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

S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. A. Razborov, J. Sgall
1996 Computational Complexity  
We prove a lower bound of the form N Ω(1) on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Z m , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N Ω(1) design for the Count q principle over Z m . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 N Ω(1) for the number of formulas in a constant-depth Frege proof of the modular counting principle Count N q from instances of the
more » ... nting principle Count 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 show that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective MOD 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) , Smolensky (1987) (in fact, these two proof systems are basically equivalent).
doi:10.1007/bf01294258 fatcat:pa354ea7sbatdc4vfzudhizm6i