Converting NPDAs to Context-Free Grammars

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Converting NPDAstoContext-Free Grammars

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For any NPDA
we will construct a context-free grammar
with
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Intuition
The grammar simulates the machine
A derivation in Grammar
Current configuration in NPDA
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A derivation in Grammar
terminals
variables
Input processed
Stack contents
in NPDA
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Some Necessary Modifications

• First, we modify the NPDA
• It has a single final state
• It empties the stack
• when it accepts the input

Empty Stack
Original NPDA
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Second, we modify the NPDA transitions all
transitions will have form
or
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Example of a NPDA in correct form
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The Grammar Construction
stack symbol
In grammar variables have form
states
Terminals are input symbols
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For each transition
We add production
For all states
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For each transition
We add production
For all states in the NPDA
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Stack bottom symbol
Start Variable
Start state
final state
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Example
Grammar production
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Example
Grammar productions
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Example
Grammar production
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Resulting Grammar
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(No Transcript)
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Derivation of string
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In general, in grammar
if and only if
is accepted by the NPDA
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Explanation
By construction of grammar
If and only if
in the NPDA going from to the stack
doesnt change below and is removed from
stack
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Deterministic PDAs(DPDAs)

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DPDAs

• Allowed Transitions

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Not allowed
Not allowed even when is
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Allowed
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Not allowed
Not allowed even when is
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DPDA example
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The language
is deterministic context-free
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In general
A language is deterministic context-free if there
is some NPDA that accepts it
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Example of Non-DPDA
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Not allowed transitions for DPDAs
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NPDAs have more power thanDPDAs

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We will show
There is
(accepted by a NPDA)
a context-free language
which is not
(not accepted by a DPDA)
deterministic context-free
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The language is
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The language is context-free
Context-free grammar for
(there is an NPDA that accepts )
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Theorem
The language
is not deterministic context-free
(there is no DPDA that accepts )
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Proof
Assume for contradiction that
is deterministic context free
Therefore
There exists a DPDA that accepts
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The DPDA with
accepts
accepts
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A fact
The language
is not context-free
(we will prove it at the next class)
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Another fact
The language
is not context-free
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We will construct a NPDA that accepts
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First, we modify
Replace with
Modified
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The NPDA that accepts
Modified
Original
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is accepted by a
NPDA
Therefore
is context-free
Contradiction!
( is not context-free)
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Therefore
There is no DPDA that accepts
End of Proof