On Variables with Few Occurrences in Conjunctive Normal Forms

We consider the question of the existence of variables with few occurrences in boolean conjunctive normal forms (clause-sets). Let μvd(F ) for a clause-set F denote the minimal variable-degree, the minimum of the number of occurrences of variables. Our main result is an upper bound μvd(F ) ≤ nM(σ(F )) ≤ σ(F ) + 1 + log 2 (σ(F )) for lean clause-sets F in dependency on the surplus σ(F ). Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets. For the surplus we have σ(F ) ≤ δ(F ) = c(F )− n(F ), using the deficiency δ(F ) of clause-sets, the difference between the number of clauses and the number of variables. nM(k) is the k-th “non-Mersenne” number, skipping in the sequence of natural numbers all numbers of the form 2n − 1. As an application of the upper bound we obtain that clause-sets F violating μvd(F ) ≤ nM(σ(F )) must have a nontrivial autarky (so clauses can be removed satisfiability-equivalently by an assignment satisfying some clauses and not touching the other clauses). It is open whether such an autarky can be found in polynomial time.