Hitting formulas have been studied in many different contexts at least since [Iwama 1989]. A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. [Peitl and Szeider 2022] conjectured that the family of unsatisfiable hitting formulas should contain the hardest formulas for resolution. They have supported their conjecture with experimental findings. Using the fact that hitting formulas are easy to check for satisfiability we use them as a foundation of a static proof system Hitting: a refutation of a CNF in Hitting is an unsatisfiable hitting formula such that each of its clauses is a weakening of a clause of the refuted CNF. Comparing this system to resolution and other proof systems is equivalent to studying the hardness of hitting formulas.
Our first result is that Hitting is quasi-polynomially simulated by tree-like resolution, which means that hitting formulas cannot be exponentially hard for resolution, so Peitl and Szeider's conjecture is partially refuted. We show that tree-like resolution and Hitting are quasi-polynomially separated, but for resolution, this question remains open. For a system that is only quasi-polynomially stronger than tree-like resolution, Hitting is surprisingly difficult to *polynomially* simulate in another proof system. Using the ideas of PIT for noncommutative circuits [Raz, Spilka 2005] we show that Hitting is simulated by Extended Frege, but we conjecture that much more efficient simulations exist. As a byproduct, we show that a number of static (semi)algebraic systems are verifiable in a deterministic polynomial time.
We consider multiple extensions of Hitting. In particular, refutations in a proof system Hitting(?) are conjunctions of clauses containing affine equations instead of just literals, and every assignment falsifies exactly one of the clauses. This system is related to Res(?) proof system for which no superpolynomial-size lower bounds are known: Hitting(?) simulates the tree-like version of Res(?) and is at least quasi-polynomially stronger. We show that formulas expressing the non-existence of perfect matchings in the graphs K_{n,n+2} are exponentially hard for Hitting(?).
