Several alternative definitions of Turing machines exist, all of which produce machines that are equivalent in computational ability to the Turing machine defined in this chapter. One of these alternative definitions is the multitrack Turing machine. In a multitrack Turing machine, there are multiple tapes. The machine reads a cell from each of the tapes and, on the basis of what it reads, it writes a symbol on each tape, changes state, and moves left or right. Figure 12.13 shows a two-track Turing machine currently in state 1 reading a 1 on the first tape and a 0 on the second tape.
An instruction for this Turing machine has the following form:
(current state, current first tape symbol, next first tape symbol, current second tape symbol, next second tape symbol, next state, direction of move)
An instruction of the form (1,1,0,0,0,2,R) applied to the machine configuration of Figure 12.13 results in the configuration shown in Figure 12.14.
As in the original Turing machine definition, some
conventions apply. Each tape can contain only a finite number of nonblank symbols, and the leftmost nonblank symbols must initially “line up” on the two tapes. The read head begins in this leftmost nonblank position in state 1. At any time, if no instruction applies to the current machine configuration, the machine halts.
a. Design a two-track Turing machine that compares two binary strings and decides whether they are equal. If the strings are equal, the machine halts in some fixed state; if they are not equal, the machine halts in some other fixed state.
b. Solve this same problem using the Turing machine defined in this chapter.
c. Prove the following statement: Any computation that can be carried out using a regular Turing machine can be done using a two-track Turing machine.
d. On the basis of parts (a) and (b), make an argument for the following statement: Any computation that can be carried out using a two-track Turing machine can be done using a regular Turing machine.
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