We prove that any Resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of ), (for some global constant ffl ? 0). One corollary is that a certain propositional formulation of the statement NP 6ae P=poly does not have short Resolution proofs.

For an arbitrary hypergraph H, let PM(H) be the propositional formula asserting that contains a perfect matching. We show that every resolution refutation of PM(H) must have size #(H)r(H)(log n(H))(r(H) + log n(H)) where n(H) is the number of vertices, #(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and #(H) is the maximal number of edges incident to two di#erent vertices.

A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef- ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing k-DNFs instead of clauses.

We call a pseudorandom generator Gn : {0, 1}^n → {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) ≠ b for any string b ∈ {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by the Nisan-Wigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. We prove that under certain circumstances these generators are hard for such proof systems as Resolution, Polynomial Calculus and Polynomial Calculus with Resolution (PCR).

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 constant-depth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss

This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for the polynomial calculus for k-CNF formulas. As a

this paper we are interested in systems that use uses polynomials instead of boolean formulas. From the previous list this includes the Nullstellensatz refutations. Recently a stronger system using polynomials was proposed, the polynomial calculus, also called the Groebner calculus [9]. The proof systems form a similar hierarchy as the complexity classes or classes of circuits in the computational complexity, but there is no direct relation between the two hierarchies. Recently a new method was found which makes it possible to prove lower bounds on the length of proofs for some propositional proof systems using lower bounds on circuit complexity. This method is based on proving computationally efficient versions of Craig's interpolation theorem for the proof system in question [14, 18]. For appropriate tautologies the interpolation theorem

It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The well-studied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the Lovasz-Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS^d of LS that operate with polynomial inequalities of degree at most d. It turns out

Let f0 ; f1 ; : : : ; fk be n-variable 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 Fp-linear 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 ...

Recently, Raz [Raz01] established exponential lower bounds on the size of resolution proofs of the weak pigeonhole principle. We give another proof of this result which leads to better numerical bounds. Specifically, we show that every resolution proof of PHP