Journal of the ACM (JACM)
Many hard examples for resolution
Journal of the ACM (JACM)
Short proofs are narrow—resolution made simple
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Resolution Proofs, Exponential Bounds, and Kolmogorov Complexity
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Resolution and the Weak Pigeonhole Principle
CSL '97 Selected Papers from the11th International Workshop on Computer Science Logic
Simplified and improved resolution lower bounds
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
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Resolution space measures the maximum number of clauses that need to be simultaneously active in a resolution refutation. This complexity measure was defined by Kleine Büning and Lettmann in [8] and slightly modified recently [6] to make it suitable for comparisons with other measures. Since its definition, only trivial lower bound for the resolution space, measured in terms of the number of initial clauses were known. In this paper we prove optimal lower bounds for the space needed in the resolution refutation of two important families of formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n - c for some constant c. Measured on the number of clauses, this result is best possible since the mentioned formulas have O(n) clauses, and the number of clauses is an upper bound for the resolution space. We also show that the formulas expressing the general Pigeonhole Principle with n holes and more than n pigeons, need space n + 1 independently of the number of pigeons. Since a matching space upper bound of n + 1 for these formulas exist, the obtained bound is exact. These results point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations.