Consider a variant of a Turing machine called an empty Turing machine. An empty

Turing machine is identical to a standard deterministic single$-$tape Turing machine

with one difference. An empty Turing machine has only a single halting state instead of

two. When the empty Turing machine enters this state if the tape is empty it accepts,

otherwise it rejects.

a$)$ Prove that an empty Turing machine can simulate a standard deterministic single$-$

tape Turing machine.

b$)$ Prove that a standard deterministic single$-$tape Turing machine can simulate an

empty Turing machine.