Turing machine that erases its input

Question

I have got this question :

Consider a Turing machine 𝐶_𝑤 which erases its input, writes 𝑤 on the tape, and halts while scanning the leftmost character of 𝑤. Design the Turing machine 𝐶₀₁₁

I need explanation what actually is the question and what 𝐶_𝑤 does. I kind of understand it writes empty symbols on every input it gets, but the rest is unclear to me. Hope someone can help me understand the question and what is required me to do.

Answer 1

In your case w = 011.

Indeed, the TM should first overwrite the entire input. I think we can assume that the input does not have gaps. So as soon as the TM reads an empty space on the input tape, it should begin writing 011.

When writing the second 1, enter into a state for which there do not exist any transitions. This way you ensure that the machine halts on that position. Nothing is said explicitely about whether this state should be accepting, but it would make sense to have it as the unique accepting state.