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Given a TM $M$ that halts on all inputs, and a general string $w$, consider the most trivial algorithm (Call it $A$) to decide whether $M$ accepts $w$:

$A$ simply simulates $M$ on $w$ and answer what $M$ answers.

The question here is, can this be proven to be the fastest algorithm to do the job?

(I mean, it's quite clear there could not be a faster one. Or could it?)

And more formally and clear:

Is there an algorithm $A'$, that for every input $\langle M,w\rangle$ satisfies:

  1. If $M$ is a TM that halts on all inputs, $A'$ will return what $M$ returns with input $w$.

  2. $A'$ is faster than $A$.

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    $\begingroup$ There are (theoretically) infinitely many algorithms faster than that, each faster than the previous one. See, for example, this. $\endgroup$ – dkaeae Apr 1 at 14:02
  • $\begingroup$ @dkaeae Does that mean we can infinitely make any algorithm faster? $\endgroup$ – FireCubez Apr 1 at 16:41
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    $\begingroup$ @FireCubez In the technical sense of TMs, and for a particular meaning of infinity, yes. In the sense of algorithms running on real computers, no. $\endgroup$ – rlms Apr 1 at 19:19
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Is there an algorithm $A′$, that for every input $\langle M,w\rangle$ satisfies:

1) If $M$ is a TM that halts on all inputs, $A′$ will return what $M$ returns with input $w$.

2) $A′$ is faster than $A$ (In worst case terms)

It's not possible to be asymptotically faster by more than a log factor. By the time hierarchy theorem, for any reasonable function $f$, there are problems that can be solved in $f(n)$ steps that cannot be solved in $o(f(n)/\log n)$ steps.

Other answers point out that you can get faster by any constant factor by the linear speedup theorem which, roughly speaking, simulates a factor of $c$ faster by simulating $c$ steps of the Turing machine's operation at once.

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  • $\begingroup$ So for input M that runs in exponential time in worst case, does this mean that the (asymptotically) fastest algorithm (or family of algorithms) to improve A must be exponential two in worst case on the set of inputs that include this M with general string $w$? $\endgroup$ – Oren Apr 1 at 16:36
  • $\begingroup$ @Oren Exactly, yes. In particular, this is how we know that $\mathrm{EXP}\neq\mathrm{P}$: it tells us there can be no polynomial-time algorithm for an $\mathrm{EXP}$-complete problem. $\endgroup$ – David Richerby Apr 1 at 17:00
  • $\begingroup$ hey can you extend of the use of the time hierarchy theorem? It is still unclear to me as to why the specific algorithm can be reduced only by log factor as the theorem states only that exists such algorithms, though it doesn't follow immediately that A is one of them (those who can be improved only by a log factor) $\endgroup$ – Oren Apr 2 at 12:53
  • $\begingroup$ As I recall, we believe that the time hierarchy theorem ought to be strict, in the sense that there are things you can do in time $f(n)$ that can't be done in time $o(f(n))$ (analogous to the space hierarchy theorem), but that $o(f(n)/\log n)$ is the best anyone's managed to prove. You can't keep shaving off log-factors since, if you managed to do it even twice, you'd be at roughly $f(n)/(\log n)^2$, which is less than $f(n)/\log n$. $\endgroup$ – David Richerby Apr 2 at 13:11
  • $\begingroup$ Ok. still why is $A$ one of those algorithms (those that you can do in time $f(n)$ but not in time $o(f(n)/logn)$)? $\endgroup$ – Oren Apr 2 at 13:25
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Dkaeae brought up a very useful trick in the comments: the Linear Speedup Theorem. Effectively, it says:

For any positive $k$, there's a mechanical transformation you can do to any Turing machine, which makes it run $k$ times faster.

(There's a bit more to it than that, but that's not really relevant here. Wikipedia has more details.)

So I propose the following family of algorithms (with hyperparameter $k$):

def decide(M, w):
    use the Linear Speedup Theorem to turn M into M', which is k times faster
    run M' on w and return the result

You can make this as fast as you want by increasing $k$: there's theoretically no limit on this. No matter how fast it runs, you can always make it faster by just making $k$ bigger.

This is why time complexity is always given in asymptotic terms (big-O and all that): constant factors are extremely easy to add and remove, so they don't really tell us anything useful about the algorithm itself. If you have an algorithm that runs in $n^5+C$ time, I can turn that into $\frac{1}{1,000,000} n^5+C$, but it'll still end up slower than $1,000,000n+C$ for large enough $n$.

P.S. You might be wondering, "what's the catch?" The answer is, the Linear Speedup construction makes a machine with more states and a more convoluted instruction set. But that doesn't matter when you're talking about time complexity.

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Of course there is.

Consider, for instance, a TM $T$ which reads its entire input (of length $n$) $10^{100n}$ times and then accepts. Then the TM $T'$ which instantly accepts any input is at least $10^{100n}$ times faster than any (step for step) simulation of $T$. (You may replace $10^{100n}$ with your favorite largest computable number.)

Hence, the following algorithm $A'$ would do it:

  1. Check whether $\langle M \rangle = \langle T \rangle$. If so, then set $\langle M \rangle$ to $\langle T' \rangle$; otherwise, leave $\langle M \rangle$ intact.
  2. Do what $A$ does.

It is easy to see $A'$ will now be $10^{100n}$ faster than $A$ if given $\langle T, w \rangle$ as input. This qualifies as a (strict) asymptotic improvement since there are infinitely many values for $w$. $A'$ only needs $O(n)$ extra steps (in step 1) before doing what $A$ does, but $A$ takes $\Omega(n)$ time anyway (because it necessarily reads its entire input at least once), so $A'$ is asymptotically just as fast as $A$ on all other inputs.

The above construction provides an improvement for one particular TM (i.e., $T$) but can be extended to be faster than $A$ for infinitely many TMs. This can be done, for instance, by defining a series $T_k$ of TMs with a parameter $k$ such that $T_k$ reads its entire input $k^{100n}$ times and accepts. The description of $T_k$ can be made such that it is recognizable by $A'$ in $O(n)$ time, as above (imagine, for instance, $\langle T_k \rangle$ being the exact same piece of code where $k$ is declared as a constant).

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  • $\begingroup$ I get the answer from the comment above, really cool by the way, i didn't know that.. but this example is for a specific TM. I mean if $T'$ gets input of, for example, a TM that reject all inputs immediately, it won't work. or am I getting this wrong? $\endgroup$ – Oren Apr 1 at 14:13
  • $\begingroup$ If you are trying to prove a statement of the form $\forall x: A(x)$ wrong, then you only need to provide an $x$ which falsifies $A(x)$. (Here, $x$ is $T$ and $A(x)$ is the statement that simulating $T$ step for step is the fastest possible algorithm.) $\endgroup$ – dkaeae Apr 1 at 14:16
  • $\begingroup$ I get the logic, but this one example is what I can't see working. can you clarify the roles of $T$ and $T'$ in the algorithm based on $M$'s role? $\endgroup$ – Oren Apr 1 at 14:20
  • $\begingroup$ $M = T$, whereas computing $T'$ (or just answering "yes") is a faster algorithm than directly simulating $T$. $\endgroup$ – dkaeae Apr 1 at 14:23
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    $\begingroup$ Though in this example it doesn't check whether $M$ accepts $w$ so it won't always be correct. I mean $T'$ will answer correctly for $T$ and will do it faster than $M$ but for other inputs that are different from $T'$, for example with input of $T''$ that rejects all inputs immediately, it will return a wrong answer, so it is an example for a fast algorithm though is one that isn't always correct $\endgroup$ – Oren Apr 1 at 14:32

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