Many hard examples for resolution
Journal of the ACM (JACM)
Lower bounds to the size of constant-depth propositional proofs
Journal of Symbolic Logic
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
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On Interpolation and Automatization for Frege Systems
SIAM Journal on Computing
Short proofs are narrow—resolution made simple
Journal of the ACM (JACM)
The Efficiency of Resolution and Davis--Putnam Procedures
SIAM Journal on Computing
Efficient Recognition of Random Unsatisfiable k-SAT Instances by Spectral Methods
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Lower bounds for the weak Pigeonhole principle and random formulas beyond resolution
Information and Computation
Lower Bounds for Polynomial Calculus: Non-Binomial Case
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On the lengths of proofs in the propositional calculus.
On the lengths of proofs in the propositional calculus.
A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution
SIAM Journal on Computing
Lower bounds for k-DNF resolution on random 3-CNFs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Recognizing More Unsatisfiable Random k-SAT Instances Efficiently
SIAM Journal on Computing
Witnesses for non-satisfiability of dense random 3CNF formulas
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Refuting Smoothed 3CNF Formulas
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
The complexity of the pigeonhole principle
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Logical Foundations of Proof Complexity
Logical Foundations of Proof Complexity
Optimality of size-degree tradeoffs for polynomial calculus
ACM Transactions on Computational Logic (TOCL)
Random Cnf’s are Hard for the Polynomial Calculus
Computational Complexity
Short proofs for the determinant identities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Omega(n^(1.5-epsilon)) clauses (Chvatal and Szemeredi (1988), Ben-Sasson and Wigderson (1999)). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Omega((n^2)/log n) clauses, shown by Beame et al. (2002). In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC0-formulas (i.e., TC0-Frege proofs) for random 3CNF formulas with n variables and Omega(n^1.4) clauses. The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek (2006). Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.