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57
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 50 (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.
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 46 (2 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 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 40 (6 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).
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 37 (9 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
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 37 (5 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 33 (11 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
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 26 (3 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
Linear Lower Bound on Degrees of Positivstellensatz Calculus Proofs for the Parity
 THEORETICAL COMPUTER SCIENCE
"... It is established a linear (thereby, sharp) lower bound on degrees of Positivstellensatz calculus refutations over a real field introduced in [GV99], for the Tseitin tautologies and for the parity (the mod 2 principle). We use the machinery of the Laurent proofs developped for binomial systems i ..."
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Cited by 23 (3 self)
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It is established a linear (thereby, sharp) lower bound on degrees of Positivstellensatz calculus refutations over a real field introduced in [GV99], for the Tseitin tautologies and for the parity (the mod 2 principle). We use the machinery of the Laurent proofs developped for binomial systems in [BuGI 98], [BuGI 99].
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 (3 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