1. Pushdown Automata PDAs

2. Pushdown Automaton -- PDA
Input String
Stack
States

3. Initial Stack Symbol Stack Stack stack head top bottom special symbol
Appears at time 0

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. Empty Stack
input
stack
empty
Pop
top
The automaton HALTS
No possible transition after

10. A Possible Transition
input
stack
Pop
top

11. PDAs are non-deterministic
Non-Determinism
PDAs are non-deterministic
Allowed non-deterministic transitions

12. Example PDA
PDA

13. Basic Idea:
Push the a’s
on the stack
2. Match the b’s on input
with a’s on stack
3. Match
found

14. Execution Example:
Time 0
Input
Stack
current
state

15. Time 1
Input
Stack

16. Time 2
Input
Stack

17. Time 3
Input
Stack

18. Time 4
Input
Stack

19. Time 5
Input
Stack

20. Time 6
Input
Stack

21. Time 7
Input
Stack

22. Time 8
Input
Stack
accept

23. A string is accepted if there is
a computation such that:
All the input is consumed
AND
The last state is an accepting state
At the end of the computation,
we do not care about the stack contents
(the stack can be empty at the last state)

24. The input string
is accepted by the PDA:

25. In general,
is the language accepted by the PDA:

26. Rejection Example:
Time 0
Input
Stack
current
state

27. Rejection Example:
Time 1
Input
Stack
current
state

28. Rejection Example:
Time 2
Input
Stack
current
state

29. Rejection Example:
Time 3
Input
Stack
current
state

30. Rejection Example:
Time 4
Input
Stack
current
state

31. Rejection Example:
Time 4
Input
Stack
reject
current
state

32. The input string
is rejected by the PDA:

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

34. Another PDA example
PDA

35. Basic Idea:
Push v
on stack
3. Match on input
with v on stack
2. Guess
middle
of input
4. Match
found

36. Execution 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
Input
Stack

42. Time 6
Input
Stack
accept

43. Rejection Example:
Time 0
Input
Stack

44. Time 1
Input
Stack

45. Time 2
Input
Stack

46. Time 3
Input
Guess the middle
of string
Stack

47. Time 4
Input
Stack

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

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

50. Time 1
Input
Stack

51. Time 2
Input
Stack

52. Time 3
Input
Stack

53. Time 4
Input
Stack

54. Time 5
Input
No final state
is reached
Stack

55. There is no computation
that accepts string

56. Another PDA example
PDA

57. Execution Example:
Time 0
Input
Stack

58. Time 1
Input
Stack

59. Time 2
Input
Stack

60. Time 3
Input
Stack
accept

61. Rejection example:
Time 0
Input
Stack

62. Time 1
Input
Stack

63. Time 2
Input
Stack

64. Time 3
Input
Stack

65. Time 4
Input
Stack
Halt and Reject

66. Pushing Strings
Pop
symbol
Input
symbol
Push
string

67. Example:
input
pushed
string
stack
top
Push

68. Another PDA example
PDA

69. Execution Example:
Time 0
Input
Stack
current
state

70. Time 1
Input
Stack

71. Time 3
Input
Stack

72. Time 4
Input
Stack

73. Time 5
Input
Stack

74. Time 6
Input
Stack

75. Time 7
Input
Stack

76. Time 8
Input
Stack
accept

77. Formalities for PDAs

78. Transition function:

79. Transition function:

80. Formal Definition Pushdown Automaton (PDA) Final states States Input
alphabet
Stack
start
symbol
Transition
function
Initial
state
Stack
alphabet

81. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input

82. Example:
Instantaneous Description
Input
Time 4:
Stack

83. Example:
Instantaneous Description
Input
Time 5:
Stack

84. We write:
Time 4
Time 5

85. A computation:

86. For convenience we write:

87. Formal Definition
Language of PDA :
Initial state
Final state

88. Example:
PDA :

89. PDA :

90. Therefore:
PDA :