Can you please help me understand whether or not the following the problem is recursive, recursively enumerable, or co-recursively enumerable?
A Turing machine $M$ is said to be obsessive if on every input $w$ $M$ goes through all of its states (except possibly the reject and accept states).
Input: (an encoding of) a deterministic Turing machine $M$.
Question: Is $M$ obsessive?
Thank you.