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

DNF tautologies with a limited number of occurrences of every variable

It is known that every DNF tautology with all monomials of length k contains a variable with at least \Omega\Gamma 2 k =k) occurrences. It is not known, however, if this bound is tight, i.e. if there are tautologies with at most O(2 k =k) occurrences of every variable. DNF tautologies with 2 k monomials of length k and with at most 2 k =k ff occurrences of every variable, where ff = log 3 4 \Gamma 1 0:26 are presented. This has the following consequence. Let (k; s)-SAT be k-SAT restricted to instances with at most s occurrences of every variable. It is known that for every k, there is an s k such that (k; s k )-SAT is NPcomplete and (k; s k \Gamma 1)-SAT is trivial in the sense that every instance has positive answer. The above result implies that s k 2 k =k ff . This improves the previously known bound s k 11 32 2 k . 1 Introduction Let a (k; s)-formula be a formula in conjunctive normal form whose clauses consist of exactly k literals and such that every variabl..