1 Applications of Regular Closure

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

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

4 We know: is context-free

5 is regular We also know:

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

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

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

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: 1.Remove useless variables 2.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: Dependency graph Infinite language

14

15 The Pumping Lemma for Context-Free Languages

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

17 A derivation:

18 Derivation treestring

19 repeated Derivation treestring

20

21 Repeated Part

22 Another possible derivation

23

24

25

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 Therefore, this string is also generated: We know:

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 Some variable must be repeated in the derivation of (Number of productions) x (Largest right side of a production) > Consequence:

34 Last repeated variable String repeated

35 Possible derivations:

36 We know: This string is also generated:

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

38 This string is also generated: We know:

39 This string is also generated: We know:

40 This string is also generated: We know:

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

42 knowing that we also know that Therefore,

43 Observation: Since is the last repeated variable

44 Observation: Since there are no unit or productions

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

46 Applications of The Pumping Lemma

47 Context-free languages 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

65 Case 4: Possibility 1: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