1. NPDAs Accept Context-Free Languages
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2. Theorem: Context-Free Languages Languages Accepted by (Grammars) NPDAs
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3. Convert any context-free grammar to a NPDA with:
Proof - Step 1:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any context-free grammar
to a NPDA with:
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4. Convert any NPDA to a context-free grammar with:
Proof - Step 2:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any NPDA to a context-free
grammar with:
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5. Converting Context-Free Grammars to NPDAs
Proof - step 1
Converting Context-Free Grammars to NPDAs
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6. We will convert any context-free grammar
to an NPDA automaton
Such that:
Simulates leftmost derivations of
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7. Simulation of derivation
Leftmost derivation
Input
processed
Stack
contents
leftmost variable
Stack
Simulation of derivation
Input
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8. Simulation of derivation
Leftmost derivation
string of terminals
Stack
Simulation of derivation
Input
end of input is reached
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9. What is the equivalent NPDA?
An example grammar:
What is the equivalent NPDA?
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10. Grammar:
NPDA:
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11. A leftmost derivation:
Grammar:
A leftmost derivation:
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12. Derivation:
Input
Time 0
Stack
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13. Derivation:
Input
Time 0
Stack
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14. Derivation:
Input
Time 1
Stack
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15. Derivation:
Input
Time 2
Stack
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16. Derivation:
Input
Time 3
Stack
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17. Derivation:
Input
Time 4
Stack
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18. Derivation:
Input
Time 5
Stack
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19. Derivation:
Input
Time 6
Stack
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20. Derivation:
Input
Time 7
Stack
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21. Derivation:
Input
Time 8
Stack
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22. Derivation:
Input
Time 9
Stack
accept
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23. In general: Given any grammar We can construct a NPDA With
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24. Constructing NPDA from grammar :
For any production
For any terminal
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25. Grammar generates string
if and only if
NPDA accepts
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26. For any context-free language there is a NPDA
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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27. Converting NPDAs to Context-Free Grammars
Proof - step 2
Converting NPDAs to Context-Free Grammars
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28. a context-free grammar with
For any NPDA
we will construct
a context-free grammar with
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29. The grammar simulates the machine
Intuition:
The grammar simulates the machine
A derivation in Grammar :
terminals
variables
Input processed
Stack contents
Current configuration in NPDA
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30. 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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31. the stack is never empty
Modify the NPDA so that
the stack is never empty
Stack OK OK
NOT OK
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32. Introduce the new symbol to denote the bottom of the stack
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33. At the beginning push into the stack
Original NPDA
new
initial state
original
initial state
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34. replace every instance of with
In transitions:
replace every instance of with
Example:
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35. Convert all transitions so that:
if the automaton attempts to pop
or replace it will halt
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36. l , ® $ Convert transitions as follows: halting state
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37. 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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38. 3) modify the NPDA so that transitions have the following forms:
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39. Convert:
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40. Convert:
symbols
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41. Convert:
symbols
Convert recursively
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42. Example of a NPDA in correct form:
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43. The Grammar Construction
In grammar :
Stack symbol
Variables:
states
Terminals:
Input symbols of NPDA
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44. For each transition
We add production
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45. For all possible states in the automaton
For each transition
We add productions
For all possible states
in the automaton
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46. Stack bottom symbol Start Variable: Start state final state
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47. Example:
Grammar production:
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48. Example:
Grammar productions:
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49. Example:
Grammar production:
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50. Resulting Grammar:
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51. Costas Busch - RPI

52. Derivation of string
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53. the stack doesn’t change below and then is removed from stack
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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54. Therefore:
if and only if
is accepted by the NPDA
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55. there is a context-free grammar that accepts the same language
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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56. Deterministic PDA DPDA
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57. Deterministic PDA: DPDA
Allowed transitions:
(deterministic choices)
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58. (deterministic choices)
Allowed transitions:
(deterministic choices)
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59. (non deterministic choices)
Not allowed:
(non deterministic choices)
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60. DPDA example
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61. is deterministic context-free
The language
is deterministic context-free
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62. A language is deterministic context-free
Definition:
A language is deterministic context-free
if there exists some DPDA that accepts it
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63. Example of Non-DPDA (NPDA)
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64. Not allowed in DPDAs
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65. NPDAs Have More Power than DPDAs
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66. Since every DPDA is also a NPDA
It holds that:
Deterministic
Context-Free
Languages
(DPDA)
Context-Free
Languages
NPDAs
Since every DPDA is also a NPDA
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67. We will show that there exists a context-free language which is not
We will actually show:
Deterministic
Context-Free
Languages
(DPDA)
Context-Free
Languages
(NPDA)
We will show that there exists
a context-free language which is not
accepted by any DPDA
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68. is not deterministic context-free
The language is:
We will show:
is context-free
is not deterministic context-free
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69. Language is context-free
Context-free grammar for :
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70. Theorem: The language is not deterministic context-free
(there is no DPDA that accepts )
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71. Proof: Assume for contradiction that is deterministic context free
Therefore:
there is a DPDA that accepts
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72. DPDA with
accepts
accepts
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73. Such a path exists because of the determinism
DPDA with
Such a path exists because of the determinism
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74. Context-free languages
Fact 1:
The language
is not context-free
Regular languages
Context-free languages
(we will prove this at a later class using
pumping lemma for context-free languages)
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75. Fact 2: The language is not context-free
(we can prove this using pumping lemma
for context-free languages)
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76. We will construct a NPDA that accepts:
which is a contradiction!
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77. Replace
with
Modify
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78. Connect final states of with final states of
The NPDA that accepts
Connect final states of
with final states of
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79. Since is accepted by a NPDA
it is context-free
Contradiction!
(since is not context-free)
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80. Not deterministic context free
Therefore:
Not deterministic context free
There is no DPDA that accepts
End of Proof
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