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After Reading again the answer to this question and the answer to this question,

I am wondering if the language $L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in $M(x)$running $M$ doesn't move left $\}$ can be decide as well.

Maybe we can use the same method as in the last question above, and somehow locate a path which will lead us to this conclusion, by examining the transitions and etc.

What do you think?

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    After having received and understood answers to your other questions, you should be able to come with something more here.2012-08-12

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