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How can I prove that for a natural number K, a language that accepted by a Turing machine with K cells after the input word ends, belongs to R (which R is the set of languages that there is Turing machine that accept them while for each input the run is finite)?

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  • $\begingroup$ after the input word ends I take this to apply to both "ends". $\endgroup$
    – greybeard
    May 23, 2020 at 10:34

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Given the length of the input $n$, you can come up with a bound $N(n)$ on the number of configurations that the Turing machine can have. If the machine hasn't stopped on an input within $N(n)$ time steps, then it will never stop (why?), and you can use this to complete the proof. Details left to you.

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  • $\begingroup$ got it, thanks a lot! $\endgroup$
    – Stephan
    May 22, 2020 at 15:36

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