CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 8 Mälardalen University

1. CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 8 Mälardalen University 2010

2. ContentThe Pumping Lemma for CFLApplications of the Pumping Lemma for CFLMidterm Exam 2: Context-Free Languages

3. Pumping Lemma for CFL’s

4. Comparison to Regular Language Pumping Lemma/Condition

5. ...... ...... What’s Difference between CFL’s and Regular Languages? In regular languages, a single substring “pumps”

6. What’s Difference between CFL’s and Regular Languages? In CFL’s, multiple substrings can be “pumped” • Consider the language {anbn | n > 0} • No single substring can be pumped and allow us to stay in the language • However, there do exist pairs of substrings which can be pumped resulting in strings which stay in the language Thus, a modified pumping lemma applies.

7. A language L satisfies the RLpumping condition if: there exists an integer m > 0 such that for all strings x in L of length at least m there exist strings u, v, wsuch that x = uvwand |uv| ≤ mand |v| ≥ 1and For all i ≥ 0, uviw is in L A language L satisfies theCFL pumping condition if: there exists an integer m > 0such that for all strings x in L of length at least m there exist strings u, v, w, y, z such that x = uvwyz and |vwy| ≤ m and |vy| ≥ 1 and For all i ≥ 0, uviwyiz is in L Pumping Conditions for RL and CFL

8. CFL’s “Pumping Languages” All languages over {a,b} Pumping Lemma All CFL’s satisfy the CFL pumping condition

9. Implications CFL’s “Pumping Languages” All languages over {a,b} • We can use the pumping lemma to prove a language L is not a CFL • Show L does not satisfy the CFL pumping condition • We cannot use the pumping lemma to prove a language is CFL • Showing L satisfies the pumping condition does not guarantee that L is context-free

10. Pumping Lemma What does it mean?

11. Pumping Condition • A language L satisfies the CFL pumping condition if: • there exists an integer m > 0 such that • for all strings x in L of length at least m • there exist strings u, v, w, y, z such that • x = uvwyz and • |vwy| ≤ m and • |vy| ≥ 1 and • For all i ≥ 0, uviwyiz is in L

12. v and y can be pumped 1) 2) x = uvwyz3) For all i ≥ 0, uviwyiz is in L • Let x = abcdefg be in L • Then there exist substrings v and y in x such that v and y can be repeated (pumped) and the resulting string is still in L • uviwyiz is in L for all i ≥ 0

13. For example x =abcdefg v = cd and y = f uv0wy0z = uwz =abeg is in L uv1wy1z = uvwyz = abcdefgis in L uv2wy2z = uvvwyyz = abcdcdeffgis in L uv3wy3z = uvvvwyyyz = abcdcdcdefffg is in L …

14. What the other parts mean A language L satisfies the CFL pumping condition if: • there exists an integer m > 0 such that • for all strings x in L of length at least m • x must be in L and have sufficient length

15. What the other parts mean • there exist strings u, v, w, y, z such that • x = uvwyz and • |vwy| ≤ m and • v and y are contained within m characters of x • Note: these are NOT necessarily the first m characters of x • |vy| ≥ 1 and • v and y cannot both be λ, • One of them might be λ, but not both • For all i ≥ 0, uviwyiz is in L

16. Pumping Lemma Applying it to prove a specific language L is not context-free

17. How we use the Pumping Lemma • We choose a specific language L For example {aj bj cj | j > 0} • We show that L does not satisfy the pumping condition • We conclude that L is not context-free

18. A language L satisfies the CFL pumping condition if: there exists an integer m > 0 such that for all strings x in L of length at least m there exist strings u, v, w, y, z such that x = uvwyz and |vwy| ≤ m and |vy| ≥ 1 and For all i ≥ 0, uviwyiz is in L A language L does not satisfy the CFL pumping condition if: for all integers m of sufficient size there exists a string x in L of length at least m such that for all strings u, v, w, y, z such that x = uvwyz and |vwy| ≤ m and |vy| ≥ 1 There exists a i ≥ 0 such that uviwyiz is not in L Showing L “does not pump”

19. Example Languages • TWOCOPIES = {ww | w is in {a,b}* } • abbabbis in TWOCOPIES but abaabb is not • EQUAL3 = {the set of strings over {a, b, c} such that the number of a’s equals the number of b’s equals the number of c’s} • {ai bj ck | i < j < k}

20. More Pumping Lemma Applications

21. For infinite context-free language there exists an integer such that for any string we can write with lengths and The Pumping Lemma for CFL

22. Let be a context free grammar. There exists an integer such that can be written with lengths and The Pumping Lemma for CFL

23. Unrestricted grammarlanguages Non-regular languages Context-Free Languages Regular Languages

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

25. Assume the contrary - that is context-free Since is context-free and infinite we can apply the pumping lemma

26. Pumping Lemma gives a number such that: Pick any string of with length at least we pick:

27. and with lengths We can write: Pumping Lemma says: for all

28. We examine all the possible locations of string in

29. Case 1: is within the first

30. Case 1: is within the first

31. Case 1: is within the first

32. Case 1: is within the first However, from Pumping Lemma: Contradiction!

33. Case 2: is in the first is in the first

34. Case 2: is in the first is in the first

35. Case 2: is in the first is in the first

36. Case 2: is in the first is in the first However, from Pumping Lemma: Contradiction!

37. Case 3: overlaps the first is in the first

38. Case 3: overlaps the first is in the first

39. Case 3: overlaps the first is in the first

40. Case 3: overlaps the first is in the first However, from Pumping Lemma: Contradiction!

41. Case 4: in the first Overlaps the first Analysis is similar to case 3

42. or or Other cases: is within Analysis is similar to case 1:

43. More cases: overlaps or Analysis is similar to cases 2,3,4:

44. There are no other cases to consider Since , it is impossible for to overlap: neither nor nor

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

46. Unrestricted grammarlanguages Non-regular languages Context-Free Languages Regular Languages

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

48. Since is context-free and infinite we can apply the pumping lemma Assume to the contrary that is context-free

49. Pumping Lemma gurantees a number such that: Pick any string of with length at least we pick:

50. We can write: with lengths and Pumping Lemma says: for all