# Non-trivial reduction form SAT to $3$-SAT

Looking for any idea for reduction from $$SAT \leq 3-SAT$$ where $$SAT$$ is known to have $$d$$ variables at most in each clause. I am looking for a reduction in which the resulting formula will not depend on the value of $$d$$. The classical reduction adds $$O(d)$$ new variables per clause, so this is not good to my case.

Do such reductions even exist?

Edit: Even if it's impossible to find such reductions, maybe there are other interesting reductions which are not the classical one, they can also be interesting?

• You can't have it not dependent on $d$. Consider just a single clause with $d$ different variables. After the reduction, it must have at least $\frac{d}{3}$ clauses, since each clause can "hold" only up to $3$ variables, and there are in total $d$ of them. Dec 20 '21 at 20:56