# Does a given Turing machine works in time limited by $5n$?

There doesn't exist an algorithm which decides on the following problem: Does a given Turing machine work in time limited by $5n$? $n$ is the length of $w$. $w$ is an input word.

No, because if it was true then we would decide on problem: Does an universal machine accept a word $w$.
• What do you mean by "...Turing machine works in time..."? Works for at least $5n$ steps? Also please define what $n$ is. – fade2black Aug 27 '17 at 2:20
• @Evil The set of Turing machines which halt in time (at most) $5n$ steps on all inputs is decidable, for some fixed $n$. No? – fade2black Aug 27 '17 at 3:02
• @fade2black yes, you are absolutely right, it is decidable. I have parsed the question differently, 5n not as the numbers of steps it performs, so given that I retract my comment, it does matter what $n$ denotes, thank you. – Evil Aug 27 '17 at 3:32