[theGnuMe]

That jives with my intuition. I had forgotten about the connection between Gaussian elimination and xor sat. That does sound like a good direction to reason from.

How do you prevent an exponential increase in linear terms though?

Good luck!

[js8]

I am still not sure about the exact role of Gaussian elimination. I thought that it will be somehow necessary, but it seems that if you only break linear equations to 3 variables per equation (by adding extra variables), you can get away without it. Although it certainly doesn't hurt to do it, I think for efficient algorithms, it will be required (and so the whole endeavor will be at least O(n^3)).

To prevent exponential increase in linear terms, I work within a sort of limited logic that only allows formulas that are ORs of (implications between) two linear equations of up to 2 variables. This logic accommodates both 2-SAT clauses and linear clauses of 3 variables, and seems to be able to emulate Gaussian elimination to some extent, but I am not yet completely sure how. Last week, I think I was able to generalize Krom's proof of refutable-completeness for 2-SAT logic (that has linear equations on 1 variable) to this logic, but I am in the process of double checking my result for gotchas like this.

There are other ideas how to control the linear terms, once you have the linear equations solved, it's easy to verify whether some variables are linearly dependent, because you have the basis, and modify the ordinary 2-SAT algorithm accordingly (2-SAT works by propagation via transitive dependency of literals, but if the 3 variables involved are linearly dependent, then you can derive lot more). These are mainly ideas how to make the polynomial algorithm even more efficient.

And thanks for encouragement!