Decidable? If a Turing Machine M, on input w, will M ever move its
read/write head to the left?
I think the problem is decidable. We simulate M for |w| steps. Either it has moved to the left (accept) or otherwise, M never moves left on w (reject). In the latter case, M is making a single pass on the string. It eventually finishes reading w and starts reading blanks/spaces. We simply look at which state q M is in when it finishes reading w and look at the subgraph of states that M visits on reading spaces. None of these should ever involve moving to the left. Thus, we reject it.
Does my logic make sense?