Journal of the ACM (JACM)
The monotone circuit complexity of Boolean functions
Combinatorica
On the complexity of cutting-plane proofs
Discrete Applied Mathematics
The complexity of Boolean functions
The complexity of Boolean functions
Many hard examples for resolution
Journal of the ACM (JACM)
Resolution proofs of generalized pigeonhole principles. (Note)
Theoretical Computer Science
Sorting in c log n parallel steps
Combinatorica
Exponential lower bounds for the pigeonhole principle
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Lower bounds for cutting planes proofs with small coefficients
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
Lower bounds for the polynomial calculus
Computational Complexity
A new proof of the weak pigeonhole principle
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
How to Lie Without Being (Easily) Convicted and the Length of Proofs in Propositional Calculus
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
Resolution and the Weak Pigeonhole Principle
CSL '97 Selected Papers from the11th International Workshop on Computer Science Logic
Non-Automatizability of Bounded-Depth Frege Proofs
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Simplified and improved resolution lower bounds
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
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We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We show that the standard encoding of the PHP as a monotone sequent admits quasipolynomial-size proofs in this system. This result is a consequence of deriving the basic properties of certain quasipolynomial-size monotone formulas computing the boolean threshold functions. Since it is known that the shortest proofs of the PHP in systems such as Resolution or Bounded Depth Frege are exponentially long, it follows from our result that these systems are exponentially separated from the monotone Gentzen Calculus. We also consider the monotone sequent (CLIQUE) expressing the clique-coclique principle defined by Bonet, Pitassi and Raz (1997). We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one-to-one and onto PHP, and so CLIQUE also has quasipolynomial-size monotone proofs. As a consequence, Cutting Planes with polynomially bounded coefficients is also exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudlák.