# Is 3-SAT is more difficult than the SAT? (the same problem - the conversion: SAT => 3-SAT)

I have a question. SAT in which all the clauses have three variables (3-SAT) is more difficult than a traditional SAT which may also include clauses one and two variables? (Of course there are also clauses of the three variables but not all have three variables. More than 3 variables in clauses not exist.)

After my converted a traditional SAT to 3-SAT, analysis of the problem 3-SAT is more difficult for SAT solvers? Meybe it not change anything?

It may seem strange but I want to get after the conversion specified function was difficult as possible.

• If there was a polynomial-time algorithm for 3SAT we could use the reduction to get a polynomial-time algorithm for SAT. They're not really "of equal hardness" though: the degree of the polynomial would almost certainly be different. But since we don't have a polynomial-time algorithm for 3SAT, things are even worse: an exponential-time algorithm for 3SAT might take vastly more time (say $2^{n^2}$ as opposed to $2^n$) on the reduction of a size-$n$ SAT problem than it would on a size-$n$ 3SAT problem. Dec 9, 2019 at 17:00
• yes thats fine. My point was I was unaware of the fact that generic problems are / can be also reduced to specific problem. Till now I was of opinion that only specific (comparatively easier) problems are reduced to generic (comparatively harder) problems. It seems that I was incorrectly interpreting reduction symbol as $<$ and thinking that problem on left hand side of $<$ must be easier than the one on right hand side. But the reduction symbol is $\leq$, So, I guess in $SAT \leq 3SAT$, that $=$ in $\leq$ is getting into play.