Proving Unsatisfiability with Hitting Formulas

A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. Hitting formulas have been studied in many different contexts at least since [45] and, based on experimental evidence, Peitl and Szeider [53] conjectured that unsatisfiable hitting formulas are among the hardest for resolution. Using the fact that hitting formulas are easy to check for satisfiability we make them the foundation of a new 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 and partially refutes the conjecture of Peitl and Szeider. We show that tree-like resolution and Hitting are quasipolynomially separated, while 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 Raz Shpilka s polynomial identity testing for noncommutative circuits [57] we show that Hitting is p-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 deterministic polynomial time. We consider multiple extensions of Hitting, and in particular a proof system Hitting(⊕) related to the Res(⊕) proof system for which no superpolynomial-size lower bounds are known. Hitting(⊕) p-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 Kn,n+2 are exponentially hard for Hitting(⊕) via a reduction to the partition bound for communication complexity.