Unsatisfiable hitting clause-sets with three more clauses than variables

Hitting clause-sets (as CNFs), known in DNF language as “disjoint” or “orthogonal”, are clause-sets F , such that any C,D ∈ F , C 6= D, have a literal x ∈ C with x ∈ D. The set of unsatisfiable such F is denoted by UHIT ⊂ MU (minimal unsatisfiability). A basic fact is δ(F ) ≥ 1 for F ∈ MU , where the deficiency δ(F ) := c(F ) − n(F ) is the difference between the number of clauses and the number of variables. Via the known singular DP-reduction, generalising unit-clause propagation, every F ∈ UHIT can be reduced to its (unique) “nonsingular normal form” sNF(F ) ∈ UHIT ′, where δ(sNF(F )) = δ(F ), and UHIT ′ ⊂ UHIT is the subset of non-singular elements, i.e., every variable occurs positively as well as negatively at least twice. The Finiteness Conjecture (FC) is that for every k ∈ N the number n(F ) of variables for F ∈ UHIT ′ with δ(F ) = k is bounded. This conjecture is part of the project of classifying UHITδ=k. In this report we prove FC for k = 3 (known for k ≤ 2). For this, a central novel concept is transferred from number theory (Berger et al 1990 [2]), namely the fundamental notion of clause-irreducible clause-sets F , having no non-trivial clausefactors F ′, which are F ′ ⊆ F logically equivalent to some clause. The derived factorisations allow to reduce FC to the clause-irreducible case. Another new tool is nearly-full-subsumption resolution (nfs-resolution), which allows to change certain pairs C,D of clauses. Clause-sets which become clause-reducible after a series of nfs-resolutions are called nfsreducible, and we can furthermore reduce FC to the nfs-irreducible case.