The exercise I'm trying to solve is: Is the language $L = \{<M>,w\ |\ M \text{ moves its head on input } $w$ \text{ at every step}\}$ decidable or undecidable. We work with a version of the TM that doesn't just move its head to the left or right but can also stay in place. My feeling is that the problem is undecidable and I'm trying to find a way to reduce the special Halting problem to this problem, but until now without any luck. Any tips would be very appreciated, thanks in advance!
2026-02-23 22:51:41.1771887101
Given a Turing Machine $M$ and a word $w$, can we tell if the Turing Machine will move its head at every step?
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