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Thus, the transition function of M indicates a new contents to be printed in at most two of the three blocks, and the new place for the tape head, which is in one of these two blocks. In the third phase, the machine actually updates the contents of the tape in two moves, one to the left and one to the right. These moves must be done in the appropriate order to update the blocks. If only the middle block is to be updated, then one move suffices. In this case the machine performs an extra useless move to complete the dance.

Therefore, this machine accepts the same set as the nondeterministic machine M. We simulate in this way the nondeterminism by oracle queries, and construct the full accepting computation ofM. Of course, the next question that can be asked is whether the converse holds. Is it possible to simulate in a nondeterministic manner an oracle machine? An immediate negative answer follows from the fact that oracles are a much more powerful mechanism than suggested by the above simulation. For example, we could have L(M) itself as an oracle, and program the oracle machine just to copy the input into the query tape, query the oracle, and accept if and only if the answer is YES.

In this model there are no work tapes, and the input tape head is allowed to move in either direction and to change the symbols on its single tape. There exist many other modifications of the Turing machines which are equivalent to the basic model described above, in the sense that. they accept the same class of languages (the recursively enumerable sets). None is used in this book. We list some of them here: Turing machines with two-sided infinite tapes; Turing machines with several heads on each tape; and Turing machines with multidimensional "tapes".

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