Constant-Depth Frege Systems with Counting Axioms Polynomially Simulate Nullstellensatz Refutations
<|reference_start|>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 formulas to systems of polynomials with the property that, for most previously studied translations of formulas to systems of polynomials, a formula reduces to its translation. 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, small refutations for certain CNFs by constant-depth Frege systems with counting axioms.<|reference_end|>
@article{impagliazzo2003constant-depth,
    title={Constant-Depth Frege Systems with Counting Axioms Polynomially Simulate
  Nullstellensatz Refutations},
    author={Russell Impagliazzo and Nathan Segerlind},
    journal={arXiv preprint arXiv:cs/0308012},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0308012},
    primaryClass={cs.CC cs.LO}
}