[js8]

I think this is exactly the unhelpful tactics that have prevented people figuring out the problem. I have both theoretically and practically verified the 2XSAT reduction, and I believe it's a step towards P=NP. But, it's being dismissed out of hand because I don't have a practical, fully polynomial, algorithm.

So I cannot publish that (I am well aware of the unfortunate situation that only a practical implementation will now convince people that P=NP).

Add to it, why should I? What if it's not that far from a full solution, and somebody else will get the prize?

I came to understand why Perelman refused the prize. Mathematics should be about collaborative understanding of the universe, not about people working in isolation until they have fully working superoprimized implementation that can crack Bitcoins.

[klyrs]

> I have both theoretically and practically verified the 2XSAT reduction, and I believe it's a step towards P=NP.

NP is generally thought to be harder than factoring, so I'm not sure that your reduction is a "reduction" in the sense that you've restated factoring in a (potentially-)harder-than-native problem space. Proving that factoring is polynomial would be a huge result indeed, but if your strategy requires you to prove P=NP along the way, you're focused on the wrong problem and I wouldn't expect you to get much traction.

[js8]

No, I am not reducing to factoring, I only tested some on it because it's a relatively easy way to get hard instances.

I am reducing to 2XSAT, which is a name for instances that are intersections of 2-SAT and XORSAT instances.

Both 2-SAT and XORSAT have polynomial algorithms, why is it hard to believe that their intersection has one too?

[adgjlsfhk1]

If your reduction can solve factoring, what are the factors of 22112825529529666435281085255026230927612089502470015394413748319128822941402001986512729726569746599085900330031400051170742204560859276357953757185954298838958709229238491006703034124620545784566413664540684214361293017694020846391065875914794251435144458199?

[js8]

I already said, I don't have an efficient algorithm. Once I will have, in my estimation, at best, it will be O(n^3) (n is number of variables), which still means a lot - to factor 16-bit integer, my (linear) 2XSAT reduction requires about 4000 variables, so the number of variables for a cryptographically-strong problem will be in millions. You need to have a very efficient parallel algorithm to deal with that, and that's very low on the list of priorities - first I need to understand how to actually make the algorithm efficient (so far I think I proved there is a polynomial bound - around O(n^8) or so, but I know for sure it's very inefficient, because I am doing it very naively).

There are other ways to improve the method, which (if indeed P=NP) are incredibly interesting - you can directly compose presolved general instances and specialize on them. Kinda like if you need to compute many solutions to linear equations, you only need to factor the matrix once.

[tgflynn]

Why don't you try testing your algorithm on some 1000 variable or so SAT instances ? There are thousands of such problems that have been created for the SAT competitions. If your code can solve all of them and is really P-time then I think there are many people who would be interested in looking at your algorithm, myself included.

[js8]

That's what I am generally doing and planning to do, however right now, the theory had priority (there is still couple weak spots in my proof which I need to patch up). But I will return to testing once I will have a better idea what I want the algorithm to do (as I mentioned, a more naive version of the method that combined solving 2-SAT and XORSAT failed with a bug, which I think I now understand).

I think testing O(n^8) algorithm is pointless, so it needs more polishing (naive algorithm for 2-SAT (that follows from Krom) is O(n^3) or so, but the best methods are linear; so I feel there is a lot of room for improvement, but obviously my method is a little bit more complicated than 2-SAT, which it generalizes).

[FartyMcFarter]

The intersection of two sets is a subset of both sets. So that means you're reducing to 2SAT?

[js8]

No, you misunderstand, the intersection is in the solution to the instance. The 2XSAT are problems that can contain both 2-SAT clauses (two literals per OR clause) and XORSAT clauses (linear equations). 2-SAT (as well as XORSAT) are just special cases of that. You can also think of it as 2-SAT, but confined into a linear subspace of Z_2^n.

[klyrs]

That isn't the intersection of 2SAT and XORSAT, it's the union. Problems in the intersection would be solvable by either type of solver. I don't think it's "obvious" that your problem class should be polynomial; 2XSAT as you've described it (is it your own invention? I haven't found a reference) appears to be a strictly more powerful problem class.

[js8]

It's not a union of those classes, it's a different class, and as you say, it's more powerful, because it can be projected (my reduction adds additional variables) into 3-SAT and SAT instance.

Yes, 2XSAT is the name I gave it, and I couldn't find it anywhere. The reduction is surprisingly simple, yet nobody mentions it. That's why I am warning people here - just based on this alone, I 80% believe that P=NP with a practical algorithm (which either way involves solving linear equations). And I wouldn't be surprised somebody coming up with the algorithm.

The reason why I say it's an intersection is because that's how the set of solutions of an instance looks like. That's what we need to figure out - how to characterize the sets of solutions described by SAT instance (i.e. sets of assignments to boolean variables that satisfy the instance).

However, it's not that easy, even if you characterize them as interesections of 2-SAT and XORSAT instances, set of solutions to 2-SAT is notoriously hard to characterize too, for example, #2SAT is not known. And polynomial algorithms for 2-SAT and XORSAT are doing very different things, and it's not at all obvious how to generalize them into a common algorithm that can do both.

[FartyMcFarter]

> somebody else will get the prize?

> Mathematics should be about collaborative understanding of the universe

If you really think mathematics should be about collaboration and not competition, why do you worry about who gets the prize?

You should pick what's more important to you - if it's collaboration, why not publish the steps you've accomplished? If someone else uses it to achieve a full solution, you'll still have a claim to part of the credit towards the solution.

[js8]

Because I am also a human. I think I could still use the money (if anything, very few people want to be principled idiots who refuse $1m), and if I spent several years determined to understand it, perhaps it's worth spending some more to understand it fully. As I explained, it creates an atmosphere that prevents collaboration - it's incredibly hard to find anyone to talk about the problem with. Anyway, I am writing about my approach here, and I am about to write about it more. Still, people dismiss it outright.