# Is it decidable to know the number of positions used by a Turing machine for a fixed input?

I'm having trouble proving if the following language is recursive, recursively enumerable, or not r.e. at all: the set of all encodings of Turing machines $$M$$ such that the number of positions in the tape used by $$M$$ when the input is 0011 is less than 10.

I'm suspicious the problem is unrecognizable but I can't come up with any reduction to prove it. Any help would be appreciated.

• You can run $M$ on $0011$ until it either loops or uses more than $10$ positions. – Yuval Filmus Apr 8 '20 at 16:18
• Hint, the above comment tells you the way to show the problem is decidable. – John L. Apr 8 '20 at 16:44
• @JohnL. I have to know what happens if M loops forever. If I "bound" the number of steps looping within the same 10 positions, it is guaranteed that, if M loops forever, it will do it without visiting more than those positions. But I can't find a pattern. – Fran Castro Apr 8 '20 at 17:11
• @YuvalFilmus but M might loop and use less than 10 positions. – Fran Castro Apr 8 '20 at 17:13
• If it uses more than 10 positions, you can immediately stop. Otherwise, either it halts at some point, or it loops at some point (we even have an a priori bound on when that must happen, though this is not necessary). – Yuval Filmus Apr 8 '20 at 17:16