Tangential, but for some reason P vs. NP attracts an ungodly amount of cranks, probably because as far as open problems of that importance go it's easy to understand the question.
It's easy to get a superficial understanding of the problem, and then very easy to subtly mis-understand it.
I've reviewed published papers by respectable people where they've made one of several easy mistakes:
* Be careful about how you encode your problem, it's easy to make it "too large", at which point your problem can be solved in P in the input size. For example, don't represent a sudoku as triples "x-pos,y-pos,value", where those 3 numbers are encoded in binary, because now if I give you a problem with only 2 filled in values, you can't take the solution as your 'certificate', as your input is about size 6 * log(n), but the solution will be n * n * log(n).
* It's also easy if you write your problem as a little language to accidentally make it impossible to check in P time.
* When proving a reduction (say turning SAT into a Sudoku, to prove Sudoku is NP-complete), it's (usually) easy to show how solutions map to solutions (so you show how the SAT instance's solution turns into a particular solution to the Sudoku). It's usually much harder, and easy to make mistakes, showing the Sudoku can't possible have any other solutions.
Basically, they show that you can use SAT to solve Sudoku, and then claim that this makes Sudoku NP-complete. (All it shows is that Sudoku is in NP.)
People make similar mistakes often when they want to show that a certain problem isn't solvable in linear time, and they try to show that sorting can solve your problem. But it's the wrong direction.
Wait, did you mess up the direction here too, or am I confused? If you reduce problem A to B, then it means B is at least as hard as A, because solving it would solve A. Which certainly means in this case that Sudoku is NP-hard. And it doesn't (without introducing additional facts) imply Sudoku is in NP either. I don't see anything wrong here, do you?
Reducing unknown-complexity to NP-complete means you can bound the complexity by above, but not by below. I can reduce binary integer addition to SAT, which means that binary integer addition is no harder than SAT... but we also know by other algorithms that it is in fact easier than SAT.
To bound by below, you have to reduce NP-complete (or NP-hard suffices) to unknown-complexity.
That is, you've shown that an oracle that solves any SAT problem in constant time can solve any Sudoku in polynomial time.
The more difficult side is to show that for any SAT instance, you can reduce it to a Sudoku.
Really proving that you can use SAT to solve Sudoku is not a great or interesting result; since Sudoku is a decision problem it is very clear that it is in NP. Or see that verifying that a Sudoku solution is correct is achievable in polynomial time.
> Wait, did you mess up the direction here too, or am I confused? If you reduce problem A to B, then it means B is at least as hard as A, because solving it would solve A.
Using SAT to solve Sudoku is a reduction of Sudoku to SAT. The order of the problems names switches depending on how you write it.
Lololol
https://www.youtube.com/watch?v=IuX8QMgy4qE
Algorithmic isomorphism practically ensures most CS approaches will fail to formally model the problem clearly. To be blunt, that million dollar prize will be waiting around a long time.
An infinite amount of Papers do not necessarily have to converge on a correct solution. =3
Why would Gödel's Incompleteness Theorem stop them for P=NP in particular?
Why is the current definition of P or NP insufficiently formal or clear?
PS: Citing YouTube Videos in mathematical discussions is a big red flag indicating you have not really understood things.
> "not really understood things"
Something currently impossible to prove is by definition confusing. lol =3
PS: I read through the transcript of the YouTube video you linked (and also know the material from my CS degree education) so I do in fact know what Gödel's Incompleteness Theorem is. Note that the video is really not about P=NP at all, not any more than it is about 1+1=2. The use of P=NP in the video was just to give an example of "what is a statement."
People can twist it anyway they like, but stating they can "sort of" disprove or prove P=NP is 100% obfuscated BS. That is why the million dollar prize remains unclaimed.
Have a great day, =3
What? That's not even wrong. What do you mean by that?
https://www.youtube.com/watch?v=6vgoEhsJORU
Best of luck =3
The insane thing is that non-deterministic Turing machines are computable at all! It really feels like they belong to a different class of "magical" computers, or an axiom-of-choice / Banach-Tarski naval-gazing infeasible trick. You mean, you just "guess" the answers and your program will get the right answer?
But they are computable; the Church-Turing model of computation is fantastically powerful. Now the problem is just "feasibility" and "complexity". It seems to the initiate that there must be an answer hiding just around the next corner because it's SO OBVIOUS but in the end if you give someone n^100 time they can solve any problem that you care to pose but that still counts as P, so you're not going to stumble upon some grand insight.