Final answer:
The student inquires about a Turing Machine (TM) description that verifies if a unary number is a power of 2. A multi-tape TM approach with separate tapes for input, generating powers of 2, and comparison offers a solution. The TM doubles a unary 1 to represent successive powers of 2 and then compares it with the input.
Step-by-step explanation:
The student is asking for an implementation-level description of a Turing Machine (TM) that accepts if the unary representation of a natural number i is equal to 2ⁿ for some natural number n, and rejects otherwise.
To solve this problem, we can consider using a multi-tape Turing Machine, particularly a 3-tape TM. The first tape would contain the input number in unary representation. The second tape would be used to generate the powers of 2 in unary representation, and the third tape would be used for comparison.
Our TM would start by writing a single 1 on the second tape, representing the number 2⁰ (which is 1 in unary). Then, for each 1 on the first tape, the TM would double the number on the second tape by copying it again.
This effectively calculates 2ⁿ where n is the count of 1's on the first tape. Finally, a comparison is done between the first and second tapes. If they match exactly, the input number i is indeed a power of 2, and the TM accepts. Otherwise, it rejects.
This method aligns with the hint provided in the question, suggesting setting M=2 and solving for n. The TM doubles the value on the second tape until it either matches the input number or surpasses it. This operational method is an effective strategy to determine if the number is a power of 2.