1. 1 … NPDAs continued

2. 2 Pushing Strings Input symbol Pop symbol Push string

3. 3 top input stack Push pushed string Example:

4. 4 Another NPDA example NPDA

5. 5 Time 0 Input current state Stack Execution Example:

6. 6 Time 1 Input Stack

7. 7 Time 3 Input Stack

8. 8 Time 4 Input Stack

9. 9 Time 5 Input Stack

10. 10 Time 6 Input Stack

11. 11 Time 7 Input Stack

12. 12 Time 8 Input Stack accept

13. 13 Formalities for NPDAs

14. 14 Transition function:

15. 15 Transition function:

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

17. 17 Instantaneous Description Current state Remaining input Current stack contents

18. 18 Input Stack Time 4: Example:Instantaneous Description

19. 19 Input Stack Time 5: Example:Instantaneous Description

20. 20 We write: Time 4 Time 5

21. 21 A computation:

22. 22 For convenience we write:

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

24. 24 Example: NPDA :

25. 25 NPDA :

26. 26 NPDA : Therefore:

27. 27 NPDAs Accept Context-Free Languages

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

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

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

31. 31 Converting Context-Free Grammars to NPDAs

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

33. 33 Grammar: NPDA:

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

35. 35 Grammar: A leftmost derivation:

36. 36 NPDA execution: Input Stack Time 0 current state

37. 37 Input Stack Time 1

38. 38 Input Stack Time 2

39. 39 Input Stack Time 3

40. 40 Input Stack Time 4

41. 41 Input Stack Time 5

42. 42 Input Stack Time 6

43. 43 Input Stack Time 7

44. 44 Input Stack Time 8

45. 45 Input Stack Time 9

46. 46 Input Stack Time 10 accept

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

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

49. 49 Grammar generates string if and only if NPDA accepts

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

51. 51 Converting NPDAs to Context-Free Grammars

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

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

54. 54 in NPDA Input processedStack contents terminalsvariables A derivation in Grammar :

55. 55 Some Necessary Modifications First, we modify the NPDA: It has a single final state It empties the stack when it accepts the input Original NPDA Empty Stack

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

57. 57 Example of a NPDA in correct form:

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

59. 59 For each transition We add production

60. 60 For each transition We add production For all states

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

62. 62 Example: Grammar production:

63. 63 Example: Grammar productions:

64. 64 Example: Grammar production:

65. 65 Resulting Grammar:

66. 66

67. 67 Derivation of string

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

69. 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