Results 1 
6 of
6
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 ..."
Abstract

Cited by 32 (11 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
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 ...
bounds for a proof system with an exponential speedup over constantdepth Frege systems and over polynomial calculus, in: Eds
 I.Pr'ivara, P. R*u^zi^cka, 22nd Inter. Symp. Mathematical Foundations of Computer Science (Bratislava, August '97), Lecture Notes in Computer Science 1295
, 1997
"... ..."
unknown title
"... Abstract We define a map g: {0, 1}n! {0, 1}n+1 such that all output bits aredefined by 2DNF formulas in the input bits, and such that g has the following hardness property. For any b 2 {0, 1}n+1 \ Rng(g), formula o / (g)bnaturally expressing that b /2 Rng(g) requires exponential size proofsin any p ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract We define a map g: {0, 1}n! {0, 1}n+1 such that all output bits aredefined by 2DNF formulas in the input bits, and such that g has the following hardness property. For any b 2 {0, 1}n+1 \ Rng(g), formula o / (g)bnaturally expressing that b /2 Rng(g) requires exponential size proofsin any proof system for which the pigeonhole principle is exponentially hard.We define a class of generators generalizing g and show that there isa universal one in this class. Consider a map g: x 2 {0, 1}n! y 2 {0, 1}m defined by conditions yi j 'i(x) where 'i(x) are propositional formulas in x = (x1,..., xn) and m> n. As the domain of g is smaller than {0, 1}m there are b 2 {0, 1}m \ Rng(g). For any such b the formula o / (g)b(x):. i2[m] bi 6j 'i(x) expresses that b /2 Rng(g) in the sense that o / (g)b is a tautology iff b /2 Rng(g).
A proof complexity generator
"... We define a map g: {0, 1} n → {0, 1} n+1 such that all output bits are defined by 2DNF formulas in the input bits, and such that g has the following hardness property. For any b ∈ {0, 1} n+1 \ Rng(g), formula τ(g)b naturally expressing that b / ∈ Rng(g) requires exponential size proofs in any proof ..."
Abstract
 Add to MetaCart
(Show Context)
We define a map g: {0, 1} n → {0, 1} n+1 such that all output bits are defined by 2DNF formulas in the input bits, and such that g has the following hardness property. For any b ∈ {0, 1} n+1 \ Rng(g), formula τ(g)b naturally expressing that b / ∈ Rng(g) requires exponential size proofs in any proof system for which the pigeonhole principle is exponentially hard. We define a class of generators generalizing g and show that there is a universal one in this class. Consider a map g: x ∈ {0, 1} n → y ∈ {0, 1} m defined by conditions yi ≡ ϕi(x) where ϕi(x) are propositional formulas in x = (x1,..., xn) and m> n. As the domain of g is smaller than {0, 1} m there are b ∈ {0, 1} m \ Rng(g). For any such b the formula τ(g)b(x): i∈[m] bi � ≡ ϕi(x) expresses that b / ∈ Rng(g) in the sense that τ(g)b is a tautology iff b / ∈ Rng(g). Our aim is to define g for which the τformulas are hard to prove. When all τ(g)b require superpolynomial (resp. exponential) size proofs in a proof system P we say (following [22]) that g is hard (resp. exponentially hard) proof complexity generator for P. The τformulas have been defined in [7] and independently in [2], and their theory is being developed (see [8, 21, 9, 22, 10, 11, 14]); the introductions to [9] or [22] offer a more comprehensive exposition. The property ”b / ∈ Rng(g) ” can be expressed by a tautology even for maps g with output bits defined by nonuniform N P ∩ coN P conditions on the input bits. Such a generality allowed Razborov [22] to formulate an intriguing conjecture about Extended Frege system EF (see also [10]). We do not need such a generality here. ∗ Keywords: propositional proof complexity, pigeonhole principle.