On variables with few occurrences in conjunctive normal forms

TL;DR

This paper establishes an upper bound on the minimal variable-degree in lean clause-sets based on surplus, linking structural properties to variable occurrences, and explores implications for autarkies and minimally unsatisfiable clause-sets.

Contribution

It introduces a new upper bound on variable occurrences in lean clause-sets depending on surplus, advancing understanding of their structure and autarky properties.

Findings

Upper bound mvd(F) <= surp(F) + 1 + log_2(surp(F)) for lean clause-sets.

Clause-sets with mvd(F) > nM(surp(F)) have non-trivial autarkies.

Conjecture that the bound is nearly precise for minimally unsatisfiable clause-sets.

Abstract

We consider the question of the existence of variables with few occurrences in boolean conjunctive normal forms (clause-sets). Let mvd(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 mvd(F) <= nM(surp(F)) <= surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the surplus surp(F). - Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets. - For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the deficiency delta(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 2^n - 1. We conjecture that this bound is nearly precise for minimally unsatisfiable clause-sets. As an application of the upper bound we obtain that (arbitrary!) clause-sets F with mvd(F) > nM(surp(F)) must have a non-trivial 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. As a future application we discuss the classification of minimally unsatisfiable clause-sets depending on the deficiency.