:
However:
Contradiction!!!

68. Case 4:
Possibility 2:
contains and
contains only

69. Case 4:
Possibility 2:
contains and
contains only

70. Case 4:
From Pumping Lemma:

71. Case 4:
From Pumping Lemma:
However:
Contradiction!!!

72. Case 4:
Possibility 3:
contains only
contains and

73. Case 4: Possibility 3: contains only contains and
Similar analysis with Possibility 2

74. Case 5:
overlaps and

75. Case 5:
Similar analysis with case 4

76. There are no other cases to consider
(since , string cannot
overlap , and at the same time)

77. In all cases we obtained a contradiction
Therefore:
The original assumption that
is context-free must be wrong
Conclusion:
is not context-free