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… NPDAs continued

… NPDAs continued. Pushing Strings. Pop symbol. Input symbol. Push string. Example:. input. pushed string. stack. top. Push. Another NPDA example. NPDA. Execution Example:. Time 0. Input. Stack. current state. Time 1. Input. Stack. Time 3. Input. Stack. Time 4. Input.

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… NPDAs continued

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  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 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. ConvertingContext-Free Grammarsto 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

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