1. 1 Pushdown Automata PDAs

2. 2 Pushdown Automaton -- PDA Input String Stack States

3. 3 Initial Stack Symbol Stack bottom special symbol

4. 4 The States Input symbol Pop symbol Push symbol

5. 5 top input stack

6. 6 top input stack

7. 7 top input stack

8. 8 top input stack

9. 9 Non-Determinism

10. 10 NPDA: Non-Deterministic PDA Example:

11. 11 Execution Example: Input current state Time 0 Stack

12. 12 Input Time 1 Stack

13. 13 Input Stack Time 2

14. 14 Input Stack Time 3

15. 15 Input Stack Time 4

16. 16 Input Stack Time 5

17. 17 Input Stack Time 6

18. 18 Input Stack Time 7

19. 19 Input Time 8 accept Stack

20. 20 A string is accepted if: All the input is consumed The last state is a final state We do not care about the stack contents

21. 21 The input string is accepted by the NPDA:

22. 22 is the language accepted by the NPDA: In general,

23. 23 Another NPDA example NPDA

24. 24

25. 25 Execution Example: Input Time 0 Stack

26. 26 Input Time 1 Stack

27. 27 Input Time 2 Stack

28. 28 Input Time 3 Stack

29. 29 Input Time 4 Stack

30. 30 Input Time 5 Stack

31. 31 Input Time 6 Stack accept

32. 32 Pushing Strings Input symbol Pop symbol Push string

33. 33 top input stack pushed string Example:

34. 34 Another NPDA example NPDA

35. 35

36. 36 Execution: Time 0 Input current state Stack

37. 37 Time 1 Input Stack

38. 38 Time 3 Input Stack

39. 39 Time 4 Input Stack

40. 40 Time 5 Input Stack

41. 41 Time 6 Input Stack

42. 42 Time 7 Input Stack

43. 43 Time 8 Input Stack accept

44. 44 Formalities for NPDAs

45. 45 Transition function:

46. 46 Transition function:

47. 47 Formal Definition Non-Deterministic Pushdown Automaton NPDA States Input alphabet Stack alphabet Transition function Final states Stack start symbol

48. 48 Instantaneous Description Current state Remaining input Current stack contents

49. 49 Input Stack Time 4: Example:Instantaneous Description

50. 50 Input Stack Time 5: Example:Instantaneous Description

51. 51 We write: (Time 4)(Time 5)

52. 52 A computation:

53. 53 For convenience we write:

54. 54 Formal Definition Language of NPDA : Start state Final state

55. 55 Example: NPDA : thus

56. 56 NPDA : thus

57. 57 NPDA : Therefore:

58. 58 Converting Context-Free Grammars to NPDAs

59. 59 An example grammar: What is the equivalent NPDA?

60. 60 Grammar: NPDA:

61. 61 The NPDA simulates leftmost derivations of the grammar L(grammar) = L(NPDA)

62. 62 Grammar: A leftmost derivation:

63. 63 NPDA execution: Input Stack Time 0 current state

64. 64 Input Stack Time 1

65. 65 Input Stack Time 2

66. 66 Input Stack Time 3

67. 67 Input Stack Time 4

68. 68 Input Stack Time 5

69. 69 Input Stack Time 6

70. 70 Input Stack Time 7

71. 71 Input Stack Time 8

72. 72 Input Stack Time 9

73. 73 Input Stack Time 10 accept

74. 74 In general: Given any grammar We can construct a NPDA With

75. 75 Constructing NPDA from grammar : For any production For any terminal

76. 76 Grammar generates string if and only if NPDA accepts

77. 77 Therefore: For any context-free language there is an NPDA that accepts the same language