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Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting
"... We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, us ..."
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We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constantdepth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss
On the Degree of Ideal Membership Proofs From Uniform Families of Polynomials Over a Finite Field
"... Let f0 ; f1 ; : : : ; fk be nvariable polynomials over a finite prime field Fp . A proof of the ideal membership f0 2 hf1 ; : : : ; fk i in polynomial calculus is a sequence of polynomials h1 ; : : : ; h t such that h t = f0 , and such that every h i is either an f j , j 1, or obtained from h1 ..."
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Cited by 17 (2 self)
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Let f0 ; f1 ; : : : ; fk be nvariable polynomials over a finite prime field Fp . A proof of the ideal membership f0 2 hf1 ; : : : ; fk i in polynomial calculus is a sequence of polynomials h1 ; : : : ; h t such that h t = f0 , and such that every h i is either an f j , j 1, or obtained from h1 ; : : : ; h i\Gamma1 by one of the two inference rules: g1 and g2 entail any Fplinear combination of g1 , g2 , and g entails g \Delta g 0 , for any polynomial g 0 . The degree of the proof is the maximum degree of h i 's. We give a condition on families ffN;0 ; : : : ; fN;k N gN!! of nN variable polynomials of bounded degree implying that the minimum degree of polynomial calculus proofs of fN;0 from fN;1 ; : : : ; fN;k N cannot be bounded by an independent constant and, in fact, is\Omega\Gamma/31 (log(N))). In particular, we obtain an\Omega\Gamma/19 (log(N))) lower bound for the degrees of proofs of 1 (so called refutations) of the (N; m)  system (defined in [4]) formalizing ...
Lower bounds for a proof system with an exponential speedup over constantdepth Frege systems and over polynomial calculus
, 1997
"... . We prove lower bounds for a proof system having exponential speedup over both polynomial calculus and constantdepth Frege systems in DeMorgan language. Introduction An interesting open problem is to prove a lower bound for constantdepth subsystems of F (MOD p ) (cf. [5, Definition 12.6.1] or [ ..."
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. We prove lower bounds for a proof system having exponential speedup over both polynomial calculus and constantdepth Frege systems in DeMorgan language. Introduction An interesting open problem is to prove a lower bound for constantdepth subsystems of F (MOD p ) (cf. [5, Definition 12.6.1] or [3]), a system that combines boolean reasoning (DeMorgan language) with algebraic reasoning (counting modulo p). In this note we show that the lower bound for the degree of Nullstellensatz proofs for the onto pigeonhole principle PHP n+m n from [2], as well as the bound for the polynomial calculus proofs of the counting principles Count n q from [8], imply lower bounds for a weak subsystem of F (MOD p ) that has nevertheless an exponential speedup over constantdepth Frege systems in DeMorgan language and over polynomial calculus. Namely, the proofs in the system, call it F c d (MOD p ) here, are F (MOD p )  proofs that may use only formulas that can be obtained by substituting DeMorgan...