Theory of computing, part 4

1. Theory of computing, part 4

2. Course outline

3. Turing machines

4. The language hierarchy Context-Free Languages Regular Languages

5. The language hierarchy Languages accepted by Turing Machines Context-Free Languages Regular Languages

6. A Turing machine Tape ...... ...... Read-Write head Control Unit

7. The tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right

8. The tape ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

9. Example Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left

10. Example Time 1 ...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Left

11. The input string Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string

12. States and transitions Write Read Move Left Move Right

13. Example Time 1 ...... ...... current state

14. Example Time 1 ...... ...... Time 2 ...... ......

15. Example Time 1 ...... ...... Time 2 ...... ......

16. Example Time 1 ...... ...... Time 2 ...... ......

17. Determinism Turing Machines are deterministic Not Allowed Allowed

18. Example: partial transition function ...... ...... Allowed: No transition for input symbol

19. Halting The machine haltsif there are no possible transitions to follow

20. Example ...... ...... No possible transition HALT

21. Final states Allowed Not Allowed • Final states have no outgoing transitions • In a final state the machine halts

22. Acceptance If machine halts in a final state Accept Input If machine halts in a non-final state or If machine enters an infinite loop Reject Input

23. Turing machine example A Turing machine that accepts language a*

24. Turing machine example Time 0

25. Turing machine example Time 1

26. Turing machine example Time 2

27. Turing machine example Time 3

28. Turing machine example Time 4 Halt & Accept

29. Rejection example Time 0

30. Rejection example Time 1 No possible Transition Halt & Reject

31. Infinite loop example Another Turing machine for language a*and is this one correct???

32. Infinite loop example Time 0

33. Infinite loop example Time 1

34. Infinite loop example Time 2

35. Infinite loop example Time 2 Time 3 Time 4 Time 5 ... Infinite Loop

36. Infinite loop example Because of the infinite loop: • The final state cannot be reached • The machine never halts • The input is not accepted

37. Another Turing machine example Turing machine for the language

38. Another Turing machine example Time 0

39. Another Turing machine example Time 1

40. Another Turing machine example Time 2

41. Another Turing machine example Time 3

42. Another Turing machine example Time 4

43. Another Turing machine example Time 5

44. Another Turing machine example Time 6

45. Another Turing machine example Time 7

46. Another Turing machine example Time 8

47. Another Turing machine example Time 9

48. Another Turing machine example Time 10

49. Another Turing machine example Time 11

50. Another Turing machine example Time 12