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
bottom
special symbol
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The States
Pop symbol
Input symbol
Push symbol
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input
stack
top
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input
stack
top
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input
stack
top
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input
stack
top
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Non-Determinism
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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

• All the input is consumed
• The last state is a final state

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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Another NPDA example
NPDA
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(No Transcript)
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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
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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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Pushing Strings
Pop symbol
Input symbol
Push string
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Example
input
pushed string
stack
top
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Another NPDA example
NPDA
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(No Transcript)
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Execution
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 start symbol
Transition function
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 NPDA
Start state
Final state
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Example
thus
NPDA
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thus
NPDA
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Therefore
NPDA
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Converting Context-Free Grammarsto NPDAs

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An example grammar
What is the equivalent NPDA?
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Grammar
NPDA
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The NPDA simulates leftmost derivations of the
grammar
L(grammar) L(NPDA)
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Grammar
A leftmost derivation
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NPDA execution
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
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Time 9
Input
Stack
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Time 10
Input
Stack
accept
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In general
Given any grammar
We can construct a NPDA
With
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Constructing NPDA from grammar
For any production
For any terminal
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Grammar generates string
if and only if
NPDA accepts
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Therefore
For any context-free language there is an
NPDA that accepts the same language