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Is the following language decidable? $$ L = \{ \langle M,w \rangle \mid \text{$M$ moves its head left at least once when run on $w$}\}. $$

I feel like this is a decidable language. But I don't know the algorithm to prove that it is decidable. Any help would be appreciated.

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  • $\begingroup$ $L$ is clearly acceptable. Suppose that $\langle M, w \rangle$ never moves its head left, how many different configurations $<q_0, s>$, where $q_0$ is a state, and $s$ is the suffix of the tape starting at the current head position, can $M$ be in? $\endgroup$ – Steven Apr 24 '20 at 10:51
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Let's assume for simplicity that at each step, the head moves either left or right (but doesn't stay in place).

Suppose that $M$ never moves its head left when run on $w$. After the machine traverses the length of $w$, it reaches a blank cell, and from this point on, will always fall on a blank cell. Therefore the only information that changes from step to step is the current state of the machine. After at most $|Q|$ steps (where $Q$ is the set of states), the machine will either halt or repeat a state, in which case it enters an infinite loop.

This gives a simple algorithm for deciding your language, that I'll let you spell out.

If the head is also allowed to stay put, the argument becomes a bit more complicated, but the conclusion is identical.

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  • $\begingroup$ It means that all the possible states of the turing machine M on which it is reading the blank symbol after w will be rejected as none of these involve moving to the left and rest are accepted. am I correct? $\endgroup$ – jumpy123 Apr 24 '20 at 12:34
  • $\begingroup$ I'm not sure what it means for a state to be rejected. $\endgroup$ – Yuval Filmus Apr 24 '20 at 12:35
  • $\begingroup$ by rejected I mean that it will not be accepted on those states. $\endgroup$ – jumpy123 Apr 24 '20 at 12:41
  • $\begingroup$ I don't see why. The machine $M$ can perfectly choose to halt, accepting or rejecting, at a fixed number of steps after finishing scanning $w$. $\endgroup$ – Yuval Filmus Apr 24 '20 at 12:42

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