Question

Prove or Disprove

1. Suppose we construct arrays of integers. Let S be the set of all arrays which are arranged in sorted order. The set S is decidble.
2. A Turing machine with two tapes is no more powerful than a Turing machine with one tape. (That is, both types of machines can compute the same set of functions.)

Answer 1

Solution 1: Yes, the set of all arrays that are sorted within the collection of all the arrays would always be decidable. This is because not all the arrays are going to be sorted and only some arrays out of a number of arrays would be sorted and hence their number can be determined using the counting measurement.

Solution 2: A two-tape Turing machine has two tapes and each tape has its own head that can be used for the reading as well as writing. No, a two-tape Turing machine is not powerful than a single tape Turing machine and this is due to the fact that a single tape Turing machine can simulate any number of tapes by only increasing the time by a quadratic factor.

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