CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 6 Mälardalen University

1. CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 6 Mälardalen University 2010

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

3. Midterm • 5 out of 13 passed… • Many failed on basic definitions • Is something wrong with the course format? • Scheduled exercises • Don’t hesitate to ask for help! • Do labs, exercises etc. • Review lecture material before lectures • Strongly recommended to get a book if you don’t have one • It doesn’t get easier…

4. Concerning examination in the course: Exercises  are voluntary Labs are voluntary Midterms are voluntary Lectures are voluntary… All of them are recommended! JFLAP demo http://www.cs.duke.edu/csed/jflap/movies

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

6. Context-Free Languages

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

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

9. Pushdown AutomataPDAs

10. Pushdown Automaton - PDA Input String Stack States

11. 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.

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

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

14. stack top Push input

15. stack top Pop input

16. input stack top No Change

17. NPDAsNon-deterministic Push-Down Automata

18. Non-Determinism

19. 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)

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

21. Example NPDA NPDAM (Even-length palindromes)

22. Pushing Strings Pop symbol Input symbol Push string

23. Example input pushed string stack top Push

24. Another NPDA example NPDAM

25. Execution Example Time 0 Input Stack Current state

26. Time 1 Input Stack

27. Time 2 Input Stack

28. Time 3 Input Stack

29. Time 4 Input Stack

30. Time 5 Input Stack

31. Time 6 Input Stack

32. Time 7 Input Stack accept

33. Formal Definitions for NPDAs

34. Transition function

35. 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.

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

37. Instantaneous Description Current stack contents Current state Remaining input

38. Example Instantaneous Description Input Time 4: Stack

39. Example Instantaneous Description Input Time 5: Stack

40. We write Time 4 Time 5

41. A computation example

42. A computation example

43. A computation example

44. A computation example

45. A computation example

46. A computation example

47. A computation example

48. A computation example

49. For convenience we write

50. Formal Definition Language of NPDAM Initial state Final state