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I would like to know if there is a turing machine which can do the following:

take as input a turing machine T and integer n:
return true if the turing machine halts before time n and false otherwise

Clearly such a TM (turning machine) exists. Take the TM that simply runs T for n steps, then if it has halted return true and return false otherwise, however what I would like to know is if there exists a TM that can do better than this. I'm not sure how to formalize this question exactly. so hopefully it is clear what I am asking. Thanks

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    $\begingroup$ What do you mean by "better"? Do you mean "faster"? $\endgroup$ – xskxzr Feb 26 at 3:57
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Maybe the Turing Machine Linear Speed Up Theorem will answer your doubt (https://en.wikipedia.org/wiki/Linear_speedup_theorem)

Counter-intuitively or maybe intuitively, this states that given a Turing Machine which does something in say $n$ steps there are Turing Machines which can solve the same problem in $n/k$ steps for any constant $k$ independent of $n$. These Turing Machines would basically run $k$ steps of the original TM as $1$ step, hence having more state space, but linearly faster.

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