Here is a good article you might be able to understand that pretty much provides an answer to your question.
https://www.quantamagazine.org/the-busy-beaver-game-illumina...
https://www.quantamagazine.org/the-busy-beaver-game-illumina...
The busy beaver problem, again, assumes it is running on a Turing machine with infinite memory.
Don't confuse a program defined using a finite number of states with a machine that can only have a finite number of states. It's just another example of the points I'm making about how a theoretical mathematical problem is an interesting curiosity and might even be an interesting problem to solve but it does not necessarily apply to the real world. And how everyone seems to think that this mathematical problem applies to programs running in real-world computers.
Someone else mentioned the busy beaver problem and I've given a more complete answer there:
In the real world, a program will either halt, loop, or crash, but this doesn’t help us if a 6 instruction program can take longer than the age of the universe to run while still terminating. No matter how fast you go through its states. 64mb leaves a lot of room for unique states without a cycle.
Why do you assume that you have to go through all the states of the program to determine whether it will run into a cycle?
To not believe my assumption would require you to believe that such algorithms do not exist.
Since the assumption has not been proven, this is why the foundation of cryptography is still on a bit of a shaky ground, as none of those algorithms are proven to be unbeatable (the one-time pad being the exception, although it's not very useful in practice).
It's probably also why weaknesses in cryptographic algorithms are still being discovered every now and then, unfortunately.
However, it is simple to show that this is at least as hard as NP complete. For instance, you can map the Hamiltonian cycle problem to a halting problem over programs that do an infinite walk on graphs without a Hamiltonian cycle and those that stop whenever they complete a walk along a Ham cycle.
At this time P vs NP is still wide open and, in general, statements in complexity theory are usually dependent on some unproven hypothesis so I still don't understand what is your point exactly.
Among other things, because the P vs NP issue is still wide open, yes.
I'd just like to add three more minor points:
1. Many people believe the above to be false, because they believe that the problem is undecidable also for programs with a bounded number of states.
2. That I would like if more people focused on finding such an efficient algorithm.
3. And that I think that even some people who believe that the problem is decidable think that it's impossible to have an efficient algorithm because of the Halting problem and Rice's theorem, which formally speaking, neither of them say anything about whether such efficient algorithms exist or not.
The P vs NP question is open but the conditional results are not merely novelties. There are many results that show very weird consequences for having P=NP. So either the nature of computing is utterly bizarre or in fact there are many common problems which are simply impossible to solve exactly in a reasonable amount of time.
You can of course try to find good heuristics that cover some common cases or even approximations of NP hard problems in some cases. There are literally thousands of researchers doing that every day.
I hope you are correct :-)
> There are many results that show very weird consequences for having P=NP. So either the nature of computing is utterly bizarre or in fact there are many common problems which are simply impossible to solve exactly in a reasonable amount of time.
Wow, that's very interesting! Can you recall an example or two of such consequences? I would be really interested in reading about that.
Edit: Donald Knuth seems to believe that P=NP and also says there would be no major implications if we found such a proof...
> You can of course try to find good heuristics that cover some common cases or even approximations of NP hard problems in some cases. There are literally thousands of researchers doing that every day.
Completely agreed!
Perhaps you are right and my worries are unfounded.
A nice presentation by Wigderson may also help to see consequences of P=NP in a less rigorous phrasing ("'Creativity' can be efficiently automated [if P=NP]"). https://view.officeapps.live.com/op/view.aspx?src=https%3A%2... There is a more formal text as well: https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/W06/w06....
Another class of results that I find interesting are inapproximability results. Namely, for certain NP hard problems, it is impossible to have a polynomial algorithm that computes the right value (for all instances) within a certain approximation factor unless P=NP. In https://dl.acm.org/doi/10.1145/1132516.1132612 Zuckerman shows that approximating MAX-CLIQUE is hopeless unless P=NP.
This is of course just a mountain of evidence and not a proof. It does mean that there are potentially many ways in which one could prove P=NP without necessarily achieving anything practical. This is what I think Knuth means: you could have a non-constructive proof of P=NP or perhaps an algorithm with complexity n^10^10^10 that solves an NP-complete problem.