Journal of the ACM (JACM)
Many hard examples for resolution
Journal of the ACM (JACM)
A separator theorem for graphs with an excluded minor and its applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Approximation and small-depth Frege proofs
SIAM Journal on Computing
Exponential lower bounds for the pigeonhole principle
Computational Complexity
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
Short proofs are narrow—resolution made simple
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Linear gaps between degrees for the polynomial calculus modulo distinct primes
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Tseitin's Tautologies and Lower Bounds for Nullstellensatz Proofs
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Random CNF's are Hard for the Polynomial Calculus
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
An Optimal Lower Bound for Resolution with 2-Conjunctions
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
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We prove exponential lower bounds on the size of a bounded depth Frege proof of a Tseitin graph-based contradiction, whenever the underlying graph is an expander. This is the first example of a contradiction, naturally formalized as a 3-CNF, that has no short bounded depth Frege proofs. Previously, lower bounds of this type were known only for the pigeonhole principle [18, 17], and for Tseitin contradictions based on complete graphs [19].Our proof is a novel reduction of a Tseitin formula of an expander graph to the pigeonhole principle, in a manner resembling that done by Fu and Urquhart [19] for complete graphs.In the proof we introduce a general method for removing extension variables without significantly increasing the proof size, which may be interesting in its own right.