1. 1 Variations of the Turing Machine part2

2. 2 Standard Machine--Multiple Track Tape track 1 track 2 one symbol

3. 3 track 1 track 2 track 1 track 2

4. 4 Semi-Infinite Tape.........

5. 5 Standard Turing machines simulate Semi-infinite tape machines Trivial

6. 6 Semi-infinite tape machines simulate Standard Turing machines Turing machine......... Semi-infinite tape machine.........

7. 7 Turing machine......... Semi-infinite tape with two tracks......... reference point Right part Left part

8. 8 Right part Turing machine Semi-infinite tape machine

9. 9 Turing machine Semi-infinite tape machine Left part Right part For all symbols x

10. 10 Turing machine......... Semi-infinite tape Right part Left part Time 1

11. 11 Time 2 Turing machine......... Semi-infinite tape Right part Left part

12. 12 Semi-infinite tape machine Left part At the border: Right part

13. 13......... Semi-infinite tape Right part Left part......... Right part Left part Time 1 Time 2

14. 14 Theorem: Semi-infinite tape machines have the same power with Standard Turing machines

15. 15 The Off-Line Machine Control Unit Input File Tape read-only read-write

16. 16 Off-line Machines simulate Turing Machines Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine

17. 17 1. Copy input file to tape Input File Tape Turing Machine Off-line Machine

18. 18 2. Do computations as in Turing machine Input File Tape Turing Machine Off-line Machine

19. 19 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

20. 20 Input File Tape Off-line Machine Four track tape -- Standard Machine input file head position tape head position

21. 21 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

22. 22 Off-line machines have the same power with Stansard machines Theorem:

23. 23 Multitape Turing Machines Control unit tape 1tape 2 Input

24. 24 Time 1 Time 2

25. 25 Multitape machines simulate Standard Machines Just use one tape

26. 26 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:

27. 27 Multitape Machine Tape 1Tape 2 Four track tape -- Standard Machine Tape 1 head position Tape 2 head position

28. 28 Repeat for each state transition: Return to reference point Find current symbol on Tape 1 Find current symbol on Tape 2 make transition Tape 1 head position Tape 2 head position Reference point

29. 29 Theorem: Multi-tape machines have the same power with Standard Turing Machines

30. 30 Same power doesn’t mean same speed: Language Acceptance Time Standard machine Two-tape machine

31. 31 Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 Leave to tape 1 Compare tape 1 and tape 2 ( steps)

32. 32 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down

33. 33 Multidimensional machines simulate Standard machines Just use one dimension

34. 34 Standard machines simulate Multidimensional machines Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2

35. 35 symbols coordinates Two-dimensional machine Standard Machine

36. 36 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Simulation:

37. 37 Theorem: MultiDimensional Machines have the same power with Turing Machines

38. 38 NonDeterministic Turing Machines Non Deterministic Choice

39. 39 TIME 0 TIME 1 Choice 1 Choice 2

40. 40 Input is accepted if this a possible computation Initial configurationFinal Configuration Final state

41. 41 NonDeterministic Machines simulate Standard (deterministic) Machines Every deterministic machine is also a nondeterministic machine

42. 42 Deterministic machines simulate NonDeterministic machines Keeps track of all possible computations Deterministic machine:

43. 43 Non-Deterministic Choices Computation 1

44. 44 Non-Deterministic Choices Computation 2

45. 45 Keeps track of all possible computations Deterministic machine: Simulation Stores computations in a two-dimensional tape

46. 46 TIME 0NonDeterministic Deterministic Computation 1

47. 47 TIME 1NonDeterministic Deterministic Computation 1 Choice 1 Choice 2 Computation 2

48. 48 Repeat Execute a step in each computation: If there is a choice in current computation: replicate configuration change the state in the replica

49. 49 Theorem: NonDeterministic Machines have the same power with deterministic machines

50. 50 Remark: The simulation in the deterministic Machine takes time exponential time compared to NonDeterministic machines

51. 51 Polynomial Time in NonDeterministic Machine NP-Time Polynomial Time in Deterministic Machine P-Time Fundamental Problem:P = NP ?