1. … NPDAs continued

2. Pushing Strings
Pop
symbol
Input
symbol
Push
string

3. Example:
input
pushed
string
stack
top
Push

4. Another NPDA example
NPDA

5. Execution Example:
Time 0
Input
Stack
current
state

6. Time 1
Input
Stack

7. Time 3
Input
Stack

8. Time 4
Input
Stack

9. Time 5
Input
Stack

10. Time 6
Input
Stack

11. Time 7
Input
Stack

12. Time 8
Input
Stack
accept

13. Formalities for NPDAs

14. Transition function:

15. Transition function:

16. Formal Definition Non-Deterministic Pushdown Automaton NPDA Final
states
States
Input
alphabet
Stack
start
symbol
Transition
function
Initial
state
Stack
alphabet

17. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input

18. Example:
Instantaneous Description
Input
Time 4:
Stack

19. Example:
Instantaneous Description
Input
Time 5:
Stack

20. We write:
Time 4
Time 5

21. A computation:

22. For convenience we write:

23. Formal Definition
Language of NPDA :
Initial state
Final state

24. Example:
NPDA :

25. NPDA :

26. Therefore:
NPDA :

27. NPDAs Accept Context-Free Languages

28. Theorem:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs

29. Proof - Step 1:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any context-free grammar
to a NPDA with:

30. Proof - Step 2:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any NPDA to a context-free
grammar with:

31. Converting Context-Free Grammars to NPDAs

32. An example grammar:
What is the equivalent NPDA?

33. Grammar:
NPDA:

34. The NPDA simulates
leftmost derivations of the grammar
L(Grammar) = L(NPDA)

35. Grammar:
A leftmost derivation:

36. NPDA execution:
Time 0
Input
Stack
current
state

37. Time 1
Input
Stack

38. Time 2
Input
Stack

39. Time 3
Input
Stack

40. Time 4
Input
Stack

41. Time 5
Input
Stack

42. Time 6
Input
Stack

43. Time 7
Input
Stack

44. Time 8
Input
Stack

45. Time 9
Input
Stack

46. Time 10
Input
Stack
accept

47. In general:
Given any grammar
We can construct a NPDA
With

48. Constructing NPDA from grammar :
For any production
For any terminal

49. Grammar generates string
if and only if
NPDA accepts

50. Therefore:
For any context-free language
there is an NPDA
that accepts the same language

51. Converting NPDAs to Context-Free Grammars

52. For any NPDA
we will construct
a context-free grammar with

53. Intuition:
The grammar simulates the machine
A derivation in Grammar :
Current configuration in NPDA

54. A derivation in Grammar :
terminals
variables
Input processed
Stack contents
in NPDA

55. 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

56. Second, we modify the NPDA transitions:
all transitions will have form or

57. Example of a NPDA in correct form:

58. The Grammar Construction
In grammar :
Stack symbol
Variables:
states
Terminals:
Input symbols of NPDA

59. For each transition
We add production

60. For each transition
We add production
For all states

61. Stack bottom symbol
Start Variable:
Start state
final state

62. Example:
Grammar production:

63. Example:
Grammar productions:

64. Example:
Grammar production:

65. Resulting Grammar:

67. Derivation of string

68. In general, in Grammar:
if and only if
is accepted by the NPDA

69. Explanation:
By construction of Grammar:
if and only if
in the NPDA going from to
the stack doesn’t change below
and is removed from stack