1. Pushdown Automata PDAs
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Lecture 8

2. Pushdown Automaton -- PDA
Input String
Stack
States
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Lecture 8

3. Initial Stack Symbol Stack Stack stack head top bottom special symbol
Appears at time 0
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4. The States
Pop
symbol
Input
symbol
Push
symbol
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5. input
stack
top
Replace
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6. input
stack
top
Push
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7. input
stack
top
Pop
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8. input
stack
top
No Change
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9. If the automaton attempts to pop from
Pop from Empty Stack
input
Pop
Automaton halts!
stack
top
If the automaton attempts to pop from
empty stack then it halts and rejects input
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Lecture 8

10. PDAs are non-deterministic
Non-Determinism
PDAs are non-deterministic
Allowed non-deterministic transitions
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11. Example PDA
PDA :
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12. Basic Idea: Push the a’s 2. Match the b’s on input on the stack
with a’s on stack
3. Match
found
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13. Execution Example:
Time 0
Input
Stack
current
state
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14. Time 1
Input
Stack
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15. Time 2
Input
Stack
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16. Time 3
Input
Stack
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17. Time 4
Input
Stack
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18. Time 5
Input
Stack
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19. Time 6
Input
Stack
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20. Time 7
Input
Stack
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21. Time 8
Input
Stack
accept
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22. A string is accepted if there is a computation such that:
All the input is consumed
AND
The last state is an accepting state
we do not care about the stack contents
at the end of the accepting computation
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Lecture 8

23. Rejection Example:
Time 0
Input
Stack
current
state
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24. Rejection Example:
Time 1
Input
Stack
current
state
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25. Rejection Example:
Time 2
Input
Stack
current
state
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26. Rejection Example:
Time 3
Input
Stack
current
state
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27. Rejection Example:
Time 4
Input
Stack
current
state
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28. reject Rejection Example: Time 4 Input Stack current state 2nd 2017
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29. There is no accepting computation for
The string is rejected by the PDA
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Lecture 8

30. Another PDA example
PDA :
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31. Basic Idea: Push v on stack 3. Match on input with v on stack 2. Guess
middle
of input
4. Match
found
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32. Execution Example:
Time 0
Input
Stack
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33. Time 1
Input
Stack
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34. Time 2
Input
Stack
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35. Time 3
Input
Guess the middle
of string
Stack
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36. Time 4
Input
Stack
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37. Time 5
Input
Stack
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38. Time 6
Input
Stack
accept
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39. Rejection Example:
Time 0
Input
Stack
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40. Time 1
Input
Stack
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41. Time 2
Input
Stack
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42. Time 3
Input
Guess the middle
of string
Stack
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43. Time 4
Input
Stack
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44. There is no possible transition. Input
Time 5
There is no possible transition.
Input
Input is not consumed
Stack
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45. Another computation on same string:
Input
Time 0
Stack
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46. Time 1
Input
Stack
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47. Time 2
Input
Stack
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48. Time 3
Input
Stack
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49. Time 4
Input
Stack
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50. Time 5
Input
No accept state
is reached
Stack
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51. There is no computation that accepts string
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52. Pushing & Popping Strings
Input
symbol
Push
string
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53. Example: input stack top push string top pop string Replace 2nd 2017
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54. Equivalent
transitions
pop
push
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55. Another PDA example
PDA
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56. Execution Example:
Time 0
Input
Stack
current
state
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57. Time 1
Input
Stack
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58. Time 3
Input
Stack
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59. Time 4
Input
Stack
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60. Time 5
Input
Stack
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61. Time 6
Input
Stack
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62. Time 7
Input
Stack
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63. Time 8
Input
Stack
accept
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64. Formalities for PDAs
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65. Transition function:
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66. Transition function:
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67. Formal Definition Pushdown Automaton (PDA) Accept states States Input
alphabet
Stack
start
symbol
Transition
function
Initial
state
Stack
alphabet
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Lecture 8

68. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input
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69. Instantaneous Description
Example:
Instantaneous Description
Input
Time 4:
Stack
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70. Instantaneous Description
Example:
Instantaneous Description
Input
Time 5:
Stack
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71. We write:
Time 4
Time 5
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72. A computation:
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73. For convenience we write:
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74. Language of PDA Language accepted by PDA : Initial state Accept state
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75. Example:
PDA :
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76. PDA :
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77. Therefore:
PDA :
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