1. Fall 2006Costas Busch - RPI1 PDAs Accept Context-Free Languages

2. Fall 2006Costas Busch - RPI2 Context-Free Languages (Grammars) Languages Accepted by PDAs Theorem:

3. Fall 2006Costas Busch - RPI3 Context-Free Languages (Grammars) Languages Accepted by PDAs Proof - Step 1: Convert any context-free grammar to a PDA with:

4. Fall 2006Costas Busch - RPI4 Context-Free Languages (Grammars) Languages Accepted by PDAs Proof - Step 2: Convert any PDA to a context-free grammar with:

5. Fall 2006Costas Busch - RPI5 Convert Context-Free Grammars to PDAs Proof - step 1

6. Fall 2006Costas Busch - RPI6 We will convert to a PDA such that: Take an arbitrary context-free grammar

7. Fall 2006Costas Busch - RPI7 production in terminal in Conversion Procedure: For each Add transitions

8. Fall 2006Costas Busch - RPI8 Grammar PDA Example

9. Fall 2006Costas Busch - RPI9 PDA simulates leftmost derivations Grammar Leftmost Derivation PDA Computation Scanned symbols Stack contents

10. Fall 2006Costas Busch - RPI10 Grammar Leftmost Derivation Production applied Terminals Leftmost variable Variables or terminals Terminals Variable Variables or terminals

11. Fall 2006Costas Busch - RPI11 Grammar Leftmost Derivation PDA Computation Production appliedTransition applied

12. Fall 2006Costas Busch - RPI12 Grammar Leftmost Derivation PDA Computation Transition applied Read from input and remove it from stack

13. Fall 2006Costas Busch - RPI13 Grammar Leftmost Derivation PDA Computation Last Transition applied All symbols have been removed from top of stack

14. Fall 2006Costas Busch - RPI14 Production applied The process repeats with the next leftmost variable And so on……

15. Fall 2006Costas Busch - RPI15 Input Stack Time 0 Example:

16. Fall 2006Costas Busch - RPI16 Input Stack Time 1 Derivation:

17. Fall 2006Costas Busch - RPI17 Input Stack Time 2 Derivation:

18. Fall 2006Costas Busch - RPI18 Input Stack Time 3 Derivation:

19. Fall 2006Costas Busch - RPI19 Input Stack Time 4 Derivation:

20. Fall 2006Costas Busch - RPI20 Input Stack Time 5 Derivation:

21. Fall 2006Costas Busch - RPI21 Input Stack Time 6 Derivation:

22. Fall 2006Costas Busch - RPI22 Input Stack Time 7 Derivation:

23. Fall 2006Costas Busch - RPI23 Input Stack Time 8 Derivation:

24. Fall 2006Costas Busch - RPI24 Input Stack Time 9 Derivation:

25. Fall 2006Costas Busch - RPI25 Input Stack accept Time 10 Derivation:

26. Fall 2006Costas Busch - RPI26 Grammar Leftmost Derivation PDA Computation

27. Fall 2006Costas Busch - RPI27 Grammar generates string PDA accepts In general, it can be shown that: If and Only if Therefore

28. Fall 2006Costas Busch - RPI28 Convert PDAs to Context-Free Grammars Proof - step 2

29. Fall 2006Costas Busch - RPI29 We will convert to a context-free grammar such that: Take an arbitrary PDA

30. Fall 2006Costas Busch - RPI30 First modify PDA so that: 1. The PDA has a single accept state 4. Each transition either pushes a symbol or pops a symbol but not both together 3. On acceptance the stack contains only stack symbol # 2. Use new initial stack symbol #

31. Fall 2006Costas Busch - RPI31 PDA , , , Old accept states New accept state 1. The PDA has a single accept state PDA

32. Fall 2006Costas Busch - RPI32 PDA 2. Use new initial stack symbol # new initial stack symbol old initial stack symbol Top of stack still thinks that Z is the initial stack PDA auxiliary stack symbol

33. Fall 2006Costas Busch - RPI33 Empty stack New accept state PDA Old accept state PDA 3. On acceptance the stack contains only stack symbol #

34. Fall 2006Costas Busch - RPI34 4. Each transition either pushes a symbol or pops a symbol but not both together PDA

35. Fall 2006Costas Busch - RPI35 PDA Where is a symbol of the stack alphabet

36. Fall 2006Costas Busch - RPI36 PDA is the final modified PDA Note that the new initial stack symbol # is never used in any transition

37. Fall 2006Costas Busch - RPI37 Example:

38. Fall 2006Costas Busch - RPI38 Grammar Construction Variables: States of PDA

39. Fall 2006Costas Busch - RPI39 PDA Grammar Kind 1: for each state

40. Fall 2006Costas Busch - RPI40 PDA Grammar Kind 2: for every three states

41. Fall 2006Costas Busch - RPI41 PDA Grammar Kind 3: for every pair of such transitions

42. Fall 2006Costas Busch - RPI42 PDA Grammar Initial stateAccept state Start variable

43. Fall 2006Costas Busch - RPI43 Example: PDA

44. Fall 2006Costas Busch - RPI44 Grammar Kind 1: from single states

45. Fall 2006Costas Busch - RPI45 Start variable Kind 2: from triplets of states

46. Fall 2006Costas Busch - RPI46 Kind 3: from pairs of transitions

47. Fall 2006Costas Busch - RPI47 We need to prove that Suppose that a PDA is converted to a context-free grammar or equivalently

48. Fall 2006Costas Busch - RPI48 We need to show that if has derivation: Then there is an accepting computation in : (string of terminals) with input string

49. Fall 2006Costas Busch - RPI49 We will actually show that if has derivation: Then there is a computation in :

50. Fall 2006Costas Busch - RPI50 Therefore: Since there is no transition with the # symbol

51. Fall 2006Costas Busch - RPI51 Lemma: If (string of terminals) then there is a computation from state to state on string which leaves the stack empty:

52. Fall 2006Costas Busch - RPI52 Proof Intuition: Case 1: Case 2: Type 2 Type 3

53. Fall 2006Costas Busch - RPI53 Case 1: Type 2 Stack height Input string Generated by

54. Fall 2006Costas Busch - RPI54 Stack height Input string Generated by Case 2: Type 3

55. Fall 2006Costas Busch - RPI55 We formally prove this claim by induction on the number of steps in derivation: Formal Proof: number of steps

56. Fall 2006Costas Busch - RPI56 Induction Basis: (one derivation step) A Kind 1 production must have been used: Therefore, and This computation of PDA trivially exists:

57. Fall 2006Costas Busch - RPI57 Induction Hypothesis: derivation steps suppose it holds:

58. Fall 2006Costas Busch - RPI58 Induction Step: derivation steps We have to show:

59. Fall 2006Costas Busch - RPI59 derivation steps Case 1: Case 2: Type 2 Type 3

60. Fall 2006Costas Busch - RPI60 Case 1: Type 2 We can write At most steps steps

61. Fall 2006Costas Busch - RPI61 At most steps From induction hypothesis, in PDA: From induction hypothesis, in PDA:

62. Fall 2006Costas Busch - RPI62 since

63. Fall 2006Costas Busch - RPI63 Case 2: Type 3 We can write At most steps steps

64. Fall 2006Costas Busch - RPI64 At most steps From induction hypothesis, the PDA has computation:

65. Fall 2006Costas Busch - RPI65 And PDA Contains transitions Type 3 Grammar contains production

66. Fall 2006Costas Busch - RPI66

67. Fall 2006Costas Busch - RPI67 We know We also know Therefore:

68. Fall 2006Costas Busch - RPI68 since END OF PROOF

69. Fall 2006Costas Busch - RPI69 So far we have shown: With a similar proof we can show Therefore: