Many hard examples for resolution
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Using the Groebner basis algorithm to find proofs of unsatisfiability
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
GRASP—a new search algorithm for satisfiability
Proceedings of the 1996 IEEE/ACM international conference on Computer-aided design
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting
Computational Complexity
A Computing Procedure for Quantification Theory
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Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
Lower bounds for the polynomial calculus
Computational Complexity
A machine program for theorem-proving
Communications of the ACM
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Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Short proofs are narrow—resolution made simple
Journal of the ACM (JACM)
Information and Computation
Space Complexity in Propositional Calculus
SIAM Journal on Computing
Random CNF's are Hard for the Polynomial Calculus
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Simplified and improved resolution lower bounds
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Lower Bounds for Polynomial Calculus: Non-Binomial Case
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Space complexity of random formulae in resolution
Random Structures & Algorithms
On sufficient conditions for unsatisfiability of random formulas
Journal of the ACM (JACM)
On the complexity of resolution with bounded conjunctions
Theoretical Computer Science
Narrow proofs may be spacious: separating space and width in resolution
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
A combinatorial characterization of resolution width
Journal of Computer and System Sciences
Towards an optimal separation of space and length in resolution
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
PolyBoRi: A framework for Gröbner-basis computations with Boolean polynomials
Journal of Symbolic Computation
Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Towards understanding and harnessing the potential of clause learning
Journal of Artificial Intelligence Research
On the Automatizability of Polynomial Calculus
Theory of Computing Systems
Optimality of size-degree tradeoffs for polynomial calculus
ACM Transactions on Computational Logic (TOCL)
Lower bounds for width-restricted clause learning on small width formulas
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
Space Complexity in Polynomial Calculus
CCC '12 Proceedings of the 2012 IEEE Conference on Computational Complexity (CCC)
Some trade-off results for polynomial calculus: extended abstract
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Towards an understanding of polynomial calculus: new separations and lower bounds
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We devise a new combinatorial framework for proving space lower bounds in algebraic proof systems like Polynomial Calculus (Pc) and Polynomial Calculus with Resolution (Pcr). Our method can be thought as a Spoiler-Duplicator game, which is capturing boolean reasoning on polynomials instead that clauses as in the case of Resolution. Hence, for the first time, we move the problem of studying the space complexity for algebraic proof systems in the range of 2-players games, as is the case for Resolution. A very simple case of our method allows us to obtain all the currently known space lower bounds for Pc/Pcr (CTn, PHPmn, BIT-PHPmn, XOR-PHPmn). The way our method applies to all these examples explains how and why all the known examples of space lower bounds for Pc/Pcr are an application of the method originally given by [Alekhnovich et al 2002] that holds for set of contradictory polynomials having high degree. Our approach unifies in a clear way under a common combinatorial framework and language the proofs of the space lower bounds known so far for Pc/Pcr. More importantly, using our approach in its full potentiality, we answer to the open problem [Alekhnovich et al 2002, Filmus et al 2012] of proving space lower bounds in Polynomial Calculus and Polynomials Calculus with Resolution for the polynomial encoding of randomly chosen k-CNF formulas. Our result holds for k= 4. Then, as proved for Resolution in [BenSasson and Galesi 2003], also in Pc and in Pcr refuting a random k-CNF over n variables requires high space measure of the order of Omega(n). Our method also applies to the Graph-PHPmn, which is a PHPmn defined over a constant (left) degree bipartite expander graph. We develop a common language for the two examples.