$L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in $M(x)$ running $M$ moves its head left at least once $\}$
Is $L$ decidable?
2026-03-26 02:34:48.1774492488
Is it decidable: is there an input for which turing machine will move its head left?
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Yes, make tree diagram of possible states and values being read $0,1$. If it moves left its in the set and you have your input. Continue the tree until all current values have occured before, if it hasnt gone left it never will since now it will just loop. The fact that it can write new entries is irrelevant since it cannot go back and read them.