1. PDAs Accept Context-Free Languages
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2. Theorem: Context-Free Languages Languages Accepted by (Grammars) PDAs
Prof. Busch - LSU

3. Convert any context-free grammar to a PDA with:
Proof - Step 1:
Context-Free
Languages
(Grammars)
Languages
Accepted by
PDAs
Convert any context-free grammar
to a PDA with:
Prof. Busch - LSU

4. Convert any PDA to a context-free grammar with:
Proof - Step 2:
Context-Free
Languages
(Grammars)
Languages
Accepted by
PDAs
Convert any PDA to a context-free
grammar with:
Prof. Busch - LSU

5. Convert Context-Free Grammars to PDAs
Proof - step 1
Convert Context-Free Grammars to PDAs
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6. Take an arbitrary context-free grammar
We will convert to a PDA such that:
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7. Conversion Procedure:
For each
For each
production in
terminal in
Add transitions
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8. Example
Grammar
PDA
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9. Example:
Input
Time 0
Stack
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10. Derivation:
Input
Time 1
Stack
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11. Derivation:
Input
Time 2
Stack
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12. Derivation:
Input
Time 3
Stack
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13. Derivation:
Input
Time 4
Stack
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14. Derivation:
Input
Time 5
Stack
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15. Derivation:
Input
Time 6
Stack
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16. Derivation:
Input
Time 7
Stack
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17. Derivation:
Input
Time 8
Stack
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18. Derivation:
Input
Time 9
Stack
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19. Derivation:
Input
Time 10
Stack
accept
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20. Convert PDAs to Context-Free Grammars
Proof - step 2
Convert PDAs to Context-Free Grammars
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21. Courtesy Costas Busch - RPI
For any NPDA
we will construct
a context-free grammar with
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22. Courtesy Costas Busch - RPI
Intuition:
The grammar simulates the machine
A derivation in Grammar :
terminals
variables
Input processed
Stack contents
Current configuration in NPDA
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23. Courtesy Costas Busch - RPI
From NPDA to CFG
Lets look at how a PDA can consume
and empty the stack.
We shall define a grammar with variables of the
form [pi-1 Yi pi] that would represent going from pi-1 to pi with the net effect of popping Yi.
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24. To generate all those strings w that cause P to pop Z0 from its stack while going from q0 to p.

25. Courtesy Costas Busch - RPI

29. Some Necessary Modifications
Modify (if necessary) the NPDA (accepting by reaching final state) 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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30. Courtesy Costas Busch - RPI
Modify the NPDA so that
the stack is never empty
Stack OK OK
NOT OK
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31. Courtesy Costas Busch - RPI
Introduce the new symbol to denote
the bottom of the stack
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32. Courtesy Costas Busch - RPI
At the beginning push into the stack
Original NPDA
new
initial state
original
initial state
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33. Courtesy Costas Busch - RPI
In transitions:
replace every instance of with
Example:
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34. Courtesy Costas Busch - RPI
Convert all transitions so that:
if the automaton attempts to pop
or replace it will halt
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35. Courtesy Costas Busch - RPI
Convert transitions as follows: $ , l
halting state
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36. Courtesy Costas Busch - RPI
2) Modify the NPDA so that
it empties the stack
and has a unique final state
Empty the stack
NPDA l , l , l ,
Old final states
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37. Courtesy Costas Busch - RPI
3) modify the NPDA so that
transitions have the following forms: OR
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38. Courtesy Costas Busch - RPI
Convert:
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39. Courtesy Costas Busch - RPI
Convert:
symbols
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40. Courtesy Costas Busch - RPI
Convert:
symbols
Convert recursively
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41. Courtesy Costas Busch - RPI
Example of a NPDA in correct form:
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42. The Grammar Construction
In grammar :
Stack symbol
Variables:
states
Terminals:
Input symbols of NPDA
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43. Courtesy Costas Busch - RPI
For each transition
We add production
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44. Courtesy Costas Busch - RPI
For each transition
We add productions
For all possible states
in the automaton
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45. Courtesy Costas Busch - RPI
Stack bottom symbol
Start Variable:
Start state
final state
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46. Courtesy Costas Busch - RPI
Example:
Grammar production:
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47. Courtesy Costas Busch - RPI
Example:
Grammar productions:
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48. Courtesy Costas Busch - RPI
Example:
Grammar production:
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49. Courtesy Costas Busch - RPI
Resulting Grammar:
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50. Courtesy Costas Busch - RPI

51. Courtesy Costas Busch - RPI
Derivation of string
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52. Courtesy Costas Busch - RPI
In general:
if and only if
the NPDA goes from to
by reading string and
the stack doesn’t change below
and then is removed from stack
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53. Courtesy Costas Busch - RPI
Therefore:
if and only if
is accepted by the NPDA
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54. Courtesy Costas Busch - RPI
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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