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In the proof of the time hierarchy theorem given on page 69 of Arora-Barak, we define a TM D as follows:

"On input $x$, run for $|x|^{1.4}$ steps the Universal TM $\ \mathcal{U}$ of Theorem 1.9 to simulate the execution of $M_x$ on $x$." Here $M_x$ is the machine encoded by string $x$.

Since we are simulating another TM on $D$, this would implicitly make $D$ a universal TM, right? If this is the case, then why is it necessary to state that $D$ will simulate $\mathcal{U}$ in order to simulate $M_x$ when $D$ is already capable of simulating $M_x$ directly?

On the other hand, if it is not the case that $D$ is a UTM, then why can't we "run" (whatever that means) $M_x$ directly on $x$ instead of running $M_x$ through $\mathcal{U}$ (in the same way that we "run" $\mathcal{U}$)? Why do we need to simulate a machine capable of simulating an arbitrary TM if we only need to simulate exactly one TM?

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The difference between $\mathcal{U}$ and $M_x$ is that $\mathcal{U}$ can be hardcoded while $x$ is an input to the machine.

In other words, when Arora and Barak write "run $\mathcal{U}$ to simulate $M_x$", what they mean is to switch to a part of the Turing machine which behaves like $\mathcal{U}$ (you can think of it as a function), and afterwards switch to some other part of the machine. The states of $\mathcal{U}$ do the work of simulating $M_x$.

You are not simulating $\mathcal{U}$. You incorporate its code as part of your Turing machine. (In programming languages, this is sometimes known as inlining.)

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  • $\begingroup$ Makes much more sense, thank you. It's strange that they don't mention this explicitly anywhere in the sections on Universal simulation. $\endgroup$ – kotu Jun 10 at 20:21

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