Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

@article{Buss1997ProofCI,
  title={Proof complexity in algebraic systems and bounded depth Frege systems with modular counting},
  author={Samuel R. Buss and Russell Impagliazzo and Jan Kraj{\'i}cek and Pavel Pudl{\'a}k and Alexander A. Razborov and Jir{\'i} Sgall},
  journal={computational complexity},
  year={1997},
  volume={6},
  pages={256-298}
}
We prove a lower bound of the formN Ω(1) on the degree of polynomials in a Nullstellensatz refutation of theCount q polynomials over ℤ m , whereq is a prime not dividingm. In addition, we give an explicit construction of a degreeN Ω(1) design for theCount q principle over ℤ m . As a corollary, using Beameet al. (1994) we obtain a lower bound of the form 2 NΩ(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCount q N from instances of the counting… CONTINUE READING
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