1. Turing Machines

2. A Turing Machine Tape ...... ...... Read-Write head Control Unit

3. The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right

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

5. Example: Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left

6. Time 1 ...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Right

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

8. Input string Blank symbol ...... ...... head Remark: the input string is never empty

9. States & Transitions Write Read Move Left Move Right

10. Example: Time 1 ...... ...... current state

11. Time 1 ...... ...... Time 2 ...... ......

12. Example: Time 1 ...... ...... Time 2 ...... ......

13. Example: Time 1 ...... ...... Time 2 ...... ......

14. Determinism Turing Machines are (for now) deterministic Not Allowed Allowed

15. Partial Transition Function Example: ...... ...... Allowed: No transition for input symbol

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

17. Example: ...... ...... No possible transition HALT!!!

18. Final (accepting) States Allowed Not Allowed • Final states have no outgoing transitions • In a final state the machine halts

19. 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

20. Turing Machine Example A Turing machine that accepts the language:

21. Time 0

22. Time 1

23. Time 2

24. Time 3

25. Time 4 Halt & Accept

26. Rejection Example Time 0

27. Time 1 No possible Transition Halt & Reject

28. Infinite Loop Example A Turing machine for the language that includes all strings an with n>0 or strings that start with b.

29. Time 0

30. Time 1

31. Time 2

32. Time 2 Time 3 Infinite loop Time 4 Time 5

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

34. Standard Turing Machine The machine we described is the standard: • Deterministic • Infinite tape in both directions • Tape is the input/output file

35. Turing’s Thesis

36. Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

37. Computer Science Law: A computation is mechanical if and only if it can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines

38. Algorithms are Turing Machines When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm

39. Variationsof theTuring Machine

40. The Standard Model Infinite Tape Read-Write Head (Left or Right) Control Unit Deterministic

41. The variations form different Turing MachineClasses We want to prove: Each Class has the same power with the Standard Model

42. Same Power of two classes means: Both classes of Turing machines accept the same languages

43. Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa

44. Multitape Turing Machines Control unit Tape 1 Tape 2 Input

45. Tape 1 Time 1 Tape 2 Time 2

46. Multitape machines simulate Standard Machines: Use just one tape We can also prove that standard machines simulate Multitape machines.

47. Same power doesn’t imply same speed: Language Acceptance Time Standard machine Two-tape machine

48. Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 ( steps) ( steps) Leave on tape 1 Compare tape 1 and tape 2 ( steps)

49. NonDeterministic Turing Machines Non Deterministic Choice

50. Time 0 Time 1 Choice 1 Choice 2