Abstract
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 2NΩ(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCount N q from instances of the counting principleCount M m .
We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations.
Further we shwo that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connectiveMOD p follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987) and Smolensky (1987) (in fact, these two proof systems are basically equivalent).
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Buss, S., Impagliazzo, R., Krajíček, J. et al. Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. Comput Complexity 6, 256–298 (1996). https://doi.org/10.1007/BF01294258
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DOI: https://doi.org/10.1007/BF01294258