Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
Many hard examples for resolution
Journal of the ACM (JACM)
Short proofs are narrow—resolution made simple
Journal of the ACM (JACM)
Space Complexity in Propositional Calculus
SIAM Journal on Computing
Lower bounds for the weak Pigeonhole principle and random formulas beyond resolution
Information and Computation
An Application of Matroid Theory to the SAT Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Space complexity of random formulae in resolution
Random Structures & Algorithms
On the automatizability of resolution and related propositional proof systems
Information and Computation
A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution
SIAM Journal on Computing
On the complexity of resolution with bounded conjunctions
Theoretical Computer Science
Lower bounds for k-DNF resolution on random 3-CNFs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Towards an optimal separation of space and length in resolution
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Narrow Proofs May Be Spacious:Separating Space and Width in Resolution
SIAM Journal on Computing
Exponential separation between Res (k) and Res(k+1) for k≤εlogn
Information Processing Letters
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A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m-1 variables, and this bound is exact. In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordström 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most (mk)k+1 variables. This result is far from tight, however, since they could only present explicit constructions of minimally unsatisfiable sets with Ω(mk2) variables. In the current paper, we revisit this combinatorial problem and significantly improve the lower bound to (Ω(m))k, which almost matches the upper bound above. Furthermore, using similar ideas we show that the analysis of the technique in [Ben-Sasson and Nordström 2009] for proving time-space separations and trade-offs for k-DNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordström 2009] should hold, a fundamentally different approach would be needed to obtain such results.