On Davis-Putnam Reductions for Minimally Unsatisfiable Clause-Sets

DP-reduction F ❀ DPv(F ), applied to a clause-set F and a variable v, replaces all clauses containing v by their resolvents (on v). A basic case, where the number of clauses is decreased (i.e., c(DPv(F )) < c(F )), is singular DP-reduction (sDP-reduction), where v must occur in one polarity only once. For minimally unsatisfiable F ∈ MU , sDPreduction produces another F ′ := DPv(F ) ∈ MU with the same deficiency, that is, δ(F ′) = δ(F ); recall δ(F ) = c(F )− n(F ), using n(F ) for the number of variables. Let sDP(F ) for F ∈ MU be the set of results of complete sDP-reduction for F ; so F ′ ∈ sDP(F ) fulfil F ′ ∈ MU , are nonsingular (every literal occurs at least twice), and we have δ(F ′) = δ(F ). We show that for F ∈ MU all complete reductions by sDP must have the same length, establishing the singularity index of F . In other words, for F ′, F ′′ ∈ sDP(F ) we have n(F ′) = n(F ′′). In general the elements of sDP(F ) are not even (pairwise) isomorphic. Using the fundamental characterisation by Kleine Büning, we obtain as application of the singularity index, that we have confluence modulo isomorphism (all elements of sDP(F ) are pairwise isomorphic) in case δ(F ) = 2. In general we prove that we have confluence (i.e., |sDP(F )| = 1) for saturated F (i.e., F ∈ SMU). More generally, we show confluence modulo isomorphism for eventually saturated F , that is, where we have sDP(F ) ⊆ SMU, yielding another proof for confluence modulo isomorphism in case of δ(F ) = 2.