Pushdown Automata PDAs

1. Pushdown AutomataPDAs Courtesy Costas Busch - RPI

2. Pushdown Automaton -- PDA Input String Stack States Courtesy Costas Busch - RPI

3. Initial Stack Symbol Stack Stack stack head top bottom special symbol Courtesy Costas Busch - RPI

4. The States Pop symbol Input symbol Push symbol Alternatively Courtesy Costas Busch - RPI

5. input stack top Replace Courtesy Costas Busch - RPI

6. input stack top Push Courtesy Costas Busch - RPI

7. input stack top Pop Courtesy Costas Busch - RPI

8. input stack top No Change Courtesy Costas Busch - RPI

9. A Possible Transition input stack empty Pop top Courtesy Costas Busch - RPI

10. A Bad Transition input Empty stack HALT The automaton Halts in state and Rejects the input string Courtesy Costas Busch - RPI

11. A Bad Transition input Empty stack HALT The automaton Halts in state and Rejects the input string Courtesy Costas Busch - RPI

12. No transition is allowed to be followed When the stack is empty Empty stack Courtesy Costas Busch - RPI

13. A Good Transition input stack Pop top Courtesy Costas Busch - RPI

14. Non-Determinism These are allowed transitions in a Non-deterministic PDA (NPDA) Courtesy Costas Busch - RPI

15. NPDA: Non-Deterministic PDA Example: Courtesy Costas Busch - RPI

16. Execution Example: Time 0 Input Stack current state Courtesy Costas Busch - RPI

17. Time 1 Input Stack Courtesy Costas Busch - RPI

18. Time 2 Input Stack Courtesy Costas Busch - RPI

19. Time 3 Input Stack Courtesy Costas Busch - RPI

20. Time 4 Input Stack Courtesy Costas Busch - RPI

21. Time 5 Input Stack Courtesy Costas Busch - RPI

22. Time 6 Input Stack Courtesy Costas Busch - RPI

23. Time 7 Input Stack Courtesy Costas Busch - RPI

24. Time 8 Input Stack accept Courtesy Costas Busch - RPI

25. A string is accepted if there is a computation such that: All the input is consumed AND The last state is a final state At the end of the computation, we do not care about the stack contents Courtesy Costas Busch - RPI

26. The input string is accepted by the NPDA: Courtesy Costas Busch - RPI

27. In general, is the language accepted by the NPDA: Courtesy Costas Busch - RPI

28. Another NPDA example NPDA Courtesy Costas Busch - RPI

29. Execution Example: Time 0 Input Stack Courtesy Costas Busch - RPI

30. Time 1 Input Stack Courtesy Costas Busch - RPI

31. Time 2 Input Stack Courtesy Costas Busch - RPI

32. Time 3 Input Guess the middle of string Stack Courtesy Costas Busch - RPI

33. Time 4 Input Stack Courtesy Costas Busch - RPI

34. Time 5 Input Stack Courtesy Costas Busch - RPI

35. Time 6 Input Stack accept Courtesy Costas Busch - RPI

36. Rejection Example: Time 0 Input Stack Courtesy Costas Busch - RPI

37. Time 1 Input Stack Courtesy Costas Busch - RPI

38. Time 2 Input Stack Courtesy Costas Busch - RPI

39. Time 3 Input Guess the middle of string Stack Courtesy Costas Busch - RPI

40. Time 4 Input Stack Courtesy Costas Busch - RPI

41. Time 5 There is no possible transition. Input Input is not consumed Stack Courtesy Costas Busch - RPI

42. Another computation on same string: Input Time 0 Stack Courtesy Costas Busch - RPI

43. Time 1 Input Stack Courtesy Costas Busch - RPI

44. Time 2 Input Stack Courtesy Costas Busch - RPI

45. Time 3 Input Stack Courtesy Costas Busch - RPI

46. Time 4 Input Stack Courtesy Costas Busch - RPI

47. Time 5 Input No final state is reached Stack Courtesy Costas Busch - RPI

48. There is no computation that accepts string Courtesy Costas Busch - RPI

49. A string is rejected if there is no computation such that: All the input is consumed AND The last state is a final state At the end of the computation, we do not care about the stack contents Courtesy Costas Busch - RPI

50. In other words, a string is rejected if in every computation with this string: The input cannot be consumed OR The input is consumed and the last state is not a final state OR The stack head moves below the bottom of the stack Courtesy Costas Busch - RPI