For each action below give the full specifications (formal description) for a Turing machine that computes it. Assume = {a, b} and = {a, b, B, X, Y, $}.

a) From any configuration (tape, current state), the TM will find the next B to the right of the tape head.

b) From the initial configuration (tape, q₀), the TM will put a $ as the first tape symbol and move the entire input one place right.

c) From any configuration (tape, current state), the TM will rewind to the left end of the tape where there is a $.