Pushdown Automata PDAs

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Pushdown AutomataPDAs

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Pushdown Automaton -- PDA
Input String
Stack
States
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Initial Stack Symbol
Stack
Stack
stack head
top
bottom
special symbol
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The States
Pop symbol
Input symbol
Push symbol
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input
stack
top
Replace
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input
stack
top
Push
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input
stack
top
Pop
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input
stack
top
No Change
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A Possible Transition
input
stack
empty
Pop
top
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A Bad Transition
input
Empty stack

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

HALT
The automaton Halts in state and Rejects the
input string
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No transition is allowed to be followed when the
stack is empty
Empty stack
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A Good Transition
input
stack
Pop
top
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Non-Determinism
These are allowed transitions in a
Non-deterministic PDA (NPDA)
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NPDA Non-Deterministic PDA
Example
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Execution Example
Time 0
Input
Stack
current state
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Stack
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Time 4
Input
Stack
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Time 5
Input
Stack
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Time 6
Input
Stack
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Time 7
Input
Stack
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Time 8
Input
Stack
accept
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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
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The input string is accepted by the NPDA
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In general,
is the language accepted by the NPDA
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NPDA -- Another example
NPDA
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Execution Example
Time 0
Input
Stack
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Guess the middle of string
Stack
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Time 4
Input
Stack
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Time 5
Input
Stack
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Time 6
Input
Stack
accept
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Rejection Example
Time 0
Input
Stack
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Guess the middle of string
Stack
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Time 4
Input
Stack
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Time 5
There is no possible transition.
Input
Input is not consumed
Stack
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Another computation on the same string
Input
Time 0
Stack
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Stack
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Time 4
Input
Stack
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Time 5
Input
No final state is reached
Stack
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There is no computation that accepts string
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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
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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 OR The stack head moves below the bottom of
the stack
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NPDA Another Example
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Pushing Strings
Pop symbol
Input symbol
Push string
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Example
input
pushed string
stack
top
Push
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Another NPDA example
NPDA
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Execution Example
Time 0
Input
Stack
current state
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Time 1
Input
Stack
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Time 3
Input
Stack
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Time 4
Input
Stack
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Time 5
Input
Stack
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Time 6
Input
Stack
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Time 7
Input
Stack
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Time 8
Input
Stack
accept
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Formalities for NPDAs

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Transition function
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Transition function
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Formal Definition
Non-Deterministic Pushdown Automaton
NPDA
Final states
States
Input alphabet
Stack end symbol
Transition function
Initial state
Stack alphabet
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Instantaneous Description (ID)
Current state
Current stack contents
Remaining unread input
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Example
Instantaneous Description (ID)
Input
Time 4
Stack
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Example
Instantaneous Description (ID)
Input
Time 5
Stack
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We write
Time 4
Time 5
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A computation
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For convenience we write
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Formal Definition
Language of NPDA
Initial ID
Final ID
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Example
NPDA
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NPDA
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Therefore
NPDA