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51
Resolution Lower Bounds for the Weak Pigeonhole Principle
, 2001
"... 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. ..."
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Cited by 51 (3 self)
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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.
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus Resolution
, 2003
"... 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 ..."
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Cited by 42 (4 self)
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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 kDNFs instead of clauses.
Pseudorandom Generators in Propositional Proof Complexity
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REP. NO.23
, 2000
"... 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 g ..."
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Cited by 41 (7 self)
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We call a pseudorandom generator Gn : {0, 1}^n &rarr; {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) &ne; b for any string b &isin; {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by the NisanWigderson 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).
Resolution lower bounds for perfect matching principles
 Journal of Computer and System Sciences
"... For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, ..."
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Cited by 41 (4 self)
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For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and λ(H) is the maximal number of edges incident to two different vertices. For ordinary graphs G our general bound considerably simplifies to exp Ω (implying an exp(Ω(δ(G) 1/3)) lower bound that depends on the minimal degree only). As a direct corollary, every resolution proof of the functional ( ( onto)) version of must have size exp Ω (which the pigeonhole principle onto − FPHP m n n (log m) 2 δ(G) (log n(G)) 2 becomes exp ( Ω(n 1/3) ) when the number of pigeons m is unbounded). This in turn immediately implies an exp(Ω(t/n 3)) lower bound on the size of resolution proofs of the principle asserting that the circuit size of the Boolean function fn in n variables is greater than t. Inparticular,Resolution does not possess efficient proofs of NP ⊆ P/poly. These results relativize, in a natural way, to a more general principle M(UH) asserting that H contains a matching covering all vertices in U ⊆ V (H).
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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Cited by 32 (9 self)
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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
Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1999
"... 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 ..."
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Cited by 29 (8 self)
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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 kCNF formulas. As a
Complexity of SemiAlgebraic Proofs
, 2001
"... It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver calculi ..."
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Cited by 24 (2 self)
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It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver 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
Algebraic Models of Computation and Interpolation for Algebraic Proof Systems
, 1998
"... 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 sy ..."
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Cited by 23 (2 self)
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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
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 ...
Improved Resolution Lower Bounds for the Weak Pigeonhole Principle
 Electronic Colloquium on Computational Complexity
, 2001
"... 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 ..."
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Cited by 16 (2 self)
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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