1. Pushdown Automata PDAs

3. Autómatas a Pila
Los Autómatas a Pila permiten analizar palabras para reconocer si pertenecen o no a lenguajes de tipo 2, independientes del contexto.
Tienen la misma estructura que los Autómatas Finitos, pero se añade una pila (memoria auxiliar) para poder guardar información que podrá ser útil en momentos posteriores del análisis.
Definición
Se definen los Autómatas a pila como:

7. Pushdown Automaton -- PDA
Input String
Stack
States

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

9. The States
Pop
symbol
Input
symbol
Push
symbol

10. input
stack
top
Replace

11. input
stack
top
Push

12. input
stack
top
Pop

13. input
stack
top
No Change

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

15. A Possible Transition
input
stack
Pop
top

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

17. Example PDA
PDA

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

19. Execution Example:
Time 0
Input
Stack
current
state

20. Time 1
Input
Stack

21. Time 2
Input
Stack

22. Time 3
Input
Stack

23. Time 4
Input
Stack

24. Time 5
Input
Stack

25. Time 6
Input
Stack

26. Time 7
Input
Stack

27. Time 8
Input
Stack
accept

28. 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)

29. The input string
is accepted by the PDA:

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

31. Rejection Example:
Time 0
Input
Stack
current
state

32. Rejection Example:
Time 1
Input
Stack
current
state

33. Rejection Example:
Time 2
Input
Stack
current
state

34. Rejection Example:
Time 3
Input
Stack
current
state

35. Rejection Example:
Time 4
Input
Stack
current
state

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

37. The input string
is rejected by the PDA:

38. 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

39. Another PDA example
PDA

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

41. Execution Example:
Time 0
Input
Stack

42. Time 1
Input
Stack

43. Time 2
Input
Stack

44. Time 3
Input
Guess the middle
of string
Stack

45. Time 4
Input
Stack

46. Time 5
Input
Stack

47. Time 6
Input
Stack
accept

48. Rejection Example:
Time 0
Input
Stack

49. Time 1
Input
Stack

50. Time 2
Input
Stack

51. Time 3
Input
Guess the middle
of string
Stack

52. Time 4
Input
Stack

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

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

55. Time 1
Input
Stack

56. Time 2
Input
Stack

57. Time 3
Input
Stack

58. Time 4
Input
Stack

59. Time 5
Input
No final state
is reached
Stack

60. There is no computation
that accepts string

61. Another PDA example
PDA

62. Execution Example:
Time 0
Input
Stack

63. Time 1
Input
Stack

64. Time 2
Input
Stack

65. Time 3
Input
Stack
accept

66. Rejection example:
Time 0
Input
Stack

67. Time 1
Input
Stack

68. Time 2
Input
Stack

69. Time 3
Input
Stack

70. Time 4
Input
Stack
Halt and Reject

71. Pushing Strings
Pop
symbol
Input
symbol
Push
string

72. Example:
input
pushed
string
stack
top
Push

73. Another PDA example
PDA

74. Execution Example:
Time 0
Input
Stack
current
state

75. Time 1
Input
Stack

76. Time 3
Input
Stack

77. Time 4
Input
Stack

78. Time 5
Input
Stack

79. Time 6
Input
Stack

80. Time 7
Input
Stack

81. Time 8
Input
Stack
accept

82. Formalities for PDAs

83. Transition function:

84. Transition function:

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

86. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input

87. Example:
Instantaneous Description
Input
Time 4:
Stack

88. Example:
Instantaneous Description
Input
Time 5:
Stack

89. We write:
Time 4
Time 5

90. A computation:

91. For convenience we write:

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

93. Example:
PDA :

94. PDA :

95. Therefore:
PDA :