Determining if a TM decidable or not, with limited information

Question

Were being asked to determine the whether this Turing Machine is decidable or not

"Given a two way, one-tape DTM $M$ whose tape set is $\Gamma=${$a,b,B$} and a string $x\in${$a,b$}*, determine whether $M$ will ever overwrite a symbol $a$ by a symbol $b$ in its computation on input $x$"

I'm not looking for answers, just hints on how to even approach this problem. Like, would we declare a language which is a subset of that language, and then apply reducibility?

Answer 1

A TM cannot be "decidable". What you are asking is whether the language that you describe is decidable.

Hint: the language $A_{TM}=\{<M,w>: M \text{ accepts }w\}$ is undecidable even if we restrict to TMs with a binary work alphabet. Try to reduce from this problem, by utilizing only the letters $\{a,B\}$ from $\Gamma$. Then, you can use $b$ only when you need it.

Please comment if you need further hints.