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A function $T: \mathbf{N} \rightarrow \mathbf{N}$ is time constructible if $T(n) \ge n$ and there is a Turing Machine $M$ that computes the function $x \mapsto \lfloor T(|x|) \rfloor$ in time $T(n).$

Here $\lfloor m \rfloor$ denotes the binary representation of $m$ for a natural number $m$.

I understand that $T(n) \ge n$ is required in order to give the $M$ sufficient time to read its input. About the mapping $x \mapsto \lfloor T(|x|) \rfloor$: since $x$ is a natural number, does this mean $|x|$ is the number of digits in the decimal representation of $x$?

As an example, let's forget about the the requirement that $T(n) \ge n$. Suppose $T(n) = \log_2 n$. Then, for example, $T(2^{100}) = 100$, but what is $T(|2^{100}|)$ in this case?

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$|x|$ here means the length of the string $x$ over an alphabet $\Sigma$. Note that $x$ does not need to be a natural number.

So, $|2^{100}|$ is $100$ (but do remember that you MUST understand that you are encoding the natural number $2^{100}$ into string over finite alphabet) if we want to encode a natural number in binary. Therefore, $T(|2^{100}|) = T(100) = \log_2100$.

It would be type-safer to write $|\lfloor{2^{100}}\rfloor|$

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