CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 8 Mälardalen University

1. CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 8 Mälardalen University 2010

2. ContentContext-Free LanguagesPush-Down Automata, PDANPDA: Non-Deterministic PDAFormal Definitions for NPDAs NPDAs Accept Context-Free LanguagesConverting NPDA to Context-Free Grammar

3. Non-regular languages Context-Free Languages Regular Languages

4. Context-Free Languages

5. stack automaton Context-Free Languages Context-Free Grammars Pushdown Automata (CF grammars are defined as generalized Regular Grammars)

6. is string of variables and terminals Definition: Context-Free Grammars Grammar Variables Terminal symbols Start variables Productions of the form:

7. is string of terminals Definition: Regular Grammars Grammar Variables Terminal symbols Start variables Right or Left Linear Grammars. Productions of the form: or

8. Pushdown AutomataPDAs

9. Pushdown Automaton - PDA Input String Stack States

10. The Stack A PDA can write symbols on stack and read them later on. POP reading symbol PUSH writing symbol All access to the stack only on the top! (Stack top is written leftmost in the string, e.g. yxz) A stack is valuable as it can hold an unlimitedamount of (certain kind of) information. The stack allows pushdown automata to recognize some non-regular languages.

11. The States Pop old- reading stack symbol Push new - writing stacksymbol Input symbol

12. input stack top Replace (An alternative is to start and finish with empty stack or a stack bottom symbol such as $)

13. stack top Push input

14. stack top Pop input

15. input stack top No Change

16. Example 3.7 Salling: A PDA for simple nested parenthesis strings Time 0 Input Stack

17. Example 3.7 Time 1 Input Stack

18. Example 3.7 Time 2 Input Stack

19. Stack Example 3.7 Time 3 Input

20. Stack Example 3.7 Time 4 Input

21. Stack Example 3.7 Time 5 Input

22. Example 3.7 Time 6 Input Stack

23. Example 3.7 Time 7 Input Stack

24. NPDAsNon-deterministic Push-Down Automata

25. Non-Determinism

26. A string is accepted if: • All the input is consumed • The last state is a final state • Stack is in the initial condition • (either: empty (when we started with empty stack), • or: bottom symbol reached, or similar)

27. Example NPDA is the language accepted by the NPDA:

28. Example NPDA NPDAM (Even-length palindromes)

29. Pushing Strings Pop symbol Input symbol Push string

30. Example input pushed string stack top Push

31. Another NPDA example NPDAM

32. Execution Example Time 0 Input Stack Current state

33. Time 1 Input Stack

34. Time 2 Input Stack

35. Time 3 Input Stack

36. Time 4 Input Stack

37. Time 5 Input Stack

38. Time 6 Input Stack

39. Time 7 Input Stack accept

40. Formal Definitions for NPDAs

41. Transition function

42. Transition function new state current state current stack top new stack top current input symbol An unspecified transition function is to the null set and represents a dead configuration for the NPDA.

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

44. Instantaneous Description Current stack contents Current state Remaining input

45. Example Instantaneous Description Input Time 4: Stack

46. Example Instantaneous Description Input Time 5: Stack

47. We write Time 4 Time 5

48. A computation example

49. A computation example

50. A computation example