Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutations
We show that constant-depth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constant-depth Frege with counting axioms systems.
nullstellensatz refutations theory modular counting axioms propositional proof complexity