NPDAs Accept ContextFree Languages

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NPDAs Accept Context-Free Languages
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Theorem
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
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Proof - Step 1
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
Convert any context-free grammar to a NPDA
with
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Proof - Step 2
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
Convert any NPDA to a context-free
grammar with
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Converting Context-Free Grammarsto NPDAs
Proof - step 1

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We will convert any context-free grammar
to an NPDA automaton
Such that
Simulates leftmost derivations of
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Leftmost derivation
Input processed
Stack contents
leftmost variable
Stack
Simulation of derivation
Input
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Leftmost derivation
string of terminals
Stack
Simulation of derivation
Input
end of input is reached
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An example grammar
What is the equivalent NPDA?
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Grammar
NPDA
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Grammar
A leftmost derivation
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Derivation
Input
Time 0
Stack
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Derivation
Input
Time 0
Stack
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Derivation
Input
Time 1
Stack
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Derivation
Input
Time 2
Stack
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Derivation
Input
Time 3
Stack
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Derivation
Input
Time 4
Stack
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Derivation
Input
Time 5
Stack
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Derivation
Input
Time 6
Stack
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Derivation
Input
Time 7
Stack
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Derivation
Input
Time 8
Stack
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Derivation
Input
Time 9
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 a
NPDA that accepts the same language
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
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Converting NPDAstoContext-Free Grammars
Proof - step 2

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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
terminals
variables
Input processed
Stack contents
Current configuration in NPDA
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Some Necessary Modifications

• Modify (if necessary) the NPDA so that
• 1) The stack is never empty
• 2) It has a single final state
• and empties the stack when it accepts a
  string
• 3) Has transitions in a special form

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• Modify the NPDA so that
• the stack is never empty

Stack
OK
OK
NOT OK
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Introduce the new symbol to denote the
bottom of the stack
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At the beginning push into the stack
Original NPDA
new initial state
original initial state
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In transitions replace every instance of
with
Example
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Convert all transitions so that
if the automaton attempts to pop or replace
it will halt
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Convert transitions as follows
halting state
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2) Modify the NPDA so that it empties the
stack and has a unique final state
Empty the stack
NPDA
Old final states
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3) modify the NPDA so that transitions have
the following forms
OR
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Convert
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Convert
symbols
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Convert
symbols
Convert recursively
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Example of a NPDA in correct form
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The Grammar Construction
In grammar
Stack symbol
Variables
states
Terminals Input symbols of NPDA
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For each transition
We add production
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For each transition
We add productions
For all possible states in the automaton
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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
if and only if
the NPDA goes from to by reading string
and the stack doesnt change below and
then is removed from stack
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Therefore
if and only if
is accepted by the NPDA
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Therefore
For any NPDA there is a context-free grammar that
accepts the same language
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
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Deterministic PDADPDA

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Deterministic PDA DPDA
Allowed transitions
(deterministic choices)
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Allowed transitions
(deterministic choices)
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Not allowed
(non deterministic choices)
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DPDA example
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The language
is deterministic context-free
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Definition
A language is deterministic context-free if
there exists some DPDA that accepts it
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Example of Non-DPDA (NPDA)
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Not allowed in DPDAs
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NPDAs Have More Power thanDPDAs

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It holds that
Deterministic Context-Free Languages (DPDA)
Context-Free Languages NPDAs
Since every DPDA is also a NPDA
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We will actually show
Deterministic Context-Free Languages (DPDA)
Context-Free Languages (NPDA)
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The language is
We will show

• is context-free

• is not deterministic context-free

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Language is context-free
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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 is a DPDA that accepts
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DPDA with
accepts
accepts
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DPDA with
Such a path exists because of the determinism
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Fact 1
The language is not
context-free
(we will prove this at a later class
using pumping lemma for context-free languages)
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Fact 2
The language is not
context-free
(we can prove this using pumping lemma for
context-free languages)
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We will construct a NPDA that accepts
which is a contradiction!
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Replace with
Modify
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The NPDA that accepts
Connect final states of with final states
of
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Since is accepted by a
NPDA
it is context-free
Contradiction!
(since is not
context-free)
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Therefore
Not deterministic context free
There is no DPDA that accepts
End of Proof