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
Appears at time 0
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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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Empty Stack
input
stack
empty
Pop
top
The automaton HALTS No possible
transition after
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A Possible Transition
input
stack
Pop
top
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Non-Determinism
PDAs are non-deterministic
Allowed non-deterministic transitions
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Example PDA
PDA
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Basic Idea

• Push the as
• on the stack

2. Match the bs on input with as on stack
3. Match found
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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 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)
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The input string is accepted by the PDA
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In general,
is the language accepted by the PDA
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Rejection Example
Time 0
Input
Stack
current state
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Rejection Example
Time 1
Input
Stack
current state
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Rejection Example
Time 2
Input
Stack
current state
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Rejection Example
Time 3
Input
Stack
current state
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Rejection Example
Time 4
Input
Stack
current state
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Rejection Example
Time 4
Input
Stack
reject
current state
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The input string is rejected by the PDA
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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 an accept state
At the end of the computation, we do not care
about the stack contents
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Another PDA example
PDA
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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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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 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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Another PDA example
PDA
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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
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
Stack
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Time 4
Input
Stack
Halt and Reject
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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 PDA example
PDA
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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 PDAs

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Transition function
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Transition function
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Formal Definition
Pushdown Automaton (PDA)
Final states
States
Input alphabet
Stack start symbol
Transition function
Initial state
Stack alphabet
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Instantaneous Description
Current state
Current stack contents
Remaining input
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Example
Instantaneous Description
Input
Time 4
Stack
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Example
Instantaneous Description
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 PDA
Initial state
Final state
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Example
PDA
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PDA
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Therefore
PDA