# When the tree-like resolution size is the same with general(regular) resolution size？

Background:
For an unsatisfiable SAT formulas, the length of a resolution refutaion means the number of clauses in it. It's well known that there exist exponential separation between tree-like and general resolution length.
i.e. there are some unsatisfiable SAT formulas whose shortest general resolution refutation length is $$S$$, while the shortest tree-like resolution refutation length is $$2^{\Omega(S/logS)}$$. see

Problem:
Conversely, I am wondering for what kind of unsatifiable SAT formulas, the shortest tree-like and general resolution refutation length is the same/polynomially?

My purpose:
My purpose is to get the lower bound of general(regular) resolution for an unsatisfiable SAT formula. By far I have get the lower bound of tree-like resolution.and I want to show there's no big separation in my SAT instance to get the lower bound of general(regular) resolution.
• @notautogenerated Exactly lower bound for regular resolution. Since general size is always $\leq$ regular size. If I can get lower bound for general resolution, it makes sense.
• They are the same up to a polynomial speedup e.g. for all CNF that require size $2^{\Omega(n)}$ in general resolution, or that have polynomial-size tree-like resolution refutations. However, I guess this is not very helpful; I am not aware of any easily recognizable class of CNF where the speedup is only polynomial. Apr 21, 2023 at 6:53