1. Pushdown Automata PDAs

2. Pushdown Automaton -- PDA
Input String
Stack
States

3. Initial Stack Symbol
Stack
Stack
stack
head
top
bottom
special symbol

4. The States
Pop
symbol
Input
symbol
Push
symbol

5. input
stack
top
Replace

6. input
stack
top
Push

7. input
stack
top
Pop

8. input
stack
top
No Change

9. A Possible Transition
input
stack
empty
Pop
top

10. A Bad Transition
input
Empty stack
HALT
The automaton Halts in state
and Rejects the input string

11. A Bad Transition
input
Empty stack
HALT
The automaton Halts in state
and Rejects the input string

12. No transition is allowed to be followed
When the stack is empty
Empty stack

13. A Good Transition
input
stack
Pop
top

14. Non-Determinism These are allowed transitions in a
Non-deterministic PDA (NPDA)

15. NPDA: Non-Deterministic PDA
Example:

16. Execution Example:
Time 0
Input
Stack
current
state

17. Time 1
Input
Stack

18. Time 2
Input
Stack

19. Time 3
Input
Stack

20. Time 4
Input
Stack

21. Time 5
Input
Stack

22. Time 6
Input
Stack

23. Time 7
Input
Stack

24. Time 8
Input
Stack
accept

25. A string is accepted if there is
a computation such that:
All the input is consumed
AND
The last state is a final state
At the end of the computation,
we do not care about the stack contents

26. The input string
is accepted by the NPDA:

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

28. Another NPDA example
NPDA

29. Execution Example:
Time 0
Input
Stack

30. Time 1
Input
Stack

31. Time 2
Input
Stack

32. Time 3
Input
Guess the middle
of string
Stack

33. Time 4
Input
Stack

34. Time 5
Input
Stack

35. Time 6
Input
Stack
accept

36. Rejection Example:
Time 0
Input
Stack

37. Time 1
Input
Stack

38. Time 2
Input
Stack

39. Time 3
Input
Guess the middle
of string
Stack

40. Time 4
Input
Stack

41. Time 5
There is no possible transition.
Input
Input is not consumed
Stack

42. Another computation on same string:
Input
Time 0
Stack

43. Time 1
Input
Stack

44. Time 2
Input
Stack

45. Time 3
Input
Stack

46. Time 4
Input
Stack

47. Time 5
Input
No final state
is reached
Stack

48. There is no computation
that accepts string

49. A string is rejected if there is
no computation such that:
All the input is consumed
AND
The last state is a final state
At the end of the computation,
we do not care about the stack contents

50. In other words, a string is rejected
if in every computation with this string:
The input cannot be consumed OR
The input is consumed and the last state is not a final state
The stack head moves below the bottom of the stack

51. Another NPDA example
NPDA

52. Execution Example:
Time 0
Input
Stack

53. Time 1
Input
Stack

54. Time 2
Input
Stack

55. Time 3
Input
Stack
accept

56. Rejection example:
Time 0
Input
Stack

57. Time 1
Input
Stack

58. Time 2
Input
Stack

59. Time 3
Input
Stack

60. Time 4
Input
Stack
Halt and Reject

61. Pushing Strings
Pop
symbol
Input
symbol
Push
string

62. Example:
input
pushed
string
stack
top
Push

63. Another NPDA example
NPDA

64. Execution Example:
Time 0
Input
Stack
current
state

65. Time 1
Input
Stack

66. Time 3
Input
Stack

67. Time 4
Input
Stack

68. Time 5
Input
Stack

69. Time 6
Input
Stack

70. Time 7
Input
Stack

71. Time 8
Input
Stack
accept

72. Formalities for NPDAs

73. Transition function:

74. Transition function:

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

76. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input

77. Example:
Instantaneous Description
Input
Time 4:
Stack

78. Example:
Instantaneous Description
Input
Time 5:
Stack

79. We write:
Time 4
Time 5

80. A computation:

81. For convenience we write:

82. Formal Definition
Language of NPDA :
Initial state
Final state

83. Example:
NPDA :

84. NPDA :

85. Therefore:
NPDA :