1. Multi-tape TM’s
Often it’s useful to have several tapes when carrying out a computations. For example, consider a two tape I/O TM for adding numbers (we show only how it acts on a typical input)

2. Multi Tape TM Addition Example$ 1
Input string

3. Multi Tape TM Addition Example$ 1

4. Multi Tape TM Addition Example$ 1

5. Multi Tape TM Addition Example$ 1

6. Multi Tape TM Addition Example$ 1

7. Multi Tape TM Addition Example$ 1

8. Multi Tape TM Addition Example$ 1

9. Multi Tape TM Addition Example$ 1

10. Multi Tape TM Addition Example$ 1

11. Multi Tape TM Addition Example$ 1

12. Multi Tape TM Addition Example$ 1

13. Multi Tape TM Addition Example$ 1

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17. Multi Tape TM Addition Example$ 1

18. Multi Tape TM Addition Example$ 1

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22. Multi Tape TM Addition Example1

23. Multi Tape TM Addition Example1

24. Multi Tape TM Addition Example1

25. Multi Tape TM Addition Example1

26. Multi Tape TM Addition Example1
HALT!
Output
string

27. Multitape TM’s Formal Notation
NOTE: Sipser’s multitape machines cannot pause on one of the tapes as above example. This isn’t a problem since pausing 1-tape machines can simulate pausing k-tape machines, and non-pausing 1-tape machines can simulate 1-tape pausing machines by adding dummy R-L moves for each pause.
Formally, the d-function of a k-tape machine:

28. Multitape TM’s Conventions
Input always put on the first tape
If I/O machine, output also on first tape
Can consider machines as “string-vector” generators. E.g., a 4 tape machine could be considered as outputting in (S*)4

29. Equivalence of k-tape vs. 1-tape
The argument that k-tape TM’s are universal, is fairly convincing. After all, the computers we use daily split their memory into several locations: RAM, Hard-Disc, Floppy, CD-ROM. It is plausible that each memory unit could be simulated by a separate tape. The following theorem implies that the computer could still function fully (though probably with a loss of efficiency) if all the memory was merged onto a single tape:

30. Equivalence of k-tape vs. 1-tape
THM: Every k-tape TM can be simulated by a 1-tape TM, and vice versa.
Proof. It’s clear that every 1-tape TM can be simulated by a k-tape TM. Simply ignore tapes no. 2 through k.
On the other hand, to show that any k-tape TM can be simulated by a 1-tape TM, we’ll show two parts:
Any k-tape TM can be simulated by a 2k-track TM.
Any k-track TM can be simulated by a 1-tape TM.

31. k-track TM’s
A k-track machine contains a tape whose cells are broken into k vertical sub-cells. As opposed to k-tape machines, there is only one reading head1, but it may read all k tracks and base its actions on these values.
EG. A 2-tape machine:
vs. a 2-track machine:
1Equivalently, we may think of k-heads which are forced to move in lock-step, so can only see one column of the tapes at a time. $ 1 $ 1

32. k-track TM’s
A k-track machine contains a tape whose cells are broken into k vertical sub-cells. As opposed to k-tape machines, there is only one reading head1, but it may read all k tracks and base its actions on these values.
EG. A 2-tape machine:
vs. a 2-track machine:
1Equivalently, we may think of k-heads which are forced to move in lock-step, so can only see one column of the tapes at a time. $ 1 $ 1

33. 2k-tracks Simulating k-tapes
We can simulate any k-tape TM by a k-track TM as follows:
Each tape of the k-tape TM is represented by one track.
Each track representing a tape, is underneath a track which consists of all blanks, except for a single X which tells us which cells of the k-tape TM are active: X $ 1 $ 1

34. 2k-tracks Simulating k-tapes
For each move of k-tape TM, the 2k-track TM sweeps to right and then to left.
During right-sweep, 2k-track TM notes the current state of the k-tape machine as well as the symbols read by each head, keeping this information in its finite control. Head back when passed all X’s.
During the left-sweep, every X encountered causes corresponding track active cell to change in accordance to the memorized configuration of the k-tape machine.
EG:

35. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

36. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

37. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

38. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

39. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

40. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: 2nd head reads  $ 1 X $ 1
k-tape TM transition $ 1 3 2

41. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1
k-tape TM transition $ 1 3 2

42. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: 1st head reads 1 passed all X’s! $ 1 X $ 1
k-tape TM transition $ 1 3 2

43. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: nd phase, on 1st tape: 13,R $ 1 $ 1 3 X
k-tape TM transition $ 1 3 2

44. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: nd phase, on 1st tape: 13,R $ 1 X $ 1 3
k-tape TM transition $ 1 3 2

45. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1 3
k-tape TM transition $ 1 3 2

46. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: on 2nd tape: 2,L $ 1 X $ 1 3
k-tape TM transition $ 1 3 2

47. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: on 2nd tape: 2,L $ 1 X $ 1 3 2
k-tape TM transition $ 1 3 2

48. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: on 2nd tape: 2,L $ 1 X $ 1 3 2
k-tape TM transition $ 1 3 2

49. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1 3 2
k-tape TM transition $ 1 3 2

50. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: $ 1 X $ 1 3 2
k-tape TM transition $ 1 3 2

51. 2k-tracks Simulating k-tapes
Corresponding simulation sequence:
Notable: ready for next k-tape TM transition $ 1 X $ 1 3 2
k-tape TM transition $ 1 3 2

52. 1 tape simulating k-tracks
Finally, convert the 2k-track machine to a single tape machine. In a sense, we are already done, as there is only one head.
Q: How can the information about a column of tracks be recorded by a single cell? 1 X

53. 1 tape simulating k-tracks
A: Just expand the tape alphabet G! In this case, the expanded tape alphabet would consist of 4-tuples, and the 4-tuple (,1,X,) would represent the column on the right.
Thus a k-track TM is already a 1-tape TM if we look at it the right way. No further modifications are necessary.
This complete the simulation of k-tape TM’s by 1-tape TM’s.  1 X