1. Fall 2004COMP 3351 Turing’s Thesis

2. Fall 2004COMP 3352 Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

3. Fall 2004COMP 3353 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

4. Fall 2004COMP 3354 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

5. Fall 2004COMP 3355 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine that executes the algorithm

6. Fall 2004COMP 3356 Variations of the Turing Machine

7. Fall 2004COMP 3357 Read-Write Head Control Unit Deterministic The Standard Model Infinite Tape (Left or Right)

8. Fall 2004COMP 3358 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with:

9. Fall 2004COMP 3359 We want to prove: Each Class has the same power with the Standard Model The variations form different Turing Machine Classes

10. Fall 2004COMP 33510 Same Power of two classes means: Both classes of Turing machines accept the same languages

11. Fall 2004COMP 33511 Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa

12. Fall 2004COMP 33512 a technique to prove same powerSimulation: Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine

13. Fall 2004COMP 33513 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine:

14. Fall 2004COMP 33514 The Simulation Machine and the Original Machine accept the same language Original Machine: Simulation Machine: Final Configuration

15. Fall 2004COMP 33515 Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: moves

16. Fall 2004COMP 33516 Example: Time 1 Time 2

17. Fall 2004COMP 33517 Stay-Option Machines have the same power with Standard Turing machines Theorem:

18. Fall 2004COMP 33518 Proof: Part 1: Stay-Option Machines are at least as powerful as Standard machines Proof:a Standard machine is also a Stay-Option machine (that never uses the S move)

19. Fall 2004COMP 33519 Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof:a standard machine can simulate a Stay-Option machine Proof:

20. Fall 2004COMP 33520 Stay-Option Machine Simulation in Standard Machine Similar for Right moves

21. Fall 2004COMP 33521 Stay-Option Machine Simulation in Standard Machine For every symbol

22. Fall 2004COMP 33522 Example Stay-Option Machine: 12 Simulation in Standard Machine: 123

23. Fall 2004COMP 33523 Standard Machine--Multiple Track Tape track 1 track 2 one symbol

24. Fall 2004COMP 33524 track 1 track 2 track 1 track 2

25. Fall 2004COMP 33525 Semi-Infinite Tape.........

26. Fall 2004COMP 33526 Standard Turing machines simulate Semi-infinite tape machines: Trivial

27. Fall 2004COMP 33527 Semi-infinite tape machines simulate Standard Turing machines: Standard machine......... Semi-infinite tape machine.........

28. Fall 2004COMP 33528 Standard machine......... Semi-infinite tape machine with two tracks......... reference point Right part Left part

29. Fall 2004COMP 33529 Left partRight part Standard machine Semi-infinite tape machine

30. Fall 2004COMP 33530 Standard machine Semi-infinite tape machine Left part Right part For all symbols

31. Fall 2004COMP 33531 Standard machine......... Semi-infinite tape machine Right part Left part Time 1

32. Fall 2004COMP 33532 Time 2 Right part Left part Standard machine......... Semi-infinite tape machine

33. Fall 2004COMP 33533 Semi-infinite tape machine Left part At the border: Right part

34. Fall 2004COMP 33534......... Semi-infinite tape machine Right part Left part......... Right part Left part Time 1 Time 2

35. Fall 2004COMP 33535 Theorem:Semi-infinite tape machines have the same power with Standard Turing machines

36. Fall 2004COMP 33536 The Off-Line Machine Control Unit Input File Tape read-only read-write

37. Fall 2004COMP 33537 Off-line machines simulate Standard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine

38. Fall 2004COMP 33538 1. Copy input file to tape Input File Tape Standard machine Off-line machine

39. Fall 2004COMP 33539 2. Do computations as in Turing machine Input File Tape Standard machine Off-line machine

40. Fall 2004COMP 33540 Standard Turing machines simulate Off-line machines: Use a Standard machine with four track tape to keep track of the Off-line input file and tape contents

41. Fall 2004COMP 33541 Input File Tape Off-line Machine Four track tape -- Standard Machine Input File head position Tape head position

42. Fall 2004COMP 33542 Input File head position Tape head position Repeat for each state transition: Return to reference point Find current input file symbol Find current tape symbol Make transition Reference point

43. Fall 2004COMP 33543 Off-line machines have the same power with Stansard machines Theorem:

44. Fall 2004COMP 33544 Multitape Turing Machines Control unit Tape 1Tape 2 Input

45. Fall 2004COMP 33545 Time 1 Time 2 Tape 1Tape 2

46. Fall 2004COMP 33546 Multitape machines simulate Standard Machines: Use just one tape

47. Fall 2004COMP 33547 Standard machines simulate Multitape machines: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks Standard machine:

48. Fall 2004COMP 33548 Multitape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position

49. Fall 2004COMP 33549 Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition Tape 1 head position Tape 2 head position Reference point

50. Fall 2004COMP 33550 Theorem:Multi-tape machines have the same power with Standard Turing Machines

51. Fall 2004COMP 33551 Same power doesn’t imply same speed: Language Acceptance Time Standard machine Two-tape machine

52. Fall 2004COMP 33552 Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 Leave on tape 1 Compare tape 1 and tape 2 ( steps)

53. Fall 2004COMP 33553 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down

54. Fall 2004COMP 33554 Multidimensional machines simulate Standard machines: Use one dimension

55. Fall 2004COMP 33555 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2

56. Fall 2004COMP 33556 symbols coordinates Two-dimensional machine Standard Machine

57. Fall 2004COMP 33557 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Standard machine:

58. Fall 2004COMP 33558 MultiDimensional Machines have the same power with Standard Turing Machines Theorem:

59. Fall 2004COMP 33559 NonDeterministic Turing Machines Non Deterministic Choice

60. Fall 2004COMP 33560 Time 0 Time 1 Choice 1 Choice 2

61. Fall 2004COMP 33561 Input string is accepted if this a possible computation Initial configurationFinal Configuration Final state

62. Fall 2004COMP 33562 NonDeterministic Machines simulate Standard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine

63. Fall 2004COMP 33563 Deterministic machines simulate NonDeterministic machines: Keeps track of all possible computations Deterministic machine:

64. Fall 2004COMP 33564 Non-Deterministic Choices Computation 1

65. Fall 2004COMP 33565 Non-Deterministic Choices Computation 2

66. Fall 2004COMP 33566 Keeps track of all possible computations Deterministic machine: Simulation Stores computations in a 2D tape

67. Fall 2004COMP 33567 Time 0 NonDeterministic machine Deterministic machine Computation 1

68. Fall 2004COMP 33568 Computation 1 Choice 1 Choice 2 Computation 2 NonDeterministic machine Deterministic machine Time 1

69. Fall 2004COMP 33569 Repeat Execute a step in each computation: If there are two or more choices in current computation: 1. Replicate configuration 2. Change the state in the replica

70. Fall 2004COMP 33570 Theorem: NonDeterministic Machines have the same power with Deterministic machines

71. Fall 2004COMP 33571 Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine