# Time constructible function T in equivalence of Turing Machines

I'm reading Computational Complexity by Arora and Barak and I had a doubt regarding a statement made about the equivalence of Turing machines:

For every $$f\colon \{0, 1\}^∗ → \{0, 1\}$$ and time-constructible $$T\colon \mathbb N → \mathbb N$$, if $$f$$ is computable in time $$T (n)$$ by a Turing machine $$M$$ using alphabet $$\Gamma$$ then it is computable in time $$4\log|\Gamma|T (n)$$ by a Turing machine $$\tilde{M}$$ using the alphabet $$\{0, 1, \square, B\}$$.

Specifically, what if any role $$T$$ being time constructible has here?

The proof (sketch) encodes $$\Gamma$$ in binary, and uses a larger register to simulate $$M$$'s transition function, and also a counter. But I don't think $$T$$ being time constructible comes into play. Could someone confirm or tell me if I'm missing something?

• It doesn't seem to come into play. But since we never encounter non-time-constructible functions in practice, the difference doesn't really matter. Feb 13, 2021 at 8:49