# Is a TM that is simulated by a universal TM theoretically inherently slower than the TM itself?

When a CPU simulates a certain program, as they do all the time, this is inherently slower than if the program would have been "baked in" into the hardware and computed directly. We know this from practical experience.

My question is: Is there some theoretical relationship that shows that a program that is simulated by a universal program, is inherently slower than that program executed by itself?

• The question is not well-posed. You need to fix a particular hardware model, or else we can make the answer "yes" or "no", as we wish. – Andrej Bauer Aug 7 '16 at 8:54
• @Andrej Bauer, But my question is precisely whether there is a theoretical relationship that holds accross all imaginable computer models. for example there could be a theorem that states that for ANY arbitrary TM, a UTM simulating that TM is always slower than the TM itself by some minimum factor or some order of magnitude. – user56834 Aug 7 '16 at 9:34
• So you've fixed your computational model to Turing machines? How about a model in which there is a universal machines which runs faster than other machines? Do you see my point? – Andrej Bauer Aug 7 '16 at 19:42
• @Andrej Bauer, but you can still compare that universal machine with the program running directly on a machine of similar design, right? As a very simple example: ofcourse a program running within a simulated virtual computer on a 2016 CPU (a TM running on a UTM running on a UTM) will run faster than the same program running directly on a 1990 CPU, but that doesn't mean that we can still compare the program on the 2016 CPU versus the program on a virtual computer on a 2016 CPU. I want to know if there is a theoretical upper limit to the efficiency of simulating a program. – user56834 Aug 7 '16 at 19:56
• For example, a UTM will need to make more tape head moves to simulate a TM than the TM itself would ... but each state-transition lookup of the UTM has a fixed, small constant size while the TM's state-transition table might be massive. – usul Aug 8 '16 at 10:27

No. There is no such proof. There exists a universal Turing machine $U$ and a machine $M_0$ such that $U$ simulating $M_0$ is faster than running $M_0$ directly.

For instance, $M_0$ might implement sorting using a bubble sort. $U$ might be a universal Turing machine that has an extra check: it checks its input, and if its input is exactly $M_0$ (it has the source code of $M_0$ hardcoded and it checks for that one particular string on its input), then instead of simulating $M_0$ step-by-step, it instead branches to execute Turing machine $M_1$, which is a faster version of $M_0$ (e.g., using mergesort instead of bubble sort) that otherwise has the same behavior. Such a $U$ is universal and has the property I articulated above.

Bottom line: No, simulation by a universal Turing machine is not necessarily slower than running the Turing machine itself. There is no such theorem.

That said, I don't see why you need such a theorem. As you said, we all know that simulation is typically slower than direct execution. That ought to be enough.

• In this case, have you actually simulated $M_0$? Or have you just started with a simulation of $M_0$ then modified it? – jmite Aug 7 '16 at 16:05
• I would not call this simulation, though perhaps that is the way the word is used. Anyway, the reason why I'm asking the question is because I want to know whether there is some "upper limit" on the efficiency of simulating a program using another program. if such an upper limit exists/does not exist, this would be good to know, just like it is good to know the upper limit efficiency of a Carnot Cycle in thermodynamics. – user56834 Aug 7 '16 at 19:49
• This is the Turing Machine equivalent of the Volkswagen emissions scandal :-) – yatima2975 Aug 9 '16 at 15:42

The way i see it, talking about slowdowns on simulations of a specific Turing machine $M_0$ doesn't make much sense. I could always just run $M_0$ and call this a simulation, which will result in no slowdown. I could also hardwire the code of $M_0$ , and in case the input was $M_0$, use some better algorithm (as D.W. did in his answer).

The more interesting question here is (in my opinion at least), what is the optimal slowdown achievable when simulating an arbitrary Turing machine $M$ on some input $x$ ? (asymptotically, in terms of $|x|$ and the length of the description of $M$)

We look at all possible inputs, and examine the worst case slowdown (perhaps for some machine $M_0$ you can do a better job, but here we consider the worst case running time).

More formally, Let $\mathcal{U}(\langle M\rangle,x)$ denote the universal Turing machine, which takes as input an encoding of a machine $M$ and some string $x$, and outputs $M(x)$, or does not halt in the case that the computation of $M$ on $x$ does not halt. We know that we can implement $\mathcal{U}$ in such a way that if the computation $M(x)$ requires time $T$, then $\mathcal{U}(\langle M\rangle,x)$ requires time $O\left(T\log T\right)$. Here the $O$ notation hides constants which depend on the number of states and the alphabet size of $M$ (but independent of $|x|$). Your question then translates to whether we can implement $\mathcal{U}$ such that the computation of $\mathcal{U}(\langle M\rangle,x)$ requires only $O(T)$ time?

It seems that for single tape machines, it is not known whether this $\log T$ factor is necessary, however for $k\ge 2$ tapes machines we can avoid it (proved by Furer, 1982). See this post by Kaveh for a detailed discussion and related quotes.