On Davis–Putnam reductions for minimally unsatisfiable clause-sets

Abstract

DP-reduction $F\rightsquigarrow {DP}_v\left(F\right)$, 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., $c\left({DP}_v\left(F\right)\right)<c\left(F\right)$), is singular DP-reduction (sDP-reduction), where $v$ must occur in one polarity only once. For minimally unsatisfiable $F\in \mathcal{MU}$, sDP-reduction produces another $F^{\prime }≔{DP}_v\left(F\right)\in \mathcal{MU}$ with the same deficiency, that is, $\delta \left(F^{\prime }\right)=\delta \left(F\right)$; recall $\delta \left(F\right)=c\left(F\right)-n\left(F\right)$, using $n\left(F\right)$ for the number of variables. Let $\mathrm{sDP}\left(F\right)$ for $F\in \mathcal{MU}$ be the set of results of complete sDP-reduction for $F$; so $F^{\prime }\in \mathrm{sDP}\left(F\right)$ fulfil $F^{\prime }\in \mathcal{MU}$, are nonsingular (every literal occurs at least twice), and we have $\delta \left(F^{\prime }\right)=\delta \left(F\right)$. We show that for $F\in \mathcal{MU}$ all complete reductions by sDP must have the same length, establishing the singularity index of $F$. In other words, for $F^{\prime },F^{″}\in \mathrm{sDP}\left(F\right)$ we have $n\left(F^{\prime }\right)=n\left(F^{″}\right)$. In general the elements of $\mathrm{sDP}\left(F\right)$ 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 $\mathrm{sDP}\left(F\right)$ are pairwise isomorphic) in case $\delta \left(F\right)=2$. In general we prove that we have confluence (i.e., $\left|\mathrm{sDP}\left(F\right)\right|=1$) for saturated $F$ (i.e., $F\in \mathcal{SMU}$). More generally, we show confluence modulo isomorphism for eventually saturated$F$, that is, where we have $\mathrm{sDP}\left(F\right)\subseteq \mathcal{SMU}$, yielding another proof for confluence modulo isomorphism in case of $\delta \left(F\right)=2$.