Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least a^a^...^a clauses, where a is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between title of the paper and title of the arxiv submissio

On Davis–Putnam reductions for minimally unsatisfiable clause-sets

"Minimally unsatisfiable clause-sets" are fundamental building blocks of satisfiability (SAT) theory. In order to establish a structural theory about them,elimination of certain types of degenerations via "Davis-Putnam (DP) reductions" are essential. These DP-reductions have been used at many placessince more than 50 years, and we now show that we have certain forms of confluence, that is, that the applications of DP-reductions are independent oftheir implementation, to a certain degree