Disjoint DNF Tautologies with Conflict Bound Two

Abstract

Many aspects of the relation of different decision tree and DNF complexity measures of Boolean functions have been more or less substantially explored. This paper adds a new detail to the picture: we prove that DNF tautologies with terms conflicting in one or two variables pairwise possess a tree-like structure. An equivalent reformulation of this result (adopting the terminology of [7, 8, 9]) is the following. Call a clause-set (or CNF) a hitting clause-set if any two distinct clauses of it clash in at least one literal, and call a hitting clause-set an at-most-d hitting clause-set if any two clauses of it clash in at most d variables. If now an at-most-2 hitting clause-set Φ is unsatisfiable (as a CNF), then, by the above result, there must exist a variable occurring (negated or unnegated) in each clause of Φ.