The independence of the modulo p counting principles
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Exponential lower bounds for the pigeonhole principle
Computational Complexity
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
Using the Groebner basis algorithm to find proofs of unsatisfiability
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting
Computational Complexity
Good degree bounds on Nullstellensatz refutations of the induction principle
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
Lower bounds for the polynomial calculus
Computational Complexity
Linear gaps between degrees for the polynomial calculus modulo distinct primes
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Random CNF's are Hard for the Polynomial Calculus
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A Study of Proof Search Algorithms for Resolution and Polynomial Calculus
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Pseudorandom generators in propositional proof complexity
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Counting Axioms Do Not Polynomially Simulate Counting Gates
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Homogenization and the polynomial calculus
Computational Complexity
Hard examples for the bounded depth Frege proof system
Computational Complexity
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We show that constant-depth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constant-depth Frege with counting axioms systems.